On the Induced Circularity of Orbits through Collision
Since the moment of momentum is the velocity into the perpendicular upon its direction, in the time it is:—
The whole moment of momentum from perihelion to perihelion is therefore:—
which is twice the area of the ellipse.
The energy in the ellipse during an interval is
from the well-known equation for the velocity in a focal conic. The integral of this for the whole ellipse is
and is given above.
By collision a part of this energy is lost, being converted into heat. The major axis, a, is, therefore, shortened. But from the expression for the moment of momentum we see that this is greatest when e is least. If, therefore, a is diminished, e must also be diminished, or the moment of momentum would be lessened, which is impossible.