All Problems Universal Gravitation
Problem 1.209
A planet \(A\) moves along an elliptical arbit 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 \(\mathrm{r}_ {0}\) and the velocity vector \(\mathbf{v}_ {0}\) was equal to \(\alpha .\) Find the maximum and minimum distances that will separate this planet from the Sun during its orbital motion.
\[ \text { 1.209. } r_{m}=\frac{r_{0}}{2-\eta}\left[1 \pm \sqrt{1-(2-\eta) \eta \sin ^{2} \alpha}\right], \quad \text { where } \quad \eta= \] \(=r_{0} v_{0}^{2} / \gamma m_{S}, m_{S}\) being the mass of the Sun.