$A$ satellite orbits the Earth in a circular orbit of radius $R$. Another satellite orbits in an orbit of radius $1.02 R$. What is the percentage increase in the time period of the second satellite compared to the first (in $\%$)?

$A$ planet orbits the sun in an elliptical path as shown in the figure. Let $v_P$ and $v_A$ be the speed of the planet when at perihelion and aphelion respectively. Which of the following relations is correct?

Describe the model and theory of planetary motions presented in the earliest of times.

How will you 'weigh the sun',that is estimate its mass? The mean orbital radius of the earth around the sun is $1.5 \times 10^{8} \text{ km}$.

$A$ planet revolves around the Sun in an elliptical orbit,where the semi-major axis $a$ is double the semi-minor axis $b$ $(a = 2b)$. The Sun is at the focus. Given that the planet takes $24$ hours to travel through the path $bed$ as shown in the figure,find the time taken by the planet to travel along the path $dab$.

$A$ 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: