Figure 35. Relative width of ecosphere as a function of inclination of planet's equator.

Using data on the equatorial inclinations of the planets of the solar system, an assumed cumulative probability curve was chosen of the form,/5 = 1 — (1 — //180°)", with/1 being the probability that a randomly selected planet will have an equatorial inclination less than a given value i. A satisfactorily good fit was obtained for n = 4.5, as shown in Figure 34.

This general relationship was applied to the habitability regions on the illumination-versus-inclination diagram (see Figure 24 on page 66) to produce the probability-of-occurrence pattern of Figure 35. For simplification, the curve-bounded region of Figure 24 was replaced by a rectangle having the same area. For illumination limits of E1 (level at inner edge of ecosphere) and E2 (level at outer edge), related to each other so that the required value of P is about 0.81. This corresponds to an inclination of about 54 degrees as an upper limit.

Choosing Pj = 0.81 over the whole range of tolerable illumination levels as an approximation, rather than letting Pf be a function of illumination level, greatly simplifies the subsequent calculations without introducing any great errors. Thus, it will be assumed that = 0.81.

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