MNRAS 000, 1–6 (2020) Preprint 7 August 2020 Compiled using MNRAS LATEX style file v3.0 Black Dwarf Supernova in the Far Future M. E. Caplan,1⋆ 1Department of Physics, Illinois State University, Normal, IL 61761 USA Accepted XXX. Received YYY; in original form ZZZ ABSTRACT In the far future long after star formation has ceased the universe will be populated by sparse degenerate remnants, mostly white dwarfs, though their ultimate fate is an open question. These white dwarfs will cool and freeze solid into black dwarfs while pycnonuclear fusion will slowly process their composition to iron-56. However, due to the declining electron fraction the Chandrasekhar limit of these stars will be decreasing and will eventually be below that of the most massive black dwarfs. As such, isolated dwarf stars with masses greater than 1.2M will collapse in the far future due to the slow accumulation of iron-56 in their cores. If∼ proton⊙ decay does not occur then this is the ultimate fate of about 1021 stars, approximately one percent of all stars in the observable universe. We present calculations of the internal structure of black dwarfswith iron cores as a model for progenitors. From pycnonuclear fusion rates we estimate their lifetime and thus delay time to be 101100 years. We speculate that high mass black dwarf supernovae resemble accretion induced collapse of O/Ne/Mg white dwarfs while later low mass transients will be similar to stripped-envelope core-collapse supernova, and may be the last interesting astrophysical transients to occur prior to heat death. Key words: dense matter – supernovae: general – white dwarfs – cosmology: miscellaneous 1 INTRODUCTION that a large amount of baryonic matter will be found in degenerate remnants. While it is now expected that all stars will evolve toward degenerate Low mass stars, comprising the bulk of all stars today, will be remnants (such as neutron stars (NS) or white dwarfs (WD)) or black abundant among these remnants having evolved to WDs. van Horn holes, the fate of these objects in the far future is an open question. (1968) argues that as WDs cool their cores freeze solid and re- Observations of the accelerating expansion and ΛCDM now sug- lease latent heat which has recently been observationally confirmed gest that our universe will expand forever, becoming increasingly with Gaia (Tremblay et al. 2019). On cosmological timescales WDs dark energy dominated while the temperature and matter density without a heat source will fully crystallize and cool to equilibrium asymptotically approach zero. In such a future it is expected that with the cosmic background radiation, at sub-Kelvin temperatures the universe will exhaust all gas for star formation and that almost in the far future, though there may be a period of heating due to halo all stars will become degenerate WDs when they exhaust their fuel dark matter annihilation of 1025 up to 1047 years depending on the over the next 1014 years (Dyson 1979; Adams & Laughlin 1997). arXiv:2008.02296v1 [astro-ph.HE] 5 Aug 2020 exact dark matter candidate considered (Adams & Laughlin 1997). The ultimate fate of these 1023 WDs is an open question.1 ∼ Any future evolution of these near absolute zero ‘black dwarfs’ then Dyson (1979) and Adams & Laughlin (1997) both describe a far fu- depends on the stability of the proton. ture in which all galaxies evaporate through gravitational scattering If the proton decays then Adams & Laughlin (1997) expect which ejects most objects, while occasional collisions or encounters these black dwarfs to decay on timescales of 1032 to 1049 years. In with black holes destroy a minority of remnants. Those in binaries the alternative scenario, where the proton is stable, we can expect merge due to gravitational wave radiation if close, but are more black dwarfs to become iron rich in the far future. Pycnonuclear likely disrupted by encounters during the period of scattering that fusion reactions, driven by quantum tunneling of adjacent nuclei evaporates galaxies. Therefore, virtually all surviving degenerate on the crystal lattice within black dwarfs, will tend to process the remnants are eventually isolated on sufficiently long timescales (ap- matter toward iron-56 (which is possibly the ground state of baryonic proximately 1020 years). In the far future it therefore seems likely matter (Page 1992)). Dyson (1979) estimates a tunneling timescale of ⋆ E-mail: [email protected] 1 Assuming order 1012 galaxies and 1011 stars per galaxy in the observable S universe. T = e T0 (1) © 2020 The Authors 2 M. E. Caplan et al. 2 25 where T is a characteristic nuclear timescale (~ mpc 10 s) 2 THE MAXIMUM MASS OF ELECTRON 0 / ≈ − and S is an action integral approximated for fusion by DEGENERATE STARS The Chandrasekhar mass can be calculated easily from the ultra- S A1 2 Z5 6 relativistic equation of state, P ρ4 3. Integrating the structure 30 / / (2) ∝ / ≈ profileofaWD(i.e. its density radially outward from the core) yields with A and Z the mass and charge of the fusion product. Dyson its mass (Chandrasekhar 1931, 1935; Maoz 2016; Koonin 2018). (1979) obtains these expressions by approximating the action For this equation of state one finds that there is a fixed mass for 2 2 1 2 S 8MUd ~ / using mean barrier height U and width d all WDs, regardless of the core density (i.e. the boundary condition for≈ a tunneling( / particle) of mass M. This barrier is then taken to be of integration). This equation of state will yield arbitrarily small the Coulomb barrier screened over a distance d = Z 1 3 ~ me2 radii for ever increasing core densities, and the convergence to zero − / ( / ) with a reduced mass M = Amp 4 for two A 2 and Z 2 nuclei. radius at infinite central density is clearly shown by Chandrasekhar This treatment excludes the sensitive/ density dependence/ / for py- (1935). This of course is not physical and other physics must set an cnonuclear fusion, which occurs most quickly at higher densities upper limit on the central density. Schramm & Koonin (1990); Afanasjev et al. (2012); Meisel et al. As a practical matter there is an upper limit on possible core (2018). For silicon-28 nuclei fusing to iron group elements, Dyson’s pressures and densities which is set by the condition for electron scheme gives an approximate 101500 year timescale for tunneling to capture onto iron-56 nuclei (Cardall 2008; Warren et al. 2019). This process black dwarfs to iron-56. capture produces manganese-56 (Z = 25) and a neutrino and due WDs (and black dwarfs) are electron degenerate which sup- to odd-even staggering this capture is almost immediately followed ports matter up to a finite mass limit, the Chandrasekhar mass, given by another electron capture to chromium-56 (Z = 24) whenever it by (Chandrasekhar 1931, 1935; Maoz 2016): occurs. So while the ultrarelativistic equation of state gives an upper limit on the mass of a degenerate electron gas of fixed Ye, reactions exist in the cores of degenerate remnants which, if the threshold is 2 MCh 1.44 2Ye M . (3) met, further reduce Y thereby triggering collapse. These reactions ≈ ( ) ⊙ e give us a maximum realizable mass of a cold and isolated degener- Y = For a canonical WD ( e 0.5) we obtain the known Chandrasekhar ate remnant which is slightly below the Chandrasekhar mass. This mass limit. If a WD exceeds the Chandrasekhar mass it will collapse condition is one of several possible triggers for a supernova pro- (Howell 2011; Maoz 2016). genitor today, though given the low temperatures considered here it As a white dwarf evolves toward a black dwarf, and eventually is our only relevant mechanism (Bethe et al. 1979; Nomoto 1984b; an iron black dwarf, its equation of state is always that of a rela- Langanke & Martínez-Pinedo 2014; Warren et al. 2019). tivistic electron gas. However, in the far future when light nuclides The threshold electron Fermi energy needed to spontaneously are converted to iron-56 the electron fraction of the core will have drive electron capture can be determined by following Bahcall Y = = decreased to e 26 56 0.464. Thus, by Eq. 3 the maximum (1964): mass will be smaller by/ a factor 0.464 0.5 2 0.86. Therefore, the ( / ) ≈ effective upper mass limit of these degenerates remnants is decreas- 2 e 2 2 ing with time down toward about 1.2M as they become increas- m Z, A c + ǫ( ) = m Z 1, A c + mec (4) ( ) F ( − ) ingly iron rich. Indeed, it is long known⊙ in the supernova literature m Z A Z A e that iron cores above 1.12 M will collapse Baron & Cooperstein where , is the mass of the nuclide with , , ǫF( ) is the ⊙ ( ) ( ) (1990). As black dwarfs above this mass limit accumulate iron in electron Fermi energy, and me is the electron mass. This is a typi- their cores they will necessarily exceed their Chandrasekhar mass cal criteria for finding capture layers in accreted neutron star crusts precisely because the Chandrasekhar mass is decreasing with Ye. As (Haensel & Zdunik 1990a,b; Fantina et al. 2018). For iron-56 and e = a result, one may expect catastrophic collapse of these black dwarfs manganese-56 we find ǫF( ) 4.207 MeV. This Fermi energy corre- in the far future, powering transients similar to supernova today. sponds to pressures of P = 5.57 1026 dyn/cm2, or equivalently a In this work we consider this fate for the most massive isolated central mass density of 1.19 10×9 g/cm3. Thus, iron black dwarfs × white dwarfs, between approximately 1.2 and 1.4 solar masses. with core densities exceeding this will collapse.