Kepler's second law is based on

Options

A. Newton's first law
B. Newton's second law
C. special theory of relativity
D. conservation of angular momentum.

Answer

**Step-by-Step Solution:**

1. **Understanding Kepler's Second Law**: Kepler's second law, also known as the law of areas, states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This means that the speed of a planet in its orbit varies; it moves faster when it is closer to the Sun and slower when it is farther away.

2. **Mathematical Representation**: The law can be mathematically expressed as:
\[ \frac{dA}{dt} = \text{constant} \]
where \(dA\) is the area swept out in a small time interval \(dt\).

3. **Relation to Angular Momentum**: The area swept out by the planet can also be related to its angular momentum. The angular momentum \(L\) of a planet in orbit is given by:
\[ L = mvr \]
where \(m\) is the mass of the planet, \(v\) is its tangential velocity, and \(r\) is the distance from the Sun.

4. **Deriving the Conservation of Angular Momentum**: From the expression for area swept out, we can derive that:
\[ \frac{dA}{dt} = \frac{L}{2m} \]
Since \(dA/dt\) is constant, it implies that \(L\) (angular momentum) must also be constant, as \(m\) (mass) is constant.

5. **Conclusion**: Therefore, Kepler's second law is fundamentally based on the principle of conservation of angular momentum. Hence, the correct answer is that Kepler's second law is based on conservation of angular momentum.

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