Two asteroids, A and B, orbit the Sun with periods $T_A$ and $T_B$ respectively, where $T_A = 2T_B$. Let $a_A$ and $a_B$ be their respective semi-major axes. If asteroid A has an orbital eccentricity of 0.2, what is the maximum possible eccentricity of asteroid B?

A satellite orbits just above Earth's surface with period $T$. If Earth's density is $d$ and $G$ is the gravitational constant, what does $\frac{3\pi}{Gd}$ represent?

Assertion: For circular orbits, the law of periods is ${T}^2 \propto {r}^3$, where ${M}$ is the mass of sun and $r$ is the radius of orbit. Reason: The square of the period $T$ of any planet about the sun is proportional to the cube of the semi-major axis $a$ of the orbit.

If distance between two masses is increased by a

factor of 3, then the gravitational force changes

by a factor

Assertion: If the earth expands in size without any change in mass, the length of the day would increase. Reason: The rotation of earth about its axis follows the law of conservation of angular momentum.

Assertion: If gravitational force is proportional to $\frac{1}{\mathrm{R}^3}$ then Keplar's second law would not be affected. Reason: For Keplar's second law, central nature of force is the only condition.

Assertion: Generally, the path of a projectile from the Earth is parabolic but it is elliptical for projectiles going to a very large height. Reason: The path of a projectile is independent of the gravitational force of Earth.

A satellite is in an elliptical orbit around the earth with aphelion of 6R and perihelion of 2R, where R is the radius of the earth. The eccentricity of the orbit is

The period of revolution of a planet around a star is 8 years. What is the semi-major axis of its orbit in astronomical units (AU)?

A planet of mass '$m$' is in an elliptical orbit about the sun $\left[ {m < < M} \right]$ with an orbital time period '$T$'. If A is the area of the orbit then its angular momentum is

The time period of a satellite of earth is 24 hours. If the separation between the earth and the satellite is decreased to one fourth of the previous value, then its new time period will become.