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imaging capabilities with an African +extension to the Event Horizon Telescope +Noemi La Bella1, Sara Issaoun2, 3, 1, Freek Roelofs2, 4, 1, Christian Fromm5, 6, 7, and Heino Falcke1 +1 Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.O. Box +9010, 6500 GL Nijmegen, The Netherlands +e-mail: n.labella@astro.ru.nl +2 Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA +3 NASA Hubble Fellowship Program, Einstein Fellow +4 Black Hole Initiative, Harvard University, 20 Garden Street, Cambridge, MA 02138, USA +5 Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Emil-Fischer-Strasse 31, 97074 Würzburg, Germany +6 Institut für Theoretische Physik, Goethe Universität, Max-von-Laue-Str. 1, D-60438 Frankfurt, Germany +7 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany +January 30, 2023 +ABSTRACT +Context. The Event Horizon Telescope (EHT) has recently published the first images of the supermassive black hole at the center of +our Galaxy, Sagittarius A* (Sgr A*). Imaging Sgr A* is plagued by two major challenges: variability on short (approximately minutes) +timescales and interstellar scattering along our line of sight. While the scattering is well studied, the source variability continues to +push the limits of current imaging algorithms. In particular, movie reconstructions are hindered by the sparse and time-variable +coverage of the array. +Aims. In this paper, we study the impact of the planned Africa Millimetre Telescope (AMT, in Namibia) and Canary Islands telescope +(CNI) additions to the time-dependent coverage and imaging fidelity of the EHT array. This African array addition to the EHT further +increases the eastwest (u, v) coverage and provides a wider time window to perform high-fidelity movie reconstructions of Sgr A*. +Methods. We generated synthetic observations of Sgr A*’s accretion flow and used dynamical imaging techniques to create movie +reconstructions of the source. To test the fidelity of our results, we used one general-relativistic magneto-hydrodynamic model of the +accretion flow and jet to represent the quiescent state and one semi-analytic model of an orbiting hotspot to represent the flaring state. +Results. We found that the addition of the AMT alone offers a significant increase in the (u, v) coverage, leading to robust averaged +images during the first hours of the observating track. Moreover, we show that the combination of two telescopes on the African +continent, in Namibia and in the Canary Islands, produces a very sensitive array to reconstruct the variability of Sgr A* on horizon +scales. +Conclusions. We conclude that the African expansion to the EHT increases the fidelity of high-resolution movie reconstructions of +Sgr A* to study gas dynamics near the event horizon. +Key words. Black hole physics - Galaxy: center - Instrumentation: high angular resolution - interferometers - Techniques: image +processing +1. Introduction +The Event Horizon Telescope (EHT) collaboration has recently +published the first images of the black hole shadow of Sagittar- +ius A* (Sgr A*), the supermassive black hole (SMBH) at the +center of the Milky Way, characterized by an asymmetric bright +ring of (52.1 ± 0.6) µas (Event Horizon Telescope Collaboration +et al. 2022a). The ring-like morphology was recovered in over +95% of the best-fit images produced from 2017 April 6 and 7 +observations. The EHT images of Sgr A* are consistent with the +prediction of a shadow for a Kerr black hole (Falcke et al. 2000) +with a mass M ∼ 4 × 106M⊙ at a distance D ∼ 8 kpc, which +were accurately measured by high-resolution infrared studies of +stellar orbits in the Galactic Center (Gravity Collaboration et al. +2018; Do et al. 2019). In 2019, the EHT collaboration delivered +the first ever image of a black hole shadow in the giant ellipti- +cal galaxy M87 (Event Horizon Telescope Collaboration et al. +2019a). The main difference between the two SMBHs is their +mass. M87* is about 1600 times more massive than Sgr A* and +thus, it has a longer gravitational timescale. In fact, the period of +the innermost stable circular orbit (ISCO) for a nonrotating black +hole as massive as M87* is ∼30 days, while for Sgr A* it is ∼30 +minutes. As a consequence, the estimation of the ring diameter +of Sgr A* is more uncertain than in M87* and we need movies +to properly study the plasma motion surrounding the black hole +on this short orbital timescale. +The variability of Sgr A* required a reformulation of the +static source assumptions in the interferometric Earth aperture +synthesis method and imaging algorithms used for M87*. In par- +ticular, to generate a typical static image of Sgr A*, a variabil- +ity noise budget needs to be added, while a dynamical imaging +process is required to capture the evolving structure of Sgr A* +(Event Horizon Telescope Collaboration et al. 2022b). Because +of the sparsity of the EHT array, time slots with good (u, v) cover- +age were selected to perform dynamical studies on the variability +(Farah et al. 2022). +Article number, page 1 of 11 +arXiv:2301.11384v1 [astro-ph.IM] 26 Jan 2023 + +A&A proofs: manuscript no. core +The SMBH also presents flare events observed across the +electromagnetic spectrum in the last decades. An accurate study +of the millimeter light curves during the 2017 EHT campaign +was done by Wielgus et al. (2022a). In particular, the authors +found excess variability on 2017 April 11, following a flare +observed in the X-ray. Subsequent studies on polarized light +curves with the Atacama Large Millimeter/submillimeter Array +(ALMA) on the same day (Wielgus et al. 2022b) revealed the +presence of a hotspot orbiting Sgr A* clockwise. +In addition to its quiescent variability, imaging Sgr A* is a +complex process because the very long baseline interferometry +(VLBI) observations are affected by scattering in the interstel- +lar medium along our line of sight toward the Galactic Center. +The consequent diffractive and refractive effects of the scatter- +ing were mitigated by modeling their chromatic properties in the +radio band (see Psaltis et al. 2018; Johnson et al. 2018; Issaoun +et al. 2019a, 2021; Event Horizon Telescope Collaboration et al. +2022b, for more details). +Eight telescopes at six geographic locations formed the 2017 +EHT array configuration that led to the first images of Sgr A* +and M87*. Since 2017, the array has doubled in bandwidth and +increased the number of baselines with three new telescopes. +As of 2022, the EHT has consisted of eleven telescopes at +eight locations: ALMA and the Atacama Pathfinder Experiment +(APEX) on the Llano de Chajnantor in Chile; the Large Millime- +ter Telescope (LMT) Alfonso Serrano on the Volcán Sierra Ne- +gra in Mexico; the James Clerk Maxwell Telescope (JCMT) and +Submillimeter Array (SMA) on Maunakea in Hawai’i; the In- +stitut de Radioastronomie Millimétrique 30-m telescope on Pico +Veleta (PV) in Spain; the Submillimeter Telescope (SMT) on Mt. +Graham and the 12-m telescope on Kitt Peak (KP) in Arizona; +the South Pole Telescope (SPT) in Antarctica; the Northern Ex- +tended Millimeter Array (NOEMA) in France; and the Green- +land Telescope (GLT) at Thule. This new configuration offers +increased sensitivity of the array and will enable higher-fidelity +images of Sgr A* and M87*. However, all new telescopes are +in the northern hemisphere and are less effective for imaging +southern sources. Additional telescopes are being considered to +expand the capabilities of the array, especially on the African +continent, which offers prime site locations to increase the (u, v) +coverage toward Sgr A*. +In this work, we consider two additions to the EHT in the +African region: one in Namibia and one in the Canary Islands. +The Africa Millimetre Telescope (AMT), planned on Mt. Gams- +berg (2,347 m a.s.l.) in Namibia, will be the first millimeter-wave +telescope in Africa. The project to add this telescope is currently +underway, and aims to relocate the decommissioned 15-meter +SEST telescope in Chile to Gamsberg in the next years. This site +will offer low precipitable water vapor levels during the typical +fall and spring EHT campaign seasons (Backes et al. 2016) and +its strategical position in the southern hemisphere at the same lat- +itude as ALMA provides important eastwest baselines to Chile +and northsouth baselines to Europe, significantly increasing the +snapshot coverage in the first half of a typical observing night. +The island of La Palma in the Canary Islands (2,000 m a.s.l.) +has dry weather conditions throughout the year (Raymond et al. +2021) and offers a prime location to provide mid-range coverage +between Namibia and Europe that is crucial to constrain source +compactness and extent. Furthermore, the site’s established in- +frastructure from existing observatories would make an addi- +tional telescope easily feasible and well supported, making it an +ideal candidate for a telescope location in the near term. +We present simulated dynamical images of Sgr A* using +the 2022EHT array and an African extension including two new +telescopes: the 15-meter AMT, and the Canary Islands telescope, +CNI, on the island of La Palma. We assume the dish size of +CNI to be six meters, following the design concept for a next- +generation EHT facility in the long term (Doeleman et al. 2019). +We investigate the impact of the AMT and CNI stations on imag- +ing Sgr A* in both quiescent and flaring states. The methods we +use can easily be expanded to other EHT configurations. +The paper is organized as follows: in Section 2, we describe +the synthetic generation pipeline and imaging algorithms used. +In Section 3, we present the African extension to the EHT and +its contribution to snapshot and full-track (u, v) coverage. In Sec- +tion 4, we show the static and dynamical reconstructions ob- +tained with the enhanced EHT array. Finally, in Section 5, we +discuss the advantages of the African extension to the array in +producing high-fidelity movies of Sgr A*. +2. Methods +2.1. GRMHD ground truth movies +The quiescent state of the plasma flow of Sgr A* was reproduced +by generating synthetic data from general relativistic magneto- +hydrodynamic (GRMHD) simulations at 230 GHz. The typical +range of simulations used to study Sgr A* include two classes of +models: magnetically arrested disk (MAD; Igumenshchev et al. +2003; Narayan et al. 2003) and Standard And Normal Evolu- +tion (SANE; Narayan et al. 2012) models. The SANE mode is +characterized by a weak and turbulent magnetic field crossing +the hemisphere of the event horizon, while the MAD mode has +high magnetic flux. The recent EHT Sgr A* results have shown +that GRMHD simulations are more variable than the data (Event +Horizon Telescope Collaboration et al. 2022d). Because SANE +models are less variable than MAD models, they are more rep- +resentative of the degree of variability in Sgr A*. We thus used a +SANE model for our quiescent state reconstructions. +The simulation was generated with the GRMHD code BHAC +Porth et al. (2017); Olivares et al. (2020). We initialized a torus +in hydrodynamic equilibrium where the inner edge is located +at 6 M (where M is the gravitational timescale GM/c3) and the +pressure maximum is found at 13 M. We set a black hole spin +of a⋆ = 0.9375 and an adiabatic index ˆγ = 4/3 and per- +formed the simulations on spherical grid (r, θ, φ) with resolution +of 512 × 192 × 192 and three layers of adaptive mesh refinement +(AMR) using logarithmic Kerr-Schild coordinates. For more de- +tails on the simulations see Fromm et al. (2022). We evolve the +simulations until 30000 M, which ensures a quasi steady-state in +the mass accretion rate. The radiative transfer calculations were +performed with the GRRT code BHOSS Younsi et al. (2012, 2016, +2020, 2021). We used a field of view of 200 µas together with +a black hole mass of 4.14 × 106 M⊙ at a distance of 8.127 kpc +(Event Horizon Telescope Collaboration et al. 2022d). The im- +ages were created assuming a viewing angle ϑ = 10◦ and a nu- +merical resolution of 4002 pixels. Since the electron temperature +is not evolved during the GRMHD simulations we computed +their temperature using the R − β description of Mo´scibrodzka +et al. (2016) where we set Rlow = 1 and Rhigh = 5. In order to +adjust the simulations to the observations, we iterated over the +mass accretion rate to provide an average flux density of ∼2.4 +Jy at 230 GHz in a time window of 5000 M. Two time windows +were used (20-25 kM and 25-30 kM) and individually normal- +ized. +The 16-hour movie consists of 300 frames separated by 200 +seconds, with a rotation period of the plasma around the black +hole of ∼30 minutes. The simulation does not include effects +Article number, page 2 of 11 + +Noemi La Bella et al.: Sgr A* dynamical imaging with an African extension to the EHT +of interstellar scattering, therefore we characterized those effects +using a phase screen toward Sgr A* (see Psaltis et al. 2018; John- +son et al. 2018, for more details). +2.2. Synthetic data generation +The GRMHD synthetic data were produced with the SYMBA 1 +software (Roelofs et al. 2020), which reconstructs a model im- +age following the same calibration and imaging processes of a +realistic observation. Given a VLBI array configuration and a +specific model as input, the synthetic observations are generated +with MeqSilhouette (Blecher et al. 2017; Natarajan et al. 2022) +and the corrupted raw data are then processed with the VLBI +data calibration pipeline rPICARD (Janssen et al. 2019), which +is used to calibrate real EHT data (Event Horizon Telescope Col- +laboration et al. 2019b). The calibrated data set can also pass +through the network calibration step that solves gains for colo- +cated sites using the flux of the source at large scales (Fish et al. +2011; Johnson & Gwinn 2015; Blackburn 2019; Event Horizon +Telescope Collaboration et al. 2019c). Our synthetic data are +based on the antenna and weather parameters as measured in +the 2017 observations (Event Horizon Telescope Collaboration +et al. 2019d). The weather conditions were extracted from the +VLBI monitor server 2, which collects weather data (e.g., ground +pressure, ground temperature) from in situ measurements. The +weather conditions used are reported in Table 2 of Roelofs et al. +(2020), which includes the parameters for the stations that joined +the 2017 EHT campaign, and those for the enhanced array, with +GLT joining the array in 2018, NOEMA and KP in 2021, and +with the planned AMT. As described by the authors, the weather +parameter estimation for stations that did not join the 2017 obser- +vations was done using the Modern-Era Retrospective Analysis +for Research and Applications, version 2 (MERRA − 2) from the +NASA Goddard Earth Sciences Data and Information Services +Center (Gelaro et al. 2017), and the am atmospheric model soft- +ware Paine (2019). We applied the same method to obtain the +weather conditions on La Palma, in the Canary Islands. Finally, +we adopted the observing schedule of 2017 April 7 (Event Hori- +zon Telescope Collaboration et al. 2022c), encompassing scans +on Sgr A* from the 4 to 15 UT hours. +For generating movies of flares in Sgr A*, we used a sim- +ulated Gaussian flaring feature with an orbiting period of 27 +min around a ray-traced image of a semi-analytic advection- +dominated accretion flow (ADAF) model of Sgr A* (model B +of Doeleman et al. 2009). The movie at 230 GHz is composed +of 100 frames separated by 16.2 seconds. The eht-imaging +Python library 3 (Chael et al. 2016, 2018) was used to gener- +ate the hotspot synthetic data. The eht-imaging package does +not produce realistic VLBI-mm observations as SYMBA, for +instance the data are not frequency-resolved, gain effects are +not based on physical models, and there are no calibration ef- +fects added (more details about the difference between the two +pipelines can be found in Event Horizon Telescope Collabora- +tion et al. 2019c, Appendix C). As in the case of the GRMHD +movies, the synthetic data were based on the 2017 April 7 ob- +serving parameters. Unlike SYMBA, the simulated visibilities +are not scan-separated, but have a cadence of 30 seconds. +1 https://bitbucket.org/M_Janssen/symba +2 https://bitbucket.org/vlbi +3 https://github.com/achael/eht-imaging +2.3. Dynamical imaging +We +imaged +the +SYMBA +synthetic +data +set +using +the +eht-imaging library, developed specifically for the EHT. The +imaging algorithm utilizes the regularized maximum likelihood +(RML) method, which aims to find an image that minimizes a +specified objective function, consisting of data fit quality (χ2) +terms, and additional regularizer terms favoring, for example, +smooth or sparse image structures (Event Horizon Telescope +Collaboration et al. 2019e). The static assumption based on the +Earth rotation aperture synthesis technique, where the source is +assumed static during the course of the observation, is not valid +in the case of Sgr A* due to its intraday variability (Event Hori- +zon Telescope Collaboration et al. 2022b). To tackle this chal- +lenge, we use a method called “dynamical imaging." The dy- +namical imaging algorithm within the eht-imaging package is +a generalization of the standard RML method which introduces +three dynamical regularizers that enforce time-sensitive proper- +ties between snapshot frames (see Johnson et al. 2017, for more +details). To reconstruct the hotspot movies we used the R∆t reg- +ularizer, which imposes a time continuity between frames. Since +the hotspot model simulates coherent motion of a flare orbit- +ing Sgr A*, this regularizer let us reconstruct continuous motion +of structure. For the GRMHD movies, we also added the R∆I +regularizer, which enforces similarity between the reconstructed +frame and a time-averaged image. As GRMHD simulations de- +scribe the turbulent behavior of an accretion flow onto Sgr A*, +this regularizer allows us to look for turbulence on top of a static +structure. +To inspect the capability of the expanded EHT array to re- +construct dynamical motion, we selected time windows during +the observation for which coverage and filling fraction were op- +timized, as was done in Farah et al. (2022). For the GRMHD sim- +ulations, we produced movies of 5.7 hours, while for the hotspot +movies we chose optimal time windows of 1.7 hours where the +array offers the best coverage. To obtain a good reconstructed +movie, larger time windows were required for the GRMHD data +set generated with SYMBA, which includes actual scans and +gaps between the scans (more details in Section 3.1). +2.4. Movie quality metrics +Two quality metrics were selected to evaluate the fidelity of +the reconstructed images: the normalized root-mean-square er- +ror (NRMSE) and the normalized cross-correlation (NXCORR; +e.g., Event Horizon Telescope Collaboration et al. 2019e). +NRMSE is more sensitive to pixel-by-pixel differences, while +NXCORR is more sensitive to large scale structure (Issaoun et al. +2019b). We estimated values for both metrics for each frame +of the movie, quantifying the fidelity of the reconstruction as +a function of time with respect to the ground truth. +The NRMSE measures similarities per kth pixel and it is de- +fined as: +NRMSE = +�� +k(Ik − I′ +k)2 +� +k I2 +k +, +(1) +where I′ and I are the intensity of the reconstructed movie frame +and the model movie frame, respectively (e.g., Chael et al. 2018; +Issaoun et al. 2019b). An NRMSE value of zero corresponds to +identical images. +For given frames I′ and I, NXCORR is given by: +NXCORR = 1 +N +� +k +(Ik − ⟨I⟩)(I′ +k − ⟨I′⟩) +σIσI′ +, +(2) +Article number, page 3 of 11 + +A&A proofs: manuscript no. core +Fig. 1: Sgr A* (u, v) coverage of the 2017 April 7 EHT observa- +tions. Seven scans on Sgr A* were added to the original schedule +at the beginning of the observation, brought by the introduction +of the NOEMA array and the African stations. In blue, the cov- +erage obtained with the 2022EHT array. The contributions of the +AMT and CNI baselines are shown in red and in brown, respec- +tively. The AMT adds long northeast and southwest baselines in- +creasing the EHT resolution, while CNI offers shorter baselines +to detect large-scale emission and constrain the source extent. +where N is the total number of pixels per frame, ⟨I⟩ and ⟨I′⟩ are +the mean pixel values and σI, σI′ are the respective standard de- +viations. An NXCORR of 1 corresponds to a perfect correlation +between the frames, -1 for anticorrelation, and 0 for no correla- +tion (e.g., Event Horizon Telescope Collaboration et al. 2019e). +3. The African expansion to the EHT +In this section, we discuss a potential implementation of the +African expansion to the EHT, its (u, v) coverage, and Fourier +filling fraction, which let us identify potential time windows to +generate movies of Sgr A*. We also investigate the location of +the new baselines with respect to the position of the two local +minima in the correlated flux density profile of a thin ring of 54 +µas. To assess the impact of the new African stations, different +array configurations were used. We name those configurations as +follows: 2022EHT, the current EHT configuration composed of +eleven telescopes; 2022EHT + AMT, the 2022EHT with the ad- +dition of AMT; 2022EHT + Africa, the 2022EHT plus the AMT +and CNI stations; Eastern array + Africa, the 2022EHT subar- +ray until ∼9.5 UT hours (∼22.7 Greenwich Mean Sidereal Time, +GMST), after this time the AMT does not observe Sgr A*; West- +ern array, the 2022EHT subarray from ∼9.5 UT hours to ∼15 UT +hours (∼4.1 GMST). So far, the Western array has been offering +the best coverage to produce dynamic reconstructions of Sgr A*. +3.1. (u, v) coverage +Fig. 1 depicts the Sgr A* (u, v) coverage using the 2017 April +7 observing schedule as a base, enhanced by the addition of +NOEMA and KP, which joined the array post-2017, and the two +proposed African antennas. Moreover, the observation was im- +posed to start when the source is at an elevation of more than +10 degrees at NOEMA and the African telescopes, allowing us +to extend the observation by two hours. The 2022EHT baselines +are shown in blue, the AMT baselines in red and the CNI base- +lines in brown. The AMT is a potential southern site to image +Sgr A* that adds determinant baselines to the array. Specifically, +the AMT adds northsouth baselines to PV and NOEMA, east- +west baselines to Chile, and a redundancy baseline to ALMA- +SPT, since Mt. Gamsberg is at the same latitude as ALMA. +Moreover, the AMT increases the resolution in the northeast and +southwest, by adding long baselines to LMT and the Arizona +stations. On the other hand, the CNI telescope yields new short +inter-site baselines to the European sites, PV and NOEMA, con- +tributing to the measurement of the source extent, together with +the inter-sites SMT-LMT, PV-NOEMA baselines. In addition, +the baseline CNI-AMT provides further northsouth coverage to +the array. +3.2. Fourier filling fraction +The sparsity and changing coverage of the EHT array affect +the accuracy of the dynamical reconstructions of time-variable +sources. To produce VLBI movies of Sgr A*, it is thus required +to identify time periods with optimal and stable (u, v) coverage. +For the 2017 Sgr A* results, Farah et al. (2022) selected time re- +gions using three different metrics and found the best dynamical +time period to be from ∼01:30 GMST to ∼03:10 GMST, hence in +the Western array window. We utilized one of these metrics, the +(u, v) filling fraction, to inspect if new temporal regions are of- +fered by the Eastern array + Africa. The Fourier filling fraction +measures the area sampled in the (u, v) plane by the observed +visibilities. Following Farah et al. (2022), the (u, v) points were +convolved with a circle of radius 0.71/θFOV, with FOV being the +field of view adopted for imaging, representing the half-width at +half-maximum of a filled disk of uniform brightness on the sky +(see Palumbo et al. 2019, for more details). In our analysis, we +calculated the filling fraction normalized to the maximum fill- +ing fraction value of the 2022EHT array. On the left of Fig. 2, +we show the time-dependent normalized filling fraction for the +2022EHT + AMT array in red, and that of the 2022EHT array in +blue. The colored windows delimit time regions in which the fill- +ing fraction is persistently above the 70% 2022EHT maximum +threshold (dashed grey line). Time windows below this threshold +do not have sufficient coverage to produce high-fidelity movies. +The 2022EHT array provides good time regions in the Western +array. Notably, our results confirm the 01:30 GMST to 03:10 +GMST best-time window obtained from the 2017 array selective +dynamical imaging analysis (Farah et al. 2022). The AMT adds +three additional optimal time periods (red areas) in the Eastern +array, of almost 4 hours in total. Furthermore, on the right of Fig. +2 we show a further increase in the Fourier filling area achieved +by the combination of the CNI (brown) and AMT sites (i.e., with +the 2022EHT array + Africa) leading to a persistent time block +of 7.4 hours. Therefore, the African stations will provide signif- +icantly improved (u, v) coverage and stability for the Eastern ar- +ray, increasing the ability to study rapid variations of the source +at the beginning of the observing track. +Article number, page 4 of 11 + +10 +CNI baselines +AMT baselines +7.5 +2022EHT +5.0 +2.5 +[G^] +0.0 +-2.5 +-5.0 +-7.5 +-10 +-10 +-7.5 +-5.0 +-2.5 +0.0 +2.5 +5.0 +7.5 +10 +u [G入]Noemi La Bella et al.: Sgr A* dynamical imaging with an African extension to the EHT +Fig. 2: Time-dependent Fourier filling fraction normalized by the maximum Fourier filling of the 2022EHT array. The curves +represent the filling fraction of the 2022EHT array, 2022EHT + AMT array and 2022EHT + Africa array, in blue, red and brown, +respectively. The dashed gray line corresponds to the lower limit used for identifying good time windows to perform dynamical +imaging. The optimal time regions for the current EHT array are shown in blue. The 2022EHT + AMT adds three time windows +(red areas) of ∼4 hours in total, while the 2022EHT + Africa array (brown area) produces a time window of ∼7.4 hours. +3.3. Correlated flux density profile +The correlated flux density (in Jy) of Sgr A* as a function of pro- +jected baseline length was investigated for both the Eastern and +Western arrays using the network calibrated data sets obtained as +output of SYMBA. The calibrated amplitudes of April 7, shown +in Fig.3a for the Eastern array and in Fig.3b for the Western ar- +ray, resemble a Bessel function with a first null at ∼3.0 Gλ and +a second null at ∼6.5 Gλ, corresponding to a thin ring with a +54 µas diameter (Event Horizon Telescope Collaboration et al. +2022b). In Fig.3a, the African baselines, which are represented +in orange, probe the prominent secondary peak. The African +stations also provide short inter-site baselines at the same pro- +jected baseline length as the SMT-LMT baseline, highlighted in +cyan in Fig.3b. In 2017, the SMT-LMT baseline was the short- +est inter-site baseline in the EHT array, which yields the size +and the compact flux density estimation of the source (Issaoun +et al. 2019b). However, 2017 EHT observations have shown that +LMT is a challenging station to calibrate and the determination +of the compact flux is required to establish constraints on the data +(Event Horizon Telescope Collaboration et al. 2019e, 2022b). +Since 2021, NOEMA and KP have added short baselines to PV +and SMT, respectively, useful for amplitude calibration. Thus, +the African baselines shorter than 2Gλ are important for the EHT +imaging process as they can contribute to compute the size and +the total compact flux density of the source. +4. Results from imaging +From the filling fraction study with the 2022EHT array + Africa, +we estimated new time regions offered in the Eastern array to +perform dynamical imaging. Here, we show the static and dy- +namical reconstructions from the GRMHD datasets generated +with SYMBA using the Eastern array + Africa and Western ar- +ray. Moreover, we present the dynamical reconstructions ob- +tained from the hotspot model, which lets us test the capability +of the array to image coherent motion or flares in Sgr A*. Unlike +the (u, v) coverage inspection, the following images are obtained +without the additional 2 hours observing Sgr A* provided by the +African stations. In this way, we compare the capabilities of the +two subarrays to image Sgr A* for the same observing time. +4.1. GRMHD static reconstructions +Fig. 4 shows the static images reconstructed from the GRMHD +datasets for the different array configurations. The synthetic im- +ages were compared with the time-averaged image of the SANE +simulation (first column), which was convolved with a Gaus- +sian kernel with Full Width Half Maximum (FWHM) of 0.6 × +clean beam. As described in Sec. 2.3, the static images were pro- +duced using the eht-imaging package. We corrected for the ef- +fect of the diffractive scattering with the eht − imaging deblur +function (Event Horizon Telescope Collaboration et al. 2022b), +which divides the interferometric visibilities by the Sgr A* scat- +tering kernel. +Because the Eastern array without the African stations does +not have sufficient coverage toward Sgr A*, as we note from the +filling fraction analysis, it is not able to resolve its black hole +shadow. The static reconstruction of Sgr A* significantly im- +proves when the AMT is added to the Eastern array, producing +an image with a clear evidence of the ring-like structure. The im- +age robustness increases with the Eastern array + Africa, indeed +the artifacts present in the northwest and northeast of the ring +are less evident than in the Eastern array + AMT image. The av- +eraged reconstruction using the Western array is also illustrated +in the right-most side of the figure. The subarray is capable of +reconstructing the black hole shadow, but with a lower quality +than the Eastern array with the African stations. The fidelity of +Article number, page 5 of 11 + +1.4 +2022EHT+AMT +2022EHT +1.2 +Normalized filling fraction +1.0 +0.8 +0.6 +0.4 +2.4 hr +1.0hr 0.8 hr +1.0hr +2.0 hr +0.2 +17 +19 +21 +23 +1 +3 +Time GMST (hr)2022EHT+Africa +2022EHT+AMT +2022EHT +1.2 +1.0 +0.8 +0.6 +0.4 +7.4 hr +1.0hr +2.0 hr +0.2 +17 +19 +21 +23 +1 +3 +Time GMST (hr)A&A proofs: manuscript no. core +0 +2 +4 +6 +8 +Projected Baseline Length (G ) +10 +4 +10 +3 +10 +2 +10 +1 +100 +Correlated Flux Density (Jy) +Africa baselines +other baselines +(a) +0 +2 +4 +6 +8 +Projected Baseline Length (G ) +10 +4 +10 +3 +10 +2 +10 +1 +100 +Correlated Flux Density (Jy) +SMT-LMT baseline +other baselines +(b) +Fig. 3: Correlated flux density as a function of baseline length +for the Eastern (a) and Western (b) arrays. The African baselines +(in orange) will contribute to probe the secondary peak, but also +add short baselines to the array, at comparable projected baseline +lengths to the SMT-LMT baseline (cyan). The shortest inter-site +baselines are needed to estimate the extent and the total compact +flux density of the source. +the reconstructions using the different array configurations well +represents the filling fraction trend reported in Fig. 2 and dis- +cussed in Sec. 3.2. +In typical static imaging, the full observing track is used +to produce the final averaged image. In Fig. 4, we show seg- +mented time-averaged reconstructions obtained with the East- +ern and Western arrays individually with the purpose of examin- +ing the African station impact on imaging the static structure of +Sgr A*. The high-fidelity average image from the full 2022EHT ++ Africa array is illustrated on the left of Fig. 5, while on the +right we show the static reconstruction using the 2022EHT ar- +ray (see for comparison the representative model of Sgr A* in +Fig. 4). The 2022EHT + Africa average image is used as the +prior and initial image for the RML dynamical imaging of the +GRMHD data sets presented in the next section. +4.2. GRMHD dynamical reconstructions +Movies of Sgr A* were produced with the dynamical imaging +algorithm introduced in Sec. 2. Based on the candidate time re- +gions with good (u, v) coverage explored in Sec. 2, we produced +movies for the Eastern and Western arrays, separately. To per- +form dynamical imaging on the GRMHD data sets, which con- +tain visibilities on a scan basis, we chose large time periods of +∼5.7 hours, specifically from 17 GMST to 22.7 GMST for the +Eastern array and from 22.7 GMST to 4.1 GMST for the West- +ern array. The visibilities were averaged every 1 min to enhance +the signal-to-noise ratio. The GRMHD simulation movie, which +has a frame duration of 200 seconds, was synchronized to the +reconstructed movies, which have a frame separation of 1 min. +The synchronized model movie was created by averaging over +the model frames that fall between the start and the end of the +observed frame. In this way, we could estimate the NRMSE and +NXCORR between the ground truth movie and the reconstructed +movie frame by frame and select the data term and regularizer +weights that minimize the NRMSE and maximize the NXCORR. +In Fig. 6, we illustrate five snapshots of the movies recon- +structed for the Eastern array + Africa (second row) and for the +Western array (fourth row), and the corresponding frames of the +SANE model. Each snapshot timestamp is shown at the top of +the images. As for the static imaging, the reconstructions are +descattered, by deblurring the interferometric data. The model +movie was blurred using a Gaussian with a FWHM of 0.9 × +clean beam of the 2022EHT + Africa data sets, while the recon- +structions were blurred with a FWMH of 0.6 × clean beam. A +lower blurring fraction is needed for the reconstructions because +the dynamical imaging process inherently produces smoother +structure. +The dynamical reconstructions generated with the Eastern +array + Africa reproduce accurately the ring-like structure of the +GRMHD simulation, while a less solid performance is obtained +with the Western array. The reported NXCORR values in the +bottom of the images confirm the robustness of the Eastern array ++ Africa reconstructions. The NRMSE values are also consistent +with the general goodness trend of the reconstructions. +We use GRMHD simulations of Sgr A* to test if the East- +ern array + Africa is able to reconstruct the main ring structure +and its brightness distribution. GRMHD models reproduce a qui- +escent yet turbulent accretion flow and are not representative of +coherent motion of features expected in the event of flaring activ- +ity. Moreover, GRMHD models are complex and challenging to +reconstruct due to the large amplitudes in the variability (Event +Horizon Telescope Collaboration et al. 2022d), making it diffi- +cult to recognize the rotation of individual features. Dynamical +imaging using a simple hotspot model, shown in the next sec- +tion, allows us to easily investigate the capability of the array to +reconstruct coherent motion in Sgr A* in the event of flares. +4.3. Hotspot dynamical reconstructions +Fig. 7 shows five snapshots of the dynamical reconstructions +generated using as ground truth the hotspot crescent model. +In the first row, we present the synchronized model snapshots, +while the Eastern array + Africa and Western array dynamical +reconstructions are illustrated in the second and third row, re- +spectively. Similarly to the GRMHD models, we identified the +data terms and regularizer weights that maximize the similari- +ties between the model and the reconstruction snapshots, exploit- +ing the NXCORR and NRMSE metrics. Unlike the GRMHD re- +constructions, the visibilities are separated by ∼30 seconds and +the dynamical imaging was performed in narrow time regions of +about 1.7 hours. In particular, for the Eastern array + Africa this +was chosen to be from 21 to 22.7 GMST, which corresponds to +the best time window offered by the subarray. For the Western +array, the best period is given between the 1.5 and 3.2 GMST. +The five snapshots are separated by almost 0.1 hour in order to +represent the hotspot orbit, which is completed in ∼0.5 hours +(i.e., 27 minutes). As confirmed by the NXCORR (reported in +the figure) and the NRMSE, the individual frames produced in +Article number, page 6 of 11 + +Noemi La Bella et al.: Sgr A* dynamical imaging with an African extension to the EHT +Fig. 4: Time-averaged reconstructions of Sgr A* obtained from the GRMHD synthetic observations for the different array configura- +tions. The leftmost image shows the static representation of the GRMHD simulation used as ground truth movie. The Eastern array +without the AMT (second image) does not resolve the shadow of the black hole. The addition of the AMT significantly impacts +the fidelity of the reconstruction, and a further improvement is obtained with the African array (third image). The rightmost image +shows the averaged reconstruction produced using the Western array alone. The images were blurred with a Gaussian FWHM equal +to 0.6 × clean beam of the 2022EHT + Africa data set. +Fig. 5: Time-averaged reconstructions of GRMHD simulations +of Sgr A* with the 2022EHT + Africa and 2022EHT arrays. +The ground truth model is shown in the first column of Fig. 4. +The 2022EHT + Africa array produces a higher fidelity image, +which is used as the prior for the dynamical imaging. +the Eastern array + Africa time region are more accurate than +in the Western array window. Indeed, in the latter, the snapshots +present pronounced northeast and southwest imaging artifacts. +Both subarrays are capable of reconstructing the motion of the +hotspot, confirming that the addition of the African stations to +the EHT array provides a new time window in the first half of the +observation to detect rapid coherent flux variations in Sgr A*’s +accretion flow or jet. In order to effectively establish the capa- +bility of the array in reproducing the flare motion, we developed +two methods that evaluate the robustness of our dynamical im- +ages. For the two subarrays, we investigate the ability to recover +the flux density profile and the time-dependent rotational veloc- +ity of the hotspot. The two methods are described in Sec. 4.3.1 +and in Sec. 4.3.2. +4.3.1. Method 1: Flux density profile +To assess the ability of the Eastern array + Africa to reconstruct +the flux density around the crescent model, we calculated the +flux density pixel by pixel as a function of the position angle +for each snapshot. We selected the ring from which to extract +the flux using the hough_ring function in the eht-imaging +library, which finds circles in an image according to the pixel +brightness distribution. The choice was made giving as input the +time-averaged model image. Thus, for each model snapshots and +reconstructed frames, the flux density was estimated within a ra- +dius of 32µas and in sectors 10 degrees wide. In Fig. 8, we show +the flux density as a function of the angle for five snapshots of +the ground truth model (in green and also illustrated in the lower +panel of the image), of the Eastern array + Africa movie (in red) +and of the Western array movie (in blue). Because of the asym- +metry of the brightness distribution in the crescent model, the +flux profile has a peak in the snapshots when the hotspot is at its +maximum intensity (i.e., third column), while it decreases when +the hotspot is located on the opposite side. From the model snap- +shots and the corresponding flux density profile, we note that the +angular position of the hotspot is correctly determined by this +method. The flux density profiles obtained with the Eastern ar- +ray + Africa and Western array recover quite well the hotspot +motion, both in term of intensity and in position angle. +4.3.2. Method 2: Rotational velocity profile +Additionally, we computed the rotational velocity of the hotspot +as a function of time. This rotation (in degrees per minute) is de- +fined as the degree of rotation for each frame i with respect to the +fifth subsequent frame j. In order to measure it, we rotated frame +i in steps of two degrees across a range of angles. We calculated +the NXCORR (i.e., the image correspondence) with respect to +frame j at each rotation angle. The angle at which the NXCORR +is maximized between the two frames divided by the time dura- +tion between frames i and j gives us the rotational velocity. We +measured the rotational velocity of the hotspot every five frames, +which lets us reconstruct its motion. As the hotspot completes its +orbit every 27 min and the frame separation of the reconstructed +movie is ∼30 seconds, the rotation every five frames (∼33◦) is +easier to measure than the rotation per frame (∼6.6◦). +The rotational velocity obtained for the Eastern array + +Africa and the Western array movies are shown in the left and +right of Figure 9 in red and in blue, respectively. The hotspot ve- +locity measured from the model movie is represented in green. +As in the case of the flux profile, the method represents the asym- +metric brightness distribution of the crescent model. Indeed, the +frames with the maximum intensity of the hotspot have a maxi- +mum value of the rotational velocity, which drops to zero when +the hotspot is not present. The negative values of the velocity +Article number, page 7 of 11 + +Model +Eastern +Eastern + AMT +Eastern + Africa +Western +0 +0 +60 μas +0.5 +1.0 +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +1.5 +Tb[K] +1e10 +Tb[K] +1e10 +Tb[K] +1e10 +Tb[K] +1e10 +Tb[K] +1e102022EHT + Africa +2022EHT +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +T,[K] +1e10 +T,[K] +1e10A&A proofs: manuscript no. core +Fig. 6: Dynamical reconstructions obtained from the GRMHD data sets. The first row shows five snapshots of the GRMHD sim- +ulation taken in the Eastern array (17-22.7 GMST), the second row represents the respective dynamical reconstructions using the +Eastern array + Africa. In a similar way, the third row and forth row illustrate the GRMHD frame simulations and the correspondent +frame reconstructions using the Western array (22.7-4.1 GMST). The blurring utilized for the GRMHD simulation is 0.6 × clean +beam. Higher quality dynamical reconstructions are produced by the Eastern array + Africa, also confirmed by the NXCORR metric +reported at the bottom of each image. The numbers on the top of the GRMHD simulation snapshots represent the frame time. +are artifact produced by the method. In particular, these unphys- +ical features are generated for each period of the hotspot movie, +when we compare the last frame that contains the hotspot and the +fifth frame that presents only the crescent emission. Comparing +the rotational velocity curves derived from the Eastern array + +Africa and the Western array movies with the model simulation, +we find that the flare variability is most accurately recovered in +the Eastern time window. +5. Summary and conclusions +We generated synthetic data of Sgr A* with the current EHT +array and two stations in the African continent, the AMT and +the CNI telescope. We have evaluated the capability of the +EHT Eastern subarray with the African sites (17-22.7 GMST) +to produce movies of Sgr A* and compared it to the Western +subarray (22.7-4.1 GMST). The data sets were created from +ray-traced images of a SANE GRMHD simulation, which is +representative of the quiescent yet turbulent black hole accretion +Article number, page 8 of 11 + +GRMHD simulation +18.0 GMST +19.1 GMST +21.3 GMST +21.8GMST +22.3GMST +60 μas +Eastern array + Africa +NXCORR0.994 +NXCORR0.993 +NXCORR0.985 +NXCORR0.990 +NXCORR0.993 +GRMHD simulation +23.2 GMST +0.4 GMST +1.4 GMST +1.8GMST +3.1 GMST +C +Westernarray +NXCORR0.981 +NXCORR0.987 +NXCORR0.990 +NXCORR0.986 +NXCORR0.974Noemi La Bella et al.: Sgr A* dynamical imaging with an African extension to the EHT +Fig. 7: Dynamical reconstructions generated using the hotspot synthetic data. In the first row we show five snapshots of the hotspot +model movie. The hotspot performs a full rotation every 27 mins. The frames were chosen to represent a complete orbit. The +reconstructions obtained from the dynamical imaging using the Eastern array + Africa and Western array are shown in the second +and third row, respectively. The movies were generated in a time window of about 1.7 hours (21-22.7 GMST for the Eastern array, +1.5-3.2 GMST for the Western). The NXCORR values estimated for the reconstructions is reported in the bottom of each images. +The temporal evolution is available as an online movie. +flow, and from a crescent hotspot model to test the imaging +performance of the array in reconstructing coherent motion +from flaring activity in Sgr A*. +We found that the AMT increases the resolution of the EHT +array via long baselines with the Arizona and Mexico sites, while +short baselines provided by the African extension to the EHT +constrain the compactness and extent of the source on larger +scales. We estimated the Fourier filling fraction with the EHT ar- +ray and the Africa telescopes to investigate the presence of good +time regions to perform dynamical imaging. We found that the +added baselines offer an optimal time window of about 7 hours +in the Eastern array, allowing to produce high-fidelity movies of +Sgr A* from the very start of a typical observing track. This in- +creases the time in which dynamical imaging is possible by a +factor > 4. In comparison, Farah et al. (2022) demonstrated that +with the 2017 EHT array, the only time period in which we are +able to reconstruct the variability of the source is from ∼01:30 +GMST to ∼03:10 GMST, with the Western array. +Our static reconstructions of the GRMHD simulation con- +firm the importance of the AMT in imaging Sgr A*. Without +the AMT, the data set generated with the current EHT configu- +ration is not able to reproduce a physical image of the black hole +shadow in the Eastern array window. Including the African sites, +we can perform high-fidelity imaging of Sgr A* with reduced +artifacts. Additionally, we produced GRMHD dynamical recon- +structions limited to the best Eastern and Western time regions. +The African stations enable accurate frame reconstructions of +the ring-like structure when included in the Eastern array. Since +the rotation of individual features is difficult be recognized in the +turbulent flow of GRMHD simulations, we performed a hotspot +dynamical imaging analysis to test the capability of the different +arrays to reconstruct coherent motion mimicking flaring activity +in Sgr A*. Compared to the 2022EHT array, the African stations +open a new time window in the Eastern array that can be used +to reconstruct motion in the accretion disk. We developed two +methods involving the determination of the flux density profile +and the rotational velocity of the hotspot to establish the suc- +cessful performance of the enhanced Eastern array in reproduc- +ing the motion in Sgr A*. Our results show the impact of adding +stations in the African continent in increasing the time-variable +(u, v) coverage of the EHT toward Sgr A*. The African exten- +sion will be crucial for future EHT observations to study accu- +Article number, page 9 of 11 + +Hotspot movie +t=0.05hr +t=0.13hr +t=0.23hr +t=0.31 hr +t=0.54 hr +70μas +Easternarray+Africa +NXCORR0.957 +NXCORR0.968 +NXCORR0.955 +NXCORR0.970 +NXCORR0.953 +Western array +NXCORR0.906 +NXCORR0.923 +NXCORR0.892 +NXCORR0.906 +NXCORR0.952A&A proofs: manuscript no. core +Fig. 8: Flux density (Jy/pixel) in function of the angle (degrees) estimated in five snapshots of the model movie (in green), of the +Eastern array + Africa movie (in red), and of the Western array movie (in blue). The brightness distribution was estimated using a +ring with outer radius of 32 µas, divided in sectors 10 degrees wide. The five frames of the model simulation from where the flux +densities were extracted are shown in the bottom panel. +Fig. 9: Rotational velocity (degree per minute) in function of the time for the Eastern + Africa array movie (left) and for the Western +array movie (right). In green, the rotational velocity for the hotspot movie simulation. The profile were obtained by searching for the +angle that maximize the similarity between each frame and the subsequent fifth frame. The Eastern array + Africa movie presents a +more robust reconstruction of the hotspot rotation than the Western array. The negative values of the rotation are artifacts produced +by the method utilized. +rately the time-variable source at our Galactic Center through +high-fidelity movies across an observing track. +Acknowledgements. We thank Oliver Porth for performing the ray-tracing for +the GRMHD simulation used. This publication is part of the project Dutch Black +Hole Consortium (with project number 1292.19.202) of the research programme +NWA which is (partly) financed by the Dutch Research Council (NWO). SI +is supported by Hubble Fellowship grant HST-HF2-51482.001-A awarded by +the Space Telescope Science Institute, which is operated by the Association of +Universities for Research in Astronomy, Inc., for NASA, under contract NAS5- +26555. FR is supported by NSF grants AST-1935980 and AST-2034306, and the +Black Hole Initiative at Harvard University, made possible through the support +of grants from the Gordon and Betty Moore Foundation and the John Templeton +Foundation. The opinions expressed in this publication are those of the author(s) +and do not necessarily reflect the views of the Moore or Templeton Foundations. +CMF is supported by the DFG research grant “Jet physics on horizon scales and +beyond" (Grant No. FR 4069/2-1) The simulations were performed on LOEWE +at the CSC-Frankfurt, Iboga at ITP Frankfurt and Pi2.0 at Shanghai Jiao Tong +University. +References +Backes, M., Müller, C., Conway, J. 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D, 94, 084025 +Article number, page 11 of 11 + diff --git a/1tFIT4oBgHgl3EQf4CvP/content/tmp_files/load_file.txt b/1tFIT4oBgHgl3EQf4CvP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3c079f64ac8d5819c5b351561ef8f7dd71444ca8 --- /dev/null +++ b/1tFIT4oBgHgl3EQf4CvP/content/tmp_files/load_file.txt @@ -0,0 +1,883 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf,len=882 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' core ©ESO 2023 January 30, 2023 Expanding Sgr A* dynamical imaging capabilities with an African extension to the Event Horizon Telescope Noemi La Bella1, Sara Issaoun2, 3, 1, Freek Roelofs2, 4, 1, Christian Fromm5, 6, 7, and Heino Falcke1 1 Department of Astrophysics, Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Box 9010, 6500 GL Nijmegen, The Netherlands e-mail: n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='labella@astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='nl 2 Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA 3 NASA Hubble Fellowship Program, Einstein Fellow 4 Black Hole Initiative, Harvard University, 20 Garden Street, Cambridge, MA 02138, USA 5 Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Emil-Fischer-Strasse 31, 97074 Würzburg, Germany 6 Institut für Theoretische Physik, Goethe Universität, Max-von-Laue-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 1, D-60438 Frankfurt, Germany 7 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany January 30, 2023 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The Event Horizon Telescope (EHT) has recently published the first images of the supermassive black hole at the center of our Galaxy, Sagittarius A* (Sgr A*).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Imaging Sgr A* is plagued by two major challenges: variability on short (approximately minutes) timescales and interstellar scattering along our line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' While the scattering is well studied, the source variability continues to push the limits of current imaging algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In particular, movie reconstructions are hindered by the sparse and time-variable coverage of the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In this paper, we study the impact of the planned Africa Millimetre Telescope (AMT, in Namibia) and Canary Islands telescope (CNI) additions to the time-dependent coverage and imaging fidelity of the EHT array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' This African array addition to the EHT further increases the eastwest (u, v) coverage and provides a wider time window to perform high-fidelity movie reconstructions of Sgr A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We generated synthetic observations of Sgr A*’s accretion flow and used dynamical imaging techniques to create movie reconstructions of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' To test the fidelity of our results, we used one general-relativistic magneto-hydrodynamic model of the accretion flow and jet to represent the quiescent state and one semi-analytic model of an orbiting hotspot to represent the flaring state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We found that the addition of the AMT alone offers a significant increase in the (u, v) coverage, leading to robust averaged images during the first hours of the observating track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Moreover, we show that the combination of two telescopes on the African continent, in Namibia and in the Canary Islands, produces a very sensitive array to reconstruct the variability of Sgr A* on horizon scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We conclude that the African expansion to the EHT increases the fidelity of high-resolution movie reconstructions of Sgr A* to study gas dynamics near the event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Black hole physics - Galaxy: center - Instrumentation: high angular resolution - interferometers - Techniques: image processing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Introduction The Event Horizon Telescope (EHT) collaboration has recently published the first images of the black hole shadow of Sagittar- ius A* (Sgr A*), the supermassive black hole (SMBH) at the center of the Milky Way, characterized by an asymmetric bright ring of (52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='6) µas (Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The ring-like morphology was recovered in over 95% of the best-fit images produced from 2017 April 6 and 7 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The EHT images of Sgr A* are consistent with the prediction of a shadow for a Kerr black hole (Falcke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2000) with a mass M ∼ 4 × 106M⊙ at a distance D ∼ 8 kpc, which were accurately measured by high-resolution infrared studies of stellar orbits in the Galactic Center (Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Do et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In 2019, the EHT collaboration delivered the first ever image of a black hole shadow in the giant ellipti- cal galaxy M87 (Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The main difference between the two SMBHs is their mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' M87* is about 1600 times more massive than Sgr A* and thus, it has a longer gravitational timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In fact, the period of the innermost stable circular orbit (ISCO) for a nonrotating black hole as massive as M87* is ∼30 days, while for Sgr A* it is ∼30 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' As a consequence, the estimation of the ring diameter of Sgr A* is more uncertain than in M87* and we need movies to properly study the plasma motion surrounding the black hole on this short orbital timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The variability of Sgr A* required a reformulation of the static source assumptions in the interferometric Earth aperture synthesis method and imaging algorithms used for M87*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In par- ticular, to generate a typical static image of Sgr A*, a variabil- ity noise budget needs to be added, while a dynamical imaging process is required to capture the evolving structure of Sgr A* (Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Because of the sparsity of the EHT array, time slots with good (u, v) cover- age were selected to perform dynamical studies on the variability (Farah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Article number, page 1 of 11 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='11384v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='IM] 26 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' core The SMBH also presents flare events observed across the electromagnetic spectrum in the last decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' An accurate study of the millimeter light curves during the 2017 EHT campaign was done by Wielgus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In particular, the authors found excess variability on 2017 April 11, following a flare observed in the X-ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Subsequent studies on polarized light curves with the Atacama Large Millimeter/submillimeter Array (ALMA) on the same day (Wielgus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2022b) revealed the presence of a hotspot orbiting Sgr A* clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In addition to its quiescent variability, imaging Sgr A* is a complex process because the very long baseline interferometry (VLBI) observations are affected by scattering in the interstel- lar medium along our line of sight toward the Galactic Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The consequent diffractive and refractive effects of the scatter- ing were mitigated by modeling their chromatic properties in the radio band (see Psaltis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Issaoun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2019a, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2022b, for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Eight telescopes at six geographic locations formed the 2017 EHT array configuration that led to the first images of Sgr A* and M87*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Since 2017, the array has doubled in bandwidth and increased the number of baselines with three new telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' As of 2022, the EHT has consisted of eleven telescopes at eight locations: ALMA and the Atacama Pathfinder Experiment (APEX) on the Llano de Chajnantor in Chile;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' the Large Millime- ter Telescope (LMT) Alfonso Serrano on the Volcán Sierra Ne- gra in Mexico;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' the James Clerk Maxwell Telescope (JCMT) and Submillimeter Array (SMA) on Maunakea in Hawai’i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' the In- stitut de Radioastronomie Millimétrique 30-m telescope on Pico Veleta (PV) in Spain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' the Submillimeter Telescope (SMT) on Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Graham and the 12-m telescope on Kitt Peak (KP) in Arizona;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' the South Pole Telescope (SPT) in Antarctica;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' the Northern Ex- tended Millimeter Array (NOEMA) in France;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' and the Green- land Telescope (GLT) at Thule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' This new configuration offers increased sensitivity of the array and will enable higher-fidelity images of Sgr A* and M87*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' However, all new telescopes are in the northern hemisphere and are less effective for imaging southern sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Additional telescopes are being considered to expand the capabilities of the array, especially on the African continent, which offers prime site locations to increase the (u, v) coverage toward Sgr A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In this work, we consider two additions to the EHT in the African region: one in Namibia and one in the Canary Islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The Africa Millimetre Telescope (AMT), planned on Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Gams- berg (2,347 m a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=') in Namibia, will be the first millimeter-wave telescope in Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The project to add this telescope is currently underway, and aims to relocate the decommissioned 15-meter SEST telescope in Chile to Gamsberg in the next years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' This site will offer low precipitable water vapor levels during the typical fall and spring EHT campaign seasons (Backes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2016) and its strategical position in the southern hemisphere at the same lat- itude as ALMA provides important eastwest baselines to Chile and northsouth baselines to Europe, significantly increasing the snapshot coverage in the first half of a typical observing night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The island of La Palma in the Canary Islands (2,000 m a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=') has dry weather conditions throughout the year (Raymond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2021) and offers a prime location to provide mid-range coverage between Namibia and Europe that is crucial to constrain source compactness and extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Furthermore, the site’s established in- frastructure from existing observatories would make an addi- tional telescope easily feasible and well supported, making it an ideal candidate for a telescope location in the near term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We present simulated dynamical images of Sgr A* using the 2022EHT array and an African extension including two new telescopes: the 15-meter AMT, and the Canary Islands telescope, CNI, on the island of La Palma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We assume the dish size of CNI to be six meters, following the design concept for a next- generation EHT facility in the long term (Doeleman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We investigate the impact of the AMT and CNI stations on imag- ing Sgr A* in both quiescent and flaring states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The methods we use can easily be expanded to other EHT configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The paper is organized as follows: in Section 2, we describe the synthetic generation pipeline and imaging algorithms used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In Section 3, we present the African extension to the EHT and its contribution to snapshot and full-track (u, v) coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In Sec- tion 4, we show the static and dynamical reconstructions ob- tained with the enhanced EHT array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Finally, in Section 5, we discuss the advantages of the African extension to the array in producing high-fidelity movies of Sgr A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' GRMHD ground truth movies The quiescent state of the plasma flow of Sgr A* was reproduced by generating synthetic data from general relativistic magneto- hydrodynamic (GRMHD) simulations at 230 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The typical range of simulations used to study Sgr A* include two classes of models: magnetically arrested disk (MAD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Igumenshchev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Narayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2003) and Standard And Normal Evolu- tion (SANE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Narayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2012) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The SANE mode is characterized by a weak and turbulent magnetic field crossing the hemisphere of the event horizon, while the MAD mode has high magnetic flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The recent EHT Sgr A* results have shown that GRMHD simulations are more variable than the data (Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2022d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Because SANE models are less variable than MAD models, they are more rep- resentative of the degree of variability in Sgr A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We thus used a SANE model for our quiescent state reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The simulation was generated with the GRMHD code BHAC Porth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Olivares et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We initialized a torus in hydrodynamic equilibrium where the inner edge is located at 6 M (where M is the gravitational timescale GM/c3) and the pressure maximum is found at 13 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We set a black hole spin of a⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='9375 and an adiabatic index ˆγ = 4/3 and per- formed the simulations on spherical grid (r, θ, φ) with resolution of 512 × 192 × 192 and three layers of adaptive mesh refinement (AMR) using logarithmic Kerr-Schild coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' For more de- tails on the simulations see Fromm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We evolve the simulations until 30000 M, which ensures a quasi steady-state in the mass accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The radiative transfer calculations were performed with the GRRT code BHOSS Younsi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' (2012, 2016, 2020, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We used a field of view of 200 µas together with a black hole mass of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='14 × 106 M⊙ at a distance of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='127 kpc (Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2022d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The im- ages were created assuming a viewing angle ϑ = 10◦ and a nu- merical resolution of 4002 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Since the electron temperature is not evolved during the GRMHD simulations we computed their temperature using the R − β description of Mo´scibrodzka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' (2016) where we set Rlow = 1 and Rhigh = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In order to adjust the simulations to the observations, we iterated over the mass accretion rate to provide an average flux density of ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='4 Jy at 230 GHz in a time window of 5000 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Two time windows were used (20-25 kM and 25-30 kM) and individually normal- ized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The 16-hour movie consists of 300 frames separated by 200 seconds, with a rotation period of the plasma around the black hole of ∼30 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The simulation does not include effects Article number, page 2 of 11 Noemi La Bella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' : Sgr A* dynamical imaging with an African extension to the EHT of interstellar scattering, therefore we characterized those effects using a phase screen toward Sgr A* (see Psaltis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' John- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2018, for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Synthetic data generation The GRMHD synthetic data were produced with the SYMBA 1 software (Roelofs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2020), which reconstructs a model im- age following the same calibration and imaging processes of a realistic observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Given a VLBI array configuration and a specific model as input, the synthetic observations are generated with MeqSilhouette (Blecher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Natarajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2022) and the corrupted raw data are then processed with the VLBI data calibration pipeline rPICARD (Janssen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2019), which is used to calibrate real EHT data (Event Horizon Telescope Col- laboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The calibrated data set can also pass through the network calibration step that solves gains for colo- cated sites using the flux of the source at large scales (Fish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Johnson & Gwinn 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Blackburn 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2019c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Our synthetic data are based on the antenna and weather parameters as measured in the 2017 observations (Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2019d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The weather conditions were extracted from the VLBI monitor server 2, which collects weather data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', ground pressure, ground temperature) from in situ measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The weather conditions used are reported in Table 2 of Roelofs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' (2020), which includes the parameters for the stations that joined the 2017 EHT campaign, and those for the enhanced array, with GLT joining the array in 2018, NOEMA and KP in 2021, and with the planned AMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' As described by the authors, the weather parameter estimation for stations that did not join the 2017 obser- vations was done using the Modern-Era Retrospective Analysis for Research and Applications, version 2 (MERRA − 2) from the NASA Goddard Earth Sciences Data and Information Services Center (Gelaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2017), and the am atmospheric model soft- ware Paine (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We applied the same method to obtain the weather conditions on La Palma, in the Canary Islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Finally, we adopted the observing schedule of 2017 April 7 (Event Hori- zon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2022c), encompassing scans on Sgr A* from the 4 to 15 UT hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' For generating movies of flares in Sgr A*, we used a sim- ulated Gaussian flaring feature with an orbiting period of 27 min around a ray-traced image of a semi-analytic advection- dominated accretion flow (ADAF) model of Sgr A* (model B of Doeleman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The movie at 230 GHz is composed of 100 frames separated by 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='2 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The eht-imaging Python library 3 (Chael et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2016, 2018) was used to gener- ate the hotspot synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The eht-imaging package does not produce realistic VLBI-mm observations as SYMBA, for instance the data are not frequency-resolved, gain effects are not based on physical models, and there are no calibration ef- fects added (more details about the difference between the two pipelines can be found in Event Horizon Telescope Collabora- tion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2019c, Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' As in the case of the GRMHD movies, the synthetic data were based on the 2017 April 7 ob- serving parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Unlike SYMBA, the simulated visibilities are not scan-separated, but have a cadence of 30 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 1 https://bitbucket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='org/M_Janssen/symba 2 https://bitbucket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='org/vlbi 3 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='com/achael/eht-imaging 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Dynamical imaging We imaged the SYMBA synthetic data set using the eht-imaging library, developed specifically for the EHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The imaging algorithm utilizes the regularized maximum likelihood (RML) method, which aims to find an image that minimizes a specified objective function, consisting of data fit quality (χ2) terms, and additional regularizer terms favoring, for example, smooth or sparse image structures (Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2019e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The static assumption based on the Earth rotation aperture synthesis technique, where the source is assumed static during the course of the observation, is not valid in the case of Sgr A* due to its intraday variability (Event Hori- zon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' To tackle this chal- lenge, we use a method called “dynamical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='" The dy- namical imaging algorithm within the eht-imaging package is a generalization of the standard RML method which introduces three dynamical regularizers that enforce time-sensitive proper- ties between snapshot frames (see Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2017, for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' To reconstruct the hotspot movies we used the R∆t reg- ularizer, which imposes a time continuity between frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Since the hotspot model simulates coherent motion of a flare orbit- ing Sgr A*, this regularizer let us reconstruct continuous motion of structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' For the GRMHD movies, we also added the R∆I regularizer, which enforces similarity between the reconstructed frame and a time-averaged image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' As GRMHD simulations de- scribe the turbulent behavior of an accretion flow onto Sgr A*, this regularizer allows us to look for turbulence on top of a static structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' To inspect the capability of the expanded EHT array to re- construct dynamical motion, we selected time windows during the observation for which coverage and filling fraction were op- timized, as was done in Farah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' For the GRMHD sim- ulations, we produced movies of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='7 hours, while for the hotspot movies we chose optimal time windows of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='7 hours where the array offers the best coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' To obtain a good reconstructed movie, larger time windows were required for the GRMHD data set generated with SYMBA, which includes actual scans and gaps between the scans (more details in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Movie quality metrics Two quality metrics were selected to evaluate the fidelity of the reconstructed images: the normalized root-mean-square er- ror (NRMSE) and the normalized cross-correlation (NXCORR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2019e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' NRMSE is more sensitive to pixel-by-pixel differences, while NXCORR is more sensitive to large scale structure (Issaoun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We estimated values for both metrics for each frame of the movie, quantifying the fidelity of the reconstruction as a function of time with respect to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The NRMSE measures similarities per kth pixel and it is de- fined as: NRMSE = �� k(Ik − I′ k)2 � k I2 k , (1) where I′ and I are the intensity of the reconstructed movie frame and the model movie frame, respectively (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', Chael et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Issaoun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' An NRMSE value of zero corresponds to identical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' For given frames I′ and I, NXCORR is given by: NXCORR = 1 N � k (Ik − ⟨I⟩)(I′ k − ⟨I′⟩) σIσI′ , (2) Article number, page 3 of 11 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' core Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 1: Sgr A* (u, v) coverage of the 2017 April 7 EHT observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Seven scans on Sgr A* were added to the original schedule at the beginning of the observation, brought by the introduction of the NOEMA array and the African stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In blue, the cov- erage obtained with the 2022EHT array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The contributions of the AMT and CNI baselines are shown in red and in brown, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The AMT adds long northeast and southwest baselines in- creasing the EHT resolution, while CNI offers shorter baselines to detect large-scale emission and constrain the source extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' where N is the total number of pixels per frame, ⟨I⟩ and ⟨I′⟩ are the mean pixel values and σI, σI′ are the respective standard de- viations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' An NXCORR of 1 corresponds to a perfect correlation between the frames, -1 for anticorrelation, and 0 for no correla- tion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2019e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The African expansion to the EHT In this section, we discuss a potential implementation of the African expansion to the EHT, its (u, v) coverage, and Fourier filling fraction, which let us identify potential time windows to generate movies of Sgr A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We also investigate the location of the new baselines with respect to the position of the two local minima in the correlated flux density profile of a thin ring of 54 µas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' To assess the impact of the new African stations, different array configurations were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We name those configurations as follows: 2022EHT, the current EHT configuration composed of eleven telescopes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2022EHT + AMT, the 2022EHT with the ad- dition of AMT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2022EHT + Africa, the 2022EHT plus the AMT and CNI stations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Eastern array + Africa, the 2022EHT subar- ray until ∼9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5 UT hours (∼22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='7 Greenwich Mean Sidereal Time, GMST), after this time the AMT does not observe Sgr A*;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' West- ern array, the 2022EHT subarray from ∼9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5 UT hours to ∼15 UT hours (∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='1 GMST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' So far, the Western array has been offering the best coverage to produce dynamic reconstructions of Sgr A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' (u, v) coverage Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 1 depicts the Sgr A* (u, v) coverage using the 2017 April 7 observing schedule as a base, enhanced by the addition of NOEMA and KP, which joined the array post-2017, and the two proposed African antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Moreover, the observation was im- posed to start when the source is at an elevation of more than 10 degrees at NOEMA and the African telescopes, allowing us to extend the observation by two hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The 2022EHT baselines are shown in blue, the AMT baselines in red and the CNI base- lines in brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The AMT is a potential southern site to image Sgr A* that adds determinant baselines to the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Specifically, the AMT adds northsouth baselines to PV and NOEMA, east- west baselines to Chile, and a redundancy baseline to ALMA- SPT, since Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Gamsberg is at the same latitude as ALMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Moreover, the AMT increases the resolution in the northeast and southwest, by adding long baselines to LMT and the Arizona stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' On the other hand, the CNI telescope yields new short inter-site baselines to the European sites, PV and NOEMA, con- tributing to the measurement of the source extent, together with the inter-sites SMT-LMT, PV-NOEMA baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In addition, the baseline CNI-AMT provides further northsouth coverage to the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Fourier filling fraction The sparsity and changing coverage of the EHT array affect the accuracy of the dynamical reconstructions of time-variable sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' To produce VLBI movies of Sgr A*, it is thus required to identify time periods with optimal and stable (u, v) coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' For the 2017 Sgr A* results, Farah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' (2022) selected time re- gions using three different metrics and found the best dynamical time period to be from ∼01:30 GMST to ∼03:10 GMST, hence in the Western array window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We utilized one of these metrics, the (u, v) filling fraction, to inspect if new temporal regions are of- fered by the Eastern array + Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The Fourier filling fraction measures the area sampled in the (u, v) plane by the observed visibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Following Farah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' (2022), the (u, v) points were convolved with a circle of radius 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='71/θFOV, with FOV being the field of view adopted for imaging, representing the half-width at half-maximum of a filled disk of uniform brightness on the sky (see Palumbo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2019, for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In our analysis, we calculated the filling fraction normalized to the maximum fill- ing fraction value of the 2022EHT array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' On the left of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2, we show the time-dependent normalized filling fraction for the 2022EHT + AMT array in red, and that of the 2022EHT array in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The colored windows delimit time regions in which the fill- ing fraction is persistently above the 70% 2022EHT maximum threshold (dashed grey line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Time windows below this threshold do not have sufficient coverage to produce high-fidelity movies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The 2022EHT array provides good time regions in the Western array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Notably, our results confirm the 01:30 GMST to 03:10 GMST best-time window obtained from the 2017 array selective dynamical imaging analysis (Farah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The AMT adds three additional optimal time periods (red areas) in the Eastern array, of almost 4 hours in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Furthermore, on the right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2 we show a further increase in the Fourier filling area achieved by the combination of the CNI (brown) and AMT sites (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', with the 2022EHT array + Africa) leading to a persistent time block of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='4 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Therefore, the African stations will provide signif- icantly improved (u, v) coverage and stability for the Eastern ar- ray, increasing the ability to study rapid variations of the source at the beginning of the observing track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Article number, page 4 of 11 10 CNI baselines AMT baselines 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5 2022EHT 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5 [G^] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5 10 10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5 10 u [G入]Noemi La Bella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' : Sgr A* dynamical imaging with an African extension to the EHT Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2: Time-dependent Fourier filling fraction normalized by the maximum Fourier filling of the 2022EHT array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The curves represent the filling fraction of the 2022EHT array, 2022EHT + AMT array and 2022EHT + Africa array, in blue, red and brown, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The dashed gray line corresponds to the lower limit used for identifying good time windows to perform dynamical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The optimal time regions for the current EHT array are shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The 2022EHT + AMT adds three time windows (red areas) of ∼4 hours in total, while the 2022EHT + Africa array (brown area) produces a time window of ∼7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='4 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Correlated flux density profile The correlated flux density (in Jy) of Sgr A* as a function of pro- jected baseline length was investigated for both the Eastern and Western arrays using the network calibrated data sets obtained as output of SYMBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The calibrated amplitudes of April 7, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='3a for the Eastern array and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='3b for the Western ar- ray, resemble a Bessel function with a first null at ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 Gλ and a second null at ∼6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5 Gλ, corresponding to a thin ring with a 54 µas diameter (Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='3a, the African baselines, which are represented in orange, probe the prominent secondary peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The African stations also provide short inter-site baselines at the same pro- jected baseline length as the SMT-LMT baseline, highlighted in cyan in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In 2017, the SMT-LMT baseline was the short- est inter-site baseline in the EHT array, which yields the size and the compact flux density estimation of the source (Issaoun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' However, 2017 EHT observations have shown that LMT is a challenging station to calibrate and the determination of the compact flux is required to establish constraints on the data (Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2019e, 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Since 2021, NOEMA and KP have added short baselines to PV and SMT, respectively, useful for amplitude calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Thus, the African baselines shorter than 2Gλ are important for the EHT imaging process as they can contribute to compute the size and the total compact flux density of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Results from imaging From the filling fraction study with the 2022EHT array + Africa, we estimated new time regions offered in the Eastern array to perform dynamical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Here, we show the static and dy- namical reconstructions from the GRMHD datasets generated with SYMBA using the Eastern array + Africa and Western ar- ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Moreover, we present the dynamical reconstructions ob- tained from the hotspot model, which lets us test the capability of the array to image coherent motion or flares in Sgr A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Unlike the (u, v) coverage inspection, the following images are obtained without the additional 2 hours observing Sgr A* provided by the African stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In this way, we compare the capabilities of the two subarrays to image Sgr A* for the same observing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' GRMHD static reconstructions Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 4 shows the static images reconstructed from the GRMHD datasets for the different array configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The synthetic im- ages were compared with the time-averaged image of the SANE simulation (first column), which was convolved with a Gaus- sian kernel with Full Width Half Maximum (FWHM) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='6 × clean beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='3, the static images were pro- duced using the eht-imaging package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We corrected for the ef- fect of the diffractive scattering with the eht − imaging deblur function (Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2022b), which divides the interferometric visibilities by the Sgr A* scat- tering kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Because the Eastern array without the African stations does not have sufficient coverage toward Sgr A*, as we note from the filling fraction analysis, it is not able to resolve its black hole shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The static reconstruction of Sgr A* significantly im- proves when the AMT is added to the Eastern array, producing an image with a clear evidence of the ring-like structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The im- age robustness increases with the Eastern array + Africa, indeed the artifacts present in the northwest and northeast of the ring are less evident than in the Eastern array + AMT image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The av- eraged reconstruction using the Western array is also illustrated in the right-most side of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The subarray is capable of reconstructing the black hole shadow, but with a lower quality than the Eastern array with the African stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The fidelity of Article number, page 5 of 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='4 2022EHT+AMT 2022EHT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='2 Normalized filling fraction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='4 hr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0hr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='8 hr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0hr 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 hr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='2 17 19 21 23 1 3 Time GMST (hr)2022EHT+Africa 2022EHT+AMT 2022EHT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='4 hr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0hr 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 hr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='2 17 19 21 23 1 3 Time GMST (hr)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' core 0 2 4 6 8 Projected Baseline Length (G ) 10 4 10 3 10 2 10 1 100 Correlated Flux Density (Jy) Africa baselines other baselines (a) 0 2 4 6 8 Projected Baseline Length (G ) 10 4 10 3 10 2 10 1 100 Correlated Flux Density (Jy) SMT-LMT baseline other baselines (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 3: Correlated flux density as a function of baseline length for the Eastern (a) and Western (b) arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The African baselines (in orange) will contribute to probe the secondary peak, but also add short baselines to the array, at comparable projected baseline lengths to the SMT-LMT baseline (cyan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The shortest inter-site baselines are needed to estimate the extent and the total compact flux density of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' the reconstructions using the different array configurations well represents the filling fraction trend reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2 and dis- cussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In typical static imaging, the full observing track is used to produce the final averaged image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 4, we show seg- mented time-averaged reconstructions obtained with the East- ern and Western arrays individually with the purpose of examin- ing the African station impact on imaging the static structure of Sgr A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The high-fidelity average image from the full 2022EHT + Africa array is illustrated on the left of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 5, while on the right we show the static reconstruction using the 2022EHT ar- ray (see for comparison the representative model of Sgr A* in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The 2022EHT + Africa average image is used as the prior and initial image for the RML dynamical imaging of the GRMHD data sets presented in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' GRMHD dynamical reconstructions Movies of Sgr A* were produced with the dynamical imaging algorithm introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Based on the candidate time re- gions with good (u, v) coverage explored in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2, we produced movies for the Eastern and Western arrays, separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' To per- form dynamical imaging on the GRMHD data sets, which con- tain visibilities on a scan basis, we chose large time periods of ∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='7 hours, specifically from 17 GMST to 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='7 GMST for the Eastern array and from 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='7 GMST to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='1 GMST for the West- ern array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The visibilities were averaged every 1 min to enhance the signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The GRMHD simulation movie, which has a frame duration of 200 seconds, was synchronized to the reconstructed movies, which have a frame separation of 1 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The synchronized model movie was created by averaging over the model frames that fall between the start and the end of the observed frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In this way, we could estimate the NRMSE and NXCORR between the ground truth movie and the reconstructed movie frame by frame and select the data term and regularizer weights that minimize the NRMSE and maximize the NXCORR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 6, we illustrate five snapshots of the movies recon- structed for the Eastern array + Africa (second row) and for the Western array (fourth row), and the corresponding frames of the SANE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Each snapshot timestamp is shown at the top of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' As for the static imaging, the reconstructions are descattered, by deblurring the interferometric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The model movie was blurred using a Gaussian with a FWHM of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='9 × clean beam of the 2022EHT + Africa data sets, while the recon- structions were blurred with a FWMH of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='6 × clean beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' A lower blurring fraction is needed for the reconstructions because the dynamical imaging process inherently produces smoother structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The dynamical reconstructions generated with the Eastern array + Africa reproduce accurately the ring-like structure of the GRMHD simulation, while a less solid performance is obtained with the Western array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The reported NXCORR values in the bottom of the images confirm the robustness of the Eastern array + Africa reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The NRMSE values are also consistent with the general goodness trend of the reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We use GRMHD simulations of Sgr A* to test if the East- ern array + Africa is able to reconstruct the main ring structure and its brightness distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' GRMHD models reproduce a qui- escent yet turbulent accretion flow and are not representative of coherent motion of features expected in the event of flaring activ- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Moreover, GRMHD models are complex and challenging to reconstruct due to the large amplitudes in the variability (Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2022d), making it diffi- cult to recognize the rotation of individual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Dynamical imaging using a simple hotspot model, shown in the next sec- tion, allows us to easily investigate the capability of the array to reconstruct coherent motion in Sgr A* in the event of flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Hotspot dynamical reconstructions Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 7 shows five snapshots of the dynamical reconstructions generated using as ground truth the hotspot crescent model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In the first row, we present the synchronized model snapshots, while the Eastern array + Africa and Western array dynamical reconstructions are illustrated in the second and third row, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Similarly to the GRMHD models, we identified the data terms and regularizer weights that maximize the similari- ties between the model and the reconstruction snapshots, exploit- ing the NXCORR and NRMSE metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Unlike the GRMHD re- constructions, the visibilities are separated by ∼30 seconds and the dynamical imaging was performed in narrow time regions of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='7 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In particular, for the Eastern array + Africa this was chosen to be from 21 to 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='7 GMST, which corresponds to the best time window offered by the subarray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' For the Western array, the best period is given between the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='2 GMST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The five snapshots are separated by almost 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='1 hour in order to represent the hotspot orbit, which is completed in ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5 hours (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', 27 minutes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' As confirmed by the NXCORR (reported in the figure) and the NRMSE, the individual frames produced in Article number, page 6 of 11 Noemi La Bella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' : Sgr A* dynamical imaging with an African extension to the EHT Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 4: Time-averaged reconstructions of Sgr A* obtained from the GRMHD synthetic observations for the different array configura- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The leftmost image shows the static representation of the GRMHD simulation used as ground truth movie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The Eastern array without the AMT (second image) does not resolve the shadow of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The addition of the AMT significantly impacts the fidelity of the reconstruction, and a further improvement is obtained with the African array (third image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The rightmost image shows the averaged reconstruction produced using the Western array alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The images were blurred with a Gaussian FWHM equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='6 × clean beam of the 2022EHT + Africa data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 5: Time-averaged reconstructions of GRMHD simulations of Sgr A* with the 2022EHT + Africa and 2022EHT arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The ground truth model is shown in the first column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The 2022EHT + Africa array produces a higher fidelity image, which is used as the prior for the dynamical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' the Eastern array + Africa time region are more accurate than in the Western array window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Indeed, in the latter, the snapshots present pronounced northeast and southwest imaging artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Both subarrays are capable of reconstructing the motion of the hotspot, confirming that the addition of the African stations to the EHT array provides a new time window in the first half of the observation to detect rapid coherent flux variations in Sgr A*’s accretion flow or jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In order to effectively establish the capa- bility of the array in reproducing the flare motion, we developed two methods that evaluate the robustness of our dynamical im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' For the two subarrays, we investigate the ability to recover the flux density profile and the time-dependent rotational veloc- ity of the hotspot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The two methods are described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='1 and in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Method 1: Flux density profile To assess the ability of the Eastern array + Africa to reconstruct the flux density around the crescent model, we calculated the flux density pixel by pixel as a function of the position angle for each snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We selected the ring from which to extract the flux using the hough_ring function in the eht-imaging library, which finds circles in an image according to the pixel brightness distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The choice was made giving as input the time-averaged model image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Thus, for each model snapshots and reconstructed frames, the flux density was estimated within a ra- dius of 32µas and in sectors 10 degrees wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 8, we show the flux density as a function of the angle for five snapshots of the ground truth model (in green and also illustrated in the lower panel of the image), of the Eastern array + Africa movie (in red) and of the Western array movie (in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Because of the asym- metry of the brightness distribution in the crescent model, the flux profile has a peak in the snapshots when the hotspot is at its maximum intensity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', third column), while it decreases when the hotspot is located on the opposite side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' From the model snap- shots and the corresponding flux density profile, we note that the angular position of the hotspot is correctly determined by this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The flux density profiles obtained with the Eastern ar- ray + Africa and Western array recover quite well the hotspot motion, both in term of intensity and in position angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Method 2: Rotational velocity profile Additionally, we computed the rotational velocity of the hotspot as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' This rotation (in degrees per minute) is de- fined as the degree of rotation for each frame i with respect to the fifth subsequent frame j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In order to measure it, we rotated frame i in steps of two degrees across a range of angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We calculated the NXCORR (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', the image correspondence) with respect to frame j at each rotation angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The angle at which the NXCORR is maximized between the two frames divided by the time dura- tion between frames i and j gives us the rotational velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We measured the rotational velocity of the hotspot every five frames, which lets us reconstruct its motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' As the hotspot completes its orbit every 27 min and the frame separation of the reconstructed movie is ∼30 seconds, the rotation every five frames (∼33◦) is easier to measure than the rotation per frame (∼6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='6◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The rotational velocity obtained for the Eastern array + Africa and the Western array movies are shown in the left and right of Figure 9 in red and in blue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The hotspot ve- locity measured from the model movie is represented in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' As in the case of the flux profile, the method represents the asym- metric brightness distribution of the crescent model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Indeed, the frames with the maximum intensity of the hotspot have a maxi- mum value of the rotational velocity, which drops to zero when the hotspot is not present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The negative values of the velocity Article number, page 7 of 11 Model Eastern Eastern + AMT Eastern + Africa Western 0 0 60 μas 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5 Tb[K] 1e10 Tb[K] 1e10 Tb[K] 1e10 Tb[K] 1e10 Tb[K] 1e102022EHT + Africa 2022EHT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 T,[K] 1e10 T,[K] 1e10A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' core Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 6: Dynamical reconstructions obtained from the GRMHD data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The first row shows five snapshots of the GRMHD sim- ulation taken in the Eastern array (17-22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='7 GMST), the second row represents the respective dynamical reconstructions using the Eastern array + Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In a similar way, the third row and forth row illustrate the GRMHD frame simulations and the correspondent frame reconstructions using the Western array (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='7-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='1 GMST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The blurring utilized for the GRMHD simulation is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='6 × clean beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Higher quality dynamical reconstructions are produced by the Eastern array + Africa, also confirmed by the NXCORR metric reported at the bottom of each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The numbers on the top of the GRMHD simulation snapshots represent the frame time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' are artifact produced by the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In particular, these unphys- ical features are generated for each period of the hotspot movie, when we compare the last frame that contains the hotspot and the fifth frame that presents only the crescent emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Comparing the rotational velocity curves derived from the Eastern array + Africa and the Western array movies with the model simulation, we find that the flare variability is most accurately recovered in the Eastern time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Summary and conclusions We generated synthetic data of Sgr A* with the current EHT array and two stations in the African continent, the AMT and the CNI telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We have evaluated the capability of the EHT Eastern subarray with the African sites (17-22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='7 GMST) to produce movies of Sgr A* and compared it to the Western subarray (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='7-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='1 GMST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The data sets were created from ray-traced images of a SANE GRMHD simulation, which is representative of the quiescent yet turbulent black hole accretion Article number, page 8 of 11 GRMHD simulation 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 GMST 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='1 GMST 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='3 GMST 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='8GMST 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='3GMST 60 μas Eastern array + Africa NXCORR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='994 NXCORR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='993 NXCORR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='985 NXCORR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='990 NXCORR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='993 GRMHD simulation 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='2 GMST 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='4 GMST 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='4 GMST 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='8GMST 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='1 GMST C Westernarray NXCORR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='981 NXCORR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='987 NXCORR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='990 NXCORR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='986 NXCORR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='974Noemi La Bella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' : Sgr A* dynamical imaging with an African extension to the EHT Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 7: Dynamical reconstructions generated using the hotspot synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In the first row we show five snapshots of the hotspot model movie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The hotspot performs a full rotation every 27 mins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The frames were chosen to represent a complete orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The reconstructions obtained from the dynamical imaging using the Eastern array + Africa and Western array are shown in the second and third row, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The movies were generated in a time window of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='7 hours (21-22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='7 GMST for the Eastern array, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='5-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='2 GMST for the Western).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The NXCORR values estimated for the reconstructions is reported in the bottom of each images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The temporal evolution is available as an online movie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' flow, and from a crescent hotspot model to test the imaging performance of the array in reconstructing coherent motion from flaring activity in Sgr A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We found that the AMT increases the resolution of the EHT array via long baselines with the Arizona and Mexico sites, while short baselines provided by the African extension to the EHT constrain the compactness and extent of the source on larger scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We estimated the Fourier filling fraction with the EHT ar- ray and the Africa telescopes to investigate the presence of good time regions to perform dynamical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We found that the added baselines offer an optimal time window of about 7 hours in the Eastern array, allowing to produce high-fidelity movies of Sgr A* from the very start of a typical observing track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' This in- creases the time in which dynamical imaging is possible by a factor > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In comparison, Farah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' (2022) demonstrated that with the 2017 EHT array, the only time period in which we are able to reconstruct the variability of the source is from ∼01:30 GMST to ∼03:10 GMST, with the Western array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Our static reconstructions of the GRMHD simulation con- firm the importance of the AMT in imaging Sgr A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Without the AMT, the data set generated with the current EHT configu- ration is not able to reproduce a physical image of the black hole shadow in the Eastern array window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Including the African sites, we can perform high-fidelity imaging of Sgr A* with reduced artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Additionally, we produced GRMHD dynamical recon- structions limited to the best Eastern and Western time regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The African stations enable accurate frame reconstructions of the ring-like structure when included in the Eastern array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Since the rotation of individual features is difficult be recognized in the turbulent flow of GRMHD simulations, we performed a hotspot dynamical imaging analysis to test the capability of the different arrays to reconstruct coherent motion mimicking flaring activity in Sgr A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Compared to the 2022EHT array, the African stations open a new time window in the Eastern array that can be used to reconstruct motion in the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We developed two methods involving the determination of the flux density profile and the rotational velocity of the hotspot to establish the suc- cessful performance of the enhanced Eastern array in reproduc- ing the motion in Sgr A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Our results show the impact of adding stations in the African continent in increasing the time-variable (u, v) coverage of the EHT toward Sgr A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The African exten- sion will be crucial for future EHT observations to study accu- Article number, page 9 of 11 Hotspot movie t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='05hr t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='13hr t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='23hr t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='31 hr t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='54 hr 70μas Easternarray+Africa NXCORR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='957 NXCORR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='968 NXCORR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='955 NXCORR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='970 NXCORR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='953 Western array NXCORR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='906 NXCORR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='923 NXCORR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='892 NXCORR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='906 NXCORR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='952A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' core Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 8: Flux density (Jy/pixel) in function of the angle (degrees) estimated in five snapshots of the model movie (in green), of the Eastern array + Africa movie (in red), and of the Western array movie (in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The brightness distribution was estimated using a ring with outer radius of 32 µas, divided in sectors 10 degrees wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The five frames of the model simulation from where the flux densities were extracted are shown in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 9: Rotational velocity (degree per minute) in function of the time for the Eastern + Africa array movie (left) and for the Western array movie (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' In green, the rotational velocity for the hotspot movie simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The profile were obtained by searching for the angle that maximize the similarity between each frame and the subsequent fifth frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The Eastern array + Africa movie presents a more robust reconstruction of the hotspot rotation than the Western array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The negative values of the rotation are artifacts produced by the method utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' rately the time-variable source at our Galactic Center through high-fidelity movies across an observing track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' We thank Oliver Porth for performing the ray-tracing for the GRMHD simulation used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' This publication is part of the project Dutch Black Hole Consortium (with project number 1292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='202) of the research programme NWA which is (partly) financed by the Dutch Research Council (NWO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' SI is supported by Hubble Fellowship grant HST-HF2-51482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', for NASA, under contract NAS5- 26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' FR is supported by NSF grants AST-1935980 and AST-2034306, and the Black Hole Initiative at Harvard University, made possible through the support of grants from the Gordon and Betty Moore Foundation and the John Templeton Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' The opinions expressed in this publication are those of the author(s) and do not necessarily reflect the views of the Moore or Templeton Foundations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' CMF is supported by the DFG research grant “Jet physics on horizon scales and beyond" (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' FR 4069/2-1) The simulations were performed on LOEWE at the CSC-Frankfurt, Iboga at ITP Frankfurt and Pi2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='0 at Shanghai Jiao Tong University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' References Backes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', Müller, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', Conway, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2016, in The 4th Annual Conference on High Energy Astrophysics in Southern Africa (HEASA 2016), 29 Blackburn, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2019, in ALMA2019: Science Results and Cross-Facility Syner- gies, 37 Article number, page 10 of 11 Angle (degrees) Angle (degrees) Angle (degrees) Angle (degrees) Angle (degrees) 200 0 200 0 200 0 200 0 200 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='008 Model Eastern + Africa (Jy/pixel) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='006 Western 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='004 Flux ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='000 70 μas12 10 Rotational velocity (deg/min) 8 2 0 2 4 Model Eastern +Africa 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='50 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='75 20.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2022, A&A, 660, A107 Gelaro, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', McCarty, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', Suárez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2017, Journal of Climate, 30, 5419 Gravity Collaboration, Abuter, R.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', Johnson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', Blackburn, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2021, ApJ, 915, 99 Issaoun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', Johnson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2019, A&A, 626, A75 Johnson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', Bouman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', Blackburn, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2017, ApJ, 850, 172 Johnson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' & Gwinn, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2015, ApJ, 805, 180 Johnson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' D.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content='01242 Raymond, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', Palumbo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', Paine, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2021, ApJS, 253, 5 Roelofs, 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Konoplya, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=', & Mizuno, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' 2016, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} +page_content=' D, 94, 084025 Article number, page 11 of 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFIT4oBgHgl3EQf4CvP/content/2301.11384v1.pdf'} diff --git a/2NE1T4oBgHgl3EQfAAJE/content/tmp_files/2301.02833v1.pdf.txt b/2NE1T4oBgHgl3EQfAAJE/content/tmp_files/2301.02833v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..da31b2629c5ebe20b27009aa5e75efd636214c26 --- /dev/null +++ b/2NE1T4oBgHgl3EQfAAJE/content/tmp_files/2301.02833v1.pdf.txt @@ -0,0 +1,1523 @@ +On the Weihrauch degree of the additive Ramsey +theorem +Arno Pauly � � +School of Mathematics and Computer Science, Swansea University, UK +Pierre Pradic � +School of Mathematics and Computer Science, Swansea University, UK +Giovanni Soldà � � +School of Mathematics and Computer Science, Swansea University, UK 1 +Abstract +We characterize the strength, in terms of Weihrauch degrees, of certain problems related to Ramsey- +like theorems concerning colourings of the rationals and of the natural numbers. The theorems we +are chiefly interested in assert the existence of almost-homogeneous sets for colourings of pairs of +rationals respectively natural numbers satisfying properties determined by some additional algebraic +structure on the set of colours. +In the context of reverse mathematics, most of the principles we study are equivalent to Σ0 +2- +induction over RCA0. The associated problems in the Weihrauch lattice are related to TC∗ +N, (LPO′)∗ +or their product, depending on their precise formalizations. +2012 ACM Subject Classification Theory of computation → Proof theory; Theory of computation +→ Computability +Keywords and phrases Weihrauch reducibility, Reverse mathematics, additive Ramsey, Σ0 +2-induction. +Related Version This article extends the conference paper [15] by the second and third author. +Funding Giovanni Soldà: This author was supported by an LMS Early Career Fellowship. +1 +Introduction +The infinite Ramsey theorem says that for any colouring c of n-uples of a given arity of an +infinite set X, there exists a infinite subset H ⊆ X such that the set of n-tuples [H]n of +elements of H is homogeneous. This statement is non-constructive: even if the colouring c is +given by a computable function, it is not the case that we can find a computable homogeneous +subset of X. Various attempts have been made to quantify how non-computable this problem +and some of its natural restrictions are. This is in turn linked to the axiomatic strength +of the corresponding theorems, as investigated in reverse mathematics [17] where Ramsey’s +theorem is a privileged object of study [9]. +This paper is devoted to a variant of Ramsey’s theorem with the following restrictions: we +colour pairs of rational numbers and we require some additional structure on the colouring, +namely that it is additive. A similar statement first appeared in [16, Theorem 1.3] to give a +self-contained proof of decidability of the monadic second-order logic of (Q, <). We will also +analyse a simpler statement we call the shuffle principle, a related tool appearing in more +modern decidability proofs [5, Lemma 16]. The shuffle principle states that every Q-indexed +word (with letters in a finite alphabet) contains a convex subword in which every letter +appears densely or not at all. Much like the additive restriction of the Ramsey theorem for +pairs over N, studied from the point of view of reverse mathematics in [11], we obtain a neat +correspondence with Σ0 +2-induction (Σ0 +2-IND). +1 Soldà has since moved to Ghent University +arXiv:2301.02833v1 [cs.LO] 7 Jan 2023 + +2 +On the Weihrauch degree of the additive Ramsey theorem +▶ Theorem 1. In the weak second-order arithmetic RCA0, Σ0 +2-IND is equivalent to both the +shuffle principle and the additive Ramsey theorem for Q. +We take this analysis one step further in the framework of Weihrauch reducibility that al- +lows to measure the uniform strength of general multi-valued functions (also called problems) +over Baire space. Let Shuffle and ARTQ be the most obvious problems corresponding to the +shuffle principle and additive Ramsey theorem over Q respectively. We relate them, as well +as various weakenings cShuffle, cARTQ, iShuffle and iARTQ that only output sets of colours +or intervals, to the standard (incomparable) problems TCN and LPO′. We also consider the +ordered Ramsey principle, ORTQ, where the colours k come equipped with a partial order +⪯, and the colouring α : [Q]2 → k satisfying that α(r1, r2) ⪯ α(q1, q2) if q1 ≤ r1 < r2 ≤ q2. +A weakening of Shuffle is the principle (η)1 +<∞ introduced in [8] where we ask merely for an +interval where some colour is dense; respectively for a colour which is dense somewhere. +▶ Theorem 2. We have the following equivalences +Shuffle ≡W ARTQ ≡W TC∗ +N × (LPO′)∗ +cShuffle ≡W cARTQ ≡W (LPO′)∗ +iShuffle ≡W iARTQ ≡W (η)1 +<∞ ≡W i(η)1 +<∞ ≡W TC∗ +N +ORTQ ≡W LPO∗ +c(η)1 +<∞ ≡W cRT1 ++ +Finally, we turn to carrying out the analysis of those Ramseyan theorems over N in +the framework of Weihrauch reducibility. The additive Ramsey theorem over N is also an +important tool in the study of monadic second order logic over countable scattered orders. As +for the case of Q, we relate problems ARTN and ORTN as well as some natural weakenings +cARTN, cORTN, iARTN and iORTN, to TCN and LPO′ (the i variants of those principle +return, rather than an interval, some upper bound n on the first two points of some infinite +homogeneous set). +▶ Theorem 3. We have the following equivalences +ORTN ≡W ARTN ≡W TC∗ +N × (LPO′)∗ +cORTN ≡W cARTN ≡W (LPO′)∗ +iORTN ≡W iARTN ≡W TC∗ +N +2 +Background +In this section, we will introduce the necessary background for the rest of the paper, and +fix most of the notation that we will use, except for formal definitions related to weak +subsystems of second-order arithmetic, in particular RCA0 (which consists of Σ0 +1-induction +and recursive comprehension) and RCA0 + Σ0 +2-IND. A standard reference for that material +and, more generally, systems of interest in reverse mathematics, is [17]. +2.1 +Generic notations +We identify k ∈ N with the finite set {0, . . . , k − 1}. For every linear order (X, 0 cRT1 +k (denoted RT1,+ in [4]) is the disjoint union of the cRT1 +k: it can be +thought of as a problem taking as input a pair (k, f) where f ∈ N and f : N → k is a +colouring, and outputting n such that f −1(n) is infinite. +The definition of LPO′ could have been obtained by composing the one of LPO and the +definition of jump as given in [3]: we include it for convenience. Intuitively, LPO′ corresponds +to the power of answering a single binary Σ0 +2-question. In particular, LPO′ is easily seen to +be (strongly) Weihrauch equivalent to both IsFinite and IsCofinite, the problems accepting +as input an infinite binary string p and outputting 1 if p contains finitely (respectively, +cofinitely) many 1s, and 0 otherwise. We will use this fact throughout the paper. +2 Whereas LPO and CN have been widely studied, TCN is somewhat less known (and does not appear +in [3]): we refer to [13] for an account of its properties, and to [2] for a deeper study of some principles +close to it. + +A. Pauly, P. Pradic & G. Soldà +5 +Another problem of combinatorial nature, introduced in [6], will prove to be very useful +for the rest of the paper. +▶ Definition 5. ECT is the problem whose instances are pairs (n, f) ∈ N × NN such that +f : N → n is a colouring of the natural numbers with n colours, and such that, for every +instance (n, f) and b ∈ N, b ∈ ECT(n, f) if and only if +∀x > b ∃y > x (f(x) = f(y)). +Namely, ECT is the problems that, upon being given a function f of the integers with finite +range, outputs a b such that, after that b, the palette of colours used is constant (hence +its name, which stands for eventually constant palette tail). We will refer to suitable bs as +bounds for the function f. +A very important result concerning ECT and that we will use throughout the paper is +its equivalence with TC∗ +N. +▶ Lemma 6 ([6, Theorem 9]). ECT ≡W TC∗ +N +Another interesting result concerning ECT is the following: if we see it as a statement of +second-order arithmetic (ECT can be seen as the principle asserting that for every colouring +of the integers with finitely many colours there is a bound), then ECT and Σ0 +2-IND are +equivalent over RCA0 (actually, over RCA∗ +0). +▶ Lemma 7 ([6, Theorem 7]). Over RCA0, ECT and Σ0 +2-IND are equivalent. +Hence, thanks to the results above, it is clear why TC∗ +N appears as a natural candidate +to be a “translation” of Σ0 +2-IND in the Weihrauch degrees. +We end this section with several technical results about Weihrauch degrees. +Following [13], IsFiniteS : 2N → S is the following problem : for every p ∈ 2N, IsFiniteS(p) = +⊤ if p contains only finitely many occurrences of 1 and IsFiniteS(p) = ⊥ otherwise 3. +▶ Lemma 8. IsFiniteS ̸≤W ECT +Proof. Suppose for a contradiction that a reduction exists and is witnessed by functionals +H and K. We build an instance p of IsFiniteS contradicting this. +Let us consider the colouring H(0N), and let b0 ∈ ECT(H(0N)) be a bound for it. Since +IsFiniteS(0N) = ⊤, the outer reduction witness will commit to answering ⊤ after having read +a sufficiently long prefix of 0N together with b0, say of length n0. Now consider the colouring +H(0n010N), and a bound b1 > b0 for it. Again by the fact that IsFiniteS(0n010N) = ⊤, there +is an n1 such that K commits to answering ⊤ after having read the prefix 0n010n1 together +with b1. We iterate this process indefinitely and obtain an instance p = 0n010n110n21 . . . +such that IsFiniteS(p) = ⊥. +However, for the colouring H(p) there must be some bk which is a valid bound, as the +sequence b0 < b1 < b2 < . . . is unbounded. +But K will commit to ⊤ upon reading a +sufficiently long prefix of p together with bk by construction, thereby answering incorrectly. +◀ +We can now assert that the two main problems that we use as benchmarks in the sequel, +namely (LPO′)∗ and TC∗ +N, are incomparable in the Weihrauch lattice. +3 S is the Sierpinski space {⊤, ⊥}, where ⊤ is coded by the binary strings containing at least one 1, and +⊥ is coded by 0N. IsFiniteS is strictly weaker than IsFinite + +6 +On the Weihrauch degree of the additive Ramsey theorem +▶ Lemma 9. (LPO′)∗ and TC∗ +N are Weihrauch incomparable. Thus (LPO′)∗ s, Ijis = Iji +t. This means that Ijis is c-homogeneous, and thus, in particular, a +c-shuffle. Hence, Ci is a valid solution. +If instead for all is such that Ci is a singleton σ(i) = 0: then, we know that no interval I +is c-monochromatic, otherwise we would be in the previous case. We move to consider the +is such that |Ci| = 2. Suppose that for one such i, σ(i) = 1: again, this means that, for a +sufficiently large stage s, the current interval Ijis is such that, for every q ∈ Ijis, c(q) ∈ Ci, +since the second case in the construction is triggered only finitely many times. But since +we know that there are no c-monochromatic intervals, the two colours of Ci occur densely +in Ijis, which then is a c-shuffle for the colours in Ci. Hence, any Ci such that σ(i) = 1 is a +valid solution for c. +This argument can be iterated for every number of colours. +Since, by the theory, a +c-shuffle exists, at least one of the pi instances above contains only finitely many 1s. To +compute a solution to c, it is thus sufficient to look for the minimal k such that, for some i, +σ(i) = 1 and |Ci| = k, and output Ci. +◀ +Putting the previous lemmas together, we have the following: +▶ Theorem 24. (LPO′)∗ ≡W cShuffle +Proof. (LPO′)∗ ≤W cShuffle is given by Lemmas 21 and 22. For the other direction, notice +that cShuffle ≡W +� +n∈N cShufflen. The result then follows from Lemma 23. +◀ +While Theorem 24 tells us that for any finite number of parallel LPO′-instances can be +reduced to cShuffle for m-colourings for a suitable choice of m, and vice versa, a sufficiently +large number of LPO′-instances can solve cShuffle for m-colourings, both directions of our +proof involved an exponential increase in the parameter. Before moving on to iShuffle, we +thus raise the open question of whether this gap can be narrowed: +▶ Question 25. What is the relationship between (LPO′)n and cShufflem for individual +n, m ∈ N? +▶ Lemma 26. Let iShufflen be the restriction of iShuffle to the instances of the form (n, c). +For n ≥ 1, it holds that iShufflen ≤sW TCn−1 +N +. +Proof. Fix an enumeration Ij of the intervals of Q, an enumeration qh of Q, a computable +bijection ⟨·, ·⟩: N × N → N, and let (n, c) be an instance of iShufflen. +The idea of the reduction is the following: with the first instance en−1 of TCN, we look +for an interval Ij on which c takes only n − 1 colours: if no such interval exists, then this +means that every colour is dense in every interval, and so every Ij is a valid solution to +c. Hence, we can suppose that such an interval is eventually found: we will then use the +second instance en−2 of TCN to look for a subinterval of Ij where c takes only n − 2 values. +Again, we can suppose that such an interval is found. We proceed like this for n − 1 steps, +so that in the end the last instance e1 of TCN is used to find an interval I′ inside an interval +I on which we know that at most two colours appear: again, we look for c-monochromatic +intervals: if we do not find any, then I′ is already a c-shuffle, whereas if we do find one, then +that interval is now a solution to c, since c-monochromatic intervals are trivially c-shuffles.. +Although not apparent in the sketch given above, an important part of the proof is that +the n − 1 searches we described can be performed in parallel: the fact that this can be +accomplished relies on the fact that any subinterval of a shuffle is a shuffle. More formally, +we proceed as follows: we define n − 1 instances e1, . . . , en−1 of TCN as follows. For every + +12 +On the Weihrauch degree of the additive Ramsey theorem +stage s, every instance ei will have a “current interval” Ijis and a “current point” qhis and a +“current list of colours” Lkis. We start the construction by the setting the current interval +equal to I0, the current point equal to q0 and the current list of points equal to ∅ for every +i. +At stage s, there are two cases: +if, for every i, qhis ̸∈ Ijis or |Lkis ∪ {c(qhis)}| ≤ i, we set Iji +s+1 = Ijis, qhi +s+1 = qhis+1 and +Lki +s+1 = Lkis ∪ {c(qhis)}. Moreover, we let every ei enumerate every number of the form +⟨s, a⟩, for every a ∈ N, except for ⟨s, ji +s⟩. We then move to stage s + 1. +In practice, this means that if the set of colours of the points of the current interval seen +so far does not have cardinality larger than i, no particular action is required, and we +can move to check the next point on the list. +otherwise: let i′ be maximal such that qhis ∈ Ijis and |Lkis ∪ {c(qhis)}| > i. Then, for +every i > i′ we proceed as in the previous case (i.e., the current interval, current point, +current list of colours and enumeration are defined as above). For the other components, +we proceed as follows: we first look for the minimal ℓ > ji′ +s such that Iℓ ⊆ Iji′+1 +s +(if +i′ = n − 1, just pick ℓ = jn−1 +s ++ 1). Then, for every i ≤ i′, we set Iji +s+1 = Iℓ, qhi +s+1 = q0 +and Lki +s+1 = ∅. Moreover, we let ei enumerate every number of the form ⟨t, a⟩ with t < s +that had not been enumerated so far, and also every number of the form ⟨s, a⟩, with the +exception of ⟨s, ji +s⟩. We then move to stage s + 1. +In practice, this means that if, for a certain component i′, we found that the current +interval has too many colours, then, for all the components i ≤ i′, we move to consider +intervals strictly contained in the current interval of component i′. +We iterate the procedure for every s ∈ N, thus obtaining the TCn−1 +N +-instance ⟨e1, . . . , en−1⟩. +Let σ ∈ Nn−1 be such that σ ∈ TCn−1 +N +(⟨e1, . . . , en−1⟩). Then, we look for the minimal +i such that Iπ2(σ(i)) ⊆ Iπ2(σ(i+1)) ⊆ · · · ⊆ Iπ2(σ(n−1)) (by πi we denote the projection on +the ith component, so ⟨π1(x), π2(x)⟩ = x)). We claim that Iπ2(σ(i)) is a c-shuffle, which is +sufficient to conclude that iShufflen ≤sW TCn−1 +N +. +We now prove the claim. First, suppose that en−1 enumerates all of N. Then, the second +case of the construction was triggered infinitely many times with i′ = n − 1: hence, no +interval contains just n − 1 colours, and so, as we said at the start of the proof, this means +that every interval is a c-shuffle. In particular, this imples that Iπ2(σ(i)) is a valid solution. +Hence we can suppose that en−1 does not enumerate all of N. +Next, we notice that for every m > 1, if em enumerates all of N, the so does em−1, by +inspecting the second case of the construction. Let m be minimal such that em does not +enumerate all of N. For such an m, it is easy to see that Iπ2(σ(m)) is a valid solution to c: +indeed, we know from the construction that c takes m colours on Ipi2(σ(m)), and that for no +interval contained in Iπ2(σ(m)) c takes m−1 colours, which means that Iπ2(σ(m)) is a c-shuffle. +Moreover, it is easy to see that Iπ2(σ(m)) ⊆ Iπ2(σ(m+1)) ⊆ . . . Iπ2(σ(n−1)), which implies that +i ≤ m. Since every subinterval of a c-shuffle is a c-shuffle, Iπ2(σ(i)) is a valid solution to c, +as we wanted. +◀ +▶ Lemma 27. Let ECTn be the restriction of ECT to the instances of the form (n, f). It +holds that ECTn ≤sW iShufflen. +Proof. Let (n, f) be an instance of ECTn. +We define c: Q → n by c( a +b ) = f(b) where +gcd(a, b) = 1. Hence, all the points of the same denominator have the same colour according +to c. Let ( u +k , v +ℓ ) ∈ iShufflen(n, c). Let b be such that 1 +b < v +ℓ − u +k . We claim that b is a bound +for f. Suppose not, then there is a colour i < n and a number x ∈ N such that x > b and +f(x) = i, but for no y > x it holds that f(y) = i. Hence, all the reduced of the form w +x are + +A. Pauly, P. Pradic & G. Soldà +13 +given colour i, but i does not appear densely often in any interval of Q. But by choice of b, +there is a z ∈ Z such that z +b ∈ +� u +k , v +ℓ +� +, which is a contradiction. Hence b is a bound for f. +◀ +Putting things together, we finally have a characterization of iShuffle. We even get a +precise characterization for each fixed number of colours. +▶ Theorem 28. We have the Weihrauch equivalence +ECTn ≡W iShufflen ≡W TCn−1 +N +whence +ECT ≡W iShuffle ≡W TC∗ +N +Proof. We get TCn−1 +N +≤W ECTn by inspecting the second half of [6, Theorem 9]. Then +Lemma 27 gives us ECTn ≤W iShufflen. Lemma 26 closes the cycle by establishing iShufflen ≡W +TCn−1 +N +. +◀ +3.3 +The full shuffle problem +The main result of this section is that Shuffle ≡W TC∗ +N × (LPO′)∗, which will be proved +in Theorem 31. For one direction, we merely need to combine our results for the weaker +versions: +▶ Lemma 29. TC∗ +N × (LPO′)∗ ≤W Shuffle +Proof. From Theorem 24 and Theorem 28, we have that TC∗ +N × (LPO′)∗ ≤W iShuffle × +cShuffle, and since clearly iShuffle ≤W Shuffle and cShuffle ≤W Shuffle, by Lemma 21 we +have that TC∗ +N × (LPO′)∗ ≤W Shuffle. +◀ +For the other direction, again, we want to be precise as to the number of TCN- and +(LPO′)-instances we use to solve an instance of Shuffle. Note that we will use a far larger +number of TCN-instances to obtain a suitable interval than we used in Lemma 26. +▶ Lemma 30. Let Shufflen be the restriction of Shuffle to the instances of the form (n, c). +Then, Shufflen ≤W (TCN × LPO′)2n−1 +Proof. Let (n, c) be an instance of Shuffle. The idea of the proof of Shufflen ≤W (TCN × +LPO′)2n−1 is, in essence, to combine the proofs of Lemma 26 and of Lemma 23: we want to +use TCN to find a candidate interval for a certain subset C of n, and on the side we use LPO′ +(or equivalently, IsFinite) to check for every such set C whether a c-shuffle for the colours of +C actually exists. The main difficulty with the idea described above is that the two proofs +must be intertwined, in order to be able to find both a c-shuffle and the set of colours that +appears on it. +We proceed as follows: let Ci be an enumeration of the non-empty subsets of n. Moreover, +let us fix some computable enumeration Ij of the intervals of Q, some computable enumer- +ation qh of the points of Q, and some computable bijection ⟨·, ·⟩: N × N → N. For every +Ci, we will define an instance ⟨pi, ei⟩ of IsFinite × TCN in stages as follows: at every stage s, +for every index i, there will be a “current interval” Ijis and a “current point” qhis. We begin +stage 0 by setting the current interval to I0 and the current point to q0 for every index i. +At stage s, for every component i, there are two cases: +if qhis ̸∈ Ijis or if c(qhis) ∈ Ci, we set Iji +s+1 = Ijis and qhi +s+1 = qhi +s+1. Moreover, we set +pi(s) = 0 and we let ei enumerate all the integers of the form ⟨s, a⟩, except ⟨s, ji +s+1⟩. We +then move to stage s + 1. +In plain words, for every component i, we check if the colour of the current point is in +Ci, or if the current point is not in the current interval: if this happens, no special action +is required. + +14 +On the Weihrauch degree of the additive Ramsey theorem +If instead qhis ∈ Ijis and c(qhis) ̸∈ Ci, we set Iji +s+1 = Ijis+1 and qhi +s+1 = q0. Moreover, we +set pi(s) = 1, and we let ei enumerate all the numbers of the form ⟨t, a⟩, for t < s, that +had not been enumerated at a previous stage, and also all the numbers of the form ⟨s, a⟩, +with the exception of ⟨s, ji +s+1⟩. We then move to stage s + 1. +In plain words: if we find that for some component i the colour of the current point is +not in Ci, then, from the next stage, we start considering another interval, namely the +next one in the fixed enumeration. We then reset the current point to q0 (so that all +rationals are checked again), and we record the event by letting pi(s) = 1 and changing +the form of the points that ei is enumerating. +We iterate the procedure for every integer s. Let σ ∈ (2 × N)2n−1 be such that +σ ∈ (IsFinite × TCN)2n−1(⟨⟨p1, e1⟩ . . . , ⟨p2n−1, e2n−1⟩)⟩ +Let k be the minimal cardinality of a subset Ci ⊆ n such that IsFinite(pi) = 1: notice that +such a k must exist, because c-shuffle exist. Then, we claim that the corresponding Iπ2(σ(i)) +is a c-shuffle (by πi we denote the projection on the ith component, so ⟨π1(x), π2(x)⟩ = x)). +If we do this, it immediately follows that Shuffle ≤W ((LPO′) × TCN)2n−1. +Hence, all that is left to be done is to prove the claim. By the fact that IsFinite(pi) = 1, we +know that the second case of the construction is triggered only finitely many times. Hence, +ei does not enumerate all of N, and so Iπ2(σ(i)) is an interval containing only colours from +Ci. Moreover, by the minimality of |Ci|, we know that no subinterval of Iπ2(σ(i)) contains +fewer colours, which proves that Iπ2(σ(i)) is a c-shuffle. +◀ +Putting the previous results together, we obtain the following. +▶ Theorem 31. Shuffle ≡W TC∗ +N × (LPO′)∗ +3.4 +The (η)1 +<∞-problem +A weakening of the shuffle principle was studied in [8] under the name (η)1 +<∞. The principle +(η)1 +<∞ asserts that for any colouring of Q in finitely many colours, some colour will be dense +somewhere. We formalize it here as follows: +▶ Definition 32. The principle (η)1 +<∞ takes as input a pair (k, α) where k ∈ N and α : Q → k +is a colouring, and returns an interval I and a colour n < k such that α−1(n) is dense in +I. The principle i(η)1 +<∞ returns only the interval I, c(η)1 +<∞ only the dense colour n. Let +(c(η)1 +<∞)k be the restriction of c(η)1 +<∞ to k-colourings. +An important aspect of the definition above to notice is that we require a bound on the +number of colours used to be declared in the instance of (η)1 +<∞. +While (η)1 +<∞ also exhibits the pattern that we can neither compute a suitable interval +from knowing the dense colour nor vice versa, we shall see that as far as the Weihrauch +degree is concerned, finding the interval is as hard as finding both interval and colour. Our +proof does not preserve the number of colours though. +▶ Proposition 33. (η)1 +<∞ ≡W i(η)1 +<∞ ≡W TC∗ +N ≡W iShuffle +Proof. Taking into account Theorem 28, it suffices for us to show that (η)1 +<∞ ≤W iShuffle +and that ECT ≤W i(η)1 +<∞. For (η)1 +<∞ ≤W iShuffle we observe that an interval which is +a shuffle not only has a dense colour in it, but every colour that appears is dense. Thus, +we return the interval obtained from iShuffle on the same colouring, together with the first +colour we spot in that interval. + +A. Pauly, P. Pradic & G. Soldà +15 +It remains to prove that ECT ≤W i(η)1 +<∞. Given a k-colouring c of N, we will compute +a 2k-colouring α of Q. We view the 2k-colouring as a colouring by subsets of k, i.e. each +rational gets assigned a set of the original colours. To determine whether the n-th rational qn +should be assigned the colour j < k, we consider the number mn,j = |{s | s ≤ n ∧ c(s) = j}| +of prior ocurrences of the colour j in c. If the integer part of qn ∗ 2mn,j is odd, qn is assigned +colour j, otherwise not. +If j appears only finitely many times in c, then mn,j is eventually constant, and the +distribution of j in α follows (with finitely many exceptions) the pattern of alternating +intervals of width 2−mn,j. This ensures that none of the 2k-many colours for α can be dense +on an interval wider than 2−mn,j. Subsequently, we find that the width of the interval having +a dense colour returned by i(η)1 +<∞ provide a suitable bound to return for ECT. +◀ +The Proposition above implies that c(η)1 +<∞ has to be weaker than (LPO′)∗, since it is +immediate to see that it is computed by both (η)1 +<∞ and cShuffle. We now give more bounds +on its strength. +▶ Lemma 34. (c(η)1 +<∞)k+1 ≤W cRT1 +k+1 × (c(η)1 +<∞)k +Proof. Fix some enumeration (In)n∈N of all rational intervals. +The forwards reduction +witness is constructed as follows. +We keep track of an interval index n and a colour c, +starting with n = 0 and c = 0. We keep writing the current value of c to the input of +cRT1 +k+1, and we construct a colouring β : Q → {0, 1, . . . , k − 1} by scaling the colouring α +restricting to In up to Q, while excluding c and subtracting 1 from every colour d > c. The +fact that we may have already assigned β-colours to finitely many points in a different way +before is immaterial. +If we ever find a rational q ∈ In with α(q) = c < k, we increment c. If we find q ∈ In +with α(q) = c = k, we set c = 0 and increment n. In particular, we stick with any particular +In until we have found points of all different colours inside it. +The backwards reduction witness receives two colours, c ∈ {0, 1, . . . , k} and d ∈ {0, 1, . . . , k− +1}. If d < c, it returns d. If d ≥ c, it returns d + 1. +To see that the reduction works correctly, first consider the case where every colour +is dense everywhere. +In this case, everything is a correct answer, and the reduction is +trivially correct. Otherwise, there has to be some interval In and some colour c such that +α−1(c) ∩ In = ∅. In this case, our updating of n and c will eventually stabilize at such a +pair. The answer we will receive from cRT1 +k+1 is c. Apart from finitely many points, β will +be look like the restriction of α to In with c skipped. Thus, any colour d which is dense +somewhere for β will be dense somewhere inside In for α if d < c, or if d ≥ c, then d + 1 will +be dense. Thus, the reduction works. +◀ +▶ Corollary 35. +(c(η)1 +<∞)k ≤W cRT1 +k × cRT1 +k−1 × . . . × cRT1 +2 +≤W (cRT1 +2)k−1 × (cRT1 +2)k−2 × . . . (cRT)1 +2 +≡W (cRT1 +2)k(k−1)/2 +These bounds allow us to characterize the stregth of c(η)1 +<∞. +▶ Corollary 36. c(η)1 +<∞ ≡W cRT1 ++ +Proof. The direction c(η)1 +<∞ ≤W cRT1 ++ is provided by Corollary 35. For the other direction +we show cRT1 +k ≤W (c(η)1 +<∞)k. Fix a computable bijection ν : N → Q. Given a colouring + +16 +On the Weihrauch degree of the additive Ramsey theorem +f : N → k as input for cRT1 +k, we define the colour αf : Q → k by αf(q) = f(ν−1(q)). Clearly, +any colour appearing somewhere dense in αf must have appeared infinitely often in f. +◀ +Our result that c(η)1 +<∞ ≡W cRT1 ++ stands in contrast to the reverse mathematics results +obtained in [8]. In reverse mathematics, RT1 +N is equivalent to BΣ0 +2 [10], yet [8, Theorem 3.5] +shows that BΣ0 +2 does not imply (η)1 +<∞ over RCA0. +4 +ARTQ and related problems +We now analyse the logical strength of the principle ARTQ. As in the case of Shuffle, we +start with a proof of ARTQ in RCA0 + Σ0 +2-IND. This will give us enough insights to assess +the strength of the corresponding Weihrauch problems. +4.1 +Additive Ramsey over Q in reverse mathematics +As a preliminary step, we figure out the strength of ORTQ, the ordered Ramsey theorem over +Q. It is readily provable from RCA0 and is thus much weaker than most other principles we +analyse. We can be a bit more precise by considering RCA∗ +0 which is basically the weakening +of RCA0 where induction is restricted to ∆0 +1 formulas (see [17, Definition X.4.1] for a nice +formal definition). +▶ Lemma 37. RCA∗ +0 ⊢ RCA0 ⇔ ORTQ +We now show that the shuffle principle is equivalent to ARTQ. So overall, much like the +Ramsey-like theorems of [11], they are equivalent to Σ0 +2-induction. +▶ Lemma 38. RCA0 + Shuffle ⊢ ARTQ. Hence, RCA0 + Σ0 +2-IND ⊢ ARTQ. +Proof. Fix a finite semigroup (S, ·) and an additive colouring c : [Q]2 → S. Say a colour +s ∈ S occurs in X ⊆ Q if there exists (x, y) ∈ [X]2 such that c(x, y) = s. +We proceed in two stages: first, we find an interval ]u, v[ such that all colours occurring +in ]u, v[ are J -equivalent to one another. Then we find a subinterval of ]u, v[ partitioned +into finitely many dense homogeneous sets. For the first step, we apply the following lemma +to obtain a subinterval I1 = ]u, v[ of Q where all colours lie in a single J -class. +▶ Lemma 39. For every additive colouring c, there exists (u, v) ∈ [Q]2 such that all colours +of c +�� +]u,v[ are J -equivalent to one another. +Proof. If we post-compose c with a map taking a semigroup element to its J -class, we get +an ordered colouring. Applying ORTQ yields a suitable interval. +◀ +Moving on to stage two of the proof, we want to look for a subinterval of I1 partitioned +into finitely many dense homogeneous sets. To this end, define a colouring γ : I1 → S2 by +setting γ(z) = (c(u, z), c(z, v)). +By Shuffle, there exist x, y ∈ I1 with x < y such that ]x, y[ is a γ-shuffle. For l, r ∈ S, +define Hl,r : = γ−1({(l, r)}) ⊆ ]x, y[; note that this is a set by bounded recursive compre- +hension. Clearly, all Hl,r are either empty or dense in ]x, y[, and moreover ]x, y[ = � +l,r Hl,r. +Since there are finitely many pairs (l, r), all we have to prove is that each non-empty Hl,r is +homogeneous for c. +Let s = c(w, z) such that w, z ∈ Hl,r with w < z. By additivity of c and the definition +of Hl,r, +s · r = c(w, z) · c(z, v) = c(w, v) = r. +(1) + +A. Pauly, P. Pradic & G. Soldà +17 +In particular r ≤R s. But we also have r J s, which gives r R s by Lemma 15. This shows +that all the colours occurring in Hl,r are R-equivalent to one another. A dual argument +shows that they are all L-equivalent, so they are all H-equivalent. +The assumptions of +Lemma 14 are satisfied, so their H-class is actually a group. +All that remains to be proved is that any colour s occurring in Hl,r is actually the +(necessarily unique) idempotent of this H-class. +Since r R s, there exists a such that +s = r ·a. But then by (1), s·s = s·r ·a = r ·a = s, so s is necessarily the idempotent. Thus, +all sets Hl,r are homogeneous and we are done. +◀ +We conclude this section by showing that the implication proved in the Lemma above +reverses., thus giving the precise strength of ARTQ over RCA0. +▶ Theorem 40. RCA0 + ARTQ ⊢ Shuffle. Hence, RCA0 ⊢ ARTQ ↔ Σ0 +2-IND. +Proof. Let f : Q → n be a colouring of the rationals. Let (Sn, ·) be the finite semigroup +defined by Sn = n and a · b = a for every a, b ∈ Sn. Define the colouring c: [Q]2 → Sn +by setting c(x, y) = f(x) for every x, y ∈ Q. Since for every x < y < z, c(x, z) = f(x) = +c(x, y) · c(y, z), c is additive. +By additive Ramsey, there exists ]u, v[ which is c-densely +homogeneous and thus a f-shuffle. +◀ +4.2 +Weihrauch complexity of additive Ramsey +We now start the analysis of ARTQ in the context of Weihrauch reducibility. We will mostly +summarize results, relying on the intuitions we built up so far. First off, we determine the +Weihrauch degree of the ordered Ramsey theorem over Q. +▶ Theorem 41. Let ORTQ be the problem whose instances are ordered colourings c : [Q]2 → +P, for some finite poset (P, ≺), and whose possible outputs on input c are intervals on which +c is constant. We have that ORTQ ≡W LPO∗. +Proof. LPO∗ ≤sW ORTQ: let ⟨n, p0, . . . , pn−1⟩ be an instance of LPO∗. Let (P, ≺) be the +poset such that P = 2n, the set of subsets of n, and ≺ = ⊃, i.e. ≺ is reverse inclusion. +We define an ordered colouring c : [Q]2 → P in stages by deciding, at stage s, the colour +of all the pairs of points (x, y) ∈ [Q]2 such that |x − y| > 2−s. +At stage 0, we set c(x, y) = ∅ for every (x, y) ∈ [Q]2 such that |x−y| > 1. At stage s > 0, +we check pi +�� +s for every i < n (i.e., for every i, we check the sequence pi up to pi(s − 1)), and +for every (x, y) ∈ [Q]2 with 2−s+1 ≥ |x − y| > 2−s, we let +c(x, y) = {i < n : ∃t < s(pi(t) = 1)}. +It is easily seen that c defined as above is an ordered colouring: if x ≤ x′ < y′ ≤ y′, then +|x′ − y′| ≤ |x − y|, which means that to determine the colour of (x′, y′) we need to examine +a longer initial segment of the pis. Let I ∈ ORTQ(P, c), and let ℓ ∈ N be least such that +the length of I is larger that 2−ℓ: since I is c-homogeneous, we know that for every i < n, +∃t(pi(t) = 1) ⇔ ∃t < ℓ(pi(t) = 1). Hence, for every pair of points (x, y) ∈ [I]2, the colour of +c(x, y) is exactly the set of i such that LPO(pi) = 1. +ORTQ ≤W LPO∗: Let (P, c) be an instance of ORTQ, for some finite poset (P, ≺P ). Let + ns such +that aℓ ⊆ ams (i.e., we look for a pair of points contained in the current interval), and +set ans+1 = aℓ, and ams+1 = ams. We set pi(s) = 0 and no component is set to inactive. +We then move to stage s + 1. +suppose instead there is an active component i such that c(ans) ns such that aℓ ⊂ ans: we set ans+1 = aℓ, and we set pk(s) = 0 for every active +component k < |P|. We then move to stage s + 1. +We iterate the procedure above for every integer s. +Let σ ∈ 2|P | be such that σ ∈ LPO∗(⟨|P|, p0, . . . , p|P |−1⟩). Notice that σ(0) = 0, since no +pair of points can attain colour i have been set inactive by step t. Hence, after step t, +the current interval I never changes, and thus we eventually check the colour of all the pairs +in that interval. Since the second case of the construction is never triggered, it follows that +I is a c-homogeneous interval. Hence, in order to find it, we know we just have to repeat +the construction above until all the components of index larger than i are set inactive. This +proves that ORTQ ≤W LPO∗. +◀ +Now let us discuss Weihrauch problems corresponding to ARTQ. +▶ Definition 42. Regard ARTQ as the following Weihrauch problem: the instances are pairs +(S, c) where S is a finite semigroup and c : [Q]2 → S is an additive colouring of [Q]2, and +such that, for every C ⊆ S and every interval I of Q, (I, C) ∈ ARTQ if and only if I is +c-densely homogeneous for the colours of C. +Similarly to what we did in Definition 20, we also introduce the problems cARTQ and iARTQ +that only return the set of colours and the interval respectively. +We start by noticing that the proof of Theorem 40 can be readily adapted to show the +following. +▶ Lemma 43. +cShuffle ≤sW cARTQ, hence (LPO′)∗ ≤W cARTQ. + +A. Pauly, P. Pradic & G. Soldà +19 +iShuffle ≤sW iARTQ, hence TC∗ +N ≤W iARTQ. +Shuffle ≤sW ARTQ, hence (LPO′)∗ × TC∗ +N ≤W ARTQ. +The rest of the section is devoted to find upper bounds for cARTQ, iARTQ and ARTQ. +The first step to take is a careful analysis of the proof of Lemma 38. For an additive colouring +c: [Q]2 → S, the proof can be summarized as follows: +we start with an application of ORTQ to find an interval ]u, v[ such that all the colours +of c +�� +]u,v[ are all J -equivalent (Lemma 39). +define the colouring γ : Q → S2 and apply Shuffle to it, thus obtaining the interval ]x, y[. +the rest of the proof consists simply in showing that ]x, y[ is a c-densely homogeneous +interval. +Hence, from the uniform point of view, this shows that ARTQ can be computed via a +composition of Shuffle and ORTQ. Whence the next theorem. +▶ Theorem 44. +cARTQ ≤W (LPO′)∗ × LPO∗, therefore cARTQ ≡W (LPO′)∗. +iARTQ ≤W TC∗ +N × LPO∗, therefore iARTQ ≡W TC∗ +N. +ARTQ ≤W (LPO′)∗ × TC∗ +N × LPO∗, therefore ARTQ ≡W (LPO′)∗ × TC∗ +N. +Proof. The three results are all proved in a similar manner. We recall that LPO∗ ≤W CN +and observe that LPO∗ is single-valued. This enables us to use Lemma 10 with LPO∗ in +place of P. +For x ∈ {c, i, s} and every n ∈ N, let xARTQ,n be the restriction of xARTQ to instances of +the form (S, c) with S of cardinality n. Hence, by the considerations preceding the statement +of the theorem in the body of the paper, we have the following facts: +cARTQ,n ≤W cShufflen2 ∗ ORTQ, hence, by Lemma 23 and Theorem 41, we have that +cARTQ,n ≤W (LPO′)2n2−1∗LPO∗. By Lemma 10, we have that cARTQ,n ≤W (LPO′)2n2−1× +LPO∗, from which the claim follows. +iARTQ,n ≤W iShufflen2 ∗ ORTQ, hence, by Lemma 26 and Theorem 41, we have that +iARTQ,n ≤W TCn2−1 +N +∗ LPO∗. By Lemma 10, we have that iARTQ,n ≤W TCn2−1 +N +× LPO∗, +from which the claim follows. +ARTQ,n ≤W Shufflen2 ∗ ORTQ, hence, by Lemma 30 and Lemma 10, we have that +ARTQ,n ≤W (LPO′ × TCN)2n2−1 ∗ LPO∗. +By Lemma 10, we have that ARTQ,n ≤W +(LPO′ × TCN)2n2−1 × LPO∗, from which the claim follows. +◀ +5 +ARTN and ORTN +We finally turn to the case of the additive and ordered theorems over N and prove Theorem 3. +We obtain results which are completely analogous to the case of Q when it comes to the +additive Ramsey theorem. However, in contrast to Theorem 41, the ordered Ramsey theorem +for N exhibits the same behaviour as the additive Ramsey theorem. +That the principles ORTN and ARTN are equivalent to Σ0 +2-induction +was established +in [11], so we only focus on the analysis of the Weihrauch degrees below. We first start by +defining properly the principles involved, and then we give the proof that TC∗ +N, (LPO′)∗ or +their product reduces to them. We then give the converse reductions, first for the principles +pertaining to the ordered colourings, and then we handle the additive colourings. +The +proof for the ordered colouring is a simple elaboration on [11, Lemma 4.3]. For the additive +colouring, formally the corresponding statement in that paper, [11, Proposition 4.1], depends + +20 +On the Weihrauch degree of the additive Ramsey theorem +on the ordered version in a way that would translate to a composition in the setting of +Weihrauch degrees. It turns out that we can avoid invoking the composition by carefully +interleaving the two steps in our analysis. +5.1 +Definitions +We have already covered the principles ORTN and ARTN in Section 2. The corresponding +Weihrauch problems are relatively clear: given a colouring as input, as well as the finite +semigroup or finite ordered structure, output an infinite homogeneous set. The principles +cORTN and cARTN instead only output a possible colour for an infinite homogeneous set – +much like in the case for Q. However, the principles iORTN and iARTN will require some +more attention; now it is rather meaningless to ask for a containing interval. Nevertheless, +the analogous principle will also output some information regarding the possible location of +an homogeneous set, without giving away a whole set or a candidate colour, so we keep a +similar naming convention. +▶ Definition 45. Define the following Weihrauch problems: +ORTN takes as input a finite poset (P, ⪯P ) and a right-ordered colouring c : [N]2 → P, +and outputs an infinite c-homogeneous set ⊆ N. +ARTN takes as input a finite semigroup S and an additive colouring c : [N]2 → S, and +outputs an infinite c-homogeneous set ⊆ N. +cORTN takes as input a finite poset (P, ⪯P ) and a right-ordered colouring c : [N]2 → P, +and outputs a colour p ∈ P such that there exists an infinite c-homogeneous set ⊆ N with +colour p. +cARTN takes as input a finite semigroup S and an additive colouring c : [N]2 → S, and +outputs a colour s ∈ S such that there exists an infinite c-homogeneous set ⊆ N with +colour s. +iORTN takes as input a finite poset (P, ⪯P ) and a right-ordered colouring c : [N]2 → P, +and outputs a n0 ∈ N such that there is an infinite c-homogeneous set X ⊆ N with two +elements ≤ n0. +iARTN takes as input a finite semigroup S and an additive colouring c : [N]2 → S, and +outputs a n0 ∈ N such that there is an infinite c-homogeneous set X ⊆ N with two +elements ≤ n0. +5.2 +Reversals +▶ Lemma 46. We have ECT ≤sW iORTN and ECT ≤sW iARTN. +Proof. Let f : N → k be a would-be instance of ECT. Then one may define the colouring +˜f : [N]2 → P(k) by setting a ∈ ˜f(n, m) if and only if there is n′ with n ≤ n′ ≤ m and +f(n′) = a. +This colouring is both additive for the semigroup (P(k), ∪) and ordered by +⊆, and can be fed to either iORTN or iARTN. Let n0 be such that there is an infinite ˜f- +homogeneous set with first two elements k0 < k1 ≤ n0. Clearly, every colour occuring in f +after n0 needs to occur in ˜f(k0, k1); so n0 is a solution of the given instance for ECT. +◀ +▶ Lemma 47. cORT∗ +N ≡W cORTN and cART∗ +N ≡W cARTN +Proof. The non-trivial reductions are easily made by amalgamating finite sequences of col- +ouring via a pointwise product, which will always still carry an additive or ordered struc- +ture. +◀ + +A. Pauly, P. Pradic & G. Soldà +21 +▶ Lemma 48. LPO′ ≤sW cORTN and LPO′ ≤sW cARTN. +Proof. We use IsFinite in place of LPO′. We start with an input f : N → 2 for IsFinite. We +compute ˜f : [N]2 → 2 where ˜f(n, m) = 1 iff 1 ∈ f −1([n, m]). This yields an additive and +ordered colouring. The colour of any given ˜f-homogeneous set indicates if f has infinitely +many ones or not, thus answering IsFinite for f. +◀ +5.3 +Reducing the ordered Ramsey theorem over N to (LPO′)∗ and ECT +We now explain how to bound the Weihrauch degree of ORTN and its weakenings. To do so, +it will be helpful to consider a construction approximating would-be homogeneous sets for a +given right-ordered colouring c : [N]2 → P and a target colour p ∈ P. With these parameters, +we build a recursive sequence of finite sets Y (p) : N → Pfin(N) meant to approximate a p- +homogeneous set (we shall simply write Y instead of Y (p) when p may be inferred from +context). If the construction succeeds, lim sup(Y ) will be an infinite homogeneous set with +colour p, otherwise lim sup(Y ) will be finite. But the important aspect will be that a fixed +number of calls to (LPO′)∗ will let us know if the construction was successful or not, while +ECT can indicate after which indices n we shall have Yn ⊆ lim sup(Y ) when it succeeds. +Now let us describe this construction for a fixed c and p. We begin with Y0 = ∅ and will +maintain the invariant that max(Yn) < n and for every (k, k′) ∈ [Yn]2, c(k, k′) = p. Then, +for Yn+1, we have several possibilities; +If min(Yn) exists and for any min(Yn) ≤ k < n, we have that p ≺P c(k, n), we set +Yn+1 = ∅ and say that the construction was injured at stage n. +Otherwise, if we have some k′ < n such that c(k′, n) = p and, for every k ∈ Yn, k < k′ +and c(k, k′) = p, then we set Yn+1 = Yn ∪ {k} and say that the construction progressed +at stage n. +Otherwise, set Yn+1 = Yn and say that the construction stagnated. +Clearly, we can also define recursive sequences injury(p) +n +: N → 2 and progress(p) +n +: N → 2 +that witness whether the construction was injured or progressed, and we have that lim sup(Y ) +is infinite if and only if injury contains finitely many 1 and progress contains infinitely many +ones. lim sup(Y ) is moreover always c-homogeneous with colour p. +▶ Lemma 49. For any ordered colouring c, there is p such that lim sup(Y ) is infinite +Proof. The suitable p may be found as follows: say that a colour p occurs after n in c if there +is k > m ≥ n with c(m, k) = p. There is a n0 such that every colour occuring after n0 in c +occurs arbitrarily far. For the ⪯P -maximal such colour occuring after n0, the construction +above will succeed with no injuries after stage n0 and infinitely many progressing steps (this +is exactly the same argument as for [11, Lemma 4.3]). +◀ +▶ Lemma 50. We have that cORTN ≤sW (LPO′)∗. +Proof. Given an input colouring c, compute in parallell all injury(p) and progress(p) for every +colour p and feed each sequence to an instance of LPO′. By Lemma 49, there is going to be +some p for which there is going to be finitey many injuries and infinitely many progressing +steps, and that p is the colour of some homogeneous set. +◀ +▶ Lemma 51. We have that ORTN ≤W (LPO′)∗ × ECT. +Proof. Given an input colouring c, compute in parallell all injury(p) and progress(p) for +every colour p and feed each sequence to an instance of LPO′ and all injury(p) to ECT. As + +22 +On the Weihrauch degree of the additive Ramsey theorem +before, use LPO′ to find out some p for which the construction succeed. For that p, ECT +will yield some n0 such that injury(p) +n += 0 for every n ≥ n0, so in particular, lim sup(Y (p)) = +� +n≥n0 Y (p) +n +, which is computable from n0. +◀ +▶ Lemma 52. We have that iORTN ≤sW ECT. +Proof. Given an input colouring c, consider for every colour p the sequence u(p) : N → +{0, 1, 2} defined by u(p) +n += min(3, |Yn|). Clearly it is computable from c. Applying ECT we +get some np such that +either there are infinitely many injuries after np +or u(p) +k += u(p) +np for every k ≥ np +By Lemma 49, we even know there is a p0 such that lim sup(Y (p0)) is infinite; additionally we +defined u in such a way that necessarily, np0 bounds two elements of lim sup(Y (p0)) because +we shall have u(p0) +k += 2 for every k ≥ np0. So we may simply take the maximum of all np to +solve our instance of iORTN. +◀ +This concludes our analysis of the ordered Ramsey theorem. +5.4 +Reducing the additive Ramsey theorem over N to (LPO′)∗ and ECT +We now turn to ARTN. The basic idea is that, given an additive colouring c, it is useful +to define the composite colouring L ◦ c, with L being a map from a finite semigroup to its +L-classes. ≤R then induces a right-ordered structure on the colouring. Constructing a L ◦ c +homogeneous set X such that we additionally have that c(min X, x) = c(min X, y) for every +x, y ∈ X \ {min X} ensures that X is c-homogeneous by Lemma 16. So we will give a recipe +to construct exactly such an approximation, similarly to what we have done in the previous +section. +So this time around, assume a semigroup S and a colouring c : [N]2 → S to be fixed. +For every s ∈ S, we shall define a recursive sequence of sets Y (s) : N → Pfin(N) (we omit +the superscript when clear from context) such that max(Yn) < n and Yn \ {min(Yn)} be +homogeneous, with, if Yn ̸= ∅, c(min(Yn), k) = s for k ∈ Yn \ {min(Yn)}. +For n = 0, we define Y0 = ∅. For Yn+1, we have a couple of options: +If Yn is empty and there is k < n such that c(k, n) = s and there is no k ≤ k′ < n′ ≤ n +with c(k′, n′) i0 such that ki1 is defined, and then +continue to produce the colouring αi1 + k0; either forever or until i1 gets enumerated into +A. We repeat this process until some iℓ is reached which is never enumerated into A (this +has to happen). + +24 +On the Weihrauch degree of the additive Ramsey theorem +Given a colour c that appears infinitely often in the resulting colouring, we can retrace +our steps and identify what iℓ was. We can then un-shift c to obtain a colour appearing +infinitely often in αiℓ, and thereby answer cRT1 ++ ⋆ CN. +For the converse direction, we observe that CN can compute from a colouring α : N → N +with finite range some k ∈ N such that α is a k-colouring. +◀ +The very same relationship holds for all our principles, i.e. the Weihrauch degree of the +version without finite information on the colours is just the composition of the usual version +with CN. The core idea, as in the proposition above, is that we can always start over by +moving to a fresh finite set of colours. For the interval versions we may have to do a little +bit more work to encode the CN-output by ensuring that all “large” intervals can never be +a valid answer. +To see that this observation already fully characterizes the Weihrauch reductions and +non-reductions between the usual and the relaxed principles, the notion of a (closed) fractal +from [1, 12] is useful. +▶ Definition 57. A Weihrauch degree f is called a fractal, if there is some F :⊆ NN ⇒ NN +with f ≡W F such that for any w ∈ N∗ either wNN ∩ dom(F) = ∅ or F|wNN ≡W f. If we +can chose F to be total, we call the Weihrauch degree a closed fractal. +If f is a fractal and f ≤W +� +i∈N gi, then there has to be some n ∈ N with f ≤W gn. +If f is a closed fractal and f ≤W g ⋆ CN, then already f ≤W g. Of our principles, the +versions with a fixed number of colours are closed fractals, the versions with a given-but- +not-fixed number of colours are not fractals at all, and the versions without explicit colour +information are fractals, but not closed fractals. From this, it follows that the versions with +no explicit colour information are never Weihrauch equivalent to our studied principles, and +that versions without explicit colour information are equivalent to one-another if and only +if their counterparts with explicit colour information are equivalent. +7 +Conclusion and future work +Summary +We have analysed the strength of an additive Ramseyan theorem over the ra- +tionals from the point of view of reverse mathematics and found it to be equivalent to Σ0 +2- +induction, and then refined that analysis to a Weihrauch equivalence with TC∗ +N × (LPO′)∗. +We have also shown that the problem decomposes nicely: we get the distinct complexities +(LPO′)∗ or TC∗ +N if we only require either the set of colours or the location of the homogeneous +set respectively. The same holds true for another equally and arguably more fundamental +shuffle principle, as well as the additive Ramsey theorem over N that was already studied +from the point of view of reverse mathematics in [11]. +Perpectives +It would be interesting to study further mathematical theorems that are +known to be equivalent to Σ0 +2-IND in reverse mathematics: this can be considered to contrib- +ute to the larger endeavour of studying principles already analyzed in reverse mathematics +in the framework of the Weihrauch degrees. In the particular case of Σ0 +2-IND, it can be +interesting to see which degrees are necessary for such an analysis. We refer to [4] for more +on this topic, and for a more comprehensive study of Ramsey’s theorem in the Weihrauch +degrees. + +A. Pauly, P. Pradic & G. Soldà +25 +Acknowledgements +The second author warmly thanks Leszek Kołodziejczyk for the proof of Lemma 17 as well as +Henryk Michalewski and Michał Skrzypczak for numerous discussions on a related project. +References +1 +Vasco Brattka, +Matthew de Brecht, +and Arno Pauly. +Closed choice and a uni- +form low basis theorem. +Annals of Pure and Applied Logic, 163(8):968–1008, 2012. +doi:10.1016/j.apal.2011.12.020. +2 +Vasco Brattka and Guido Gherardi. Completion of choice. Annals of Pure and Applied Logic, +172(3):102914, 2021. doi:10.1016/j.apal.2020.102914. +3 +Vasco Brattka, +Guido Gherardi, +and Arno Pauly. +Weihrauch Complexity in Com- +putable +Analysis, +pages +367–417. +Springer +International +Publishing, +Cham, +2021. +doi:10.1007/978-3-030-59234-9_11. +4 +Vasco Brattka and Tahina Rakotoniaina. On the uniform computational content of Ramsey’s +theorem. The Journal of Symbolic Logic, 82, 08 2015. doi:10.1017/jsl.2017.43. +5 +Olivier Carton, Thomas Colcombet, and Gabriele Puppis. +Regular languages of words +over countable linear orderings. In ICALP 2011 proceedings, Part II, pages 125–136, 2011. +doi:10.1007/978-3-642-22012-8_9. +6 +Caleb Davis, Denis R. Hirschfeldt, Jeffry L. Hirst, Jake Pardo, Arno Pauly, and Keita Yokoy- +ama. Combinatorial principles equivalent to weak induction. Comput., 9(3-4):219–229, 2020. +doi:10.3233/COM-180244. +7 +Damir D. Dzhafarov, Jun Le Goh, Denis. R. Hirschfeldt, Ludovic. Patey, and Arno Pauly. +Ramsey’s theorem and products in the Weihrauch degrees. +Computability, 9(2), 2020. +doi:10.3233/COM-180203. +8 +Emanuele Frittaion and Ludovic Patey. Coloring the rationals in reverse mathematics. Com- +putability, 6(4):319–331, 2017. doi:10.3233/COM-160067. +9 +Denis R. Hirschfeldt. Slicing the Truth. World Scientific, 2014. doi:10.1142/9208. +10 +Jeffry L. Hirst. +Combinatorics in subsystems of second order arithmetic. +Phd thesis, +Pennsylvania State University, 1987. +11 +Leszek Aleksander Kolodziejczyk, Henryk Michalewski, Pierre Pradic, and Michal Skrzypczak. +The logical strength of Büchi’s decidability theorem. Log. Methods Comput. Sci., 15(2), 2019. +doi:10.23638/LMCS-15(2:16)2019. +12 +Stéphane Le Roux and Arno Pauly. Finite choice, convex choice and finding roots. Logical +Methods in Computer Science, 2015. doi:10.2168/LMCS-11(4:6)2015. +13 +Eike Neumann and Arno Pauly. A topological view on algebraic computation models. J. +Complex., 44:1–22, 2018. doi:10.1016/j.jco.2017.08.003. +14 +Dominique Perrin and Jean-’Eric Pin. Infinite words : automata, semigroups, logic and games. +Pure and applied mathematics. 2004. +15 +Pierre Pradic and Giovanni Soldá. +On the Weihrauch degree of the additive Ramsey +theorem over the rationals. +In Ulrich Berger, Johanna N. Y. Franklin, Florin Manea, +and Arno Pauly, editors, Revolutions and Revelations in Computability - 18th Confer- +ence on Computability in Europe, CiE 2022, Swansea, UK, July 11-15, 2022, Proceed- +ings, volume 13359 of Lecture Notes in Computer Science, pages 259–271. Springer, 2022. +doi:10.1007/978-3-031-08740-0\_22. +16 +Saharon Shelah. The monadic theory of order. Ann. of Math. (2), 102(3):379–419, 1975. +17 +Stephen G. Simpson. Subsystems of second order arithmetic. Perspectives in Mathematical +Logic. 1999. doi:10.1007/978-3-642-59971-2. +18 +Giovanni Solda and Manlio Valenti. Algebraic properties of the first-order part of a problem, +2022. URL: https://arxiv.org/abs/2203.16298, doi:10.48550/ARXIV.2203.16298. + diff --git a/2NE1T4oBgHgl3EQfAAJE/content/tmp_files/load_file.txt b/2NE1T4oBgHgl3EQfAAJE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4dd7f092276f7d81870cd17e43f007ff08a7a6c7 --- /dev/null +++ b/2NE1T4oBgHgl3EQfAAJE/content/tmp_files/load_file.txt @@ -0,0 +1,1024 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf,len=1023 +page_content='On the Weihrauch degree of the additive Ramsey theorem Arno Pauly � � School of Mathematics and Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' Swansea University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' UK Pierre Pradic � School of Mathematics and Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' Swansea University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' UK Giovanni Soldà � � School of Mathematics and Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' Swansea University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' UK 1 Abstract We characterize the strength,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' in terms of Weihrauch degrees,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' of certain problems related to Ramsey- like theorems concerning colourings of the rationals and of the natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' The theorems we are chiefly interested in assert the existence of almost-homogeneous sets for colourings of pairs of rationals respectively natural numbers satisfying properties determined by some additional algebraic structure on the set of colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' In the context of reverse mathematics, most of the principles we study are equivalent to Σ0 2- induction over RCA0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' The associated problems in the Weihrauch lattice are related to TC∗ N, (LPO′)∗ or their product, depending on their precise formalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' 2012 ACM Subject Classification Theory of computation → Proof theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' Theory of computation → Computability Keywords and phrases Weihrauch reducibility, Reverse mathematics, additive Ramsey, Σ0 2-induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' Related Version This article extends the conference paper [15] by the second and third author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' Funding Giovanni Soldà: This author was supported by an LMS Early Career Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' 1 Introduction The infinite Ramsey theorem says that for any colouring c of n-uples of a given arity of an infinite set X, there exists a infinite subset H ⊆ X such that the set of n-tuples [H]n of elements of H is homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' This statement is non-constructive: even if the colouring c is given by a computable function, it is not the case that we can find a computable homogeneous subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' Various attempts have been made to quantify how non-computable this problem and some of its natural restrictions are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' This is in turn linked to the axiomatic strength of the corresponding theorems, as investigated in reverse mathematics [17] where Ramsey’s theorem is a privileged object of study [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' This paper is devoted to a variant of Ramsey’s theorem with the following restrictions: we colour pairs of rational numbers and we require some additional structure on the colouring, namely that it is additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' A similar statement first appeared in [16, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content='3] to give a self-contained proof of decidability of the monadic second-order logic of (Q, <).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' We will also analyse a simpler statement we call the shuffle principle, a related tool appearing in more modern decidability proofs [5, Lemma 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' The shuffle principle states that every Q-indexed word (with letters in a finite alphabet) contains a convex subword in which every letter appears densely or not at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' Much like the additive restriction of the Ramsey theorem for pairs over N, studied from the point of view of reverse mathematics in [11], we obtain a neat correspondence with Σ0 2-induction (Σ0 2-IND).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' 1 Soldà has since moved to Ghent University arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content='02833v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content='LO] 7 Jan 2023 2 On the Weihrauch degree of the additive Ramsey theorem ▶ Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' In the weak second-order arithmetic RCA0, Σ0 2-IND is equivalent to both the shuffle principle and the additive Ramsey theorem for Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' We take this analysis one step further in the framework of Weihrauch reducibility that al- lows to measure the uniform strength of general multi-valued functions (also called problems) over Baire space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' Let Shuffle and ARTQ be the most obvious problems corresponding to the shuffle principle and additive Ramsey theorem over Q respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' We relate them, as well as various weakenings cShuffle, cARTQ, iShuffle and iARTQ that only output sets of colours or intervals, to the standard (incomparable) problems TCN and LPO′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' We also consider the ordered Ramsey principle, ORTQ, where the colours k come equipped with a partial order ⪯, and the colouring α : [Q]2 → k satisfying that α(r1, r2) ⪯ α(q1, q2) if q1 ≤ r1 < r2 ≤ q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' A weakening of Shuffle is the principle (η)1 <∞ introduced in [8] where we ask merely for an interval where some colour is dense;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' respectively for a colour which is dense somewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' ▶ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' We have the following equivalences Shuffle ≡W ARTQ ≡W TC∗ N × (LPO′)∗ cShuffle ≡W cARTQ ≡W (LPO′)∗ iShuffle ≡W iARTQ ≡W (η)1 <∞ ≡W i(η)1 <∞ ≡W TC∗ N ORTQ ≡W LPO∗ c(η)1 <∞ ≡W cRT1 + Finally, we turn to carrying out the analysis of those Ramseyan theorems over N in the framework of Weihrauch reducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' The additive Ramsey theorem over N is also an important tool in the study of monadic second order logic over countable scattered orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' As for the case of Q, we relate problems ARTN and ORTN as well as some natural weakenings cARTN, cORTN, iARTN and iORTN, to TCN and LPO′ (the i variants of those principle return, rather than an interval, some upper bound n on the first two points of some infinite homogeneous set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' ▶ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' We have the following equivalences ORTN ≡W ARTN ≡W TC∗ N × (LPO′)∗ cORTN ≡W cARTN ≡W (LPO′)∗ iORTN ≡W iARTN ≡W TC∗ N 2 Background In this section, we will introduce the necessary background for the rest of the paper, and fix most of the notation that we will use, except for formal definitions related to weak subsystems of second-order arithmetic, in particular RCA0 (which consists of Σ0 1-induction and recursive comprehension) and RCA0 + Σ0 2-IND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' A standard reference for that material and, more generally, systems of interest in reverse mathematics, is [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content='1 Generic notations We identify k ∈ N with the finite set {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' , k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfAAJE/content/2301.02833v1.pdf'} +page_content=' For every linear order (X,