# How fast would a gas giant, or very close moon, need to orbit to maintain distance?

I have noticed that there is a common theme in science fiction art to depict one or more very large planetary bodies in very close proximity to the surface of the other world being depicted. (you can see this in the header artwork on this site)

My question is, (and I realize I am talking about fiction…) how realistic is this given the pull of gravity? It would seem that a binary planet or large moon of the relative size typically depicted would have to be orbiting extraordinarily fast in order to avoid collapsing into each other. Maybe this works in a still painting or computer generated image, but I have seen it in motion pictures where a giant orb is just hanging there, seemingly motionless in real time. Shouldn’t there be enough relative motion to be able to perceive movement?

Is there a way calculate how rapidly a moon sized object would need to move across the sky if it were close enough to earth to appear as large as some common Sci-Fi scenes? Just for the sake of discussion, what if our current moon was close enough to appear 10 times larger, (probably still much smaller than some depictions I have seen) how long might a single orbit take?

• It all depends on how fast the primary rotates around its axis, doesn't it? To give a well-known example, seen from the Moon the Earth appears about 4 times larger than the Moon seen from the Earth; and the Earth doesn't move at all on the Moon sky: because the Moon is tidally locked, so that a rotation around the axis takes the same time as a revolution around the center of mass of the Earth-Moon system. – AlexP Feb 18 at 0:51
• Good point, I didn't consider rotation of the observing planet. Still, many Sci-Fi fantasy pictures don't show a moon 4 times larger in appearance than our moon, or even 10 times... They are HUGE, spanning up to 1/4 of the sky from horizon to horizon, and this with some amount of mass obscured below the horizon. – Michael Hall Feb 18 at 1:25
• @Michael Hall in many cases, the "moon" depicted in the illustration may be the planet, and the "planet's" surface may be that of a moon of that planet. It is theoretically possible for a giant moon of a gas giant planet to be habitable, and the planet might appear many times the apparent size of Earth's moon in the sky of a habitable moon. It is also possible for a neighboring planet to appear larger than the moon in the sky - see TRAPPIST-1 en.wikipedia.org/wiki/TRAPPIST-1 – M. A. Golding Feb 18 at 17:48
• Understood. Moon or planet, the main point was in questioning the apparent distance and the rotation speed needed to maintain an orbit so close. – Michael Hall Feb 18 at 19:14

If you have some moon orbiting in roughly a circle around a planet of mass $$m$$ (much greater than the moon's mass) at a distance $$r$$, its orbital velocity will be very roughly $$v = \sqrt{\frac{Gm}{r}}$$ (where $$G$$ is the gravitational constant). The naive solution to this is that bringing the Moon ten times closer to the Earth-Moon barycenter will multiply its orbital velocity by about $$\sqrt{10}$$, and since the path it follows is ten times shorter, it'll make $$10\sqrt{10}$$ orbits in the original orbital period. This would bring Earth's Moon's orbital period down to about 21 hours.

Note that to increase orbital velocity while maintaining a circular orbit and a constant planet mass, you have to actually bring the moon in closer. If it is faster than this $$v$$ at your given distance $$r$$, then you don't have a circular orbit; either the moon is following an elliptical orbit that takes it out much farther than this, or it is on a parabolic or hyperbolic trajectory that sends it out of the planet's sphere of influence and into deep space.

In our case, a Moon with a 21-hour orbital period (and an orbital direction still matching the Earth's rotation) would be quite noticeable in its motion. Unlike the stars, it wouldn't make a full circle around the celestial sphere every 24 (sidereal) hours; it'd only move about an eighth of the way around. This would put it very close to tidally locking with the Earth, which would result in a total lack of apparent motion through the sky. It is thought that planets with close moons typically end up tidally locked to each other since they also influence each other's rotations; one example can be found with Pluto and Charon, which are relatively closer to each other than the Earth and the Moon.

Bringing a moon too close to a planet runs the risk of crossing it over the Roche limit, at which point gravitational forces on the moon's far and near sides are so different that the net "stretching" overcomes the moon's own inward gravity and rips it apart. This limit depends on the densities of the planet and moon in question, but generally the biggest issue with the super-close moons we see in science fiction is that they might be inside their planet's Roche limit and liable to getting destroyed.

EDIT: I'll clarify that similarly-sized objects can get very close; the Roche limit for two identical bodies is around 2.4 times the radius of one. This would cause some serious stretching effects, of course.

With planetary-scale masses, I think you could bring the orbital period of one of these binary systems down to a few hours. A minute-long shot of such a system in a film probably still wouldn't show significant motion, but you'd definitely be able to track it over time from the ground.

• Thanks, I hadn't heard of this Roche limit before, but I would guess that to me a major factor, and flaw in this sort of depiction. – Michael Hall Feb 18 at 1:26