Question

It is theoretically possible to place a satellite at a position between the earth and the sun on the line joining them, where the gravitational forces of the sun and the earth on the satellite combine in such a way that the satellite will execute a circular orbit around the sun that is synchronous with the earth's orbit around the sun. (In other words, the satellite and the earth have the same orbital period about the sun, even though they are at different distances from the sun. The satellite always remains on the line joining the earth and the sun.) Write an expression that relates the appropriate circular orbital speed v of a satellite in such a situation to its distance r from the sun. Your expression may also contain quantities shown in Fig. 4.16 plus the gravitational constant G.

Answer 1

For any satellite to orbit the centripetal force must be equal to the net gravitational force on the satellite

Total gravitational force on the satellite= Force due to sun-force due to earth

$=\frac{G{M}_{s}m}{r^2}-\frac{G{M}_{E}m}{(D-r{)}^2}$

And centripetal force= $\frac{m{v}^2}{r}$

These two forces must be equal,

$\frac{m{v}^2}{r}=\frac{G{M}_{s}m}{r^2}-\frac{G{M}_{E}m}{(D-r{)}^2}$

we get,

$v=\sqrt{\frac{G{M}_s}{r}-\frac{G{M}_{E}r}{(D-r{)}^2}}$