A planet of mass $m$ moves around the sun in an elliptical orbit. Find the total mechanical energy using conservation laws.

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Solution

A simple expression for the total mechanical energy of the orbit can be obtained by evaluating the mechanical energy at the apoapsis and periapsis. At these extrema, the mechanical energy and angular momentum have a very simple relationship, which can be seen by multiplying the mechanical energy by $r^{2}$ (mass of sun = M)
$E r^{2} = \frac{1}{2} M v_{e x}^{2} r_{e x}^{2} - G M m r_{e x} = \frac{L^{2}}{2 m} - G M m r_{e x}$
Now, angular momentum is a constant of motion.
the mechanical equation for the angular momentum can be solved at the periapsis and apoapsis and set these equal to obtain.
$E r_{a}^{2} + G M m r_{a} = E r_{p}^{2} + G M m r_{p}$
solving for E-
E= $\frac{G M_{m} (r_{p} - r_{a})}{r_{a}^{2} - r_{p}^{2}} = \frac{- G M m}{r_{a} + r_{p}}$
$r_{a} + r_{p}$ = full major axis distance = 2a
E= $\frac{- G M m}{2 a}$

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