Page:The Solar System - Six Lectures - Lowell.djvu/120



Jovian hyperbolic velocity, and its relative orbit about the planet will be an hyperbola. Jupiter, therefore, cannot completely possess itself of the comet.

The general equation of the relative motion I shall not bother you with. But certain deductions from it I think you will find of interest. In the first place, it appears that the comet will be accelerated or retarded, according as it passes behind or in front of the planet. This may be seen directly from the consideration that if it pass in front of the planet, it accelerates the latter, and since action and reaction are equal and opposite, it must itself be retarded; contrariwise, if it pass behind the planet.

Suppose now the comet to have been pursuing a parabolic path before the encounter; then the least retardation will make of its orbit an ellipse; for whether a body move in an ellipse, a parabola, or an hyperbola is a question simply of its speed at a given distance, shown by the well-known equation,— $$\textstyle v^{2}=\mu(\frac{2}{r}-\frac{1}{a}).$$

Similarly, the least acceleration will throw it into an hyperbola, and it will pass out of the solar system, never to return.