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A planet of mass m orbits the Sun in an elliptical path with eccentricity e. Its speed at aphelion (farthest point) is v1 and at perihelion (nearest point) is v2. The semi-major axis of the orbit is a. Which of the following statements are correct?
- The Sun occupies one of the two foci of the ellipse.
- v1/v2 = (1 - e)/(1 + e)
- The orbital period is T = (m * pi * a² * sqrt(1 - e²)) / L, where L is the angular momentum of the planet about the Sun.
- The angular momentum of the planet about the Sun is conserved throughout the orbit.
Correct answer: The Sun occupies one of the two foci of the ellipse.
Solution
Options A, B, and D are correct. Kepler's first law (A), angular-momentum conservation at extreme points giving v1/v2 = (1-e)/(1+e) (B), and conservation of angular momentum for a central force (D) are all valid. Option C is off by a factor of 2: the correct expression is T = 2m*pi*a²*sqrt(1-e²)/L.
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