Question

problem set #3 - makeup begin date: 9/22/2025 12:01:00 am due date: 12/12/2025 11:57:00 pm end date: 12/12/2025 11:59:00 pm problem 8: (6% of assignment value) the figure to the right illustrates a planet revolving around its star on an elliptical orbit in the direction shown. at the point where the planet is shown, which of the following statements is false? the planets semimajor axis is decreasing. the planets orbital speed is increasing. the gravitational force between the planet and star is increasing. the planet is approaching perihelion (technically called \periastron\). the planets angular momentum remains conserved. hints: 1% deduction per hint. hints remaining: 1 feedback: 1% deduction per feedback.

Explanation:

1. Semi - major axis: The semi - major axis of an elliptical orbit is a fixed property for a given orbit and does not change during the normal course of the planet's revolution around the star. It is determined by the initial conditions of the orbit and the total energy of the system.
2. Orbital speed: According to Kepler's second law, a planet sweeps out equal areas in equal times. As the planet approaches the star (periastron), its distance from the star decreases and its orbital speed increases.
3. Gravitational force: The gravitational force \(F = G\frac{Mm}{r^{2}}\) between the planet and the star, where \(G\) is the gravitational constant, \(M\) and \(m\) are the masses of the star and planet respectively, and \(r\) is the distance between them. As the planet approaches the star, \(r\) decreases and the gravitational force increases.
4. Approaching periastron: From the direction of the planet's motion in the figure, it is moving towards the closest point to the star, which is the periastron.
5. Angular momentum conservation: In the absence of external torques (which is the case for a planet - star system in isolation), the planet's angular momentum \(L = mvr\sin\theta\) is conserved.

Answer:

The planet's semimajor axis is decreasing.