sin p

The contribution FA of the large triangle area in Fig. E.7 is c n/2 Ymax

9s 0


Wilson computes F3 with a GauB integration as J2Li W fi. Based on the computed quantities, it is now possible to define the penumbra function P = P(rc,

Cases 1-4 represent four geometric possibilities:

1. the irradiating star is completely above the local horizon;

2. it is more than half above the local horizon;

3. it is less than half above the local horizon; or

4. it is completely below the horizon.

The quantities P1 and P2 are defined by

Pi = Fsec + fa , P2 = F - ( fF + FA = 1 - Pi (E.29.26)

The contribution Fsec is

2 Jo Jo 2 n and eventually

Now the dependence of the factor P on rc and R according to (E.29.22), (E.29.23), (E.29.24), and (E.29.28) is transformed into a dependence on p defined in (E.29.17). As can be seen in Fig. E.9, it is also possible to define the fractional radius p above or below the horizon as

the visible part of the irradiating star

where additional auxiliary quantities a

and cosh

appear. The replacement of the approximation (E.29.17) by (E.29.29) is only useful for D > a. In contact systems we have D = 3a; in detached systems the approximation is even better.

If (3.2.45) is to be used instead of (3.2.44), cos e is replaced by cos e. In the case of a point source the ratio F1/F2 depends on the angle defined in (E.29.31), i.e., on cos e = cos h. As shown in Fig. E.10, e can be defined as p c p max

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