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DENSITY EFFECTS ON THE OPACITY OF COOL HELIUM WHITE DWARF ATMOSPHERES ... White dwarf cosmochronology is a potentially powerful tool.
The Astrophysical Journal, 569:L111–L114, 2002 April 20 䉷 2002. The American Astronomical Society. All rights reserved. Printed in U.S.A.

DENSITY EFFECTS ON THE OPACITY OF COOL HELIUM WHITE DWARF ATMOSPHERES Carlos A. Iglesias and Forrest J. Rogers Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94550

and Didier Saumon Department of Physics and Astronomy, Vanderbilt University, P.O. Box 1803, Station B, Nashville, TN 37235 Received 2002 February 13; accepted 2002 March 14; published 2002 March 26

ABSTRACT Density effects on two important processes contributing to the opacity of a weakly ionized helium gas are considered. The correction to free-free absorption by the negative helium ion is treated in the Born approximation using analytical approximations for the electron-atom interaction. The correction to Rayleigh scattering uses wellknown formulae obtained from density fluctuation theory. In both processes, the main density effect appears through the static structure factor, which accounts for spatial correlations between the helium atoms. These correlations significantly reduce the cross sections at the high densities encountered in cool, He white dwarf atmospheres. Subject headings: atomic processes — scattering — stars: atmospheres — white dwarfs scattering transport cross section (Chiu 1968),

1. INTRODUCTION



⫹1

White dwarf cosmochronology is a potentially powerful tool to determine the age of galactic and globular clusters as well as the ages of the thin disk, thick disk, and halo of the Galaxy (e.g., Fontaine, Brassard, & Bergeron 2001). The method is based on the age determination of observed white dwarf populations and is sensitive to the modeled white dwarf cooling. For old white dwarf populations, a significant source of uncertainty is the modeling of the atmospheres, which largely controls the cooling rate once the envelope convective zone reaches the degenerate core (Lamb & Van Horn 1975). Even though these atmospheres are relatively dense, present models use photon absorption cross sections derived for isolated atoms and ions. The purpose here is to investigate many-body effects on processes that determine the atmosphere opacities. The large surface gravity of white dwarfs causes stratification, leading to nearly pure hydrogen or helium atmospheres. Mixed H-He compositions can occur in cool white dwarfs owing to convective mixing and accretion from the interstellar medium. Although density effects could affect either pure H or H-He atmospheres, the pure He atmospheres have lower opacities than those with even a modest admixture of H, resulting in much higher photospheric pressures. Furthermore, the pure He cases are more amenable to calculations better illustrating the many-body effects and potential impact on white dwarf models. This is because the helium atom has a large, ground-state photoionization energy, leaving only Rayleigh scattering by the atom and free-free absorption by electronatom collisions as the dominant opacity sources.

0 jsct (q) p j Ray (q) 38

dm(1 ⫺ m)(1 ⫹ m2 )

⫺1

冕 [ ⬁

#

dQ

⫺⬁

(q ⫹ Q) N(q ⫹ Q) 2 S(k, Q), q N(q)

]

(2.1)

where m p cos v, with v the angle between initial and final photon momenta, Q the energy transfer, N(q) the refractive 0 index, and j Ray (q) the Rayleigh scattering cross section for an isolated helium atom. The momentum transfer is given by

kp

{

qN(q) (q ⫹ Q) N(q ⫹ Q) 1⫹ c q N(q)

[

2

]

1/2

⫺ 2m

[(q ⫹q Q) N(qN(q)⫹ Q)]}

.

(2.2)

For the conditions of interest here, the photon energy is much greater than the energy shifts in the Brillouin components (Berne & Pecora 1976); thus, equations (2.1) and (2.2) simplify to



⫹1

j (q) ≈ j t sc

0 Ray

(q)

3 8

dm(1 ⫺ m)(1 ⫹ m2 )S(k),

(2.3)

⫺1

k≈ 2. RAYLEIGH SCATTERING

qN(q) 冑2(1 ⫺ m), c

(2.4)

where the exact sum rule (Berne & Pecora 1976)

Scattering of a photon with energy បq from a collection of atoms is given by the ideal gas result modified by the dynamic structure factor, S(k, Q) , which represents the spectrum of density fluctuations of wavevector k (Berne & Pecora 1976). Furthermore, the quantity of interest in radiation transport is the





S(k) p

⫺⬁

L111

dQ S(k, Q)

(2.5)

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OPACITY OF COOL He WHITE DWARF ATMOSPHERES

was used and S(k) is the static structure factor. In the zero density limit, S(k) p 1 and equation (2.3) reduces to the ideal gas result. 3. FREE-FREE ABSORPTION

The free-free absorption cross section per atom can be written as the product of the classical Kramers result times a Gaunt factor (Cox & Giuli 1968), 2 6

16p e ne j f f (q) p 3mបcq 3



g

冑3 (q) p

p



4pe 2 n m

(3.1)

Here m and e are the electron mass and electric charge, respectively, k B is the Boltzmann constant, T the temperature, ne the free electron number density, and c the speed of light in a vacuum. The Gaunt factor in the Born approximation and nondegenerate electrons is given by (Kawakami et al. 1988) Born ff

interaction, S(k) can be computed in the hypernetted-chain approximation (McQuarrie 1976). The atom-atom interaction is taken from an analytic fit to the energy potential curve (Cvetko 1994). In cool, white dwarf atmospheres the helium is weakly ionized. Consequently, the gas is treated as a collection of atoms and the dielectric function approximated by (Jackson 1975) e(q) ≈ 1 ⫹

2p g (q). 3mk B T f f



dk I(k),

(3.2)

0

e(q) ≈ 1 ⫹ S(k) I(k) p I0 (k) , Fe(k, q)F2

1 ប2 k mq I0 (k) p exp ⫺ ⫺ k 2mk B T 2 បk

(

[

)]F 2

(3.3)

k 2V˜ ea(k) 4pe 2

F

2

,

) (

)

(3.5)

where a 0 is the Bohr radius and Q p 1.685 is chosen to minimize the bound electron energy. The second term is a polarization interaction,

2qn2 q ⫺ q2 2 0

e2 rp4 2r ⫺ e⫺3r/rp 1 ⫹ 2 rp (rp ⫹ r 2 ) 2 rp

[

(

(3.7)

⫺1

2 n

)(1 ⫺ q q⫺ q ) 2 0

,

2

(3.8)

where only the j p 0 resonant line was included in the sum and qn2 p 4pe 2 f0 n/m. Finally, the index of refraction can be readily obtained (Jackson 1975) from equation (3.8) by N(q) p [e(q)]1/2.

Numerical results are presented for a sample He atmosphere. The density effects on the photon absorption and scattering are given in terms of multiplicative factors to the ideal gas results. Similarly, multiplicative corrections to the Rosseland mean opacity are reported. The Born approximation for the free-free absorption by the negative helium ion is not accurate for the conditions of interest. Consequently, it is only used to compute the collective effects correction. That is, we write the free-free absorption cross section as j f f (q) p d f f (q)j f0f (q) p

)] .

(3.6)

The interactions in equations (3.5) and (3.6) were chosen to approximate earlier potentials (Geltman 1973) but still retain analytical Fourier transforms. The values of the free parameters, rp p 0.5a 0 and a p p 5.532, follow the choice in Geltman (1973). For a spherically symmetric, pair-wise additive atom-atom

gfBorn (q) 0 f j (q), 0, Born gf f (q) f f

(4.1)

where j f0f (q) is the ideal gas result and the multiplicative correction is estimated by the ratio of the Born approximation with and without density effects. From equations (3.2)–(3.4), we get

[冕



d f f (q) ≈

0

Vp (r) p ⫺a p

fj . (qj2 ⫺ q 2 ) ⫺ iqGj

4. RESULTS

2e 2 Qr 2Qr 1⫹ exp ⫺ , r a0 a0

(

j

(3.4)

and V˜ ea(k) the Fourier transform of the electron-atom potential, S(k) the static structure factor, and e(k, q) the dielectric function. The latter two quantities describe collective effects; that is, S(k) accounts for spatial correlations between the atoms and e(k, q) for time displacements of currents due to the gas polarizability. For an isolated electron-atom pair, the collective effects vanish, S(k) p 1 p e(k, q), and equation (3.2) reduces to the ideal gas result, gf0,f Born (q). For simplicity, the electron-atom potential is approximated by the sum of two terms. The first is the interaction of an electron with a frozen ground-state helium atom. Assuming hydrogenic bound wave functions with effective charge Q yields Vc (r) p ⫺



Here បqj, fj, and Gj are the transition energy, oscillator strength, and decay constant for the jth transition in the helium atom, respectively, and n is the atom number density. The result in equation (3.7) implies that a local field polarizing the atom is the same as a macroscopic field applied to the gas. At high densities, however, the surrounding atoms are similarly polarized, introducing a correction to the local field leading to the well-known Clausius-Mossotti relation (Jackson 1975). Furthermore, for the cases of interest Gj K q K qj, so that equation (3.7) becomes

(

with

Vol. 569

][冕



dk I(k)

0

⫺1

dk I0 (k)

]

,

(4.2)

with the expectation that errors will tend to cancel. The correction to scattering can be written in a similar form, 0 jsct (q) p dsc (q)j Ray (q) p

jsct (q) 0 j Ray (q), 0 j Ray (q)

(4.3)

where the multiplicative correction, dsc (q) , is readily obtained from equation (2.3).

No. 2, 2002

IGLESIAS, ROGERS, & SAUMON

Fig. 1.—Multiplicative correction to the Rosseland mean opacity (left y-axis) and pressure in units of the ideal gas result, Pideal (right y-axis) for a He white dwarf atmosphere with Teff p 4000 K and surface gravity of 108 cm s⫺2 as a function of Rosseland mean optical depth, tR, from the stellar model. The arrow corresponds to the conditions in Fig. 2.

The correction to the Rosseland mean opacity, d R , is defined by dR p

kR , k R0

(4.4)

where kR and k R0 are the Rosseland mean opacities with and without density effects, respectively. That is (Cox & Giuli 1968), 1 p k R0 1 p kR





0

du





0

du

WR(u) , 0 n[(1 ⫺ e )j f0f (u) ⫹ j Ray (u)] ⫺u

L113

Fig. 2.—Multiplicative corrections to the ideal gas cross sections as a function of photon energy for T p 3240 K and mass density r p 1.23 g cm⫺3.

far from ideal, resulting in more than an order of magnitude reduction in opacity. The opacity near the surface is dominated by scattering. However, deeper in the star as the free electron density increases with increasing pressure, the opacity is dominated by free-free absorption. Figure 2 shows d f f (q) and dsc (q) for conditions where both photon processes are comparable. The behavior of these corrections follows from Figure 3, which presents S(k). Also displayed is the integrand I0 (k) in equation (3.4) for photon energies near the peak and in the tail of the Rosseland weighting function. It follows that the main contribution to the integrals comes for small k-values well below the first peak in S(k). The peak in I0 (k) does shift as a function of photon energy,

(4.5)

N 2 (u)WR(u) , 0 n{(1 ⫺ e⫺u )[d f f (u)/N(u)]j f0f (u) ⫹ dsc (u)j Ray (u)} (4.6)

and the Rosseland weighting function in terms of the variable u p បq/k B T is WR(u) p

15 u 4e⫺u . 4p 4 (1 ⫺ e⫺u ) 2

(4.7)

0 The values of j f0f (q)/ne and j Ray (q) in these calculations are from Sommerville (1967) and Kurucz (1970), respectively. Also necessary are the results from an ionization equilibrium model to provide n and ne (Bergeron, Saumon, & Wesemael 1995) used for both k R and k R0 avoiding discrepancies in number densities between ideal and nonideal models. Results are presented in Figure 1 for a cool, He atmosphere white dwarf. The figure displays the Rosseland mean opacity reduction. Also plotted is the gas pressure obtained from the stellar model (Bergeron et al. 1995). Clearly, the gas can be

Fig. 3.—Plots of S(k) as well as I0(k) for u p 4 and u p 16 as a function of q p kai for the same conditions as in Fig. 2. The particle sphere radius is defined by 4pna3i /3 p 1. For plotting convenience, I0(k) has been normalized to a peak value of unity.

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OPACITY OF COOL He WHITE DWARF ATMOSPHERES

thus reducing or enhancing the density effects. On the other hand, for scattering the values of k sampled by equation (2.3) are given by equation (2.4), which for relevant photon energies only involve the S(k ≈ 0) limit.

Vol. 569

5. CONCLUSION

much as an order of magnitude. Most of the correction is described by the small k behavior of the atom static structure factor. Although only cool He atmospheres were considered, it follows that similar photon scattering and free-free absorption processes involving hydrogen species found in very cool pure H and mixed H-He atmospheres could be similarly affected.

The radiative opacity in the helium atmospheres of white dwarfs can be significantly smaller than the ideal gas results used in present stellar models. At the highest densities, the collective effects reduce the Rosseland mean opacity by as

Work by C. A. I. and F. J. R. was performed under the auspices of the Department of Energy by Lawrence Livermore National Laboratory under contract W-7405-Eng-48. D. S. acknowledges support from NSF grant AST 97-31438.

REFERENCES Bergeron, P., Saumon, D., & Wesemael, F. 1995, ApJ, 443, 764 Berne, B. J., & Pecora, R. 1976, Dynamic Light Scattering (New York: Wiley) Chiu, H.-Y. 1968, Stellar Physics (Waltham: Blaisdell) Cox, J. P., & Giuli, R. T. 1968, Principles of Stellar Structure (London: Gordon & Breach) Cvetko, V., et al. 1994, J. Chem. Phys., 100, 2052 Fontaine, G., Brassard, P., & Bergeron, P. 2001, PASP, 113, 409 Geltman, S. 1973, J. Quant. Spectrosc. Radiat. Transfer, 13, 601

Jackson, W. D. 1975, Classical Electrodynamics (New York: Wiley) Kawakami, R., Mima, K., Totsuji, H., & Yokoyama, Y. 1988, Phys. Rev. A, 38, 3618 Kurucz, R. L. 1970, SAO Spec. Rep., 308 Lamb, D. Q., & Van Horn, H. M. 1975, ApJ, 200, 306 McQuarrie, D. A. 1976, Statistical Mechanics (New York: Harper & Row) Sommerville, W. B. 1967, ApJ, 149, 811