Page:The Solar System - Six Lectures - Lowell.djvu/25

Our Solar System

7

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Which of the two orbits a body is pursuing may be determined either by actually finding the body's path or by finding the distance of the body from the Sun and its speed at the moment. For an interesting equation connects the speed with the distance, giving the major axis of the orbit, upon which alone the class of curve depends. This equation is

$\textstyle v^{2}=\mu \left({\frac {2}{r}}\mp {\frac {1}{a}}\right)\,\!$,

in which the – sign betokens the ellipse, the + sign the hyperbola.

Suppose a body at p moving along the curve whose tangent is pt with acceleration f always directed to s. Then $\textstyle {\dot {v}}\,\!$, the resolved part of the acceleration along the tangent, is f cos ψ.