You need to think about whether you're averaging with respect to time or position. In an elliptical orbit the planet travels faster when it is nearer the center-of-mass so it spends more time near the farther focus. Brent On 7/14/2016 3:31 PM, John Aspinall wrote:
On 07/14/2016 04:35 PM, James Propp wrote:
Ignoring the effects of bodies other than the Earth and the Sun, and treating the Earth as a point mass, what is the average position of the Earth over the course of a year as it travels an elliptical orbit with the Sun at one focus?
If you ask me "Are you interested in the average position of Earth relative to the the Sun, or relative to the position of the center of mass of the two-body system?", my answer is "I'm interested in both". The equivalence between the two frames of reference is fairly simple, and expressed by the reduced mass: https://en.wikipedia.org/wiki/Reduced_mass
As to the average position, I would expect it to be the center of mass. The equation of motion is decomposable into two one-dimensional equations in orthogonal coordinates, reducing the question to "what is the average of a sinusoid".
In this view ("scaffolding", to borrow Henry's term from another recent email) the ellipse is not the fundamental solution. The ellipse is assembled from two more basic bits: two one-dimensional sinusoids. Various ellipses are possible by varying amplitudes and relative phase, but neither amplitude nor phase affects the average value.
- John
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