A planet A moves along an elliptical orbit around the Sun. At the moment when it was at the distance `r_0` from the Sun its velocity was equal to `v_0` and the angle between the radius vector `r_0` and the velocity vector `v_0` was equal to `alpha`. Find the maximum and minimum distances that will separate this planet from the Sun during its orbital motion.

Explanation

From the conservation of angular momentum about the Sun.
`mv_0r_0sin alpha=mv_1r_1=mv_2r_2` or, `v_1r_1=v_2r_2=v_0r_2sinalpha` (1)
From conservation of mechanical energy,
`1/2mv_0^2-(gammam_sm)/(r_0)=1/2mv_1^2-(gammam_sm)/(r_1)`
or, `v_0^2/2-(gammam_s)/(r_0)=(v_0^2r_0^2sin^2alpha)/(2r_1^2)-(gammam_s)/(r_1)` (Using 1)
or, `(v_0^2-(2gammam_s)/(r_0))r_1^2+2gammam_sr_1-v_0^2r_0^2sin alpha=0`
So, `r_(1) =(-2gammam_(s)+-sqrt(4gamma^(2)m_(s)^(2)+4(v_(0)^(2)r_(0)^(2)sin^(2)alpha)(v_(0)^(2)-(2gammam_(s))/(r_(0)))))/(2(v_(0)^(2)-(2gammam_(s))/(r_(0))))`
`=(1+-sqrt(1-(v_0^2r_0^2sin^2alpha)/(gammam_s)(2/r_0-(v_0^2)/(rm_s))))/((2/r_0-(v_0)/(gammam_s)))=(r_0[1+-sqrt(1-(2-eta)etasin^2alpha)])/((2-eta))`
where `eta=v_0^2r_0//gammam_s`, (`m_s` is the mass of the Sun).