A satellite of mass m is in circular orbit of radius `3R_(E)` above earth (mass of earth `M_(E)` radius of earth `R_(E)` How much additional energy is required to transfer the satellite to ac orbit of radius `9 R_(E)` ?

Options

A. `(GM_(E) m)/(3R_(E))`
B. `(GM_(E)m)/(18 R_(E))`
C. `(2GM_(E)m)/(2R_(E))`
D. `(GM_(E) m)/(9R_(E))`

Explanation

To solve the problem of how much additional energy is required to transfer a satellite from a circular orbit of radius \(3R_E\) to a circular orbit of radius \(9R_E\), we will follow these steps:

### Step 1: Calculate the Total Energy in the Initial Orbit
The total mechanical energy \(E\) of a satellite in a circular orbit is given by the formula:

\[ E = -\frac{GM_E m}{2r} \]

For the initial orbit at radius \(r_1 = 3R_E\):

\[ E_1 = -\frac{GM_E m}{2 \times 3R_E} = -\frac{GM_E m}{6R_E} \]

### Step 2: Calculate the Total Energy in the Final Orbit
Now, we calculate the total energy for the final orbit at radius \(r_2 = 9R_E\):

\[ E_2 = -\frac{GM_E m}{2 \times 9R_E} = -\frac{GM_E m}{18R_E} \]

### Step 3: Calculate the Change in Energy
The additional energy required to transfer the satellite from the initial orbit to the final orbit is the difference in total energy between the two orbits:

\[ \Delta E = E_2 - E_1 \]

Substituting the values we calculated:

\[ \Delta E = \left(-\frac{GM_E m}{18R_E}\right) - \left(-\frac{GM_E m}{6R_E}\right) \]

### Step 4: Simplifying the Change in Energy
Now we simplify the expression:

\[ \Delta E = -\frac{GM_E m}{18R_E} + \frac{GM_E m}{6R_E} \]

To combine these fractions, we need a common denominator, which is \(18R_E\):

\[ \Delta E = -\frac{GM_E m}{18R_E} + \frac{3GM_E m}{18R_E} \]

\[ \Delta E = \frac{2GM_E m}{18R_E} = \frac{GM_E m}{9R_E} \]

### Final Answer
The additional energy required to transfer the satellite to an orbit of radius \(9R_E\) is:

\[ \Delta E = \frac{GM_E m}{9R_E} \]

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