Express the constant $k$ of $T^{2}=k(R_{E}+h)^{3}$ in days and kilometres. Given $k = 10^{-13} \; s^{2} \; m^{-3}$. The moon is at a distance of $3.84 \times 10^{5} \; km$ from the earth. Obtain its time period of revolution in days.

What are the conditions necessary for a satellite to appear stationary? What is the direction of rotation of a geostationary satellite?

Two satellites revolve around a planet in coplanar circular orbits in an anticlockwise direction. Their periods of revolution are $1\, h$ and $8\, h$ respectively. The radius of the orbit of the nearer satellite is $2 \times 10^{3}\, km$. The angular speed of the farther satellite as observed from the nearer satellite at the instant when both the satellites are closest is $\frac{\pi}{x}\, rad\, h^{-1}$ where $x$ is ..... .

The orbital speed of a satellite revolving around a planet in a circular orbit is $v_0$. If its speed is increased by $10 \%$,then ..........

Consider a star of mass $m_2 \ kg$ revolving in a circular orbit around another star of mass $m_1 \ kg$ with $m_1 \gg m_2$. The heavier star slowly acquires mass from the lighter star at a constant rate of $\gamma \ kg/s$. In this transfer process,there is no other loss of mass. If the separation between the centers of the stars is $r$,then its relative rate of change $\frac{1}{r} \frac{dr}{dt} \ (\text{in } s^{-1})$ is given by:

Imagine a light planet revolving around a very massive star in a circular orbit of radius $R$ with a period of revolution $T$. If the gravitational force of attraction between the planet and the star is proportional to $R^{-5/2}$,then ${T^2}$ is proportional to: