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29 April 1996

Physics Letters A 213 (1996) 313-315

PHYSICS LETTERS A

Comment on ‘ ‘Relativistic corrections to inverse bremsstrahlung in the solar interior” by Tsytovich et al.

Carlos A. Iglesias Lawrence Liuermore National Laboratory, P.O. Box 808, Livermore, CA 94551, USA

Received 8 November 1995; revised manuscript received 12 January 1996; accepted for publication 29 January 1996

Communicated by M. Porkolab

Abstract

A recent examination of inverse bremsstrahlung in the solar interior by Tsytovich et al. [Phys. Lett. A 205 (1995) 1991 is shown to predict incorrectly both the sign and size of the relativistic corrections.

PACS: 52.25; 96.60

Keywords: Opacity; Solar interior

Helioseismology and neutrino flux measurements probe the deep solar interior and can provide tests for theories describing generation and transport of energy in hot dense matter, neutrino physics, and other issues difficult to explore in the laboratory. In particular, solar models depend on input opacities for plasma conditions that cannot be achieved by experiments. Recently, Tsytovich et al. [ll considered relativistic corrections to the inverse bremsstrahlung cross section. They found a decrease of 7.5% in the Rosseland mean opacity (neglecting bound-bound and bound-free transitions) at solar center conditions. Such an opacity decrease is significant to

solar model calculations. The purpose here is to show that the Tsytovich et al. [l] result not only overestimates the relativistic corrections by about an order of magnitude but is in the wrong direction.

‘The relativistic bremsstrahlung cross section for an electron-ion pair in the Born approximation was first obtained by Bethe and Heitler [2]. The thermal averaged inverse bremsstrahlung results including both relativistic momentum distribution and cross section have also been investigated [3,4]. More recently, Itoh et al. [5,6] performed relativistic calculations including Coulomb corrections, electron degeneracy, and positron

contributions. All these works show that the relativistic corrections increase the free-free opacity in the Sun by less than 1%. In particular, Table 2 of Ref. [6] for 77 = (chemical potential/k,T) = - 2 and log(y ‘I= log(Ry/k,T) = - 2 with k,T the temperature in energy units, which approximates solar center conditions, shows small increases for the electron-proton inverse bremsstrahlung cross section. Thus, the conclusions by Tsytovich et al. [l] contradict previous work.

Using well known results [2-61 it is possible to compute the relativistic corrections to inverse bremsstrahlung

explicitly without expanding to order u2/c2 where u is the electron velocity and c the speed of light in

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314 CA. Iglrsicrs/Physics Letters A 213 (1996) 313-315

vacuum. The results can then be compared to those from Tsytovich et al. [ I]. The derivation starts from the

thermal averaged inverse bremsstrahlung cross section for a photon with energy hw by electron-proton pairs

[51

%(4=%(4d~JL (1) where ux is the Kramers [7] classical cross section and g(w, T) is the gaunt factor averaged over an equilibrium distribution and represents corrections to the classical, non-relativistic result. In the relativistic Born

approximation the Gaunt factor is given by [5]

(2)

where K, is a modified Bessel function [8] and

+ m2c2 EE EE -

--+?_E PC PC PF

with

m2c2h o + p2E2c4) f ~

Ez + p2c2

2PF p3c3 z-

EE + p2c2

p32 &+-

(3) (4)

In Eqs. (2)-(4) p and E (jj and _@ are the initial (final) electron momentum and energy, respectively, and m the electron rest mass. Form energy conservation

hw=E-E. (5)

In the non-relativistic limit Eq. (2) becomes [9]

fi go( w, T) = ye-“/*K,( u/2), (6)

where u = h w/k,T. Following Eqs. (6) and (15) of Ref. [ll, the results in Eqs. (2) and (6) above neglect plasma effects (e.g., degeneracy, Coulomb correlations, and dispersion).

The impact of the relativistic corrections to inverse bremsstrablung on the Rosseland mean opacity in the solar interior can be obtained from

A(T) G Kre’ - Ko , Ko

with

,Ic+“d,?! 1

4?r4 I) LIT (1 -e-“)o,g,+a;’ (8)

where B is the Planckian distribution and o, is the Thomson scattering cross section [lo]. In Eq. (8) Kn( g,) stands for either K,, or ~~ (g,, or go). Eq. (8) should also include contributions from heavier elements, however, for simplicity these are neglected. The required integrations were evaluated numerically for conditions relevant to the solar center, log(y2) = - 2, giving A(T) = 0.16% in agreement with earlier work [3-61.

CA. Iglesias/Physics Letters A 213 (1996) 313-315 315

There is clear, independent evidence that relativistic corrections increase the inverse bremsstrahlung Rosseland mean opacity in the solar interior by less than 1%. Since free-free absorption plus photon scattering contributes approximately two-thirds of the total opacity near the center of the Sun, the relativistic corrections to inverse bremsstrahlung should be negligible.

The explicit identification of the errors in Ref. [l] would involve the expansion of Eq. (2) to order v2/c2. Nevertheless, without performing this laborious exercise it is clear that Eq. (12) of Ref. [l] is incorrect since it contains an unphysical divergence at small initial electron kinetic energy absent in either the non-relativistic or fully relativistic expressions. Finally, the lower limit in the integration over initial velocities in Eq. (6) of Ref. [l] is incorrectly given as dm. The correct lower limit for photon absorption is zero [lo]. Similarly, the energy conservation expressions immediately following should be for absorption rather than emission.

This work was performed under the auspices of the Department of Energy by the Lawrence Livermore National Laboratory under contract W-7405Eng-48.

References

[l] V.N. Tsytovich, R. Bingham, U. de Angelis and A. Forlani, Phys. Lett. A 205 (1995) 199.

[2] H.A. Bethe and W. Heitler, Proc. R. Sot. (London) A 146 (1934) 83.

[3] C. Quigg, Phys. Fluids 11 (1968) 461.

[4] V.L. Ginzburg, High energy astrophysics, eds. C. Dewitt, E. Schatzman and P. V&on (Gordon and Breach, New York, 1967);

This result also appears in Eq. (5.20) of G.B. Rybicky and A.P. Lightman, Radiation processes in astrophysics (Wiley, New York,

1979).

[5] N. Itoh, M. Nakagawa and Y. Kohyma, Astrophys. 294 (1985) 17.

[6] M. Nakagawa, Y. Kohyma and N. Itoh, Astrophys. J. Suppl. 63 (1987) 661.

[7] H.A. Kramers, Philos. Mag. 46 (1923) 836.

]8] M. Abramowitz and L.A. Smgun, Handbook of mathematical functions (National Bureau of Standards, Washington, DC).

[9] G. Bekefi, Radiation processes in plasmas (Wiley, New York, 1966).

[lo] J.P. Cox and R.T. Giuli, Stellar structure (Gordon and Breach, New York, 1968).