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
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Relativistic Degenerate Electron Equation of State
The RFD01 documents are software and documentation for evaluating both the classical and relativistic degenerate electron equation of state for all degrees of degeneracy.
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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.
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
Mα
+ 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)
2
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).
3
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.
4
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)
5
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.
6
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].
7
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.
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
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
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
24
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)
25
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.