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This question may be a little lazy, but can anybody give me a proof of the Hill sphere formula? Acording to wikipedia, the formula for the radius, $r$, is

$$r\approx a(1-e)\left(\frac{m}{3M}\right)^{1/3}$$

where a body of mass $m$ is orbiting a much more massive body of mass $M$ with a semi-major axis $a$ and eccentricty $e$.

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    $\begingroup$ Look at the introduction in this paper. $\endgroup$ – Dave Oct 24 '16 at 22:26
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    $\begingroup$ Place a test mass between two masses, assume the origin is in the bigger mass and calculate where the magnitudes to both forces are equal? $\endgroup$ – AtmosphericPrisonEscape Oct 24 '16 at 23:03
  • $\begingroup$ @Dave that's a pretty cool paper (I'd planned on getting something done today, but now...), and I am sure it's in there; $R_H=3^{-1/3}$ and "unit of length is scaled by the factor µ${}^{1/3}$" but I don't see how to get the (1-e) in the front so easily. $\endgroup$ – uhoh Oct 30 '16 at 12:49
  • $\begingroup$ Because a(1-e) is periastron? $\endgroup$ – chris Jul 8 '17 at 9:32
  • $\begingroup$ The scaling of the hill radius is such that the effect is so dominant at periapse as to be a good approximation to the limiting radius over the whole orbit (i.e. even though it's only at periapse for a short while it's where most mass loss occurs) $\endgroup$ – Zephyr Sep 17 at 10:03
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Hill sphere is named after John William Hill (1812–1879) and its simple logic follows from the presence of three bodies (let's assume Sun is the largest mass with Earth as the secondary mass and a satellite of negligible mass orbiting the Earth as the third mass), where the radius of the Hill sphere will be the largest radius at which a satellite could orbit the secondary mass (Earth in this case). If its orbit exceeds the Hills radius, then it will fall to the gravitational influence of the first body (sun) and hence will no longer be a satellite of the secondary body.

One could write Newton's equations using the idea that the satellite has the same angular velocity as the secondary object. This is that, the angular velocity of the Earth around the sun equals to the angular velocity of the satellite around the sun. A demonstration about the derivation is given in the following link as well as that of the Roche limit:

http://www.jgiesen.de/astro/stars/roche.htm

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