As R-r = a₁(1 + e) for the point of impact if the particle be wholly within the orbit of the planet and e the eccentricity of its orbit, we find
| e = | 2 | √ | MR/r | - RM/r | approx. |
for the case of no action, the other terms being insensible for the satellites in the table, since in all r < R/400.
Supposing, now, the particles within the orbit of the planet to be equally distributed according to their major axes, then as the velocity of any one of them, taking R-r = R approx. as unity, is
v₁ = (2/1 - 1/a₁)½,
the mean velocity of all of those which may encounter the satellite is, at the point of collision,
| ⌠¹ | |
| ⎮ | ((2a₁ - 1)½ / a₁½) da₁ |
| ⌡½ |
⸻⸻⸻⸻⸻
| ⌠¹ | |
| ⎮ | da₁ |
| ⌡½ |
| 1 | ||
| = 2 [ | (2a₁² - a₁)½ - 1/√(2) log{(2a₁ - 1)½ + √(2a₁)} | ] |
| ½ | ||
| = 0.754; | ||