The distance of a planet from the sun is $5$ times the distance between the earth and the sun. The time period of the planet is

$A$ star of mass $M$ (equal to the solar mass) with a planet (much smaller than the star) revolves around the star in a circular orbit. The velocity of the star with respect to the centre of mass of the star-planet system is shown below. The radius of the planet's orbit is closest to .......... $AU$ ($1 \, AU =$ Earth-Sun distance).

The figure shows an elliptical orbit of a planet of mass $m$ about the sun $S.$ The shaded area $SCD$ is twice the shaded area $SAB.$ If $t_1$ is the time taken for the planet to move from $C$ to $D$ and $t_2$ is the time taken to move from $A$ to $B,$ then:

In planetary motion,the areal velocity of the position vector of a planet depends on the angular velocity $\omega$ and the distance of the planet from the sun $r$. The correct relation for areal velocity is:

If the distance between the earth and the sun becomes half its present value,the number of days in a year would have been

$A$ geostationary satellite is orbiting the earth at a height of $6R$ above the surface of the earth ($R$ is the radius of the earth). The time period of another satellite at a height of $2.5R$ from the surface of the earth is: