Consider a planet moving around a star in an elliptical orbit with period $T$. The area of the elliptical orbit is proportional to ………..

If the earth is at one-fourth of its present distance from the sun, the duration of the year will be

A planet moving along an elliptical orbit is closest to the sun at a distance $r_1$ and farthest away at a distance of $r_2$. If $v_1$ and $v_2$ are the linear velocities at these points respectively, then the ratio $\frac{v_1}{v_2}$ is

The maximum and minimum distances of a comet from the Sun are $1.6 \times 10^{12}\, m$ and $8.0 \times 10^{10}\, m$ respectively. If the speed of the comet at the nearest point is $6 \times 10^{4}\, ms ^{-1},$ the speed at the farthest point is ……… $\times 10^{3}\, m / s$

The planet Mars has two moons, phobos and delmos.

$(i)$ phobos has a period $7$ hours, $39$ minutes and an orbital radius of $9.4 \times 10^{3} \;km .$ Calculate the mass of mars.

$(ii)$ Assume that earth and mars move in circular orbits around the sun. with the martian orbit being $1.52$ times the orbital radius of the earth. What is the length of the martian year in days?