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Astron. Astrophys. 360, 969-990 (2000)

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Appendix A: lateral heat diffusion

Here we show that in a realistic situation the lateral temperature gradient caused by irradiating the secondary anisotropically is negligible compared to the radial temperature gradient in the subphotospheric layers and that, therefore, lateral energy transport by radiation is unimportant. We demonstrate this in the framework of the point source model. In that model we estimate the lateral gradient of [FORMULA] to be

[EQUATION]

On the other hand, the radial temperature stratification is given by the Eddington approximation (e.g. Tout et al. 1989)

[EQUATION]

where [FORMULA] is the optical depth. Therefore

[EQUATION]

and

[EQUATION]

From the definition of the optical depth we have

[EQUATION]

Inserting (A.5) into (A.4) yields the minimum radial temperature gradient in the presence of irradiation:

[EQUATION]

Combining now (A.1) and (A.6) we obtain

[EQUATION]

Now, because [FORMULA] and in the subphotospheric layers [FORMULA], we have

[EQUATION]

Because in the stars in question [FORMULA] is so small, the lateral temperature gradient is always much smaller than the radial one unless [FORMULA] and thus [FORMULA] [FORMULA]. However, in the range of interest of x and [FORMULA], i.e. where [FORMULA] is near its maximum and [FORMULA] to a few, (A.8) yields that the lateral temperature gradient is of order [FORMULA].

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© European Southern Observatory (ESO) 2000

Online publication: August 23, 2000
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