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1 + b2/c2 tan2 i' Wilson approximates the integral in expression (E.29.11) by

o sin i

c2 â– tan2 i where z has been eliminated using (E.29.12) and eventually gets

The required quantities cot i = cot 0+ and cos2 0 = cos2 < have already been computed in (E.29.6) and (E.29.7).

Fig. E.7 Geometry of the reflection effect 3. Here we see the quantities rc and rH involved in describing the visible part of the irradiating star

The bolometric flux F1(a) received at a point a is, after application of the correction for limb darkening, modified by a multiplicative correction factor E(0+, <,)

The step from an irradiating point source to an extended source star requires an explicit treatment of penumbral regions. Modeling penumbral effects forces us also to include limb darkening. Derived in Chapter 3 for a linear limb-darkening law, formula (3.2.31) gives the flux received from a unit disk with unit intensity at disk center

horizon

Fig. E.8 Geometry of the reflection effect 4. The local coordinates on the stellar disk are illustrated

Figure E.7 illustrates the geometry of the visible part of the irradiating star seen from a point x of the irradiated star. The horizon is assumed to be a straight line. In the first step, ps = ps(rc, R) is computed according to rc . rc p := sinps = r ^ ps = arcsin r = arcsin p.

for given values rc and R. To each value p we can assign a value rH

As illustrated in Fig. E.8, we can compute sin ymax as a function of rH

sin Ymax =

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