Energy of a satellite in circular orbit is `E_(0)`. The energy required to move the satellite to a circular orbit of 3 times the radius of the initial orbit is

Options

A. `(2)/(3)E_(0)`
B. `2E_(0)`
C. `(E_(0))/(3)`
D. `(3)/(2)E_(0)`

Explanation

To solve the problem, we need to determine the energy required to move a satellite from its initial circular orbit to a new circular orbit that is three times the radius of the initial orbit.

### Step-by-Step Solution:

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

2. **Initial Energy \( E_0 \)**:
Let the initial radius of the orbit be \( r \). Therefore, the energy of the satellite in the initial orbit is:
\[ E_0 = -\frac{GMm}{2r} \]

3. **Final Energy in the New Orbit**:
The new orbit has a radius of \( 3r \). The energy of the satellite in this new orbit is:
\[ E_{\text{final}} = -\frac{GMm}{2(3r)} = -\frac{GMm}{6r} \]

4. **Calculating the Change in Energy**:
The work done (or energy required) to move the satellite to the new orbit is the change in energy, which can be calculated as:
\[ \Delta E = E_{\text{final}} - E_0 \]
Substituting the values we calculated:
\[ \Delta E = \left(-\frac{GMm}{6r}\right) - \left(-\frac{GMm}{2r}\right) \]
\[ \Delta E = -\frac{GMm}{6r} + \frac{GMm}{2r} \]

5. **Finding a Common Denominator**:
To combine these fractions, we need a common denominator. The common denominator between 6 and 2 is 6:
\[ \Delta E = -\frac{GMm}{6r} + \frac{3GMm}{6r} \]
\[ \Delta E = \frac{2GMm}{6r} = \frac{GMm}{3r} \]

6. **Expressing in Terms of \( E_0 \)**:
We know that \( E_0 = -\frac{GMm}{2r} \). To express \( \Delta E \) in terms of \( E_0 \):
\[ \Delta E = \frac{GMm}{3r} = \frac{2}{3} \left(-\frac{GMm}{2r}\right) = \frac{2}{3} E_0 \]

### Final Answer:
The energy required to move the satellite to a circular orbit of three times the radius of the initial orbit is:
\[ \Delta E = \frac{2}{3} E_0 \]