A satellite of mass `m` orbits the earth in an elliptical orbit having aphelion distance `r_(a)` ad perihelion distance `r_(p)`. The period of the orbit is T. The semi-major and semi-minor axes of the ellipse are a and b respectively. The angular momentum of the satellite is

Options

A. `mpir_(a)//T`
B. `2mpir_(a)//T`
C. `mpiab//T`
D. `2mpiab//T`

Explanation

To find the angular momentum of a satellite of mass `m` orbiting the Earth in an elliptical orbit, we can follow these steps:

### Step 1: Understand the definition of angular momentum
Angular momentum (L) of a satellite in orbit can be expressed in terms of its mass (m) and the area it sweeps out over time. The formula for angular momentum is given by:
\[ L = m \cdot v \cdot r \]
where \(v\) is the tangential velocity and \(r\) is the distance from the center of the Earth.

### Step 2: Use the concept of areal velocity
In an elliptical orbit, the satellite sweeps out equal areas in equal times. The area swept out per unit time (areal velocity) is given by:
\[ \frac{dA}{dt} = \frac{1}{2} r^2 \omega \]
where \( \omega \) is the angular velocity. For elliptical orbits, we can also express the areal velocity in terms of the semi-major axis (a) and semi-minor axis (b).

### Step 3: Calculate the area of the ellipse
The area \(A\) of an ellipse is given by:
\[ A = \pi a b \]
where \(a\) is the semi-major axis and \(b\) is the semi-minor axis.

### Step 4: Relate area to time period
The total area swept out in one complete orbit is equal to the area of the ellipse, and the time taken to sweep this area is the period \(T\). Therefore, the average areal velocity can be expressed as:
\[ \frac{dA}{dt} = \frac{\pi a b}{T} \]

### Step 5: Substitute into the angular momentum formula
Now, substituting the expression for areal velocity into the angular momentum formula, we get:
\[ L = m \cdot \frac{dA}{dt} = m \cdot \frac{\pi a b}{T} \]

### Step 6: Final expression for angular momentum
Thus, the angular momentum of the satellite is:
\[ L = \frac{m \pi a b}{T} \]

### Conclusion
The final expression for the angular momentum of the satellite is:
\[ L = \frac{m \pi a b}{T} \]