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How can the gravitational force of Bennu (being so weak) hold the spacecraft in orbit?

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A body of mass can keep a spacecraft in orbit, if the spacecraft is moving slow enough. If the velocity of the spacecraft is too high, it escapes. This limit is the escape velocity.

Escape velocity depends on both distance and the mass of the body:

$$v_e = \sqrt{\frac{2GM}{r}}$$

Where $G$ is the universal gravitational constant, $M$ is the mass of the asteroid, and $r$ is the distance between the centre of the asteroid and the spacecraft.

From the Scheeres et al. 2019 mass estimate astronsnapper linked ($7.329 ± 0.009 × 10^{10} kg$), when OSIRIS-REx was 1 km form the centre of Bennu, it could orbit no faster than 10cm/s to stay in orbit.

Weak gravity means slow orbits.


Another consideration is that when far away enough from Bennu, the spacecraft will tend to orbit the Sun instead of the asteroid. A first order approximation for when it's close enough for orbits to be stable is the Hill sphere:

$$r_H \approx a_{bennu}\sqrt{\frac{M_{bennu}}{3M_{sun}}}$$

Where $a_{bennu}$ is the semi-major axis of the asteroids orbit. This works out to a stable region of about 40km. This is a somewhat fuzzy limit as "stable" depends on the timescale you are looking at.

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    $\begingroup$ Maybe include estimated GM value or mass of 73.29 million tonnes from Scheeres et al. 2019 in answer ? $\endgroup$ – astrosnapper Nov 10 '20 at 3:04
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Along with escape velocity and sphere of influence considerations, one also needs to account for solar radiation pressure and the body's gravity field when analyzing orbital mechanics around small bodies.

enter image description here

Due to the relatively large ratio of magnitude of solar radiation pressure acceleration vs. gravity (since the gravity is so small), the only ~stable orbits around these bodies are terminator orbits. A terminator line on a body is the line separating light from dark on the surface (due to incident sunlight), so a terminator orbit is aligned orbits above that line. Another way to describe it is that the angular momentum vector of the orbit is either aligned with the sun vector or 180 degrees away (pointing in the exact opposite direction). In the plot, the blue is a terminator orbit, where the sun is in the positive x direction. The purple is another spacecraft who had the same initial magnitude of velocity but a different inclination, thus is wildly perturbed by solar radiation pressure in this orbit. I chose a border line inclination and semi-major axis to get this chaotic motion, but only another few degrees on inclination would make it escape within 2-3 orbits

When accounting for solar radiation pressure, there is a limit to the semi-major axis of the orbit, which is also a function of the inclination. This also means that you don't need to have escape velocity to actually escape, you can also be "blown away" by the solar wind.

D.J. Scheeres has a great paper on this analysis, called Orbital Mechanics about Small Asteroids: https://www.researchgate.net/publication/265973243_Orbit_mechanics_about_small_asteroids

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