Can Low Level Nuclear Reactions, Or Dark matter Annihilation, Heat Massive White Dwarf Stars (WD)?

Can low level nuclear reactions, or dark matter annihilation, heat massive white dwarf stars (WD)? Recently Cheng and colleagues identified a number of WD, with masses between 1.08 and 1.23M, that appear to have an additional heat source. This extra heat may maintain the star’s luminosity near ≈ 10-³L for multiple billion years. Latent heat from crystallization and gravitational energy released from conventional ²²Ne sedimentation do not appear to be large enough to explain this luminosity. Note that ²²Ne sedimentation is significantly slowed down by C/O crystallization however Blouin and colleagues speculate that Ne phase separation could enhance the heating from conventional Ne sedimentation.

Now, in this study, Horowitz and colleagues, explored heating from electron capture and pycnonuclear reactions.

They assume isolated stars that are not in binary systems. They are interested in reactions that may take place preferentially at the very high central densities of massive WD and may be less important at lower densities in less massive stars. In principle, even relatively slow nuclear reactions could contribute noticeable heat. This is because, in the absence of nuclear reactions, there is only a modest luminosity from WD cooling. Alternatively, dark matter annihilation in massive WD could produce additional heating. Dark matter can produce noticeable heating even when the dark matter is made of particles with properties, such as scattering cross sections and masses that may be difficult to observe in laboratory experiments. Furthermore, massive WD have large escape velocities. These stars may trap lower mass, higher velocity, dark matter particles that can escape less massive stars.

Finally, dark matter could collect in massive WD. If this dark matter concentrates to very high densities, its gravity can modify the structure of a WD and increase the star’s central density. This in turn could further increase the rate of electron capture and or pycnonuclear fusion reactions.

The central density ρC of massive WD follows from hydrostatic equilibrium and an equation of state dominated by relativistic electrons. In Fig. 1 Horowitz plot ρC of a WD with electron fraction Ye = 0.5, this could be made of C and O or O and Ne. They also showed ρC for a possible Fe WD with Ye ≈ 0.464. They assume a simple relativistic free Fermi gas equation of state and neglect Coulomb corrections. The central density of a C/O WD can exceed 10^9 g/cm³ for star masses above 1.35M.

FIG. 1. Central density of carbon / oxygen (solid black line) and Fe (dashed red line) white dwarfs versus mass. © Horowitz et al.

High densities can drive electron capture reactions. In Table I they list the threshold densities ρT for a variety of electron capture reactions. This density is where the electron Fermi energy is high enough to provide for the reaction Q value. They calculate ρT from atomic masses. In general, the threshold density is seen to decrease as the mass number increases. For C/O or O/Ne stars, electron capture (at zero temperature) is not expected until ρC > ρT ≈ 6×10^9 g/cm³ and this density is not reached until the mass of the star is above 1.40M, see Table I.

The threshold densities in Table I are for ground state to ground state transitions. These transitions may be forbidden by the high spin of the daughter nucleus. However, a large forbidden matrix element was recently observed for the transition corresponding to electron capture from the (0+) 20Ne ground state to the (2+) 20F ground state. If the reaction must proceed via an excited state of the daughter nucleus, to obtain a significant rate, ρT will be even higher than the value in Table I.

TABLE I. Threshold density ρT for the indicated nuclear reaction in a star of electron fraction Ye. The final column is the mass M of a WD with central density equal to ρT . ©Horowitz et al.

Cheng and colleagues considered C/O or O/Ne WD with masses between 1.08 and 1.23M. They inferred these masses by comparing the stars absolute magnitudes and colors to WD cooling models. Note that the absolute magnitudes were determined by recent Gaia parallax measurements. The central density of a 1.23M WD (assuming Ye = 0.5) is only 1.9 × 10^8 g/cm³. This is too low for electron capture on ¹²C, 16^O, or 20^Ne, see Table I. Therefore conventional electron capture reactions are likely not significantly heating the stars Cheng considers.

It may be possible to form WD with Fe cores where the electron fraction is Ye ≈26/56=0.464. For example a failed SN could leave behind an Fe core. Not only
do these stars have higher central densities, see Fig. 1, but the threshold density for electron capture on Fe is also lower ≈ 1.1 × 10^9 g/cm³, see Table I. This density is reached in a 1.16M Fe WD. Furthermore, impurities could have even lower threshold densities. For example, 54^Fe has a sizable isotopic abundance on Earth ≈ 6% and a very low threshold density for electron capture, see Table I. They concluded that electron capture could very well be significant in massive Fe WD.

In addition to electron capture, pycnonuclear, or density driven, fusion reactions can also take place. In pycnonuclear fusion, quantum zero point motion allows two nuclei to approach and tunnel through the coulomb barrier. The pycnonuclear fusion reaction that occurs first, at the lowest density, is likely to be ¹²C + ¹²C. This is because heavier nuclei, in general, will need to tunnel through larger coulomb barriers. Pycnonuclear reactions are greatly aided by the strong screening of the coulomb barrier by other nearby ions. At present there are significant uncertainties in pycnonuclear reaction rates because they depend very sensitively on the exact distribution of ions within the crystal lattice. In addition, there is some uncertainty in the nuclear S factor at very low energies.

Nevertheless, there are useful estimates of pycnonuclear rates either in the pure pycnonuclear regime near zero temperature or in the thermally enhanced pycnonuclear regime at somewhat higher temperatures. In order to have a luminosity near 10-³ L from ¹²C+¹²C fusion Horowitz and colleagues estimated needing a reaction rate of roughly R ≈ 5 × 10¹¹ cm³ ss-¹. Using the rate shown in the insert to Fig. 2, this requires a density of very roughly ρT ≈ 3 × 10^9g/cm³ as listed in Table I. This density is an order of magnitude larger than the 2 × 10^8 g/cm³ central density of a 1.25M (C/O) WD. Although there is considerable uncertainty in the pycnonuclear rate, it is unlikely the uncertainty is this large. Furthermore, the pycnonuclear rate depends strongly on the density. Therefore, if pycnonuclear fusion were to provide 10-³L for a 1.25M star, the luminosity would likely be very much smaller for even slightly smaller stars and very much larger for even slightly more massive stars. They concluded that pycnonuclear fusion is unlikely to provide significant heating for many of the massive WD that Cheng et al. considers.

FIG. 2. Density profile of an originally 1.25 M WD vs radius r. The dashed blue line has no dark matter while the red lines assume Md = 10^-6M/and the black line 10^-4M. The density of the dark matter core is 10^15 (dotted), 10^17 (dashed) and 10^19 g/cm³ (solid). Note the log scales. ©Horowitz et al.

They also explored heating from
dark matter annihilation and found that WD appear to be too small to capture enough dark matter for this to be important. Finally, if dark matter condenses to very high densities inside a WD then this will also increase the density of conventional matter and could start pycnonuclear or electron capture reactions. What happens next may depend on the dynamical scenario. One possibility is the ignition of a Type Ia supernova and the complete destruction of the star. Another possibility, if the high density region is very small indeed, is that the tiny amount of material in this region is burned to Fe without releasing enough heat to start material burning at the lower densities outside the small dark matter core. In this case the dark matter may become encased in a more or less inert Fe core with little overall change to the star. In neither case would there be a modest amount of heat for billions of years.

References: (1) Sihao Cheng, Jeffrey D. Cummings and Brice Ménard, “A Cooling Anomaly of High-mass White Dwarfs”, The Astrophysical Journal 886 (2) 100 ( 2019) DOI: 10.3847/1538-4357/ab4989 link: (2) S. Horowitz, “Nuclear and dark matter heating in massive white dwarf stars”, ArXiv, 2020.

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