A planet of mass m moves along an ellipse around the Sun so that its maximum and minimum distances from the Sun are equal to `r_1` and `r_2` respectively. Find the angular momentum M of this planet relative to the centre of the Sun.

Explanation

As the planet is under central force (gravitational interaction), its angular momentum is conserved about the Sun (which is situated at one of the focii of the ellipse)
So, `mv_1r_1=mv_2r_2` or, `v_1^2=(v_2^2r_2^2)/(r_1^2)` (1)
From the conservation of mechanical energy of the system (Sun+planet),
`-(gammam_sm)/(r_1)+1/2mv_1^2=-(gammam_sm)/(r_2)+1/2mv_2^2`
or, `-(gammam_s)/(r_1)+1/2v_2^2(r_2^2)/(r_1^2)=-((gammam_s)/(r_2))+1/2v_2^2` [Using (1)]
Thus, `v_2=sqrt(2gammam_sr_1//r_2(r_1+r_2))` (2)
Hence `M=mv_2r_2=msqrt(2gammam_sr_1r_2//(r_1+r_2))`