If two planets are at mean distances \(d_1\) and \(d_2\) from the sun and their frequencies are \(n_1\) and \(n_2\) respectively, then:
1. \(n^2_1d^2_1= n_2d^2_2\)
2. \(n^2_2d^3_2= n^2_1d^3_1\)
3. \(n_1d^2_1= n_2d^2_2\)
4. \(n^2_1d_1= n^2_2d_2\)

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: 

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

Two satellites of Earth, \(S_1\)_, and \(S_2\), are moving in the same orbit. The mass of \(S_1\) is four times the mass of \(S_2\). Which one of the following statements is true?

When a planet revolves around the sun in an elliptical orbit, then which of the following remains constant?

If \(R\) is the radius of the orbit of a planet and \(T\) is the time period of the planet, then which of the following graphs correctly shows the motion of a planet revolving around the sun?

1.		2.
3.		4.

Two satellites \(S_1\) and \(S_2\) are revolving around a planet in coplanar and concentric circular orbits of radii \(R_1\) and \(R_2\) in the same direction respectively. Their respective periods of revolution are \(1~\text{hr}\) and \(8~\text{hr}.\) The radius of the orbit of satellite \(S_1\) is equal to \(10^4~\text{km}.\) Find the relative speed when they are closest to each other. 
1. \(2\pi \times 10^4~\text{kmph}\)
2. \(\pi \times 10^4~\text{kmph}\)
3. \(\frac{\pi}{2} \times 10^4~\text{kmph}\)
4. \(\frac{\pi}{3} \times 10^4~\text{kmph}\)

Two planets orbit a star in circular paths with radii \(R\) and \(4R,\) respectively. At a specific time, the two planets and the star are aligned in a straight line. If the orbital period of the planet closest to the star is \(T,\) what is the minimum time after which the star and the planets will again be aligned in a straight line?

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:
1. \(\frac{dA}{dt}\propto \omega r\)
2. \(\frac{dA}{dt}\propto \omega^2 r\)
3. \(\frac{dA}{dt}\propto \omega r^2\)
4. \(\frac{dA}{dt}\propto \sqrt{\omega r}\)

If \(A\) is the areal velocity of a planet of mass \(M,\) then its angular momentum is: