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.

Abstract

The effect of nonlinear ultra-relativistic electron dispersion on the mass–radius relation of high-mass white dwarfs is studied. The dispersion is described by a permeability tensor in the Dirac equation, generated by the ionized high-density stellar matter, which constitutes the neutralizing background of the nearly degenerate electron plasma. The electron dispersion results in a stable mass–radius relation for high-mass white dwarfs, in contrast to a mass limit in the case of vacuum permeabilities. In the ultra-relativistic regime, the dispersion relation is a power law whose amplitude and scaling exponent is inferred from mass and radius estimates of two high-mass white dwarfs, Sirius B and LHS 4033. Evidence for the existence of super-Chandrasekhar mass white dwarfs is provided by several Type Ia supernovae (e.g., SN 2013cv, SN 2003fg, SN 2007if and SN 2009dc), whose mass ejecta exceed the Chandrasekhar limit by up to a factor of two. The dispersive mass–radius relation is used to estimate the radii, central densities, Fermi temperatures, bulk and compression moduli and sound velocities of their white dwarf progenitors.

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 $M_{⊙}$ $M_{⊙}$ 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 $M_{⊙}$ $M_{⊙}$ 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 $M_{⊙}$ $M_{⊙}$ 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 $E \sim p$ $E \sim p$ into a power law, $◂∼▸ E (p) \sim ◂+▸ m ◂◽˙▸ {(p ∕ m)}^{η} ∕ (◂⋅▸ ε_{0} μ_{0})$ $E (p) \sim m {(p ∕ m)}^{η} ∕ (ε_{0} μ_{0})$ , where $ε_{0}$ $ε_{0}$ and $μ_{0}$ $μ_{0}$ are permeability amplitudes, $m$ $m$ denotes the electron mass and $η$ $η$ is a positive scaling exponent. The spectral decay of the Fermi distribution $◂...▸ d, ◂+▸ ρ (p) \propto p^{2}, d, ◂+▸ p ∕ (◂◽˙▸ e^{α + ◂⋅▸ β E (p)} + 1)$ $d ρ (p) \propto p^{2} d p ∕ (e^{α + β E (p)} + 1)$ (where $α$ $α$ and $β$ $β$ are fugacity and temperature parameters defining the chemical potential $μ = - α ∕ β$ $μ = - α ∕ β$ ) is of Weibull type [5], the decay factor $exp (- ◂⋅▸ β E (p \to \infty))$ $exp (- β E (p \to \infty))$ being a stretched ( $η < 1$ $η < 1$ ) or compressed ( $η > 1$ $η > 1$ ) 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 $η < 1$ $η < 1$ and $η > 1$ $η > 1$ , 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 $(α, β, V)$ $(α, β, V)$ 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 $(N, β, V)$ $(N, β, V)$ representation of the energy, pressure and entropy variables, we derive the thermal equation of state, the isochoric and isobaric heat capacities $◂◽.▸ C_{V, P}$ $C_{V, P}$ and the isobaric expansion coefficient, as well as the isothermal and adiabatic compressibilities $◂◽.▸ κ_{T, S}$ $κ_{T, S}$ . We demonstrate, by explicit calculation, that the equilibrium stability conditions $◂>⋯▸ κ_{T} > κ_{S} > 0$ $κ_{T} > κ_{S} > 0$ and $◂>⋯▸ C_{P} > C_{V} > 0$ $C_{P} > C_{V} > 0$ are satisfied for positive scaling exponents $η$ $η$ 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, $P \propto ◂◽˙▸ {(N ∕ V)}^{1 + η ∕ 3}$ $P \propto {(N ∕ V)}^{1 + η ∕ 3}$ , where $η$ $η$ is the scaling exponent of the electronic dispersion relation $E \sim ◂+▸ m ◂◽˙▸ {(p ∕ m)}^{η} ∕ (◂⋅▸ ε_{0} μ_{0})$ $E \sim m {(p ∕ m)}^{η} ∕ (ε_{0} μ_{0})$ . 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 $η > 1$ $η > 1$ 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 $η$ $η$ and the product $◂⋅▸ ε_{0} μ_{0}$ $ε_{0} μ_{0}$ of the permeability amplitudes in the electronic dispersion relation. These are two additional (fitting) parameters as compared with the linear ultra-relativistic dispersion relation $E \sim p$ $E \sim p$ in vacuum ( $◂⋅▸ ε_{0} μ_{0} = η = 1$ $ε_{0} μ_{0} = η = 1$ ) which gives a mass limit instead of a mass–radius relation. The Lane–Emden equation does not define a mass limit for dispersion exponents $η > 1$ $η > 1$ , 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 $η = 1 . 240$ $η = 1.240$ and the amplitude $◂,▸ ◂⋅▸ ε_{0} μ_{0} = 4 . 85 ∕ ◂◽:▸ μ_{n}^{1 + η ∕ 3}$ $ε_{0} μ_{0} = 4.85 ∕ μ_{n}^{1 + η ∕ 3}$ of the electronic dispersion relation. $μ_{n}$ $μ_{n}$ denotes the molecular weight per electron (nucleon–electron ratio, approximately $μ_{n} \approx 2$ $μ_{n} \approx 2$ for white dwarfs, unless neutronization by electron capture sets in, which increases $μ_{n}$ $μ_{n}$ , cf. Section 6.3). The scaling exponent $η = 1 . 240$ $η = 1.240$ is safely in the stability domain $η > 1$ $η > 1$ 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, $ψ = exp (i p_{μ} x^{μ}) u (p)$ $ψ = exp (i p_{μ} x^{μ}) u (p)$ , $◂=▸ p_{μ} = (- E, p)$ $p_{μ} = (- E, p)$ . The spinor $u (p)$ $u (p)$ satisfies the Dirac equation in momentum space coupled to a dispersive isotropic permeability tensor $◂◽˙▸ g^{μ ν} (p),$ $g^{μ ν} (p),$

(2.1)

◂,▸ ◂⋅▸ (i, ◂⋅▸ γ_{μ} ◂◽˙▸ g^{μ ν} p_{ν} + m) u (p) = 0, ◂=▸ g^{00} = - ε (p), ◂◽˙▸ g^{i k} = ◂/▸ \frac{◂◽˙▸ δ^{i k}}{μ (p)},

$(i γ_{μ} g^{μ ν} p_{ν} + m) u (p) = 0, g^{00} = - ε (p), g^{i k} = \frac{δ^{i k}}{μ (p)},$

with vanishing flanks

◂◽˙▸ g^{0 i} = 0

$g^{0 i} = 0$ . The sign convention for the Minkowski metric is

◂...▸ ◂◽.▸ η_{μ ν} =, diag, (◂,▸ - 1, 1, 1, 1)

$η_{μ ν} = diag (- 1, 1, 1, 1)$ , the Dirac matrices satisfy

◂=▸ ◂⋅▸ γ_{μ} γ_{ν} + ◂⋅▸ γ_{ν} γ_{μ} = 2 ◂◽.▸ η_{μ ν}

$γ_{μ} γ_{ν} + γ_{ν} γ_{μ} = 2 η_{μ ν}$ , and

m

$m$ 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

◂=▸ ◂◽˙▸ h^{μ ν} = ◂◽˙▸ g^{μ α} ◂◽.▸ η_{α β} ◂◽˙▸ g^{β ν}

$h^{μ ν} = g^{μ α} η_{α β} g^{β ν}$ :

(2.2)

◂,▸ ◂⋅▸ (◂⋅▸ ◂◽˙▸ h^{μ ν} p_{μ} p_{ν} + m^{2}) u (p) = 0, ◂=▸ h^{00} = - ε^{2}, ◂=▸ ◂◽˙▸ h^{i k} = \frac{◂◽˙▸ δ^{i k}}{μ^{2}},

$(h^{μ ν} p_{μ} p_{ν} + m^{2}) u (p) = 0, h^{00} = - ε^{2}, h^{i k} = \frac{δ^{i k}}{μ^{2}},$

and

◂◽˙▸ h^{0 i} = 0

$h^{0 i} = 0$ , which defines the electronic dispersion relation

(2.3)

◂=▸ E (p) = \frac{◂√▸ \sqrt{p^{2} + ◂⋅▸ μ^{2} (p) m^{2}}}{◂⋅▸ ε (p) μ (p)} .

$E (p) = \frac{\sqrt{p^{2} + μ^{2} (p) m^{2}}}{ε (p) μ (p)} .$

(

ℏ = c = 1

$ℏ = c = 1$ .) We will focus on the ultra-relativistic limit,

p ∕ m ≫ 1

$p ∕ m ≫ 1$ , and assume power-law asymptotics of the permeabilities,

◂∼▸ ε \sim ε_{0} ◂◽˙▸ {(p ∕ m)}^{χ}

$ε \sim ε_{0} {(p ∕ m)}^{χ}$ ,

◂∼▸ μ \sim μ_{0} ◂◽˙▸ {(p ∕ m)}^{φ}

$μ \sim μ_{0} {(p ∕ m)}^{φ}$ , with positive dimensionless amplitudes

◂,▸ ε_{0}, μ_{0}

$ε_{0}, μ_{0}$ and real exponents

χ

$χ$ and

φ

$φ$ . For exponents

φ < 1

$φ < 1$ , the ultra-relativistic limit of the dispersion relation (2.3) reads

(2.4)

◂,▸ ◂=▸ E = \frac{m}{◂⋅▸ ε_{0} μ_{0}} ◂◽˙▸ {(p ∕ m)}^{η}, η = 1 - χ - φ,

$E = \frac{m}{ε_{0} μ_{0}} {(p ∕ m)}^{η}, η = 1 - χ - φ,$

since the mass term

◂⋅▸ μ^{2} (p) m^{2}

$μ^{2} (p) m^{2}$ in (2.3) drops out in leading order. (The electron mass in (2.4) is just a convenient scale parameter.) The group velocity

η E ∕ p

$η E ∕ p$ can approach zero (

0 < η < 1

$0 < η < 1$ ) or become superluminal (

η > 1

$η > 1$ ) in the ultra-relativistic limit; the electromagnetic counterpart is ‘slow’ or ‘fast’ light in highly dispersive media [37–39]. If

η < 0

$η < 0$ , the group velocity is negative, pointing in the opposite direction of the energy transfer [40]. We will use the shortcut

E = ◂+▸ p^{η} ∕ a^{η}

$E = p^{η} ∕ a^{η}$ , with amplitude

a^{η} = ◂⋅▸ ◂◽˙▸ m^{η - 1} ε_{0} μ_{0}

$a^{η} = m^{η - 1} ε_{0} μ_{0}$ , and also restrict the exponent

η

$η$ to be positive. In the case of vacuum permeabilities,

χ = φ = 0

$χ = φ = 0$ ,

η = 1

$η = 1$ ,

◂=⋯▸ ε_{0} = μ_{0} = 1

$ε_{0} = μ_{0} = 1$ , 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, $p ∕ m ≪ 1$ $p ∕ m ≪ 1$ , we assume constant positive permeabilities, $ε \sim ε_{nr}$ $ε \sim ε_{nr}$ , $μ \sim μ_{nr}$ $μ \sim μ_{nr}$ instead of power laws and find, by expanding (2.3), the dispersion relation $◂∼▸ E \sim ◂+▸ m ∕ ε_{nr} + p^{2} ∕ (◂⋅▸ 2 ε_{nr} μ_{nr}^{2} m)$ $E \sim m ∕ ε_{nr} + p^{2} ∕ (2 ε_{nr} μ_{nr}^{2} m)$ . This resembles the dispersion relation in electronic band theory, where $◂⋅▸ ε_{nr} μ_{nr}^{2} m$ $ε_{nr} μ_{nr}^{2} m$ is the effective mass and $m ∕ ε_{nr}$ $m ∕ ε_{nr}$ 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, $p ∕ m ≫ 1$ $p ∕ m ≫ 1$ , 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)

d, ◂=▸ ρ (p) = ◂⋅▸ ◂/▸ \frac{4 π s}{◂◽˙▸ {(2 π)}^{3}} \frac{◂...▸ p^{2}, d, p}{◂◽˙▸ e^{α + ◂⋅▸ β E (p)} + 1},

$d ρ (p) = \frac{4 π s}{{(2 π)}^{3}} \frac{p^{2} d p}{e^{α + β E (p)} + 1},$

where

E (p)

$E (p)$ is the ultra-relativistic dispersion relation (2.4),

s = 2

$s = 2$ is the electronic spin degeneracy,

f = e^{- α}

$f = e^{- α}$ the fugacity, and

β = ◂+▸ 1 ∕ (k_{B} T)

$β = 1 ∕ (k_{B} T)$ the temperature parameter. We will mostly put

◂=⋯▸ ℏ = c = k_{B} = 1

$ℏ = c = k_{B} = 1$ and use the shortcut

E = ◂+▸ p^{η} ∕ a^{η}

$E = p^{η} ∕ a^{η}$ for the dispersion relation (2.4), with power-law exponent

η > 0

$η > 0$ and amplitude

a^{η} = ◂⋅▸ ◂◽˙▸ m^{η - 1} ε_{0} μ_{0}

$a^{η} = m^{η - 1} ε_{0} μ_{0}$ , where

ε_{0}

$ε_{0}$ and

μ_{0}

$μ_{0}$ are positive permeability constants and

m

$m$ is the electron mass. We will also write

◂=▸ ◂⋅▸ β E (p) = \hat{β} p^{η}

$β E (p) = \hat{β} p^{η}$ , with the rescaled temperature parameter

◂=▸ \hat{β} = β ∕ a^{η}

$\hat{β} = β ∕ a^{η}$ . The classical spectral density is recovered if the fugacity is small,

◂...▸ d, ρ \sim ◂⋅▸ 4 π s ◂◽˙▸ {(2 π)}^{- 3} ◂◽˙▸ e^{- α - ◂⋅▸ β E (p)} p^{2}, d, p

$d ρ \sim 4 π s {(2 π)}^{- 3} e^{- α - β E (p)} p^{2} d p$ . This is also the high-energy limit of (2.5). The Weibull exponential

◂f()▸ exp ◂()▸ (- \hat{β} p^{η})

$exp (- \hat{β} p^{η})$ generates a sub- or super-exponential spectral cutoff for

0 < η < 1

$0 < η < 1$ and

η > 1

$η > 1$ , respectively [41–44]. In the following, we will consider the opposite nearly degenerate quantum limit [45,46] of density (2.5),

- α ≫ 1

$- α ≫ 1$ , and rename the fugacity parameter as

λ = - α

$λ = - α$ .

Table 1. Mass, radius and density parameters of Sirius B and LHS 4033 and of the white dwarf progenitors of the super-Chandrasekhar mass supernovae SN 2013cv, SN 2003fg, SN 2007if and SN 2009dc. The mass and radius estimates of Sirius B and LHS 4033 are taken from Refs. [32,33] and the progenitor masses from Refs. [1–4]. The progenitor radii are obtained by applying the dispersive mass–radius relation (5.11) with permeability constants inferred from the high-mass white dwarfs Sirius B and LHS 4033, cf. Section 6.1. Also recorded are the average and central mass densities $ρ_{av}$ $ρ_{av}$ and $ρ_{cent}$ $ρ_{cent}$ in solar units $(◂,▸ ρ_{⊙} = 1 . ◂+▸ 410 g ∕ {cm}^{3})$ $(ρ_{⊙} = 1.410 g ∕ {cm}^{3})$ and in units of the critical density $◂,▸ ρ_{crit} = 9 . ◂+▸ 810 \times ◂⋅▸ 1 0^{5} μ_{n} g ∕ {cm}^{3}$ $ρ_{crit} = 9.810 \times 1 0^{5} μ_{n} g ∕ {cm}^{3}$ (see (5.5)) which defines the ultra-relativistic regime $◂≫▸ ρ ∕ ρ_{crit} ≫ 1$ $ρ ∕ ρ_{crit} ≫ 1$ . $μ_{n}$ $μ_{n}$ denotes the molecular weight per electron (nucleon–electron ratio).

	$M ∕ M_{⊙}$ $M ∕ M_{⊙}$	$R ∕ R_{⊙}$ $R ∕ R_{⊙}$	$◂+▸ ρ_{av} ∕ ρ_{⊙}$ $ρ_{av} ∕ ρ_{⊙}$	$◂+▸ ρ_{cent} ∕ ρ_{⊙}$ $ρ_{cent} ∕ ρ_{⊙}$	$◂⋅▸ μ_{n} ρ_{av} ∕ ρ_{crit}$ $μ_{n} ρ_{av} ∕ ρ_{crit}$	$◂⋅▸ μ_{n} ρ_{cent} ∕ ρ_{crit}$ $μ_{n} ρ_{cent} ∕ ρ_{crit}$
Sirius B	$◂,▸ 0 . 94 \pm 0 . 05$ $0.94 \pm 0.05$	$◂+▸ (◂,▸ 8 . 4 \pm 2 . 5) \times 1 0^{- 3}$ $(8.4 \pm 2.5) \times 1 0^{- 3}$	$1 . ◂+▸ 59 \times 1 0^{6}$ $1.59 \times 1 0^{6}$	$3 . ◂+▸ 28 \times 1 0^{7}$ $3.28 \times 1 0^{7}$	2.28	47.2
LHS 4033	$◂,▸ 1 . 33 \pm 0 . 02$ $1.33 \pm 0.02$	$◂+▸ (◂,▸ 3 . 6 \pm 0 . 2) \times 1 0^{- 3}$ $(3.6 \pm 0.2) \times 1 0^{- 3}$	$2 . ◂+▸ 85 \times 1 0^{7}$ $2.85 \times 1 0^{7}$	$5 . ◂+▸ 90 \times 1 0^{8}$ $5.90 \times 1 0^{8}$	41.0	848
progen. SN 2013cv	1.6	$2 . ◂+▸ 29 \times 1 0^{- 3}$ $2.29 \times 1 0^{- 3}$	$1 . ◂+▸ 33 \times 1 0^{8}$ $1.33 \times 1 0^{8}$	$2 . ◂+▸ 75 \times 1 0^{9}$ $2.75 \times 1 0^{9}$	191	$3 . ◂+▸ 95 \times 1 0^{3}$ $3.95 \times 1 0^{3}$
progen. SN 2003fg	2.1	$1 . ◂+▸ 18 \times 1 0^{- 3}$ $1.18 \times 1 0^{- 3}$	$1 . ◂+▸ 28 \times 1 0^{9}$ $1.28 \times 1 0^{9}$	$2 . ◂+▸ 64 \times 1 0^{10}$ $2.64 \times 1 0^{10}$	$1 . ◂+▸ 84 \times 1 0^{3}$ $1.84 \times 1 0^{3}$	$3 . ◂+▸ 80 \times 1 0^{4}$ $3.80 \times 1 0^{4}$
progen. SN 2007if	2.4	$8 . ◂+▸ 52 \times 1 0^{- 4}$ $8.52 \times 1 0^{- 4}$	$3 . ◂+▸ 88 \times 1 0^{9}$ $3.88 \times 1 0^{9}$	$8 . ◂+▸ 03 \times 1 0^{10}$ $8.03 \times 1 0^{10}$	$5 . ◂+▸ 58 \times 1 0^{3}$ $5.58 \times 1 0^{3}$	$1 . ◂+▸ 15 \times 1 0^{5}$ $1.15 \times 1 0^{5}$
progen. SN 2009dc	2.8	$5 . ◂+▸ 85 \times 1 0^{- 4}$ $5.85 \times 1 0^{- 4}$	$1 . ◂+▸ 40 \times 1 0^{10}$ $1.40 \times 1 0^{10}$	$2 . ◂+▸ 90 \times 1 0^{11}$ $2.90 \times 1 0^{11}$	$2 . ◂+▸ 01 \times 1 0^{4}$ $2.01 \times 1 0^{4}$	$4 . ◂+▸ 16 \times 1 0^{5}$ $4.16 \times 1 0^{5}$

Table 2. Electronic number density, Fermi momentum/energy/temperature, gravitational surface potential and surface gravity of Sirius B, LHS 4033 and the super-Chandrasekhar mass progenitors of SN 2013cv, SN 2003fg, SN 2007if and SN 2009dc. The electron density $◂=▸ n (ρ) = N ∕ V$ $n (ρ) = N ∕ V$ and Fermi momentum, energy and temperature $◂,▸ p_{F} (ρ), E_{F} (ρ), T_{F} (ρ)$ $p_{F} (ρ), E_{F} (ρ), T_{F} (ρ)$ (see (6.3) and (6.4)) are calculated at the average and central mass densities $ρ_{av}$ $ρ_{av}$ and $ρ_{cent}$ $ρ_{cent}$ listed in Table 1. These quantities scale with the nucleon–electron ratio, $◂+▸ n \propto 1 ∕ μ_{n}$ $n \propto 1 ∕ μ_{n}$ , $p_{F} \propto ◂◽:▸ μ_{n}^{- 1 ∕ 3}$ $p_{F} \propto μ_{n}^{- 1 ∕ 3}$ , $◂+▸ E_{F} \propto T_{F} \propto μ_{n}$ $E_{F} \propto T_{F} \propto μ_{n}$ , since the fit parameter $◂,▸ ◂⋅▸ ◂◽:▸ μ_{n}^{1 + η ∕ 3} ε_{0} μ_{0} = 4 . 85$ $μ_{n}^{1 + η ∕ 3} ε_{0} μ_{0} = 4.85$ is kept fixed, see Section 6.1, and they are listed here for $μ_{n} = 2$ $μ_{n} = 2$ . (The permeability amplitudes $ε_{0}$ $ε_{0}$ and $μ_{0}$ $μ_{0}$ and the scaling exponent $η$ $η$ define the electronic dispersion relation (2.4).) The surface potential $◂+▸ G M ∕ (c^{2} R)$ $G M ∕ (c^{2} R)$ and surface gravity $◂+▸ G M ∕ R^{2}$ $G M ∕ R^{2}$ , cf. after (6.9), are based on the mass and radius estimates in Table 1.

	$n (ρ_{av})$ $n (ρ_{av})$ $[{cm}^{- 3}]$ $[{cm}^{- 3}]$	$n (ρ_{cent})$ $n (ρ_{cent})$ $[{cm}^{- 3}]$ $[{cm}^{- 3}]$	$p_{F} (ρ_{av})$ $p_{F} (ρ_{av})$ $[MeV/, c]$ $[MeV/ c]$	$p_{F} (ρ_{cent})$ $p_{F} (ρ_{cent})$ $[MeV/, c]$ $[MeV/ c]$	$E_{F} (ρ_{av})$ $E_{F} (ρ_{av})$ $[MeV]$ $[MeV]$	$E_{F} (ρ_{cent})$ $E_{F} (ρ_{cent})$ [MeV]	$T_{F} (ρ_{av})$ $T_{F} (ρ_{av})$ $[◂...▸ 1 0^{10}, K]$ $[1 0^{10} K]$	$T_{F} (ρ_{cent})$ $T_{F} (ρ_{cent})$ $[◂...▸ 1 0^{10}, K]$ $[1 0^{10} K]$	$◂+▸ G M ∕ (c^{2} R)$ $G M ∕ (c^{2} R)$	$◂+▸ G M ∕ R^{2}$ $G M ∕ R^{2}$ $[◂...▸ cm, ∕ s^{2}]$ $[cm ∕ s^{2}]$
Sirius B	$6 . ◂+▸ 68 \times 1 0^{29}$ $6.68 \times 1 0^{29}$	$1 . ◂+▸ 38 \times 1 0^{31}$ $1.38 \times 1 0^{31}$	0.534	1.47	0.297	1.03	$0.344$ $0.344$	$1.20$ $1.20$	$2 . ◂+▸ 38 \times 1 0^{- 4}$ $2.38 \times 1 0^{- 4}$	$◂...▸ 3.65, \times 1 0^{8}$ $3.65 \times 1 0^{8}$
LHS 4033	$1 . ◂+▸ 20 \times 1 0^{31}$ $1.20 \times 1 0^{31}$	$2 . ◂+▸ 49 \times 1 0^{32}$ $2.49 \times 1 0^{32}$	1.40	3.84	0.983	3.42	$1.14$ $1.14$	$3.97$ $3.97$	$7 . ◂+▸ 84 \times 1 0^{- 4}$ $7.84 \times 1 0^{- 4}$	$◂...▸ 2.81, \times 1 0^{9}$ $2.81 \times 1 0^{9}$
progen. SN 2013cv	$5 . ◂+▸ 60 \times 1 0^{31}$ $5.60 \times 1 0^{31}$	$1 . ◂+▸ 16 \times 1 0^{33}$ $1.16 \times 1 0^{33}$	2.34	6.41	1.85	6.47	$2.15$ $2.15$	$7.51$ $7.51$	$1 . ◂+▸ 48 \times 1 0^{- 3}$ $1.48 \times 1 0^{- 3}$	$◂...▸ 8.37, \times 1 0^{9}$ $8.37 \times 1 0^{9}$
progen. SN 2003fg	$5 . ◂+▸ 38 \times 1 0^{32}$ $5.38 \times 1 0^{32}$	$1 . ◂+▸ 11 \times 1 0^{34}$ $1.11 \times 1 0^{34}$	4.97	13.6	4.72	16.5	$5.47$ $5.47$	$19.1$ $19.1$	$3 . ◂+▸ 78 \times 1 0^{- 3}$ $3.78 \times 1 0^{- 3}$	$◂...▸ 4.14, \times 1 0^{10}$ $4.14 \times 1 0^{10}$
progen. SN 2007if	$1 . ◂+▸ 64 \times 1 0^{33}$ $1.64 \times 1 0^{33}$	$3 . ◂+▸ 39 \times 1 0^{34}$ $3.39 \times 1 0^{34}$	7.19	19.7	7.46	26.1	$8.66$ $8.66$	$30.3$ $30.3$	$5 . ◂+▸ 98 \times 1 0^{- 3}$ $5.98 \times 1 0^{- 3}$	$◂...▸ 9.07, \times 1 0^{10}$ $9.07 \times 1 0^{10}$
progen. SN 2009dc	$5 . ◂+▸ 90 \times 1 0^{33}$ $5.90 \times 1 0^{33}$	$1 . ◂+▸ 22 \times 1 0^{35}$ $1.22 \times 1 0^{35}$	11.0	30.3	12.7	44.4	$1, 4 . 7$ $14.7$	$51.5$ $51.5$	$1 . ◂+▸ 02 \times 1 0^{- 2}$ $1.02 \times 1 0^{- 2}$	$◂...▸ 2.24, \times 1 0^{11}$ $2.24 \times 1 0^{11}$

Table 3. Electron degeneracy pressure $P$ $P$ , bulk modulus $K_{S}$ $K_{S}$ , compression modulus (volume incompressibility) $P_{, n}$ $P_{, n}$ and speed of sound $υ_{s}$ $υ_{s}$ . These quantities depend on the mass density $ρ$ $ρ$ , cf. (6.5), (6.7) and (6.8), and are listed for the average and central densities $ρ_{av}$ $ρ_{av}$ and $ρ_{cent}$ $ρ_{cent}$ (see Table 1). The compression modulus linearly scales with the molecular weight per electron, $◂+▸ P_{, n} \propto μ_{n}$ $P_{, n} \propto μ_{n}$ , and is recorded here for $μ_{n} = 2$ $μ_{n} = 2$ , see the remarks after (6.8) and the caption to Table 2. The pressure is proportional to the bulk modulus, $◂,▸ P = 0 . 7075 K_{S}$ $P = 0.7075 K_{S}$ , cf. (6.7), due to the polytropic equation of state; the conversion to cgs pressure units is $◂,▸ ◂=▸ 1 {MeV/cm}^{3} = 1 . 602 \times ◂⋅▸ 1 0^{- 6} {dyn/cm}^{2}$ $1 {MeV/cm}^{3} = 1.602 \times 1 0^{- 6} {dyn/cm}^{2}$ .

	$P (ρ_{av})$ $P (ρ_{av})$ $[{dyn/cm}^{2}]$ $[{dyn/cm}^{2}]$	$P (ρ_{cent})$ $P (ρ_{cent})$ $[{dyn/cm}^{2}]$ $[{dyn/cm}^{2}]$	$K_{S} (ρ_{av})$ $K_{S} (ρ_{av})$ $[◂...▸ MeV, ∕ {cm}^{3}]$ $[MeV ∕ {cm}^{3}]$	$K_{S} (ρ_{cent})$ $K_{S} (ρ_{cent})$ $[◂...▸ MeV, ∕ {cm}^{3}]$ $[MeV ∕ {cm}^{3}]$	$P_{, n} (ρ_{av})$ $P_{, n} (ρ_{av})$ $[MeV]$ $[MeV]$	$P_{, n} (ρ_{cent})$ $P_{, n} (ρ_{cent})$ $[MeV]$ $[MeV]$	$◂+▸ υ_{s} (ρ_{av}) ∕ c$ $υ_{s} (ρ_{av}) ∕ c$	$◂+▸ υ_{s} (ρ_{cent}) ∕ c$ $υ_{s} (ρ_{cent}) ∕ c$
Sirius B	$9 . ◂+▸ 28 \times 1 0^{22}$ $9.28 \times 1 0^{22}$	$6 . ◂+▸ 72 \times 1 0^{24}$ $6.72 \times 1 0^{24}$	$8 . ◂+▸ 19 \times 1 0^{28}$ $8.19 \times 1 0^{28}$	$5 . ◂+▸ 93 \times 1 0^{30}$ $5.93 \times 1 0^{30}$	0.123	0.429	$8 . ◂+▸ 08 \times 1 0^{- 3}$ $8.08 \times 1 0^{- 3}$	$1 . ◂+▸ 51 \times 1 0^{- 2}$ $1.51 \times 1 0^{- 2}$
LHS 4033	$5 . ◂+▸ 51 \times 1 0^{24}$ $5.51 \times 1 0^{24}$	$3 . ◂+▸ 99 \times 1 0^{26}$ $3.99 \times 1 0^{26}$	$4 . ◂+▸ 86 \times 1 0^{30}$ $4.86 \times 1 0^{30}$	$3 . ◂+▸ 52 \times 1 0^{32}$ $3.52 \times 1 0^{32}$	0.405	1.42	$1 . ◂+▸ 47 \times 1 0^{- 2}$ $1.47 \times 1 0^{- 2}$	$2 . ◂+▸ 75 \times 1 0^{- 2}$ $2.75 \times 1 0^{- 2}$
progen. SN 2013cv	$4 . ◂+▸ 85 \times 1 0^{25}$ $4.85 \times 1 0^{25}$	$3 . ◂+▸ 50 \times 1 0^{27}$ $3.50 \times 1 0^{27}$	$4 . ◂+▸ 28 \times 1 0^{31}$ $4.28 \times 1 0^{31}$	$3 . ◂+▸ 09 \times 1 0^{33}$ $3.09 \times 1 0^{33}$	0.764	2.67	$2 . ◂+▸ 02 \times 1 0^{- 2}$ $2.02 \times 1 0^{- 2}$	$3 . ◂+▸ 77 \times 1 0^{- 2}$ $3.77 \times 1 0^{- 2}$
progen. SN 2003fg	$1 . ◂+▸ 19 \times 1 0^{27}$ $1.19 \times 1 0^{27}$	$8 . ◂+▸ 60 \times 1 0^{28}$ $8.60 \times 1 0^{28}$	$1 . ◂+▸ 05 \times 1 0^{33}$ $1.05 \times 1 0^{33}$	$7 . ◂+▸ 59 \times 1 0^{34}$ $7.59 \times 1 0^{34}$	1.95	6.82	$3 . ◂+▸ 22 \times 1 0^{- 2}$ $3.22 \times 1 0^{- 2}$	$6 . ◂+▸ 03 \times 1 0^{- 2}$ $6.03 \times 1 0^{- 2}$
progen. SN 2007if	$5 . ◂+▸ 72 \times 1 0^{27}$ $5.72 \times 1 0^{27}$	$4 . ◂+▸ 14 \times 1 0^{29}$ $4.14 \times 1 0^{29}$	$5 . ◂+▸ 05 \times 1 0^{33}$ $5.05 \times 1 0^{33}$	$3 . ◂+▸ 65 \times 1 0^{35}$ $3.65 \times 1 0^{35}$	3.09	10.8	$4 . ◂+▸ 05 \times 1 0^{- 2}$ $4.05 \times 1 0^{- 2}$	$7 . ◂+▸ 58 \times 1 0^{- 2}$ $7.58 \times 1 0^{- 2}$
progen. SN 2009dc	$3 . ◂+▸ 50 \times 1 0^{28}$ $3.50 \times 1 0^{28}$	$2 . ◂+▸ 54 \times 1 0^{30}$ $2.54 \times 1 0^{30}$	$3 . ◂+▸ 09 \times 1 0^{34}$ $3.09 \times 1 0^{34}$	$2 . ◂+▸ 24 \times 1 0^{36}$ $2.24 \times 1 0^{36}$	5.24	18.3	$5 . ◂+▸ 29 \times 1 0^{- 2}$ $5.29 \times 1 0^{- 2}$	$9 . ◂+▸ 89 \times 1 0^{- 2}$ $9.89 \times 1 0^{- 2}$

The electronic number count $N$ $N$ , internal energy $U$ $U$ and pressure $P$ $P$ read

(2.6)

◂,▸ ◂...▸ N = V ◂∫▸ \int_{0}^{\infty}, d, ρ (p), ◂...▸ U = V ◂∫▸ \int_{0}^{\infty} E, (, p,)d, ρ (p), ◂...▸ ◂=▸ P = \frac{1}{3} ◂∫▸ \int_{0}^{\infty} E^{'}, (, p,), p, d, ρ (p) .

$N = V \int_{0}^{\infty} d ρ (p), U = V \int_{0}^{\infty} E (p)d ρ (p), P = \frac{1}{3} \int_{0}^{\infty} E^{'} (p) p d ρ (p) .$

Since the partition function is related to the pressure by

◂=▸ log Z = β P V

$log Z = β P V$ in an equilibrium system, we use integration by parts to find

(2.7)

◂=▸ log Z = ◂/▸ \frac{4 π s V}{◂◽˙▸ {(2 π)}^{3}} ◂∫▸ \int_{0}^{\infty} ◂⋅▸ log (1 + ◂◽˙▸ e^{- α - ◂⋅▸ β E (p)}) p^{2}, d, p,

$log Z = \frac{4 π s V}{{(2 π)}^{3}} \int_{0}^{\infty} log (1 + e^{- α - β E (p)}) p^{2} d p,$

which can also be obtained by a fermionic trace calculation [16,17]. The entropy is assembled as

◂=▸ S (α, β, V) = ◂−▸ β P V - λ N + β U

$S (α, β, V) = β P V - λ N + β U$ , with the integral representations (2.6) of particle count, energy and pressure substituted and

λ = - α

$λ = - α$ , cf. after (2.5).

3. Quantifying dispersion in the nearly degenerate quantum regime: fugacity expansion

The integral representations (2.6) of the quantized thermodynamic variables are of type

(3.1)

◂=▸ I [g] = ◂∫▸ \int_{0}^{\infty} \frac{◂...▸ g (p), d, p}{◂◽˙▸ e^{F (p) - λ} + 1},

$I [g] = \int_{0}^{\infty} \frac{g (p) d p}{e^{F (p) - λ} + 1},$

where

◂=⋯▸ F (p) = ◂⋅▸ β E (p) = \hat{β} p^{η}

$F (p) = β E (p) = \hat{β} p^{η}$ , see after (2.5), and

g (p)

$g (p)$ is a power-law function. (We have put

λ = - α

$λ = - α$ and

◂=▸ \hat{β} = β ∕ a^{η}

$\hat{β} = β ∕ a^{η}$ .) The integrals (4.1) can be evaluated by employing a Sommerfeld expansion valid for large

λ ≫ 1

$λ ≫ 1$ ,

(3.2)

◂...▸ ◂=▸ I [g] = ◂∫▸ \int_{0}^{F^{- 1} (λ)} g (p), d, p + ◂⋅▸ \frac{π^{2}}{6} g_{F}^{'} (λ) + ◂⋅▸ ◂/▸ \frac{7 π^{4}}{360} ◂◽:▸ g_{F}^{(3)} (λ) +, O, (◂◽:▸ g_{F}^{(5)} (λ)),

$I [g] = \int_{0}^{F^{- 1} (λ)} g (p) d p + \frac{π^{2}}{6} g_{F}^{'} (λ) + \frac{7 π^{4}}{360} g_{F}^{(3)} (λ) + O (g_{F}^{(5)} (λ)),$

(3.3)

◂=▸ g_{F} (λ) = ◂⋅▸ ◂f()▸ g ◂()▸ (F^{- 1} (λ)) ◂◽˙▸ {(F^{- 1})}^{'} (λ),

$g_{F} (λ) = g (F^{- 1} (λ)) {(F^{- 1})}^{'} (λ),$

where

◂=▸ F^{- 1} (λ) = ◂◽˙▸ λ^{1 ∕ η} ∕ ◂◽˙▸ {\hat{β}}^{1 ∕ η}

$F^{- 1} (λ) = λ^{1 ∕ η} ∕ {\hat{β}}^{1 ∕ η}$ . The power-law functions

g (p)

$g (p)$ in (3.1) defining the particle count, internal energy and pressure can be read off from (2.6),

\frac{N}{V} = ◂⋅▸ ◂/▸ \frac{4 π s}{◂◽˙▸ {(2 π)}^{3}} I [◂=▸ g_{N} (p) = p^{2}],

$\frac{N}{V} = \frac{4 π s}{{(2 π)}^{3}} I [g_{N} (p) = p^{2}],$

(3.4)

◂,▸ \frac{U}{V} = ◂⋅▸ ◂/▸ \frac{4 π s}{◂◽˙▸ {(2 π)}^{3}} I [◂=▸ g_{U} (p) = ◂◽˙▸ p^{η + 2} ∕ a^{η}], ◂=▸ P = \frac{η}{3} \frac{U}{V} .

$\frac{U}{V} = \frac{4 π s}{{(2 π)}^{3}} I [g_{U} (p) = p^{η + 2} ∕ a^{η}], P = \frac{η}{3} \frac{U}{V} .$

The entropy is thus

◂=▸ S (α, β, V) = ◂⋅▸ (1 + η ∕ 3) β U - λ N

$S (α, β, V) = (1 + η ∕ 3) β U - λ N$ .

It suffices to calculate integral $I [g]$ $I [g]$ in (3.1) with a power-law kernel $◂=▸ g (p) = p^{κ}$ $g (p) = p^{κ}$ , $κ > 0$ $κ > 0$ , so that $◂=▸ g_{F} (λ) = ◂+▸ ◂◽˙▸ λ^{(1 + κ - η) ∕ η} ∕ (η ◂◽˙▸ {\hat{β}}^{(1 + κ) ∕ η})$ $g_{F} (λ) = λ^{(1 + κ - η) ∕ η} ∕ (η {\hat{β}}^{(1 + κ) ∕ η})$ as defined in (3.3). It will also be convenient to introduce the parameter $◂=▸ ξ = F^{- 1} (λ)$ $ξ = F^{- 1} (λ)$ or inversely $◂=▸ λ = \hat{β} ξ^{η}$ $λ = \hat{β} ξ^{η}$ , and to express the derivatives $g_{F}^{'} (λ)$ $g_{F}^{'} (λ)$ and $◂◽:▸ g_{F}^{(3)} (λ)$ $g_{F}^{(3)} (λ)$ arising in the Sommerfeld expansion (3.2) of $I [◂=▸ g (p) = p^{κ}]$ $I [g (p) = p^{κ}]$ in terms of $ξ$ $ξ$ :

◂=▸ g_{F}^{'} (λ) = ◂⋅▸ (1 + κ - η) \frac{◂◽˙▸ ξ^{1 + κ}}{◂⋅▸ η^{2} {◂()▸ (\hat{β} ξ^{η})}^{2}},

$g_{F}^{'} (λ) = (1 + κ - η) \frac{ξ^{1 + κ}}{η^{2} {(\hat{β} ξ^{η})}^{2}},$

(3.5)

◂=▸ ◂◽:▸ g_{F}^{(3)} (λ) = ◂⋅▸ (1 + κ - η) (◂−▸ 1 + κ - 2 η) (◂−▸ 1 + κ - 3 η) \frac{◂◽˙▸ ξ^{1 + κ}}{◂⋅▸ η^{4} {◂()▸ (\hat{β} ξ^{η})}^{4}} .

$g_{F}^{(3)} (λ) = (1 + κ - η) (1 + κ - 2 η) (1 + κ - 3 η) \frac{ξ^{1 + κ}}{η^{4} {(\hat{β} ξ^{η})}^{4}} .$

The integral determining the leading order in expansion (3.2) gives

◂+▸ ◂◽˙▸ ξ^{κ + 1} ∕ (κ + 1)

$ξ^{κ + 1} ∕ (κ + 1)$ . The fugacity expansions of the number count and internal energy in (3.4) are found by specifying

κ = 2

$κ = 2$ and

κ = 2 + η

$κ = 2 + η$ , respectively,

(3.6)

\frac{N}{V} = ◂⋅▸ ◂/▸ \frac{4 π s}{◂◽˙▸ {(2 π)}^{3}} \frac{ξ^{3}}{3} [◂+▸ 1 + ◂⋅▸ \frac{π^{2}}{6} \frac{◂⋅▸ 3 (3 - η)}{◂⋅▸ η^{2} {◂()▸ (\hat{β} ξ^{η})}^{2}} + ◂/▸ \frac{7 π^{4}}{360} \frac{◂⋅▸ 3 (3 - η) (3 - 2 η) (3 - 3 η)}{◂⋅▸ η^{4} {◂()▸ (\hat{β} ξ^{η})}^{4}} + \dots],

$\frac{N}{V} = \frac{4 π s}{{(2 π)}^{3}} \frac{ξ^{3}}{3} [1 + \frac{π^{2}}{6} \frac{3 (3 - η)}{η^{2} {(\hat{β} ξ^{η})}^{2}} + \frac{7 π^{4}}{360} \frac{3 (3 - η) (3 - 2 η) (3 - 3 η)}{η^{4} {(\hat{β} ξ^{η})}^{4}} + \dots],$

(3.7)

\frac{U}{V} = ◂⋅▸ ◂/▸ \frac{4 π s}{◂◽˙▸ {(2 π)}^{3}} \frac{1}{a^{η}} \frac{◂◽˙▸ ξ^{3 + η}}{3 + η} [◂+▸ 1 + ◂⋅▸ \frac{π^{2}}{6} \frac{◂⋅▸ (3 + η) 3}{◂⋅▸ η^{2} {◂()▸ (\hat{β} ξ^{η})}^{2}} + ◂/▸ \frac{7 π^{4}}{360} \frac{◂⋅▸ (3 + η) 3 (3 - η) (3 - 2 η)}{◂⋅▸ η^{4} {◂()▸ (\hat{β} ξ^{η})}^{4}} + \dots],

$\frac{U}{V} = \frac{4 π s}{{(2 π)}^{3}} \frac{1}{a^{η}} \frac{ξ^{3 + η}}{3 + η} [1 + \frac{π^{2}}{6} \frac{(3 + η) 3}{η^{2} {(\hat{β} ξ^{η})}^{2}} + \frac{7 π^{4}}{360} \frac{(3 + η) 3 (3 - η) (3 - 2 η)}{η^{4} {(\hat{β} ξ^{η})}^{4}} + \dots],$

The expansions of the pressure and partition function are obtained via the internal energy expansion (3.7), since

P = ◂+▸ η U ∕ (3 V)

$P = η U ∕ (3 V)$ and

◂=▸ log Z = β U η ∕ 3

$log Z = β U η ∕ 3$ , cf. (3.4). The entropy is assembled by substituting the series (3.6) and (3.7) into

◂=▸ S (α, β, V) = ◂⋅▸ (1 + η ∕ 3) \hat{β} a^{η} U - ◂⋅▸ \hat{β} ξ^{η} N

$S (α, β, V) = (1 + η ∕ 3) \hat{β} a^{η} U - \hat{β} ξ^{η} N$ , see after (3.4). In this case, the leading orders in (3.6) and (3.7) cancel each other,

(3.8)

\frac{S}{V} = ◂⋅▸ ◂/▸ \frac{4 π s}{◂◽˙▸ {(2 π)}^{3}} ◂/▸ \frac{ξ^{3}}{3 η} ◂/▸ \frac{π^{2}}{\hat{β} ξ^{η}} [◂+▸ 1 + ◂/▸ \frac{7 π^{2}}{30} \frac{◂⋅▸ (3 - η) (3 - 2 η)}{◂⋅▸ η^{2} {◂()▸ (\hat{β} ξ^{η})}^{2}} + \dots] .

$\frac{S}{V} = \frac{4 π s}{{(2 π)}^{3}} \frac{ξ^{3}}{3 η} \frac{π^{2}}{\hat{β} ξ^{η}} [1 + \frac{7 π^{2}}{30} \frac{(3 - η) (3 - 2 η)}{η^{2} {(\hat{β} ξ^{η})}^{2}} + \dots] .$

The indicated second-order term is defined by the third-order terms in (3.6) and (3.7); apart from that, we will only need the second order in (3.6) and (3.7). We note that

◂=⋯▸ \hat{β} ξ^{η} = λ = - α

$\hat{β} ξ^{η} = λ = - α$ is the fugacity parameter, cf. after (3.4), and

η > 0

$η > 0$ is the exponent of the electronic dispersion relation (2.4).

◂=▸ \hat{β} = β ∕ a^{η}

$\hat{β} = β ∕ a^{η}$ is the temperature parameter rescaled with the amplitude of the dispersion relation, see after (2.5). The expansions (3.6)–(3.8) give the thermodynamic variables in

(α, β, V)

$(α, β, V)$ representation; they are based on the ultra-relativistic dispersion relation

◂=▸ E (p) = ◂+▸ p^{η} ∕ a^{η}

$E (p) = p^{η} ∕ a^{η}$ in (2.4) and apply in the low-temperature high-density regime,

◂≫▸ \hat{β} ξ^{η} ≫ 1

$\hat{β} ξ^{η} ≫ 1$ . In Section 4, we will identify

ξ

$ξ$ as density parameter, cf. (4.2).

4. Effect of electron dispersion on thermodynamic variables at low temperature and high density

4.1. Internal energy, entropy and isochoric heat capacity

We start by inverting the fugacity expansion (3.6) of the number density, solving for $ξ$ $ξ$ . Using the shortcut

(4.1)

◂=▸ \hat{N} = \frac{N}{V} ◂/▸ \frac{3 ◂◽˙▸ {(2 π)}^{3}}{4 π s},

$\hat{N} = \frac{N}{V} \frac{3 {(2 π)}^{3}}{4 π s},$

we find

(4.2)

◂=▸ ξ (β, N, V) = ◂⋅▸ ◂◽˙▸ {\hat{N}}^{1 ∕ 3} (1 - ◂⋅▸ \frac{π^{2}}{6} ◂/▸ \frac{(3 - η)}{η^{2} ◂◽˙▸ {(\hat{β} ◂◽˙▸ {\hat{N}}^{η ∕ 3})}^{2}} + \dots) .

$ξ (β, N, V) = {\hat{N}}^{1 ∕ 3} (1 - \frac{π^{2}}{6} \frac{(3 - η)}{η^{2} {(\hat{β} {\hat{N}}^{η ∕ 3})}^{2}} + \dots) .$

The leading order thereof is the Fermi momentum,

p_{F} = ◂◽˙▸ {\hat{N}}^{1 ∕ 3}

$p_{F} = {\hat{N}}^{1 ∕ 3}$ , and the ultra-relativistic Fermi energy/temperature is

◂=⋯▸ E_{F} = ◂⋅▸ k_{B} T_{F} = ◂+▸ p_{F}^{η} ∕ a^{η}

$E_{F} = k_{B} T_{F} = p_{F}^{η} ∕ a^{η}$ with

a^{η} = ◂⋅▸ ◂◽˙▸ m^{η - 1} ε_{0} μ_{0}

$a^{η} = m^{η - 1} ε_{0} μ_{0}$ according to the dispersion relation (2.4). Substituting

ξ (β, N, V)

$ξ (β, N, V)$ into the series expansions (3.7) and (3.8), we obtain the

(T, N, V)

$(T, N, V)$ parametrization of the internal energy, that is the caloric equation of state of the electron gas,

(4.3)

\frac{U}{V} = ◂⋅▸ ◂/▸ \frac{4 π s}{◂◽˙▸ {(2 π)}^{3} a^{η}} \frac{◂◽˙▸ {\hat{N}}^{η ∕ 3 + 1}}{η + 3} ◂()▸ (1 + ◂⋅▸ \frac{π^{2}}{6} ◂/▸ \frac{η + 3}{η ◂◽˙▸ {(\hat{β} ◂◽˙▸ {\hat{N}}^{η ∕ 3})}^{2}} + \dots),

$\frac{U}{V} = \frac{4 π s}{{(2 π)}^{3} a^{η}} \frac{{\hat{N}}^{η ∕ 3 + 1}}{η + 3} (1 + \frac{π^{2}}{6} \frac{η + 3}{η {(\hat{β} {\hat{N}}^{η ∕ 3})}^{2}} + \dots),$

and the entropy

S (T, N, V)

$S (T, N, V)$ ,

(4.4)

\frac{S}{V} = ◂⋅▸ ◂/▸ \frac{4 π s}{◂◽˙▸ {(2 π)}^{3}} \frac{1}{3 η} ◂/▸ \frac{π^{2}}{\hat{β}} ◂◽˙▸ {\hat{N}}^{(3 - η) ∕ 3} ◂()▸ (1 + ◂⋅▸ \frac{π^{2}}{10} ◂/▸ \frac{◂⋅▸ (2 - 3 η) (3 - η)}{η^{2} ◂◽˙▸ {(\hat{β} ◂◽˙▸ {\hat{N}}^{η ∕ 3})}^{2}} + \dots) .

$\frac{S}{V} = \frac{4 π s}{{(2 π)}^{3}} \frac{1}{3 η} \frac{π^{2}}{\hat{β}} {\hat{N}}^{(3 - η) ∕ 3} (1 + \frac{π^{2}}{10} \frac{(2 - 3 η) (3 - η)}{η^{2} {(\hat{β} {\hat{N}}^{η ∕ 3})}^{2}} + \dots) .$

The leading order thereof can be written as

S ∕ N \sim ◂+▸ π^{2} ∕ (◂⋅▸ η β E_{F})

$S ∕ N \sim π^{2} ∕ (η β E_{F})$ , with Fermi energy

◂=▸ E_{F} = ◂◽˙▸ {\hat{N}}^{η ∕ 3} ∕ a^{η}

$E_{F} = {\hat{N}}^{η ∕ 3} ∕ a^{η}$ . The expansion is in powers of

1 ∕ ◂◽˙▸ {(\hat{β} ◂◽˙▸ {\hat{N}}^{η ∕ 3})}^{2}

$1 ∕ {(\hat{β} {\hat{N}}^{η ∕ 3})}^{2}$ , where

◂=⋯▸ \hat{β} ◂◽˙▸ {\hat{N}}^{η ∕ 3} = β E_{F} = T_{F} ∕ T

$\hat{β} {\hat{N}}^{η ∕ 3} = β E_{F} = T_{F} ∕ T$ . The leading order of the internal energy (4.3) is independent of temperature and defines the equipartition ratio

U ∕ N \sim ◂+▸ 3 E_{F} ∕ (3 + η)

$U ∕ N \sim 3 E_{F} ∕ (3 + η)$ , cf. (4.1). The isochoric heat capacity

◂=▸ C_{V} = T S_{, T}

$C_{V} = T S_{, T}$ reads like

S

$S$ in (4.4), with the second-order term multiplied by a factor of 3.

4.2. Mechanical and thermal stability: thermal equation of state, isobaric heat capacity, isobaric expansion coefficient, isothermal and adiabatic compressibility

The thermal equation of state $P (T, N, V)$ $P (T, N, V)$ is obtained by substituting the Sommerfeld expansion (4.3) of the internal energy into the ultra-relativistic identity $P = ◂+▸ η U ∕ (3 V)$ $P = η U ∕ (3 V)$ , cf. (3.4),

(4.5)

◂=▸ \hat{P} = ◂◽˙▸ {\hat{N}}^{η ∕ 3 + 1} ◂()▸ (1 + ◂⋅▸ \frac{π^{2}}{6} ◂/▸ \frac{η + 3}{η ◂◽˙▸ {(\hat{β} ◂◽˙▸ {\hat{N}}^{η ∕ 3})}^{2}} + \dots),

$\hat{P} = {\hat{N}}^{η ∕ 3 + 1} (1 + \frac{π^{2}}{6} \frac{η + 3}{η {(\hat{β} {\hat{N}}^{η ∕ 3})}^{2}} + \dots),$

where we have introduced the rescaled pressure variable

(4.6)

\hat{P} = ◂⋅▸ \frac{◂⋅▸ 3 (η + 3) a^{η}}{η} ◂/▸ \frac{◂◽˙▸ {(2 π)}^{3}}{4 π s} P .

$\hat{P} = \frac{3 (η + 3) a^{η}}{η} \frac{{(2 π)}^{3}}{4 π s} P .$

To obtain the volume in

V (T, P, N)

$V (T, P, N)$ parametrization, we solve (4.5) for

V

$V$ (which appears in the rescaled number density

\hat{N}

$\hat{N}$ , cf. (4.1)),

(4.7)

V = ◂⋅▸ ◂/▸ \frac{3 ◂◽˙▸ {(2 π)}^{3}}{4 π s} N ◂◽˙▸ {\hat{P}}^{- 3 ∕ (η + 3)} ◂()▸ (1 + ◂⋅▸ \frac{π^{2}}{2} \frac{1}{η ◂◽˙▸ {(◂◽˙▸ {\hat{P}}^{η ∕ (η + 3)} \hat{β})}^{2}} + \dots) .

$V = \frac{3 {(2 π)}^{3}}{4 π s} N {\hat{P}}^{- 3 ∕ (η + 3)} (1 + \frac{π^{2}}{2} \frac{1}{η {({\hat{P}}^{η ∕ (η + 3)} \hat{β})}^{2}} + \dots) .$

We substitute this into

S (T, V, N)

$S (T, V, N)$ in (4.4) to find the

S (T, P, N)

$S (T, P, N)$ representation of entropy,

(4.8)

S = ◂⋅▸ ◂◽˙▸ {\hat{P}}^{- η ∕ (η + 3)} \frac{1}{η} ◂/▸ \frac{π^{2}}{\hat{β}} N ◂()▸ (1 + ◂⋅▸ \frac{π^{2}}{30} ◂/▸ \frac{◂⋅▸ (2 η - 3) (7 η - 6)}{η^{2} ◂◽˙▸ {(\hat{β} ◂◽˙▸ {\hat{P}}^{η ∕ (η + 3)})}^{2}} + \dots) .

$S = {\hat{P}}^{- η ∕ (η + 3)} \frac{1}{η} \frac{π^{2}}{\hat{β}} N (1 + \frac{π^{2}}{30} \frac{(2 η - 3) (7 η - 6)}{η^{2} {(\hat{β} {\hat{P}}^{η ∕ (η + 3)})}^{2}} + \dots) .$

The isobaric heat capacity $◂=▸ C_{P} (T, P, N) = T S_{, T}$ $C_{P} (T, P, N) = T S_{, T}$ reads like $S$ $S$ in (4.8), with the second-order term multiplied by a factor of 3. By substituting the thermal Eq. (4.5) into $C_{P} (T, P, N)$ $C_{P} (T, P, N)$ , we find the $(T, V, N)$ $(T, V, N)$ parametrization of the isobaric heat capacity,

(4.9)

C_{P} = ◂⋅▸ V ◂◽˙▸ {\hat{N}}^{1 - η ∕ 3} ◂/▸ \frac{4 π s}{◂◽˙▸ {(2 π)}^{3}} \frac{1}{3 η} ◂/▸ \frac{π^{2}}{\hat{β}} (1 + ◂⋅▸ \frac{π^{2}}{30} ◂/▸ \frac{◂+▸ 54 - 99 η + 37 η^{2}}{η^{2} ◂◽˙▸ {(\hat{β} ◂◽˙▸ {\hat{N}}^{η ∕ 3})}^{2}} + \dots) .

$C_{P} = V {\hat{N}}^{1 - η ∕ 3} \frac{4 π s}{{(2 π)}^{3}} \frac{1}{3 η} \frac{π^{2}}{\hat{β}} (1 + \frac{π^{2}}{30} \frac{54 - 99 η + 37 η^{2}}{η^{2} {(\hat{β} {\hat{N}}^{η ∕ 3})}^{2}} + \dots) .$

By comparing this with the isochoric heat capacity, see after (4.4), we find

(4.10)

◂∼▸ C_{P} (T, V, N) \sim C_{V} + ◂⋅▸ V \frac{π^{4}}{9} ◂/▸ \frac{4 π s}{◂◽˙▸ {(2 π)}^{3}} ◂/▸ \frac{1}{η ◂◽˙▸ {\hat{N}}^{η - 1} ◂◽˙▸ {\hat{β}}^{3}} .

$C_{P} (T, V, N) \sim C_{V} + V \frac{π^{4}}{9} \frac{4 π s}{{(2 π)}^{3}} \frac{1}{η {\hat{N}}^{η - 1} {\hat{β}}^{3}} .$

The isothermal compressibility

κ_{T} = ◂+▸ - V_{, P} ∕ V

$κ_{T} = - V_{, P} ∕ V$ and isobaric expansion coefficient

◂=▸ α_{exp} = V_{, T} ∕ V

$α_{exp} = V_{, T} ∕ V$ are calculated from the series

V (T, P, N)

$V (T, P, N)$ in (4.7),

(4.11)

◂,▸ κ_{T} = ◂⋅▸ \frac{3}{η + 3} \frac{1}{P} ◂()▸ (1 + ◂⋅▸ \frac{π^{2}}{3} \frac{1}{◂◽˙▸ {(◂◽˙▸ {\hat{P}}^{η ∕ (η + 3)} \hat{β})}^{2}} + \dots), α_{exp} \sim ◂/▸ \frac{π^{2} ◂◽˙▸ a^{2 η}}{η ◂◽˙▸ {\hat{P}}^{2 η ∕ (η + 3)} β} .

$κ_{T} = \frac{3}{η + 3} \frac{1}{P} (1 + \frac{π^{2}}{3} \frac{1}{{({\hat{P}}^{η ∕ (η + 3)} \hat{β})}^{2}} + \dots), α_{exp} \sim \frac{π^{2} a^{2 η}}{η {\hat{P}}^{2 η ∕ (η + 3)} β} .$

Using the leading order of the thermal equation in (4.5) and (4.6), we find

(4.12)

◂/▸ \frac{α_{exp}^{2}}{β κ_{T}} \sim ◂⋅▸ \frac{π^{4}}{9} ◂/▸ \frac{4 π s}{◂◽˙▸ {(2 π)}^{3}} ◂/▸ \frac{1}{η ◂◽˙▸ {\hat{N}}^{η - 1} ◂◽˙▸ {\hat{β}}^{3}},

$\frac{α_{exp}^{2}}{β κ_{T}} \sim \frac{π^{4}}{9} \frac{4 π s}{{(2 π)}^{3}} \frac{1}{η {\hat{N}}^{η - 1} {\hat{β}}^{3}},$

so that the equilibrium relation

C_{P} = ◂+▸ C_{V} + ◂⋅▸ V T α_{exp}^{2} ∕ κ_{T}

$C_{P} = C_{V} + V T α_{exp}^{2} ∕ κ_{T}$ is satisfied, cf. (4.10).

To obtain the adiabatic compressibility, $κ_{S} = ◂+▸ - V_{, P} ∕ V$ $κ_{S} = - V_{, P} ∕ V$ , we need the $V (S, P, N)$ $V (S, P, N)$ representation of the volume factor. To this end, we solve $S (T, P, N)$ $S (T, P, N)$ in (4.8) for $\hat{β}$ $\hat{β}$ in leading order, $\hat{β} \sim ◂+▸ π^{2} N ∕ (η ◂◽˙▸ {\hat{P}}^{η ∕ (3 + η)} S)$ $\hat{β} \sim π^{2} N ∕ (η {\hat{P}}^{η ∕ (3 + η)} S)$ , and substitute this into $V (T, P, N)$ $V (T, P, N)$ in (4.7),

(4.13)

Outline

Tables (3)

Physica A: Statistical Mechanics and its Applications

White dwarf stars exceeding the Chandrasekhar mass limit

Highlights

Abstract

Keywords

1. Introduction

2. Fermionic spectral densities in a dispersive medium

2.1. Nonlinear ultra-relativistic dispersion relation

2.2. Quantized thermodynamic variables

3. Quantifying dispersion in the nearly degenerate quantum regime: fugacity expansion

4. Effect of electron dispersion on thermodynamic variables at low temperature and high density

4.1. Internal energy, entropy and isochoric heat capacity

4.2. Mechanical and thermal stability: thermal equation of state, isobaric heat capacity, isobaric expansion coefficient, isothermal and adiabatic compressibility