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I'm considering a problem involving star S, planet A, and a comet B. The orbits of the planet and comet are eccentric, with eccentricities $e_A$ and $e_B$. I'm trying to calculate the average change in position that A should cause in B. I'm assuming $M_A \gg M_B$, so B should have negligible effect on A.

If I start using a coordinate system and assume S is at the origin, and A and B have semimajor axes $a_A$ and $a_B$ then A's orbit has polar equation $$r_A(\alpha) = \frac{a_A(1-e_A^2)}{1 - e_A\cos(\alpha)}$$

And if the angle between the major axes of A and B (the axes connecting their foci) at S is $\theta$ then B's orbit has polar equation $$r_B(\beta) = \frac{a_B(1-e_B^2)}{1 - e_B\cos(\beta-\theta)}$$

Thanks to Newton I know that the acceleration on B toward A is given by $$\text{acc}(\alpha, \beta) := \frac{GM_A (\vec{x}_A(\alpha) - \vec{x}_B(\beta))}{\mid\mid\vec{x}_A(\alpha) - \vec{x}_B(\beta)\mid\mid^3}$$ where $\vec{x}_P(\phi)$ is my shorthand for the position of object P, $(r_P(\phi), \phi)$.

My initial thought was the position change is given by $$\int_0^{2\pi}\int_0^{2\pi}\text{acc}(\alpha, \beta) \; \mathrm{d}\alpha \; \mathrm{d}\beta$$ and so the average position change would be $$\frac{1}{4\pi^2}\int_0^{2\pi}\int_0^{2\pi}\text{acc}(\alpha, \beta) \; \mathrm{d}\alpha \; \mathrm{d}\beta$$ however a colleague suggested that this assumes that the distribution of A and B in their orbits is uniform, which it isn't. So I did some calculations and worked out that object P has probability distribution $$\mu_{P}(\theta) = \frac{(1 - e_{P}^{2})^{3/2}}{2\pi (1-e\cos(\theta))^2}.$$

Armed with this knowledge I now think that the average effect of A on B is the change in position given by $$\frac{1}{4\pi^2}\int_0^{2\pi}\int_0^{2\pi}\text{acc}(\alpha, \beta)\mu_A(\alpha)\mu_B(\beta) \; \mathrm{d}\alpha \; \mathrm{d}\beta.$$

That said I'm not sure if I'm missing anything or if my whole approach is off, and so I thought I'd ask here if my reasoning and ultimately my concluding integral seem correct. I'm a pure mathematician so I'm not very comfortable sorting out whether an application of theory has been properly applied to a real world situation. Any input is most welcome.

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  • $\begingroup$ FYI, your equation for the orbits, $r(\alpha)$ and $r(\beta)$, is not the equation for when the sun sits at the center of the ellipse. That equation applies when the Sun sits at one of the foci. $\endgroup$ – zephyr Feb 10 '17 at 21:34
  • $\begingroup$ But the sun must be at a focus. That's one of Kepler's laws. $\endgroup$ – Allen O'Hara Feb 10 '17 at 21:36
  • $\begingroup$ Yes, that's exactly what I said. The sun is the origin of your coordinate system and your equation assumes the sun is at the focus, not the center of the ellipse. Just making sure that is known as some people mistakenly think that equation assumes the sun is at the center of the ellipse. $\endgroup$ – zephyr Feb 10 '17 at 22:00
  • $\begingroup$ Oh ok, now I understand. I thought you were correcting the equations. My mistake, and my apologies. $\endgroup$ – Allen O'Hara Feb 10 '17 at 22:22
  • $\begingroup$ Without spending the time to get into the details of your math, I'd say my first instinct is that the double integral over acceleration doesn't give you average position change but rather average acceleration change. $\endgroup$ – zephyr Feb 10 '17 at 22:28

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