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Fig. E.10 Geometry of the reflection effect 6. The geometry at the horizon da

Fig. E.10 Geometry of the reflection effect 6. The geometry at the horizon

where emean can be interpreted as the mean height above the horizon. However, it is easier to define a mean radius or mean (linear) height a. From

^mean — a pmax the angle, emean can be computed. Then, a follows as a =

with and

we eventually find a =

4 21

The height aH above the horizon is identical with p computed from (E.29.29).

More detailed computations of the reflection effect and also multiple reflection are done in subroutine LUMP (see Appendix E.25).

This subroutine, similar to Wilson's subroutine VOLUME, computes the Roche potential Q for a given (dimensionless) mean radius r+ by solving the equation r V (Q) = (E.30.1)

where rV(Q) is calculated according to (6.3.21). For given values q and Q, a modified version of Wilson's subroutine SURFAS yields not only the surface points but also the volume and thus the mean radius r Vaccording to (6.3.21). Thus, the implicit condition (E.30.1) can be solved w.r.t. Q by applying Newton's method, i.e.,

The derivative in the denominator of (E.30.2) can be approximated by the finite difference expression dr rV(Q+^Q)- rV(Q)

The initial guess Q(0) is computed by the approximations

based on Kopal's (1959, p. 129, formula 2-3) approximation

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