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

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

Let the speed of the planet at the perihelion \(P\) in figure shown below be \(v_{_P}\) and the Sun-planet distance \(\mathrm{SP}\) be \(r_{_P}.\) Relation between \((r_{_P},~v_{_P})\) to the corresponding quantities at the aphelion \((r_{_A},~v_{_A})\) is:

              
1. \(v_{_P} r_{_P} =v_{_A} r_{_A}\)
2. \(v_{_A} r_{_P} =v_{_P} r_{_A}\)
3. \(v_{_A} v_{_P} = r_{_A}r_{_P}\)
4. none of these

Which of the following quantities remain constant in a planetary motion (consider elliptical orbits) as seen from the sun?
1. speed
2. angular speed
3. kinetic energy
4. angular momentum

Two planets are in a circular orbit of radius \(R\) and \(4R\) about a star. At a specific time, the two planets and the star are in a straight line. If the period of the closest planet is \(T,\) then the star and planets will again be in a straight line after a minimum time: