If a planet of radius R and density D is revolving around the sun in a circular orbit of radius r with time period T, then the mass of the sun is

Options

A. `(4pi^2r^3)/(GT)`
B. `(4pi^2r^3)/(GT^2)`
C. `(4pi^2r)/(GT)`
D. `(4pi^2r^3)/(G^2T^2)`

Answer

To find the mass of the Sun (M) given a planet of radius R and density D revolving around it in a circular orbit of radius r with time period T, we can follow these steps:

### Step-by-Step Solution:

1. **Use Kepler's Third Law**: According to Kepler's Third Law, the square of the time period (T) of a planet is directly proportional to the cube of the radius (r) of its orbit around the Sun. This can be expressed mathematically as:
\[ T^2 \propto r^3 \]
This implies:
\[ T^2 = k \cdot r^3 \]
where k is a constant.

2. **Apply Newton's Law of Gravitation**: The gravitational force acting on the planet can be expressed using Newton's law of gravitation:
\[ F = \frac{G \cdot M \cdot m}{r^2} \]
where G is the gravitational constant, M is the mass of the Sun, and m is the mass of the planet.

3. **Centripetal Force**: For a planet moving in a circular orbit, the centripetal force required to keep it in that orbit is given by:
\[ F = \frac{m v^2}{r} \]
where v is the orbital velocity of the planet.

4. **Equate the Forces**: Since the gravitational force provides the necessary centripetal force, we can set the two expressions for force equal to each other:
\[ \frac{m v^2}{r} = \frac{G \cdot M \cdot m}{r^2} \]
We can cancel m from both sides (assuming m ≠ 0):
\[ \frac{v^2}{r} = \frac{G \cdot M}{r^2} \]
Rearranging gives:
\[ v^2 = \frac{G \cdot M}{r} \]

5. **Relate Velocity to Time Period**: The velocity (v) of the planet can also be expressed in terms of the time period (T):
\[ v = \frac{2 \pi r}{T} \]
Squaring this gives:
\[ v^2 = \left(\frac{2 \pi r}{T}\right)^2 = \frac{4 \pi^2 r^2}{T^2} \]

6. **Substitute into the Equation**: Now we can substitute this expression for \(v^2\) into the equation we derived from equating the forces:
\[ \frac{4 \pi^2 r^2}{T^2} = \frac{G \cdot M}{r} \]

7. **Solve for Mass of the Sun (M)**: Rearranging this equation to solve for M gives:
\[ M = \frac{4 \pi^2 r^3}{G T^2} \]

### Final Expression:
Thus, the mass of the Sun (M) is given by:
\[ M = \frac{4 \pi^2 r^3}{G T^2} \]