Relativistic Degenerate Electron Equation of State

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Preprint LDC-2009-001 4/16/2009 ROUGH DRAFT ON THE DEGENERATE ELECTRON EQUATION OF STATE Lawrence D. Cloutman [email protected] Abstract We present a set of informal notes on the equation of state appropriate to deep stellar interiors of ordinary stars. We assume the ions are treated as ideal gases, the radiation is treated in the gray one-temperature (1T) approximation, and the electrons are degenerate. We consider both the thermal and caloric equations of state for arbitrary degrees of degeneracy. Non-relativistic and relativistic formalisms are considered for a small selection of zero-dimensional problems where there is no flow and no chemistry. These cases can be helpful in validating equation of state software. c 2009 by Lawrence D. Cloutman. All rights reserved. 1

description

The RFD01 documents are software and documentation for evaluating both the classical and relativistic degenerate electron equation of state for all degrees of degeneracy.

Transcript of Relativistic Degenerate Electron Equation of State

Page 1: Relativistic Degenerate Electron Equation of State

Preprint LDC-2009-0014/16/2009ROUGH DRAFT

ON THE DEGENERATE ELECTRON EQUATION OF STATE

Lawrence D. Cloutman

[email protected]

Abstract

We present a set of informal notes on the equation of state appropriate to deepstellar interiors of ordinary stars. We assume the ions are treated as ideal gases,the radiation is treated in the gray one-temperature (1T) approximation, and theelectrons are degenerate. We consider both the thermal and caloric equations of statefor arbitrary degrees of degeneracy. Non-relativistic and relativistic formalisms areconsidered for a small selection of zero-dimensional problems where there is no flowand no chemistry. These cases can be helpful in validating equation of state software.

c©2009 by Lawrence D. Cloutman. All rights reserved.

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1 Equations of State

We consider thermal and caloric equations of state appropriate to the interiors of ordinary

stars. These have been incorporated into an updated version of the COYOTE computer

program [1], which was used to produce the numerical results presented here. A short table

of physical constants is given in the appendix. All units are CGS and the temperature T is

in K.

The thermal equation of state is the sum of partial pressures for each species, with

the ionic species treated as ideal gases:

P =∑α 6=e

RραT

+ Pe +aT 4

3, (1)

where P is the total pressure, R is the universal gas constant, ρα is the density of species

α, a is the radiation energy density constant, and Mα is the molecular weight of species α.

The electron pressure Pe can be calculated from the usual degenerate gas equation of state

or as just another ideal gas, depending upon the thermodynamic conditions. The radiation

pressure term assumes local thermodynamic equilibrium between the gas and radiation field.

The caloric equation of state provides the relationship among temperature, density,

internal energy, and composition and is given by

ρI =∑α 6=e

ραIα(T ) + ρeIe + aT 4, (2)

where Iα is the ionic species specific thermal internal energy, and Ie is the electron internal

energy. In the present application, we assume

Iα = CvαT =RT

(γα − 1)Mα

(α 6= e). (3)

As in the case of the electron pressure, the electron internal energy can be calculated from

the ideal gas case (3) in the nondegenerate limit.

The degenerate non-relativistic electron equations of state are [2, 3]

ne(η, T ) =4π

h3(2mekBT )3/2 F1/2(η) = 5.44940154× 1015 T 3/2F1/2(η), (4)

Pe(η, T ) =8πkBT

3h3(2mekBT )3/2 F3/2(η) = 0.501583989 T 5/2F3/2(η), (5)

Fn(η) =∫ ∞0

un

exp(u− η) + 1du, (6)

and

ρeIe = 1.5Pe, (7)

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where h is Planck’s constant, me is the electron mass (not to be confused with the electron

molecular weight Me), kB is the Boltzmann constant, and the integration variable is u =

p2/2mekBT . The degeneracy parameter is the chemical potential in units of kBT , η =

µ/kBT . Given ne and T , first F1/2(η), then η are computed. Then F3/2(η) and finally Pe

may be computed. This electron equation of state is correct unless relativistic effects become

important, which we shall discuss in the next section.

The degenerate electron equation of state requires numerical evaluation of the Fermi-

Dirac integrals and their inverses. A simple but useful approximation that avoids the calcu-

lation of η is [4]

F3/2(η) = F1/2(η)3[1 + 0.1938F1/2(η)

]5/3

2[1 + 0.12398F1/2(η)

] , (8)

which is accurate to within 0.02 percent for η < 30. Accurate tables of the Fermi-Dirac

integrals are given by Cloutman [5], and Antia [6] provides accurate rational approximations

to the integrals and their inverses.

For problems in the ideal gas limit, use of the ideal gas approximation for electrons

is computationally less complex and more efficient than the degenerate electron equation of

state. For η < −3 (or F1/2(η) < 4.3366× 10−2), the difference between the two is less than

one percent. Using this accuracy criterion, the ideal gas limit is valid for

ne < 2.36× 1014T 3/2 cm−3, (9)

or

ρe < 2.15× 10−13T 3/2 g/cm3. (10)

Less restrictive criteria are found commonly in the literature. We define the mean

molecular weight per free electron µe by

1

µe

=∑Z

YZZ

AZ

. (11)

The value of µe is usually between 1 and 2. Cox and Giuli [7] (p. 844) show that for η = 0,

which is where degeneracy definitely becomes significant,

ρ

µe

= 6.12× 10−9T 3/2 g/cm3. (12)

Slightly different choices of accuracy criterion give slightly different numerical factors (for

example, see Clayton [3], p. 88).

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2 Relativistic Degeneracy

At the high densities and temperatures where relativistic effects must be included, equa-

tions (4) through (7) must be generalized. The general case is given by equations (161) and

(163) of chapter X of Chandrasekhar’s classic book [2]: 1

ne =8π

h3

∫ ∞0

p2 dp

exp (E/kBT − η) + 1,

ρeIe =8π

h3

∫ ∞0

Ep2 dp

exp (E/kBT − η) + 1, (13)

and

Pe =8π

3h3

∫ ∞0

p3

exp (E/kBT − η) + 1

∂E

∂pdp, (14)

where p is the electron momentum and E is its kinetic energy. Equations (4) through (7)

are derived from these general equations using the Newtonian approximation E = p2/2me.

Following Cox and Giuli [7], we note the following assumptions are made in the above

equations:

1. Thermodynamic equilibrium

2. Isotropic momentum distribution

3. Fermi-Dirac distribution function dNe(p) = 8πp2 dp/{h3 [exp(E/kBT − η) + 1]}

4. Weakly interacting electrons

5. No e−-e+ pair creation

To generalize to the relativistic case, we note that

p =mev

[1−( v/c )2]1/2=

vEt

c2(15)

for a single electron with total energy Et (rest energy mec2 plus kinetic E). This is equivalent

to

E2t = p2c2 + m2

ec4 =

mec2

1− (v/c)2 (16)

Chadrasekhar’s equations (48) and (65) in chapter X give the kinetic energy of a single free

electron as

E =(p2c2 + m2

ec4)1/2

−mec2 (17)

1Note that Chandrasekhar uses a different convention than me for the chemical potential: His α = −η.

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Note that this equation may be rearranged algebraically to give Chandrasekhar’s equa-

tion (160),E2

c2+ 2Eme = p2. (18)

At low energies, this reduces to the classical Newtonian approximation. The derivative

∂E/∂p is the electron speed,

v =∂E

∂p=

p

me

(1 +

p2

m2ec

2

)−1/2

. (19)

This reduces to p/me and c in the classical and extreme relativistic regimes, respectively.

Divine [8] recasts the degenerate electron equation of state into a form valid for all

speeds, from the slow classical limit to the extreme relativistic limit. To do this, the Fermi-

Dirac integrals are generalized to the following three cases:

IN1/2(η, β) =∫ ∞0

u1/2 du

exp(u− η) + 1

(1 +

5

2βu + 2β2u2 +

β3u3

2

)1/2

(20)

IP 3/2(η, β) =∫ ∞0

u3/2 du

exp(u− η) + 1

(1 +

1

2βu)3/2

(21)

IU3/2(η, β) =∫ ∞0

u3/2 du

exp(u− η) + 1

(1 +

5

2βu + 2β2u2 +

β3u3

2

)1/2

. (22)

The integration variable is the scaled energy of an individual electron u = E/kBT , and

β = kBT/mec2 =

T

5.93× 109. (23)

The degenerate electron equations of state are

ne(η, T ) =4π

h3(2mekBT )3/2 IN1/2(η, β), (24)

Pe(η, T ) =8πkBT

3h3(2mekBT )3/2 IP 3/2(η, β), (25)

and

ρeIe = Ue(η, T ) =4πkBT

h3(2mekBT )3/2 IU3/2(η, β), (26)

In the nonrelativistic limit (β → 0), we recover equations (4) through (7). See also Blinnikov,

et al. [9] for expansions appropriate to all levels of degeneracy and relativistic effects.

We note that most of the literature uses a form of the generalized Fermi-Dirac inte-

grals (20) through (22) written in terms of another generalized Fermi-Dirac integral,

Fn(η, β) =∫ ∞0

un (1 + 0.5βu)1/2

exp(u− η) + 1du (n > −1). (27)

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Then

IN1/2(η, β) = F1/2(η, β) + βF3/2(η, β) (28)

IP 3/2(η, β) = F3/2(η, β) +β

2F5/2(η, β) (29)

IU3/2(η, β) = F3/2(η, β) + βF5/2(η, β). (30)

While this form is mathematically elegant, it has no clear computational advantage over

Divine’s form.

When are relativistic effects important? Part of the conventional wisdom has it that

relativistic effects are important for densities greater than 106 − 107 g/cm3. Clayton [3]

derives this limit assuming that relativistic effects are important once the Fermi energy is

twice the rest mass energy of an electron. This limit is far too lax if one needs pressure and

internal energy to accuracies of a few percent or less. That is, this criterion is a sufficient

condition for relativistic effects to be significant. However, it is clear from the definitions

of the generalized Fermi-Dirac integrals that relativistic effects are important if β is large,

regardless of the density.

There is an additional limit emphasized by Mitalas [10], and a derivation of this limit

may be found on page 807 of Cox and Giuli. For relativistic effects to be negligible for

η > 0, βη << 1. The product βη is the chemical potential in units of the electron rest mass,

which clearly is a physically meaningful relativity parameter. Figure 24.4 of Cox and Giuli

(p. 847) is a useful version of the traditional ρ − T plane showing where relativistic effects

are important for the electron equation of state.

Mitalas [10] suggests a slightly different approximation,

Fn(η, β) = Fn(η) +β

4Fn+1(η). (31)

He also notes F5/2(η) ≈ 5ηF3/2(η)/7 for η >> 1, in which case F3/2(η, β) ≈ F3/2(η)(1 +

15 β η / 28). Given the ease with which the integrals in equations (20) through (22) may

be evaulated and tabulated using the methods of Cloutman [5], we shall not investigate the

accuracy of such approximations at the present time.

The integrals obey the following recursion relation (for example, Cox and Giuli, p.

810):

(n + 1)Fn(η, β) =dFn+1(η, β)

dη− β

dFn(η, β)

dβ(k > −1) (32)

For large β, Fn(η, β) ≈ (β/2)1/2Fn+1/2(η, β) for all η.

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3 Partial and Pressure Ionization, Coulomb Correc-

tions, Pair Production

There are several effects that we have not yet included that affect the equation of state under

certain conditions. We shall discuss some of these in this section. Clayton [3] (pp. 139-155)

provides an elementary discussion of these issues.

The first issue is partial and pressure ionization. In this study, we assumed total

ionization. We argue on energetic grounds that this should be a good approximation for

the conditions of deep stellar interiors where the thermal energy per particle exceeds the

ionization potential. In principle, partial ionization can be included in my program by using

either the equilibrium chemistry package to solve the Saha equation or the kinetic chemistry

package using ionization and recombination rates. There is presently no mechanism in the

code to account for pressure ionization (lowering of the continuum). Eggleton, Faulkner

and Flannery [11] provide a first pass at computing variable ionization, including pressure

ionization. Updates to this model are presented by Pols, et al. [12]. Weaver, Zimmerman,

and Woosley [13] briefly describe an average atom model for partial and pressure ionization

of a multicomponent mixture.

Another physical effect is the change to pressure and internal energy due to the

Coulomb interactions between charged particles in the plasma:

“Coulomb interactions between charged particles provide the major non-ideal

correction to the pressure at the densities and temperatures encountered in stars

of around a solar mass or less, while they also crucially influence the properties

of matter at high densities and low temperatures.” Pols, et al. [12], p. 964.

Clayton provides an introduction to this effect, and Iben, Fujimoto, and MacDonald [14]

provide a more detailed and complete model. Pols, et al. also discuss Coulomb corrections.

For very high temperatures, pair creation of electron-positron pairs must be included

in the equation of state. This occurs when the thermal energy of fluid particles approaches

or exceeds the rest mass energy of an electron, or kBT > mec2 [15].

Timmes and Arnett [16] also provide expressions for arbitrary levels of degeneracy

and relativistic effects, including electron-positron pair creation. Timmes and Swesty [17]

describe a numerical method for implementing tables of such equations of state while insuring

thermodynamic consistency.

Researchers at LLNL created the more detailed OPAL equation of state tables [18, 19,

20]. Similar work has been reported by Mihalas, Hummer, and Dappen (MHD) [21, 22, 23].

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4 Non-Relativistic Numerical Examples

The COYOTE computer program was used to run several zero-dimensional test cases in

order to check out the software that evaluates the equations of state described in the previous

section. We shall consider four sets of physics options and two densities: one nondegenerate,

one degenerate.

The first example is conditions for helium burning via the triple alpha process. Here

the temperature is above 108 K, which corresponds to an average thermal energy per particle

of 104 ev. 2 Here is a short list of the ionization energies of ions with a single electron: 3

1. H I (Z=1): 13.59844 ev

2. He II (Z=2): 54.41778

3. O VIII (Z=8): 871.4101

4. S XVI (Z=16): 3494.1892

5. Cr XXIV (Z=24): 7894.81

6. Cu XXIX (Z=29): 11567.617

We should expect that if the thermal energy (temperature) is larger than the ionization

energy, then the electron will be stripped from most atoms. In the cases I will be considering

first, helium is burned to carbon and oxygen. My temperatures will always exceed 104 ev, but

the ionization energy of O VIII is less than 103 ev. Therefore, the approximation of total

ionization should be adequate, and that approximation will be assumed in the following

examples.

4.1 Ideal Gas Case Without Radiation

We consider a mixture of fully ionized 4He and its electrons at a temperature of 5.0× 108 K

and densities of 103 and 106 g/cm3. Here is selected output from the program.

isp label wt gamma charge C_v [#1]

1 e- N 5.485798959D-04 1.666666667D+00 -1 2.273463736d+11

2 4He III N 4.001502840D+00 1.666666667D+00 2 3.116770247d+07

3 12C VII N 1.199670852D+01 1.666666667D+00 6 1.039598901d+07

4 16O IX N 1.599052636D+01 1.666666667D+00 8 7.799471207d+06

idealg = 1, eosform = 2.0, rad1T = 0.0

2An average thermal energy per particle of 1 ev corresponds to a temperature of 11,605.9 K.3A note on nomenclature: The roman numeral following a chemical symbol is the number of electrons

stripped from the ion, plus 1. Therefore, H I is a neutral hydrogen atom and He II is singly-ionized helium.

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mixture mean molecular weight = 1.3342D+00

mixture C_v = 9.3483D+07 [#2]

mixture sound speed = 2.2788D+08 cm/s

i j P rho I T

3 3 3.11677D+19 1.00027D+03 4.67415D+16 5.00000D+08

3 3 3.11677D+22 1.00027D+06 4.67415D+16 5.00000D+08

Mass Fractions

i j 1 2 3 4

3 3 2.74112D-04 9.99726D-01 0.00000D+00 0.00000D+00

3 3 2.74112D-04 9.99726D-01 0.00000D+00 0.00000D+00

Hand calculation produces these results for the ρ = 1000 case:

• ρe = 0.27418601, ρHe = 9.99995926× 102 g/cm3

• ne = ρe/me = 3.009927328× 1026 cm−3

• CvT = 4.67415× 1016 erg/g using Cv at [#2]

• Pe = 2.077839836 × 1019, PHe = 1.038919184 × 1019, P = Pe + PHe = 3.11676 × 1019

dynes/cm2

• ρeIe = 3.116759753 × 1019, ρHeIHe = 1.558378775 × 1019, ρI = ρeIe + ρHeIHe =4.675138528× 1019 ergs/cm3

• Ie = 1.136731868 × 1020, IHe = 1.558385124 × 1016, I = (ρI)/ρ = 4.673876581 × 1016

ergs/g

Note that I have presented 10 digits in some of the above numbers. This runs counter to

standard practice of reporting results only to the number of significant figures determined by

the least accurate input number. In the present example, we are limited to 4 or 5 significant

digits by the accuracy of the physical constants. The computer doesn’t care how many digits

of R or Mα are significant, it carries all our double precision arithmetic out to about 16 digits.

I did the hand calculations with a 10-digit scientific calculator using the full significance, just

as the computer did, in an attempt to duplicate the computer output. Therefore, differences

between my hand calculations and the computational results should be limited by rounding

in the hand calculator or in the printout if it is has fewer than 10 digits.

Note that the pressure calculations look correct, but the internal energy is off in the

fourth digit. This small discrepancy appears to be due to the number of significant figures

carried in the physical constants. The hand calculation used Cvα for each species computed

from equation (3). These are listed in the printout above in the column labelled [#1]. The

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enthalpy tables generated by gasnuc and used in the cfd program to compute the mixture Cv

at [#2] uses a specific heat at constant presssure CP = 4.968 kcal/mol-K = 20.786 kJ/mol-K.

The ρ = 106 g/cm3 case also appears to be working correctly. The pressure scales

linearly with density, and I is independent of density.

4.2 Degenerate Gas Case Without Radiation

This is the same problem as in the previous subsection except that idealg has been changed

from 1 to 2 (electrons treated as degenerate rather than as an ideal gas).

i j p rho I T

3 3 3.11881D+19 1.00027D+03 4.67703D+16 5.00000D+08

3 3 4.98584D+22 1.00027D+06 7.47681D+16 5.00000D+08

Mass Fractions

i j 1 2 3 4

3 3 2.74112D-04 9.99726D-01 0.00000D+00 0.00000D+00

3 3 2.74112D-04 9.99726D-01 0.00000D+00 0.00000D+00

Hand calculation produces these results for the ρ = 1000 case:

• ρe = 0.27418601, ρHe = 9.99995926× 102 g/cm3

• ne = ρe/me = 3.009927328× 1026 cm−3

• F1/2(η) = 4.940287164× 10−3, η ≈ −5.2, F3/2(η) = 7.417716111× 10−3

• Pe = 2.079882898×1019, PHe = 1.038919184×1019, P = Pe+PHe = 3.118802082×1019

dynes/cm2

• ρeIe = 3.119824347 × 1019, ρHeIHe = 1.558378775 × 1019, ρI = ρeIe + ρHeIHe =

4.678203122× 1019 ergs/cm3

• I = (ρI)/ρ = 4.676940348× 1016 ergs/g

The pressure is correctly computed by the code. We are at a low enough density that

degeneracy effects are not important. The discrepancy in the internal energy is the same

small value as in the previous case and is undoubtedly due to the same cause.

For ρ = 106, electron degeneracy is important, which can be seen by comparing P

and I for the two densities. Here are the hand calculations:

• ρe = 2.7418601× 102, ρHe = 9.99995926× 105 g/cm3

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• ne = ρe/me = 3.009927328× 1029 cm−3

• F1/2(η) = 4.940287164, η ≈ 3.55, F3/2(η) = 14.07626571

• Pe = 3.946900081×1022, PHe = 1.038919184×1022, P = Pe+PHe = 4.985819265×1022

dynes/cm2

• ρeIe = 5.920350122 × 1022, ρHeIHe = 1.558378775 × 1022, ρI = ρeIe + ρHeIHe =

7.478728897× 1022 ergs/cm3

• I = (ρI)/ρ = 7.476710185× 1016 ergs/g

4.3 Degenerate Gas Case With Radiation

This is the same problem as in the previous subsection, except radiation pressure has been

included (idealg = 2, rad1T = 0.0 changed to 1.0).

i j p rho I T

3 3 1.88751D+20 1.00027D+03 5.19328D+17 5.00000D+08

3 3 5.00160D+22 1.00027D+06 7.52406D+16 5.00000D+08

Mass Fractions

i j 1 2 3 4

3 3 2.74112D-04 9.99726D-01 0.00000D+00 0.00000D+00

3 3 2.74112D-04 9.99726D-01 0.00000D+00 0.00000D+00

The electron and ion contributions are the same as for the case in the previous sub-

section. However, comparison with those results shows that the radiation dominates in the

lower density case and is not very important in the higher density case. For ρ = 103 g/cm3,

• Pr = 1.575625× 1020, P = 1.887505208× 1020 dynes/cm3

• ρIr = 4.726875× 1020, ρI = 5.194695312× 1020 ergs/cm3

• Ir = 4.725599× 1017, I = 5.193293123× 1017 ergs/g

For ρ = 106 g/cm3,

• P = 5.001575515× 1022 dynes/cm3

• ρI = 7.525997647× 1022 ergs/cm3

• I = 7.523966176× 1016 ergs/g

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We get good agreement between the code and the hand calculation, so we conclude that

the equations of state have been implemented correctly. Note one peculiarity: The internal

energy density I decreases with density while ρI increases. This feature is due to the radiation

pressure term, for which I does indeed decrease with increasing density.

4.4 Ideal Gas Case Without Radiation Revisited

The ideal gas equation of state has been extensively tested and used for combustion research

over many years, but this is the first serious test of the astrophysical version. The results in

the previous three subsections used the input file gasdata generated by the physics library

program gasnuc. The results in this subsection used an input file generated with the gasmak

library, which is intended for use under less extreme conditions of density and temperature.

However, it should produce the same results as in subsection 4.1.

To test this idea, we reran the case in subsection 4.1 with the code “switches” all set

to typical combustion physics values, and produced a new gasdata from gasmak. I had to

include the wt and gamma arrays in the input file citape and use eosform(k)=1.0. These

results are very close to those in subsection 4.1. The difference is that gasnuc assumes all

nuclei are totally ionized, whereas gasmak does not.

gasdata input

isp label wt htform gamma charge

1 e- 5.485800000D-04 0.000000000D+00 1.666667000D+00 -1

2 He++ 4.000763040D+00 1.821902486D+03 1.666667000D+00 2

3 C 1.201120000D+01 1.699772945D+02 1.666667000D+00 0

4 O+ 1.599880000D+00 3.730241396D+02 1.666667000D+00 1

idealg = 1 rad1T = 0.0D+00 rgas = 8.31451D+07

species input

k=1 wt=5.485800D-04 gamma=1.66667D+00 eosform=1.0D+00 ze=-1.0

k=2 wt=4.000763D+00 gamma=1.66667D+00 eosform=1.0D+00 ze= 2.0

k=3 wt=1.201120D+01 gamma=1.66667D+00 eosform=1.0D+00 ze= 0.0

k=4 wt=1.599880D+00 gamma=1.66667D+00 eosform=1.0D+00 ze= 1.0

nreg = 1 mmw, P, C_v = 1.3340D+00 3.1173D+19 9.3495D+07

nreg = 1 gamma, sound spd = 1.6667D+00 2.2791D+08

i j P rho I T

3 3 3.11735D+19 1.00027D+03 4.67474D+16 5.00000D+08

3 3 3.11735D+22 1.00027D+06 4.67474D+16 5.00000D+08

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Mass Fractions

i j 1 2 3 4

3 3 2.74163D-04 9.99726D-01 0.00000D+00 0.00000D+00

3 3 2.74163D-04 9.99726D-01 0.00000D+00 0.00000D+00

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5 Relativistic Numerical Examples

Tables of relativistic Fermi-Dirac integrals were calculated that were used to develop software

for the relativistic equation of state. The numerical method is the same as in [5] and is

summarized in appendix B. The tables cover the ranges −30 ≤ η ≤ 70 and 0.0 ≤ β ≤ 1.2.

The software is based on bilinear interpolation, which is easy to implement and has the

advantage that it is reversible. That is, if one interpolates a table to find η for a given I3/2N

and β, then interpolation for that η and β will produce the original value of I3/2N to within

rounding errors. The question of thermodynamic consistency, however, needs examination.

It may be necessary to change to another algorithm in the future.

Non-Relativistic Fermi-Dirac Integrals, beta = 0.000000D+00

eta F_1/2(eta) F_3/2(eta) F_5/2(eta) F_3/2/F_1/2

-2.0D+01 1.8266498364D-09 2.7399747556D-09 6.8499368901D-09 1.500000

-1.0D+01 4.0233994367D-05 6.0351475898D-05 1.5087929519D-04 1.500012

-5.0D+00 5.9571769052D-03 8.9463822604D-03 2.2379248359D-02 1.501782

0.0D+00 6.7809389515D-01 1.1528038371D+00 3.0825860828D+00 1.700065

5.0D+00 7.8379760573D+00 2.7802446216D+01 1.2748954491D+02 3.547146

1.0D+01 2.1344471492D+01 1.3427015996D+02 1.0346842542D+03 6.290629

2.0D+01 5.9812795370D+01 7.2656828397D+02 1.0590639177D+04 12.14737

3.0D+01 1.0969481834D+02 1.9853113777D+03 4.2929257585D+04 18.09850

7.0D+01 3.9053966669D+02 1.6419179065D+04 8.2233568906D+05 42.04228

The above table contains selected values of the non-relativistic Fermi-Dirac inte-

grals from non-degenerate to highly degenerate conditions. The last column is the ratio

F3/2(η)/F1/2(η), which has a value of 1.5 in the non-degenerate regime. Degeneracy effects

begin to be important in the range −5 ≤ η ≤ 0.

Relativistic Fermi-Dirac Integrals, beta = 1.000000D-03 (T = 5.93d+06 K)

eta I_N^1/2 I_P^3/2 I_U^3/2 I_P / I_N

F_1/2(eta,beta) F_3/2(eta,beta) F_5/2(eta,beta) I_U / I_P

-2.0D+01 1.8300763027D-09 2.7451144550D-09 2.7485424186D-09 1.50000

F_n 1.8273346162D-09 2.7416864914D-09 6.8559272181D-09 1.00125

-1.0D+01 4.0309466704D-05 6.0464684858D-05 6.0540190479D-05 1.50001

F_n 4.0249077525D-05 6.0389179238D-05 1.5101124068D-04 1.00125

-5.0D+00 5.9683647766D-03 8.9631740393D-03 8.9743734518D-03 1.50178

F_n 5.9594128020D-03 8.9519746268D-03 2.2398825018D-02 1.00125

0.0D+00 6.7953557400D-01 1.1551168247D+00 1.1566595149D+00 1.69986

F_n 6.7838199987D-01 1.1535741345D+00 3.0853804114D+00 1.00134

5.0D+00 7.8727569873D+00 2.7898127777D+01 2.7961958398D+01 3.54363

F_n 7.8449226902D+00 2.7834297156D+01 1.2766124123D+02 1.00229

1.0D+01 2.1512535321D+01 1.3504700934D+02 1.3556546596D+02 6.06484

14

Page 15: Relativistic Degenerate Electron Equation of State

F_n 2.1378006768D+01 1.3452855273D+02 1.0369132308D+03 1.00384

2.0D+01 6.0723318472D+01 7.3452712403D+02 7.3984357633D+02 12.0963

F_n 5.9994107800D+01 7.2921067173D+02 1.0632904599D+04 1.00724

3.0D+01 1.1218582469D+02 2.0176032258D+03 2.0391942639D+03 17.9845

F_n 1.1018981251D+02 1.9960121877D+03 4.3182076140D+04 1.01070

7.0D+01 4.1124248826D+02 1.7040118103D+04 1.7456855841D+04 41.4357

F_n 3.9461910790D+02 1.6623380365D+04 8.3347547647D+05 1.02446

The above table contains selected values of the relativistic Fermi-Dirac integrals from

non-degenerate to highly degenerate conditions for β = 0.001, which corresponds to a tem-

perature of only 5.93 × 106 K. There are two kinds of lines in this table. Lines with a

numerical value of η in the first column contain values of IN1/2 , IP 3/2 , IU3/2 , and the ratio

IP 3/2/IN1/2 . This ratio should be 1.5 in the non-degenerate, non-relativistic (ideal gas) limit,

and that is the case.

Lines that begin with Fn contain the generalized Fermi-Dirac integrals F1/2(η, β),

F3/2(η, β), F5/2(η, β), and the ratio IU3/2/IP 3/2 . This ratio should be 1.0 in the non-relativistic

ideal gas limit. These values should be non-relativistic to a high degree of accuracy. In-

deed, even the most extreme table entry F5/2(70, 0.001) = 8.33 × 105 is very close to the

non-relativistic value of F5/2(70) = 8.22 × 105. As pointed out by Cox and Giuli [7] and

Mitalas [10], βη is the relevant relativity parameter if η > 0, and this parameter has a value

of 0.07 in this case. Relativistic corrections of a few percent are therefore the most that we

should expect unless the degeneracy becomes even stronger.

Relativistic Fermi-Dirac Integrals, beta = 1.000000D-02

eta I_N^1/2 I_P^3/2 I_U^3/2 I_P/I_N I_U/I_P

-2.0D+01 1.8610488070D-09 2.7915732115D-09 2.8261209177D-09 1.5000 1.0124

-1.0D+01 4.0991676049D-05 6.1488002972D-05 6.2248963800D-05 1.5000 1.0124

-5.0D+00 6.0694944118D-03 9.1149584335D-03 9.2278286230D-03 1.5018 1.0124

0.0D+00 6.9257111528D-01 1.1760276835D+00 1.1915796198D+00 1.6981 1.0132

5.0D+00 8.1882795811D+00 2.8765028924D+01 2.9411003478D+01 3.5130 1.0225

1.0D+01 2.3045277346D+01 1.4211333702D+02 1.4739706693D+02 6.1667 1.0372

2.0D+01 6.9122745126D+01 8.0756464659D+02 8.6259256375D+02 11.6831 1.0681

3.0D+01 1.3542777639D+02 2.3166022714D+03 2.5435570133D+03 17.1058 1.0980

4.0D+01 2.2204918983D+02 4.9718830712D+03 5.5992962195D+03 22.3909 1.1262

7.0D+01 6.1283442222D+02 2.2989230017D+04 2.7626633947D+04 37.5130 1.2017

The above table contains selected values of the relativistic Fermi-Dirac integrals from

non-degenerate to highly degenerate conditions for β = 0.01 (T = 5.9× 107 K). The ratios

IP 3/2(η, β)/IN1/2(η, β) = 1.5 and IU3/2(η, β)/IP 3/2(η, β) = 1.0124 for η = −30 already show

15

Page 16: Relativistic Degenerate Electron Equation of State

a small correction to the internal energy even at this low value of β. Both ratios show a

deviation from the non-relativistic equation of state for η ≥ 0. The deviations in the pressure

and internal energy are in opposite directions: Relativistic effects lower the pressure from

the non-relativistic value, but raise the internal energy.

The ratio IU3/2(η, β)/IP 3/2(η, β) is unity in the non-relativistic limit. If one needs the

equation of state to be highly accurate, say to one or two percent, then the relativistic form

of the Fermi-Dirac integrals must be used for all advanced stages of thermonuclear burning

in stars. That is the case for our He-burning deflagrations, which have β values ranging from

0.02 to 0.2 and η ranging from around 0 on up to (more typically) very large values.

Relativistic Fermi-Dirac Integrals, beta = 8.000000D-02

eta I_N^1/2 I_P^3/2 I_U^3/2 I_P/I_N I_U/I_P

-2.0D+01 2.1099706215D-09 3.1649559333D-09 3.4573522515D-09 1.5000 1.0924

-1.0D+01 4.6474500680D-05 6.9712272321D-05 7.6152704776D-05 1.5000 1.0924

-5.0D+00 6.8822748209D-03 1.0334840467D-02 1.1290138286D-02 1.5017 1.0924

0.0D+00 7.9756571221D-01 1.3442792541D+00 1.4761587432D+00 1.6855 1.0981

5.0D+00 1.0789258068D+01 3.5848901929D+01 4.1468602335D+01 3.3226 1.1568

1.0D+01 3.6127876763D+01 2.0141514315D+02 2.4938290334D+02 5.5751 1.2382

2.0D+01 1.4571631007D+02 1.4544288748D+03 1.9962686266D+03 9.9812 1.3725

3.0D+01 3.5969711021D+02 5.0947576926D+03 7.4855216620D+03 14.1640 1.4693

4.0D+01 7.0982871156D+02 1.2926075918D+04 1.9916917487D+04 18.2101 1.5408

7.0D+01 2.8965620597D+03 8.6778679108D+04 1.4526126426D+05 29.9592 1.6739

Relativistic Fermi-Dirac Integrals, beta = 5.000000D-01

eta I_N^1/2 I_P^3/2 I_U^3/2 I_P/I_N I_U/I_P

-2.0D+01 3.8647559200D-09 5.7971338816D-09 8.1250650143D-09 1.5000 1.4016

-1.0D+01 8.5125935008D-05 1.2768963684D-04 1.7896555141D-04 1.5000 1.4016

-5.0D+00 1.2612456657D-02 1.8934790886D-02 2.6540859491D-02 1.5013 1.4017

0.0D+00 1.5444667734D+00 2.5360735014D+00 3.5928741185D+00 1.6420 1.4167

5.0D+00 3.0920462644D+01 8.9053253616D+01 1.3762655541D+02 2.8801 1.5454

1.0D+01 1.4791258199D+02 6.8542850125D+02 1.1427387425D+03 4.6340 1.6672

2.0D+01 8.9445052736D+02 7.4406200481D+03 1.3328810905D+04 8.3186 1.7914

3.0D+01 2.7410012980D+03 3.3019001872D+04 6.1061718833D+04 12.0463 1.8493

4.0D+01 6.1875555471D+03 9.7670675641D+04 1.8384141223D+05 15.7850 1.8823

7.0D+01 3.1127224576D+04 8.4106570839D+05 1.6224610601D+06 27.0203 1.9291

The values of β = 0.08 and 0.5 were chosen to be in the middle of the range of T

encountered in He burning and near the upper limit of T found all but the most advanced

stages of nuclear burning. The ratios clearly show the importance of relativistic effects at

these temperatures regardless of the density (which determines the degeneracy parameter η

for a given T ). Note also that the ratio IU3/2(η, β)/IP 3/2(η, β) is approaching the extreme

16

Page 17: Relativistic Degenerate Electron Equation of State

relativistic limit of 2.0 as η and β increase. Note also that relativistic effects become impor-

tant at lower values of η and β for the caloric equation of state than for the thermal equation

of state.

The question of accuracy of the numerical integrations used to make these tables

is legitimate. The code is the same one used to make extensive tables for β = 0 that

are accurate to at least 10 digits [5]. The numerical method is based on Simpson’s rule

with Aitken extrapolation to obtain values for finite integration intervals, followed by two

extrapolation techniques to obtain the value on the infinite interval. Internal checks indicate

that the tables in this appendix were computed to an accuracy of 12-15 digits. This is

consistent with the 16 digit accuracy of double precision artithmetic on my computer.

A cross-check was made by using Gauss-Laguerre quadrature with 8, 16, 32, and

64 nodes to compute some of these integrals, an approach proposed by Kippenhahn and

Thomas [24]. We obtained agreement to about 10 digits in the non-degenerate limit. How-

ever, the Gauss-Laguerre quadrature failed miserably at high degeneracy, with the values

of the integrals not converging smoothly as the number of integration nodes was increased.

While the source of this behavior was not investigated, there are three obvious suspects:

Round-off errors, coding errors, and (most likely) the fact that the error term is proportional

to the n-th order derivative of the integrand, where n is the number of nodes. If the integrand

is such that derivatives grow without bound as the order increases, then the truncation error

is also unbounded. My code has been checked in the limits of large and small η by comparing

to analytic series expansions of the integrals and by comparing to earlier published values.

Now let us consider the relativistic effects by examining the governing equations.

Consider equation (4) rewritten as

F1/2(η) = 1.105× 108 ρ

µe

T−3/2 (33)

as a measure of the importance of degeneracy effects. If F1/2(η) < 6.0 × 10−3, or η < −5,

then degeneracy is negligible (approximately 0.1 percent effect on the pressure). Relativistic

effects are important when β, defined by equation (23), is much less than unity. For β = 0.01,

we have T = 5.93×107 K. We saw from the table above that even at this low (by the standards

of He burning) temperature, relativistic effects are present at the one percent level. Even for

η ≈ −5, the effect is about 2 percent, and it gets larger the more degenerate the electrons.

The density for these conditions is ρ/µe = 24.6 g/cm3. The molecular weight per electron is

a weak function of composition for H-depleted mixtures, and its value is approximately 2.0.

17

Page 18: Relativistic Degenerate Electron Equation of State

6 Conclusions

The results of this study suggest the following conclusions:

1. It is possible, with relatively elementary numerical techniques, to provide Fermi-Dirac

integrals, both relativistic and non-relativistic, with an accuracy of at least one part

in 1010.

2. It is easy to generate extensive tables of the required generalized Fermi-Dirac integrals.

Only a few selected values are presented here.

3. Relativistic electron degeneracy is important in all advanced phases of stellar evolution,

including helium burning.

4. The degenerate electron equation of state for arbitrary degrees of degeneracy and

relativity parameter is a straightforward generalization of the non-relativistic equation

of state and is little more difficult to implement into a fluid dynamics code.

5. Some simple Pade approximants or similar numerical device would be convenient for

computing the integrals as functions of η and β, as well as for their inverses.

18

Page 19: Relativistic Degenerate Electron Equation of State

References

[1] L. D. Cloutman, COYOTE: A Computer Program for 2D Reactive Flow Simulations,

Lawrence Livermore National Laboratory Report UCRL-ID-103611, 1990.

[2] S. Chandrasekhar, An Introduction to the Study of Stellar Structure (University of

Chicago Press, Chicago, 1939).

[3] D. D. Clayton, Principles of Stellar Evolution and Nucleosynthesis (McGraw-Hill, New

York, 1968).

[4] R. B. Larson and P. R. Demarque, “An application of Henyey’s approach to the inte-

gration of the equations of stellar structure,” Ap. J. 140, 524 (1964).

[5] L. D. Cloutman, “Numerical evaluation of the Fermi-Dirac integrals,” Ap. J. Suppl. 71,

677 (1989).

[6] H. M. Antia, “Rational function approximations for Fermi-Dirac integrals,” Ap. J.

Suppl. 84, 101 (1993).

[7] J. P. Cox and R. T. Giuli, 1968, Principles of Stellar Structure, Vol. 2. Physical Prin-

ciples (New York: Gordon and Breach); chapt. 24.

[8] N. Divine, “Numerical evaluation of the degenerate equation of state,” Ap. J. 142, 1652

(1965).

[9] S. I. Blinnikov, N. V. Dunina-Barkovskaya, and D. K. Nadyozhin, “Equation of state

of a Fermi gas: Approximations for various degrees of relativism and degeneracy,” Ap.

J. Suppl. 106, 171 (1996).

[10] R. Mitalas, “Relativity parameter at high degeneracy,” Ap. J. 172, 179 (1972).

[11] P. P. Eggleton, J. Faulkner, and B. P. Flannery, “An approximate equation of state for

stellar material,” Astron. Astrophys. 23, 325 (1973).

[12] O. R. Pols, C. A. Tout, P. P. Eggleton, and Z. Han, “Approximate input physics for

stellar modelling,” MNRAS 274, 964 (1995).

[13] T. A. Weaver, G. B. Zimmerman, and S. E. Woosley, “Presupernova evolution of massive

stars,” Ap. J. 225, 1021 (1978).

[14] I. Iben, Jr., M. Y. Fujimoto, and J. MacDonald, “Diffusion and mixing in accreting

white dwarfs,” Ap. J. 388, 521 (1992).

19

Page 20: Relativistic Degenerate Electron Equation of State

[15] H.-Y. Chiu, 1968, Stellar Physics (Waltham, MA: Baisdell).

[16] F. X. Timmes and D. Arnett, “The accuracy, consistency, and speed of five equations

of state for stellar hydrodynamics,” Ap. J. Suppl. 125, 277 (1999).

[17] F. X. Timmes and F. D. Swesty, “The accuracy, consistency, and speed of an electron-

positron equation of state based on table interpolation of the Helmholtz free energy,”

Ap. J. Suppl. 126, 501 (2000).

[18] C. A. Iglesias, F. J. Rogers, and B. G. Wilson, “Spin-orbit interaction effects on the

Rosseland mean opacity,” Ap. J. 397, 717 (1992). (OPAL)

[19] F. J. Rogers and C. A. Iglesias, “Radiative atomic Rosseland mean opacity tables,” Ap.

J. Suppl. 79, 507 (1992). (OPAL)

[20] F. J. Rogers, 1994, in G. Chabrier and E. Schatzman, eds., Proc. IAU Colloq. 147, The

Equation of State in Astrophysics. Cambridge Univ. Press, Cambridge. (OPAL)

[21] D. G. Hummer and D. Mihalas, “The equation of state for stellar envelopes. I. An

occupation probability formalism for the truncation of internal partition functions,”

Ap. J. 331, 794 (1988). (MHD)

[22] D. Mihalas, W. Dappen, and D. G. Hummer, “The equation of state for stellar envelopes.

II. Algorithm and selected results,” Ap. J. 331, 815 (1988). (MHD)

[23] W. Dappen, D. Mihalas, D. G. Hummer, and B. W. Mihalas, “The equation of state

for stellar envelopes. III. Thermodynamic quantities,” Ap. J. 332, 261 (1988). (MHD)

[24] R. Kippenhahn and H.-C. Thomas, “Integralapproximationen fur die Zustandsgleichung

eines entarteten Gases,” Zs. f. Ap. 60, 19 (1964).

[25] E. Isaacson and H. B. Keller, 1966, Analysis of Numerical Methods (New York: John

Wiley).

[26] H. L. Gray and T. A. Atchison, 1968, in Proc. ACM 23rd National Conf. (Princeton:

Brandon/Systems Press), p. 73.

[27] M. Abramowitz and I. A. Stegun, 1965, Handbook of Mathematical Functions (New

York: Dover).

[28] A. Sommerfeld, Z. Phys. 47, 1 (1928).

[29] J. McDougall and E. C. Stoner, Phil. Trans. 237, 67 (1939).

20

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A Physical Constants

• Radiation energy density constant: a = 7.563×10−15 erg cm−3 K−4 (astef in the code)

• Atomic mass unit: 1.6605402× 10−24 g (atmass)

• Universal gas constant: R = 8.314510× 107 erg mol−1 K−1 (rgas)

• Planck’s constant: h = 6.6260755× 10−27 erg s (hplanck)

• Reduced Planck’s constant: h = 1.05457266× 10−27 erg s (hbar)

• Avogadro’s number: NA = 6.0221367× 1023 mol−1 (avogad)

• Speed of light: c = 2.99792458× 1010 cm/s (clight)

• Boltzmann’s constant: kB = 1.380658× 10−16 erg/K (boltzk)

• Electron mass: me = 9.1093897× 10−28 g (emass)

• Proton charge: e = 4.8029× 10−10 esu (echarge)

• π = 3.141592654 (pi)

21

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B Excerpts from ApJS Paper [5]

B.1 Numerical Integration Techniques

In general numerical quadrature must be used to evaluate the Fermi-Dirac integrals for condi-

tions of partial degeneracy (η near zero), and attention must be paid to several complications.

For n < 0, the integrand has a singularity at the origin. For nonintegral n, certain deriva-

tives of the integrand have a singularity at the origin. Finally, the interval of integration is

infinite. This section describes the techniques and strategies that avoid numerical difficulties

associated with these features and allow one to obtain excellent accuracy with only a small

computational effort. The algorithm we adopt is based on a pair of extrapolation procedures

and readily produces accurate results without manual input beyond setting up the program

and checking the output for consistency of the extrapolations.

The difficulty with singularities in the derivatives is due to the truncation error term of

the chosen quadrature rule. Consider the example of Simpson’s rule applied to the evaluation

of a definite integral. Simpson’s rule is derived by fitting a parabola to three equidistant

points and integrating the parabola. For m (an odd integer) equally spaced points, m ≥ 5,

the three-point rule may be applied to a sequence of three-point subintervals to obtain

∫ b

af(x) dx =

H

3

{f(a)− f(b) + 4

(m−1)/2∑j=1

f(a + [2j − 1] H) + 2(m−1)/2∑

j=1

f(a + 2jH)}

− (b− a)

180H4 f (4)(ξ), (34)

where H is the integration step size, and ξ is some point in the open interval (a, b) =

(a, a + [m− 1] H) [25]. The truncation error term is valid only if the first four derivatives of

the integrand are bounded in the interval of integration. For n = 1/2 and 3/2, the first and

second derivatives, respectively, are singular at the origin, and Simpson’s rule is less accurate

than shown in equation (34). This is because, for n = 1/2, the slope of the integrand at

the origin is infinite, and the parabola used to fit the integrand at the first three integration

nodes must have a finite slope and therefore cannot make a very accurate fit. The n = 3/2

case is less accurate because the parabolic fit has similar trouble with the infinite second and

higher derivatives of the finite integrand.

This difficulty is not peculiar to Simpson’s rule. For example, the trapezoidal rule

∫ b

af(x) dx =

H

2

f(a) + f(b) + 2m−2∑j=1

f(a + jH)

− (b− a)

12H2f (2)(ξ) (35)

requires the first two derivatives to be finite. Gaussian quadrature of order N requires that

the first 2N derivatives be bounded, which is even more restrictive.

22

Page 23: Relativistic Degenerate Electron Equation of State

The difficulty with singular integrands and derivatives for half-integer values of n ≥−1/2 may be eliminated by the simple change of variables z2 = x applied to equation (6):

Fn(η) = 2∫ ∞0

z2n+1 dz

1 + exp(z2 − η).) (36)

This same change of variables is used for the relativistic integrals. Although this transfor-

mation can be used for integral values of n, it is not necessary.

Evaluation of equation (6) or equation (36) begins with application of Simpson’s rule

over the finite interval [0, t] for two values of t. If the values of t are properly chosen, then

we can extrapolate to the integral over [0, ∞]. Consider the general case of numerically

evaluating improper integrals of the form

S(p) =∫ ∞

af(p, x) dx = lim

t→∞St(p), (37)

where

St(p) =∫ t

af(p, x) dx, (38)

and where a is a finite constant. The limit in equation (37) is obtained by means of ex-

trapolation procedures. This may be done by means of any of several transforms that were

originally devised for doing Laplace transforms numerically.

Define the B and G transforms [26] as

G[S(p); t, k] =St+k(p)−Rt(p, k) St(p)

1−Rt(p, k), Rt 6= 1, (39)

and

B[S(p); t, k] =Skt(p)− ρt(p, k) St(p)

1− ρt(p, k), ρt 6= 1, (40)

where

Rt(p, k) =f(p, t + k)

f(p, t), k > 0, (41)

and

ρt(p, k) =k f(p, kt)

f(p, t), k > 1. (42)

These transforms have the property that

limt→∞

G[S(p); t, k] = limt→∞

B[S(p); t, k] = S(p). (43)

Gray and Atchison [26] show that if limt→∞Rt(p, k) 6= 0 or 1, G converges to S(p) faster

than St+k(p). A similar theorem was given for the B transform. It can be shown that, if

St is evaluated exactly, the G transformation is exact for f = exp(−x), and the B trans-

formation is exact for f = x−s, s > 1 for finite t and k. Experience has shown that these

23

Page 24: Relativistic Degenerate Electron Equation of State

transforms do help the convergence in evaluating the Fermi-Dirac integrals. Because the

Fermi-Dirac integrand decays exponentially, the G transformation is more accurate than the

B transformation.

The St(p) are evaluated by numerical integration and contain truncation and round-

off errors. Aitken extrapolation is used to improve the accuracy of the St(p) and is based

on the assumption that, for sufficiently large m, the truncation error vanishes smoothly and

monotonically as m increases. If we use µ, 2µ, and 4µ subintervals, where µ = m− 1, then

we assume

I = Iµ + c Hp,

I = I2µ + c (H/2)p, (44)

and

I = I4µ + c (H/4)p,

where I is the exact value of the integral, Iµ is the approximate value using µ subintervals,

and c and p are free parameters. Equations (15) can be solved for I, c, and p given H and

the Im:

I = I4µ −(I4µ − I2µ)2

I4µ − 2I2µ + Iµ

, (45)

p = log10[(I − Iµ)/(I − I2µ)]/ log10(2), (46)

and

c = (I − Iµ)/Hp. (47)

For integrands with finite fourth derivatives, p ≈ 4 for Simpson’s rule.

Because extrapolations are usually considered risky at best, this algorithm for eval-

uating Fermi-Dirac integrals has been extensively tested. A sample of the test results are

detailed by Cloutman [5]. Included in the tests were the single known analytic expression

for a Fermi-Dirac integral [28]. If we make the change of variables w = 1 + exp(x− η) and

set n = 0, equation (6) becomes

F0(η) =∫ ∞1+exp(−η)

dw

w(w − 1)= ln

(w − 1

w

) ∣∣∣∞1+exp(−η)

= ln[exp(η) + 1]. (48)

We obtained 12-digit accuracy with only 121 nodes in the Simpson quadratures.

B.2 Fermi-Dirac Integral Evaluation in Applications Programs

It would be inefficient to use the techniques of the previous section to evaluate a Fermi-

Dirac integral every time an applications program needs a value. Therefore the numerical

integration program was used to generate accurate tables to be used with a separate function

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subprogram to evaluate Fn in applications programs. Tables were published for the non-

relativistic case for n = -1/2, 1/2, 3/2, and 5/2 and for −5 ≤ η ≤ 25 in steps of 0.05 [5].

Outside that range, series expansions provide good accuracy.

Cox and Giuli [7] show that for η ≤ 0, a good approximation is

Fn(η) = Γ(n + 1) exp(η)∞∑

r=0

(−1)r exp(r η)

(r + 1)n+1, n > −1, (49)

where Γ is the gamma function. Using Γ(z + 1) = zΓ(z) and Γ(1/2) = π1/2 (for example,

Abramowitz and Stegun [27]), we specialize this series to

F1/2(η) =π1/2

2

∞∑j=1

(−1)j+1 exp(j η)

j3/2, (50)

and

F3/2(η) =3π1/2

4

∞∑j=1

(−1)j+1 exp(j η)

j5/2. (51)

Only five terms are used for η ≤ −5, which provide 12 digit accuracy for the worst cases,

F1/2(−5) and F3/2(−5).

For η > 25, another asymptotic form is used. For integrals of the form

I(η) =∫ ∞0

φ′(u) du

exp(u− η) + 1

≈ φ(η) + 2∞∑

j=1

C2jφ(2j)(η), (52)

for large η. This series expansion has errors of order exp(−η). The C2j are given by

C2j =∞∑i=1

(−1)i+1 i−2j = [1 − 2−2j+1] ζ(2j), (53)

where ζ is the Riemann zeta function. The first five coefficients are C2 = π2/12, C4 =

7π4/720, C6 = 31π6/30240, C8 = 127π8/1209600, and C10 = 511π10/47900160. After

evaluating the necessary derivatives, the final expansions are

Fn(η) =ηn+1

n + 1

1 +∞∑

r=1

2 C2r

n+1∏k=n−2r+2

k

η−2r

, n > 0, η >> 1. (54)

This expression specializes to

F1/2(η) ≈ 2

3η3/2 +

π2

12η−1/2 +

7π4

960η−5/2 +

31π6

4608η−9/2 +

1397π8

81920η−13/2 (55)

and

F3/2(η) ≈ 2

5η5/2 +

π2

4η1/2 − 7π4

960η−3/2 − 31π6

10752η−7/2 − 381π8

81920η−11/2. (56)

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Page 26: Relativistic Degenerate Electron Equation of State

For intermediate values of η, we interpolate in the tables. Hermite interpolation (for

example, Isaacson and Keller [25]) is highly accurate and fits both function values and first

derivatives at the table’s nodes. The Fermi-Dirac integrals obey the relation [29]

dFn

dη= n Fn−1(η), n > 0, (57)

so we need the table for F−1/2 for a routine that evaluates F1/2 and F3/2. Cubic Hermite

interpolation, which fits the function values and derivatives at two nodes, provides seven

digits accuracy near η = -5 and 12 digits near η = 20. We finally adopted fifth order Hermite

interpolation, which fits function values and derivatives at three nodes. The accuracy is at

least ten digits near η = -5 and least 12 near η = 25.

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