Using the conservation laws, demonstrate that the total mechanical energy of a planet of mass m moving around the Sun along an ellipse depends only on its semi-major axis a. Find this energy as a function of a.

Explanation

From the previous problem, if `r_1`, `r_2` are the maximum and minimum distances from the sun to the planet and `v_1`, `v_2` are the corresponding velocities, then, say,
`E=1/2mv_2^2-(gammamm_s)/(r_2)`
`=(gammamm_s)/(r_1+r_2)*r_1/r_2-(gammamm_s)/(r_2)=-(gammamm_s)/(r_1+r_2)=-(gammamm_s)/(2a)`
where `2a=` major axis `=r_1+r_2`. The same result can also be obtained directly by writing an equation analogous to Eq(1) of problem
`E=1/2moverset(.)r^2+(M^2)/(2mr^2)-(gammamm_s)/(r)`
(Here M is angular momentum of the planet and m is its mass). For position `overset(.)r=0` and we get the quardratic
`Er^2+gammamm_sr-(M^2)/(2m)=0`
The sum of the two roots of this equation are
`r_1+r_2=-(gammamm_s)/(E)=2a`
Thus `E=-(gammamm_s)/(2a)=const ant`