The variation in the speed of the planet in their orbits about the sun can be explained on the basic of the conservation of :-

Options

A. Angular kinetic energy
B. Linear momentum
C. Angular momentum
D. None of these

Answer

To solve the question regarding the variation in the speed of planets in their orbits around the sun, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding Planetary Motion**:
- Planets move in elliptical orbits around the sun, with the sun located at one of the foci of the ellipse.

2. **Identifying Key Points**:
- In the elliptical orbit, there are two significant points:
- **Perigee**: The point where the planet is closest to the sun.
- **Apogee**: The point where the planet is farthest from the sun.

3. **Speed Variation**:
- At perigee, the speed of the planet (let's denote it as \( v_p \)) is maximum.
- At apogee, the speed of the planet (let's denote it as \( v_a \)) is minimum.

4. **Central Forces and Torque**:
- The gravitational force between the sun and the planet acts as a central force directed towards the sun.
- Since the force is central, the net torque about the sun is zero. This is because the torque \( \tau \) is given by \( \tau = r \times F \), where \( r \) is the distance vector from the sun to the planet and \( F \) is the gravitational force. Since these vectors are anti-parallel at any point in the orbit, \( \tau = 0 \).

5. **Conservation of Angular Momentum**:
- The fact that the net torque is zero implies that angular momentum is conserved.
- Angular momentum \( L \) is given by \( L = mvr \), where \( m \) is the mass of the planet, \( v \) is its speed, and \( r \) is the distance from the sun.

6. **Relationship Between Speed and Distance**:
- Since angular momentum is conserved, we can express this as \( m v_p r_p = m v_a r_a \), where \( r_p \) and \( r_a \) are the distances from the sun at perigee and apogee, respectively.
- This leads to the relationship \( v_p r_p = v_a r_a \), which implies \( v \propto \frac{1}{r} \). Hence, as the distance \( r \) decreases (at perigee), the speed \( v \) increases, and vice versa.

7. **Conclusion**:
- The variation in the speed of the planets in their orbits can be explained by the **conservation of angular momentum**.

### Final Answer:
The variation in the speed of the planets in their orbits about the sun can be explained on the basis of the conservation of **angular momentum**.

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