# Practical limits on the size of orbiting objects: could two pebbles orbit each other

As I understand it, gravity is inherent to mass and therefore even a small rock has its own gravitational pull. It seems entirely plausible then that a rock 1" in diameter could orbit a bigger rock, say 10" in diameter. Is this actually possible? Are there any practical limits on the size of objects that could have another object orbiting them?

What kind of environment would be necessary to facilitate something like this occurring? Could it happen in a high or low earth orbit? Would it have to be far away from any stars, planets, or other large bodies? It seems to me that the speed of the objects would have to be very, very slow.

I have no background in physics and don't know a ton of math so answers without a lot of complicated equations would be appreciated. Thanks.

• For very small bodies, the Yarkovsky effect could be important. Electromagnetic effects might be important, e.g., if the bodies are ferromagnetic or acquire static charges. They could be disrupted by tidal forces. They could be knocked around by micrometeorite impacts. In low earth orbit, air friction would eventually bring them down completely. – Ben Crowell May 30 '13 at 22:56
• Man, picking an answer was hard as both answers (and all the comments) are fantastic. – popthestack Jun 7 '13 at 21:33

As Brandon mentioned, two small objects couldn't orbit each other near a significant gravitational field. The Hill Sphere "approximates the gravitational sphere of influence of a smaller body in the face of perturbations from a more massive body." Therefore, your pebble's Hill Sphere would be too small to permit orbits near Earth. The Wiki article has a calculation showing that an astronaut couldn't orbit the 104 tonne space shuttle 300 km above the Earth since the shuttle's Hill Sphere was only 120 cm.

• Nice answer, +1. It would be interesting to have some way of estimating the time required for the satellite's orbit to accumulate a significant disruption, maybe becoming unbound. For an object orbiting the Space Shuttle at a distance of 40 m, the period is about a week. If it can complete, say, thousands of orbits before being disrupted by tidal forces from the earth, then for many purposes we might not care about the disruption. In fact, atmospheric drag is probably enough to guarantee that you can't complete even one orbit. – Ben Crowell May 31 '13 at 2:23
• @BenCrowell Well, the tidal acceleration is of the order of $5\times 10^{-5}\ \mathrm{m/s^2}$. So tidal disruption takes about 15 or 20 minutes. This accords with my experience trying to rendezvous & dock in simulators. – Michael Brown May 31 '13 at 3:53

If there were no external influences like the gravity from the Earth and stars and no light radiation to push things, then even tiny objects could orbit each other.

Assuming somewhat constant densities (which is generally true for small objects), the mass of an object grows as the cube of the radius: $Mass \propto r_{}^3$. The gravitational strength falls as the square of the distance which must be greater than the radius: $Force \propto \frac{1}{r^2}$. Following this reasoning you can expect the attraction and therefore the orbital speed to roughly increase proportionally as you increase the size of the objects. Small objects means a low attractive force and therefor a very slow orbital speed.

This will break down for microscopic objects though. At those sizes the uncertainty principle would start to have an effect (not to mention other forces like Coulomb repulsion).

I haven't done the calculation but I suspect if small objects on the order of a few inches in diameter would need to be outside of our galaxy for the gravity between them to dominate over the gravity of the galaxy.

Anything with mass will orbit around anything else with mass. Gravity is of infinite range - a proton on your nose knows about a proton in the Andromeda Galaxy.

What's interesting is to see if the orbits produced mean anything sensible. This spreadsheet extract shows some orbital periods (T) for some common and hypopthetical pairs of objects. The point to note is that the orbital period depends strongly on the separation - the closer they are, the faster the orbit. The last row is the two rocks from the question. If they were 1m apart, they would take two days to orbit each other at about 0.4 mm/s.

G=6.67E-11
m1 (kg)     m2 (kg)     r (m)       mu          F (N)       v (m/s)     T (secs)    T(hrs)  T(days)
7.00E+22    6.00E+24    3.84E+08    4.05E+14    1.90E+20    1.02E+03    2.35E+06    650     27.2    earth-moon
6.00E+24    2.00E+30    1.50E+11    1.33E+20    3.56E+22    2.98E+04    3.16E+07    8780    365     sun earth
1.50E+21    1.30E+22    2.00E+07    9.67E+11    3.25E+18    2.08E+02    5.71E+05    158     6.61    pluto charon
1.00E+09    1.00E+09    1.00E+03    1.33E-01    6.67E+01    8.17E-03    5.44E+05    151     6.30    million tons 1km apart
1000        1000        100         1.33E-07    6.67E-09    2.58E-05    1.72E+07    4780    199     1 ton 100m apart
10          10          100         1.33E-09    6.67E-13    2.58E-06    1.72E+08    47800   1990    10kg 100m apart
10          10          10          1.33E-09    6.67E-11    8.17E-06    5.44E+06    1510    63      10kg 10m apart
10          10          1           1.33E-09    6.67E-09    2.58E-05    1.72E+05    47.8    1.99    10kg 1m apart
0.02        20          1           1.34E-09    2.67E-11    3.65E-05    1.72E+05    47.8    1.99    1"-10" rocks 1m apart

There is a lower limit to size. Specifically, that imposed by light pressure. While the sun is very bright, it is also very big. A meter-wide object the density of the sun at room temperature has a considerably higher power density than the sun, and so exerts a lot more light pressure per gravitational force. A sufficiently tiny object orbiting another such object will have to compensate for this, meaning something a few centimeters across is probably the lower limit at room temperature.