What orbit is the smallest or takes the least time to complete?

What orbit in our solar system would take the least time to perform 1 orbit?

What is the fastest possible orbit to complete in the universe?

• @CarlWitthoft that's a 100% unhelpful, unproductive, and gratuitous comment.
– uhoh
Mar 27, 2019 at 0:24

In general, the closer you are to the primary, the shorter the orbital period, except of course that you can't orbit inside the primary. So, taking the primary to be a sphere of radius $$R$$ and density $$\rho$$ we find that the mass is $$\frac{4}{3}\pi R^3\rho$$ so that the acceleration due to gravity at the surface is $$G\frac{4}{3}\pi R\rho$$ For a surface-grazing orbit, this has to equal $$R\omega^2$$ where $$\omega$$ is the angular velocity, so we get $$\omega^2 = \frac{4}{3}\pi G\rho$$ Now the orbital period $$P$$ is $$2\pi/\omega$$ so we get $$P = \sqrt{\frac{3\pi}{G\rho}}$$ So the answer, at least for spherical primaries) is going to be a close orbit around the densest object in the solar system. The densest planet is Earth! There are denser objects, such as the LAGEOS satellites, but I'm not sure if it is possible to orbit LAGEOS under its own gravity, because of the perturbing effects of Earth's gravity.

(This is a partial answer only addressing the second question; ideally this question should be split into at least two separate questions)

Theoretically, fastest possible orbits in terms of orbital velocity can be found around black holes. If we consider photons, the orbital speed would be equal to c in photon sphere. For massive particles, innermost stable circular orbit should be the fastest, both in terms of average speed as well as time it takes to complete one full orbit. For the latter, you’d want the smallest possible black hole.

I have however no clear understanding what is the smallest black hole currently observed and whether we are able to detect distinct objects orbiting in order to establish a solid measurement.

One good reference point for this could be derived from LIGO observations of binary black hole mergers: I believe the last orbits take fractions of seconds as gravitational wave frequencies are peaking at around 700Hz which could mean 700 or 350 orbits per second if I understand the mechanics correctly. In this case, orbital radius being in the range of tens or hundreds km, the orbital speed is very close to the speed of light; can’t recall exact estimates right now though.

• Also, photon sphere is largely theoretical construct, as I believe in theory no photon could enter the photon sphere, and as any external source of gravity will distort the space enough to destroy the sphere. Mar 25, 2019 at 9:35

I guess you only meant gravity based orbit but since you asked smallest, technically, other forces which are more stronger but act on small distances, would give much smaller orbit. For example electron orbit. But they are harder to think as circular and better explained by quantum number. (Sorry - maybe better as comment)

• It is good for me, and they would orbit each other.
– Muze
Mar 28, 2019 at 14:05