Astron. Astrophys. 323, 415-428 (1997)

Appendix: fitting formulae

A.1. Electron-electron collisions

The effective collision frequency of non-relativistic degenerate electrons (, ) was analyzed by Lampe (1968) using the formalism of the dynamic screening of the electron-electron interaction. The expression of for the relativistic degenerate electrons at was obtained by Flowers & Itoh (1976). Here, is the electron plasma temperature determined by the electron plasma frequency ,

. The degeneracy temperature in the NS envelopes is typically higher than . Urpin & Yakovlev (1980) extended the results of Flowers & Itoh (1976) to higher temperatures, . In the approximation of static electron screening of the Coulomb interaction, Urpin & Yakovlev (1980) obtained

where , , and (see Eq. (15)). Now it is sufficient to calculate the function , presented by Urpin and Yakovlev (1980) as a 2D integral which depends parametrically on the relativistic parameter x. Lampe (1968) analyzed this function in the static screening approximation at . The asymptotes of J were obtained by Lampe (1968) for at and , by Flowers & Itoh (1976) for at any , and by Urpin & Yakovlev (1980) for and . Timmes (1992) calculated numerically in the limit of . However, the unified expression of at valid equally for relativistic and non-relativistic electrons has been absent. Note that the fit expression of obtained by Timmes (1992) (his Eq. (10)) is valid only at .

We have calculated J numerically for a dense grid of x and y in the intervals and . The results are fitted by the expression:

which reproduces also all the asymptotic limits mentioned above. The mean error of the fits is 3.7%, and the maximum error of 11% takes place at and .

A.2. Coulomb logarithms

The tables of the Coulomb logarithms calculated from Eqs. (14) and (15) as described in Sect. 2.3.3 are fitted by

where and

For , the fit parameters are given by:
,
,
,
,
,
,
.

For , we obtain , , , , , .

If , the rms fit error over all our data set is about 3.2%, and the maximum error of 7.2% takes place at g cm^-3 and . If we have 3.7%, 7.8% at g cm^-3 and . For , we obtain 2.4%, 7.0% at and . For higher Z, the rms error remains about 1.5-1.7%, and the maximum error mainly decreases. For instance, we have 5.1% at and for ; 3.7% at and for ; and 3.4% at and for . This fit accuracy is quite sufficient for studying the thermal structure of NSs.

A.3. Relation between internal and effective temperatures

In this section, we derive a fitting formula for as a function of , valid for , , where is the surface gravity in units of cm s^-2.

Let us define  K,  K, and

where M is the NS mass and is the accreted mass. According to GPE, where is the pressure at the bottom of the accreted envelope.

For a purely iron (non-accreted) envelope, a very crude estimate (with an error 30%) yields

Then is approximately an increase of the temperature through the iron envelope (in  K). A more accurate fitting formula reads

The typical fit error of is about 2%, with maximum 4.2%, over the domain indicated above.

For a fully accreted envelope, we obtain

which is valid at not too high temperature (  K).

Finally, for the partially accreted envelopes at any temperatures within the indicated range, we have

where

The typical fit error of Eq. (A9) for is about 3%, with maximum 5.2%, for all possible values of and any values of g and within the indicated range.

The dependence (A7) is recovered not only at sufficiently low accreted mass (), but also at sufficiently high . The latter result reflects the fact (first demonstrated by GPE) that at high the thermal insulation is mostly produced by the conductive opacities in the deep and hot layers of the envelope, in which light elements (H, He) burn into heavier ones. On the other hand, even at very low accreted mass (), the approximation of fully accreted crust is good enough at sufficiently low temperature because in this case the thermal insulation is provided mainly by the low-density accreted surface layers.

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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