Speed at perihelion and aphelion?

[Master J]

SO the equations for the speed at the aphelion and perihelion are, respectively:

v=SQRT[GM( (2/r_p) -(1/a) )]

v=SQRT[GM( (2/r_a) - (1/a) )]

where M is mass of sun, r_a & r_p are distances from sun at aphelion and perihelion , and a is length of semi major axis.

How do you derive them? I am having trouble seeing where they come from, and a quick Google turns up NOTHING on them unfortunately.

ANy help?

[D H]

Those equations result simply from inserting the perifocal and apofocal distances into the vis-viva equation,

v^2 = GM\left(\frac 2 r - \frac 1 a\right)

This equation follows directly from conservation of energy. The total energy (kinetic plus potential) of a point mass m separated by a distance r from some other point mass M and moving with a velocity v relative to that other point is

E = \frac 1 2 m v^2 - \frac {G M m}{r}

The total energy of a point mass in an elliptical orbit is also given (see any intermediate-level classical mechanics text) by

E = - \frac {G M m}{2a}

Equating the two expressions leads directly to the vis-viva equation.