A satellite of mass `m` is in a circular orbit of radius `2R_(E)` about the earth. The energy required to transfer it to a circular orbit of radius `4R_(E)` is (where `M_(E)` and `R_(E)` is the mass and radius of the earth respectively)

Options

A. `(GM_(E)m)/(2R_(E))`
B. `(GM_(E)m)/(4R_(E))`
C. `(GM_(E)m)/(8R_(E))`
D. `(GM_(E)m)/(16R_(E))`

Explanation

To find the energy required to transfer a satellite of mass `m` from a circular orbit of radius `2R_E` to a circular orbit of radius `4R_E`, we need to calculate the total energy in both orbits and then find the difference.

### Step-by-Step Solution:

1. **Understand the Total Energy in Orbit**:
The total mechanical energy \( E \) of a satellite in a circular orbit is given by the formula:
\[ E = -\frac{G M m}{2R} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the satellite,
- \( R \) is the radius of the orbit.

2. **Calculate Total Energy at Radius \( 4R_E \)**:
For the orbit at radius \( 4R_E \):
\[ E_{4R} = -\frac{G M m}{2 \times 4R_E} = -\frac{G M m}{8R_E} \]

3. **Calculate Total Energy at Radius \( 2R_E \)**:
For the orbit at radius \( 2R_E \):
\[ E_{2R} = -\frac{G M m}{2 \times 2R_E} = -\frac{G M m}{4R_E} \]

4. **Find the Energy Difference**:
The energy required to transfer the satellite from the orbit at \( 2R_E \) to \( 4R_E \) is the difference in total energy:
\[ \Delta E = E_{4R} - E_{2R} \]
Substituting the values we calculated:
\[ \Delta E = \left(-\frac{G M m}{8R_E}\right) - \left(-\frac{G M m}{4R_E}\right) \]
Simplifying this:
\[ \Delta E = -\frac{G M m}{8R_E} + \frac{G M m}{4R_E} \]
\[ \Delta E = -\frac{G M m}{8R_E} + \frac{2G M m}{8R_E} = \frac{G M m}{8R_E} \]

5. **Final Result**:
The energy required to transfer the satellite from the orbit of radius \( 2R_E \) to \( 4R_E \) is:
\[ \Delta E = \frac{G M m}{8R_E} \]