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2024PhRvD.110f3023L

We propose a simple fit function inlineformulammlmath displayinlinemmlmsubmmlmiLmmlmimmlmsubmmlmimmlmimmlmiimmlmimmlmsubmmlmsubmmlmo stretchyfalsemmlmommlmitmmlmimmlmo stretchyfalsemmlmommlmommlmommlmiCmmlmimmlmsupmmlmitmmlmimmlmrowmmlmommlmommlmimmlmimmlmrowmmlmsupmmlmsupmmlmiemmlmimmlmrowmmlmommlmommlmo stretchyfalsemmlmommlmitmmlmimmlmommlmommlmimmlmimmlmsupmmlmo stretchyfalsemmlmommlminmmlmimmlmsupmmlmrowmmlmsupmmlmathinlineformula to parametrize the luminosities of neutrinos and antineutrinos of all flavors during the protoneutron star PNS cooling phase at postbounce times inlineformulammlmath displayinlinemmlmitmmlmimmlmommlmommlmn1mmlmnmmlmtext mmlmtextmmlmtext mmlmtextmmlmi mathvariantnormalsmmlmimmlmathinlineformula. This fit is based on results from a set of neutrinohydrodynamics simulations of corecollapse supernovae in spherical symmetry. The simulations were performed with an energydependent transport for six neutrino species and took into account the effects of convection and muons in the dense and hot PNS interior. We provide values of the fit parameters inlineformulammlmath displayinlinemmlmiCmmlmimmlmathinlineformula inlineformulammlmath displayinlinemmlmimmlmimmlmathinlineformula inlineformulammlmath displayinlinemmlmimmlmimmlmathinlineformula and inlineformulammlmath displayinlinemmlminmmlmimmlmathinlineformula for different neutron star masses and equations of state as well as correlations between these fit parameters. Our functional description is useful for analytic supernova modeling for characterizing the neutrino light curves in large underground neutrino detectors and as a tool to extract information from measured signals on the mass and equation of state of the PNS and on secondary signal components on top of the PNSs neutrino emission.

2024-09-01T00:00:00Z

['2024PhRvD.110f3023L', '10.48550/arXiv.2405.00769', '10.1103/PhysRevD.110.063023', 'arXiv:2405.00769', '2024arXiv240500769L']

['Astrophysics and astroparticle physics', 'Astrophysics - High Energy Astrophysical Phenomena', 'High Energy Physics - Phenomenology']

Simple fits for the neutrino luminosities from protoneutron star cooling

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML']

https://arxiv.org/pdf/2405.00769.pdf

{'Simple fits for the neutrino luminosities from protoneutron star cooling': "Giuseppe Lucente, 1, 2, 3, 4, ∗ Malte Heinlein, 5, 6, † Hans-Thomas Janka, 5, ‡ and Alessandro Mirizzi 1, 2, § \n1 Dipartimento Interateneo di Fisica 'Michelangelo Merlin', Via Amendola 173, 70126 Bari, Italy. \n2 \nIstituto Nazionale di Fisica Nucleare - Sezione di Bari, Via Orabona 4, 70126 Bari, Italy. \n3 Kirchhoff-Institut fur Physik, Universitat Heidelberg, \nIm Neuenheimer Feld 227, 69120 Heidelberg, Germany \n4 Institut fur Theoretische Physik, Universitat Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany 5 Max-Planck-Institut fur Astrophysik, Karl-Schwarzschild-Str. 1, 85748 Garching, Germany 6 Technische Universitat Munchen, TUM School of Natural Sciences, \nPhysics Department, James-Franck-Str. 1, 85748 Garching, Germany \nWe propose a simple fit function, L ν i ( t ) = Ct -α e -( t/τ ) n , to parametrize the luminosities of neutrinos and antineutrinos of all flavors during the protoneutron star (PNS) cooling phase at post-bounce times t ≳ 1 s. This fit is based on results from a set of neutrino-hydrodynamics simulations of core-collapse supernovae in spherical symmetry. The simulations were performed with an energy-dependent transport for six neutrino species and took into account the effects of convection and muons in the dense and hot PNS interior. We provide values of the fit parameters C , α , τ , and n for different neutron star masses and equations of state as well as correlations between these fit parameters. Our functional description is useful for analytic supernova modeling, for characterizing the neutrino light curves in large underground neutrino detectors, and as a tool to extract information from measured signals on the mass and equation of state of the PNS and on secondary signal components on top of the PNS's neutrino emission.", 'I. INTRODUCTION': "The neutrino signal from a future Galactic supernova (SN) explosion represents one of the next frontiers of neutrino astrophysics (see, e.g., Refs. [1-5]). Existing and planned large underground neutrino detectors guarantee that a high-statistics neutrino burst will be collected during such an event (see, e.g., Refs. [1, 2, 6-11]). This detection will be extremely important to probe the SN explosion mechanism [12-19], neutrino flavor conversions [20-24], and particle physics [24-26] occurring in the deepest stellar regions. \nIn order to characterize the response of a detector to a SN neutrino burst, one has to rely on the outcome of state-of-the-art numerical SN simulations. Different from the case of solar neutrinos, a standard prediction for SN neutrino fluxes does not exist. Therefore, many current studies employ a parametric approach with a suitable range of variation guided by the results of the simulations. For this purpose, it has been shown that the time-integrated SN neutrino fluxes are well represented by a spectrum with the functional form [27, 28] \nf ν ( E ) ∝ E β e -( β +1) E/ ⟨ E ⟩ , (1) \nwhere ⟨ E ⟩ is the average energy of a given neutrino species. \nConcerning the time evolution of the neutrino luminosities, most of the detector characterizations use, for comparison, results obtained from numerical tables provided by simulations (see, e.g., [1, 8, 29-32]). However, in order to extend parametric studies also to the temporal evolution of the neutrino signal, it would be useful to have a simple functional prescription also for the luminosities, based on a few parameters that permit one to cover the range of variation obtained in the numerical SN simulations. \nIn this respect, the seminal work of Loredo and Lamb [33], concerned with an analysis of SN 1987A data, assumed an exponential cooling model. 1 This simple recipe was often used (see, e.g., Refs. [35-37]) for schematic estimations. However, as noticed already in Ref. [38], protoneutron star (PNS) cooling calculations do not yield exponential neutrino light curves but instead suggest that the neutrino luminosity is better described by a power-law decline [39]. A further step was taken in the recent work of Ref. [40], where on the grounds of an analysis of recent spherically symmetric PNS cooling simulations covering post-bounce times up to t ∼ 70 s, a combined ansatz of power-law and exponential behavior of the form L ν i ( t ) ∝ t -1 e -( t/τ ) n was proposed for the long-time behavior of the neutrino light \ncurve, where τ is a characteristic timescale of the PNS cooling. 2 In that work τ ∼ 30 s was adopted to handle the transition to the regime of neutrino transparency, and therefore the functional description of the neutrino light curve follows a simple decline according to t -1 in the first 10 s after core bounce. However, the models considered in Ref. [40] did not include the effect of convection in the PNS. This has important consequences for the shape of the neutrino light curve, because the Kelvin-Helmholtz neutrino cooling of the PNS is strongly accelerated if convection persists for many seconds. \nIn Ref. [42] it was shown that a kink in the neutrino light curve, when displayed in a doubly logarithmic form, signals the end of convection in the mantle layer of the PNS, while convection still continues in the deeper, high-density PNS core. The duration of PNS mantle convection and therefore the time of the kink depends on the nuclear equation of state (EoS), in particular on the behavior of the nuclear symmetry energy as a function of the density at and above nuclear saturation. The kink is also present in the count rate of a neutrino detection and thus could be easily observed. When it occurs late during the PNS cooling, it resembles a 'knee' since it is followed by a steep decline when the PNS cools off and gradually becomes transparent to neutrinos. Otherwise the kink transitions into a longer, flatter tail of the light curve. Typically in modern models the knee is witnessed to show up at t < 10 s [43, 44]. \nInspired by these previous findings, we perform a detailed analysis of the neutrino luminosities obtained in a set of spherically symmetric PNS cooling models recently described in Ref. [43]. These simulations were performed with sixspecies neutrino transport based on a fully energy-dependent two-moment scheme with a variable Eddington closure obtained from the solution of the Boltzmann transport equation. Moreover, the effects of PNS convection and muons were taken into account. After this analysis we propose an accurate fit for the neutrino luminosities during the PNS cooling evolution at post-bounce times t ≳ 1 s, based on the following functional form: \nL ν i ( t ) = C t -α e -( t/τ ) n , (2) \nwhere the parameter C is a normalization constant, α describes the power-law behavior in the early cooling phase, τ is a characteristic cooling time of the exponential drop at later times, and n determines the steepness of the exponential decline at these late times. Importantly, for models including PNS convection, the value of τ is much shorter, τ < 10 s, than assumed in Ref. [40]. Therefore, this fit function can be useful as a sensitive probe of the presence and duration of convection in the PNS mantle during the long-time neutrino cooling. Such a probe is complementary to the method of analysis considered in Ref. [42], where the ratio of neutrino events at early and late times was proposed as a diagnostic measure. \nIn the present paper, we report our analysis and fitting procedure and provide extended tables with the parameter values for the fit functions of all our models and for interesting correlations between these fit parameters. The outline of the paper is as follows. In Section II we show the neutrino luminosities from our 1D PNS cooling simulations for different PNS masses and nuclear EoSs. In Section III we describe our analytical fitting of the neutrino and antineutrino luminosities of all flavors based on Eq. (2) for the models discussed before. In Section IV we discuss the dependence of the values of the fitting parameters on PNS mass and nuclear EoS and how their determination can help in deducing information on these latter quantities from a neutrino measurement. In Section V we discuss tight correlations of the parameter values on the one hand with the PNS mass (for fixed EoS) and, on the other hand, between each other (for fixed PNS mass). Then, in Section VI, we consider how the previous results change for models that neglect the effects of PNS convection and of muons in the PNS cooling models. In Section VII we show how our analytical recipe can be used to fit the event rate in a large neutrino underground detector. Finally, in Section VIII we discuss possible applications of our fitting formula and finish with conclusions. In several appendices we provide extended tables with the values obtained by our numerical fitting and needed for practical use.", 'II. NEUTRINO LUMINOSITIES': "Our work is based on results from 1D hydrodynamical PNS cooling simulations employing the PROMETHEUS-VERTEX neutrino-hydrodynamics code [45], which solves the fully energy and velocity-dependent neutrino transport for all six species of neutrinos and antineutrinos with a state-of-the-art implementation of the neutrino interactions [46-48]. The models we will use are taken from the Garching Core-collapse Supernova Archive [49] (all data are available upon request) and they are the same as those considered in the recent study of Ref. [43], to which we refer the interested readers for more information on the model evolution and a discussion of the neutrino emission properties. \nA concise and comprehensive summary of the physics inputs in the latest version of the PROMETHEUS-VERTEX code is also provided by Ref. [43]. In particular, the discussed models include a 1D treatment of PNS convection via a \nTABLE I. Final times t fin and ratios X fin ν i [see Eq. (3)] for all PNS simulations and neutrino species. Values of X fin ν i larger than 0.15 are printed in bold and asterisks mark the largest values, X fin ν µ = 0 . 219 and X fin ν µ = 0 . 185, for models 1.62-SFHx-c and for 1.61-LS220-c, respectively. The detailed numerical neutrino outputs for all the simulations listed here are available at the Garching Core-Collapse Supernova Archive [49]. \nmixing-length description of the convective fluxes [2] and take into account the presence of muons in the hot PNS, including the corresponding muonic neutrino interactions [48], although the differences in the luminosities and spectra of µ and τ neutrinos turn out to be relatively small during most of the PNS cooling evolution (unless the PNS is very massive). \nWe consider simulations that yield baryonic PNS masses 3 of 1.36 M ⊙ , 1.44 M ⊙ , 1.62 M ⊙ , 1.77 M ⊙ , and 1.93 M ⊙ , in each case computed with four different nuclear EoSs, namely DD2 [50-52], SFHo, SFHx [51, 53], and LS220 with a nuclear incompressibility at saturation density of K = 220MeV [54]. Correspondingly, we follow Ref. [43] in our naming convention of the simulations, specifying the PNS mass and the EoS in the model names, e.g. 1.62-DD2. Our standard simulations include PNS convection and muons; those without convection are denoted by a suffix '-c' to their names, and those without muons by a suffix '-m'. In Table I we list all of the discussed simulations and the final post-bounce times t fin when they were stopped. \nIn this work we are interested in the post-accretion phase of the PNSs born in core-collapse SNe, for which reason we focus on the evolution of the neutrino signal only at post-bounce times t ≳ 1 s. The discussed PNS models result from 1D simulations with initial conditions from several 1D models of stellar progenitors with different pre-collapse masses. These models are artificially exploded at different instants after bounce, which are chosen such that the PNS mass after the end of the post-bounce accretion attains the desired value. The neutrino-driven explosions are triggered by a sufficiently strong reduction of the density and thus of the ram pressure in the infall region upstream of the \nFIG. 1. Time evolution of the luminosities L ν i for neutrinos (left panels) and antineutrinos (right panels) of all flavors for a PNS baryonic mass of M NS = 1 . 62 M ⊙ and all considered EoSs: DD2 (blue), SFHo (orange), SFHx (green) and LS220 (red). \n<!-- image --> \nstalled SN shock (for more details, see Ref. [43]). Although it is common knowledge now that SN explosions are a 3D phenomenon and 3D simulations are needed to investigate the explosion mechanism and the post-bounce accretion phase as well as the associated neutrino signal, the late evolution times considered in our study, t ≳ 1 s, are sufficiently well represented by our 1D simulations if the PNS does not experience continued accretion that extends beyond 1 s after bounce. Another prerequisite is that our mixing-length treatment of PNS convection is a good approximation of the 3D hydrodynamics and associated energy and lepton transport that takes place inside the PNS over timescales of many seconds. \nFigure 1 displays the time evolution of the luminosities L ν ( t ) for all the neutrino (left panels) and antineutrino (right panels) flavors in units of B / s (1 B = 10 51 erg) in the time interval [1,10] s for the 1.62 M ⊙ models, computed with the set of considered EoSs cases: DD2 in blue, SFHo in orange, SFHx in green, and LS220 in red. From this figure it is evident that the neutrino luminosities exhibit a change of their slope in the time interval considered. Since we expect a power-law behavior of the neutrino luminosities in the early cooling phase and we also want to enhance the visibility of the change in the steepness of the luminosity decline at later times, we take inspiration from Ref. [40] and show in Fig. 2 the product of the post-bounce time and the luminosities, t L ν ( t ), for all of the neutrino (left panels) and antineutrino (right panels) flavors in the same time interval and for the same simulations as in Fig. 1. One witnesses an interesting difference of the results for the employed EoSs. For DD2, SFHo and SFHx EoSs and all \nFIG. 2. Time evolution of the product of time and luminosities, t L ν i , for neutrinos (left panels) and antineutrinos (right panels) of all flavors for M NS = 1 . 62 M ⊙ and all considered EoSs: DD2 (blue), SFHo (orange), SFHx (green), and LS220 (red). \n<!-- image --> \nof the neutrino species the quantity t L ν increases as a power-law at t ≳ 1 s to peak around t ≈ 4 s, followed by a steep suppression at later times. This behavior produces a prominent knee at about 5-6 s after bounce. In contrast, for LS220 t L ν reaches its peak earlier, namely at t ≲ 3 s, and it features a slower decline later on. Since for each model the luminosities of all heavy-lepton neutrino species show a similar evolution, we will focus exclusively on the ¯ ν µ signal as representative of all non-electron neutrino species, unless otherwise noted. \nIn order to compare the mass dependence for the different EoS cases, we show the time evolution of t L ν ( t ) for the different PNS masses keeping the EoS fixed. Specifically, in Fig. 3 we consider the results for DD2 (left) and SFHo (right), and in Fig. 4 we present those for SFHx (left) and LS220 (right). In each panel the 1 . 36 M ⊙ (blue), 1 . 44 M ⊙ (orange), 1 . 62 M ⊙ (green), 1 . 77 M ⊙ (red) and 1 . 93 M ⊙ (purple) models are plotted for ν e (top), ¯ ν e (middle) and ¯ ν µ (bottom). It is obvious that, at fixed time t , the luminosity becomes larger as the PNS mass increases. This dependence is similar for all EoSs and all neutrino species. Additionally, the maximum of t L ν is shifted to later times and the subsequent decline starts correspondingly later for higher PNS masses. These findings, which are consistent with multi-D results in [17], can be explained by the fact that more gravitational binding energy is released in neutrinos when the PNS has a bigger mass. \nFor each case shown in Figs. 2, 3, and 4, the quantity t L ν has a peak at t ν i , max ≲ 6 s, when we define L ν i , max ≡ L ν i ( t ν i max ). The corresponding values of t ν i , max are listed in Table II for neutrinos and in Table III \nFIG. 3. Time evolution of the product of time and luminosity, t L ν i , for ν e (upper panels), ¯ ν e (middle panels) and ¯ ν µ (lower panels) for DD2 (left) and SFHo (right) and all investigated PNS masses: M NS = 1 . 36 M ⊙ (blue), M NS = 1 . 44 M ⊙ (orange), M NS = 1 . 62 M ⊙ (green), M NS = 1 . 77 M ⊙ (red) and M NS = 1 . 93 M ⊙ (purple). \n<!-- image --> \nfor antineutrinos in Appendix A. In Table I we provide the values of the quantity \nX fin ν i ≡ t fin L ν i , fin t ν i , max L ν i , max , (3) \nwhich is a measure for how long our simulations followed the late-time luminosity decline until they were stopped at t fin . For our benchmark PNS models (all models including muons and convection) X fin ν i ≲ 0 . 15 holds for all neutrinos and antineutrinos (the largest value is X fin ν e = 0 . 123 for 1.62-LS220). In order to adopt a well-defined, common final time for all of the neutrino species and simulations when fitting the luminosities in the following, we cut the data at an instant t ν i , c when X c ν i ≡ t ν i , c L ν i , c / ( t ν i , max L ν i , max ) = 0 . 15. In addition to the values of t ν i , max we also provide those of t ν i , c for all neutrino species and simulations in Tables II and III in Appendix A. It can be seen that for a fixed EoS and neutrino species, the values of both of these quantities increase with the PNS mass. In contrast, given the PNS mass and EoS, t ν i , max and t ν i , c do not vary much among the different kinds of neutrinos and antineutrinos (e.g., for 1.62-DD2 t ν i , max ≈ 4 s and t ν i , c ≈ 8 s for all species). However, at fixed PNS mass, t ν i , max and t ν i , c feature a strong dependence on the considered EoS. In particular, t ν i , c ≲ 10 s for all the benchmark models with DD2, whereas SFHo and SFHx lead to slightly larger values of t ν i , c , with t ν i , c ≲ 12 . 5 s for all models, and in the cases with LS220 we find the highest values of t ν i , c , with t ν i , c ≲ 15 s for the largest PNS mass. \nFIG. 4. Time evolution of the product of time and luminosity, t L ν i , for ν e (upper panels), ¯ ν e (middle panels) and ¯ ν µ (lower panels) for SFHx (left) and LS220 (right) and all investigated PNS masses: M NS = 1 . 36 M ⊙ (blue), M NS = 1 . 44 M ⊙ (orange), M NS = 1 . 62 M ⊙ (green), M NS = 1 . 77 M ⊙ (red) and M NS = 1 . 93 M ⊙ (purple). \n<!-- image -->", 'III. FIT FUNCTIONS': "After an analysis of the models described above, we propose Eq. (2) as fit function for the neutrino luminosities in the considered time window: \nL ν i ( t ) = C t -α e -( t/τ ) n , \nwhere C , α , τ and n are free parameters and t and τ are measured in seconds. 4 In Eq. (2) the parameter C is a normalization constant, α describes the power-law behavior at early cooling time deviating from the simple t -1 behavior, τ is a characteristic cooling time for the exponential drop after the peak in t L ν , with n representing the strength of the suppression at late times. We mention here that at the moment there is no straightforward argument explaining the power-law behavior at early cooling time, given that PNS convection plays an important role and it cannot be treated in simple ways. As an example, we show for the 1 . 62 M ⊙ model and all EoSs the time evolution of t L ν i from simulations (blue lines) and its best fit (orange) between t = 1s and t ν i , c , for ν e in Fig. 5, for ¯ ν e in Fig. 6 and for ¯ ν µ in Fig. 7. For DD2, SFHo and SFHx the fit well reproduces the results from the simulations in \nFIG. 5. Time evolution between 1 s and t ν e , c of the product of time and ν e luminosity, t L ν e , for simulation data (blue) and their fits (orange) for M NS = 1 . 62 M ⊙ and different EoSs: DD2 (upper left), SFHo (upper right), SFHx (lower left) and LS220 (lower right). \n<!-- image --> \nthe considered time interval, with best-fit parameters in a similar range. For each EoS, the highest values of the fit parameters C and τ are obtained for ¯ ν µ , while ν e show the lowest ones. A similar behavior is found for the other PNS masses. \nThe fit for models with EoS LS220 shows worse agreement with the data from simulations (see the lower-right panels in Figs. 5, 6, and 7). In particular, for all neutrino species the fit overestimates the luminosity at t ≈ 1 s and exhibits a lower value of the peak, whereas it well reproduces the luminosity at later times. In this case, the values of the best-fit parameters are in a completely different range compared to those of the models with the other EoSs. In particular, for each neutrino species C ≈ O (100) B / s and α < 0, whereas τ and n adopt much smaller values than for the other considered EoSs, with τ ≲ 0 . 1 s and n ≲ 0 . 5, due to the more shallow decline of the luminosity at late times. The differences in the time dependence of the neutrino luminosities between models with different EoSs can be understood by the different behavior of the nuclear symmetry energy as a function of baryon density for the considered EoS. In the LS220 EoS, the symmetry energy exhibits a steeper increase with baryon density than in all other cases. As discussed in Ref. [42], a larger positive derivative of the symmetry energy with baryon density can lead to suppressed convection in the PNS mantle at high densities and low electron fraction. This effect happens in the simulations with the LS220 EoS after about 3 s and gradually quenches PNS mantle convection, thus delaying subsequent PNS neutrino cooling due to reduced neutrino luminosities. In contrast, PNS convection continues to be active in a spatially more extended region including the PNS mantle in simulations with the other three EoSs, for which reason convectively enhanced neutrino luminosities are maintained for a longer period of time until a late, steep decline follows when the PNS has deleptonized and cools off. These differences explain the early kink in the \nFIG. 6. Time evolution between 1 s and t ¯ ν e , c of the product of time and ¯ ν e luminosity, t L ¯ ν e , for simulation data (blue) and their fits (orange) for M NS = 1 . 62 M ⊙ and different EoSs: DD2 (upper left), SFHo (upper right), SFHx (lower left) and LS220 (lower right). \n<!-- image --> \nneutrino luminosities for models with the LS220 EoS, whereas a prominent knee-like shape of L ν ( t ) can be witnessed for the DD2, SFHo, and SFHx models (see Fig. 1). These differences motivate us to define DD2, SFHo, and SFHx as members of an EoS-class that we call 'Class A', whereas LS220 is a representative of a 'Class B'. The parameters of the symmetry energies for all the EoS cases used in our study are listed in Table XII in Appendix F. \nIn Appendix B we report the values of the best-fit parameters and their 1 σ uncertainties, 5 for neutrinos and antineutrinos of all flavors and all considered models with different PNS masses and EoSs. For each species, the 1 σ errors on the best-fit parameters are within ∼ O (0 . 1 -1)% for Class A EoSs, whereas for LS220 the maximum errors are ∼ O (10)% or even larger, and tend to increase at higher PNS masses, reflecting the lower quality of the fit.", 'IV. DEPENDENCE ON PNS MASS AND EQUATION OF STATE': 'The best-fit parameters are plotted as functions of the PNS mass in Fig. 8 for ν e , in Fig. 9 for ¯ ν e and in Fig. 10 for ¯ ν µ , separately for Class A EoSs in the left panels and the LS220 case of Class B in the right panels. \nFIG. 7. Time evolution between 1 s and t ¯ ν µ , c of the product of time and ¯ ν µ luminosity, t L ¯ ν µ , for simulation data (blue) and their fits (orange) for M NS = 1 . 62 M ⊙ and different EoSs: DD2 (upper left), SFHo (upper right), SFHx (lower left) and LS220 (lower right). \n<!-- image -->', 'A. Class A EoS': "For Class A EoSs and all species of neutrinos and antineutrinos, the behavior of all of the fitting parameters can be well reproduced with linear dependences on the PNS mass, as shown by the black (DD2), red (SFHo), and blue (SFHx) lines in the left panels of Figs. 8, 9, and 10. The shaded area around the linear interpolation represents the 1 σ confidence band for each linear fit, obtained from the standard mean square uncertainties associated with the linear regression. In Table VIII of Appendix C we provide the best-fit parameter values that describe the linear dependencies of the fitting parameters C , α , τ , and n on the PNS mass for all neutrino and antineutrino species and all of the considered EoS. \n- (i) The 'normalization parameter C ' (top panels) adopts values of 5 B / s ≲ C ≲ 15 B / s for all neutrino species and it increases with the PNS mass. In particular, DD2 leads to the lowest values and SFHo and SFHx feature a similar dependence, with SFHx providing the largest values of C .\n- (ii) The 'power-law index' α (second panels from top) shows a mild dependence on the PNS mass, featuring values of 0 . 50 ≲ α ≲ 0 . 70 for all neutrino species, with an increase in the case of DD2 and a slight decrease for SFHo and SFHx. Similar to the normalization parameter, for each kind of neutrino and PNS mass, the largest value of α is obtained with SFHx and the smallest one with DD2.\n- (iii) The 'late-time suppression parameters' τ and n (third and fourth panels from top) tend to increase with the \nFIG. 8. Best-fit parameters C , α , τ , and n as functions of the PNS mass for ν e for Class A EoSs (left panels) and for LS220 (right panels), obtained with data between 1 s and t ν e , c . The shaded areas represent the 1 σ confidence bands. \n<!-- image --> \nPNS mass for all neutrino species, with values of 5 s ≲ τ ≲ 10 s and 3 ≲ n ≲ 5. In particular, for all neutrino kinds and PNS masses, DD2 leads to the lowest values of τ and SFHx to the largest ones. On the other hand, DD2 yields the largest n values for ν and ¯ ν of all flavors and all PNS masses, whereas the 1 σ confidence bands of n for SFHo and SFHx models overlap in the mass range of 1 . 4 M ⊙ ≲ M NS ≲ 1 . 7 M ⊙ , with SFHx showing larger values of n than SFHo at higher PNS masses. \n12 \n10 \n8 \n6 \n0.70 \n0.65 \n0.60 \n0.55 \n10 \n9 \n8 \n7 \n6 \n5.0 \n4.8 \n4.6 \n4.4 \n4.2 \n4.0 \n3.8 \nFIG. 9. Best-fit parameters C , α , τ , and n as functions of the PNS mass for ¯ ν e for Class A EoSs (left panels) and for LS220 (right panels), obtained with data between 1 s and t ¯ ν e , c . The shaded areas represent the 1 σ confidence bands. \n<!-- image --> \n1.4 \n1.5 \n1.6 \n1.7 \n1.8 \n1.9 \nA given value of C , τ , and n can be obtained with different combinations of EoS and neutron star (NS) mass. For instance, for ¯ ν e , τ = 7 s corresponds to M NS ≈ 1 . 38 M ⊙ and SFHx, M NS ≈ 1 . 45 M ⊙ and SFHo, as well as M NS ≈ 1 . 77 M ⊙ and DD2. However, the best-fit values of α lie in different ranges for different EoSs. For instance, 0 . 6 ≲ α ¯ ν e ≲ 0 . 65 would point to the SFHo EoS, irrespective of the PNS mass. Therefore, in the case of a future observation of a SN neutrino signal, the measurement of α would provide information on the EoS and the PNS mass and, combined with the measurements of the other fit parameters, would allow us to characterize the PNS mass and the EoS. More explicitly, neglecting for simplicity flavor mixing, a combined measurement of τ = 7 s, α ≈ 0 . 63, \nDD2 \nSFHo \nSFHx \nFIG. 10. Best-fit parameters C , α , τ , and n as functions of the PNS mass for ¯ ν µ for Class A EoSs (left panels) and for LS220 (right panels), obtained with data between 1 s and t ¯ ν µ , c . The shaded areas represent the 1 σ confidence bands. \n<!-- image --> \nand n ≈ 3 . 9 for the ¯ ν e luminosity would suggest a SN explosion leading to a PNS of mass 1 . 44 M ⊙ , whose interior properties are described by the SFHo EoS. As further discussed in Sec. VII, due to the similarities of the spectra and luminosities of electron antineutrinos and heavy-lepton antineutrinos in our models at times t ≳ 1 s, flavor conversions are not a major effect, although they would make the reconstruction of the fit parameters less straightforward, but without spoiling our result. The accurate reconstruction of the parameters is left for future work. \nA different discussion is required by the LS220 models. In this case, as shown in the right panels of Figs. 8, 9, and 10 and in Table VIII of Appendix C, the lower quality of the fit leads to a slightly different dependence of the fit parameters on the PNS mass. \n- (i) The 'normalization parameter' C (top panels) is C ∼ O (10 -1000) B / s for all of the neutrino species and it increases linearly with the PNS mass.\n- (ii) The 'power-law index' α (second panels from top) tends to decrease as the PNS mass increases. It is negative for all models and neutrino species except for ν e and ¯ ν e in the simulation with M NS = 1 . 36 M ⊙ , where α > 0.\n- (iii) The 'late-time suppression parameters' τ and n (third and fourth panels from top) tend to decrease with higher PNS mass. For the 1.36-LS220 model we get τ ∼ 4 s and n > 1 for ν e and ¯ ν e (see Table V in Appendix B), whereas τ ≲ 0 . 25 s and n ≲ 0 . 5 in all the other cases (Tables VI and VII). \nEven though the linear interpolations of the mass dependence have a lower quality for the LS220 EoS, the best-fit parameters for this EoS possess values in completely different ranges than for the Class A EoSs. Thus, the observation of such values would clearly point to a Class B EoS for the PNS in the discovered SN.", 'V. CORRELATION BETWEEN τ AND α': 'Since the fit parameters of all Class A EoSs have values in similar ranges, it is useful to search for possible relations between them. We disregard the case of LS220 here, because its parameter values lie in completely different regimes. For fixed PNS mass and given EoS, we find that the power-law index α and the suppression time τ for the Class A EoSs exhibit correlations that can be fairly well described by linear functions, \nτ (s) = A + Bα, (4) \nfor all neutrino species except ¯ ν e . This is shown in Fig. 11 for neutrinos (left) and antineutrinos (right) of all flavors and all Class A EoSs. In Appendix D, in Table IX we report the best-fit values with errors obtained from linear regression for A and B for all neutrino species. The pairs of values ( τ, α ) are represented by filled circles for the DD2 EoS, filled squares for SFHo, and filled diamonds for SFHx in Fig. 11. Different colors correspond to the different PNS masses: black for M NS = 1 . 36 M ⊙ , red for M NS = 1 . 44 M ⊙ , blue for M NS = 1 . 62 M ⊙ , green for M NS = 1 . 77 M ⊙ , and orange for M NS = 1 . 93 M ⊙ . \nAs a general trend for fixed NS mass, SFHx leads to the largest values of τ and α for ν and ¯ ν of all flavors, whereas DD2 yields the smallest values of τ and α . The linear increase of τ as a function of α could be understood in simple terms by the fact that a larger value of α causes a faster decline of the luminosity in the first seconds, and therefore, if the initial luminosity (i.e., specifically at 1 s in our context) were the same, less energy is carried away by neutrinos during this early power-law phase of the luminosity. Thus, if the total energy released in neutrinos were fixed, one would expect that a higher value of α leads to a stretching of the subsequent exponential luminosity decrease with a final drop only at later times, implying a larger τ . Although this explanation sounds plausible, it is an oversimplification of the real situation. First, at t = 1s the luminosities are not the same for a given PNS mass, but smallest for DD2 and largest for SFHx (see Fig. 1). Second, also the total energy (individually for all neutrino species as well as summed up) is different, namely smallest for DD2 and largest for SFHx (see Tables I and VII in [43]), which correlates with the final gravitational binding energy of the cold NS (but not strictly with the final NS radius, which is largest for models with the DD2 EoS and smallest with SFHo when M NS ≳ 1 . 2 M ⊙ ). The true reasons for the tight correlation between α and τ values are therefore more subtle than suggested by the simple argument given above. \nMoreover, the quality of the linear relation, Eq. (4), differs between different neutrino species. Figure 11 and the values of the coefficients A and B in Table IX in Appendix D reveal that the linear relation works better for neutrinos than for antineutrinos. The good quality of the linear fit especially for ν e could be useful to disentangle the NS mass and EoS with a measured pair of values ( τ, α ). In this context, it should be mentioned that the linear fits for ¯ ν e are considerably worse than for all other neutrino species (upper right panel in Fig. 11); the 1 σ confidence bands display substantial overlap in the whole range of α values. In contrast, the linear fit function works better for the antineutrinos of the non-electron flavors, with ¯ ν τ (bottom right panel) showing slightly narrower 1 σ confidence bands than ¯ ν µ (middle right panel). The reason for the particularly poor quality of the linear fit for ¯ ν e is the relatively wide separation of the α values for the SFHo and SFHx EoSs, in contrast to ν e , where the corresponding values are very close to each other and the linear fit is very good. The lower quality of linear τ -α relations for the heavy-lepton \nFIG. 11. Correlations of the fitting parameters τ (in units of seconds) and α for ν e (top left panel), ¯ ν e (top right panel), ν µ (central left panel), ¯ ν µ (central right panel), ν τ (bottom left panel), ¯ ν τ (bottom right panel), for all Class A EoSs and all considered PNS masses, obtained from simulation output data between 1 s and t ν i , c for each neutrino species ν i . \n<!-- image --> \nneutrinos also confirms this general tendency that such a fit function is less suitable to describe the correlation of both parameters when the distance between α for SFHo and SFHx grows. This leaves the possibility that the linear function works well for ν e just because of favorable properties of the SFHo and SFHx EoSs. More investigation with larger sets of different nuclear EoS cases is therefore required before one can rely on the validity of linear τ -α relations. \nFIG. 12. Time evolution of the product of time and luminosity, t L ν i , for ν e (top panels), ¯ ν e (middle) and ¯ ν µ (bottom) for a simulations without convection (left panels) and without muons (right panels), with M NS = 1 . 62 M ⊙ and different EoS: DD2 (blue), SFHo (orange), SFHx (green), and LS220 (red). Simulations using SFHx and LS220 without muons are not available. \n<!-- image -->', 'VI. IMPACT OF MUONS AND CONVECTION': "In order to assess the impact of some of our physics inputs of the simulations, we investigate additional PNS cooling calculations for a PNS mass of 1.62 M ⊙ now, where we either omitted convection or muons. Non-convective models were computed for all considered EoSs and are denoted by a suffix '-c' appended to their names (e.g., 1.62-DD2-c), whereas only two models are considered without muons (suffix '-m'), namely 1.62-DD2-m and 1.62-SFHo-m [49]. The bottom two data blocks of Table I provide the final simulation times and the corresponding reduction factors X fin ν i for all neutrino species in these additional models. Bold numbers in Table I for models 1.62-SFHx-c and 1.61LS220-c signal that the simulations were stopped when X fin ν i > 0 . 15, implying that for these models our standard cutoff time t ν i , c is larger than t fin . In both of the simulations, the weakest suppression is obtained for ν µ , with X fin ν µ = 0 . 219 for 1.62-SFHx-c and X fin ν µ = 0 . 185 for 1.61-LS220-c (see the values marked by a star in Table I). Therefore, to test the impact of convection for the SFHx and LS220 EoSs we cut our luminosity data at t ν i , Co , when X Co ν i = t ν i , Co L ν i , Co /t ν i , max L ν i , max = 0 . 22 for simulations with SFHx and X Co ν i = 0 . 19 for simulations with LS220. The values of t ν i , Co for all neutrino species of these simulations are given in Table IV of Appendix A. \nFigure 12 presents neutrino and antineutrino signals at t > 1 s for the mentioned M NS = 1 . 62 M ⊙ simulations with \nFIG. 13. Time evolution of the product of time and ¯ ν e luminosity, t L ¯ ν e , for M NS = 1 . 62 M ⊙ and different EoS, namely DD2 and SFHo (left panels) and SFHx and LS220 (right panels), comparing models with both convection and muons (blue), without convection (orange), and without muons (green). Simulations using SFHx and LS220 without muons are not available. \n<!-- image --> \nmodified input physics and the different EoS previously considered. The left panels display the results for our models without convection in the time interval [1,20] s, whereas the right panels show our cases without muons in the time interval [1,10] s, for ν e (upper panels), ¯ ν e (central) and ¯ ν µ (lower). \nIn the absence of convection the quantity t L ν declines steeply only at t ≳ 10 s for all neutrino species, with different characteristic features depending on the EoS (see the second data block from the bottom of Tables II and III in Appendix A). In particular, we witness the following: \n- (i) DD2 (blue lines) has the shortest cooling time, with t L ν peaking at t ≈ 4 s and being reduced by a factor 0 . 15 of the maximum values at t ≈ 12 s.\n- (ii) SFHo (orange lines) shows a peak of t L ν at t ≈ 4-6 s and a later, steep decline, beginning roughly at t ≈ 15 s.\n- (iii) SFHx (green lines) is similar to SFHo, displaying a peak of t L ν at a slightly earlier time and with a final decrease that is slightly delayed compared to SFHo.\n- (iv) LS220 (red lines) leads to a peak in t L ν at t ≈ 2 s and shows a more shallow decline afterward, forming a plateau-like shape in the time interval 2 s ≲ t ≲ 10 s (i.e., L ν follows approximately L ν ∝ t -1 ) before a steeper decline sets at t ≳ 15 s. \nIn simulations without muons (see the right panels in Fig. 12 and the data block at the bottom of Tables II-III in Appendix A), the product of time and luminosity for all ν species starts to become exponentially suppressed already at t < 10 s, with DD2 (blue lines) leading to a faster cooling than SFHo (orange lines). \nTo explicitly demonstrate the impact of convection and muons by means of the ¯ ν e luminosity, Fig. 13 displays the time evolution of tL ¯ ν e for the 1 . 62 M ⊙ models including both convection and muons (blue lines) compared to the corresponding results without convection (orange) and without muons (green, if available) for the DD2 EoS (upper left panel), SFHo EoS (lower left panel), SFHx EoS (upper right panel) and LS220 EoS (lower right panel). For all the cases \nFIG. 14. Time evolution of the product of time and ¯ ν e luminosity, t L ¯ ν e , for simulation data (blue) and their fits (orange). The different panels show results for simulations without convection or without muons, namely of models 1.62-DD2-c (top left panel), 1.62-SFHo-c (top right), 1.62-SFHx-c (middle left), 1.61-LS220-c (middle right), 1.62-DD2-m (bottom left) and 1.62SFHo-m (bottom right). We consider data up to t ¯ ν e , c for simulations with DD2 and SFHo and up to t ¯ ν e , Co for 1-62-SFHx-c and 1.61-LS220-c (see Appendix A for more details). \n<!-- image --> \n- (i) the 'absence of convection' leads to a considerable stretching of the PNS Kelvin-Helmholtz neutrino cooling time, with the most moderate change for DD2, \n- (ii) the 'omission of muons' has a relatively mild effect on the evolution of the neutrino signals for the displayed 1.62 M ⊙ models, making the suppression in the luminosity only slightly faster (because the NS becomes less compact with a lower binding energy), as visible by the green lines in the left panels. \nTo quantitatively assess the impact of these variations of the input physics of our models, we also fit the neutrino and antineutrino signals of the additional simulations with the expression of Eq. (2) and compare the best-fit parameters with those obtained in our benchmark simulations. As an example, Fig. 14 presents the simulation data (blue) and their best fits (orange) in the time interval of interest for ¯ ν e results from the simulations without convection (DD2 and SFHo in the top panels and SFHx and LS220 in the middle panels) and from our two simulations without muons (bottom panels). For the simulations without muons, where the neutrino signal does not experience major changes, the agreement between data and fits is similarly good as for our benchmark simulations. For the simulations without convection, the fit is still of excellent quality for DD2, although the omission of convection has altered the shape of the curve of tL ¯ ν e (see Fig. 13). In contrast, we obtain visibly larger discrepancies between fits and data for the nonconvective simulations with SFHo and SFHx, for which, in particular, the shape of tL ¯ ν e in the power-law dominated early phase cannot be reproduced as well as for models that include convection. Notably, the fits for ¯ ν e (and similarly for all other neutrino species) slightly overestimate tL ν at t ≈ 1 s and tend to peak only at somewhat later times. Finally, the plateau-like region of tL ν in the simulation results for LS220 implies a best-fit value of α ≈ 1, with the luminosity fit overestimating the data at t ≈ 1 s and following well a t -1 power law before being exponentially suppressed at t ≳ 10 s. Our findings for all other neutrino species are analogous. In Tables X and XI of Appendix E we provide the best-fit parameter values and their 1 σ errors for electron and muon neutrinos and antineutrinos, for our 1 . 62 M ⊙ models with different EoSs and varied input physics. \nAs further discussed in Appendix E, the parameters of non-convective models adopt best-fit values that are well outside the 1 σ confidence bands found for the benchmark simulations. This fact underlines the strong impact of convection on the neutrino signal. As general trends, we find in simulations with Class A EoSs in the absence of convection that \n- (i) C decreases because of the lack of convective enhancement of the luminosities at early times,\n- (ii) τ becomes larger because of the extended PNS neutrino cooling time without convective energy transport,\n- (iii) n becomes smaller to account for the considerable signal stretching at late times, \nwhich implies that the exponential luminosity decline starts at later times and also proceeds more slowly. For ¯ ν e in model 1.62-DD2-c we notice an exception from the described general trends with respect to τ ¯ ν e , which is slightly smaller than the value of the corresponding model with convection (see Table X and Fig. 16 in Appendix E). In this case the mild decrease of τ ¯ ν e seems to be compensated by a reduction of n ¯ ν e by a factor ∼ 3 compared to the nonconvective model, which is by far the largest relative change for any neutrino species in all models with vs. without convection. Interestingly, the change in α depends on the EoS and neutrino species, showing, for instance, a decrease in 1.62-DD2-c compared to 1.62-DD2, an increase in 1.62-SFHo-c compared to 1.62-SFHo, and a decrease or slight increase in 1.62-SFHx-c compared to 1.62-SFHx depending on the type of neutrino. This nonuniform behavior points to differences in the influence of the EoS on PNS convection and the associated effects on the emission of different kinds of neutrinos during the early PNS cooling phase. \nFor simulations with the LS220 EoS, partly because of the poorer quality of the fits for the benchmark models, the omission of convection leads to radical changes in the values of the best-fit parameters. Indeed, LS220 simulations without convection show positive values of α (around unity), much larger values of τ and n compared to the fullphysics cases, and values of C that are well compatible with those of simulations with the other EoSs including and excluding convection, i.e., the C values are close to the luminosity values at 1 s instead of being several 100 B/s for our benchmark models. Finally, the weaker impact of muons on the neutrino signal is highlighted by the small changes in the best-fit parameters obtained for simulations without muons, as further detailed in Appendix E.", 'VII. COUNTING RATE IN NEUTRINO DETECTORS': "In order to exemplify a possible application of our luminosity fits, we discuss in this section the time evolution of the counting rate tR ν ( t ) in a neutrino detector that will monitor the tL ν ( t ) evolution in the case of a future Galactic SN explosion. We will demonstrate that our fitting recipe is also useful for fitting the observed neutrino signal. For this purpose, we consider as a reference case a SN at a distance of D = 10kpc and evaluate the predicted signal in the water Cherenkov detector of Super-Kamiokande (SK) [55, 56], inspired by the analysis in Ref. [57]. \n<!-- image --> \nFIG. 15. Left panel: Simulation results for the average energy ⟨ E ¯ ν e ⟩ (blue), rms energy √ ⟨ E 2 ¯ ν e ⟩ (orange) and spectral shape parameter β ¯ ν e (green) from model 1.62-DD2. Right panel: ¯ ν e -induced counting rate tR ¯ ν e in SK computed from simulation data (blue), the fit of the counting rate (orange), and the fit of tL ¯ ν e (dashed line). \n<!-- image --> \nWe consider the following ν differential flux per unit energy in MeV -1 s -1 cm -2 : \nF 0 ν ( E ν ) = dF ν dE ν = L ν 4 π D 2 ⟨ E ν ⟩ (1 + β ν ) 1+ β ν Γ(1 + β ν ) ⟨ E ν ⟩ ( E ν ⟨ E ν ⟩ ) β ν e -(1+ β ν ) E ν / ⟨ E ν ⟩ , (5) \nwhere the shape parameter β ν is given by \nβ ν = ⟨ E 2 ν ⟩ -2 ⟨ E ν ⟩ 2 ⟨ E ν ⟩ 2 -⟨ E 2 ν ⟩ , (6) \nwith ⟨ E ν ⟩ and ⟨ E 2 ν ⟩ being the average neutrino energy and the average squared neutrino energy, respectively. \nIn SK, the main detection process is inverse β decay, ¯ ν e p → ne + , where the final-state positron shows up by its Cherenkov radiation. Because of the similarity of the electron and non-electron antineutrino luminosities and spectra in our models during PNS Kelvin-Helmholtz cooling at times t ≳ 1 s after bounce, flavor conversions are not a major effect and can be neglected in our simplified analysis. Therefore the expected rate can be written as \nR ¯ ν e = N p ∫ dE e ∫ dE ν F ¯ ν e ( E ν ) σ ' ( E e , E ν ) , (7) \nwhere N p = 1 . 51 × 10 33 is the number of protons for a 22.5 kton Cherenkov detector. Here, we follow Ref. [58] for the limits of integration in dE ν and we integrate the positron energies above the energy threshold E th , SK = 5 MeV. We mention that SK is essentially background free. Estimates for the future Hyper-Kamiokande detector with fiducial mass of 187 kton [59] can be obtained by rescaling the counting rate computed for SK by a factor ∼ 8 . 3, without affecting the temporal evolution of the signal. \nTo obtain a numerical estimate, we use data from model 1.62-DD2 as an example, but the same analysis is valid for all of the models. In the left panel of Fig. 15 we plot the average energy ⟨ E ¯ ν e ⟩ , the root-mean-square (rms) energy √ ⟨ E 2 ¯ ν e ⟩ , and the shape parameter β ¯ ν e as a function of the time, between 1 s and t ¯ ν e , c = 7 . 94 s. These quantities \nexhibit a weak time dependence, with ⟨ E ¯ ν e ⟩ ≈ (12 -13 . 5) MeV, √ ⟨ E 2 ¯ ν e ⟩ ≈ (14 -15) MeV and β ¯ ν e ≈ 2 . 5 -3 at t ≲ 5 s and a decrease at later times. Here we focus on the 1.62-DD2 simulation as a representative case, but qualitatively similar results can be obtained with all the other models. Basic information on the time evolution of the mean neutrino energies for all considered EoSs can be found in Ref. [43] (see Fig. 3 therein, where the time evolution of the average neutrino energies for the 1 . 44 M ⊙ with different nuclear EoSs are shown). \nThe rate in SK can be simply estimated as \nR ¯ ν e ≈ L ¯ ν e 4 π D 2 ⟨ E ¯ ν e ⟩ N SK ⟨ σ ⟩ , (8) \nwhere the average cross section is [57] \n⟨ σ ⟩ = 7 . 37 × 10 -46 cm 2 2 + β ¯ ν e 1 + β ¯ ν e ⟨ E ¯ ν e ⟩ 2 . 15 × ( 76 . 64 β 0 . 021 ¯ ν e -⟨ E ¯ ν e ⟩ β 0 . 24 ¯ ν e ) × [ 1 -exp ( -0 . 25 + 0 . 55 ⟨ E ¯ ν e ⟩ 2 . 2 + β ¯ ν e -1 + 1 . 6 β ¯ ν e 1 + 4 β ¯ ν e ⟨ E ¯ ν e ⟩ )] , (9) \ndependent on ⟨ E ¯ ν e ⟩ and β ¯ ν e [and also on √ ⟨ E 2 ¯ ν e ⟩ via Eq. (6)]. Therefore, we expect that the observed rate R ¯ ν e will follow the time dependence of the neutrino luminosity because of the weak time dependence of the other parameters, and that we can also fit it with our analytical formula in Eq. (2). In the right panel of Fig. 15 we show the quantity tR ¯ ν e obtained from Eq. (7) using data from model 1.62-DD2 (blue line), its fit (orange), and the fitted tL ¯ ν e (dashed line). As shown by the best-fit parameters for t R ¯ ν e given in the plot, the fit leads to C = 8 . 28 × 10 2 s -1 , α = 0 . 54, τ = 6 . 17 s and n = 4 . 61, to be compared with the ones for tL ¯ ν e . The parameters α ¯ ν e , τ ¯ ν e and n ¯ ν e are well reconstructed, while one could get information on the normalization factor C ¯ ν e by inverting Eq. (8). The slight difference between the parameters α , τ and n reconstructed from the rate and the original ones for L ¯ ν e can be explained by the weak time dependence of ⟨ E ν ⟩ , ⟨ E 2 ν ⟩ , and β ν entering in the computation of the rate. Therefore, one can fit the detected event rate with the same functional form used for the luminosity, and from the reconstructed fitting parameters one can get information on the PNS mass and on the EoS, as discussed in the previous Sections. However, the accurate reconstruction of the parameters is beyond the scope of this analysis and will be the subject of future work dealing with their possible reconstruction using current and future neutrino detectors. There, more comprehensive information on the time evolution of ⟨ E ν ⟩ , ⟨ E 2 ν ⟩ and β ν will be provided.", 'VIII. CONCLUSIONS': 'In this paper we have investigated whether the simple analytical function of Eq. (2) can be used as a parametric fit to the SN neutrino luminosities during the Kelvin-Helmholtz cooling phase of the PNS. For this purpose we have considered a set of several 1D simulations for different NS masses and EoSs. Our benchmark models account for PNS convection, which has a strong impact on the cooling evolution and its associated neutrino signal. In particular, we presented fits for the time-dependent neutrino luminosities from numerical PNS cooling simulations and reported systematic dependences of the fitting-parameter values as functions of the NS mass. Future work is desirable where these fits are connected to analytic descriptions and basic PNS and EoS properties in a more formal way, e.g., similar to what was done for PNS cooling models that did not include the effects of PNS convection (see, e.g., [41, 60]). \nOur fit function employs four free parameters, namely a normalization factor C , a power-law exponent α for the time, an exponential cooling timescale τ , and an exponent n of ( t/τ ) in the exponential function. Their characteristic dependence on the PNS mass and on the EoS can be used to draw inferences on these latter properties, if the parameters are deduced from the neutrino signal of a future Galactic SN explosion. For this purpose, we plan to investigate in future work how one can infer the parameter values of the neutrino luminosity from the SN neutrino signal measured in large underground detectors. We have demonstrated that this possibility is facilitated by the fact that the time evolution of the detected event rate depends on the neutrino luminosity L ν , the average neutrino energy ⟨ E ν ⟩ , and the rms energy √ ⟨ E 2 ν ⟩ , but the time dependence of ⟨ E ν ⟩ and √ ⟨ E 2 ν ⟩ is weak. This allowed us to show that Eq. (2) provides a good functional form to also fit the time evolution of the observed neutrino signal. Therefore, for a first estimation of the L ν -fit parameters from the event rate measured by a SN neutrino detector, one can simply apply our analytical expression for the luminosity fit and make use of the assumption that the average neutrino energy and the pinching parameter characterizing the spectral shape are constant in time. \nA number of caveats of our study reported here need to be mentioned. First, the present analysis and our proposed luminosity fit are based on 1D SN and PNS cooling simulations using a fairly limited set of cases for the NS EoS. The general applicability of the fit function of Eq. (2) needs confirmation by testing a much larger variety of EoS models with a wide range of fundamental nuclear physics inputs that are compatible with all experimental, theoretical, and astrophysical constraints on the properties of nuclear matter and observed NSs. In particular, possible correlations of some of the fitting parameters [e.g., the relation in Eq. (4)] require confirmation based on a wider spectrum of nuclear EoS representations. Second, our 1D SN and PNS calculations disregard 3D effects such as long-lasting accretion onto the PNS (continuing also after the onset of the explosion, when in 1D models accretion abruptly stops) and fallback of some initial explosion ejecta during the late PNS evolution [43, 61, 62]. Moreover, the mixing-length treatment of PNS convection in our 1D models will have to be validated by long-time 3D simulations of PNS cooling once such calculations with good spatial resolution become available. In particular, this will also provide a test whether the fit function of Eq. (2) and our best-fit parameter values are compatible with 3D results for PNS cooling. If so, any deviation from the luminosity evolution described by our fit function would signal additional contributions to the neutrino emission added on top of the cooling component from the PNS. Thus, our L ν -fit could help to diagnose, disentangle, and describe such secondary neutrino emission phenomena in the neutrino measurement for a future \nGalactic SN. Finally, it will have to be seen how our fitting function reacts to additional, so far disregarded effects of potential importance in neutrino-cooling SN cores, for example fast flavor conversion of neutrino-antineutrino pairs, which could have a major impact on the neutrino emission properties [63, 64], or extra cooling associated with the emission of light, weakly interacting beyond-standard-model particles (e.g., axions [65]). Again, our neutrino luminosity fits could help to diagnose such effects beyond current standard SN modeling, once simulations including this new physics become available to be analyzed for long-time fitting. \nIn conclusion, we are confident that the simplicity of our fitting procedure will make it a useful tool for the neutrino community to describe the SN neutrino signal expected in a high-statistics detection, to probe a future SN neutrino measurement, and to infer valuable information on the PNS mass, nuclear EoS, and different signal components (see Ref. [66] for a recent approach in this direction). \nThe considered model results are adopted from Ref. [43] and are available in the Garching Core-collapse Supernova Archive [49] upon request.', 'ACKNOWLEDGMENTS': "The work of AM was partially supported by the research grant number 2022E2J4RK 'PANTHEON: Perspectives in Astroparticle and Neutrino THEory with Old and New messengers' under the program PRIN 2022 funded by the Italian Ministero dell'Universit'a e della Ricerca (MUR). This work is (partially) supported by ICSC - Centro Nazionale di Ricerca in High Performance Computing, Big Data and Quantum Computing, funded by European Union-NextGenerationEU. GL acknowledges support by the European Union's Horizon 2020 Europe research and innovation programme under the Marie Skglyph[suppress]lodowska-Curie grant agreement No 860881-HIDDeN. 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Typel, Equations of state for supernovae and compact stars, Reviews of Modern Physics 89 , 015007 (2017), arXiv:1610.03361 [astro-ph.HE].\n- [68] CompOSE - CompStar Online Supernovae Equations of State.", 'Appendix A: Tables for the final-time parameters': 'Here we provide the values of the final simulation time t fin , the time t ν i , max when the quantity tL ν i reaches its maximum, t ν i , max L ν i ( t ν i , max ) ≡ t ν i , max L ν i , max , and the selected cut time t ν i , c , for neutrinos in Table II and for antineutrinos in Table III for all the simulations considered in our study. For each (anti)neutrino species, the selected cut time t ν i , c is the time when the quantity t L ν i is reduced relative to its maximum value by a factor X c ν i = t ν i , c L ν i , c /t ν i , max L ν i , max = 0 . 15, in order to take a common final time for all neutrino species. As shown by the quantities in bold print in Table II and Table III, t ν i , c > t fin for 1.62-SFHx-c and 1.61-LS220-c, i.e. these simulations stop before reaching t ν i , c . Therefore, to check the impact of convection in these two cases we cut our simulation outputs at earlier times, t ν i , Co , when X Co ν i = t ν i , Co L ν i , Co /t ν i , max L ν i , max = 0 . 22 for simulations with SFHx and X Co ν i = 0 . 19 for LS220. We show t fin , t ν i , max and t ν i , Co used to check the impact of convection for electron and muon (anti)neutrinos in Table IV. \nTABLE II. Times t ν i , max when t L ν i adopts its maximum t ν i , max L ν i , max and times t ν i , c when X c ν i = 0 . 15 for all neutrino species ν i . Bold print marks values corresponding to t ν i , c > t fin , i.e., cases when the simulation was stopped before t ν i , c was reached.', 'Appendix B: Tables for best-fit parameters of luminosities for neutrinos and antineutrinos of all flavors': 'Here we report the best-fit values with 1 σ errors of the parameters characterizing the fit for the time evolution of all neutrino and antineutrino luminosities in the time interval from 1 s to t ν i , c (see Appendix A for more details). The fit function is given by Eq. (2), \nL ν i ( t ) = C t -α e -( t/τ ) n , \nwith C , α , τ and n being free parameters. We show values of the fit parameters for ν e and ¯ ν e in Table V, for ν µ and ¯ ν µ in Table VI and for ν τ and ¯ ν τ in Table VII, obtained with the NonlinearModelFit function in Mathematica .', 'Appendix C: Tables for the linear relations between luminosity-fitting parameters and PNS mass': 'In Table VIII we provide the best-fit values and their 1 σ errors for the parameters K 0 and K 1 that describe the linear dependencies of the parameter values in the L ν -fit of Eq. (2) on the PNS mass M NS at fixed EoS, for all neutrino and antineutrino species: \nK = K 0 + K 1 M NS M ⊙ , (C1) \nwhere K = C, α, τ, n . The larger values of the relative uncertainties on K 0 and K 1 in Table VIII and the widths of the confidence bands in Figs. 8, 9, and 10 suggest that the linear fits work better for Class A EoSs than for LS220. \nTABLE III. Times t ¯ ν i , max when t L ¯ ν i adopts its maximum t ¯ ν i , max L ¯ ν i , max and times t ¯ ν i , c when X c ¯ ν i = 0 . 15 for all antineutrino species ¯ ν i . Bold print marks values corresponding to t ¯ ν i , c > t fin , i.e., cases when the simulation was stopped before t ¯ ν i , c was reached. \nTABLE IV. The time t ν i , max when t L ν i is maximum and the time t ν i , Co for ν e , ¯ ν e , ν µ and ¯ ν µ adopted to test the impact of convection in simulations where X fin ν i > 0 . 15. For simulations with the SFHx EoS, t ν i , Co is the time when X Co ν i = 0 . 22, while for LS220 is the time when X Co ν i = 0 . 19. \nAt fixed EoS (in particular for Class A EoSs), the linear fits are excellent for C and τ and slightly worse for α and n , featuring larger relative errors of the best-fit values of the parameters in Eq. (C1) (see, e.g., the values of α 1 and n 1 ).', 'Appendix D: Tables for parameter values of the correlations between τ and α': 'We report in Table IX the best-fit values and the 1 σ errors for the parameters A and B of the linear functions used for describing the correlations between τ (in seconds) and α [see Eq. (4)]: \nτ ( s ) = A + Bα. \nAs shown in Table IX, the fit works better for neutrinos than for antineutrinos. Indeed, for ¯ ν e the error on the parameter A is larger than its best-fit value (thus, A is compatible with zero) for all the NS masses, and the same \nTABLE V. Best-fit parameters with 1 σ errors for L ν i = Ct -α e -( t/τ ) n in the time interval between 1 s and t ν i , c for ν e (left) and ¯ ν e (right) for our benchmark models. \nTABLE VI. Best-fit parameters with 1 σ errors for L ν i = Ct -α e -( t/τ ) n in the time interval between 1 s and t ν i , c for ν µ (left) and ¯ ν µ (right) for our benchmark models. \nis true for ¯ ν µ for the largest NS mass. Additionally, the quality of the fit is similar for ν µ and ν τ , while it is slightly better for ¯ ν τ compared to ¯ ν µ , since the relative error of the best-fit parameters for ¯ ν τ is smaller. This reveals a small \nTABLE VII. Best-fit parameters with 1 σ errors for L ν i = Ct -α e -( t/τ ) n in the time interval between 1 s and t ν i , c for ν τ (left) and ¯ ν τ (right) for our benchmark models. \ndifference between the non-electron flavors. As a common trend, the parameter B increases with the NS mass for all of the neutrino species.', 'Appendix E: Further details on the impact of convection and muons': "Here we give further details on the impact of convection and muons. Since models 1.62-SFHx-c and 1.61-LS220-c stop before the product tL ν i for all neutrino species is reduced to a value of 0.15 of the maximum, to make the comparison on a solid ground in this appendix we consider data up to t ν i , c for simulations with DD2 and SFHo and up to t ν i , Co for simulations with SFHx and LS220 (see Appendix A for more details). In this way, we take into account results for neutrinos and antineutrinos of all flavors up to the time when they reach the same reduction factor. Moreover, since non-electron flavors show, in general, a similar behavior and in simulations without muons, τ and µ neutrinos behave exactly in the same way, for all the simulations considered in this Section we report values related only to ν e , ¯ ν e , ν µ and ¯ ν µ . \nWe list the best-fit parameter values and their 1 σ errors for the 1 . 62 M ⊙ models with different EoS and different ingredients of the input physics for the luminosities of neutrinos in Table X and of antineutrinos in Table XI. A better visualization of the change in the best-fit parameters can be obtained by plotting them as a function of the PNS mass for the different EoSs. As an example, we show in Fig. 16 the best-fit parameters for ¯ ν e as a function of M NS , for DD2 (left panels) and SFHo (right panels). Here, black dots are the values of the best-fit parameters obtained from simulations including both convection and muons, red dots are related to simulations without convection and blue dots to simulations without muons. \nTables X and XI, as well as Fig. 16, show that, as a general trend, in simulations without convection τ becomes larger and n smaller, i.e., the luminosity suppression starts at later times and it is slower. The only exception is found for τ ¯ ν e in model 1.62-DD2-c, which is smaller than τ ¯ ν e in 1.62-DD2 (see the red dot in the third panel from top on the left of Fig. 16). This behavior is confirmed by inspecting the upper left panel in Fig. 13, where the orange line (without convection) is peaked at earlier times compared to the blue (benchmark case) and the green (without muons) lines. As shown by the upper left panel in Fig. 14, even if τ ¯ ν e is smaller than in the benchmark case, the fit well reproduces the data, because the interplay between τ ¯ ν e and a much smaller n ¯ ν e (compared to the complete case) well describes the slightly longer cooling time. Therefore, for ¯ ν e we observe a mathematical peculiarity connected to the fit function, reacting to the fact that tL ¯ ν e is peaked at earlier times in the absence of convection (see the upper left \nTABLE VIII. Coefficients K 0 and K 1 with errors for the linear dependence on the PNS mass K = K 0 + K 1 M NS /M ⊙ at fixed EoS, for electron (upper data block), muon (central data block) and tau (lower data block) neutrinos and antineutrinos, obtained on grounds of simulation data in the time interval between 1 s and t ν i , c . \nTABLE IX. Coefficients A and B with errors for the relation τ [s] = A + Bα , for neutrinos (left) and antineutrinos (right), for all flavors and NS masses, obtained on grounds of simulation data in the time interval between 1 s and t ν i , c . \npanel in Fig. 13), i.e. smaller τ ¯ ν e , and it is characterized by a milder exponential suppression, i.e. smaller n , leading to a longer cooling time. On the other hand, in the absence of convection α becomes smaller in the case of DD2 (see \nTABLE X. Best-fit parameter values for L ν i = Ct -α e -( t/τ ) n for ν e (left) and ¯ ν e (right) for M NS = 1 . 62 M ⊙ and different EoS, considering both convection and muons (upper lines), without convection (labeled with the suffix '-c') and without muons (labeled with the suffix '-m'). We consider data up to t ν i , c for simulations with DD2 and SFHo and up to t ν i , Co for simulations with SFHx and LS220 (see Appendix A for more details). Simulations with SFHx and LS220 without muons are not available. \nthe red dot in the second panel from top on the left of Fig. 16) and larger in the case of SFHo (see the red dot in the second panel from top on the right of Fig. 16), describing a change in the power-law behavior in the early cooling phase. In all the cases, the best-fit parameters in the absence of convection lie well outside the 1 σ confidence band found for benchmark simulations, stressing the strong impact of convection on the neutrino signal. \nThe weaker impact of muons on the neutrino signal is highlighted by the small changes in the best-fit parameters obtained from simulations without muons. In this case, for both DD2 and SFHo, in simulations without muons τ becomes slightly lower and n is approximately equal or slightly larger for neutrinos and antineutrinos of all flavors (compare the first with the third line in the first two data blocks of Table X and Table XI, as well as the black and blue dots in Fig. 16). This means that in simulations without muons the suppression in the luminosity starts slightly before and it is a bit faster than in the full-physics cases. In contrast, α tends to increase for DD2 and to decrease for SFHo, even if the change in all cases is much smaller compared to the changes induced by the absence of convection. \nSince simulations without muons are not available for SFHx and LS220 and for them X fin ν i > 0 . 15 in the absence of convection, we do not show the best-fit parameter values as a function of the NS mass in these two cases, but we only summarize the values of the best-fit parameters and their errors in the last two data blocks of Table X (for the electron flavor) and Table XI (for the muon flavor), obtained by considering simulation data up to t ν i , Co . Even if the nominal values of the best-fit parameters in the benchmark simulations slightly change when switching from t ν i , c to t ν i , Co , the impact of convection on simulations with SFHx and SFHo is similar, with an increase in τ , and a decrease in C and in n in absence of convection. As expected, convection strongly affects also simulations with LS220. In this case, given the worse quality of the fit, neglecting convection leads to completely different values of the best-fit parameters compared to the benchmark case. Indeed, simulations without convection show positive values of α , much larger values of τ and n , and drastically reduced values of C compared to the complete-physics case, with all of these parameter values more closely related to the true magnitude and exponential decay time of the neutrino luminosities.", 'Appendix F: Equation-of-state parameters': "We report in Table XII the parameters for the symmetry energies for the EoS cases used in our work. With the customary definitions of x = ( n -n 0 ) / (3 n 0 ) and the asymmetry parameter δ = 1 -2 Y e , the energy per nucleon can be expressed as \nE ( n ) = -E 0 + 1 2 Kx 2 + δ 2 ( J + Lx + 1 2 K s x 2 ) + ... , (F1) \nwith E 0 being the binding energy of symmetric matter at saturation density, K the incompressibility, J the symmetry energy, L the slope of the symmetry energy, and K s the curvature of the symmetry energy. \nTABLE XI. Best-fit parameter values for L ν i = Ct -α e -( t/τ ) n for ν µ (left) and ¯ ν µ (right) for M NS = 1 . 62 M ⊙ and different EoS, considering both convection and muons (upper lines), without convection (labeled with the suffix '-c') and without muons (labeled with the suffix '-m'). We consider data up to t ν i , c for simulations with DD2 and SFHo and up to t ν i , Co for simulations with SFHx and LS220 (see Appendix A for more details). Simulations with SFHx and LS220 without muons are not available. \nTABLE XII. Parameter values for the energy per nucleon around the nuclear saturation density n 0 according to Eq. (F1) for the EoSs used in the model simulations in our work. The values in this table are taken from Table IV in [67] and from entries for the respective EoS in the CompOSE database [68]. \nFIG. 16. Best-fit parameters C , α , τ and n as functions of the PNS mass for ¯ ν e and DD2 (left) respectively SFHo (right), with data up to t ¯ ν e , c . The shaded areas represent the 1 σ confidence bands. The black dots are obtained with simulations considering both convection and muons, red dots neglect convection, whereas blue dots correspond to simulations without muons. \n<!-- image -->"}

2024MNRAS.534.1060H

Red giant stars host solarlike oscillations which have mixed character being sensitive to conditions both in the outer convection zone and deep within the interior. The properties of these modes are sensitive to both core rotation and magnetic fields. While asteroseismic studies of the former have been done on a large scale studies of the latter are currently limited to tens of stars. We aim to produce the first large catalogue of both magnetic and rotational perturbations. We jointly constrain these parameters by devising an automated method for fitting the power spectra directly. We successfully apply the method to 302 lowluminosity red giants. We find a clear bimodality in core rotation rate. The primary peak is at inlineformulatexmath idTM0001 notationLaTeXdelta nu mathrmrottexmathinlineformula 0.32 inlineformulatexmath idTM0002 notationLaTeXmutexmathinlineformulaHz and the secondary at inlineformulatexmath idTM0003 notationLaTeXdelta nu mathrmrottexmathinlineformula 0.47 inlineformulatexmath idTM0004 notationLaTeXmutexmathinlineformulaHz. Combining our results with literature values we find that the percentage of stars rotating much more rapidly than the population average increases with evolutionary state. We measure magnetic splittings of 2inlineformulatexmath idTM0005 notationLaTeXsigmatexmathinlineformula significance in 23 stars. While the most extreme magnetic splitting values appear in stars with masses inlineformulatexmath idTM0006 notationLaTeXgt texmathinlineformula1.1 Minlineformulatexmath idTM0007 notationLaTeXodot texmathinlineformula implying they formerly hosted a convective core a small but statistically significant magnetic splitting is measured at lower masses. Asymmetry between the frequencies of a rotationally split multiplet has previously been used to diagnose the presence of a magnetic perturbation. We find that of the stars with a significant detection of magnetic perturbation 43 per cent do not show strong asymmetry. We find no strong evidence of correlation between the rotation and magnetic parameters.

2024-10-01T00:00:00Z

['2024MNRAS.tmp.2015H', '10.1093/mnras/stae2053', '10.48550/arXiv.2409.01157', 'arXiv:2409.01157', '2024MNRAS.534.1060H', '2024arXiv240901157H']

['Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Instrumentation and Methods for Astrophysics']

Asteroseismic signatures of core magnetism and rotation in hundreds of lowluminosity red giants

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2409.01157.pdf

{'Asteroseismic Signatures of Core Magnetism and Rotation in Hundreds of Low-Luminosity Red Giants.': "Emily J. Hatt, 1 ★ , J. M. Joel Ong 2 † , Martin B. Nielsen 1 , William J. Chaplin 1 , Guy R. Davies 1 , Sébastien Deheuvels 3 , Jérôme Ballot 3 , Gang Li 3 , 4 , Lisa Bugnet 5 \n- 1 Royal Astronomical Society, Burlington House, Piccadilly, London W1J 0BQ, UK\n- 2 Institute for Astronomy, University of Hawai'i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA\n- 3 IRAP, Université de Toulouse, CNRS, CNES, UPS, 14 avenue Edouard Belin, 31400 Toulouse, France\n- 4 Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium\n- 5 Institute of Science and Technology Austria (IST Austria), Am Campus 1, Klosterneuburg, Austria \nAccepted XXX. Received YYY; in original form ZZZ", 'ABSTRACT': 'Red Giant stars host solar-like oscillations which have mixed character, being sensitive to conditions both in the outer convection zone and deep within the interior. The properties of these modes are sensitive to both core rotation and magnetic fields. While asteroseismic studies of the former have been done on a large scale, studies of the latter are currently limited to tens of stars. We aim to produce the first large catalogue of both magnetic and rotational perturbations. We jointly constrain these parameters by devising an automated method for fitting the power spectra directly. We successfully apply the method to 302 low-luminosity red giants. We find a clear bimodality in core rotation rate. The primary peak is at 𝛿𝜈 rot = 0.32 𝜇 Hz, and the secondary at 𝛿𝜈 rot = 0.47 𝜇 Hz. Combining our results with literature values, we find that the percentage of stars rotating much more rapidly than the population average increases with evolutionary state. We measure magnetic splittings of 2 𝜎 significance in 23 stars. While the most extreme magnetic splitting values appear in stars with masses > 1.1M ⊙ , implying they formerly hosted a convective core, a small but statistically significant magnetic splitting is measured at lower masses. Asymmetry between the frequencies of a rotationally split multiplet has previously been used to diagnose the presence of a magnetic perturbation. We find that of the stars with a significant detection of magnetic perturbation, 43% do not show strong asymmetry. We find no strong evidence of correlation between the rotation and magnetic parameters. \nKey words: asteroseismology', '1 INTRODUCTION': 'Magnetic fields play a critical role in stellar evolution. The prevailing theory regarding those which are observed in the Sun and other mainsequence solar-type stars is that they are generated by a dynamo process. This mechanism is, crucially, dependent on the interplay between turbulent convection and differential rotation (Noyes et al. 1984). Convection is supported at various phases in the lifecycle of low to intermediate mass stars, providing several avenues for the formation of a magnetic field. Magnetic fields have been invoked as a possible solution to many open problems in stellar evolution (see Brun & Browning 2017 for a review). Notably, a magnetic field could transport angular momentum from the core to the outer envelope, reducing the degree of differential rotation occurring in evolved stars (Cantiello et al. 2014; Spada et al. 2016; Eggenberger et al. 2019; Fuller et al. 2019; Gouhier et al. 2022; Eggenberger et al. 2022; Moyano et al. 2023). This could resolve the observed discrepancy between predicted rotation rates in the cores of evolved stars and \n- ★ E-mail: [email protected]\n- † Hubble Fellow \nthose observed, the latter being (at best) two orders of magnitude too small (Eggenberger et al. 2012; Marques et al. 2013; Ceillier et al. 2013). \nGiven magnetic fields in the outer layers of stars can be detected in light curves due to the manifestation of this field as star-spots, most detected stellar magnetic fields in convection zones are sun-like in nature. However, a magnetic field capable of producing observed red giant branch (RGB) rotation rates would need to operate near the stellar core (Maeder & Meynet 2014). Core convection is expected in stars with masses above ≈ 1.1M ⊙ during the main sequence (Kippenhahn & Weigert 1990). It is possible for these fields to remain stable as the star evolves off of the main sequence, where the interior becomes radiative (Emeriau-Viard & Brun 2017; Villebrun et al. 2019; Becerra et al. 2022). Even without the presence of a convective core on the main sequence, it is possible to create a magnetic field in stably-stratified zones via a Tayler-Spruit dynamo or related processes (Spruit 2002; Fuller et al. 2019; Eggenberger et al. 2022; Petitdemange et al. 2023). \nAsteroseismology, the study of stellar pulsations, offers the only probe sensitive to near core regions. Solar-like oscillations come in two types, propagating in two largely distinct regions. In the sur- \nface convection zones, turbulent motion drives acoustic oscillations knownaspressureorp-modes.Closer to the core, strong density stratification supports bouyancy oscillations, gravity or g-modes. When the frequencies of p and g-modes approach each other, the two types of modes can couple forming what is known as a mixed-mode (Unno et al. 1989). Sharing properties of both the pure p and g modes, mixed-modes are both sensitive to conditions near the stellar core and the surface. Mode amplitudes reach a maximum about a characteristic oscillation frequency ( 𝜈 max) that scales with the acoustic cut off. On the main sequence, the maximum g-mode frequency is significantly lower than 𝜈 max, such that mixed modes are not excited to observable amplitudes. As a star evolves off of the main sequence onto the RGB, 𝜈 max decreases in response to the expansion of the outer layers. Concurrently the core contracts, increasing the density of g-modes. As 𝜈 max approaches the maximum frequency of the gmodes, mixed modes become increasingly observable. The Kepler telescope (Borucki et al. 2010) observed a large number of evolved stars with high-precision photometry, such that we are able to measure mixed modes in thousands of stars (Mosser et al. 2014; Vrard et al. 2016; Kuszlewicz et al. 2023). \nThe presence of a magnetic field within the region where modes propagate has been shown to perturb mode frequencies (Gough & Thompson 1990; Goode & Thompson 1992; Takata & Shibahashi 1994; Hasan et al. 2005; Mathis et al. 2021; Bugnet et al. 2021; Li et al. 2022a; Mathis & Bugnet 2023). This owes both directly to the introduction of the Lorentz force into the equations of stellar oscillation and indirectly by impacting the properties of mode cavities. Unlike rotation, magnetic fields are not, in general, azimuthally symmetric. As such, the degree to which a mode is perturbed is dependent both on field strength and its geometry and topology (Gomes & Lopes 2020; Mathis et al. 2021; Bugnet et al. 2021; Loi 2021). Alongside identifying the presence of a magnetic field, measurements of perturbations to mode frequencies can put key constraints on field strength and structure. The theoretical tools required to exploit the perturbed spectra of evolved stars in such a way have only just been established. Furthermore, the size of the parameter space involved with fitting even unperturbed mixed modes makes the problem computationally expensive and contingent on well-informed priors (Kuszlewicz et al. 2023). Even more free parameters are required when considering perturbations, amplifying the issue. As such, only the cases with the strongest magnetic signatures have thus far been analysed. Accordingly current catalogues of core magnetic fields identified through perturbations to mode frequencies are very limited, numbering 24 stars at time of writing (Li et al. 2022a, 2023; Deheuvels et al. 2023). \nIn addition to perturbing frequencies, it has been shown that once the field strength exceeds a critical value (typically in the range of 10 5 - 10 7 G for a low-luminosity red giant) it can act to suppress the amplitudes of mixed dipole modes (Fuller et al. 2015). A number of red giants observed by Kepler exhibit mode amplitudes that suggest they have been altered by such a field (García et al. 2014; Fuller et al. 2015). The identification of these depressed dipole modes implies a field strength in the core which must exceed the critical value. In this work we investigate perturbations to the mode frequencies, such that we do not select stars in which the dipole modes are substantially depressed. Accordingly we are sensitive to intermediate field strengths, likely below the critical value defined in Fuller et al. (2015). \nWe investigate the perturbations caused by core rotation and magnetic fields using a sample of 302 low luminosity RGB stars observed by Kepler . In section 2 we describe how the sample is selected from the > 16000 RGB stars in Yu et al. (2018) (hereafter Y18). We then go on to fit a perturbed asymptotic expression to the power spectra. To enable a large scale fitting without the problem becoming compu- \nFigure 1. 𝜈 max and 𝑇 eff values for the selected sample (pink) compared to those from the catalogue by Yu et al. (2018) (green). For reference we show the MIST (Choi et al. 2016; Dotter 2016) evolutionary tracks for 1 M ⊙ , 1 . 5 M ⊙ , 2 M ⊙ , and 2 . 5 M ⊙ stars at [ Fe / H ] = -0 . 25 which approximately corresponds to the median metallicity of the selected sample, as reported in table 2 of Yu et al. (2018). \n<!-- image --> \ntationally intractable, we construct priors on the perturbed quantities. This is done via a novel method of exploiting stretched period échelle diagrams calculated using the spectrum directly, a tool so far only used on previously measured mode frequencies. In section 3 we detail the resulting measurements, before discussing correlations with fundamental stellar properties in section 4.', '2.1 Target Selection': "We used Kepler long-cadence light curves, calculating the power spectral density via a Lomb-Scargle periodogram (Lomb 1976; Scargle 1982) using the lightkurve package (Lightkurve Collaboration et al. 2018). \nOf the observable mixed modes present in a given spectrum, those which are gravity-dominated are the most sensitive to core conditions. As the widths of these modes are smaller than their pressuredominated counterparts (Mosser et al. 2011, 2015, 2018; Vrard et al. 2016), they set an upper limit on the frequency resolution we required. For such modes, the linewidths are on the order of 0.01 𝜇 Hz (Yu et al. 2018; Li et al. 2020) necessitating time series that exceed three years in length. Therefore we restrict ourselves to stars that were observed by Kepler for a full 4-years. \nFor a given radial order and ℓ , rotation breaks the degeneracy between modes of differing azimuthal order, 𝑚 . To leading order in perturbation theory (i.e. for slow rotation), the degree to which a modeis perturbed by rotation is proportional to 𝑚 , such that modes of order 𝑚 = 0 are not affected, remaining at the unperturbed frequency. For modes with non-zero 𝑚 , the magnitude of the perturbation is shared between two modes having azimuthal orders 𝑚 and 𝑚 ' if | 𝑚 | = | 𝑚 ' | . The sign of 𝑚 then determines whether the mode increases or decreases in frequency. \nMagnetic fields will perturb all components of a multiplet, with (generally) a different shift for all | 𝑚 | components. The resulting asymmetry in the spectrum of a magnetically perturbed multiplet is distinct, and as such is commonly used to establish a detection. However, the degree to which the shift differs among components is dependent on the topology of the field (see section 2.3) and certain configurations result in zero asymmetry regardless of field strength (Loi 2021; Mathis & Bugnet 2023; Li et al. 2022a). Therefore, we makenoselectionbasedonasymmetrybutratherfocusontheaverage shift of the multiplet. In the following we used modes of angular degree ℓ = 1 (dipole modes, see section 2.3), such that multiplets consist of 3 components ( 𝑚 = 0 , ± 1). Given all components of the multiplet contain information about the magnetism, we restricted ourselves to targets where-in we could visually identify all three peaks for a given ℓ = 1 mode. This occurs at intermediate inclination, approximately in the range 30 ° < 𝑖 < 60 ° . \nFinally, as a star evolves along the RGB, the mixed mode density ( N = Δ 𝜈 / ΔΠ 1 𝜈 2 max ) increases. Once the rotational splitting is of the order of the separation between adjacent mixed modes, identification of multiplet components becomes difficult. Therefore we restrict ourselves to targets with 𝜈 max > 100 𝜇 Hz (implying mixed mode densities in the region of N < 10). The upper limit on 𝜈 max is set by the Nyquist limit in the data (277 𝜇 Hz). Applying these constraints to the > 16000 targets identified in Y18, we construct a list of 334 stars (see Fig. 1).", '2.2 Data pre-processing': 'Rather than measuring mode frequencies prior to fitting for the perturbed quantities, we fit the power spectra directly. Given p-modes of angular degree ℓ = 0 do not couple to g-modes, they will not provide information about core magnetism or rotation. Additionally, mode coupling in modes of angular degree ℓ = 2 is orders of magnitude smaller than that in ℓ = 1 modes. As such, we removed the additional power from the ℓ = 0 and 2 modes (see sections 2.2.1, 2.2.2) prior to fitting a perturbed expression to the spectrum of the dipole modes (section 2.3).', '2.2.1 Computing the S/N spectrum': 'The first step in the process is to estimate the background noise level. Here we define the background noise as any power that is not directly attributed to the oscillation modes. For stars on the red giant branch (RGB) the background typically consists of a frequency independent term due to photon noise, two frequency dependent terms due to granulation on the stellar surface, and finally a third frequency dependent term which accounts for any residual, longterm, instrumental variability (see, e.g, Kallinger et al. 2014). \nHere we model the photon noise as a frequency independent, random, variable. Three frequency-dependent terms as Harvey-like profiles (Harvey 1985), following Kallinger et al. (2014), are introduced to model the signature of granulation. All the terms in the \nbackground noise model vary slowly with frequency, and so we bin the spectrum in log-frequency, after which the model parameters are sampled. \nWe evaluate a set of 100 draws from the model posterior distribution to compute a mean background level on the unbinned frequency grid. We divide the power spectral density (PSD) by the mean background model to obtain a residual S/N spectrum which now only contains the oscillation envelope.', '2.2.2 Establishing a mean ℓ = 2 , 0 model': 'The next step is to remove the contribution of the ℓ = 0 and ℓ = 2 modes to the S/N spectrum. This is done by computing a mean ℓ = 2 , 0 model which is then used to obtain a residual S/N spectrum which notionally only contains the ℓ = 1 modes. The ℓ = 2 , 0 model that we use is consistent with that of Nielsen et al. (2021) which, to summarize, consists of a set of mode frequencies determined by the asymptotic p-mode relation for the ℓ = 0 and ℓ = 2 modes. The spectrum is then approximated as a sum of Lorentzian profiles at these frequencies, modulated by a Gaussian envelope in power setting the mode heights. For simplicity, we set a single width for all modes. To construct the mean ℓ = 2 , 0 model we draw 50 samples from the model posterior distribution, and average the resulting model spectra. The sampling is performed using a principal component based dimensionality reduction method presented in Nielsen et al. (2023). \nDividing the S/N spectrum by the mean ℓ = 2 , 0 model leaves us with a residual spectrum which consists primarily of power due to the ℓ = 1 modes, any potential ℓ = 3 modes, and to a lesser extent any residual power remaining due to errors in the ℓ = 2 , 0 model. While this simplifies the sampling of the ℓ = 1 model posterior distribution, the inference on the core rotation and magnetism is now conditional on the background and ℓ = 2 , 0 models. This means that in the following we neglect any errors due to uncertainty in these models. However, for high S/N red giant stars where the background and ℓ = 2 , 0 model parameters can be precisely estimated, this is not expected to be a significant contribution to uncertainty on the rotation and magnetic field terms. We also cannot capture correlations between perturbed quantities and the background.', '2.3 Estimating the ℓ = 1 model parameters': "We then used the remaining S/N spectrum to estimate the posterior distribution of the ℓ = 1 model parameters. At high radial order,pure 𝑔 -modes approximately satisfy an asymptotic eigenvalue equation, \n1 \n/ 𝜈 𝑔 ≈ ΔΠ 1 ( 𝑛 𝑔 + 𝜖 𝑔 ) , (1) \nwhere ΔΠ 1 is the period spacing for ℓ = 1 modes, 𝑛 𝑔 is the g-mode radial order and 𝜖 𝑔 is the phase offset. To calculate the frequencies of mixed-modes, these pure g-modes must be coupled to the pure p-modes. For this purpose we used the matrix construction of Ong et al. (2021a) (see also Deheuvels & Michel 2010). That is, mixed modes emerge as the eigenvalues ( 𝜔 ) of the following, \nGLYPH<18> GLYPH<20> -𝛀 2 𝑝 L L † -𝛀 2 𝑔 GLYPH<21> + 𝝎 2 GLYPH<20> I D D † I GLYPH<21> GLYPH<19> v = 0 , (2) \nwhere 𝛀 𝑝 = 2 𝜋 𝝂 𝑝 and 𝛀 𝑔 = 2 𝜋 𝝂 𝑔 are diagonal matrices containing the angular frequencies of the pure p- and g-mode frequencies ( 𝝂 𝑝 and 𝝂 𝑔 ), 𝒗 is the eigenvector specifying mixed-modes as a combination of pure p- and g-modes, L and D are coupling matrices (with elements 𝐿 𝑖 𝑗 and 𝐷 𝑖 𝑗 respectively). In general, the \nFigure 2. Stretched échelle power diagrams for two red giants showing characteristic features of rotation and magnetism. (a) : KIC 10006097, showing symmetric rotational splitting. (b) : KIC 8684542, from the sample of Li et al. (2022a), showing pronounced asymmetric rotational splitting indicative of core magnetism. The power spectrum indicates excess power along the gmode ridges that do not correspond to identified and fitted modes in, e.g., their Fig. 3. \n<!-- image --> \nelements of these coupling matrices vary with their associated pand g-mode frequencies. We parameterised this frequency dependence as 𝐿 𝑖 𝑗 ∼ 𝜔 2 𝑔, 𝑗 · 𝑝 𝐿 , where 𝑝 𝐿 is a scalar, and similarly 𝐷 𝑖 𝑗 ∼ 𝜔 𝑔, 𝑗 / 𝜔 𝑝 𝑖 · 𝑝 𝐷 , where 𝑝 𝐷 is a scalar. A full motivation for this parameterisation will be provided in Nielsen et al. (in prep.). Therefore, for a given set of pure p- and g-mode frequencies, mixed mode frequencies can be described using the introduction of two parameters, 𝑝 𝐿 and 𝑝 𝐷 . For each star we sample these as random, independent variables. \nThe parameters ΔΠ 1 , 𝜖 𝑔 , 𝑝 𝐿 and 𝑝 𝐷 were used to provide unperturbed 𝑚 = 0 mode frequencies, to which we introduced the perturbations due to core rotation and a core magnetic field. In the presence of slow rotation, as is the case in these red giants, modes are perturbed linearly. Here, we approximate the rotation as happening in the core, ignoring the much slower envelope rotation (Goupil et al. 2013). Under such assumptions, pure g-modes are perturbed according to, \n𝜈 ' 𝑚,𝑔 = 𝜈 𝑔 + 𝑚𝛿𝜈 rot ,𝑔 , (3) \nwhere 𝜈 𝑔 is the unperturbed frequency and 𝛿𝜈 rot ,𝑔 is the rotational splitting of the pure g-modes. In the following we will drop the subscript g for simplicity. \nA magnetic field in the core will also perturb the pure g-mode frequencies. In the following we used models consistent with those \nestablished in Li et al. (2022a), which are subject to the constraint that effects of non-axisymmetry of the magnetic field are negligible. Unlike the case of rotation, a magnetic field will also impact the 𝑚 =0 modes, such that we have, \n𝛿𝜈 mag ,𝑚 = 0 = ( 1 -𝑎 ) 𝛿𝜈 mag GLYPH<18> 𝜈 max 𝜈 GLYPH<19> 3 , (4) \n𝛿𝜈 mag ,𝑚 = ± 1 = ( 1 + 𝑎 / 2 ) 𝛿𝜈 mag GLYPH<18> 𝜈 max 𝜈 GLYPH<19> 3 , (5) \nwhere 𝑎 is a parameter dependent on the field topology, dependent on an average of the radial field strength weighted by a second order Legendre polynomial. As such, the value ranges between -0 . 5 < 𝑎 < 1, with the maximum negative value corresponding to a field concentrated about the equator and maximum positive values corresponding to a field concentrated at the poles. A full inversion for the structure of the field is not possible, given the degeneracy between fields of different spatial scales (see Loi 2021; Mathis & Bugnet 2023). The parameter 𝛿𝜈 mag is dependent on an average of the radial field strength (see section 4.2.1). It should be noted that the assumption of non-axisymmetry are met only when the ratio of the magnetic to the rotational splitting is less than one. \nOur total model for the perturbed pure g-mode frequencies is thus, \n𝜈 ' 𝑚 = 𝜈 + 𝑚𝛿𝜈 rot + 𝛿𝜈 mag ,𝑚 . (6) \nGiven values for the parameters ΔΠ 1 , 𝜖 𝑔 , 𝑝 𝐿 and 𝑝 𝐷 , we may then calculate the resulting mixed-mode frequencies by performing modecoupling calculations via equation 2 for each 𝑚 separately, linearly perturbing the pure g-mode frequencies as above to describe core rotation and a core magnetic field. For this purpose, we neglect the effects of rotation and magnetism on the pure p-modes, which are largely insensitive to the core. \nThe final additional parameter required to describe the mode frequencies is the ℓ = 1 small frequency spacing ( 𝛿𝜈 01 ), which describes the deviation between the pure p-mode ℓ = 1 frequencies and the midpoint of the adjacent ℓ = 0 mode frequencies. The complete set of parameters required to describe the frequencies numbers 8 ( ΔΠ 1 , 𝑝 𝐿 , 𝑝 𝐷 , 𝜖 g, 𝛿𝜈 01 , 𝛿𝜈 mag, 𝑎 , 𝛿𝜈 rot). \nSimilar to the analysis in section 2.2.2, we used these frequencies to fit a forward model to the power spectrum, described by a sum of Lorentzian profiles. Linewidths are fixed at the value of the ℓ = 0 linewidth modified by the 𝜁 function, Γ ℓ = 1 = Γ ℓ = 0 ( 1 -𝜁 ) . This accounts for the reduction in linewidth for g-dominated mixed modes, where the mode inertia is large. We assume mode heights can be approximated by the product of the envelope height from the ℓ = 0 , 2 model with the relative mode visibility of ℓ = 1 modes (V ℓ = 1 ), which for Kepler is V ℓ = 1 = 1.505 (Mosser et al. 2012a; Lund et al. 2017). Additionally, the relative power between modes in a multiplet depends on stellar inclination, 𝑖 . This is frequently accounted for by multiplying mode heights by the factor E ℓ, | 𝑚 | ( 𝑖 ) . For dipole modes with 𝑚 = 0 this is given by E 1 , 0 ( 𝑖 ) = cos 2 ( 𝑖 ) , and for 𝑚 = ± 1 by E 1 , | 1 | ( 𝑖 ) = 1 / 2 sin 2 ( 𝑖 ) . We included this in our model, allowing 𝑖 to vary as a free parameter, setting a uniform prior in the range 0 ° to 90 ° . \nTo estimate the posterior distribution on the parameters of the ℓ = 1 model we use the Dynesty nested sampling package (Speagle 2020). This relies on establishing a set of prior distributions for each of the model parameters. \nTo construct priors on these parameters, we exploited the so-called 'stretched' échelle diagram construction (Vrard et al. 2016). In the asymptotic approximation, mixed modes are the roots of the characteristic equation \ntan 𝜃 𝑝 ( 𝜈 ) tan 𝜃 𝑔 ( 𝜈 ) -𝑞 ( 𝜈 ) = 0 , (7) \nwhere 𝑞 is a coupling strength (in most cases approximated as a constant), and 𝜃 𝑝 , 𝜃 𝑔 are smooth functions of frequency constructed such that at pure p- and g-mode frequencies 𝜈 𝑝 and 𝜈 𝑔 , \n𝜃 𝑝 ( 𝜈 𝑝 ) = 𝜋𝑛 𝑝 ; and 𝜃 𝑔 ( 𝜈 𝑔 ) = 𝜋𝑛 𝑔 . (8) \nGiven observational access to only mixed modes, but also inferences of notional p-mode frequencies, g-mode period spacings, and coupling strengths consistent with Eqs. (1), (7) and (8), one may invert Eq (7) to produce 'stretched' frequencies 𝜈 𝑔 associated with each mixed mode 𝜈 . While several numerical formulations for doing this exist (e.g. Mosser et al. 2012b, 2015, 2017, 2018; Gehan et al. 2018, 2021), Ong & Gehan (2023) prescribe an analytic expression, \n1 𝜈 𝑔 ≡ 𝜏 ( 𝜈 ) ∼ 1 𝜈 + ΔΠ 𝑙 𝜋 arctan GLYPH<18> 𝑞 tan 𝜃 𝑝 GLYPH<19> , (9) \nassuming that the pure p-modes are affected by neither rotation nor magnetism. Traditionally, these stretching functions are applied to mixed-mode frequencies fitted in advance from the power spectrum. In this work, we instead apply the stretching directly to the frequency coordinate of the power spectrum. Having done so, the morphology of the resulting stretched period-échelle power diagrams correspond directly to the linear expressions Eqs (3) to (5). \nTo exploit this diagram to construct priors we note two features: \n(i) If the correct values of ΔΠ 1 and 𝑞 are used to construct the diagram, modes of given azimuthal order should sit in distinct ridges. Therefore, by varying these parameters manually and identifying those which return the most well defined ridges we arrive at initial estimates of ΔΠ 1 and 𝑞 . \n(ii) In the absence of a magnetic field, 𝑚 = 0 modes would align vertically in a ridge at the value of 𝜖 g. As such, once we have settled on the combination ΔΠ and 𝑞 , we can use the central ridge as an estimate of 𝜖 g. \nWe varied these parameters by hand using the interactive tool introduced in Ong et al. (2024). We show examples of the resulting power diagrams in figure 2. \nPriors on ΔΠ 1 and 𝜖 g were set according to a normal distribution centered on our estimate from the stretched échelle. Uncertainties on ΔΠ 1 from methods exploiting stretched échelles are on the order of a few percent (Vrard et al. 2016). The only literature work using stretched échelles to measure 𝜖 g is Mosser et al. (2018). There-in the mean uncertainty on 𝜖 g is ≈ 30%. We also note that previous uncertainty estimates do not account for the presence of magnetic asymmetry. For cases where 𝑚 = 0 components have been significantly perturbed by a magnetic field, the combination of parameters constructing the most vertical 𝑚 = 0 ridge will not be an accurate representation of the true values. In an attempt to quantify this effect, we constructed a mock spectrum with values of 𝛿𝜈 mag and 𝑎 consistent with those reported for KIC8684542 by Li et al. (2023) (the full set of asymptotic values used can be found in the appendix). We found that the difference between the injected and recovered ΔΠ 1 was below the 1% level. The difference between the input 𝜖 g , input and that from the hand tuned stretched échelle was more significant, at approximately 10%, but remained below the mean uncertainty reported in Mosser et al. (2018). \nFigure 3. Distribution of summed power across ridges defined by 𝛿𝜈 rot = 0, 𝛿𝜈 mag = 0 and 𝑎 = -0.5 for a white noise spectrum stretched according to the asymptotic parameters of KIC10006097. The number of realisations has been increased from 50 to 100 for illustrative purposes. \n<!-- image --> \nWe set the width of the prior on ΔΠ 1 as 10% of the mean. As previously discussed, the average fractional uncertainty on 𝜖 g reported in the literature is ≈ 30% (Mosser et al. 2018). The computational expense associated with nested sampling scales with the volume of prior space, such that setting a very wide prior leads to the calculation becoming infeasible. As such, the width of the prior on 𝜖 g was set to 30% of the mean. To ensure our results are not prior dominated, we visually inspected the posterior versus prior distributions on 𝜖 g. \nExploiting the stretched period échelles required us to use the asymptotic expression for mixed modes, rather than the matrix formalism used in the sampling. Following Ong & Gehan (2023) the value of 𝑞 evaluated at 𝜈 max can be determined from the matrix coupling parameters as \n𝑞 ≈ 1 Δ 𝜈 ΔΠ 1 GLYPH<18> 𝐿 + 𝜔 2 𝐷 8 𝜋𝜈 2 GLYPH<19> 2 = 1 Δ 𝜈 ΔΠ 1 GLYPH<16> 𝜋 2 ( 𝑝 𝐿 + 𝑝 𝐷 ) GLYPH<17> 2 , (10) \nwith all frequencies evaluated at 𝜈 max. This expression indicates that, from a single value of 𝑞 alone, it is not possible to uniquely identify 𝑝 𝐿 and 𝑝 𝐷 . However, given they are informed by the internal profile of the star, certain values will be more physically motivated. To identify characteristic values of 𝑝 𝐿 and 𝑝 𝐷 for our stars we exploited the grid of stellar models used in Ong et al. (2021b). For a given model star in this grid, values of 𝑝 𝐿 and 𝑝 𝐷 were subsequently calculated from the corresponding coupling matrices. Stellar tracks were calculated with masses from 0.8 M ⊙ to 2 M ⊙ , and [Fe/H] from -1 . 0 dex to 0 . 5 dex. \nFor a given star, we select stellar model tracks in a mass range consistent with those reported in Y18. We then located models with 𝜈 max in the range 𝜈 max , obs ± 𝜎 ( 𝜈 max , obs ) , ΔΠ 1 in the range ΔΠ 1 , prior ± 0 . 1 ΔΠ 1 , prior and a value of 𝑞 consistent with that derived from the stretched échelle (to within 10%). The means of the distributions of 𝑝 𝐿 and 𝑝 𝐷 in the selected models were taken as the means of the normal distributions we used as priors on 𝑝 𝐿 and 𝑝 𝐷 . For all the stars in our sample, the model grid returned values of 𝑝 𝐿 and 𝑝 𝐷 in ranges spanning standard deviations that were on average 2% and 10% of the mean model values, respectively. Accordingly we set the widths of the priors on 𝑝 𝐿 and 𝑝 𝐷 for each star at 10% of their mean value.", '2.3.2 Priors on 𝛿𝜈 mag , 𝑎 , 𝛿𝜈 rot': 'Once ΔΠ 1 , 𝑞 and 𝜖 𝑔 have been set, ridges in the stretched échelle can be approximated using the three remaining parameters in our mixed mode model. Therefore, for each star we constructed a grid of templates describing possible ridges given test values of 𝛿𝜈 mag, 𝑎 and 𝛿𝜈 rot. We uniformly sample 𝛿𝜈 mag in the range 0 to 0.2 𝜇 Hz, \n<!-- image --> \nFigure 4. Top panel: Inverse of the 2-dimensional H0 likelihood space for 𝛿𝜈 mag and 𝛿𝜈 rot. Two stars are shown, KIC8684542 and KIC10006097. Bottom panel: Inverse of the 2-dimensional H0 likelihood space for 𝛿𝜈 mag and 𝑎 for KIC8684542 and KIC10006097. \n<!-- image --> \n𝛿𝜈 rot in the range 0.0 to 0.8 𝜇 Hz and 𝑎 from -0.5 to 1. Our grid had 50 points in each direction, such that the resolution on the magnetic splitting is 0.004 𝜇 Hz, on the rotation it is 0.02 𝜇 Hz and on 𝑎 is 0.03. \nTo establish the parameter values required to best describe the data we performed a null hypothesis test (H0 test). For a given star, we summed the total observed power in these ridges, and established the likelihood that we would observe the resulting power just due to white noise (the H0 likelihood). If we were summing power in N bins without performing the stretching, this would simply amount to the likelihood of drawing a given value of summed power from a 𝜒 2 distribution with 2N degrees of freedom. However, the stretching introduces correlation between bins, such that the number of degrees of freedom in the stretched spectrum is no longer 2N. An analytical definition of the correct number of degrees of freedom required to describe a stretched spectrum is yet to be established. Given the degree of stretching depends on ΔΠ 1 and 𝑞 , this will vary on a star-by-star basis. \nTo approximate the statistics for the summed stretched power we therefore used white noise simulations. For each target we performed 50 realisations of a white noise spectrum evaluated on the same \nfrequency grid as the real data. We then stretched this spectrum according to the value of ΔΠ 1 and 𝑞 used in the prior and calculated the summed power about the predicted ridges according to our grid of 𝛿𝜈 mag, 𝑎 and 𝛿𝜈 rot (as would be done for a real star). This resulted in 50 realisations of a 3 dimensional summed power array for a given star. Given each point corresponds to the sum of a large number of 𝜒 2 distributed parameters, the resulting sum should be distributed according to a Gaussian (according to the central limit theorem). Accordingly, we calculated the H0 likelihood of the real data for a combination of 𝛿𝜈 mag, a, 𝛿𝜈 rot via, \nL( Θ | 𝐻 0 ) ≈ N( 𝜇 Θ ,𝑊𝑁 , 𝜎 Θ ,𝑊𝑁 ) , (11) \nwhere 𝜇 Θ ,𝑊𝑁 and 𝜎 Θ ,𝑊𝑁 are the mean and standard deviation of the white noise realisations. The parameters are Θ =( 𝛿𝜈 mag, 𝑎 , 𝛿𝜈 rot). Figure 3 shows the distribution of summed power in a single cell of the 3-d array for a white noise spectrum stretched according to the asymptotic parameters of KIC10006097. \nThe width over which we summed the power about the predicted ridge was informed by the expected (stretched) line-width for gdominated modes. At the base of the RGB, the distribution of radial mode linewidths peaks at ≈ 0.15 𝜇 Hz (Yu et al. 2018). The dipole mode linewidth for a given mixed mode then scales as Γ 1 ( 𝜈 ) = Γ 0 ( 1 -𝜁 ( 𝜈 )) . For a mode 𝜈 𝑖 with 𝜁 ( 𝜈 𝑖 ) = 0 . 9, this implies Γ 1 ( 𝜈 ) ≈ 0.015 𝜇𝐻𝑧 . Therefore, we sum power in a width of 0.03 𝜇 Hz. Given this definition was set using an arbitrary selection of 𝜁 , we tested 10 different widths up to a maximum of 0.15 𝜇 Hz. For KIC10006097 we found the resulting value of 𝛿𝜈 rot was consistent across widths from 0.015 to 0.084 𝜇 Hz. At widths larger than 0.084 𝜇 Hz, the measured value of 𝛿𝜈 rot was consistently smaller by ≈ 0.1 𝜇 Hz. \nExamples of the 2-d distributions in likelihood for the échelle diagrams shown in figure 2 can be seen in figure 4. The H0 likelihood was then marginalised over each axis, and the minima of the 1D distributions used to inform the mean of the prior used in the sampling. \nIn a handful of cases the likelihood space was multimodal. To identify which mode best described the data, we manually vetted the associated ridges and subsequently reduced the range to exclude the spurious peak. This multimodality was a by-product of the use of H0 likelihood, as a model that is not necessarily the best descriptor of the signal can still capture power that is very unlikely to be the result of noise (for example residual power from ℓ = 0 , 2 modes). \nGiven these likelihoods are conditional on the combination of ΔΠ 1 , 𝑞 and 𝜖 𝑔 , we did not use the width of the minima in the H0 likelihood to establish the width of the prior. To establish the most appropriate width to set on the prior on rotational splitting, we compared our values to those from Gehan et al. (2018) (henceforth G18). The resulting differences in measured rotation rates give a better estimate of the uncertainty associated with varying asymptotic parameters. Of the 334 stars in our target list, 142 are also in G18. The differences between the values in that catalogue and those we measured can be well approximated using a normal distribution with 𝜇 = 0.00 𝜇 Hz and 𝜎 = 0.05 𝜇 Hz (see figure 5). As such our prior was N( 𝛿𝜈 rot , echelle , 0 . 05 𝜇 Hz ) . \nThe largest catalogue of magnetic parameters is that of Li et al. (2023) (henceforth L23, see also Li et al. 2022a, henceforth L22). Of the 13 stars listed there, 8 appear in our target list. We found our values of 𝑎 differed substantially (see figure 6), with a preference for large values, which could be a consequence of the correlation with ΔΠ 1 (noted in L23). As such we ignore the result from the summed stretched power and set a uniform prior on 𝑎 between -0.5 and 1 for all stars. \nFigure 5. Core rotational splitting from the template matching technique versus those reported in G18. Black dotted lines show the 1-1 relation ± 0.05 𝜇 Hz \n<!-- image --> \n<!-- image --> \nFigure 6. Top panel: Magnetic splitting, 𝛿𝜈 mag, measured using summed power versus that reported in L23. Black dotted lines are the 1-1 relation ± 0.05 𝜇 Hz. Bottom panel: Topology parameter, 𝑎 , measured using summed power versus that reported in L23. Black dotted line is the 1-1 relation. \n<!-- image --> \nOur values of 𝛿𝜈 mag are in better agreement (see figure 6), with a standard deviation of 0.05 𝜇 Hz. We set the prior as uniform, allowing values in the range [ 𝛿𝜈 mag , echelle - 0.15 𝜇 Hz, 𝛿𝜈 mag , echelle + 0.15 𝜇 Hz]. For cases where this width would cause the prior to allow negative values of 𝛿𝜈 mag, we set the lower limit on the prior at 0. In one outlier, the grid method has a 𝛿𝜈 mag which is smaller than that in L23 by 0.13 𝜇 Hz. This is likely a consequence of selecting the value of ΔΠ 1 that made the 𝑚 = 0 ridge appear most vertical (see section 2). Given setting a wider prior on 𝛿𝜈 mag for all stars would result in significant additional computational expense, we manually vetted posteriors and best fit models and only expanded the prior ranges where necessary. This amounted to expanding the prior range on 𝛿𝜈 mag for KIC8684542.', '3 RESULTS': 'Of the 334 targets we report the magnetic and rotational parameters in 302 stars. Those 32 stars for which we do not report results are cases where the posterior distribution was simply a replica of the prior. Such were the result of either low SNR or low inclination such that the rotational splitting was not well constrained. Example corner plots for two stars drawn from the sample of 302 (for which we report results) are shown in the appendix (figures C1 and C2). For a comparison of our values of ΔΠ 1 , 𝑞 and literature values, see appendix B. The full catalogue of asymptotic g-mode parameters, alongside 𝛿𝜈 rot, 𝛿𝜈 mag and 𝑎 is available online, with the first 10 rows presented in Table 1.', '3.1 Rotational splitting': 'The distribution of core rotational splitting is shown in the top panel of figure 7, and is bimodal. The more populous peak is located at ≈ 0.32 𝜇 Hz with the secondary peak at ≈ 0.47 𝜇 Hz. There does not appear to be any strong correlation between the rotational splitting (or associated bimodality) and the remaining asymptotic parameters (see figure B1, which shows a corner plot of the distribution of asymptotic parameters across the whole population). Notably the distributions of 𝛿𝜈 mag and 𝑎 with 𝛿𝜈 rot > 0.4 𝜇 Hz are consistent with those in the remaining catalogue. \nRevisiting the 142 stars which appear in G18 which we used to inform the width of the prior on 𝛿𝜈 rot, we found general agreement, with ≈ 80% agreeing to 10% or better. As can be seen in figure 8, we found a correlation between the difference in 𝛿𝜈 rot with 𝛿𝜈 mag. For the star with the largest value of 𝛿𝜈 mag (KIC8684542), the difference in 𝛿𝜈 rot is of order 0.23 𝜇 Hz. This was also noted in L23. Given the small values of magnetic splitting present in the vast majority of our targets, the offset in 𝛿𝜈 rot is below the scale of the uncertainties. As such, this is unlikely to impact previous conclusions regarding the distribution of rotational splittings on a population scale. We note a small number of stars (9) with differences in 𝛿𝜈 rot exceeding 0.1 despite having 𝛿𝜈 mag values below 0.001. This is a symptom of low SNR on the 𝑚 = ± 1 components of multiplets due to low stellar inclination, the average in the 9 cases being 45 ° .', '3.2 Magnetic Parameters': 'The distribution in magnetic splittings peaks about log 10 ( 𝛿𝜈 mag) = -2.54, with a standard deviation of 𝜎 (log 10 ( 𝛿𝜈 mag)) = 0.45 (see figure 9). This indicates that, from an observational perspective, perturbations to mode frequencies due to a core magnetic field of the scales reported in L22 and L23 are uncommon regardless of asymmetry. In total we find the mean on the posterior of the magnetic splitting is at least 2 𝜎 from zero in 23 stars, approximately 8% of the total sample. The corner plots and stretched échelle diagrams for these 23 stars are available as online materials. \nThe measurements of the topology parameter 𝑎 span the full range in values allowed by the prior, with a peak at zero - the value which minimizes the asymmetry (see figure 9). For stars in which the mean value of the posterior on 𝛿𝜈 mag is at least 2 𝜎 from zero, 35% have values of 𝑎 exceeding 0.5. These are inconsistent with a dipolar field, and must be identified with an architecture having the field concentrated more towards the poles than the equator. Values of 𝑎 below -0.2 occur in 30% of the stars with significant 𝛿𝜈 mag. These also cannot be the result of a dipolar field, instead being consistent with a field concentrated near the equator. \nThe asymmetry between modes in a multiplet is often used as \nFigure 7. Distribution of 𝛿𝜈 rot. The top panel shows the distribution for the measurements made in this work. The following three panels include values from G18 with 𝜈 max > 100 𝜇 Hz and are separated into three mass ranges. The black curves show KDEs of the distributions. Black dotted lines mark the locations of the two peaks identified in the lowest mass set to guide the eye. \n<!-- image --> \nFigure 8. Fractional difference in 𝛿𝜈 rot measured here and reported in G18 as a function of 𝛿𝜈 mag. \n<!-- image --> \nan identifier of the presence of a magnetic perturbation. L22 and L23 quantify this using an additional parameter, 𝛿 asym = 3 𝑎𝛿𝜈 mag. In our catalogue 𝛿 asym also peaks near zero, the distribution having a mean value log 10 ( 𝛿 asym)= -2.78 with a standard deviation of 𝜎 (log 10 ( 𝛿 asym)) = 0.67 (see figure 9). Only 12 stars have asymmetry parameters that are at least 2 𝜎 from zero, making them easily identifiable by eye. All 8 of the stars that we have in common with L23 appear in this set. The remaining targets identified as having significant 𝛿𝜈 mag values but little asymmetry have not been previously identified. \nWe find no obvious correlation between 𝛿𝜈 mag and any other asymptotic parameter, including the topology. This was also noted in L23. Bugnet (2022) noted that not accounting for a magnetic pertur- \nFigure 9. Distributions of magnetic parameters. Histograms are built using 100 draws from the posterior for each of the 302 stars. Top panel: Distribution of 𝛿𝜈 mag. Middle panel: Distribution of 𝛿 asym. Bottom panel: Distribution of 𝑎 . \n<!-- image --> \ncould produce a systematic underestimate of ΔΠ 1 when using the techniques presented in Vrard et al. (2016). They simulated a star with a magnetic splitting of 𝛿𝜈 mag = 0.4 𝜇 Hz, which is much larger than those reported in L22 and L23 (where-in the maximum reported value is 0.21 𝜇 Hz), and found this would cause a 1% difference in the measured period spacing. We do not find a clear trend between the difference in our period spacing and that recorded in Vrard et al. (2016) with the magnetic splitting. As such, for stars with the magnitudes of magnetic perturbation reported here, discrepant ΔΠ 1 alone cannot be used as an identifier. \nFor the 8 stars we have in common with Li et al. (2023), our values of 𝛿𝜈 mag are in broad agreement, with the mean absolute difference being 1.09 𝜎 . Agreement on 𝑎 is slightly worse, with a mean absolute difference of 1.5 𝜎 . Our methods do differ, Li et al. (2023) fit the asymptotic expression to mode frequencies rather than directly fitting a forward model to the spectrum. Additionally we differ in our method of coupling modes, where we used the matrix construction discussed in Ong et al. (2021a) and Deheuvels & Michel (2010), Li et al. (2023) use the JWKB construction of Unno et al. (1989). Finally, our priors on asymptotic parameters differ (see Li et al. 2023 for details).', '4.1 Rotation and stellar mass': 'As previously noted, the distribution of core rotational splitting appears bimodal. To better sample the underlying distribution, we expand our data to include stars from the G18 catalogue with 𝜈 max in the range of the stars in our sample. To that end we selected stars from G18 that both do not appear in our target list and have 𝜈 max greater \nthan the minimum in our targets at 𝜈 max , min = 135 𝜇 Hz. In total the expanded sample numbers 583 stars. Although it was not reported in G18, a bimodality in core rotation does appear to be present in their measurements for stars in this range in 𝜈 max (see the discussion in Appendix D). \nWe show the distribution of 𝛿𝜈 rot as a function of stellar mass in figure 10. Masses are from Y18, where-in they are calculated via scaling relations with 𝜈 max, Δ 𝜈 and 𝑇 eff . In the left-most panel (showing the new measurements reported here without the additional stars from G18) it appears the divide between the two populations in rotation is mass dependent, with the more rapidly rotating peak preferentially populated with less massive stars. To identify whether the two apparently distinct over-densities remain in the sample with the additional stars from G18, a KDE estimate is plotted in black in the final panel in 10. There-in we can identify the clear over-density at 𝛿𝜈 rot ≈ 0.3 𝜇 Hz, alongside the less populous peak in density at 𝛿𝜈 rot ≈ 0.5 𝜇 Hz. \nWe divided the combined catalogue into stars in ranges 1.0M ⊙ < M ∗ < 1.2M ⊙ , 1.2M ⊙ < M ∗ < 1.4M ⊙ and M > 1.4M ⊙ . The resulting distributions in 𝛿𝜈 rot can be seen in figure 7. In the lowest mass range, the secondary peak is identifiable at 𝛿𝜈 rot 0.47 𝜇 Hz. In the mid range, the peak shifts upwards in 𝛿𝜈 rot to 0.50 𝜇 Hz and is less well populated. The set with the highest masses contains the fewest total stars and it is unclear whether a secondary peak is present. In total 25% of the population have 𝛿𝜈 rot > 0.40 𝜇 Hz. \nIn figure 11 we show our measurements of 𝛿𝜈 rot and those from G18 as a function of stellar mass in three ranges of N . The left-hand panel is a reproduction of figure 10. The following panels show this distribution with increasing mixed mode density, N . As such, for a given mass, stars go from least to most evolved from the left-most to the right-most panel. G18 and Mosser et al. (2012b) concluded that core rotation rates in red giants decrease slightly as stars evolve, but do not undergo significant change. Indeed, the highest density of stars is at ≈ 0.3 𝜇 Hz in all three subsets. However, the spread in the distribution increases, such that the secondary peak identified at 0.47 𝜇 Hz appears to migrate to larger values with increasing N . For stars with N < 7, 2.4% of the population have 𝛿𝜈 rot > 0.6 𝜇 Hz, this increases to 6.1% for stars with N > 11. \nRed Clump stars are expected to have core rotation rates that differ from those on the RGB. However, given the cores are undergoing expansion due to the onset of helium burning, rotation rates are expected to decrease. This was observed to be the case in Mosser et al. (2012b). There-in the authors reported splittings ranging from 0.01 to 0.1 𝜇 Hz. Therefore, it is unlikely the subset of rapid rotators are misclassified clump stars. \nOne interpretation of this result is that the stars in the more rapidly rotating population in each range in N belong to a single population with a weaker rotational coupling between core and envelope, such that the efficiency of angular momentum transport has been reduced. Accordingly their cores would be able to more effectively spin up as they contract. As the core contracts, the envelope is undergoing expansion, such that we can use the evolution of core rotation with stellar radius to infer the sign of the dependence on core contraction. Cantiello et al. (2014) found that the increase in core rotation should scale with stellar radius as Ω core ∝ 𝑅 𝛼 ∗ , with 𝛼 taking a value of 1.32 or 0.58, depending on whether they include just rotational instabilities or fold in those due to magnetic torques in radiative regions. That is, the cores should spin up as the star evolves. This is in clear disagreement with the core rotation rates measured by Mosser et al. (2012b), who found exponents of -0.5 for stars on the RGB and -1.3 in the RC. The mean values of stellar radius in the stars with N < 7 and N > 11 are 4.89 R ⊙ and 7.14 R ⊙ respectively. For representative \nstars at these radii to spin up from ≈ 0.5 𝜇 Hz to ≈ 0.7 𝜇 Hz would imply a relation of the form Ω core ∝ 𝑅 0 . 8 ∗ . This exponent sits in between those predicted by Cantiello et al. (2014).', '4.2 Magnetic Perturbations and Stellar Properties': 'Asnoted in section 1, a core magnetic field could impact the transport of angular momentum within a star. We found no clear correlation between the bimodality in 𝛿𝜈 rot and 𝛿𝜈 mag or 𝑎 . However, given a strong core magnetic field is frequently hypothesised as a solution to the discrepancy between modelled and observed core rotation rates, constraints on the average field strength in a large sample of stars remain in demand. In the following sections we invert the observed magnetic splitting to constrain the average core magnetic field strength.', '4.2.1 Stellar Models': "According to L22, the root-mean-square (rms) of the radial field strength scales with the magnetic splitting as \n⟨ B 2 r ⟩ = ∫ 𝑟 𝑜 𝑟 𝑖 𝐾 ( 𝑟 ) 𝐵 2 𝑟 𝑑𝑟 = 16 𝜋 4 𝜇 0 𝜈 3 max 𝛿𝜈 mag I , (12) \nwhere I is a factor determined by the internal structure of the star. This is given by, \nI = ∫ 𝑟 𝑜 𝑟 𝑖 GLYPH<0> 𝑁 𝑟 GLYPH<1> 3 𝑑𝑟 𝜌 ∫ 𝑟 𝑜 𝑟 𝑖 𝑁 𝑟 𝑑𝑟 , (13) \nwhere 𝑁 is the Brunt-Väisällä frequency, 𝑟 is radius and 𝜌 the stellar density. Therefore, we require models of the internal profile of our targets to invert our measured 𝛿𝜈 mag to give an estimate of the radial field strength. \nTo that end, we used Modules for Experiments in Stellar Astrophysics (MESA, Paxton et al. 2011, 2013) to calculate a grid of stellar models with varying mass and metallicity. We calculated stellar evolution tracks with masses varying in the range of those reported in Y18 for our targets, spanning from 1M ⊙ to 2M ⊙ in increments of 0.05M ⊙ . Metallicities ranged from -1.0dex to +1.0dex with a spacing of 0.25dex. We used a mixing length of 𝛼 MLT = 2.29, which was found via calibration of a solar model. Overshoot was treated using the exponential formalism with 𝑓 1 = 0.015 and 𝑓 0 = 0.004. Mode frequencies were then calculated using GYRE. To avoid too much computational expense, we restricted the grid to just radial modes. \nAsuitable model from this grid was selected for each star using the AIMS package (Asteroseismic Inference on a Massive Scale, Rendle et al. 2019). To do this AIMS searches the grid for the region with the highest posterior probability given the observed parameters, then explores the surrounding space using an MCMC sampler ( emcee , Foreman-Mackey et al. 2013). For general use in fitting global properties, AIMS interpolates between grid points. However, we require internal profiles and are, therefore, restricted to selecting models in the grid. We choose the model with the highest posterior probability, but note this is naturally restricted by the grid resolution. AIMS provides several methods to apply surface corrections to model frequencies, of which we selected the method of Ball & Gizon (2014). We provided 𝑇 eff , [Fe/H], log ( 𝑔 ) , mass (from Y18) alongside ΔΠ 1 , 𝜈 max, Δ 𝜈 and radial mode frequencies from the ℓ = 2 , 0 model (see section 2.2.2) as observables. \nThe only model parameter which could induce additional uncertainty in equation 12 is I . Across all of the best fitting models, the \nFigure 12 shows 𝛿𝜈 mag as a function of stellar mass (mass values from Y18). We note that 𝛿𝜈 mag values in excess of 0.04 𝜇 Hz only begin to appear at masses larger than ≈ 1.1 M ⊙ . However, this is not the case for small but significant magnetic splittings at values below 0.04 𝜇 Hz. In this regime stars span the full range in stellar mass present in our population. This could be consistent with a field driven in the small main sequence convection zone caused by the burning of 3 He and 12 C outside of equilibrium. Given that these targets have not previously been published due to the lack of asymmetry, this \n<!-- image --> \n<!-- image --> \nFigure 10. Distribution of 𝛿𝜈 rot as a function of stellar mass for the combined set of stars reported here and in G18. The left-most panel shows just the measurements reported in this work. In the center panel we fold in measurements from G18 (orange). The Black contours in the right hand panel are a kernel density estimate (KDE) highlighting the bimodality. There-in the bin-width has been set according to Scott's rule of thumb (Scott 1992). \n<!-- image --> \nFigure 11. Distribution of 𝛿𝜈 rot reported in G18 (grey edgecolor) and here (black edgecolor) for stars in 3 different N ranges. The left-most panel shows targets with N < 7, the middle panel has targets in the range 7 < N < 11. The right-most shows stars with N > 11. Black dotted lines are at 𝛿𝜈 rot = 0.32 𝜇 Hz and 0.47 𝜇 Hz. Histograms on the right show the distribution in the stars with N < 7 in pink and with N >11 in green. \n<!-- image --> \nvariation in this parameter is well approximated by a Gaussian with a spread which is 30% of the mean. If we take this as a conservative proxy for the model uncertainty on this parameter (given our stars share similar observable properties), we would expect a modeling error on the inferred field strength of order 30%. We took this value forward as the model error, stating it in addition to the statistical error from the measurement of 𝛿𝜈 mag, noting this is a very conservative estimate.", '4.2.2 Mass': 'The presence of core convection during the main sequence is dependent on stellar mass, requiring the star to have a mass greater than ≈ 1.1 M ⊙ (Kippenhahn & Weigert 1990). A magnetic field could then be driven in this convection zone and remain on the RGB in fossil form. Should significant magnetic perturbations only be measured in stars of mass > 1.1M ⊙ , we may take this as evidence that core magnetic fields in red giants are the remains of those generated in a convection zone on the main sequence. However, as shown in Li \net al. (2022a) and Li et al. (2023), stars with masses below 1.1M ⊙ can still develop small convective cores on the main sequence, due to the burning of 3 He and 12 C outside of equilibrium. There-in the authors find that the core sizes are not large enough to reach the hydrogen burning shell on the red giant branch. As such, modes would be less sensitive to the presence of the field and therefore require a larger field strength to produce a shift of similar scale. Assuming field strength does not depend on stellar mass, we should then find that measurable magnetic perturbations in stars with M < 1.1 M ⊙ are broadly smaller in magnitude. \nFigure 13. Best fitting √︃ ⟨ B 2 r ⟩ as a function of stellar mass. The 23 stars with magnetic splitting values at least 2 𝜎 from zero are marked with white diamonds. The remaining stars are marked with pink circles. \n<!-- image --> \nFigure 12. 𝛿𝜈 mag as a function of stellar mass. In the central panel the 23 stars with magnetic splitting values that are at least 2 𝜎 from zero are shown in white diamonds, the remaining measurements are pink circles. The top panel shows KDEs of the mass distribution of the 23 stars in black and the remaining targets in pink. The right-most panel shows KDEs of the distribution of 𝛿𝜈 mag, with the same colour-scheme. \n<!-- image --> \nhighlights a potential detection bias when manually selecting targets. This result could also be the signature of a systematic underestimate on stellar mass for the stars with significant 𝛿𝜈 mag at M < 1.1M ⊙ . However, comparisons to masses derived from eclipsing binaries have shown masses from asteroseismic scaling relations are likely to be systematically overestimated rather than underestimated (Gaulme et al. 2016; Brogaard et al. 2018; Themeßl et al. 2018; Li et al. 2022b).', '4.2.3 Magnetic Field Strengths': 'To determine the value of ⟨ B 2 r ⟩ that would reproduce the magnetic splitting we measured, we utilized the best fitting models from the grid outlined in section 4.2.1. From these models we calculated the value of I for each star. We then drew 1000 samples from the posterior distributions on 𝛿𝜈 mag and 𝜈 max and calculated the mean field strength, ⟨ B 2 r ⟩ , according to equation 12. The reported field strength for each star is taken as the mean of the resulting distribution on ⟨ B 2 r ⟩ . Uncertainties are the standard deviation on this distribution plus the expected uncertainty from the models (see section 4.2.1) added in quadrature. \nThe distribution of ⟨ B 2 r ⟩ peaks at zero, reflecting the measured magnetic splitting. For the stars with a field value at least 2 𝜎 from zero, 30% of the population have ⟨ B 2 r ⟩ < 30kG. The distribution then tails with increasing field strength to the maximum at 169.4 ± 51 kG, occurring in KIC5696081. \nThere is no significant correlation between the measured core field strengths and stellar mass (see figure 13). At the low mass end of the distribution, the scale of the spread appears larger in ⟨ B 2 r ⟩ than 𝛿𝜈 mag. This is a consequence of the 𝜈 -3 max dependence on 𝛿𝜈 mag. For a given field strength, a larger 𝜈 max (preferentially occurring in lower mass stars) implies a smaller magnetic splitting. \nL23 identified a decrease in the core field strength as stars evolve along the RGB, but caveat this with a note that there is significant scatter. We observe the same dependence in our larger set of stars, as is shown in figure 14. There-in the authors identified the decrease follows the decrease in the critical field strength, which sets an upper limit on the observable field strength.', '5 LIMITATIONS': 'Although the aim of this work was to catalogue a large number of stars such that we could start exploring population statistics, we were still subject to various detection biases. Firstly, we restricted ourselves to spectra where-in we could clearly identify all three components of the ℓ = 1 multiplets. Therefore, we are restricted to inclination in the approximate range 30 ° < 𝑖 < 60 ° . With the assumption that stellar inclination is isotropically distributed this limits us to ≈ 40% of the possible sample of stars. \nSecondly, targets were selected by manual identification of rotational splitting. This meant that we required spectra where-in the separation between mixed modes was significantly larger than the rotational splitting. This defined both the lower limit on 𝜈 max (100 𝜇 Hz) and sets an upper limit on the 𝛿𝜈 rot for a given star. For a target with 𝜈 max = 150 𝜇 Hz, ΔΠ 1 = 80 secs, the expected separation between adjacent mixed modes is approximately 3 𝜇 Hz. Currently, the maximum recorded core rotation rate for a red giant is 0.95 𝜇 Hz (Gehan et al. 2018). Therefore, while our method is well suited to the range in core rotation previously reported in the literature, stars with rotation rates exceeding a few 𝜇 Hz would not have appeared in our initial sample selection. \nFinally, We do not treat envelope rotation, which would introduce additional splitting in the p-dominated modes. Should the envelope rotation be significant, this could lead to an overestimate in our measurement of core rotational splitting. However, surface rotation rates in red giants are observed to be orders of magnitude smaller than the core rotation (Goupil et al. 2013), such that they are unlikely to cause significant error. Non-standard stellar evolution (e.g. mergers) can cause rapid envelope rotation in red giants. However, best estimates for the prevalence of such non-standard rotators is on the order of 8% (Gaulme et al. 2020). In our catalogue, ≈ 25% of stars have core rotational splitting larger than 0.4 𝜇 Hz. A study of the relation between envelope rotational splitting, core rotational splitting and magnetic parameters is reserved for future work.', '6 CONCLUSIONS': 'Exploiting the stretched period échelle, we have demonstrated how template matching can be used to construct initial estimates of the \nFigure 14. Best fitting √︃ ⟨ B 2 r ⟩ as a function of mixed mode density, N . Stars with 𝛿𝜈 mag - 2 𝜎 ( 𝛿𝜈 mag ) > 0 are shown in orange. The remaining measurements are in grey. Blue diamonds show values reported in L23 for stars that do not appear in this work. \n<!-- image --> \nperturbations to dipole mode frequencies caused by core rotation and a magnetic field. We parameterise these using the magnetic splitting ( 𝛿𝜈 mag), a parameter dependent on field topology ( 𝑎 ) and core rotational splitting ( 𝛿𝜈 rot). This allowed us to establish wellmotivated priors for 𝛿𝜈 mag and 𝛿𝜈 rot in 334 low luminosity red giants. \nUtilizing the information gained from the stretched échelles, we performed a full fit of the perturbed asymptotic expression to the power spectrum. This allowed us to jointly constrain ΔΠ 1 , 𝑞 , 𝜖 g, 𝛿𝜈 01 , 𝛿𝜈 rot, 𝑎 and 𝛿𝜈 mag in 302 targets. We found that not accounting for the magnetic perturbation when measuring the rotational splitting can lead to biased measurements when the magnetic perturbation is large ( 𝛿𝜈 mag on the scale of 0.1 𝜇 Hz). For the star with the largest value of 𝛿𝜈 mag the value of 𝛿𝜈 rot reported in G18 is 70% smaller than the value we measured. \nWe identified a bimodality in the core rotation rates of the stars in our sample. The more populous peak is at 𝛿𝜈 rot = 0.32 𝜇 Hz, with the secondary at 0.47 𝜇 Hz. The location and size of this secondary peak appears to be mass dependent. We found the distribution also evolves with N , with the upper limit on core rotation increasing with increasing N . Assuming that in each N range the most rapidly rotating stars belong to a secondary population, the observed increase in core rotation rate would imply a relation of the form Ω core ∝ 𝑅 0 . 8 ∗ . This is much closer to the predictions in Cantiello et al. (2014), suggesting in these stars the evolution of core rotation could be reproduced using a combination of rotational and magnetic instabilities. \nWe measured a magnetic splitting that is at least 2 𝜎 from zero in 8%ofthe total sample (23 stars). Strong asymmetry was only present in 57% of these targets (4% of the full catalogue). For the stars with a clear detection of magnetic splitting, the topology parameter is not uniformly populated. A large percentage (35%) have values of 𝑎 exceeding 0.5, identifiable with an architecture with the field more concentrated at the poles than the equator. Another large group (30% of stars with significant magnetic splitting) have values of 𝑎 below -0.2, consistent with a field concentrated near the equator. \nWe did not observe any correlation between magnetic and rotational parameters, and so are unable to comment on whether the \nadditional angular momentum transport is directly related to the magnetic fields we measured. \nAlthough the largest magnetic splittings we measured were in stars with masses greater than 1.1 M ⊙ , magnetic splittings inconsistent with zero were measured in stars with masses from 1.03M ⊙ to 1.6M ⊙ . This suggests that a main sequence convective core may not be the only channel for generating stable magnetic fields that are observed in fossil form on the red giant branch. \nFor the targets in which we measured significant magnetic splittings, the field strengths are on the order of tens of kG, with the number of detections decreasing with increasing field strength. The maximum value we measured was 169.4kG in KIC5696081. We confirm the tentative conclusion made in L23 that measurable field strengths decrease as stars evolve.', 'ACKNOWLEDGEMENTS': "E.J.H., W.J.B. and G.R.D. acknowledge the support of Science and Technology Facilities Council. M.B.N. acknowledges support from the UK Space Agency. JMJO acknowledges support from NASA through the NASA Hubble Fellowship grant HST-HF2-51517.001, awarded by STScI, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. The authors acknowledge use of the BlueBEAR HPC service at the University of Birmingham. This paper includes data collected by the Kepler mission and obtained from the MAST data archive at the Space Telescope Science Institute (STScI). Funding for the Kepler mission is provided by the NASA Science Mission Directorate. This work has made use of data from the European Space Agency (ESA) mission Gaia ( https://www.cosmos.esa.int/web/gaia ), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https: //www.cosmos.esa.int/web/gaia/dpac/consortium ). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This paper has received funding from the European Research Coun- \ncil (ERC) under the European Union's Horizon 2020 research and innovation programme (CartographY GA. 804752). S.D. and J.B. acknowledge support from the Centre National d'Etudes Spatiales (CNES).", 'DATA AVAILABILITY': 'Table 1 is available at the CDS. Lightcurves used in this work are available at MAST. Other intermediate data products will be made available upon reasonable request. \n14 \nHatt E. J. et al \n𝛿 \nHz) \n𝜇 \n( \n01 \n𝜈 \n𝛿 \ng \n𝜖 \n(s) \n1 \nΠ \nΔ \n4 \n- \n/10 \n𝐷 \n𝑝 \n) \n2 \n(Hz \n4 \n- \n/10 \n𝐿 \n𝑝 \nq \nHz) \n𝜇 \n( \n𝜈 \nΔ \nHz) \n𝜇 \n( \nmax \n𝜈 \nID \nKIC \n0.0059 \n0.012 \n± \n6.391 \n0.019 \n± \n0.931 \n0.02 \n± \n80.1 \n1.0 \n± \n75.2 \n0.9 \n± \n9.1 \n0.0019 \n± \n0.1769 \n0.01 \n± \n12.38 \n1.2 \n± \n138.8 \n10001728 \nMNRAS \n000 \n, 1-18 (2024) \n0.0017 \n0.009 \n± \n7.421 \n0.009 \n± \n0.827 \n0.01 \n± \n80.94 \n0.9 \n± \n63.4 \n0.9 \n± \n11.1 \n0.0009 \n± \n0.1413 \n0.01 \n± \n11.96 \n1.0 \n± \n142.9 \n10006097 \n0.004 \n0.01 \n± \n7.979 \n0.006 \n± \n0.864 \n0.01 \n± \n85.24 \n1.2 \n± \n72.2 \n1.2 \n± \n13.2 \n0.0007 \n± \n0.1493 \n0.0 \n± \n14.15 \n0.7 \n± \n176.7 \n10014959 \n0.0025 \n0.009 \n± \n7.342 \n0.006 \n± \n0.839 \n0.01 \n± \n81.13 \n1.1 \n± \n64.4 \n1.1 \n± \n12.0 \n0.0007 \n± \n0.1381 \n0.01 \n± \n12.85 \n0.8 \n± \n161.6 \n10063214 \n0.0095 \n0.01 \n± \n7.032 \n0.015 \n± \n0.88 \n0.02 \n± \n81.48 \n1.2 \n± \n69.6 \n1.2 \n± \n11.1 \n0.001 \n± \n0.141 \n0.01 \n± \n13.97 \n0.4 \n± \n185.9 \n10068556 \n0.0019 \n0.015 \n± \n10.071 \n0.004 \n± \n0.915 \n0.01 \n± \n90.73 \n1.4 \n± \n93.7 \n1.5 \n± \n16.3 \n0.0007 \n± \n0.1825 \n0.01 \n± \n18.05 \n1.3 \n± \n236.8 \n10078979 \n0.0078 \n0.009 \n± \n7.53 \n0.007 \n± \n0.823 \n0.01 \n± \n83.3 \n1.0 \n± \n62.9 \n1.0 \n± \n12.3 \n0.0009 \n± \n0.1344 \n0.0 \n± \n12.48 \n0.7 \n± \n156.0 \n10149324 \n0.0018 \n0.013 \n± \n9.681 \n0.004 \n± \n0.892 \n0.01 \n± \n89.27 \n1.5 \n± \n86.1 \n1.6 \n± \n13.5 \n0.0006 \n± \n0.1544 \n0.01 \n± \n17.76 \n1.0 \n± \n242.2 \n10198496 \n0.0087 \n0.014 \n± \n9.995 \n0.007 \n± \n0.917 \n0.02 \n± \n91.4 \n1.4 \n± \n97.6 \n1.5 \n± \n13.9 \n0.0006 \n± \n0.1817 \n0.01 \n± \n18.49 \n1.1 \n± \n248.9 \n10199289 \n0.0012 \n0.013 \n± \n7.202 \n0.004 \n± \n0.923 \n0.01 \n± \n83.84 \n1.3 \n± \n75.8 \n1.3 \n± \n14.1 \n0.0009 \n± \n0.1601 \n0.01 \n± \n14.88 \n0.8 \n± \n198.2 \n10274410 \nonline. \nailable \nv \na \nis \ncatalogue \nfull \nThe \n. \n2 \nsection \nin \ndetailed \nmethodology \nthe \nto \naccording \ncalculated \nparameters \nseismic \nthe \nof \nample \nEx \n1. \nable \nT', 'REFERENCES': 'Table A1 contains the parameters used to generate the fake spectrum used in section 2.3.1. \nTable A1. Values of the asymptotic parameters used to construct a mock spectrum for KIC8684542.', 'B1 G-mode asymptotics': 'We note here we manually inspected both the stretched echelles and the models of the power spectrum that result from our values of ΔΠ 1 and 𝑞 for each star in our sample, to confirm the values produced models in good agreement with the data. None-the-less, we include a comparison of our values with those from the literature below. Vrard et al. (2016) (V16) published measurements of g-mode asymptotics in 6100 red giants. Mosser et al. (2017) (M17) built on this catalogue to include the mode coupling parameter, 𝑞 . In figures B2 and B3 we show comparisons between our measurements of period spacing \nFigure B1. Corner plot showing the distribution of the asymptotic parameters, rotational splitting and magnetic parameters across all 302 stars. \n<!-- image --> \nand coupling parameter (which we derive from equation 10) and those from the aforementioned catalogues. The strong gridding in 𝑞 is a result of the methods used in M17. On average, the values of 𝑞 reported here are higher, with the mean offset being 10%. This is below the average uncertainty reported in M17, which is 16%. For a test star where-in the value of q in our work differs significantly from that in M17 we have included an example of the stretched échelle according to both results in the online materials. Our period spacing measurements are consistent with those reported in V16 in all but two cases, KIC 7009365 and KIC 9945389. The former case is one with a measurable magnetic signature, which was also reported in Li et al. (2023). For the latter star, the period spacing reported in V16 is 68.8s, which is substantially lower than the average in their sam- \n83.7s). The value reported here, however, is consistent with that average, at 86.4s. Though the mean of the posterior distribution when fitting for a magnetic shift in KIC 9945389 is not significantly different from zero, we note the posterior space is bimodal. A secondary set of solutions occurs at slightly higher period spacing and a significant magnetic shift. The stretched échelle for this star using the parameters published in this work and those in V16 can be found in the online materials. We note the uncertainties reported here are smaller than those in V16 by two orders of magnitude. They are, however, consistent with those reported in more recent studies exploiting similar fitting methods (Li et al. 2022a; Kuszlewicz et al. 2023; Li et al. 2023). \nUsing a different method involving forward modelling mixed mode \nFigures C1 and C2 show the corner plots of the parameters in the perturbed dipole mode model for two example stars. KIC 7018212 is a star in which no clear magnetic signature is measured, while KIC 11515377 has a clear signature of magnetic splitting. \n<!-- image --> \nFigure B2. Difference in period spacing measured here and reported in V16. \n<!-- image --> \nFigure B3. Difference in 𝑞 measured here and reported in M17. \n<!-- image --> \nFigure B4. Difference in period spacing measured here and reported in K23. \n<!-- image --> \nfrequencies, Kuszlewicz et al. (2023) (K23) reported the g-mode asymptotic parameters in 1074 Kepler red giants. A comparison between our values and those in K23 can be seen in Fig. B4 and B5. There-in we find better agreement on both parameters than with V16 and M17. Noticeably, there is no trend or offset in q (the mean difference between the values reported here and those in K23 is 0.03%).', 'APPENDIX C: EXAMPLE CORNER PLOTS': 'Figure B5. Difference in 𝑞 measured here and reported in K23.Figure C1. Corner plot of the asymptotic parameters of KIC11515377. \n<!-- image -->', 'APPENDIX D: ESTABLISHING THE SIGNIFICANCE OF BIMODALITY IN CORE ROTATION': 'To evaluate the significance of the apparent secondary peak in the distribution of 𝛿𝜈 rot, we tested fitting different numbers of Gaussians to the distribution of core rotational splitting in G18 (for stars with 𝜈 max > 135 𝜇 Hz, the minimum in our sample) and to the measurements reported here. To quantify which number of components best represented the data, while including a penalty for models with arbitrarily large numbers of free parameters (i.e. those that are over-fitting the data), we used the Bayesian information criterion (BIC, Schwarz 1978). This is defined as, \nBIC = 𝑘𝑙𝑛 ( 𝑛 ) -2 𝑙𝑛 ( ˆ 𝐿 ) , (D1) \nwhere ˆ 𝐿 is the maximum of the likelihood for a given model, 𝑛 is the number of observations and 𝑘 is the number of parameters in the model. Models that minimize the BIC are thus preferred. We found in both the sub-sample from G18 and our measurements the value of the BIC was lower for the fit with two components, rather than one (see Fig. D1 and D2). For the models with 2 Gaussian components, the means of the components were also consistent across the sub- \nFigure C2. Corner plot of the asymptotic parameters of KIC7018212. \n<!-- image --> \n<!-- image --> \nFigure D1. Left panel: Distribution of core rotational splitting for stars with 𝜈 max > 135 𝜇 Hz, as reported in G18 in blue. Black lines represent a fit of two Gaussian distributions to the data. The individual components of this fit are represented by dotted lines. Right panel: The Bayesian information criterion (BIC) as a function of the number of Gaussian components used the fit the distribution. The minimum of this curve is for 2 components. \n<!-- image --> \nm G18 ( 𝜇 lower = 0.29 𝜇 Hz, 𝜇 higher = 0.49 𝜇 Hz) and the measurements reported here ( 𝜇 lower = 0.31 𝜇 Hz, 𝜇 higher = 0.48 𝜇 Hz). \nThis paper has been typeset from a T E X/L A T E X file prepared by the author. \n<!-- image --> \nFigure D2. As in Fig. D1, but with the core rotational splitting measured in this work. Again the 2 component fit has the lowest BIC. \n<!-- image -->'}

2024MNRAS.534..215F

"We present an analysis of the cold gas phase in a lowmetallicity starburst generated in a highresol(...TRUNCATED)

2024-10-01T00:00:00Z

"['arXiv:2408.16887', '2024MNRAS.534..215F', '10.1093/mnras/stae2072', '2024MNRAS.tmp.2029F', '2024a(...TRUNCATED)

['Astrophysics - Astrophysics of Galaxies']

"The masses structure and lifetimes of cold clouds in a highresolution simulation of a lowmetallicit(...TRUNCATED)

['EPRINT_HTML', 'EPRINT_PDF', 'PUB_HTML', 'PUB_PDF']

https://arxiv.org/pdf/2408.16887.pdf

"{'No Header': ', 1-16 (2024)', 'The masses, structure and lifetimes of cold clouds in a high-resolu(...TRUNCATED)

2023PASP..135f8001G

"Twentysix years ago a small committee report building on earlier studies expounded a compelling and(...TRUNCATED)

2023-06-01T00:00:00Z

"['2023PASP..135f8001G', '10.48550/arXiv.2304.04869', 'arXiv:2304.04869', '2023arXiv230404869G', '10(...TRUNCATED)

"['Space vehicle instruments', 'Astronomical instrumentation', 'Infrared astronomy', 'Infrared obser(...TRUNCATED)

The James Webb Space Telescope Mission

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https://arxiv.org/pdf/2304.04869.pdf

"{'The James Webb Space Telescope Mission': \"Jonathan P. Gardner, 1 John C. Mather, 1 Randy Abbott,(...TRUNCATED)

2024A&A...689A.145S

"Massive starforming galaxies in the highredshift universe host large reservoirs of cold gas in thei(...TRUNCATED)

2024-09-01T00:00:00Z

"['2024arXiv240104919S', '10.48550/arXiv.2401.04919', 'arXiv:2401.04919', '10.1051/0004-6361/2024491(...TRUNCATED)

"['galaxies: high-redshift', 'galaxies: individual: J1000+0234', 'submillimeter: galaxies', 'Astroph(...TRUNCATED)

The ALMACRISTAL survey Discovery of a 15 kpclong gas plume in a z 4.54 Lyman blob

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https://arxiv.org/pdf/2401.04919.pdf

"{'Discovery of a 15 kpc-long gas plume in a z = 4 . 54 Lymanα blob': 'M. Solimano 1 , J. González(...TRUNCATED)

2024A&A...689A.302G

"Aims. We intended to quantify the impact of stellar multiplicity on the presence and properties of (...TRUNCATED)

2024-09-01T00:00:00Z

"['arXiv:2407.20138', '10.48550/arXiv.2407.20138', '2024A&A...689A.302G', '2024arXiv240720138G', '10(...TRUNCATED)

"['astronomical databases: miscellaneous', 'virtual observatory tools', 'astrometry', 'binaries: gen(...TRUNCATED)

Multiplicity of stars with planets in the solar neighbourhood

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https://arxiv.org/pdf/2407.20138.pdf

"{'Multiplicity of stars with planets in the solar neighbourhood ⋆': 'J. González-Payo 1 , 2 , J.(...TRUNCATED)

2024AJ....168..202H

"We present the discovery of TOI 762 A b and TIC 46432937 b two giant planets transiting Mdwarf star(...TRUNCATED)

2024-11-01T00:00:00Z

"['10.3847/1538-3881/ad6f07', '2024AJ....168..202H', 'arXiv:2407.07187', '2024arXiv240707187H', '10.(...TRUNCATED)

"['Exoplanet systems', 'Exoplanet astronomy', 'Transit photometry', 'Radial velocity', 'Space telesc(...TRUNCATED)

TOI 762 A b and TIC 46432937 b Two Giant Planets Transiting Mdwarf Stars

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https://arxiv.org/pdf/2407.07187.pdf

"{'TOI 762Ab AND TIC 46432937b: TWO GIANT PLANETS TRANSITING M DWARF STARS': \"Joel D. Hartman, 1 Da(...TRUNCATED)

2024PhRvD.110h3018L

"Neutron stars NSs can capture dark matter DM particles because of their deep gravitational potentia(...TRUNCATED)

2024-10-01T00:00:00Z

"['10.1103/PhysRevD.110.083018', 'arXiv:2408.04425', '10.48550/arXiv.2408.04425', '2024PhRvD.110h301(...TRUNCATED)

['Astrophysics and astroparticle physics', 'Astrophysics - High Energy Astrophysical Phenomena']

Effects from dark matter halos on xray pulsar pulse profiles

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https://arxiv.org/pdf/2408.04425.pdf

"{'E ff ects from Dark Matter Halos on X-ray Pulsar Pulse Profiles': 'Yukun Liu, 1 Hong-Bo Li, 2 Yon(...TRUNCATED)

2024ApJ...975...59M

"We present a new grid of cloudy atmosphere and evolution models for substellar objects. These model(...TRUNCATED)

2024-11-01T00:00:00Z

"['2024arXiv240200758M', '10.3847/1538-4357/ad71d5', 'arXiv:2402.00758', '2024ApJ...975...59M', '10.(...TRUNCATED)

"['Brown dwarfs', 'L dwarfs', 'T dwarfs', 'Exoplanet atmospheres', 'Stellar atmospheres', 'Planetary(...TRUNCATED)

"The Sonora Substellar Atmosphere Models. III. Diamondback Atmospheric Properties Spectra and Evolut(...TRUNCATED)

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https://arxiv.org/pdf/2402.00758.pdf

"{'The Sonora Substellar Atmosphere Models. III. Diamondback: Atmospheric Properties, Spectra, and E(...TRUNCATED)

2024A&ARv..32....2S

"Dustobscured star formation has dominated the cosmic history of star formation since z 4 . However (...TRUNCATED)

2024-04-01T00:00:00Z

"['2024A&ARv..32....2S', 'arXiv:2310.00053', '10.48550/arXiv.2310.00053', '10.1007/s00159-024-00151-(...TRUNCATED)

"['Galaxies: high redshift', 'formation', 'evolution', 'ISM', 'ISM: dust', 'extinction', 'supernova (...TRUNCATED)

The formation and cosmic evolution of dust in the early Universe I. Dust sources

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https://arxiv.org/pdf/2310.00053.pdf

"{'The Astronomy and Astrophysics Review manuscript No.': '(will be inserted by the editor)', 'The f(...TRUNCATED)