To avoid this difficulty, we consider the scale height to be (i+αr_G) rather than i, where α is a numerical factor; α must have a value of the order of 10 to be consistent with Eq. (35). For the requirements that in the limit of i=0, <P(e,i)>_2B has to naturally tend to <P(e,0)>_2B given by Eq. (28), we put the modified collisional rate in the two-body approximation to be
<P(e,i)>_2B=Cπr_p^2{1+6/(r_p(e^2+i^2))}(e^2+i^2)^(1/2)/(2(i+ατ_G)) (36)
with
C=((2/π)^2){E(k)(1-x)+2αE(√(3/4))x}, (37)
where x is a variable which reduces to zero for i>>αr_G and to unity for i<<αr_G. The above equation reduces to Eq. (29) when i>>αr_G while it tends to the expression of the two-dimensional case (28) for i<<αr_G. Taking α to be 10 and x to be exp(-i/(αr_G)), <P(e,i)> scaled by Eq. (36) is shown in Fig. 17. Indeed, the modified <P(e,i)>_2B approximates <P(e,i)> within a factor of 5 in whole regions of the e-I plane, especially it is exact in the high energy limit (v→∞). However, two peaks remain at e≒1 and i≒3, which are closely related to the peculiar features of the three-body problem and hence cannot be reproduced by Eq. (36).
Fig. 16a and b. Behaviors of r_min(i,b): a i=0, b i=2, 2.5, and 3.0. The level of the planetary radius (r_p=0.005) is denoted by a dashed line.
Fig. 17. Contours of <P(e,i)> normalized by the modified <P(e,i)>_2B given by Eq. (36).
Fig. 16a and b.↓
http://www.fastpic.jp/images.php?file=4940423993.jpg
Fig. 17.↓
http://www.fastpic.jp/images.php?file=5825412982.jpg
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