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DOE OSTI2026

Impact of in situ nuclear networks and atomic opacities on neutron star merger ejecta dynamics, nucleosynthesis, and kilonovae

Context. Binary neutron star merger (BNSM) ejecta are key sites of rapid neutron capture (r-process) nucleosynthesis and they produce kilonovae powered by the radioactive decay of freshly synthesized nuclei. Modeling their evolution requires multi-physics simulations involving hydrodynamics, nuclear reactions, and radiative processes. The impact of nuclear burning and atomic opacity is poorly understood and often treated with simplified prescriptions. Aims. We systematically investigate different treatments of nuclear heating, particle thermalization, and atomic opacities in radiation-hydrodynamics simulations of BNSM ejecta and kilonova light curves. Methods. Ejecta profiles from long-term numerical-relativity simulations of asymmetric neutron star binaries with a massive neutron star remnant were evolved to ∼30 days using a 2D ray-by-ray approach. We compared simplified heating-rate and thermalization prescriptions with in situ Nuclear reaction Network (NN) calculations that track nuclear energy deposition and include a composition-dependent thermalization scheme. We also contrasted various gray opacity models with a frequency-dependent treatment based on atomic calculations. Results. Coupling NN and hydrodynamics significantly affects nucleosynthesis and kilonova emission. Assuming homologous expansion alters abundance evolution and produces a narrower, less populated second r-process peak and a third peak shifted to higher mass numbers. The back-reaction of nuclear heating affects the temperature evolution enough to delay and redden the early (t∼ hours) kilonova peaks. A constant thermalization efficiency underestimates and reddens the early emission while overestimating the late-time luminosity compared to the composition-dependent treatment. Analytical opacity prescriptions yield a more extended, colder photosphere, resulting in dimmer, redder kilonovae at early times (t≲ hour), while the delayed recession of the photosphere prolongs the red emission at t ≳ 5 days. Conclusions. Coupling hydrodynamics to an in situ NN is crucial for reliable nucleosynthesis and kilonova predictions. Resolving the first several hundred milliseconds of the hydrodynamics is essential for robust nucleosynthesis calculations. Composition-dependent thermalization and frequency-dependent, atomic-physics-based opacities are needed to accurately capture the temperature evolution of the ejecta and the brightness and color evolution of the kilonova. Calibrated analytic nuclear-power fits with simplified thermalization and opacity prescriptions can still reproduce the density and temperature evolution of the ejecta.

74 ATOMIC AND MOLECULAR PHYSICS↗

DOE OSTI2025

2D end-to-end modelling of kilonovae from binary neutron star merger remnants

We investigate the kilonova emission resulting from outflows produced in a 3D general-relativistic magnetohydrodynamic (GRMHD) simulation of a hypermassive neutron star (HMNS) remnant. We map the outflows into the flash hydrodynamics code to model their expansion in axisymmetry, and study the effects of employing different r-process heating rates. Except for the highest heating rate prescription, we find no significant differences with respect to overall ejecta dynamics and morphology compared to the simulation without heating. Once homologous expansion is attained, typically after ~2s for these ejecta, we map the outflows to the sedona radiative transfer code and compute the spectral evolution of the kilonova and broad-band light curves in various Legacy Survey of Space and Time (LSST) bands. The kilonova properties depend on the remnant lifetime, with peak luminosities and peak time-scales increasing for longer lived remnants that produce more massive ejecta. For all models, there is a strong dependence of both the bolometric and broad-band light curves on the viewing angle. While the short-lived (12 ms) remnant produces higher luminosities when viewed from angles closer to the pole, longer lived remnants (240 ms and 2.5 s) are more luminous when viewed from angles closer to the equator. Our results highlight the importance of self-consistent, long-term modelling of merger ejecta, and taking viewing-angle dependence into account when interpreting observed kilonova light curves. We find that magnetized outflows from an HMNS – if it survives long enough – could explain blue kilonovae, such as the blue emission seen in AT2017gfo.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS2021

Optimizing serendipitous detections of kilonovae: cadence and filter selection

The rise of multimessenger astronomy has brought with it the need to exploit all available data streams and learn more about the astrophysical objects that fall within its breadth. One possible avenue is the search for serendipitous optical/near-infrared counterparts of gamma-ray bursts (GRBs) and gravitational-wave (GW) signals, known as kilonovae. With surveys such as the Zwicky Transient Facility (ZTF), which observes the sky with a cadence of ∼3 d, the existing counterpart locations are likely to be observed; however, due to the significant amount of sky to explore, it is difficult to search for these fast-evolving candidates. Thus, it is beneficial to optimize the survey cadence for realtime kilonova identification and enable further photometric and spectroscopic observations. We explore how the cadence of wide field-of-view surveys like ZTF can be improved to facilitate such identifications. We show that with improved observational choices, e.g. the adoption of three epochs per night on a ∼ nightly basis, and the prioritization of redder photometric bands, detection efficiencies improve by about a factor of two relative to the nominal cadence. We also provide realistic hypothetical constraints on the kilonova rate as a form of comparison between strategies, assuming that no kilonovae are detected throughout the long-term execution of the respective observing plan. These results demonstrate how an optimal use of ZTF increases the likelihood of kilonova discovery independent of GWs or GRBs, thereby allowing for a sensitive search with less interruption of its nominal cadence through Target of Opportunity programs.

methods: observational↗

DOE OSTI2026

A Comparison of Three Neodymium Atomic Data Sets for Kilonova Modeling

We examine the impact of input neodymium (Nd) atomic data on the light curves and spectra of kilonovae (KNe), probing the sensitivity of kilonova observables to the atomic physics of this important lanthanide element. We use the SuperNu Monte Carlo radiative transfer code, simulating a simple semianalytic 1D kilonova (KN) with a pure Nd atmosphere, fixing the radiative transfer method while using input atomic data generated by three different codes: the LANL suite of atomic physics codes, HULLAC, and Autostructure. We see that the choice of atomic data significantly shapes the resulting light curves and spectra. Peak bolometric luminosities differ by a ratio of nearly 1.5 between HULLAC/Autostructure and LANL data sets. Moreover, we observe significant near- to mid-IR differences in the structure of the spectra. We specifically attribute these differences to the choice of atomic data for neutral Nd I. Many of the results here have been adapted from a presentation at “Radiative Transfer and Atomic Physics of Kilonovae” in Stockholm, 2023. We additionally present a LANL data set with energies calibrated to available values in the NIST Atomic Spectra Database, and demonstrate that this calibration also significantly affects IR spectral structure at late time. The substantial differences in KN observables that arise from tuning the atomic data of just one lanthanide element highlight the special attention that must be paid to atomic physics uncertainties when modeling KNe, from AT2017gfo to beyond.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2026

Collapse of magnetized white dwarfs as site of heavy-element formation and kilonova signal

We present the first end-to-end calculation connecting the accretion-induced collapse (AIC) of a magnetized, rapidly rotating white dwarf to observable kilonova signatures, combining two-dimensional (2D) general-relativistic neutrino-magnetohydrodynamic simulations, followed by radiation hydrodynamics with in-situ nuclear network and 2D Monte Carlo radiative transfer with spatially resolved heating rates. Unlike all previous unmagnetized AIC models – which predicted proton-rich, $^{56}$Ni-dominated ejecta – strong magnetic fields eject ${\approx }\, 0.2\, \mathrm{ M}_\odot$ of neutron-rich material ($\langle Y_e \rangle \sim 0.24$) on dynamical time-scales, before neutrino irradiation can raise the electron fraction, enabling strong r-process nucleosynthesis up to and beyond the third peak. The resulting kilonova is lanthanide-rich ($X_{\rm lan} \approx 8~{{\ \rm per\ cent}}$) and dominated by near-infrared emission. We compute synthetic light curves in the Large Synoptic Survey Telescope and J ames Webb Space Telescope bands and find striking agreement, without parameter tuning, between the observations of AT 2023vfi/GRB 230307A and our broadband light curves for polar viewing angles. These results establish magnetized AIC as a viable channel for heavy r-process element production and a compelling progenitor candidate for long-duration gamma-ray bursts with kilonova signatures.

MHD↗

DOE OSTI2024

Impact of Systematic Modeling Uncertainties on Kilonova Property Estimation

The precise atomic structure and therefore the wavelength-dependent opacities of lanthanides are highly uncertain. This uncertainty introduces systematic errors in modeling transients like kilonovae and estimating key properties such as mass, characteristic velocity, and heavy metal content. Here, we quantify how atomic data from across the literature as well as choices of thermalization efficiency of r-process radioactive decay heating impact the light curve and spectra of kilonovae. Specifically, we analyze the spectra of a grid of models produced by the radiative transfer code Sedona that span the expected range of kilonova properties to identify regions with the highest systematic uncertainty. Our findings indicate that differences in atomic data have a substantial impact on estimates of lanthanide mass fraction, spanning approximately 1 order of magnitude for lanthanide-rich ejecta, and demonstrate the difficulty in precisely measuring the lanthanide fraction in lanthanide-poor ejecta. Mass estimates vary typically by 25%–40% for differing atomic data. Similarly, the choice of thermalization efficiency can affect mass estimates by 20%–50%. Observational properties such as color and decay rate are highly model dependent. Velocity estimation, when fitting solely based on the light curve, can have a typical error of ∼100%. Atomic data of light r-process elements can strongly affect blue emission. Even for well-observed events like GW170817, the total lanthanide production estimated using different atomic data sets can vary by a factor of ∼6.

Gravitational wave sources↗

NASA NTRS2021

A Tale of Two Mergers: Constraints on Kilonova Detection in Two Short GRBs at z∼ 0.5

We present a detailed multiwavelength analysis of two short gamma-ray bursts (sGRBs) detected by the Neil Gehrels Swift Observatory: GRB 160624A at z = 0.483 and GRB 200522A at z = 0.554. These sGRBs demonstrate very different properties in their observed emission and environment. GRB 160624A is associated with a late-type galaxy with an old stellar population (≈3 Gyr) and moderate ongoing star formation (≈1 M⊙ yr−1). Hubble and Gemini limits on optical/near-infrared emission from GRB 160624A are among the most stringent for sGRBs, leading to tight constraints on the allowed kilonova properties. In particular, we rule out any kilonova brighter than AT2017gfo, disfavouring large masses of wind ejecta (≲0.03 M⊙). In contrast, observations of GRB 200522A uncovered a luminous (LF125W ≈ 1042 erg s−1 at 2.3 d) and red (r − H ≈ 1.3 mag) counterpart. The red colour can be explained either by bright kilonova emission powered by the radioactive decay of a large amount of wind ejecta (0.03 M⊙ ≲ M ≲ 0.1 M⊙) or moderate extinction, E(B − V) ≈ 0.1−0.2 mag, along the line of sight. The location of this sGRB in the inner regions of a young (≈0.1 Gyr) star-forming (≈2−6 M⊙ yr−1) galaxy and the limited sampling of its counterpart do not allow us to rule out dust effects as contributing, at least in part, to the red colour.

B O'Connor↗

DOE OSTI2025

Inferring Neutron Star Merger Ejecta Morphology with Kilonovae

In this study we incorporate a new grid of kilonova simulations produced by the Monte Carlo radiative transfer code SuperNu in an inference pipeline for astrophysical transients and evaluate their performance. These simulations contain four different two-component ejecta morphology classes. We analyze follow-up observational strategies by Vera Rubin Observatory in optical and James Webb Space Telescope (JWST) in mid-infrared (MIR). Our analysis suggests that, within these strategies, it is possible to discriminate between the four different morphologies only when late-time JWST observations in MIR are available. We conclude that follow-ups by the new Vera Rubin Observatory alone are not sufficient to determine ejecta morphology. Additionally, we make comparisons between surrogate models based on radiative transfer simulation grids by SuperNu and POSSIS, by analyzing the historic kilonova AT2017gfo that accompanied the gravitational wave event GW170817. We show that both SuperNu and POSSIS models provide similar fits to photometric observations but their qualitative interpretations differ.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2025

Plasma instabilities dominate radioactive transients magnetic fields: the self-confinement of leptons in Type Ia and core-collapse supernovae, and kilonovae

The light curves of radioactive transients, such as supernovae and kilonovae, are powered by the decay of radioisotopes, which release high-energy leptons through $\beta ^+$ and $\beta ^-$ decays. These leptons deposit energy into the expanding ejecta. As the ejecta density decreases during expansion, the plasma becomes collisionless, with particle motion governed by electromagnetic forces. In such environments, strong or turbulent magnetic fields are thought to confine particles, though the origin of these fields and the confinement mechanism have remained unclear. Using fully kinetic particle-in-cell (PIC) simulations, we demonstrate that plasma instabilities can naturally confine high-energy leptons. These leptons generate magnetic fields through plasma streaming instabilities, even in the absence of pre-existing fields. The self-generated magnetic fields slow lepton diffusion, enabling confinement, and transferring energy to thermal electrons and ions. Our results naturally explain the positron trapping inferred from late-time observations of thermonuclear and core-collapse supernovae. Furthermore, they suggest potential implications for electron dynamics in the ejecta of kilonovae. We also estimate synchrotron radio luminosities from positrons for Type Ia supernovae and find that such emission could only be detectable with next-generation radio observatories from a Galactic or local-group supernova in an environment without any circumstellar material.

instabilities↗

DOE OSTI2025

Long-lived neutron-star remnants from asymmetric binary neutron star mergers: element formation, kilonova signals and gravitational waves

We present 3D general-relativistic neutrino-radiation hydrodynamics simulations of two asymmetric binary neutron star mergers producing long-lived neutron stars remnants and spanning a fraction of their cooling time scale. The mergers are characterized by significant tidal disruption with neutron rich material forming a massive disc around the remnant. The latter develops one-armed dynamics that is imprinted in the emitted kilo-Hertz gravitational waves. Angular momentum transport to the disc is initially driven by spiral-density waves and enhanced by turbulent viscosity and neutrino heating on longer timescales. The mass outflows are composed by neutron-rich dynamical ejecta of mass ∼10 −3 –10 −2 M ⊙ followed by a persistent spiral-wave/neutrino-driven wind of ≳ 10 −2 M ⊙ with material spanning a wide range of electron fractions, ∼0.1–0.55. Dynamical ejecta (winds) have fast velocity tails up to ∼0.8 (∼0.4) c. The outflows are further evolved to days timescale using 2D ray-by-ray radiation-hydrodynamics simulations that include an online nuclear network. We find complete r-process yields and identify the production of 56 Ni and the subsequent decay chain to 56 Co and 56 Fe. Synthetic kilonova light curves predict an extended (near-) infrared peak a few days postmerger originating from r-process in the neutron-rich/high-opacity ejecta and UV/optical peaks at a few hours (ten minutes) postmerger originating from weak r-process (free-neutron decay) in the faster ejecta components. Additionally, the fast tail of tidal origin generates kilonova afterglows potentially detectable in radio and X band on a few to ten years time scale. Quantitative effects originating from the tidal disruption merger dynamics are reflected in the multimessenger emissions.

abundances↗

DOE OSTI2025

Nonthermal Ionization of Kilonova Ejecta: Observable Impacts

The characteristic rapid rise and decline at optical wavelengths of a kilonova is the product of the low ejecta mass (≲0.05 M ⊙ ) and high ejecta velocity (≳0.1c). We show that, even at very early times (≲2 days), regions of ejecta fall below critical density and temperature thresholds at which nonlocal thermodynamic equilibrium (NLTE) effects become important. Here, we present an approximate method for calculating the ionization state of the ejecta that accounts for the NLTE impact of high-energy electrons produced in the beta decay of freshly synthesized r-process elements. We find that incorporating ionization from high-energy electrons produces an “inverted” and “blended” ionization structure, where the most highly ionized species are located in the fastest moving homologous ejecta, and multiple ionization states coexist. In radiation transport calculations, the higher degree of ionization reduces line blanketing in optical bands, leading to improved agreement with the light-curve properties of AT 2017gfo, such as the duration, decay rates, brightness, and colors. Our quasi-NLTE implementation helps to alleviate tensions in kilonova modeling: for high-velocity (∼0.3 c) ejecta components, our models require less mass for a given peak brightness in optical bands, by as much as a factor of 3; our models can explain the presence of observed features associated to Sr II, W III, Se III, and Te III under conditions where LTE models would predict only neutral species; and we naturally predict the coexistence of species like Sr II and Ce III without the need for fine-tuning of the ejecta properties.

High energy astrophysics↗

DOE OSTI2025

Gamma-Ray Bursts and Kilonovae from the Accretion-induced Collapse of White Dwarfs

Abstract We present the first seconds-long 2D general relativistic neutrino magnetohydrodynamic simulations of accretion-induced collapse (AIC) in rapidly rotating, strongly magnetized white dwarfs (WDs), which might originate as remnants of double-WD mergers. This study examines extreme combinations of magnetic fields and rotation rates, motivated both by the need to address the limitations of 2D axisymmetric simulations and to explore the physics of AIC under rare conditions that, while yet to be observationally confirmed, may be consistent with current theoretical models and account for unusual events. Under these assumptions, our results demonstrate that, if realizable, such systems can generate relativistic jets and neutron-rich outflows with properties consistent with long gamma-ray bursts (LGRBs) accompanied by kilonovae, such as GRB 211211A and GRB 230307A. These findings highlight the potential role of AIC in heavy r -process element production and offer a framework for understanding rare LGRBs associated with kilonova emission. Longer-duration 3D simulations are needed to fully capture magnetic field amplification, resolve instabilities, and determine the fate of the energy retained by the magnetar at the end of the simulations.

Cheong 張, Patrick Chi-Kit 志杰 (ORCID:00000003144933↗

NASA NTRS2019

The Afterglow and Kilonova of the Short GRB 160821B

GRB 160821B is a short duration gamma-ray burst (GRB) detected and localized by the Neil Gehrels Swift Observatory in the outskirts of a spiral galaxy at z = 0.1613, at a projected physical offset of 16 kpc from the galaxy’s center. We present X-ray, optical/nIR, and radio observations of its counterpart and model them with two distinct components of emission: a standard afterglow, arising from the interaction of the relativistic jet with the surrounding medium, and a kilonova, powered by the radioactive decay of the sub-relativistic ejecta. Broadband modelling of the afterglow data reveals a weak reverse shock propagating backward into the jet, and a likely jet-break at 3.5 d. This is consistent with a structured jet seen slightly off-axis (θview ∼ θcore) while expanding into a low-density medium (n ≈ 10−3 cm−3). Analysis of the kilonova properties suggests a rapid evolution towards red colours, similar toAT2017gfo, and a low-nIR luminosity, possibly due to the presence of a long-lived neutron star. The global properties of the environment, the inferred low mass (Mej <~ 0.006 Msun) and velocities (vej >~ 0.05c) of lanthanide-rich ejecta are consistent with a binary neutron star merger progenitor.

Gravitational waves↗

DOE OSTI2026

Kilonovae and Long-duration Gamma-Ray Bursts

Recent detections of kilonova-like emission following long-duration gamma-ray bursts GRB 211211A and GRB 230307A have been interpreted as originating from the merger of two neutron stars. In this work, we demonstrate that these observations are also consistent with nucleosynthesis originating from a collapsar scenario. Our model is capable of reproducing the observed optical and infrared light curves using a single, weak r-process component. The absence of lanthanide-rich material in our model, consistent with the data, challenges the prevailing interpretation that a red evolution in such transients necessarily indicates the presence of heavy r-process elements.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2024

Actinide signatures in low electron fraction kilonova ejecta

ABSTRACT Neutron star (NS) mergers are known to produce heavy elements through rapid neutron capture (r-process) nucleosynthesis. Actinides are expected to be created solely by the r-process in the most neutron-rich environments. Confirming if NS mergers provide the requisite conditions for actinide creation is therefore central to determining their origin in the Universe. Actinide signatures in kilonova (KN) spectra may yield an answer, provided adequate models are available in order to interpret observational data. In this study, we investigate actinide signatures in neutron-rich merger ejecta. We use three ejecta models with different compositions and radioactive power, generated by nucleosynthesis calculations using the same initial electron fraction ($Y_e = 0.15$) but with different nuclear physics inputs and thermodynamic expansion history. These are evolved from 10 to 100 d after merger using the sumo non-local thermodynamic equilibrium (NLTE) radiative transfer code. We highlight how uncertainties in nuclear properties, as well as choices in thermodynamic trajectory, may yield entirely different outputs for equal values of $Y_e$. We consider an actinide-free model and two actinide-rich models, and find that the emergent spectra and light-curve evolution are significantly different depending on the amount of actinides present, and the overall decay properties of the models. We also present potential key actinide spectral signatures, of which doubly ionized $_{89}$Ac and $_{90}$Th may be particularly interesting as spectral indicators of actinide presence in KN ejecta.

Pognan, Quentin (ORCID:0000000152550782)↗

DOE OSTI2026

Exploring composition mixing in kilonova ejecta with ray-by-ray simulations

Binary neutron star merger (BNSM) ejecta are considered a primary repository of r-process nucleosynthesis and a source of the observed heavy-element abundances. We implement composition mixing into ray-by-ray radiation-hydrodynamic simulations of BNSM ejecta, coupled with an online nuclear network (NN). We model mixing via a gradient-based mixing approximation that evolves simultaneously with the hydrodynamics. Here, we find that mixing occurs in regions where the electron fraction changes rapidly. While mixing smooths composition gradients in transition regions, it has a negligible impact on the heavy-element yields. This is because the primary r-process site (the equatorial ejecta) is initially homogeneous in free neutrons, leaving no strong gradients for mixing to act upon. In each angular ray, the abundances of the most produced elements are robust under mixing, while the less abundant ones are more affected. The total global abundances change only slightly from mixing, since each angular ray contributes its most abundant elements. Furthermore, the predicted kilonova light curves show only minor reddening, with differences below the detectability of state-of-the-art telescopes. In general, we do not observe significant effects from mixing in the time span of the r-process. Consequently, mixing only leads to minor variations in abundances and light curves in ray-by-ray simulations.

Explosive burning↗

DOE OSTI2025

A Unified Model of Kilonovae and Gamma-Ray Bursts in Binary Mergers Establishes Neutron Stars as the Central Engines of Short GRBs

We expand the theoretical framework by O. Gottlieb et al., which connects binary merger populations with long and short binary gamma-ray bursts (lbGRBs and sbGRBs, respectively), incorporating kilonovae (KNe) as a key diagnostic tool. We show that lbGRBs, powered by massive accretion disks around black holes (BHs), should be accompanied by bright, red KNe. In contrast, sbGRBs—if also powered by BHs—would produce fainter, red KNe, potentially biasing against their detection. However, magnetized hypermassive neutron star (HMNS) remnants that precede BH formation can produce jets with power (P NS ≈ 10 51 erg s –1 ) and Lorentz factor (Γ > 10) likely compatible with sbGRB observations, and would result in distinctly bluer KNe, offering a pathway to identifying the sbGRB central engine. Recent modeling by J. C. Rastinejad et al. found luminous red KNe consistently accompany lbGRBs, supporting their origin in BH-massive disk systems, likely following a short-lived HMNS phase. The preferential association of sbGRBs with comparably luminous KNe argues against the BH engine hypothesis for sbGRBs, while the bluer hue of these KNe provides additional support for an HMNS-driven mechanism. Within this framework, BH–NS mergers likely contribute exclusively to the lbGRB population with red KNe. Our findings suggest that GW170817 may, in fact, have been an lbGRB to on-axis observers. Finally, we discuss major challenges faced by alternative lbGRB progenitor models, such as white dwarf–NS or white dwarf–BH mergers and accretion-induced collapse forming magnetars, which fail to align with observed GRB timescales, energies, and KN properties.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2025

Kilonova Emissions from Neutron Star Merger Remnants: Implications for the Nuclear Equation of State

Multimessenger observations of binary neutron star mergers can provide valuable information on the nuclear equation of state (EOS). Here, we investigate the extent to which electromagnetic observations of the associated kilonovae allow us to place constraints on the EOS. For this, we use state-of-the-art three-dimensional general-relativistic magnetohydrodynamics simulations and detailed nucleosynthesis modeling to connect properties of observed light curves to properties of the accretion disk, and hence, the EOS. Using our general approach, we use multimessenger observations of GW170817/AT2017gfo to study the impact of various sources of uncertainty on inferences of the EOS. We constrain the radius of a 1.4M ⊙ neutron star to lie within 10.30 ≤ R 1.4 ≤ 13.0 km and the maximum mass to be M TOV ≤ 3.06M ⊙ .

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗