## Physica A: Statistical Mechanics and its Applications

Volume 489, 1 January 2018, Pages 128-140

# White dwarf stars exceeding the Chandrasekhar mass limit

## Highlights

A nearly degenerate electron plasma pervading an ionized high-density background medium is studied, as it occurs in stellar matter.

The Dirac equation coupled to the permeability tensor of the medium leads to nonlinear electron dispersion in the ultra-relativistic regime.

The quantized spectral density of a low-temperature electron gas in a dispersive medium is shown to be mechanically and thermally stable.

The nonlinear electron dispersion affects the mass–radius relation of white dwarfs, whose mass can surpass the Chandrasekhar limit.

White dwarf progenitors of super-Chandrasekhar mass Type Ia supernovae: estimates of their central mass density, incompressibility and speed of sound.

## Keywords

Nearly degenerate ultra-relativistic electron plasma
Quantum densities with power-law dispersion and Weibull spectral decay
Dirac equation coupled to a permeability tensor
Mechanical and thermal stability of a dispersive Fermi gas at low temperature
Mass–radius relation of high-mass white dwarfs
Progenitors of super-Chandrasekhar mass thermonuclear supernovae

## 1. Introduction

The purpose of this paper is to study the effects of nonlinear electron dispersion in high-mass white dwarfs. We derive a stable mass–radius relation which remains valid above the Chandrasekhar mass limit of 1.44 solar masses due to nonlinear electron dispersion at ultra-relativistic energies. To this end, we determine the impact of dispersion on the thermodynamic variables of a nearly degenerate (high-density low-temperature) electron plasma. The quantized spectral density of the ultra-relativistic electrons is obtained by coupling the Dirac equation to the permeability tensor generated by the ionized stellar matter. The electronic dispersion relation defined by the permeabilities admits a power-law form in the ultra-relativistic regime, whose amplitude and scaling exponent can empirically be determined from mass and radius measurements of high-mass white dwarfs.

We calculate the entropy variable subject to electron dispersion and demonstrate thermodynamic stability, that is positive heat capacities and compressibilities. We derive the thermal equation of state of the electron plasma, which is polytropic in the totally degenerate ultra-relativistic regime, and then use the Lane–Emden equation for polytropes to derive the mass–radius relation for high-mass white dwarfs.

This mass–radius relation crucially depends on the amplitude and the scaling exponent of the electronic dispersion relation. In the case of vacuum permeabilities, the ultra-relativistic dispersion relation is linear, resulting in a mass limit instead of a mass–radius relation. In contrast, we treat the ionized stellar matter as a permeable medium pervaded by the electron gas, and infer the permeabilities from mass and radius estimates of high-mass white dwarfs. In this way, we can specify all parameters in the dispersive ultra-relativistic mass–radius relation.

The mass ejecta of several Type Ia (thermonuclear) supernovae (e.g., SN 2013cv [1], SN 2003fg [2], SN 2007if [3] and SN 2009dc [4]) substantially exceed the mass limit of 1.44 ${M}_{\odot }$ and suggest super-Chandrasekhar mass progenitors. Using estimates of the ejecta mass and applying the dispersive mass–radius relation, we derive radius and density estimates of their white dwarf progenitors. The electron dispersion is treated as a genuine nonlinear effect rather than a small perturbation of the linear vacuum dispersion relation, given that the mass of the white dwarf progenitor of supernova SN 2009dc exceeds the Chandrasekhar mass limit of 1.44 ${M}_{\mathrm{\odot }}$ by almost a factor of two. The central density of the SN 2009dc progenitor reaches the neutron drip density, so that white dwarf masses much higher than 2.8 ${M}_{\odot }$ are not attainable.

In the following, we give an outline of this paper. In Section 2, we study relativistic fermionic spectral densities in a dispersive medium. The dispersion relation is derived from the Dirac equation coupled to an isotropic energy-dependent permeability tensor in analogy to electromagnetic theory. This changes the linear ultra-relativistic vacuum relation $E\sim p$ into a power law, $\text{◂∼▸}E\left(p\right)\sim \text{◂+▸}m\text{◂◽˙▸}{\left(p∕m\right)}^{\eta }∕\left(\text{◂⋅▸}{\epsilon }_{0}{\mu }_{0}\right)$, where ${\epsilon }_{0}$ and ${\mu }_{0}$ are permeability amplitudes, $m$ denotes the electron mass and $\eta$ is a positive scaling exponent. The spectral decay of the Fermi distribution $\text{◂...▸}\text{d}⁣\text{◂+▸}\rho \left(p\right)\propto {p}^{2}⁣\text{d}⁣\text{◂+▸}p∕\left(\text{◂◽˙▸}{\text{e}}^{\alpha +\text{◂⋅▸}\beta E\left(p\right)}+1\right)$ (where $\alpha$ and $\beta$ are fugacity and temperature parameters defining the chemical potential $\mu =-\alpha ∕\beta$) is of Weibull type [5], the decay factor $exp\left(-\text{◂⋅▸}\beta E\left(p\to \infty \right)\right)$ being a stretched () or compressed ($\eta >1$) exponential.

Weibull exponentials have been extensively applied in statistical modeling. A stretched (subexponential) Weibull factor appears as Kohlrausch function in anomalous diffusion and relaxation processes [6,7]; recent examples include diffusion in magnetic resonance imaging [8], relaxation of nanoparticles in liquids [9] and diminution processes in viscous media [10]. The tensile fracture probability of fiber bundles is modeled as subexponential Weibull density in Refs. [11,12]. Astrophysical applications of subexponential Weibull factors include asteroid fragmentation statistics  [13], population decay statistics of satellite ejecta [14], cosmic ray statistics [15–18], and velocity distributions of planetary surface winds  [19–21]. The population growth and epidemic models in Refs. [22,23] exemplify interdisciplinary applications of sub- and super-exponential (compressed) Weibull factors. Densities interpolating between Weibull exponentials and power laws have been used to model wealth distributions  [24] and stock market volatility [25,26] as well as interevent times in human dynamics [27]. Network applications of Weibull densities are discussed in Refs. [28–31].

The Weibull decay of the above stated Fermi distribution is sub- or super-exponential for dispersion exponents and $\eta >1$, respectively, which affects the fugacity expansions discussed in Section 3, where we study the nearly degenerate ultra-relativistic quantum regime, subject to nonlinear electron dispersion. Starting with the integral representations of the thermodynamic variables derived in Section 2, we perform a high-density low-temperature fugacity expansion, obtaining the two leading orders of the electronic number count, internal energy, pressure and entropy in $\left(\alpha ,\beta ,V\right)$ parametrization.

In Section 4, we discuss the effect of the nonlinear dispersion relation on the mechanical and thermal stability of the electron gas in the nearly degenerate regime. By switching to the $\left(N,\beta ,V\right)$ representation of the energy, pressure and entropy variables, we derive the thermal equation of state, the isochoric and isobaric heat capacities $\text{◂◽.▸}{C}_{V,P}$ and the isobaric expansion coefficient, as well as the isothermal and adiabatic compressibilities $\text{◂◽.▸}{\kappa }_{T,S}$. We demonstrate, by explicit calculation, that the equilibrium stability conditions $\text{◂>⋯▸}{\kappa }_{T}>{\kappa }_{S}>0$ and $\text{◂>⋯▸}{C}_{P}>{C}_{V}>0$ are satisfied for positive scaling exponents $\eta$ in the dispersion relation. We also obtain fugacity expansions of the adiabatic bulk modulus, the compression modulus (adiabatic incompressibility) and the speed of sound in the ionized background medium.

In Section 5, we study the effect of electron dispersion on the mass–radius relation of high-mass white dwarfs. We employ the thermal equation of state in the totally degenerate ultra-relativistic regime, $P\propto \text{◂◽˙▸}{\left(N∕V\right)}^{1+\eta ∕3}$, where $\eta$ is the scaling exponent of the electronic dispersion relation $E\sim \text{◂+▸}m\text{◂◽˙▸}{\left(p∕m\right)}^{\eta }∕\left(\text{◂⋅▸}{\epsilon }_{0}{\mu }_{0}\right)$. As the thermal equation has a polytropic form, the stellar structure equations can be reduced to the Lane–Emden equation, which admits stable solutions for scaling exponents $\eta >1$ and allows us to derive an explicit mass–radius relation for high-mass white dwarfs. This dispersive mass–radius relation depends on the scaling exponent $\eta$ and the product $\text{◂⋅▸}{\epsilon }_{0}{\mu }_{0}$ of the permeability amplitudes in the electronic dispersion relation. These are two additional (fitting) parameters as compared with the linear ultra-relativistic dispersion relation $E\sim p$ in vacuum ($\text{◂⋅▸}{\epsilon }_{0}{\mu }_{0}=\eta =1$) which gives a mass limit instead of a mass–radius relation. The Lane–Emden equation does not define a mass limit for dispersion exponents $\eta >1$, which opens the possibility of super-Chandrasekhar mass white dwarfs discussed in Section 6.

In Section 6.1, we use the mass–radius data of two high-mass white dwarfs, Sirius B [32] and LHS 4033 [33], and the dispersive mass–radius relation derived in Section 5 to infer the scaling exponent $\eta =1.240$ and the amplitude $\text{◂,▸}\text{◂⋅▸}{\epsilon }_{0}{\mu }_{0}=4.85∕\text{◂◽:▸}{\mu }_{\text{n}}^{1+\eta ∕3}$ of the electronic dispersion relation. ${\mu }_{\text{n}}$ denotes the molecular weight per electron (nucleon–electron ratio, approximately ${\mu }_{\text{n}}\approx 2$ for white dwarfs, unless neutronization by electron capture sets in, which increases ${\mu }_{\text{n}}$, cf. Section 6.3). The scaling exponent $\eta =1.240$ is safely in the stability domain $\eta >1$ of the Lane–Emden equation. In Section 6.1, we also derive the radii and central mass densities of the white dwarf progenitors of the super-Chandrasekhar mass supernovae SN 2013cv, SN 2003fg, SN 2007if and SN 2009dc. Estimates of the central Fermi momentum and Fermi temperature of the progenitor stars are given in Section 6.2, and their bulk and compression moduli, sound velocity and gravitational surface potential are calculated in Section 6.3. In Section 7, we present our conclusions.

## 2. Fermionic spectral densities in a dispersive medium

### 2.1. Nonlinear ultra-relativistic dispersion relation

We start with the electronic Dirac equation coupled to a permeability tensor  [34–36], and consider plane wave solutions, $\psi =exp\left(\text{i}{p}_{\mu }{x}^{\mu }\right)u\left(p\right)$, $\text{◂=▸}{p}_{\mu }=\left(-E,\mathbf{p}\right)$. The spinor $u\left(p\right)$ satisfies the Dirac equation in momentum space coupled to a dispersive isotropic permeability tensor $\text{◂◽˙▸}{g}^{\mu \nu }\left(p\right),$

(2.1)$\text{◂,▸}\text{◂⋅▸}\left(\text{i}⁣\text{◂⋅▸}{\gamma }_{\mu }\text{◂◽˙▸}{g}^{\mu \nu }{p}_{\nu }+m\right)u\left(p\right)=0,\phantom{\rule{10.00002pt}{0ex}}\text{◂=▸}{g}^{00}=-\epsilon \left(p\right),\phantom{\rule{10.00002pt}{0ex}}\text{◂◽˙▸}{g}^{ik}=\text{◂/▸}\frac{\text{◂◽˙▸}{\delta }^{ik}}{\mu \left(p\right)},$
with vanishing flanks $\text{◂◽˙▸}{g}^{0i}=0$. The sign convention for the Minkowski metric is $\text{◂...▸}\text{◂◽.▸}{\eta }_{\mu \nu }=⁣\text{diag}⁣\left(\text{◂,▸}-1,1,1,1\right)$, the Dirac matrices satisfy $\text{◂=▸}\text{◂⋅▸}{\gamma }_{\mu }{\gamma }_{\nu }+\text{◂⋅▸}{\gamma }_{\nu }{\gamma }_{\mu }=2\text{◂◽.▸}{\eta }_{\mu \nu }$, and $m$ is the electron mass. By squaring the Dirac equation and using the plane-wave ansatz as stated above, we find the Klein–Gordon equation coupled to the squared permeability tensor $\text{◂=▸}\text{◂◽˙▸}{h}^{\mu \nu }=\text{◂◽˙▸}{g}^{\mu \alpha }\text{◂◽.▸}{\eta }_{\alpha \beta }\text{◂◽˙▸}{g}^{\beta \nu }$:
(2.2)$\text{◂,▸}\text{◂⋅▸}\left(\text{◂⋅▸}\text{◂◽˙▸}{h}^{\mu \nu }{p}_{\mu }{p}_{\nu }+{m}^{2}\right)u\left(p\right)=0,\phantom{\rule{10.00002pt}{0ex}}\text{◂=▸}{h}^{00}=-{\epsilon }^{2},\phantom{\rule{10.00002pt}{0ex}}\text{◂=▸}\text{◂◽˙▸}{h}^{ik}=\frac{\text{◂◽˙▸}{\delta }^{ik}}{{\mu }^{2}},$
and $\text{◂◽˙▸}{h}^{0i}=0$, which defines the electronic dispersion relation
(2.3)$\text{◂=▸}E\left(p\right)=\frac{\text{◂√▸}\sqrt{{p}^{2}+\text{◂⋅▸}{\mu }^{2}\left(p\right){m}^{2}}}{\text{◂⋅▸}\epsilon \left(p\right)\mu \left(p\right)}.$
($\hslash =c=1$.) We will focus on the ultra-relativistic limit, $p∕m\gg 1$, and assume power-law asymptotics of the permeabilities, $\text{◂∼▸}\epsilon \sim {\epsilon }_{0}\text{◂◽˙▸}{\left(p∕m\right)}^{\chi }$, $\text{◂∼▸}\mu \sim {\mu }_{0}\text{◂◽˙▸}{\left(p∕m\right)}^{\phi }$, with positive dimensionless amplitudes $\text{◂,▸}{\epsilon }_{0},{\mu }_{0}$ and real exponents $\chi$ and $\phi$. For exponents , the ultra-relativistic limit of the dispersion relation (2.3) reads
(2.4)$\text{◂,▸}\text{◂=▸}E=\frac{m}{\text{◂⋅▸}{\epsilon }_{0}{\mu }_{0}}\text{◂◽˙▸}{\left(p∕m\right)}^{\eta },\phantom{\rule{10.00002pt}{0ex}}\eta =1-\chi -\phi ,$
since the mass term $\text{◂⋅▸}{\mu }^{2}\left(p\right){m}^{2}$ in (2.3) drops out in leading order. (The electron mass in (2.4) is just a convenient scale parameter.) The group velocity $\eta E∕p$ can approach zero () or become superluminal ($\eta >1$) in the ultra-relativistic limit; the electromagnetic counterpart is ‘slow’ or ‘fast’ light in highly dispersive media [37–39]. If , the group velocity is negative, pointing in the opposite direction of the energy transfer [40]. We will use the shortcut $E=\text{◂+▸}{p}^{\eta }∕{a}^{\eta }$, with amplitude ${a}^{\eta }=\text{◂⋅▸}\text{◂◽˙▸}{m}^{\eta -1}{\epsilon }_{0}{\mu }_{0}$, and also restrict the exponent $\eta$ to be positive. In the case of vacuum permeabilities, $\chi =\phi =0$, $\eta =1$, $\text{◂=⋯▸}{\epsilon }_{0}={\mu }_{0}=1$, the dispersion relation (2.4) is linear. The permeabilities can be reparametrized by energy via (2.4), but we will use a momentum rather than energy parametrization of the spectral density, see (2.5).

In the non-relativistic regime, $p∕m\ll 1$, we assume constant positive permeabilities, $\epsilon \sim {\epsilon }_{\text{nr}}$, $\mu \sim {\mu }_{\text{nr}}$ instead of power laws and find, by expanding (2.3), the dispersion relation $\text{◂∼▸}E\sim \text{◂+▸}m∕{\epsilon }_{\text{nr}}+{p}^{2}∕\left(\text{◂⋅▸}2{\epsilon }_{\text{nr}}{\mu }_{\text{nr}}^{2}m\right)$. This resembles the dispersion relation in electronic band theory, where $\text{◂⋅▸}{\epsilon }_{\text{nr}}{\mu }_{\text{nr}}^{2}m$ is the effective mass and $m∕{\epsilon }_{\text{nr}}$ the band gap. In band theory, the effective mass is generated by adding a perturbative Bloch potential to the free electronic Lagrangian, whereas the permeability tensor in (2.1) and (2.2) affects the kinetic part of the Lagrangian [34,35]. In this paper, we will study the ultra-relativistic regime, $p∕m\gg 1$, admitting the dispersion relation (2.4).

### 2.2. Quantized thermodynamic variables

We will study an electron gas at low temperature and high density, defined by the spectral number density

(2.5)$\text{d}⁣\text{◂=▸}\rho \left(p\right)=\text{◂⋅▸}\text{◂/▸}\frac{4\pi s}{\text{◂◽˙▸}{\left(2\pi \right)}^{3}}\frac{\text{◂...▸}{p}^{2}⁣\text{d}⁣p}{\text{◂◽˙▸}{\text{e}}^{\alpha +\text{◂⋅▸}\beta E\left(p\right)}+1},$
where $E\left(p\right)$ is the ultra-relativistic dispersion relation (2.4), $s=2$ is the electronic spin degeneracy, $f={\text{e}}^{-\alpha }$ the fugacity, and $\beta =\text{◂+▸}1∕\left({k}_{\text{B}}T\right)$ the temperature parameter. We will mostly put $\text{◂=⋯▸}\hslash =c={k}_{\text{B}}=1$ and use the shortcut $E=\text{◂+▸}{p}^{\eta }∕{a}^{\eta }$ for the dispersion relation (2.4), with power-law exponent $\eta >0$