Suppose a planet, P, of mass moves around the Sun, S, which is of a larger... · PHYSICS Dataset #7564

Physics · mechanics

Problem

Suppose a planet, P, of mass

m

moves around the Sun, S, which is of a larger mass,

M

, due to the universal gravitation

F=G \frac{M m}{r^{2}}

. Here,

r

and

G

denote the distance between the planet and the Sun and the gravitational constant, respectively. Since

M

is large, the Sun can be considered at rest. (1) Show that the angular momentum

L=m r^{2} \dot{\theta}

is conserved. (2) Find the total energy

E

of the planet moving around the Sun in terms of

r

, its time derivative

\dot{r}

G

, and

L

. Find the condition for the total energy

E

that the planet continues turning around the Sun and does not move from the Sun to an infinite distance away. Finally, supposing that the planet moves on a circular orbit around the Sun, find the relation between the radius of the planet orbit and the angular momentum

L

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