Page:The Solar System - Six Lectures - Lowell.djvu/26

8

The Solar System

---

The resolved part of the velocity v along sp is $\textstyle {\dot {r}}\,\!$, and

$\textstyle {\dot {r}}=v\ cos\ \psi \,\!$;

whence

$\textstyle f{\dot {r}}=v{\dot {v}}={\frac {1}{2}}dv^{2}\,\!$.

If

$\textstyle f={\frac {\mu }{r^{2}}}\,\!$

$\textstyle {\frac {1}{2}}v^{2}=\mu \left({\frac {1}{r}}\right)+c$.

c can be determined from the actual velocity at some point in the orbit (at the end of the minor axis, for instance, in the ellipse), and from this we can find that

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

where a is the semi-major axis of the curve, the upper sign referring to the ellipse, the lower to the hyperbola.

The velocity in the hyperbola thus exceeds that in the ellipse, and the dividing line between the two classes of curves is clearly when the second term is zero.

Consequently

$\textstyle v^{2}={\frac {2\mu }{r}}\,\!$,

is the velocity which at any given distance r separates the bodies moving in ellipses from those moving in hyperbolas, the sheep from the goats.

$\textstyle v^{2}={\frac {2\mu }{r}}\,\!$,

is called the parabolic velocity, but the student should be careful to remember that the parabola is a mathematical conception, not a physical fact.