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In Newton's inverse square law of gravit...

In Newton's inverse square law of gravitation had some dependence of radial distance other than `r^(-2)` which on of kepler's three laws of planetary motion would unchanged ?

A

First law on nature of orbits

B

Second law on constant areal velocity

C

Third law on dependence of orbital time period on orbit's semi major axis

D

None of the above

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To solve the question regarding which of Kepler's three laws of planetary motion would remain unchanged if Newton's inverse square law of gravitation had some dependence on radial distance other than \( r^{-2} \), we can analyze each of Kepler's laws: ### Step 1: Understand Kepler's Laws Kepler's three laws of planetary motion are: 1. **Kepler's First Law**: The orbit of a planet is an ellipse with the Sun at one of the foci. 2. **Kepler's Second Law**: A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time (law of areas). 3. **Kepler's Third Law**: The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. ### Step 2: Analyze the Effect of Changing the Gravitational Law If we change the gravitational force from being inversely proportional to the square of the distance (\( r^{-2} \)) to some other form, we need to consider how this affects each of Kepler's laws. - **Kepler's First Law**: The shape of the orbit (ellipse) is determined by the gravitational force. If the force changes, the orbits could also change from ellipses to other shapes, thus this law may not remain unchanged. - **Kepler's Second Law**: This law states that the area swept out by a planet in a given time is constant. This law is based on the conservation of angular momentum. Angular momentum depends on the radial distance and the tangential velocity of the planet. If the gravitational force changes, it may affect the velocities and distances, potentially altering the area swept out. Therefore, this law might not remain unchanged. - **Kepler's Third Law**: This law relates the orbital period to the semi-major axis of the orbit. If the gravitational force changes, the relationship between the period and the radius will also change. Hence, this law is likely to be affected. ### Step 3: Conclusion The only law that is fundamentally based on the area swept out over time and not directly dependent on the specific form of the gravitational force is **Kepler's Second Law**. Therefore, even if the gravitational force has a different dependence on distance, the law of areas (Kepler's Second Law) would remain unchanged. ### Final Answer **Kepler's Second Law (Law of Areas) would remain unchanged.** ---
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