In our solar system, the motion of a planet in its orbit is governed by Kepler's laws of planetary motion. Specifically, Kepler's second law, also known as the law of areas, indicates that a planet travels faster when it is closer to the star it orbits (perihelion) and slower when it is farther away (aphelion).
For Gallifrey, with an orbital eccentricity of 0.3, we can analyze how its speed varies at its aphelion and perihelion based on the following principles:
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Eccentricity: An eccentricity of 0 indicates a perfect circular orbit, while an eccentricity of 1 indicates a parabolic trajectory. Gallifrey’s orbit, with an eccentricity of 0.3, suggests it has an elliptical orbit that is notably elongated, but not extremely so. This means there will be a significant difference between the distances of its closest approach (perihelion) and its farthest point (aphelion) from its star.
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Speed at Perihelion vs. Aphelion: According to Kepler's laws:
- At perihelion, where Gallifrey is closest to its sun, it will have the highest orbital speed. The gravitational force is stronger at this point, causing it to accelerate.
- At aphelion, where Gallifrey is farthest from its sun, its orbital speed decreases. The gravitational pull is weaker, resulting in a lower speed.
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Mathematical Explanation: The relationship between the distance from the sun and orbital speed can be described mathematically by the vis-viva equation: \[ v^2 = GM\left(\frac{2}{r} - \frac{1}{a}\right) \] where \(v\) is the orbital speed, \(G\) is the gravitational constant, \(M\) is the mass of the sun, \(r\) is the current distance from the sun, and \(a\) is the semi-major axis of the orbit. This indicates that as \(r\) decreases (closer distance at perihelion), \(v\) increases, and as \(r\) increases (further distance at aphelion), \(v\) decreases.
In summary, for Gallifrey, we can expect its orbital speed to be greater at perihelion than at aphelion due to the shape of its elliptical orbit and the effects of gravitational forces as described by Kepler’s laws. The difference in speed can be quite pronounced given that the eccentricity of 0.3 means Gallifrey travels significantly different distances from its sun throughout its orbit.