# Entropy and thermodynamical stability of white dwarfs

Jiří Adam, Jr.***Corresponding author †††E-mail address<EMAIL_ADDRESS>Institute of Nuclear Physics ASCR, CZ–250 68 Řež, Czech Republic  Emil
Truhlík‡‡‡E-mail address<EMAIL_ADDRESS>Institute of Nuclear Physics
ASCR, CZ–250 68 Řež, Czech Republic

###### Abstract

A structure of spherical white dwarfs is calculated for a non-zero
temperature. It is shown that the thermodynamical stability of the white dwarf
stars can be described naturally within the concept of the Helmholtz free
energy of the Coulomb fully ionized electron-ion plasma.

Entropy; Thermodynamics; Stability; White dwarfs

###### pacs:

23.40.Bw; 14.60.Cd; 67.85.Lm; 97.10.Ld; 97.20.Rp

##  I Introduction

Description of the white dwarf stars (WDs) often starts from the equation of
equilibrium in the Newtonian approximation, without the rotation and the
magnetic field included SC1 ; MC  §§§The importance of the general relativity
for a star with the mass $M$ and the radius $R$ is given by a compactness
parameter $x_{g}$ AYP ,
$x_{g}\,=\,r_{g}/R\,,\,\,\,r_{g}\,=\,2\mathrm{G}M/c^{2}\,\approx\,2.95\,M/M_{\odot}\,,$
(1) where $r_{g}$ is the Schwarzschild radius and $M_{\odot}$ is the mass of
the Sun. If one takes for $M$ the Chandrasekhar - Landau limit mass SC2 ; LDL
$M\,\approx\,1.4\,M_{\odot}$ and the radius $R\,\approx\,5\times 10^{3}$ km,
one obtains $x_{g}\,<<\,1$. However, the effects od the general relativity are
important for stabilizing very massive and fast rotating WDs SLSSAT .,

$\frac{d\,P}{d\,r}\,=\,-\frac{\mathrm{G}\,M(r)}{r^{2}}\,\rho\,,\,\,\,\frac{d\,M(r)}{d\,r}\,=\,4\,\pi\,r^{2}\,\rho\,,$
(2)

from which it follows,

$\frac{1}{r^{2}}\,\frac{d}{d\,r}\left(\frac{r^{2}}{\rho}\,\frac{d\,P}{d\,r}\right)\,=\,-4\,\pi\,\mathrm{G}\,\rho\,,$
(3)

where $P$ is the pressure, $\rho$ is the mass density and G is the Newton
gravitational constant. One gets explicitly from (3),

$\frac{1}{\rho}\,\frac{d^{2}\,P}{d\,r^{2}}\,+\,\frac{2}{\rho\,r}\,\frac{d\,P}{d\,r}\,-\,\frac{1}{\rho^{2}}\,\frac{d\,P}{d\,r}\,\frac{d\,\rho}{d\,r}\,+\,4\,\pi\,\mathrm{G}\,\rho\,=\,0\,.$
(4)

If $P$ as a function of $\rho$ (i.e., the equation of state (EoS)) is known,
the equation above can be cast into a second order differential equation for
the function $\rho(r)$.

For instance, it was shown in SC1  that writing the EoS in the form

$P\,=\,\mathrm{K}\,\rho^{(1+\frac{1}{n})}\,,$ (5)

and $\rho$ as

$\rho\,=\,\mathrm{\lambda_{c}}\,\theta^{n}\,,$ (6)

one gets EoS eq. (5) in the form,

$P\,=\,\mathrm{K}\,\mathrm{\lambda_{c}}^{(1+\frac{1}{n})}\,\theta^{(1+n)}\,,$
(7)

and from eq. (4) it then follows the famous Lane-Emden (LE) equation,

$\frac{1}{\xi^{2}}\,\frac{d}{d\,\xi}\,\left(\xi^{2}\,\frac{d\,\theta}{d\,\xi}\right)\,=\,0,$
(8)

where

$\xi\,=\,\frac{r}{a}\,,\,\,\,\,a\,=\,\sqrt{\frac{(n+1)}{4\,\pi}\,\frac{\mathrm{K}}{\mathrm{G}}\,\mathrm{\lambda_{c}}^{(\frac{1}{n}-1)}}\,.$
(9)

Further, if $\mathrm{\lambda_{c}}\,=\,\rho_{c}$, where $\rho_{c}\,=\,\rho(0)$,
then $\theta(0)\,=\,1$ and
$\left(\frac{d\,\theta}{d\,\xi}\right)\big{|}_{\xi=0}\,=\,0$.

Eq. (8) was discussed in detail in SC1 , where it was shown that for
$n\,=\,0,1,5$ one gets explicit solutions. The model with the EoS (7), known
as polytropic model, has been frequently used in the astrophysics, in
particular for treating the structure of WDs. In this case, the star is
considered as a dense Coulomb plasma of nuclei in a charge compensating
background of degenerate electron Fermi gas, where the electron - ion
interaction is taken in this polytropic approximation. However, for studying
the WDs structure, the polytropic model is of restricted use, because it
fairly approximates the equation of state only in the extreme non-relativistic
limit for $\lambda_{c}\,<<\,10^{6}$ g/cm3 (with $n$ = 3/2) and in the extreme
relativistic limit for $\lambda_{c}\,>>\,10^{6}$ g/cm3 (with $n$ = 3). Recall
that for $n$ = 3 the mass of the WDs is uniquely given by the Chandrasekhar -
Landau limit mass SC2 ; LDL .

Another problem with the polytropic model is its stability. As discussed at
length in JPM , the squared Brunt-Väisälä frequency $N^{2}\,=\,0$ in this
model. Only if $N^{2}\,>\,0$, the fluid is stable, if $N^{2}\,<\,0$, it is
convectively unstable. So the fluid described by the polytropic model is only
neutrally stable. According to Ref. JPM , no magnetic equilibrium exists in
this case.

Stable stratification has an important influence on stellar magnetic
equilibria and their stability. As discussed at length in Ref. TARMM , the
study of the magnetic equilibria in stable stratified stars has a long story.
It follows from these studies that simple magnetic field configurations are
always unstable. In their study, the authors of Ref. TARMM  constructed simple
analytic models for axially symmetric magnetic fields, compatible with
hydromagnetic equilibria in stably stratified stars, with both poloidal and
toroidal components of adjustable strength, as well as the associated
pressure, density and gravitational potential perturbations, which maked them
possible to study directly their stability. It turned out that the instability
of toroidal field can be stabilized by a poloidal field containing much less
energy than the former, as given by the condition
$E_{\mathrm{pol}}/E_{\mathrm{tor}}\,\gtrsim\,2aE_{\mathrm{pol}}/E_{\mathrm{grav}}$,
where $E_{\mathrm{pol}}$ and $E_{\mathrm{tor}}$ are the energies of the
poloidal and toroidal fields, respectively and $E_{\mathrm{grav}}$ is the
gravitational binding energy of the star. It was found in TARMM  that
$a\approx 7.4$ for main-sequence stars, which compares with $a\approx 10$
obtained by Braithwaite JB , using the method of numerical simulations. But
the results for the neutron stars differ by a factor of $\approx$ 4.

The possibility of compensation of instabilities of toroidal fields by a
relatively weak poloidal field was earlier studied by Spruit HCS .

As to the instabilities in the poloidal field, stable stratification is of
less help for eliminating them TARMM . A relatively stronger toroidal field
would be needed in order to stabilize it JB .

As it follows from the discussion in Ch. 3 of the monograph CSBK , the
necessary and sufficient condition for the thermodynamical stability of stars
with optically thick media is the positive gradient of the entropy,

$\frac{d\,S}{d\,r}\,>\,0\,.$ (10)

Respecting this criterion of the stability, the authors of Ref. JPM
considered the star as a chemically uniform, monoatomic, classical ideal gas,
described in the polytropic model with the EoS (5) as
$P\,\sim\,\rho^{\gamma}$, where $\gamma\,=\,4/3\,=\,1\,+\,1/n\,\,\,(n\,=\,3)$
and with the specific entropy

$s\,\approx\,ln(P/\rho^{\Gamma})\,+\,const\ ,\qquad{\rm
where}\quad\Gamma\,=\,\left(\frac{\partial\,P}{\partial\,ln\,\rho}\right)_{ad}\,=\,\frac{5}{3}\
,$

for which it holds

$\frac{d\,s}{d\,r}\,>0\ .$

In this model, applied to the Ap stars, the constructed magnetic equilibrium
turns out to be stable. However, for $\gamma\,=\,5/3\,\,\,(n\,=\,3/2)$, one
obtains $\gamma\,=\,\Gamma$, $d\,s/d\,r\,=\,0$ and the magnetic equilibrium is
unstable. Similar calculations have recently been done in Ref. LB .

This model cannot be applied to the study of WDs, because they consist of
plasma containing the mix of the fully ionized atoms and of the fully
degenerate electron gas.

The polytropic model was used to describe super - Chandrasekhar strongly
magnetized WDs (SMWDs) in Refs. UDM1 ; UDM2 ; UDM3  and also in Ref. BB . It
was shown in BB , that axisymmetric configurations with the poloidal or
toroidal fields are unstable and it was concluded that the long lived super -
Chandrasekhar SMWDs are unlikely to occur in nature.

In Ref. DC , the authors developed a detailed and self - consistent numerical
model with a poloidal magnetic field in the SMWDs. In their model, the
rotation, the effects of general relativity and a realistic EoS were taken
into account and extensive stability analysis of such objects was performed.
As a result, it was found that the SMWDs could potentially exist, but their
stability would be mainly limited by the onset of the electron capture and of
the picnonuclear reactions. However, it should be noted that the condition of
the thermodynamical stability (10) was not checked in Ref. DC .

In the recent paper NCLP , the authors have studied the influence of the
electron capture on the stability of the WDs. They used the EoS in the
polytropic form (5) in the ultrarelativistic limit
$P\,\approx\,K_{0}\,\rho^{4/3}$, with $A$ and $Z$ dependent constant $K_{0}$.
Besides, the electrostatic correction and the correction due to the electron-
ion interaction are also included in $K_{0}$. This allowed them to calculate
the threshold density $\rho_{\beta}$ for the capture of electrons and to set
the lower bound for the radius of the WDs. It was also found that the electron
capture reduces the mass of WDs by 3 - 13% . Solving the Einstein - Maxwell
equations, with the magnetism of the dense matter included, has shown that the
magnetized WDs with the polar magnetic field stronger than $10^{13}$ G could
be massive enough to explain overluminous type Ia supernova. It was also found
that the pure poloidal magnetic field is unstable. Actually, this result
follows from the fact that the polytropic model is only neutrally stable, as
it has already been discussed above.

In Ref. LB1 , the authors investigated the evolution of isolated massive,
highly magnetized and uniformly rotating WDs, under angular momentum loss
driven by magnetic dipole braking. They computed general relativistic
configurations of such objects using the relativistic Feynman - Metropolis -
Teller equation of state for the WD matter. One of the interesting results is
obtained for rotating magnetized WD with the mass which is not very close to
the non - rotating Chandrasekhar - Landau mass. Such a WD evolves slowing down
and behaving as an active pulsar of the type of soft gamma - repeater and
anomalous X - ray pulsar MM ; IB ; KBLI ; JAR ; JGC ; RVL ; VBB ; TRM . Let us
note that it is not clear if the condition of the thermodynamical stability
(10) is fulfilled in Ref. LB1 .

Realistic model of stars as systems of magnetized fully - ionized plasma has
been developed in CP  \- ASJ . In the model, considered in CP  \- PC3 , an
analytical EoS is derived from the Helmholtz free energy of the system of
magnetized fully - ionized atoms and of the degenerate electron gas, whereas,
in ASJ  also positrons were included. Such an EoS covers a wide range of
temperatures and densities, from low-density classical plasmas to
relativistic, quantum plasma conditions.

Starting in Sect. II.1 from the equation of equilibrium (4), and using in
Sect. II.2 the function $P(\rho)$, obtained from the EoS mentioned above, we
get the second order equation for the matter density $\rho$. Solving this
equation, we study the structure of corresponding WDs for a representative
series of values of the central density $\rho_{c}$, chosen from the interval
$10^{4}\,\mathrm{g/cm^{3}}\,<<\,\rho_{c}\,<<\,2\times\,10^{9}\,\mathrm{g/cm^{3}}$,
and simultaneously using the electron and ion entropies from Sects. III.1 and
III.2, we show that the criterion of the thermodynamical stability (10) is
fulfilled in all cases (see FIG. 2 and FIG. 3). Besides, using LE eq.(8), we
calculate for comparison for the extreme non - relativistic and extreme
relativistic values of $\rho_{c}$ the mass and radius of corresponding WDs,
which are presented in TABLE I.

In Appendix A, we discuss different ways of calculating the structure of WDs
and summarize relations for the scaling parameter $\mathrm{a_{s}}$ and for the
LE approximation. In Appendix B, we express the functions $f_{1}$ and $f_{2}$,
entering the pressure $P(\rho)$ in terms of the Fermi - Dirac integrals and in
Appendix C, we briefly describe how to decompose the thermodynamical
quantities for free electrons into series in powers of
$k_{\mathrm{B}}T/\tilde{E}_{F}$, where $\tilde{E}_{F}=\mu_{\mathrm{e}}(T=0)$
is the Fermi energy with the rest mass contribution subtracted.

Our results show that the realistic model developed in CP  \- ASJ  is a good
starting model to be applied to construct the WDs with stable magnetic fields.

##  II Methods and input

###  II.1 Modified equation of stability

Let us write eq. (4) in the form,

$\frac{d^{2}\,P}{d\,r^{2}}\,+\,\frac{2}{r}\,\frac{d\,P}{d\,r}\,-\,\frac{1}{\rho}\,\frac{d\,P}{d\,r}\,\frac{d\,\rho}{d\,r}\,+\,4\,\pi\,\mathrm{G}\,\rho^{2}\,=\,0\,.$
(11)

Considering now the pressure $P$ as a function (solely) of the density $\rho$,
eq. (11) can be transformed into the 2nd order differential equation for
$\rho$,

$\frac{d^{2}\,\rho}{d\,r^{2}}\,+\,f_{1}(\rho)\,\left(\frac{d\,\rho}{d\,r}\right)^{2}\,+\,\frac{2}{r}\,\frac{d\,\rho}{d\,r}\,+\,\frac{4\,\pi\,\mathrm{G}}{f_{2}(\rho)}\,\rho^{2}\,=\,0\,,$
(12)

where

$f_{1}(\rho)\,=\,\left(\frac{d^{2}\,P}{d\,\rho^{2}}\right)\bigg{/}\left(\frac{d\,P}{d\,r}\right)\,-\,\frac{1}{\rho}\,,\,\,\,\,\,f_{2}(\rho)\,=\,\frac{d\,P}{d\,\rho}\,.$
(13)

Next we set

$r\,=\,\mathrm{a_{s}}\,x\,,\,\,\,\rho\,=\,\lambda_{c}\,y\,.$ (14)

If $\lambda_{c}$ is the matter density in the center of the star
$\lambda_{c}=\rho(0)$, then

$y(0)\,=\,0\,,\,\,\,\frac{d\,y}{d\,r}\big{|}_{0}\,=\,0\,.$ (15)

From relations

$\frac{dP}{d\rho}\,=\,\frac{dP}{\lambda_{c}\,dy}\,,\,\,\,\frac{d^{2}P}{d\rho^{2}}\,=\,\frac{d^{2}P}{\lambda_{c}^{2}\,dy^{2}}\,,$
(16)

it follows

$f_{1}(\rho)\,=\,\frac{1}{\lambda_{c}}\,f_{1}(y)\,,\,\,\,f_{2}(\rho)\,=\,\frac{1}{\lambda_{c}}\,f_{2}(y)\,.$
(17)

Then, in terms of the new variables (14), eq. (12) becomes,

$\frac{d^{2}\,y}{d\,x^{2}}\,+\,f_{1}(y)\,\left(\frac{d\,y}{d\,x}\right)^{2}\,+\,\frac{2}{x}\,\frac{d\,y}{d\,x}\,+\,\frac{\mathrm{C}}{f_{2}(y)}\,y^{2}\,=\,0\,,$
(18)

with

$\mathrm{C}\,=\,4\,\pi\,\mathrm{G}\,\mathrm{a_{s}}^{2}\,\lambda_{c}^{2}\,.$
(19)

We solved eq. (18) by the standard 4th order Runge-Kutta method for various
values of the central matter density $\lambda_{c}$. The choice of the value of
the scaling parameter $\mathrm{a_{s}}$ is discussed in Appendix A. In our
calculations we use the value (see (70))

$\mathrm{a}_{s}\,=\,8686.26\,{\rm km}.$ (20)

###  II.2 The Helmholtz free energy and the EoS

Here we follow Ref. PC3 , where the Helmholtz free energy $F$ of the plasma is
defined as,

$\displaystyle F\,$ $\displaystyle=$
$\displaystyle\,F^{(e)}_{id}\,+\,F^{(i)}_{id}\,+\,F_{ii}\,+\,F_{ee}\,+\,F_{ie}\,,$
(21) $\displaystyle\simeq$ $\displaystyle F^{(e)}_{id}\,+\,F_{lat}\ ,\quad
F_{lat}=\,F^{(i)}_{id}\,+\,F_{ii}\ .$

On the first line, the first two terms correspond to the ideal part of the
free energy of ions and electrons, and the last three terms correspond to the
ion - ion, electron - electron, and ion - electron interactions, respectively.
The second line corresponds to the approximation adopted in this paper: the
sum of the 2nd and 3rd terms of the 1st line is denoted as $F_{lat}$ and
evaluated as in Ref. PC3  in one - component plasma model (OCP) in the regime
of Coulomb crystal. Less important and more uncertain contributions $F_{ee}$
and $F_{ie}$ are skipped. It should be noted that these contributions are less
important only in degenerate plasmas. In the non-degenerate regime, they can
be of the same order of magnitude as $F_{ii}$, especially if Z is not large
PC3 .

FIG. 1: Dependence of the electron pressure $P$ $[g/cm\cdot sec^{2}]$ on the
matter density $\rho$ $[g/cm^{3}]$ and the temperature $T$ $[K]$. In the upper
panel, somewhat wider ranges of densities and temperatures are considered,
while in the lower one, they are relevant for the carbon WDs.

The particle density $N$, internal energy $U$, the entropy $S$ and Helmholtz
free energy $F$ are related by:

$\displaystyle T\,S$ $\displaystyle=$ $\displaystyle U+P\,V-\mu\,N\ ,$ (22)
$\displaystyle F$ $\displaystyle=$ $\displaystyle U-T\,S=\mu\,N-P\,V\ .$ (23)

In the lower panel of FIG. 1, we present the dependence of the electron
pressure on $\rho$ for various values of $T$ for the carbon WDs with
$A\,=\,12\,,\ Z\,=\,6$. In this case, the dependence of the electron presure
on the matter density is $T$ \- independent ¶¶¶Similar $P^{(\mathrm{e})}-\rho$
dependence on various values of $T$ was presented in Fig. 1 of Ref. KB ..

In general, the pressure and the entropy can be calculated from $F$ by:

$P\,=\,-\left(\frac{\partial\,F}{\partial\,V}\right)_{T}\,,\qquad
S\,=\,-\left(\frac{\partial\,F}{\partial\,T}\right)_{V}\ .$

But for free electrons the number density $n_{\mathrm{e}}$, pressure
$P^{(\mathrm{e})}_{id}$ and energy $U$ are known explicitly:

$\displaystyle n_{\mathrm{e}}\,$ $\displaystyle\equiv$
$\displaystyle\frac{N_{e}}{V}=\,\mathrm{c_{n}}\,\big{[}I_{1/2}(\chi_{\mathrm{e}},\tau)+\,\tau\,I_{3/2}(\chi_{\mathrm{e}},\tau)\big{]}\,,$
(24) $\displaystyle P^{(\mathrm{e})}_{id}\,$ $\displaystyle=$
$\displaystyle\,\mathrm{c_{p}}\,\big{[}I_{3/2}(\chi_{\mathrm{e}},\tau)+\,\frac{\tau}{2}\,I_{5/2}(\chi_{\mathrm{e}},\tau)\big{]}\,,$
(25) $\displaystyle U^{(\mathrm{e})}_{id}\,$ $\displaystyle=$
$\displaystyle\,\mathrm{c_{e}}\,\big{[}I_{3/2}(\chi_{\mathrm{e}},\tau)+\,\tau\,I_{5/2}(\chi_{\mathrm{e}},\tau)\big{]}\equiv{\cal
E}_{e}\,V\,,$ (26)

where $\chi_{\mathrm{e}}\,=\,\mu_{\mathrm{e}}\beta$, $\mu_{\mathrm{e}}$ is the
electron chemical potential without the rest energy $m_{\mathrm{e}}c^{2}$ and
dimensionless $\tau\,=\,T/T_{\mathrm{r}}\,,$ with
$T_{\mathrm{r}}\,=\,m_{\mathrm{e}}\,c^{2}/k_{\mathrm{B}}\,\simeq\,5.9301\times
10^{9}\,\mathrm{K}$ (from the Boltzmann constant $k_{B}\simeq 8.617\times
10^{-11}$MeV/K). In the last relation we introduce the electron energy density
${\cal E}_{e}=U^{(\mathrm{e})}_{id}/V$. Further:

$\displaystyle\mathrm{c_{n}}\,$ $\displaystyle=$
$\displaystyle\,\frac{\sqrt{2}}{\pi^{2}\hbar^{3}}\left(\frac{m_{\mathrm{e}}}{\beta}\right)^{3/2}=3\sqrt{2}\rho_{0}\,\tau^{3/2}\,,\quad\beta\,=\,\frac{1}{k_{\mathrm{B}}T}\
,\quad\rho_{0}=\frac{1}{3\pi^{2}\,\lambda_{e}^{3}}\ ,$
$\displaystyle\mathrm{c_{p}}\,$ $\displaystyle=$
$\displaystyle\,\frac{(2m_{\mathrm{e}})^{3/2}}{3\pi^{2}\hbar^{3}\beta^{5/2}}=2\sqrt{2}\,m_{e}c^{2}\,\rho_{0}\,\tau^{5/2}\
,$ $\displaystyle\mathrm{c_{e}}=\frac{3}{2}\mathrm{c_{p}}\,$ $\displaystyle=$
$\displaystyle\,\frac{\sqrt{2}\,m_{\mathrm{e}}^{3/2}}{\pi^{2}\hbar^{3}\beta^{5/2}}=3\sqrt{2}\,m_{e}c^{2}\,\rho_{0}\,\tau^{5/2}\
,$

where $\lambda_{e}$ is the electron Compton length (69).

The generalized (relativistic) Fermi-Dirac integrals
$I_{\nu}(\chi_{\mathrm{e}}\,,\,\tau)$ are defined as follows:

$I_{\nu}(\chi_{\mathrm{e}},\tau)\,=\,\int\limits_{0}^{\infty}\frac{x^{\nu}(1\,+\,\tau
x/2)^{1/2}}{e^{(x\,-\,\chi_{\mathrm{e}})}\,+\,1}\mathrm{d}x\,.$ (27)

The functions $f_{1}(y)$ and $f_{2}(y)$ of previous section are presented in
terms of the Fermi - Dirac integrals $I_{\nu}(\chi_{\mathrm{e}}\,,\,\tau)$ in
Appendix B.

One obtains the EoS by referring to the neutrality of the plasma, which
provides the equation between the mass density $\rho$ of the ion and the
electron number density $n_{\mathrm{e}}$,

$n_{\mathrm{e}}\,=\,Zn_{i}\,=\,\frac{Z\rho}{Am_{u}}=\frac{\rho}{\mu_{u}}\,,$
(28)

where $Z$ is the ion charge number, $A$ is the mass number, $n_{i}$ is the ion
number density,
$m_{u}$ is the atomic mass unit and $\mu_{u}=A\,m_{u}/Z$ is introduced in
(68). Given values of $A\,,Z$ and $\rho$ one gets $n_{e}$ from the neutrality
condition (28) and then reversing eq. (24) for a given temperature $T$,
obtains the value of $\chi_{\mathrm{e}}$. Substituting this
$\chi_{\mathrm{e}}$ into eq. (25), one gets the value of the electron pressure
$P^{(\mathrm{e})}_{id}$ corresponding to the given value of $\rho$.

Table 1:  In the upper part of this table we present the results of calculations of the radii and masses of the WDs for representative values of the central mass densities $\lambda_{c}$. We have solved the equation of stability (18) with the scaling parameter $a_{s}$ of eq. (20). The EoS is obtained within the concept of the Helmholtz free energy of the Coulomb plasma, using eqs. (25), (24) and (28). The central fractional Fermi momentum eq. (34) $x_{rc}$ is also listed, since it indicates whether the dynamics is non-relativistic or relativistic. In the lower part of the table (below the middle double line) we present the radii and masses obtained from the LE eq. (8), which is based on the polytropic EoS eq. (7). $R_{0}$ and $M_{0}$ are radii and masses in this approximation, $r_{0}$ and $m_{0}$ the corresponding dimensionless quantities, introduced at the end of Appendix A in eqs. (78-80) and (81-83). Comparing these (LE) results with our numerical ones (which do not employ a polytropic EoS) one can see that they are close to each other in the non-relativistic case, but in the relativistic regime they approach each other only in the extreme relativistic limit of very large densities. model | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---
$\lambda_{c}[g/cm^{3}]$ | $10^{4}$ | $10^{5}$ | $10^{6}$ | $10^{7}$ | $10^{8}$ | $2\cdot 10^{9}$
$x_{rc}$ | $0.173$ | $0.372$ | $0.801$ | $1.36$ | $3.72$ | $10.1$
$R_{WD}$ [km] | $2.40\cdot 10^{4}$ | $1.63\cdot 10^{4}$ | $1.09\cdot 10^{4}$ | $7.04\cdot 10^{3}$ | $4.30\cdot 10^{3}$ | $2.05\cdot 10^{3}$
$M/M_{\odot}$ | 0.048 | 0.146 | 0.391 | 0.816 | 1.15 | 1.37
n | $3/2$ | $3/2$ |  | 3 | 3 | 3
$r_{0}$ | $2.78$ | $1.90$ |  | $1.79$ | $0.830$ | $0.306$
$m_{0}$ | $0.0185$ | $0.0583$ |  | $0.542$ | $0.542$ | $0.542$
$R_{0}$ [km] | $2.42\cdot 10^{4}$ | $1.65\cdot 10^{4}$ |  | $1.55\cdot 10^{4}$ | $7.21\cdot 10^{3}$ | $2.66\cdot 10^{3}$
$M_{0}/M_{\odot}$ | 0.050 | 0.157 |  | 1.46 | 1.46 | 1.46

Apart from the electron pressure, Chamel and Fantina CHFA  have taken into
account also the lattice pressure $P_{\mathrm{lat}}$, derived from the
dominant static-lattice (Madelung) part of the ion free energy PC3 , i.e.
approximating $F_{lat}\simeq F_{M}$:

$F_{\mathrm{M}}\,=\,N_{i}k_{B}T\mathrm{C}_{\mathrm{M}}\,\Gamma\,,\quad
N_{i}\,=\,n_{i}V\ ,$ (29)

and for the bcc crystal the Madelung constant is BPY :

$\displaystyle\mathrm{C}_{\mathrm{M}}$ $\displaystyle=$
$\displaystyle-0.89592925568\,.$ (30)

The ion coupling parameter

$\Gamma\,=\,\frac{(Ze)^{2}}{a_{i}k_{\mathrm{B}}T}\,,$ (31)

is given in terms of the ion sphere radius $a_{i}\,=\,\big{(}\frac{4}{3}\pi
n_{i}\big{)}^{-1/3}$.

As it can be seen from Table III CHFA , the effect of the pressure
$P_{\mathrm{lat}}$ on the mass of the WDs containing the light elements is
only few percents and we do not take it into account.

The results of comparative calculations are presented in Table 1.

##  III Entropy and its gradient in WDs

In this section we calculate one - electron and ion entropies and their
gradients within the above introduced Coulomb plasma theory based on the
Helmholtz free energy concept. We show on the representative set of WDs that
both entropies are positive and that their gradients satisfy the condition of
the thermodynamical stability, required by eq. (10). We will deal with a
reduced dimensional entropy defined (both for electrons and ions) as:

$\displaystyle\hat{s}$ $\displaystyle=$ $\displaystyle\frac{S}{k_{B}\,N}\ ,$
(32)

where $N$ is a number of particles and $k_{\mathrm{B}}$ is the Boltzman
constant. We will evaluate and plot a derivative of various contributions to
this reduced entropy in respect to the dimensionless radius $x$ (see (14)),
i.e., $d\hat{s}/d\,x$.

###  III.1 The electron entropy

For the free electrons it follows from (32) and relations (24-26):

$\displaystyle\hat{s}_{e}$ $\displaystyle=$
$\displaystyle\frac{1}{k_{\mathrm{B}}T}\frac{U^{(e)}_{id}+P^{(e)}_{id}\,V-N_{e}\,\mu_{e}}{V\,n_{e}}=\frac{1}{k_{\mathrm{B}}T}\,\left(\frac{{\cal
E}_{e}+P^{(e)}_{id}}{n_{e}}-\mu_{e}\right)$ (33) $\displaystyle=$
$\displaystyle\frac{5\,I_{3/2}(\chi_{e},\tau)+4\tau\,I_{5/2}(\chi_{e},\tau)}{3(I_{1/2}(\chi_{e},\tau)+\tau\,I_{3/2}(\chi_{e},\tau))}-\chi_{e}\simeq\pi^{2}\,\tau\,\frac{\epsilon_{F}}{x^{2}_{\mathrm{r}}}\,,$

where $n_{\mathrm{e}}$ is the electron density, $\tau\,=\,T/T_{\mathrm{r}}$,
$T_{\mathrm{r}}\,=\,m_{\mathrm{e}}c^{2}/k_{\mathrm{B}}$. Further,
$p_{\mathrm{F}}\,=\,\hbar(3\pi^{2}n_{\mathrm{e}})^{1/3}$ is the electron Fermi
momentum and the dimensionless Fermi momentum $x_{r}$ and energy
$\epsilon_{F}$ are

$\displaystyle x_{\mathrm{r}}=\frac{p_{\mathrm{F}}}{m_{\mathrm{e}}c}\
,\quad\epsilon_{F}=\sqrt{1+x^{2}_{\mathrm{r}}}\equiv\tilde{\epsilon}_{F}+1\ .$
(34)

The last equation in eq. (33) is obtained by the Sommerfeld expansion (see
Appendix C), it agrees with equation (6) in PC2 . We checked numerically that
for our calculations it is sufficient to take the termodynamic quantities in
the Sommerfeld approximation (SA).

As for the derivative of the electron entropy in respect to the dimensionless
WD radius $x$, let us first consider it in a more transparent SA. Using the
charge neutrality of the plasma (28), the electron Fermi momentum can be
connected to the matter density $\rho$ as follows,

$p_{\mathrm{F}}\,=\,\hbar\,\bigg{(}\frac{3\pi^{2}\rho}{\mu_{u}}\bigg{)}^{1/3}\equiv
D\,\rho^{1/3}\ ,\quad
D\,=\,\hbar\bigg{(}\frac{3\pi^{2}}{\mu_{u}}\bigg{)}^{1/3}\,.$ (35)

Since $\rho$ is decreasing to the surface of the WD, $p_{\mathrm{F}}$ and
$x_{\mathrm{r}}$ are also decreasing, and it follows from eq. (33) that
$\hat{s}_{\mathrm{e}}$ is increasing. In other words, the specific one-
electron entropy is stratified. With the help of relations:

$x_{r}=\frac{D}{m_{e}c}\,\rho^{1/3}\
,\qquad\epsilon_{F}=\frac{\sqrt{(m_{\mathrm{e}}c)^{2}\,+\,D^{2}\rho^{2/3}}}{m_{e}c}\
,$

the electron entropy (33) can be transformed to the form, suitable for
calculations,

$\hat{s}_{\mathrm{e}}\,=\,\frac{\pi^{2}k_{\mathrm{B}}T}{cD^{2}\rho^{2/3}}\,\sqrt{(m_{\mathrm{e}}c)^{2}\,+\,D^{2}\rho^{2/3}}\,,$
(36)

It is convenient to write the gradient of $\hat{s}_{\mathrm{e}}$ as a product:

$\displaystyle\frac{d\hat{s}_{\mathrm{e}}}{dx}\,$ $\displaystyle=$
$\displaystyle\,\left(\rho\,\frac{d\hat{s}_{\mathrm{e}}}{d\rho}\right)\cdot\,\frac{1}{\rho}\frac{d\rho}{dx}\
,\qquad\frac{1}{\rho}\frac{d\rho}{dx}<0\ ,$ (37)
$\displaystyle\rho\frac{d\hat{s}_{\mathrm{e}}}{d\rho}\,$ $\displaystyle=$
$\displaystyle\,-\frac{\pi^{2}k_{\mathrm{B}}T}{3cD^{2}\,\,\rho^{2/3}}\,\frac{2(m_{\mathrm{e}}c)^{2}+D^{2}\rho^{2/3}}{\sqrt{(m_{\mathrm{e}}c)^{2}\,+\,D^{2}\rho^{2/3}}}=-\frac{\pi^{2}\,\tau}{3}\,\frac{2+x_{r}^{2}}{x_{r}^{2}\,\epsilon^{2}_{F}}<0\
.$ (38)

Obviously, both terms are dimensionless and negative, hence their product is
dimensionless and positive:

$\frac{d\hat{s}_{\mathrm{e}}}{dx}\,>\,0\,.$ (39)

It should be noted, that our calculations respect the criterion of the strong
degeneracy,

$\theta\,=\,T/T_{\mathrm{F}}\,<<\,1\,,\quad{\rm where}\
T_{\mathrm{F}}\,=\,\frac{\tilde{E}_{\mathrm{F}}}{k_{\mathrm{B}}}\,,$ (40)

and
$\tilde{E}_{\mathrm{F}}=E_{\mathrm{F}}-m_{e}c^{2}=\,c[(m_{\mathrm{e}}c)^{2}\,+\,p^{2}_{\mathrm{F}}]^{1/2}\,-\,m_{\mathrm{e}}\,c^{2}=m_{e}c^{2}\,\tilde{\epsilon}_{\mathrm{F}}$
is the Fermi energy with the rest mass contribution subtracted.

Due to very good termal conductivity of the WD the temperature $T$ (and hence
$\tau=T/T_{r}$) are nearly constant inside the WD, with the exception of a
thin skip at its surface. Therefore, in our calculations we consider $\tau$ to
be independent of the radius $x$.

We have also checked that the empirical factor
$(1\,+\,\Delta\tilde{\epsilon}/\tilde{\epsilon})^{-1}$ PC2 , minimizing the
numerical jump of the transition between the fit for $\chi_{\mathrm{e}}<14$
and the Sommerfeld expansion for $\chi_{\mathrm{e}}>14$, did not lead in our
calculations to any sizeable effect.

Let us now briefly mention equation for the derivative of the electron entropy
following from the full form of (33). For easier comparison with equations
(37,38) we can again use the factorization (37), where the 2nd term is now
(with the help of (24) and (28)):

$\displaystyle\rho\frac{d\hat{s}_{\mathrm{e}}}{d\rho}\,$ $\displaystyle=$
$\displaystyle
n_{e}\,\frac{d\hat{s}_{\mathrm{e}}}{dn_{e}}=\frac{n_{e}}{n^{\prime}_{e}}\cdot\,\hat{s}^{\prime}_{e}\
,$ (41) $\displaystyle n^{\prime}_{e}$ $\displaystyle\equiv$
$\displaystyle\frac{dn_{e}}{d\chi_{e}}=I^{\prime}_{1/2}+\tau\,I^{\prime}_{3/2}\
,\qquad I^{\prime}_{\nu}\equiv\frac{d\,I_{\nu}(\chi_{e},\tau)}{d\chi_{e}}\ ,$
$\displaystyle\hat{s}^{\prime}_{e}$ $\displaystyle\equiv$
$\displaystyle\frac{d\hat{s}_{e}}{d\chi_{e}}=\frac{(I_{1/2}+\tau\,I_{3/2})(5I^{\prime}_{3/2}+4\tau
I^{\prime}_{5/2})-(I^{\prime}_{1/2}+\tau\,I^{\prime}_{3/2})(5I_{3/2}+4\tau
I_{5/2})}{3(I_{1/2}+\tau\,I_{3/2})^{2}}-1\ .$

To calculate (41) one needs derivatives of the Fermi-Dirac integrals
$I_{\nu}(\chi,\tau)$ in respect to $\chi$. It is also not obvious from the
general equation that (41) is negative. Nevertheless, we checked numerically
that in our calculations the general equation (41) agrees very well with the
approximate one (38).

###  III.2 The ion entropy

As for the ions, we consider them in the crystalline phase, in which they are
arranged in the body-centered cubic (bcc) Coulomb lattice (see Sect. 3.2.2 of
Ref. PC3 ).

In this state, $T<T_{\mathrm{m}}$, where $T_{\mathrm{m}}$ is the melting
temperature. For the one-component Coulomb plasma, it is obtained from the
relation,

$\Gamma_{\mathrm{m}}\,=\,2.2747\times
10^{5}\,\frac{Z^{5/3}}{T_{\mathrm{m}}}\,\bigg{(}\rho\frac{Z}{A}\bigg{)}^{1/3}\,,$
(42)

where $\Gamma_{\mathrm{m}}\,=\,175\,\pm\,0.4$ PC1 .

Beyond the harmonic-lattice approximation (29), the reduced dimensionless one-
ion free energy is given by:

$\displaystyle f_{\mathrm{lat}}(\Gamma,\eta)$ $\displaystyle\equiv$
$\displaystyle\frac{F_{\mathrm{lat}}}{N_{\mathrm{i}}k_{\mathrm{B}}T}=C_{\mathrm{M}}\,\Gamma\,+\,1.5\,u_{1}\,\eta\,+\,f_{\mathrm{th}}\,+\,f_{\mathrm{ah}}\
.$ (43)

The first three terms describe the harmonic lattice model BPY  and
$f_{\mathrm{ah}}$ is the anharmonic correction to the Coulomb lattice.
Further, $C_{\mathrm{M}}$ is the Madelung constant (30) and
$u_{1}\,=\,0.5113875$. The parameter $\eta$, determining the importance of the
quantum effects in a strongly coupled plasma, is PC3 :

$\eta\,\equiv\,7.835\frac{Z}{A}\cdot\frac{\sqrt{\rho}}{T}\,\times 10^{3}\,.$
(44)

The ion coupling parameter $\Gamma\sim 1/T$ is defined by (31).

For $f_{\mathrm{th}}$ we adopt the following fitting formula, used in the
Appendix B.2 of Ref. PC3 :

$f_{\mathrm{th}}(\eta)\,=\,\sum_{i=1}^{3}ln\big{(}1-e^{-\alpha_{i}\eta}\big{)}-\frac{A(\eta)}{B(\eta)}\,,$
(45)

where
$\alpha_{1}\,=\,0.932446,\,\alpha_{2}\,=\,0.334547,\,\alpha_{3}\,=\,0.265764$
and

$\displaystyle A(\eta)\,=\,\sum_{i=1}^{7}a_{i}\,\eta^{m_{i}}\ ,\quad
B(\eta)\,=\,\sum_{i=1}^{8}b_{i}\,\eta^{n_{i}}\ .$ (46)

Table 2:  In this table, the input data for eqs. (46) are presented. i | ai | mi | bi | ni
---|---|---|---|---
1 | 1.0 | 0 | 261.66 | 0
2 | 0.1839 | 1 | 7.07997 | 2
3 | 0.593 586 | 2 | 4.094 84$\times 10^{-2}$ | 4
4 | 5.4814$\times 10^{-3}$ | 3 | 3.973 55$\times 10^{-4}$ | 5
5 | 5.01813$\times 10^{-4}$ | 4 | 5.11148$\times 10^{-5}$ | 6
6 | 3.9247$\times 10^{-7}$ | 6 | 2.19749$\times 10^{-6}$ | 7
7 | 5.8356$\times 10^{-11}$ | 8 | 1.866985$\times 10^{-9}$ | 9
8 | - | - | 2.78772$\times 10^{-13}$ | 11

For the anharmonic correction of the Coulomb lattice $f_{\mathrm{ah}}$, we use
the anharmonic contribution to the one-ion entropy from the Sect. 4 of the
recent work BC .

Using eq. (43), we calculate the dimensionless one-ion entropy as,

$\displaystyle\hat{s}_{i}(\Gamma,\eta)$ $\displaystyle=$
$\displaystyle\,-\frac{1}{k_{\mathrm{B}}\,N_{i}}\,\frac{\partial
F_{lat}}{\partial T}\,=\,-\,f_{lat}-T\,\frac{\partial f_{lat}}{\partial
T}=-\,f_{lat}+\Gamma\,\frac{\partial
f_{lat}}{\partial\Gamma}+\eta\,\frac{\partial f_{lat}}{\partial\eta}\ ,$ (47)

where we used relations

$T\,\frac{\partial\Gamma}{\partial T}\,=-\Gamma\ ,\quad
T\,\frac{\partial\eta}{\partial T}\,=-\eta\ .$ (48)

It is obvious from (47) that the first two terms of (43) (linear in $\Gamma$
and $\eta$, resp.) do not contribute to the entropy (since the corresponding
contributions to $F_{lat}$ do not depend on temperature). From the last part
of harmonic contribution, i.e. from $f_{th}$, we obtain for the harmonic part
of the entropy $\hat{s}_{i}(har)$ two contributions corresponding to two terms
of (45):

$\hat{s}_{i}(har)\,=\,\hat{s}_{ths}(\eta)\,+\,\hat{s}_{thr}(\eta)\,,$ (49)

with

$\displaystyle\hat{s}_{ths}(\eta)\,$ $\displaystyle=$
$\displaystyle\,\sum_{k=1}^{3}\,\big{[}-ln(1-e^{-\alpha_{k}\,\eta})\,+\,\frac{\eta\alpha_{k}e^{-\alpha_{k}\,\eta}}{1-e^{-\alpha_{k}\,\eta}}\big{]}\,,$
(50) $\displaystyle\hat{s}_{thr}(\eta)\,$ $\displaystyle=$
$\displaystyle\,\frac{A(\eta)}{B(\eta)}\,-\,\frac{\tilde{A}(\eta)^{\prime}}{B}\,+\,A(\eta)\,\frac{\tilde{B}(\eta)^{\prime}}{B(\eta)^{2}}\
,$ (51)

where we denote

$\tilde{C}(\eta)^{\prime}\,\equiv\,\eta\,\frac{dC(\eta)}{d\eta}\,,\,\,\,\,C\,=\,A\,,B\,.$
(52)

The anharmonic part of the one-ion entropy $\hat{s}_{i}(ah)$ was parametrized
in Ref. BC . From eqs. (20) - (24) of this paper one gets:

$\displaystyle\hat{s}_{i}(ah)\,$ $\displaystyle=$
$\displaystyle\,\frac{\tilde{A}^{S}_{1}(\eta)}{\Gamma}\,+\,\frac{\tilde{A}^{S}_{2}(\eta)}{\Gamma^{2}}\,+\,\frac{\tilde{A}^{S}_{3}(\eta)}{\Gamma^{3}}\,,\qquad\tilde{A}^{S}_{i}(\eta)=\eta^{i}\,A^{S}_{i}(\eta)\,,$
(53) $\displaystyle\tilde{A}^{S}_{1}(\eta)\,$ $\displaystyle=$
$\displaystyle\,-2A_{11}\,\frac{2A_{12}\eta^{2}\,+\,1}{(1\,+\,A_{12}\eta^{2})^{2}}-2A_{13}\,\frac{2A_{14}\eta^{2}\,+\,1}{(1\,+\,A_{14}\eta^{2})^{2}}\,,$
(54) $\displaystyle\tilde{A}^{S}_{2}(\eta)\,$ $\displaystyle=$
$\displaystyle\,\frac{3A_{2cl}}{2(1\,+\,A_{21}\eta^{4})^{1/4}}\,,$ (55)
$\displaystyle\tilde{A}^{S}_{3}(\eta)\,$ $\displaystyle=$
$\displaystyle\,\frac{4A_{3cl}}{3}\,.$ (56)

The parameters entering eqs. (54)-(56) are listed in Table 1 and in eqs.
(9)-(11)
of Ref. BC :

$\displaystyle A_{1cl}\,=\,10.2\,,\quad
A_{2cl}\,=\,248\,,\quad\,A_{3cl}\,=\,2.03\times 10^{5}\,,$ $\displaystyle
A_{1q}\,=\,-0.62/6\,,\quad A_{2q}\,=\,-0.56\,,$ $\displaystyle
A_{11}\,=\,-10\,,\quad A_{12}\,=\,6\times 10^{-3}\,,\quad
A_{13}\,=\,-0.2\,,\quad A_{14}\,=\,0.2167\,,\quad A_{21}\,=\,2.9624\times
10^{-4}\ ,$

As for the derivative of the electron entropy (37), let us also for
$\hat{s}_{i}$ factorize:

$\displaystyle\frac{d\hat{s}_{\mathrm{i}}}{dx}\,$ $\displaystyle=$
$\displaystyle\,\left(\rho\,\frac{d\hat{s}_{\mathrm{i}}}{d\rho}\right)\cdot\,\frac{1}{\rho}\frac{d\rho}{dx}\
,\qquad\frac{1}{\rho}\frac{d\rho}{dx}<0\ .$ (57)

Then, using the fact that $\hat{s}_{i}=\hat{s}_{i}(\Gamma,\eta)$ and taking
into account the identities:

$\rho\,\frac{\partial\Gamma}{\partial\rho}\,=\frac{\Gamma}{3}\
,\quad\rho\,\frac{\partial\eta}{\partial\rho}\,=\frac{\eta}{2}\ ,$ (58)

we get for the ion entropy:

$\displaystyle\rho\,\frac{d\hat{s}_{i}}{d\rho}$ $\displaystyle=$
$\displaystyle\rho\,\frac{\partial\Gamma}{\partial\rho}\,\frac{\partial\hat{s}_{i}}{\partial\Gamma}+\rho\,\frac{\partial\eta}{\partial\rho}\,\frac{\partial\hat{s}_{i}}{\partial\eta}=\frac{\Gamma}{3}\,\frac{\partial\hat{s}_{i}}{\partial\Gamma}+\frac{\eta}{2}\,\frac{\partial\hat{s}_{i}}{\partial\eta}\
.$ (59)

For the harmonic part (49-51) it then follows

$\rho\frac{\partial\hat{s}_{i}(har)}{\partial\rho}\,=\frac{\eta}{2}\,\frac{\partial\hat{s}_{ths}(\eta)}{\partial\eta}+\frac{\eta}{2}\,\frac{\partial\hat{s}_{thr}(\eta)}{\partial\eta}\equiv
ds_{ths}(\eta)\,+\,ds_{thr}(\eta)\ ,$ (60)

with

$\displaystyle ds_{ths}(\eta)\,$ $\displaystyle=$
$\displaystyle\,-\frac{1}{2}\,\sum_{k=1}^{3}\,(\eta\alpha_{k})^{2}\,\frac{e^{-\alpha_{k}\eta}}{(1\,-\,e^{-\alpha_{k}\eta})^{2}}\,,$
(61) $\displaystyle ds_{thr}(\eta)\,$ $\displaystyle=$
$\displaystyle\,-\frac{A\tilde{B}^{\prime
2}}{B^{3}}\,+\,\frac{1}{2B^{2}}\,\big{(}2\tilde{A}^{\prime}\tilde{B}^{\prime}\,+\,A\tilde{B}^{\prime\prime}\big{)}-\frac{\tilde{A}^{\prime\prime}}{2B}\
,$ (62)

where

$\tilde{C}(\eta)^{\prime\prime}\,\equiv\,\eta^{2}\,\frac{d^{2}C(\eta)}{d\eta^{2}}\,,\,\,\,\,C\,=\,A\,,B\,.$
(63)

In its turn, the derivative of the anharmonic part $\hat{s}_{i}(ah)$ over
$\rho$ is,

$\rho\frac{\partial\hat{s}_{i}(ah)}{\partial\rho}\,=\,\rho\frac{\partial}{\partial\rho}\bigg{[}\frac{\tilde{A}^{S}_{1}(\eta)}{\Gamma}\bigg{]}+\rho\frac{\partial}{\partial\rho}\bigg{[}\frac{\tilde{A}^{S}_{2}(\eta)}{\Gamma^{2}}\bigg{]}+\rho\frac{\partial}{\partial\rho}\bigg{[}\frac{\tilde{A}^{S}_{3}(\eta)}{\Gamma^{3}}\bigg{]}\,,$
(64)

where we can write similar to (59),

$\displaystyle\rho\frac{\partial}{\partial\rho}\bigg{[}\frac{\tilde{A}^{S}_{n}(\eta)}{\Gamma^{n}}\bigg{]}$
$\displaystyle=$
$\displaystyle\left(\frac{\Gamma}{3}\,\frac{\partial}{\partial\Gamma}+\frac{\eta}{2}\,\frac{\partial}{\partial\eta}\right)\,\frac{\tilde{A}^{S}_{n}(\eta)}{\Gamma^{n}}=\frac{1}{\Gamma^{n}}\,\left(-\frac{n}{3}\,\tilde{A}^{S}_{n}(\eta)+\frac{\eta}{2}\,\frac{d\tilde{A}^{S}_{n}(\eta)}{d\eta}\right)\
,\quad n=1,2,3\ .$

From the explicit form (54)-(56) of factors $\tilde{A}^{S}_{n}(\eta)$ one gets
with the help of the relation above:

$\displaystyle\rho\frac{\partial}{\partial\rho}\bigg{[}\frac{\tilde{A}^{S}_{1}(\eta)}{\Gamma}\bigg{]}\,$
$\displaystyle=$
$\displaystyle\,\frac{2}{3\Gamma}\bigg{[}\frac{A_{11}}{(1\,+\,A_{12}\eta^{2})^{3}}\,(8A_{12}^{2}\eta^{4}\,+\,3A_{12}\eta^{2}\,+\,1)\bigg{]}$
(65)
$\displaystyle\,+\frac{2}{3\Gamma}\bigg{[}\frac{A_{13}}{(1\,+\,A_{14}\eta^{2})^{3}}\,(8A_{14}^{2}\eta^{4}\,+\,3A_{14}\eta^{2}\,+\,1)\bigg{]}\,,$
$\displaystyle\rho\frac{\partial}{\partial\rho}\bigg{[}\frac{\tilde{A}^{S}_{2}(\eta)}{\Gamma^{2}}\bigg{]}\,$
$\displaystyle=$
$\displaystyle\,-\frac{A_{2cl}}{4\Gamma^{2}}\frac{(4\,+\,7A_{21}\eta^{4})}{(1\,+\,A_{21}\eta^{4})^{5/4}}\,,$
(66)
$\displaystyle\rho\frac{\partial}{\partial\rho}\bigg{[}\frac{\tilde{A}^{S}_{3}(\eta)}{\Gamma^{3}}\bigg{]}\,$
$\displaystyle=$ $\displaystyle\,-\frac{4A_{3cl}}{3\Gamma^{3}}\,.$ (67)

The electron and ion entropies and their derivatives presented in this section
depend for a given WD (i.e. given $Z,\,A,\,T$ and $\lambda_{c}$) on the
chemical potential $\chi_{e}$ or on the parameters $\eta$ and $\Gamma$, resp.,
which are all determined from the density $\rho$ obtained by integrating the
equation of stability. This way we analyzed numerically the WD models 1–6 of
Table 1. The results are plotted in Figs. 2 and 3.

|
---|---
|
|
FIG. 2: Entropies (left column) and their derivatives (right column) for the
first three models.
On the horizontal axis we plot the dimensionless radial distance (14), on the
vertical axes we enter the dimensional reduced entropy $\hat{s}$ (eq. (32)) in
the left plots and $d\hat{s}/d\,x$, i.e., derivatives of reduced entropies in
respect to the dimensionless radius $x$ (eq. (14)) in the right plots. As
indicated in legends, red dotted curves are ion contributions, blue dashed
electron ones and black solid display sums (total values).

|
---|---
|
|
FIG. 3: The same as the previous figure for models 4-6.

For the light WDs (see Fig. 2) the ion entropy prevails, although the electron
contribution becomes gradually important, in particular in the inner parts of
the WDs. The ion contribution also dominates the energy gradient.

For heavier WDs (see Fig. 3) the electron entropy is more important almost in
the whole star, only close to the surface the ion part prevails. The same
trend is apparent also for entropy gradient.

As it can be seen from our calculations, the entropy is positive∥∥∥the
positivity of the entropy was noted also in ref. ASJ  for all models and the
entropy gradient satisfies the condition of the thermodynamical stability of
stars Eq.(1.10).

##  IV Conclusions

The frequently used polytropic model SC1  of the description of the WDs has
two drawbacks: a) It is of restricted use, because it is a realistic model of
the EoS only in the non-relativistic limit for $\lambda_{c}<<10^{6}g/cm^{3}$
and in the extreme relativistic limit for $\lambda_{c}>>10^{6}g/cm^{3}$.
b) The fluid, described by the polytropic model, is only neutrally stable JPM
.
In this paper, we have shown on a representative set of the carbon WDs that
their description, based on the EoS formulated in the theory of the magnetized
Coulomb plasma in Refs. CP  \- ASJ , satisfies the stability requirement,
given by eq. (10). As it is seen in Figs. (2) - (7), both the entropy and its
gradient are positive.
It would be important to investigate, if this requirement would be satisfied
also in the case of the presence of the strong magnetic field. This finding
would mean that the existence of strongly magnetized WDs would be possible.

## Acknowledgments

One of us (E. T.) thanks Dr. A.Y. Potekhin for the correspondence, discussions
and advices. The correspondence with Dr. N. Chamel is acknowledged.

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##  Appendix A Scaling And Dimensionless Equations

Let us briefly present an alternative derivation of differential equations
describing the WD, starting from the usual Newtonian formulation of the
mechanical stability for the spherical WD (which is also a starting point in
the main text (2)):

$\displaystyle\frac{d\,P_{e}(r)}{d\,r}$ $\displaystyle=$
$\displaystyle-G\,\frac{M(r)\,\rho(r)}{r^{2}}\ ,$
$\displaystyle\frac{d\,M(r)}{d\,r}$ $\displaystyle=$ $\displaystyle
4\pi\,r^{2}\,\rho(r)\ ,$

where $\rho(r)$ is a matter density:

$\displaystyle\rho(r)$ $\displaystyle=$ $\displaystyle
m_{u}\,n_{e}(r),\qquad\mu_{u}\equiv\frac{A}{Z}\,m_{u}\ ,$ (68)

and $P_{e}$ is an electron pressure. In the calculations of this paper we used
$A=12,Z=6$ and:

$\displaystyle m_{u}=931.494\,{\rm
MeV/c^{2}}\quad\rightarrow\quad\mu_{u}=1\,862.988\,{\rm MeV/c^{2}}\ .$

$M(r)$ is a mass contained inside the radius $r$. It appears convenient to re-
write the electron pressure and density in terms of dimensionless quantities
$\tilde{P}$ and $\tilde{\rho}$:

$\displaystyle n_{e}$ $\displaystyle=$ $\displaystyle\rho_{0}\,\tilde{n}_{e}\
,\quad\rho_{0}=\frac{1}{3\pi^{2}\,\lambda^{3}_{e}}\
,\quad\lambda_{e}=\frac{\hbar}{m_{e}c}\simeq 386.164\,{\rm fm}\ ,$ (69)
$\displaystyle P_{e}$ $\displaystyle=$ $\displaystyle
m_{e}c^{2}\,\rho_{0}\,\tilde{P}\ ,$

where $\lambda_{e}$ is the electron Compton length (its value is obtained from
$m_{e}\,c^{2}\simeq 0.511\,$MeV and $\hbar\,c\simeq 197.33\,{\rm MeV\cdot
fm}$).
The first equation then reads:

$\displaystyle\frac{d\,\tilde{P}(r)}{d\,r}$ $\displaystyle=$
$\displaystyle-G\,\frac{\mu_{u}}{m_{e}c^{2}}\
\frac{M(r)\,\tilde{n}_{e}(r)}{r^{2}}\ .$

Next we re-scale also the radius $r$ (as in (20)) and the mass $M(r)$:

$\displaystyle r=a_{s}\,x\,\qquad M(R)=M_{s}\,\tilde{m}(x)\ .$

In terms of $x$ and $\tilde{m}(x)$ the set of differential equations read:

$\displaystyle\frac{d\,\tilde{P}(x)}{d\,x}$ $\displaystyle=$
$\displaystyle-G\,\frac{\mu_{u}}{m_{e}c^{2}}\,\frac{M_{s}}{a^{2}_{s}}\
\frac{\tilde{m}(x)\,\tilde{n}_{e}(x)}{x^{2}}\ ,$
$\displaystyle\frac{d\,\tilde{m}(x)}{d\,x}$ $\displaystyle=$ $\displaystyle
4\pi\,\mu_{u}\,\rho_{0}\,\frac{a^{3}_{s}}{M_{s}}\ x^{2}\,\tilde{n}_{e}(x)\ .$

The dimensionless constants appearing on the r.h. sides can be fixed to our
convenience, we adopt a choice (following ref. JTPELP ):

$\displaystyle G\,\frac{\mu_{u}}{m_{e}c^{2}}\,\frac{M_{s}}{a^{2}_{s}}$
$\displaystyle=$ $\displaystyle\frac{5}{3}\ ,$ $\displaystyle
4\pi\,\mu_{u}\,\rho_{0}\,\frac{a^{3}_{s}}{M_{s}}$ $\displaystyle=$
$\displaystyle 3\ ,$

from which one gets (cp (20))

$\displaystyle a_{s}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{15\pi}}{2\mu_{u}}\,\sqrt{\frac{m_{e}c^{2}\,\lambda_{e}^{3}}{G}}=\frac{\sqrt{15\pi}}{2}\,\lambda_{e}\,\frac{m_{Pl}}{\mu_{u}}\,\simeq
8686.26\,{\rm km}\ ,$ (70) $\displaystyle M_{s}$ $\displaystyle=$
$\displaystyle a_{s}\cdot\ \frac{5m_{e}c^{2}}{3\mu m_{N}G}\simeq
2.646\,M_{\odot}\ ,$ (71)

where $m_{Pl}$ is the Planck mass (with a corresponding value of the
gravitational constant $G$):

$\displaystyle m_{Pl}$ $\displaystyle\equiv$ $\displaystyle\sqrt{\frac{\hbar
c}{G}}\simeq 1.2209\cdot 10^{22}\,{\rm MeV}\ ,\quad G\simeq 6.6742\,\cdot
10^{-8}\,{\rm cm^{3}/(g\cdot sec^{2})}\ .$ (72)

The set of the dimensionless DE is now:

$\displaystyle\frac{d\,\tilde{P}(x)}{d\,x}$ $\displaystyle=$
$\displaystyle-\frac{5}{3}\,\frac{\tilde{m}(x)\,\tilde{n}_{e}(x)}{x^{2}}\ ,$
$\displaystyle\frac{d\,\tilde{m}(x)}{d\,x}$ $\displaystyle=$ $\displaystyle
3\,x^{2}\,\tilde{n}_{e}(x)\ .$

To proceed further one has to realize that the quantities $\tilde{P}$ and
$\tilde{n}_{e}(x)$ are known functions of a temperature $T$ and electron
chemical potential $\mu_{e}$ (taken here without the electron rest mass), or
more conveniently, of the dimensionless variables (see the main text):

$\displaystyle\chi_{e}=\mu_{e}\,\beta\
,\qquad\tau=\frac{1}{\beta\,m_{e}c^{2}}\ ,\qquad\beta=\frac{1}{k_{B}\,T}\ .$

An important simplification then follows from an assumption (employed also in
an alternative formulation in the main text) that to a very good approximation
the temperature in the WD is constant, i.e. $T$ and hence $\tau$ do not depend
on $x$ and are fixed by their initial value. Then all quantities of interest
in the WD, in particular $\tilde{P}$ and $\tilde{n}_{e}(x)$, depend on the
radius $r$ – and hence on the dimensionless $x$ – implicitly just through a
single $x-$dependent function, e.g. $\chi_{e}(x)$. Thus, we can write:

$\displaystyle\frac{d\,\tilde{P}(x)}{d\,x}$ $\displaystyle=$
$\displaystyle\frac{d\,\tilde{P}(x)}{d\,\chi_{e}}\,\frac{d\,\chi_{e}(x)}{d\,x}\
,$

where the derivative $\tilde{P}(x)/d\,\chi_{e}$ can be calculated from the
explicit form of $\tilde{P}=\tilde{P}(\chi_{e},\tau)$. It is convenient to
consider instead of $\chi_{e}(x)$ a variable $\varphi(x)$:

$\displaystyle\varphi(x)$ $\displaystyle=$
$\displaystyle\chi_{e}\,\tau=\frac{\mu_{e}}{m_{e}c^{2}}\ ,$

which is a dimensionless electron chemical potential. Its advantage is that
for $T=0$ it just reduces to the dimensionless electron Fermi energy
$\tilde{\varepsilon}_{F}$:

$\displaystyle\varphi\ \xrightarrow[T\rightarrow 0]{}\
\frac{\tilde{E}_{F}}{m_{e}c^{2}}\equiv\tilde{\varepsilon}_{F}\ ,$ (73)

where $\tilde{E}_{F}$ (and $\tilde{\varepsilon}_{F}$) has a contribution of
the electron rest mass subtracted. The resulting set of the DE is then:

$\displaystyle\frac{d\,\varphi(x)}{d\,x}$ $\displaystyle=$
$\displaystyle-\frac{5}{3}\,\frac{\tilde{m}(x)}{x^{2}}\,g(\varphi)\ ,\qquad
g(\varphi)=\frac{\tilde{n}_{e}}{\frac{d\,\tilde{P}}{d\,\varphi}}=\frac{\tau\,\tilde{n}_{e}(\chi_{e},\tau)}{\frac{d\,\tilde{P}(\chi_{e},\tau)}{d\,\chi_{e}}}\
,\qquad\chi_{e}(x)=\frac{\varphi(x)}{\tau}\ ,$ (74)
$\displaystyle\frac{d\,\tilde{m}(x)}{d\,x}$ $\displaystyle=$ $\displaystyle
3\,x^{2}\,\tilde{n}_{e}(x)\ .$ (75)

As for the initial conditions for these equations:
$\tilde{m}(0)\equiv\tilde{m}_{c}=0$ (and $\tilde{m}(x)$ is an increasing
function of $x$), a value of $\varphi(0)\equiv\varphi_{c}>0$ is related to the
central matter density $\lambda_{c}$ and is discussed below ($\varphi(x)$ is a
decreasing function of $x$). For the WD the function $g(\varphi)$ is rather
close to unity.

The set of DE formulated above has several advantages which we are going to
discuss briefly: a) there is a smooth limit of $T\rightarrow 0$. In this limit
$\varphi\rightarrow\tilde{\varepsilon}_{F}=\sqrt{1+x^{2}_{r}}-1$, where
$x_{r}$ is a dimensionless electron Fermi momentum. Further, for the free
electron Fermi gas at $T=0$ it holds:

$\displaystyle\frac{\partial P_{e}}{\partial E_{F}}=n_{e}\ \Rightarrow\
\frac{\partial\tilde{P}}{\varepsilon_{F}}=\tilde{n}_{e}\ \Rightarrow\
g(\varphi)\xrightarrow[T\rightarrow 0]{}1\ \ .$

Thus, our set of coupled DEs for $T\rightarrow 0$ smoothly approaches a set of

$\displaystyle\frac{d\,\tilde{\varepsilon}_{F}(x)}{d\,x}$ $\displaystyle=$
$\displaystyle-\frac{5}{3}\,\frac{\tilde{m}(x)}{x^{2}}\ ,$
$\displaystyle\frac{d\,\tilde{m}(x)}{d\,x}$ $\displaystyle=$ $\displaystyle
3\,x^{2}\,\tilde{n}_{e}(x)=3\,x^{2}\,x^{3}_{r}\ ,$

which is equivalent to DEs considered in JTPELP . This makes comparisons of
the finite temperature solutions to $T=0$ ones very transparent.
b) A numerical solution of eqs. (74-75) is straightforward: once one specifies
the initial conditions, the equations (complemented by equations for
$\tilde{P}(\chi_{e},\tau)$ and $\tilde{n}_{e}(\chi_{e},\tau)$) are solved step
by step by appropriate numerical procedure (e.g. 4th order Runge-Kutta) and
there is no need to solve numerically at each step some transcendent equation
(cp to procedure described in a paragraph following eq. (28)). Moreover, at
some point $x_{0}$ the numerical value of the decreasing $\varphi(x)$ crosses
zero:

$\displaystyle\varphi(x_{0})=0\ \rightarrow\ r_{0}=a_{s}\,x_{0}\ .$

As in the $T=0$ limit we identify the value of $r_{0}$ with the
(dimensionless) radius of the WD. The alternative method used in the main text
does not have such a clear criterium for the radius.
c) An initial condition for $\varphi(0)=\varphi_{c}$ is expressed from the
central matter density $\lambda_{c}$ (see eq. (14)). Let us express

$\displaystyle\lambda_{c}=\mu_{u}\,\tilde{n}_{c}=\mu_{u}\,x^{3}_{rc}=\mu_{u}\,\tilde{n}_{c}(\chi_{ec},\tau)\
.$ (76)

One can invert this equation numerically to determine $\chi_{ec}$ by a
procedure mentioned below eq. (28) (and then to get
$\varphi_{c}=\tau\,\chi_{ec}$). Let us emphasize that in this formulation one
would have to solve the transcendent equation just once for the central
initial value. But even this is actually not necessary. At the center of the
WD the density is rather high and $T<<T_{F}$, hence one can use the Sommerfeld
expansion, from which it is possible to get an algebraic equation for
$\varphi_{c}$ in terms of $\varphi_{0c}=x^{2}_{rc}$ and temperature. We
checked that $\varphi_{c}$ obtained this way reproduces very accurately the
value obtained by solving eq. (76).

In last part of this appendix we briefly present equations for the WD radius
and mass in the Lame-Emden approximation. There are well known textbook
equations (see e.g. SLSSAT ) in terms of the central mass density or one can
derive very convenient representations for dimensionless radii and masses in
terms of the central fractional electron Fermi momentum $x_{rc}$. To
crosscheck numbers in our Table 1 we used both versions, so we list below for
reference corresponding equations and numerical value.

Recall that the radius $R_{0}$ and the mass of the object described by the
Lame-Emden equation (8) are defined by the first zero of its solution
$\theta(\xi_{1})=0$ and by its derivative in
$f(\xi_{1})=-\xi^{2}_{1}\,\theta^{\prime}(\xi_{1})$

$\displaystyle R_{0}=a\,\xi_{1}\ $ , $\displaystyle\quad
M_{0}=4\pi\,f(\xi_{1})\,a^{3}\,\lambda_{c}$

where according to (9) the LE scaling $a$ is

$\displaystyle
a=\sqrt{\frac{n+1}{4\pi}\,\frac{K}{G}\,\lambda_{c}^{\frac{1}{n}-1}}=\sqrt{\tilde{K}}\,\lambda_{c}^{\frac{1-n}{2n}}\
,\qquad\ .$

For the non-relativistic case with

$n=\frac{3}{2}\ ,\quad\xi_{1}^{nr}\simeq 3.65375\ ,\quad f(\xi_{1}^{nr})\simeq
2.71406\ ,$

we get:

$\displaystyle K_{nr}$ $\displaystyle=$ $\displaystyle\frac{\hbar
c\,\lambda_{e}}{15\pi^{2}}\,\left(\frac{3\pi^{2}}{\mu_{N}\,m_{u}}\right)^{5/3}\simeq
3.16119\cdot 10^{12}\,\frac{\rm cm^{4}}{\rm g^{2/3}\cdot
sec^{2}}\simeq\frac{1.00361\cdot 10^{13}}{\mu_{N}^{5/3}}\,\frac{\rm
cm^{4}}{\rm g^{2/3}\cdot sec^{2}}\ ,$

which agrees with eq. (2.3.22) of SLSSAT . Then, introducing

$\displaystyle\tilde{K}_{nr}$ $\displaystyle=$
$\displaystyle\frac{5}{8\pi}\,\frac{K_{nr}}{G}\simeq 9.42283\cdot
10^{18}\,{\rm g^{1/3}\cdot cm}\ ,$

gets (substituting $\lambda_{c}[{\rm g/cm^{3}}]$):

$\displaystyle R_{0}[{\rm km}]$ $\displaystyle=$ $\displaystyle
10^{-5}\cdot\frac{\sqrt{\tilde{K}_{nr}}\,\xi_{1}^{nr}}{\lambda_{c}^{1/6}}\simeq\frac{1.12158\cdot
10^{5}}{\lambda_{c}^{1/6}}=1.12158\cdot
10^{4}\,\left(\frac{\mu_{N}}{2}\right)^{-5/6}\,\left(\frac{\lambda_{c}}{10^{6}}\right)^{-1/6}\
,$

where the last equation agrees with eq. (3.3.13) of SLSSAT . For the WD mass
it follows:

$\displaystyle M$ $\displaystyle=$ $\displaystyle
4\pi\,\tilde{K}_{nr}^{3/2}\,f(\xi^{nr}_{1})\,\sqrt{\lambda_{c}}\simeq
9.86510\cdot 10^{29}\,\sqrt{\lambda_{c}}$ $\displaystyle=$ $\displaystyle
4.95993\cdot
10^{-4}\,M_{\odot}\,\sqrt{\lambda_{c}}=0.495993\,\left(\frac{\mu_{N}}{2}\right)^{-5/2}\,M_{\odot}\,\left(\frac{\lambda_{c}}{10^{6}}\right)^{1/2}\
,$

where we use $M_{\odot}\simeq 1.98896\cdot 10^{33}\,$g and the result fairy
agrees with eq. (3.3.14) of SLSSAT .

Alternatively, we can express from (34) and (35):

$\displaystyle x_{r}\,=\,\frac{\hbar
c}{m_{e}c^{2}}\,\bigg{(}\frac{3\pi^{2}\rho}{\mu_{u}}\bigg{)}^{1/3}\,.$ (77)

and express in terms of $x_{r}$ the reduced radius $r_{0}=R_{0}/a_{s}$ and
mass $m_{0}=M/m_{S}$, where the scaling factors $a_{s}$ and $M_{s}$ are
defined in eqs. (70,71). After some algebra one gets:

$\displaystyle r_{0}$ $\displaystyle=$
$\displaystyle\frac{R_{0}}{R_{s}}=\frac{\xi_{1}^{nr}}{\sqrt{10\,x_{rc}}}\simeq\frac{1.15542}{\sqrt{x_{rc}}}\
,$ (78) $\displaystyle m_{0}$ $\displaystyle\equiv$
$\displaystyle\frac{M}{M_{s}}=\frac{3\,f(\xi_{1}^{nr})}{5\,\sqrt{40}}\,x^{3/2}_{rc}\simeq
0.257478\,x^{3/2}_{rc}\ ,$ (79) $\displaystyle\frac{M}{M_{\odot}}$
$\displaystyle=$ $\displaystyle\frac{M_{s}}{M_{\odot}}\,\cdot
m_{0}=\frac{\sqrt{6\pi}}{8}\,f(\xi_{1}^{nr})\,\frac{m^{3}_{\rm
pl}}{(\mu_{N}m_{u})^{2}\,M_{\odot}}\,x^{3/2}_{rc}$ (80) $\displaystyle\simeq$
$\displaystyle 2.68849\,m_{0}\simeq 0.692227\,x^{3/2}_{rc}\ .$

These equations are convenient, since $r_{0}$ and $m_{0}$ depend only on
$x_{r}$ and are of natural size. The numbers in the Table 1 were calculated in
both ways, yielding identical results.

For the ultra-relativistic case with

$n=3\ ,\quad\xi_{1}^{ur}\simeq 6.89685\ ,\quad f(\xi_{1}^{ur})\simeq 2.01824\
,$

we get:

$\displaystyle K_{ur}$ $\displaystyle=$ $\displaystyle\frac{\hbar
c}{12\pi^{2}}\,\left(\frac{3\pi^{2}}{\mu_{u}}\right)^{4/3}\simeq 4.93488\cdot
10^{14}\,\frac{\rm cm^{3}}{\rm g^{1/3}\cdot sec^{2}}\simeq\frac{1.24351\cdot
10^{15}}{\mu_{N}^{4/3}}\,\frac{\rm cm^{3}}{\rm g^{1/3}\cdot sec^{2}}\ ,$

which agrees with eq. (2.3.23) of SLSSAT . We again introduce the auxiliary
constant:

$\displaystyle\tilde{K}_{ur}$ $\displaystyle=$
$\displaystyle\frac{K_{ur}}{\pi\,G}\equiv\frac{(3\pi^{2})^{1/3}}{4\pi}\,\frac{m^{2}_{Pl}}{(\mu_{N}\,m_{u})^{4/3}}\simeq
2.35357\,\cdot 10^{21}\,{\rm g^{2/3}}\ .$

Then the radius in km is:

$\displaystyle R_{0}[{\rm km}]$ $\displaystyle=$ $\displaystyle
10^{-5}\cdot\frac{\sqrt{\tilde{K}_{ur}}\,\xi_{1}^{rel}}{\lambda_{c}^{1/3}}\simeq\frac{33.4591}{\lambda_{c}^{1/3}}=0.334591\,\cdot\left(\frac{\mu_{N}}{2}\right)^{-2/3}\,\left(\frac{\lambda_{c}}{10^{6}}\right)^{-1/3}\
,$

which is consistent with eq. (3.3.16) of SLSSAT . The mass is in the ultra-
relativistic limit independent of $\lambda_{c}$ and it is known as the
Chandrasekhar limit:

$\displaystyle M$ $\displaystyle=$ $\displaystyle
4\pi\,\tilde{K}_{ur}^{3/2}\,f(\xi^{rel}_{1})\simeq 2.89584\cdot 10^{33}\,{\rm
g}\simeq
1.45595\,M_{\odot}=1.45595\,\left(\frac{\mu_{N}}{2}\right)^{-2}\,M_{\odot}\ .$

Alternatively, we can calculate $R_{0}$ and $M/M_{\odot}$ in terms of the
dimensionless:

$\displaystyle r_{0}$ $\displaystyle=$
$\displaystyle\frac{\xi_{1}^{rel}}{\sqrt{5}\,x_{rc}}\simeq\frac{3.084370}{x_{rc}}\
,\quad R_{0}=R_{s}\cdot r_{0}\ ,$ (81) $\displaystyle m_{0}$
$\displaystyle\equiv$
$\displaystyle\frac{M}{M_{s}}=\frac{3\,f(\xi_{1}^{rel})}{5\,\sqrt{5}}\simeq
0.541550\ ,$ (82)

from which one gets the same result as above:

$\displaystyle\frac{M}{M_{\odot}}$ $\displaystyle=$
$\displaystyle\frac{M_{s}}{M_{\odot}}\,\cdot
m_{0}=\frac{\sqrt{3\pi}}{2}\,f(\xi_{1}^{ur})\,\frac{m^{3}_{\rm
pl}}{(\mu_{N}m_{u})^{2}\,M_{\odot}}\simeq 1.45595\ .$ (83)

##  Appendix B Calculations of functions $f_{1}(y)$ and $f_{2}(y)$

In accord with eq. (23) PC3 , the derivative of the electron pressure
$P^{(e)}_{id}$ over $y$ is

$\frac{\partial P^{(e)}_{id}}{\partial y}\,=\,\lambda_{c}\,\frac{\partial
P^{(e)}_{id}}{\partial\rho}\,=\,\lambda_{c}\,\frac{n_{\mathrm{e}}}{\rho}\,\bigg{(}\frac{\partial
P^{(e)}_{id}}{\partial\chi_{\mathrm{e}}}\bigg{)}_{T}\bigg{/}\bigg{(}\frac{\partial
n_{\mathrm{e}}}{\partial\chi_{\mathrm{e}}}\bigg{)}_{T}\,,$ (84)

and, similarly, the second derivative is

$\frac{\partial^{2}P^{(e)}_{id}}{\partial
y^{2}}\,=\,\lambda_{c}^{2}\,\frac{n_{\mathrm{e}}^{2}}{\rho^{2}}\,\bigg{[}\,\bigg{(}\frac{\partial^{2}P^{(e)}_{id}}{\partial\chi_{\mathrm{e}}^{2}}\bigg{)}\,-\,\bigg{(}\frac{\partial
P^{(e)}_{id}}{\partial\chi_{\mathrm{e}}}\bigg{)}\,\bigg{(}\frac{\partial^{2}n_{\mathrm{e}}}{\partial\chi_{\mathrm{e}}^{2}}\bigg{)}\,\bigg{/}\,\bigg{(}\frac{\partial
n_{\mathrm{e}}}{\partial\chi_{\mathrm{e}}}\bigg{)}\,\bigg{]}_{T}\,\bigg{/}\bigg{(}\frac{\partial
n_{\mathrm{e}}}{\partial\chi_{\mathrm{e}}}\bigg{)}^{2}_{T}\,.$ (85)

So one should calculate $n_{\mathrm{e}}$ and the derivatives of $P^{(e)}_{id}$
and $n_{\mathrm{e}}$ over $\chi_{\mathrm{e}}$ in terms of the Fermi - Dirac
integrals $I_{\nu}(\chi_{\mathrm{e}}\,,\,\tau)$. Identifying

$\displaystyle I_{k+1/2}(\chi_{\mathrm{e}}\,,\,\tau)\,$ $\displaystyle\equiv$
$\displaystyle\,Wk\,,$ (86) $\displaystyle\frac{\partial
I_{k+1/2}(\chi_{\mathrm{e}}\,,\,\tau)}{\partial\chi_{\mathrm{e}}}\,$
$\displaystyle\equiv$ $\displaystyle\,WkDX\,,$ (87)
$\displaystyle\frac{\partial^{2}I_{k+1/2}(\chi_{\mathrm{e}}\,,\,\tau)}{\partial\chi_{\mathrm{e}}^{2}}\,$
$\displaystyle\equiv$ $\displaystyle\,WkDXX\,,$ (88)

one can write the $n_{\mathrm{e}}$ and the derivatives of $P^{(e)}_{id}$ and
$n_{\mathrm{e}}$ over $\chi_{\mathrm{e}}$ in terms of $Wk$, $WkDX$ and $WkDXX$
. We calculated these quantities using the program BLIN9 PC3 . We then have

$\displaystyle n_{\mathrm{e}}\,$ $\displaystyle=$
$\displaystyle\,\mathrm{c_{n}}\,[\,W0+\,\tau\,W1\,]\,\equiv\,\mathrm{c_{n}}\,Z_{5}\,,$
(89) $\displaystyle\frac{\partial
n_{\mathrm{e}}}{\partial\chi_{\mathrm{e}}}\,$ $\displaystyle=$
$\displaystyle\,\mathrm{c_{n}}\,[\,W0DX+\,\tau\,W1DX\,]\,\equiv\,\mathrm{c_{n}}\,Z_{3}\,,$
(90) $\displaystyle\frac{\partial P^{(e)}_{id}}{\partial\chi_{\mathrm{e}}}\,$
$\displaystyle=$
$\displaystyle\,\mathrm{c_{p}}\,[\,W1DX+\,\tau/2\,W2DX\,]\,\equiv\,\mathrm{c_{p}}\,Z_{4}\,,$
(91)
$\displaystyle\frac{\partial^{2}n_{\mathrm{e}}}{\partial\chi_{\mathrm{e}}^{2}}\,$
$\displaystyle=$
$\displaystyle\,\mathrm{c_{n}}\,[\,W0DXX+\,\tau\,W1DXX\,]\,\equiv\,\mathrm{c_{n}}\,Z_{2}\,,$
(92)
$\displaystyle\frac{\partial^{2}P^{(e)}_{id}}{\partial\chi_{\mathrm{e}}^{2}}\,$
$\displaystyle=$
$\displaystyle\,\mathrm{c_{p}}\,[\,W1DXX+\,\tau/2\,W2DXX\,]\,\equiv\,\mathrm{c_{p}}\,Z_{1}\,.$
(93)

In terms of $Z_{i}$ we obtain eq. (84) in the form

$\frac{\partial P^{(e)}_{id}}{\partial
y}\,=\,\frac{\mathrm{c}_{p}}{y}\,\frac{Z_{4}\,Z_{5}}{Z_{3}}\,,$ (94)

and eq. (85) will be

$\frac{\partial^{2}P^{(e)}_{id}}{\partial
y^{2}}\,=\,\frac{\mathrm{c}_{p}}{y^{2}}\,\frac{Z_{5}^{2}}{Z_{3}^{3}}\,(Z_{1}\,Z_{3}-Z_{2}\,Z_{4})\,,$
(95)

and finally,

$f_{1}(y)\,=\,\frac{1}{y}\,\bigg{[}\frac{Z_{5}}{Z_{3}^{2}\,Z_{4}}\,(Z_{1}\,Z_{3}-Z_{2}\,Z_{4})\,-\,1\,\bigg{]}\,,\,\,\,f_{2}(y)\,=\,\frac{\mathrm{c}_{p}}{y}\,\frac{Z_{4}\,Z_{5}}{Z_{3}}\,.$
(96)

##  Appendix C The Sommerfeld expansion

In this Appendix we briefly describe how to decompose the thermodynamical
quantities for free electrons into series in powers of
$k_{\mathrm{B}}T/\tilde{E}_{F}$, where $\tilde{E}_{F}=\mu_{\mathrm{e}}(T=0)$
is the Fermi energy with the rest mass contribution subtracted. We will start
from simpler non-relativistic dynamics and later extend the results to a
general case.

In the non-relativistic approximation we can write for electron density,
momentum and energy density in a conveniently normalized form:

$\displaystyle\left(\frac{n_{\mathrm{e}}(T)}{\rho_{0}}\right)_{nr}$
$\displaystyle=$
$\displaystyle=3\sqrt{2}\,\tau^{3/2}\,I_{1/2}(\chi_{\mathrm{e}})\ ,$ (97)
$\displaystyle\left(\frac{P(T)}{\rho_{0}\,m_{e}c^{2}}\right)_{nr}$
$\displaystyle=$ $\displaystyle
2\sqrt{2}\,\tau^{5/2}\,I_{3/2}\chi_{\mathrm{e}})\ ,$ (98)
$\displaystyle\left(\frac{\tilde{\cal E}(T)}{{\cal E}_{0}}\right)_{nr}$
$\displaystyle=$
$\displaystyle\sqrt{2}\,\tau^{5/2}\,I_{3/2}(\chi_{\mathrm{e}})=\frac{1}{2}\left(\frac{P(T)}{\rho_{0}\,m_{e}c^{2}}\right)_{nr}\
,\quad{\cal E}_{0}=3\rho_{0}\,m_{e}c^{2}\ ,$ (99)

where the non-relativistic Fermi-Dirac integrals are here defined as:

$\displaystyle
I_{\nu}(\chi)=\int\limits_{0}^{\infty}\,\frac{u^{\nu}}{e^{u-\chi}+1}\,du\ ,$
(100)

Substituting into the 1st line of (33) one gets the reduced entropy in the
non-relativistic limit:

$\displaystyle s_{e,nr}$ $\displaystyle=$
$\displaystyle\frac{1}{k_{\mathrm{B}}T}\,\left(\frac{5\,m_{e}c^{2}\,\tau\,I_{3/2}(\chi_{\mathrm{e}})}{3\,I_{1/2}(\chi_{\mathrm{e}})}-\mu_{\mathrm{e}}\right)=\frac{5\,I_{3/2}(\chi_{\mathrm{e}})}{3\,I_{1/2}(\chi_{\mathrm{e}})}-\chi_{\mathrm{e}}\
.$

This non-relativistic limit follows also from the general results (33) making
use of the relation:

$I_{\nu}(\chi,\tau)\ \xrightarrow[\tau\rightarrow 0]{}\ I_{\nu}(\chi)\ .$

An opposite ultra-relativistic limit is obtained from

$\displaystyle I_{\nu}(\chi,\tau)\ \xrightarrow[\tau\rightarrow\infty]{}\
\sqrt{\frac{\tau}{2}}\,I_{\nu+1/2}(\chi)\ .$

The number density, momentum and energy density in the ultra-relativistic
limit are:

$\displaystyle\left(\frac{n_{\mathrm{e}}(T)}{\rho_{0}}\right)_{ur}$
$\displaystyle=$ $\displaystyle 3\tau^{3}\,I_{2}(\chi_{\mathrm{e}})\ ,$ (101)
$\displaystyle\left(\frac{P(T)}{\rho_{0}\,m_{e}c^{2}}\right)_{ur}$
$\displaystyle=$ $\displaystyle\left(\frac{\tilde{\cal E}(T)}{{\cal
E}_{0}}\right)_{ur}=\tau^{4}\,I_{3}(\chi_{\mathrm{e}})\ .$ (102)

Equations above define the densities of the electron number, pressure and
kinetic energy as functions of $\chi_{\mathrm{e}}=\beta\,\mu_{\mathrm{e}}$ and
$\tau=k_{\mathrm{B}}T/m_{e}c^{2}$, i.e., functions of the chemical potential
$\mu_{\mathrm{e}}$ (not yet determined) and of the temperature $T$ (or of
$\beta=k_{\mathrm{B}}T$). According to ref. PC3  the chemical potential
$\mu_{\mathrm{e}}(V,T)$ is obtained by (numerically) inverting equation for
the density (24) (in its exact or non/ultra-relativistic forms). Assuming the
fixed number of electrons $N_{\mathrm{e}}$ (meaning that the electron density
$n_{\mathrm{e}}=N_{\mathrm{e}}/V$ explicitly depends only on volume $V$), one
gets the chemical potential from the condition
$n_{\mathrm{e}}(T)=n_{\mathrm{e}}(0)=N_{\mathrm{e}}/V$, where
$n_{\mathrm{e}}(0)$ is known function of the Fermi energy. At the end we get
the chemical potential dependent on the temperature $T$ and the Fermi energy,
which is its value for zero temperature:

$\displaystyle\mu_{0}$ $\displaystyle\equiv$
$\displaystyle\mu_{\mathrm{e}}(T=0)=\tilde{E}_{F}=m_{e}\,c^{2}\,\tilde{\epsilon}_{F}\
,$ (103)

where the chemical potential and energies do not include the rest mass
contributions.

For a simple non-relativistic case, for which
$\mu_{0}=\tilde{E}_{Fnr}/(m_{e}\,c^{2})=\tilde{\epsilon}_{Fnr}=x_{r}^{2}/2$
with $x_{r}=p_{F}/(m_{e}c)$, the l.h.s. of (97) at $T=0$ reads:

$\displaystyle\left(\frac{n_{e}(0)}{\rho_{0}}\right)_{nr}$ $\displaystyle=$
$\displaystyle
x^{3}_{r}=\left(2\,\tilde{\epsilon}_{Fnr}\right)^{3/2}=\left(2\,\frac{\mu_{0}}{m_{e}c^{2}}\right)^{3/2}=2\,\sqrt{2}\,\left(\frac{\mu_{0}}{m_{e}c^{2}}\right)^{3/2}\
.$

Equating this to the r.h.s and substituting for $\tau$ yields:

$\displaystyle
2\,\sqrt{2}\,\left(\frac{\mu_{0}}{m_{e}c^{2}}\right)^{3/2}=3\,\sqrt{2}\,\left(\frac{k_{\mathrm{B}}T}{m_{e}c^{2}}\right)^{3/2}\,I_{1/2}(\chi_{\mathrm{e}})\
.$

which simplifies to:

$\displaystyle
I_{1/2}(\chi_{\mathrm{e}})=\frac{2}{3}\,\left(\frac{\tilde{\mu}_{0}}{k_{\mathrm{B}}T}\right)^{3/2}=\frac{2}{3}\,\left(\frac{\tilde{E}_{Fnr}}{k_{\mathrm{B}}T}\right)^{3/2}=\frac{2}{3}\,\left(\frac{T_{F}}{T}\right)^{3/2}\
.$ (104)

Denoting by $X_{1/2}$ the inverse function to $I_{1/2}(\chi_{\mathrm{e}})$ one
gets a solution for $\chi_{\mathrm{e}}$:

$\displaystyle\chi_{\mathrm{e}}$ $\displaystyle\equiv$
$\displaystyle\frac{\mu}{k_{\mathrm{B}}T}=X_{1/2}\left(\frac{2}{3}\,\left(\frac{T_{F}}{T}\right)^{3/2}\,\right)\
,$

which is just eq. (17) of ref. CP  in our notations. In general, a similar
connection is obtained from $n_{e}$ given by (24), in the ultra-relativistic
limit from (101).

The relations above are valid for arbitrary temperature, now we will deal with
a low temperature expansion. The non-relativistic Fermi-Dirac integrals (100)
can be for small temperatures (i.e. large inverse temperatures
$\beta=1/k_{B}T$ and hence also $\chi_{\mathrm{e}}=\beta\,\mu_{\mathrm{e}}$)
approximated by a power series:

$\displaystyle I_{\nu}(\chi)$ $\displaystyle\simeq$
$\displaystyle\frac{\chi^{\nu+1}}{\nu+1}\,\Big{(}1+\frac{\pi^{2}}{6}\,\frac{(\nu+1)\nu}{\chi^{2}}+\frac{7\,\pi^{4}}{360}\,\frac{(\nu+1)\nu(\nu-1)(\nu-2)}{\chi^{4}}+\dots\Big{)}\
.$ (105)

For the non-relativistic dynamics one needs $I_{\nu}(\chi)$ with $\nu=1/2$ and
$\nu=3/2$:

$\displaystyle I_{1/2}(\chi)$ $\displaystyle\simeq$
$\displaystyle\frac{2}{3}\,\chi^{3/2}\,\Big{(}1+\frac{\pi^{2}}{8\,\chi^{2}}+\frac{7\,\pi^{4}}{640\,\chi^{4}}+\dots\Big{)}\
,$ (106) $\displaystyle I_{3/2}(\chi)$ $\displaystyle\simeq$
$\displaystyle\frac{2}{5}\,\chi^{5/2}\,\Big{(}1+\frac{5\,\pi^{2}}{8\,\chi^{2}}-\frac{7\,\pi^{4}}{384\,\chi^{4}}+\dots\Big{)}\
.$ (107)

Substituting these relations into (97)-(99) and using
$\tau\,\chi_{\mathrm{e}}=\mu_{\mathrm{e}}/(m_{e}c^{2})$ we get:

$\displaystyle\frac{1}{3\sqrt{2}}\left(\frac{n_{\mathrm{e}}(T)}{\rho_{0}}\right)_{nr}$
$\displaystyle=$
$\displaystyle\tau^{3/2}\,I_{1/2}(\chi_{\mathrm{e}})\simeq\frac{2}{3}\left(\frac{\mu_{\mathrm{e}}}{m_{e}c^{2}}\right)^{3/2}\,\left(1+\frac{\pi^{2}}{8\chi_{\mathrm{e}}^{2}}+\frac{7\pi^{4}}{640\chi_{\mathrm{e}}^{4}}+\dots\right)\
,$ (108)
$\displaystyle\frac{1}{2^{3/2}}\left(\frac{P(T)}{\rho_{0}\,m_{e}c^{2}}\right)_{nr}$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}\left(\frac{\tilde{\cal
E}(T)}{{\cal E}_{0}}\right)_{nr}=\tau^{5/2}\,I_{3/2}(\chi_{\mathrm{e}})$ (109)
$\displaystyle\simeq$
$\displaystyle\frac{2}{5}\left(\frac{\mu_{\mathrm{e}}}{m_{e}c^{2}}\right)^{5/2}\,\left(1+\frac{5\pi^{2}}{8\chi_{\mathrm{e}}^{2}}-\frac{7\pi^{4}}{384\chi_{\mathrm{e}}^{4}}+\dots\right)\
.$

These are formal power series in terms of powers of the so far unknown
$1/\chi_{\mathrm{e}}^{2}\,$. Recall that $\chi_{\mathrm{e}}$ depends on
temperature and on the chemical potential. As discussed above,
$\mu_{\mathrm{e}}$ is determined from the condition
$n_{\mathrm{e}}(T)=n_{\mathrm{e}}(0)$ (but now with $n_{\mathrm{e}}(T)$
decomposed into the power series above). Substituting
$1/\chi_{\mathrm{e}}^{2}=(kT)^{2}/\mu_{\mathrm{e}}^{2}$ (and using
$\mu_{0}=\tilde{E}_{Fnr}$ for the non-relativistic Fermi energy) leads to:

$\displaystyle\mu_{0}$ $\displaystyle=$
$\displaystyle\mu_{\mathrm{e}}\,\left(1+\frac{1}{8}\frac{(\pi
k_{\mathrm{B}}T)^{2}}{\mu_{\mathrm{e}}^{2}}+\frac{7}{640}\frac{(\pi
k_{\mathrm{B}}T)^{4}}{\mu_{\mathrm{e}}^{4}}+\dots\right)^{2/3}\ .$ (110)

This relation can be perturbatively inverted by assuming the power series for
$\mu_{\mathrm{e}}$ (in powers of $(k_{\mathrm{B}}T/\mu_{0})^{2}$):

$\displaystyle\mu_{\mathrm{e}}$ $\displaystyle=$
$\displaystyle\mu_{0}\,\left(1-\frac{\pi^{2}}{12}\frac{(k_{\mathrm{B}}T)^{2}}{\mu_{0}^{2}}-\frac{\pi^{4}}{80}\frac{(k_{\mathrm{B}}T)^{4}}{\mu_{0}^{4}}+\dots\right)\
.$ (111)

Substituting (111) into the r.h.s. of and (109) and making the Taylor
decomposition in powers of $(k_{\mathrm{B}}T)^{2}$ yields the following non-
relativistic equations for the observables:

$\displaystyle P_{nr}(T)$ $\displaystyle=$ $\displaystyle
m_{e}c^{2}\,\rho_{0}\
\frac{x^{5}_{r}}{5}\,\left(1+\frac{5\pi^{2}}{12}\frac{(k_{\mathrm{B}}T)^{2}}{\mu^{2}_{0}}-\frac{\pi^{4}}{16}\frac{(k_{\mathrm{B}}T)^{4}}{\mu^{4}_{0}}+\dots\right)\
,$ (112) $\displaystyle\tilde{\cal E}_{nr}(T)$ $\displaystyle=$
$\displaystyle\frac{3}{2}\,P_{nr}(T)\ .$ (113)

These results are consistent with the equations, presented in (Grei ) (where
slightly different notation is used).

For the ultra-relativistic dynamics one proceeds in a similar way. For sake of
briefness, we cite just the final results (using
$\mu_{0}/(m_{e}c^{2})=\epsilon_{Fur}=x_{r}$):

$\displaystyle\mu_{\mathrm{e}}$ $\displaystyle=$
$\displaystyle\mu_{0}\,\left(1-\frac{\pi^{2}}{3}\frac{(kT)^{2}}{\mu_{0}^{2}}+O(T^{6})\right)\
.$ (114) $\displaystyle n_{e,ur}(T)$ $\displaystyle=$ $\displaystyle
n_{e,ur}(0)=\rho_{0}\,\left(\frac{\mu_{0}}{m_{e}c^{2}}\right)^{3}=\rho_{0}\,\epsilon^{3}_{Fur}=\rho_{0}\,x_{r}^{3}=\frac{N_{e}}{V}\
,$ (115) $\displaystyle P_{ur}(T)$ $\displaystyle=$
$\displaystyle\rho_{0}\,m_{e}c^{2}\,\frac{x^{4}_{r}}{4}\,\left(1+\frac{2\pi^{2}}{3}\frac{(k_{\mathrm{B}}T)^{2}}{\mu^{2}_{0}}-\frac{\pi^{4}}{5}\frac{(k_{\mathrm{B}}T)^{4}}{\mu^{4}_{0}}+\dots\right)\
,$ (116) $\displaystyle\tilde{\cal E}_{ur}(T)$ $\displaystyle=$ $\displaystyle
3\,P_{ur}(T)\ .$ (117)

When the non-relativistic or ultra-relativistic limits cannot be applied, we
start from normalized equations (24-26):

$\displaystyle\frac{n_{\mathrm{e}}(T)}{\rho_{0}}$ $\displaystyle=$
$\displaystyle
3\sqrt{2}\,\tau^{3/2}\,\left[I_{1/2}(\chi_{\mathrm{e}},\tau)+\tau\,I_{3/2}(\chi_{\mathrm{e}},\tau)\right]\
,\quad\rho_{0}=\frac{1}{3\pi^{2}\,\lambda^{3}_{e}}\ ,$ (118)
$\displaystyle\frac{P(T)}{\rho_{0}\,m_{e}c^{2}}$ $\displaystyle=$
$\displaystyle
2\sqrt{2}\,\tau^{5/2}\,\left[I_{3/2}(\chi_{\mathrm{e}},\tau)+\frac{\tau}{2}\,I_{5/2}(\chi_{\mathrm{e}},\tau)\right]\
,$ (119) $\displaystyle\frac{\tilde{\cal E}(T)}{{\cal E}_{0}}$
$\displaystyle=$
$\displaystyle\sqrt{2}\,\tau^{5/2}\,\left[I_{3/2}(\chi_{\mathrm{e}},\tau)+\tau\,I_{5/2}(\chi_{\mathrm{e}},\tau)\right]\
,\quad{\cal E}_{0}=3\rho_{0}\,m_{e}c^{2}\ .$ (120)

Now, we will need the following Sommerfeld decompositions of the generalized
Fermi-Dirac integrals (denoting
$\varphi=\tau\,\chi=\mu_{\mathrm{e}}/(m_{e}c^{2})$):

$\displaystyle I_{1/2}(\chi,\tau)$ $\displaystyle\simeq$
$\displaystyle\frac{1}{\sqrt{2}\,\tau^{3/2}}\,\left\\{\frac{1}{2}\,\left[(1+\varphi)\sqrt{\varphi(2+\varphi)}-{\rm
ln}\left(1+\varphi+\sqrt{\varphi(2+\varphi)}\right)\,\right]+\right.$ (121)
$\displaystyle\hskip
56.9055pt\left.+\frac{(1+\varphi)}{\sqrt{\varphi(2+\varphi)}}\,\frac{\pi^{2}\tau^{2}}{6}+\frac{(1+\varphi)}{[\varphi(2+\varphi)]^{5/2}}\,\frac{7\pi^{4}\tau^{4}}{120}+\dots\right\\}\
,$ $\displaystyle I_{3/2}(\chi,\tau)$ $\displaystyle\simeq$
$\displaystyle\frac{1}{\sqrt{2}\,\tau^{5/2}}\,\left\\{\frac{2\varphi^{2}+\varphi-3}{6}\,\sqrt{\varphi(2+\varphi)}+\frac{1}{2}{\rm
ln}\left(1+\varphi+\sqrt{\varphi(2+\varphi)}\right)\right.$ (122)
$\displaystyle\hskip
56.9055pt\left.+\frac{\sqrt{\varphi}\,(3+2\varphi)}{\sqrt{2+\varphi}}\,\frac{\pi^{2}\tau^{2}}{6}-\frac{1}{\varphi^{3/2}\,(2+\varphi)^{5/2}}\,\frac{7\pi^{4}\tau^{4}}{120}+\dots\right\\}\
,$ $\displaystyle I_{5/2}(\chi,\tau)$ $\displaystyle\simeq$
$\displaystyle\frac{1}{\sqrt{2}\,\tau^{7/2}}\,\left\\{\frac{(6\varphi^{3}+2\varphi^{2}-5\varphi+15)\sqrt{\varphi(2+\varphi)}}{24}\,-\frac{5}{8}{\rm
ln}\left(1+\varphi+\sqrt{\varphi(2+\varphi)}\right)\right.$ (123)
$\displaystyle\hskip
14.22636pt\left.+\frac{\varphi^{2}\,(3\varphi+5)}{\sqrt{\varphi(2+\varphi)}}\,\frac{\pi^{2}\tau^{2}}{6}+\frac{\varphi^{2}(2\varphi^{3}+10\varphi^{2}+15\varphi+5)}{[\varphi(2+\varphi)]^{5/2}}\,\frac{7\pi^{4}\tau^{4}}{120}+\dots\right\\}\
.$

Now we again first find the chemical potential from the condition
$n_{\mathrm{e}}(T)=n_{\mathrm{e}}(0)$. Substituting (121,122) into equation
(118) for $n_{\mathrm{e}}(T)$ yields (the logarithmic terms are cancel each
other):

$\displaystyle\frac{n_{\mathrm{e}}(T)}{\rho_{0}}\simeq\left[\varphi(2+\varphi)\right]^{3/2}+\frac{(2\varphi^{2}+4\varphi+1)}{\sqrt{\varphi(2+\varphi)}}\,\frac{\pi^{2}\tau^{2}}{2}+\frac{1}{[\varphi(2+\varphi)]^{5/2}}\,\frac{7\pi^{4}\tau^{4}}{40}+\dots\
.$ (124)

For $T=0$ it holds $\tau=0$ and

$\displaystyle\varphi(0)\equiv\varphi_{0}=\frac{\mu_{0}}{m_{e}c^{2}}=\tilde{\epsilon}_{F}=\epsilon_{F}-1\
,\quad\epsilon_{F}=\sqrt{1+x^{2}_{r}}\ .$

This relation implies:

$\displaystyle\varphi_{0}(2+\varphi_{0})=(\epsilon_{F}-1)(\epsilon_{F}+1)=\epsilon_{F}^{2}-1=x^{2}_{r}\
,$

which reproduces the electron density at zero temperature: from the relation
(124) one gets

$\displaystyle n_{\mathrm{e}}(0)$ $\displaystyle=$
$\displaystyle\rho_{0}\,\left[\varphi_{0}(2+\varphi_{0})\right]^{3/2}=\rho_{0}\,x^{3}_{r}\equiv\rho_{F}\
.$

Using (124) one gets from $n_{\mathrm{e}}(0)=n_{\mathrm{e}}(T)$ an implicit
relation between $\varphi_{0}$ and $\varphi$:

$\displaystyle\varphi_{0}(2+\varphi_{0})=\varphi(2+\varphi)\,\left[1+\frac{(2\varphi^{2}+4\varphi+1)\,B}{2[\varphi(2+\varphi)]^{2}}+\frac{7\,B^{2}}{40[\varphi(2+\varphi)]^{4}}+\dots\right]^{2/3}\
,$ (125)

where $B=\pi^{2}\,\tau^{2}$. To solve this constraint we assume for
$\varphi=\mu_{\mathrm{e}}/(m_{e}c^{2})$ a perturbative expansion in a form:

$\displaystyle\varphi$ $\displaystyle=$
$\displaystyle\varphi_{0}\,\left(1+c_{1}\frac{B}{\varphi^{2}_{0}}+c_{2}\frac{B^{2}}{\varphi^{4}_{0}}+\dots\right)\
.$ (126)

Notice that:

$\displaystyle\frac{B}{\varphi^{2}_{0}}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}(k_{\mathrm{B}}T)^{2}}{(m_{e}c^{2})^{2}}\,\frac{(m_{e}c^{2})^{2}}{\mu_{0}^{2}}=\frac{\pi^{2}(k_{\mathrm{B}}T)^{2}}{\mu_{0}^{2}}\
,$

hence our Ansatz for $\varphi$ above is identical to the Ansatz used for
$\mu_{\mathrm{e}}$ in previous sections. Substituting (126) into (125), making
the Taylor decomposition in powers of $B$ and requiring that the coefficients
in front of $B^{n},n\geq 1$ are equal to zero, yields the coefficients $c_{i}$
in terms of $\varphi_{0}$. The first two are given by relatively simple
equations:

$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle-\frac{2\varphi^{2}_{0}+4\varphi_{0}+1}{6(\varphi^{2}_{0}+3\varphi_{0}+2)}=-\frac{2\epsilon^{2}_{F}-1}{6\epsilon_{F}(\epsilon_{F}+1)}\
,$ (127) $\displaystyle c_{2}$ $\displaystyle=$
$\displaystyle-\frac{20\varphi^{4}_{0}+80\varphi^{3}_{0}+141\varphi^{2}_{0}+122\varphi_{0}+36}{360(\varphi^{2}_{0}+3\varphi_{0}+2)^{3}}=-\frac{20\epsilon^{4}_{F}+21\epsilon^{2}_{F}-5}{360\epsilon^{3}_{F}(\epsilon_{F}+1)^{3}}\
.$ (128)

With these $c_{1}$ and $c_{2}$ in (126) the Taylor decomposition of (125) has
a first non-zero coefficients (apart from the constant
$\varphi_{0}(2+\varphi_{0})$ term) in front of the power $B^{3}$.

Let us now derive the perturbative series for the pressure and the kinetic
energy. Substituting the decompositions of $I_{3/2}(\chi_{\mathrm{e}},\tau)$
(see 122) and $I_{5/2}(\chi_{\mathrm{e}},\tau)$ (see 123) into (119) and (120)
yields:

$\displaystyle\frac{P(T)}{\rho_{0}\,m_{e}c^{2}}$ $\displaystyle=$
$\displaystyle\frac{(2\varphi^{3}+6\varphi^{2}+\varphi-3)\sqrt{\varphi(2+\varphi)}}{8}+\frac{3}{8}\,{\rm
ln}\left(1+\varphi+\sqrt{\varphi(2+\varphi)}\right)$
$\displaystyle+\frac{(\varphi^{3}+3\varphi^{2}+2\varphi)}{\sqrt{\varphi(2+\varphi)}}\,\frac{\pi^{2}\theta^{2}}{2}+\frac{(2\varphi^{5}+10\varphi^{4}+15\varphi^{3}+5\varphi^{2}-2\varphi)}{[\varphi(2+\varphi)]^{5/2}}\,\frac{7\pi^{4}\theta^{4}}{120}+\dots\
,$ $\displaystyle\frac{\tilde{\cal E}(T)}{{\cal E}_{0}}$ $\displaystyle=$
$\displaystyle\frac{(6\varphi^{3}+10\varphi^{2}-\varphi+3)\sqrt{\varphi(2+\varphi)}}{24}-\frac{1}{8}\,{\rm
ln}\left(1+\varphi+\sqrt{\varphi(2+\varphi)}\right)$
$\displaystyle+\frac{(3\varphi^{3}+7\varphi^{2}+3\varphi)}{\sqrt{\varphi(2+\varphi)}}\,\frac{\pi^{2}\theta^{2}}{6}+\frac{(2\varphi^{5}+10\varphi^{4}+15\varphi^{3}+5\varphi^{2}-\varphi)}{[\varphi(2+\varphi)]^{5/2}}\,\frac{7\pi^{4}\theta^{4}}{120}+\dots\
.$

What remains is to substitute $\varphi$ expressed in terms of $\varphi_{0}$
and powers of $B$ (see eqs. (126)-(128)) into equations for the pressure (C)
and kinetic energy (C) and decompose into the Taylor series in powers of $B$.
This is not so simple as for the limiting non-relativistic and ultra-
relativistic cases considered above, since the coefficients depend on
$\varphi_{0}$ and also $P(0)$ and ${\cal E}(0)$ are now not just simple powers
of $\varphi_{0}$ and cannot be simply factorized. Nevertheless, with the help
of _Mathematica_ the power decomposition can be performed. We made a
decomposition up to the order $B^{2}$, but the terms $\sim B^{2}$ are is
lengthy and clumsy. hence, we for the sake of briefness present just leading
and next-to-leading orders:

$\displaystyle\frac{P(T)}{\rho_{0}\,m_{e}c^{2}}$ $\displaystyle=$
$\displaystyle\frac{P(0)}{\rho_{0}\,m_{e}c^{2}}+\frac{\varphi_{0}(2+\varphi_{0})(\varphi_{0}^{2}+2\varphi_{0}+2)\,B}{6(1+\varphi_{0})\sqrt{\varphi_{0}(2+\varphi_{0})}}$
(131) $\displaystyle=$
$\displaystyle\frac{P(0)}{\rho_{0}\,m_{e}c^{2}}+\frac{x_{r}(\epsilon_{F}^{2}+1)}{6\epsilon_{F}}\,B\
,\quad B=\pi^{2}\,\tau^{2}\ ,$ $\displaystyle\frac{\tilde{\cal E}(T)}{{\cal
E}_{0}}$ $\displaystyle=$ $\displaystyle\frac{\tilde{\cal E}(0)}{{\cal
E}_{0}}+\frac{\varphi_{0}(2+\varphi_{0})(1+\varphi_{0})\,B}{6\sqrt{\varphi_{0}(2+\varphi_{0})}}=\frac{\tilde{\cal
E}(0)}{{\cal E}_{0}}+\frac{x_{r}\,\epsilon_{F}}{6}\,B\ .$ (132)

###  C.1 Sommerfeld decomposition of entropy

Recall the equation for the dimensionless reduced entropy of free electrons
(33):

$\displaystyle\hat{s}_{e}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{k_{B}}\,\frac{S_{e}}{N_{e}}\equiv\frac{1}{k_{B}\,T}\,\Sigma\
,\quad\Sigma=\frac{\tilde{\cal E}+P_{e}}{n_{e}}-\mu_{\mathrm{e}}\ ,$ (133)

where it is convenient to separate for a while a factor $\Sigma$. We re-write
this factor in a convenient form:

$\displaystyle\Sigma$ $\displaystyle=$ $\displaystyle\frac{\tilde{\cal
E}+P_{e}}{n_{e}}-\mu_{\mathrm{e}}=\left[\frac{\tilde{\cal
E}(0)+P_{e}(0)}{n_{e}}-\mu_{\mathrm{e}}\right]+\frac{\Delta\tilde{\cal
E}+\Delta P_{e}}{n_{e}}=\Sigma_{1}+\Sigma_{2}\ ,$ (134)

where $\Delta\tilde{\cal E}=\tilde{\cal E}(T)-\tilde{\cal E}(0)$ and $\Delta
P_{e}=P_{e}(T)-P_{e}(0)$. From relations (C,C) at $T=0$ and from
$\varphi(0)\equiv\varphi_{0}=\tilde{\epsilon}_{F}$ one gets:

$\displaystyle\tilde{\cal E}(0)+P_{e}(0)$ $\displaystyle=$ $\displaystyle
m_{e}c^{2}\,\rho_{0}\,x^{3}_{r}\,\tilde{\epsilon}_{F}=m_{e}c^{2}\,n_{e}\,\tilde{\epsilon}_{F}\
\rightarrow\ \frac{\tilde{\cal E}(0)+P_{e}(0)}{n_{e}}=m_{e}c^{2}\,\varphi_{0}\
.$

Making use of $\mu_{\mathrm{e}}=m_{e}c^{2}\,\varphi$ we write the first term
of $\Sigma$ in a compact form:

$\displaystyle\Sigma_{1}$ $\displaystyle=$ $\displaystyle\frac{\tilde{\cal
E}(0)+P_{e}(0)}{n_{e}}-\mu_{\mathrm{e}}=-m_{e}\,c^{2}\,(\varphi-\varphi_{0})\
.$ (135)

The low temperature decomposition of this equation follows from (126):

$\displaystyle\varphi-\varphi_{0}$ $\displaystyle\simeq$ $\displaystyle
c_{1}\,\frac{B}{\varphi_{0}}=-\frac{2\epsilon^{2}_{F}-1}{6\epsilon_{F}\,x^{2}_{r}}\,B\
,$

therefore:

$\displaystyle\Sigma_{1}$ $\displaystyle\simeq$ $\displaystyle
m_{e}\,c^{2}\,\frac{2\epsilon^{2}_{F}-1}{6\epsilon_{F}\,x^{2}_{r}}\,B\ ,\quad
B=\pi^{2}\,\tau^{2}\ .$ (136)

The non-relativistic ($\epsilon_{F}\rightarrow 1$) and ultra-relativistic
($\epsilon_{F}\rightarrow\tilde{\epsilon}_{F}\rightarrow x_{r}$) limits of
this equation read:

$\displaystyle\Sigma_{1,nr}=m_{e}\,c^{2}\,\frac{1}{6\,x^{2}_{r}}\,B\
,\quad\Sigma_{1,ur}=m_{e}\,c^{2}\,\frac{1}{3\,x_{r}}\,B\ .$ (137)

These equations can be also obtained by calculating $\varphi-\varphi_{0}$
directly from (111) and (114).

The second part of $\Sigma$ is obtained from (131,132):

$\displaystyle\Delta\tilde{\cal E}+\Delta P_{e}$ $\displaystyle=$
$\displaystyle
m_{e}c^{2}\,\rho_{0}\,\left(\frac{x_{r}\epsilon_{F}}{2}+\frac{x_{r}(\epsilon^{2}_{F}+1)}{6\epsilon_{F}}\right)\,B=m_{e}c^{2}\,\rho_{0}\,\frac{x_{r}\,(4\epsilon^{2}_{F}+1)}{6\epsilon_{F}}\,B\
,$

which implies:

$\displaystyle\Sigma_{2}$ $\displaystyle=$
$\displaystyle\frac{\Delta\tilde{\cal E}+\Delta
P_{e}}{n_{e}}=m_{e}c^{2}\,\frac{4\epsilon^{2}_{F}+1}{6\epsilon_{F}\,x^{2}_{r}}\,B\
,$ (138)

which in kinematic limits reduces to

$\displaystyle\Sigma_{2,nr}=m_{e}\,c^{2}\,\frac{5}{6\,x^{2}_{r}}\,B\
,\quad\Sigma_{2,ur}=m_{e}\,c^{2}\,\frac{2}{3\,x_{r}}\,B\ .$ (139)

Adding these results up:

$\displaystyle\Sigma$ $\displaystyle=$ $\displaystyle
k_{B}T\,\hat{s}_{e}\equiv\frac{TS_{e}}{N_{e}}=m_{e}c^{2}\,\frac{\epsilon_{F}}{x^{2}_{r}}\,B\
,\quad B=\pi^{2}\,\tau^{2}\ ,$ (140) $\displaystyle\Sigma_{nr}$
$\displaystyle=$ $\displaystyle m_{e}c^{2}\,\frac{1}{x^{2}_{r}}\,B\
,\qquad\Sigma_{ur}=m_{e}c^{2}\,\frac{1}{x_{r}}\,B\ .$

Equations on the 2nd line can be, of course, obtained directly from the non-
relativistic or ultra-relativistic results for the momentum and energy
densities.