In compliance with the wishes of several members, I have inserted in this paper the solutions of the dynamical problems involved, whose truth I had before assumed.
The agency of lessened attraction as affecting any one planet, applies only to the period which elapses while the central mass is expanding to a nebula, and it will appear that the first revolution will especially be productive of altered eccentricity on this count. The following shows the action of these forces reduced to geometrical problems:—
Problem 1. Suppose a planet to be at that part of its orbit most distant from the sun, and, while in this position, suppose the mass of the sun suddenly diminished to a given extent,—required to trace the effect of this diminution of the sun's mass upon the orbit of the planet.
At present let the sun's mass be considered constant. Let the line ax (fig. 1) be tangent to the curve at aphelion, and aa, ab, bc infinitesimals along ax in the direction of the planet's course; let aa′, bb′, cc′, be infinitesimals representing the fall of the planet during the times contained respectively in aa, ab, ac, then aa′ b′ c′ will be the path of the planet.
Now suppose the mass of the sun to be decreased, the infinitesimals aa, ab, bc will remain unaltered, but aa′, bb′, cc′, etc., will each be diminished to a″ b″ c″. Then the curve aa″ b″ c″ represents the new orbit. It falls without the old orbit, except at a where it coincides with it. Perihelion distance is therefore increased, as represented in fig. 2, by virtue of diminished attraction.
The amount of the lessening of the attractive force will depend upon the quantity of the sun's matter which expands beyond aphelion distance. The portion which so expands ceases to affect the path of the planet. As this increases the orbit will assume variously the forms of the ellipse, circle, ellipse (the foci being reversed), parabola and hyperbola. If the attraction towards the centre entirely ceased, the path would coincide with the line aa. These orbits are respectively shown in fig. 2.
In fig. 3 let p′ represent the orbit with perihelion distance increased beyond that of p, this latter representing the orbit if the sun were not to expand into a nebula. Let the dotted circle c represent the limits to which the nebula has expanded when the planet passes aphelion. As the planet is entirely in the nebula it will be subject to constantly and rapidly diminishing attraction as it approaches the centre, s, hence it will not pass along p′, but will move more slowly inwards (in agreement with the first problem), and will pass along the second dotted line p″, which shows great increase in perihelion distance.
The two actions which have now been discussed scarcely affect aphelion distance, but render the orbit more circular by increasing perihelion distance.
