that is, just over three-quarters of the planet’s speed in its orbit.
If we suppose the particles to be equally distributed in space, we shall have more with a given major axis in proportion to that axis, and our integral will become
| ⌠¹ | |
| ⎮ | (2a₁ - 1)½ a₁½ da₁ |
| ⌡½ |
⸻⸻⸻⸻⸻
| ⌠¹ | |
| ⎮ | a₁ da₁ |
| ⌡½ |
| 1 | ||
| = 8/3[ | (4a₁-1)/8 (2a₁² - a₁)½ - (1/16√2 ) log[(2a₁² - a₁)½ + √2 · a₁ - 1/2√2] | ] |
| ½ | ||
| = 0.792 of the planet’s orbital speed. | ||
The speed v, then, at which a satellite must be moving round the planet to have the same velocity as the average particle within the planet’s orbit, is
V - v₁ = v.
This velocity is, for the several planets:—
| Distribution of Particles as their Major Axes | Distribution of Particles Equal in Space | |
|---|---|---|
| Miles a second | Miles a second | |
| Jupiter | 2.0 | 1.6 |
| Saturn | 1.5 | 1.2 |
| Uranus | 1.0 | 0.9 |
| Neptune | 0.8 | 0.7 |