# What is the smallest body that has sufficient gravity for another body to orbit it?

In the question Will Rosetta have to adjust its orbit around Chury due to the comet's coma and tails? and related answers it is implied that the Rosetta spacecraft will actually orbit 67P/Churyumov–Gerasimenko. When I think of orbits I think of the high speeds and large bodies as a gravity base for the orbit; Sun, planets and such. A comet has a tail because of all of the stuff falling off and falling behind the body of the comet as it orbits the sun. Anything orbiting it seems counter intuitive.

Will Rosetta actually orbit the comet or is that just a figure of speech? Is there some minimum size of body (in our solar system) that can be orbited? Is the smallest size based on the relationship between the two bodies size?

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I believe there was a NASA mission (whether it flew or was only proposed I don't recall) that involved two satellites orbiting each other. –  Loren Pechtel Feb 13 at 3:03

There isn't any limit to how small bodies can orbit each other (gravity-wise) until you get to atomic scale where one of the remaining three fundamental forces (weak force, strong force and electromagnetism) take over and gravity becomes largely irrelevant. With smaller bodies, gravitational potential will only be that much smaller and required centrifugal force to precisely counter it and achieve orbit (orbital speed) will equally decrease.

Rosetta will orbit Churi but its orbital speed relative to the parent body won't be much. Orbital velocity of two bodies where mass of the orbiting body is non-negligible relative to the parent body it orbits around is given by:

$$v_{o}\approx {\sqrt {(M_{2})^{2}G \over (M_{1}+M_{2})r}}$$

Where $M_1$ is the mass of the orbiting body and $M_2$ mass of the parent body, $G$ is gravitational constant, and $r$ is the distance between their centers of mass (or their semi-major axis).

There might be other interactive forces at play though (mentioned weak force, strong force and electromagnetism), say atmospheric drag (or should that be exoatmospheric?) due to comet's outgassing when selected orbit is close enough to pass through its tails and coma, but in case of Churi and Rosetta we don't know to what extent yet.

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Electromagnetism also acts on our scale, as demonstrated in this relaxing video. –  Cees Timmerman Feb 12 at 17:45
@CeesTimmerman The questions is about gravitational forces, and that video was posted before by Hobbes in his answer, but later removed because that didn't rely on gravity. Which interactive forces hold an orbital system stable can be various things, yes. –  TildalWave Feb 12 at 17:50
Your answer seems to discount electromagnetism at anything greater than the atomic scale. –  Cees Timmerman Feb 12 at 17:54
@CeesTimmerman No, not really, read on. What I describe is absolute minimum at which the orbiting system held together by gravity will be disturbed by other interactive forces, not the possible range of this interaction. Of course this equilibrium can be disturbed by other forces. Even gravity anomalies themselves can cause orbital precession, especially around a non-spherical body. But the question is about minimums. ;) –  TildalWave Feb 12 at 18:02
If tidal perturbations from other bodies in the solar system reach the same scale as the orbital velocity then the orbit is doomed, so while the math workout for isolated pair down to very small scales the reality in the solar system is more limited. –  dmckee Feb 18 at 17:49

(Consider this a spaceholder until I get around to the hard part of the answer)
Principally there's no lower size above atom scale. But I would say that for stable orbiting, the gravitational force should be the dominant force compared to solar wind etc. This would depend qualitavly on these factor:

• Mass of the 'central' body - the heavier the less relative influence from solar wind and light
• Size of the orbiting body - the smaller and less dense, the bigger the impact of solar wind and light
• Position within the solar system - the closer to the sun, the bigger the influence of solar wind and light

I'll see if I quantify these assertions (or someone beats me to it!).

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Agreed, orbiting can happen at pretty much any scale. To give an idea, 67P/Churyumov–Gerasimenko has an escape velocity of about 1 m/s.

(removed reference to ISS experiment because that didn't rely on gravity)

A quick experiment: for bodies that weigh 1 kg each and a radius of 1 m, the formula given by @Tildalwave yields an orbital speed on the order of $10^{-11} m/s$.

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I would say that it is possible for an object (call it A) in the gravity well of another, more massive object (B) to be too small to allow any third object (C) to orbit A, because any object C small enough for the mutual centre of gravity of A and C to lie outside A would be captured by B.

I don't know enough about it to be able to give you the numbers though, or to calculate whether this applies to the Rosetta/Chury/Churyumov–Gerasimenko situation.

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It doesn't. Rosetta together with Philae lander weigh at 3 tones at launch (now less as it consumed some of its propellants), and the comet is roughly 4 km in diameter with a mass of $3.14(21) × 10^{12}\ kg$. Otherwise, orbits around a binary system are possible. Either opposition co-orbital ($L_3$), co-orbital at $L_4$ and $L_5$ such as e.g. Jupiter's Hilda and Trojan asteroids, horseshoe orbits, or arguably also halo orbits at $L_1$ and $L_2$ (unstable, managed). Or simply distant enough that perturbation doesn't result in orbital precession, such as Proxima Centauri and Alpha Centauri AB. –  TildalWave Feb 12 at 17:18