Suppose a planet goes around the Sun with a linear speed twice as fast as that of the Earth. What will be its orbit size compared to that of the Earth? (Radius of Earth $= R$)

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

In an elliptical orbit under gravitational force,in general

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

The kinetic energies of a planet in an elliptical orbit about the Sun,at positions $A, B$ and $C$ are $K_A, K_B$ and $K_C$,respectively. $AC$ is the major axis and $SB$ is perpendicular to $AC$ at the position of the Sun $S$ as shown in the figure. Then

Io, one of the satellites of Jupiter, has an orbital period of $1.769$ days and the radius of the orbit is $4.22 \times 10^{8} \; m$. Show that the mass of Jupiter is about one-thousandth that of the Sun.