Kepler's laws

A planet ($P_1$) is moving around the star of mass $2M$ in the orbit of radius $R$. Another planet ($P_2$) is moving around another star of mass $4M$ in a orbit of radius $2R$. Ratio of time periods of revolution of $P_2$ and $P_1$ is ________.

A geostationary satellite above the equator is orbiting around the earth at a fixed distance $r_1$ from the center of the earth. A second satellite is orbiting in the equatorial plane in the opposite direction to the earth's rotation, at a distance $r_2$ from the center of the earth, such that $r_1 = 1.21\,r_2$. The time period of the second satellite as measured from the geostationary satellite is $\dfrac{24}{p}$ hours. The value of $p$ is _____.

The distance between Sun and Earth is R. The duration of year if the distance between Sun and Earth becomes 3R will be :

Two planets A and B of equal mass are having their period of revolutions $T_A$ and $T_B$ such that $T_A=2T_B$. These planets are revolving in the circular orbits of radii $r_A$ and $r_B$ respectively. Which out of the following would be the correct relationship of their orbits?

The time period of a satellite revolving around earth in a given orbit is 7 hours. If the radius of orbit is increased to three times its previous value, then approximate new time period of the satellite will be

The time period of X planet of our solar system is 8 years. The distance of earth from the sun is $1.5\times10^{11}$ m. The distance of the X planet from the sun will be :

The distance between two stars of masses $3M_S$ and $6M_S$ is $9R$. Here $R$ is the mean distance between the centers of the Earth and the Sun, and $M_S$ is the mass of the Sun. The two stars orbit around their common center of mass in circular orbits with period $nT$, where $T$ is the period of Earth's revolution around the Sun.

The value of $n$ is ____.

A planet of mass $M$, has two natural satellites with masses $m_1$ and $m_2$. The radii of their circular orbits are $R_1$ and $R_2$ respectively. Ignore the gravitational force between the satellites. Define $v_1$, $L_1$, $K_1$ and $T_1$ to be, respectively, the orbital speed, angular momentum, kinetic energy and time period of revolution of satellite 1; and $v_2$, $L_2$, $K_2$ and $T_2$ to be the corresponding quantities of satellite 2. Given $m_1/m_2 = 2$ and $R_1/R_2 = 1/4$, match the ratios in List-I to the numbers in List-II.

Figure shows elliptical path abcd of a planet around the sun S such that the area of triangle csa is $\dfrac14$ the area of the ellipse. (See figure) With db as the semimajor axis, and ca as the semiminor axis. If $t_1$ is the time taken for planet to go over path abc and $t_2$ for path taken over cda then :

India's Mangalyan was sent to the Mars by launching it into a transfer orbit EOM around the sun. It leaves the earth at E and meets Mars at M. If the semi-major axis of Earth's orbit is $a_e=1.5\times10^{11}$ m, that of Mar's orbit $a_m=2.28\times10^{11}$ m, taken Kepler's laws give the estimate of time for Mangalyan to reach Mars from Earth to be close to :

Under the influence of the Coulomb field of charge $+Q$, a charge $-q$ is moving around it in an elliptical orbit. Find out the correct statement(s).