sin p
The contribution FA of the large triangle area in Fig. E.7 is c n/2 Ymax
9s 0
where
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
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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