Irodov Problem 1.204

As seen from problem 1.202 the time period of a planet is given by where R and r are the distance of the planet at its extremities from the sun and Ms is the sun's mass. Now scaling down the entire solar system by a factor of will have two effect, i) R and r will scale by a factor and ii) the radii of all the planets will shrink by a factor and hence their masses will shrink by a factor (since their volumes will shrink by a factor and their average densities are the same). Thus, the expression will remain unchanged, in other words the time periods of the planets will remain unchanged.

Irodov Problem 1.203

As in problem 1.202, the time period of a planet in an elliptical orbit with closest and farthest approach distances as R and r is given by,

when the mass of the sun is Ms.

Suppose that the distance from earth to sun is r and that its a circle. The rotation period is given by,



Now consider an extreme elliptical orbit that is almost like a straight line where R=0. Then the path will be almost as if the planet fell into the sun directly. The period of such a hypothetical orbit would be,

However, the planet would only be able to complete half the orbit since it will fall into the sun and so the time taken to fall would be T/2.

So we have,