₁ = 8/3 [_{½} (4_a₁_-1)/8 (2_a₁_² - _a₁_)^{½} __ __ __ - (1/16√ 2 ) log[(2_a₁_² - _a₁_)^{½} + √ 2 · _a₁_ - 1/(2√ 2 )]] = 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 | DISTRIBUTION OF | PARTICLES AS THEIR | PARTICLES EQUAL | MAJOR AXES | 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 ========+====================+================ If the satellite be moving in its orbit less fast than this, its space-speed will exceed that of the average particle; it will strike the particle at its own rear and be accelerated by the collision. If faster, the particle will strike it in front and retard it in its motion round its primary. From the table it appears that all the large satellites of all the planets have an orbital speed round their primaries exceeding those in either column. In consequence, all of them must have been retarded during their formation by the impact of interplanetary particles and forced nearer their primaries than would otherwise have been the case; and this whether the particles were distributed more densely toward the Sun, as 1/_a₁_, or were equally strewn throughout. For interplanetary particles whose orbits lie without the particular planet’s path the mean speed is the parabolic at the planet’s distance, given in the third column of the table. This is the case on either supposition of distribution. The orbital speed of the satellite which shall not be affected by collisions with them is, for the several planets:— ========+============== |MILES A SECOND --------+-------------- Jupiter | 3.4 Saturn | 2.5 Uranus | 1.7 Neptune | 1.4 ========+============== All the satellites but Iapetus have orbital speeds exceeding this, and consequently are retarded also by these particles.