# If there was a probe orbiting at the edge of Earth's Sphere of influence, how slow would it orbit?

Would it orbit very slowly? It's kinda been stuck in my head lately...

• The sphere of influence is not 'gravity stops after this', but more of a 'gravity of this object is not overly dominant anymore', e.g. for applying the patched conic approximation. So, you're not orbiting 'on the edge of gravity', but rather 'very far away'. – Sanchises Mar 29 '15 at 8:51

## 2 Answers

I define sphere of influence as: $(M_1/M_0)^{2/5} * r$

For this case $M_1$ is mass of the earth, $M_0$ is the mass of the sun, and $r$ is $149597871$ kilometers (distance from sun to earth).

According to my Hohmann spreadsheet, altitude of Sphere of Influence (SOI) is $918149$ kilometers.

Also according to my Hohmann spreadsheet a circular orbit at that altitude would move $.6565$ km/s and orbital period would be $102.4$ days.

But the spreadsheet is based on two body Newtonian mechanics. An actual orbit at that altitude might be distorted by the sun's influence.

Should you want to play with the spreadsheet here's an illustration. Typing the SOI altitude into both periapsis and apoapsis gives a circular orbit at that altitude.

Underlined is the circular orbit speed at periapsis and period in days.

It's assumed the user has already typed Earth into the departure planet cell.

• I'd bracket math expressions with dollar signs. But instead of subscripting subscripts and raising exponents, it would just draw a box around the expression. An anonymous use has Mathjaxed it for me (thank you!). Now it reads the way I want it. – HopDavid Mar 28 '15 at 18:25
• Thanks. That's odd because your answer is much different than pericynthion's... But you have more science involved, so I think you're more accurate. :) – eroettger Mar 28 '15 at 19:53
• @eroettger Pericynthion's numbers are correct. Pericynthion gave orbit speed and period for orbit having 1.5 million kilometer radius. I gave orbit speed and and period for an orbit having 924,000 km radius. – HopDavid Mar 28 '15 at 19:59

The Earth's Hill Sphere extends to a radius of approx 1.5 million km. The velocity of a circular orbit at such a distance is around 500 m/s - which is very slow compared to LEO, but very fast compared to most forms of transportation, so I suppose it depends on your perspective...

Such an orbit has a period of approx 210 days, and is usually unstable. Lunar and solar perturbations will increase the eccentricity until it (most likely) escapes into a heliocentric orbit.

• I agree with your numbers. I usually treat Hill Sphere and SOI as two different things. Still I up voted the answer because it's informative. – HopDavid Mar 28 '15 at 18:43