Explanation:

To understand why the minimum time period for a satellite in orbit around a spherical planet is independent of the radius of the planet and depends on the density of the planet, let's consider the following factors:

1. Gravitational Force:
The gravitational force between the satellite and the planet is given by Newton's law of universal gravitation:

F = (G * m * M) / r^2

where F is the gravitational force, G is the gravitational constant, m is the mass of the satellite, M is the mass of the planet, and r is the distance between the satellite and the center of the planet.

2. Centripetal Force:
The centripetal force required to keep the satellite in circular orbit is given by:

F = (m * v^2) / r

where F is the centripetal force, m is the mass of the satellite, v is the velocity of the satellite, and r is the radius of the orbit.

3. Equating Forces:
In order for the satellite to remain in orbit, the gravitational force and the centripetal force must be equal. Therefore, we can equate the two forces:

(G * m * M) / r^2 = (m * v^2) / r

4. Canceling Mass:
Notice that the mass of the satellite (m) cancels out from both sides of the equation. This means that the time period (T) of the satellite's orbit, which is equal to the time it takes to complete one revolution, is independent of the mass of the satellite.

5. Solving for Time Period:
We can rearrange the equation to solve for the time period (T):

T = 2πr / v

Substituting the expression for velocity (v) from the equation above:

T = 2πr / √[(G * M) / r]

Simplifying the expression further:

T = 2π√(r^3 / (G * M))

6. Analysis:
From the equation for the time period (T), we can see that it depends on the radius of the orbit (r) and the mass of the planet (M). However, it does not depend on the radius of the planet itself. The density of the planet, which is defined as mass divided by volume, is related to the mass and the radius of the planet. Therefore, we can say that the time period (T) depends on the density of the planet.

Conclusion:
In conclusion, the minimum time period for a satellite in orbit around a spherical planet is independent of the radius of the planet and depends on the density of the planet. This is because the gravitational force and the centripetal force depend on the mass of the planet and the radius of the orbit, while the mass of the satellite cancels out from the equations.

For minimum time period radius of orbit equals R of planet