Spinning Satellites — An Orbital Mechanics Puzzle

For satellites in space, orbital mechanics can be very counterintuitive and take some getting used to. For instance, consider a satellite in circular low-earth orbit, and apply an instantaneous thrust or force in any direction. What happens to the orbit? Note: apogee is the orbit’s farthest point from Earth, perigee is the nearest point to Earth.

Thrust forward (standard propulsion): Apogee raises, perigee stays the same.

Thrust backward (retropropulsion / deorbit burn): Apogee stays the same, perigee lowers.

Thrust upward/downward: Perigee lowers, apogee raises.

Thrust left/right: Orbital plane changes, altitude stays the same.

This last bit is quite counterintuitive. A force acting perpendicular to both gravity and the satellite’s direction of motion will not affect its orbital altitude at all, only its orbital plane.

Let’s also observe that for a satellite in a circular orbit, the upward centrifugal force of its circular motion precisely balances the downward pull of gravity.

So now, let’s consider a hypothetical scenario (above, not to scale). We have two satellites orbiting in opposite directions in the same orbital plane, at the same altitude, barely missing each other with each pass. In a 300km circular orbit, their velocity will be roughly 7.73km/s. Suppose they fly past each other above the equator with 10-meter horizontal separation, one going north and the other going south.

As they pass, the satellites encounter a 10-meter, infinitely strong, massless tether, floating in space at their altitude. Each satellite grabs one end of the tether, and they begin spinning rapidly around each other. (The satellites are also, needless to say, indestructible.)

Now, consider the forces acting on the satellites. The horizontal inward-pulling tether forces on each satellite would appear to be perpendicular to both gravity and the direction of motion, which would mean that the satellite’s “orbit” should not raise or lower; the spinning system should stay at the same altitude, with each satellite rapidly shifting orbital planes, but perpetually hovering in space. For instance, if each satellite were to hold onto the tether for a 1/4 turn (which would take about a millisecond), then disconnect, their orbits would become east-west. Or they could do a full turn around each other (4 milliseconds), disconnect, and be essentially back in the orbits where they started. Either way, it seems that the spinning satellites should remain in orbit.

On the other hand, the combined system now has a center of mass that is at a constant position above Earth’s surface. From that perspective, it would seem that gravity should simply pull the combined system downward at 1G (or slightly less since the spacecraft are farther from the center of the Earth), and that the spinning should do nothing to counteract this.

How can these two conclusions be reconciled? Which one is correct? Will the spinning satellites hover, or will they drop like a rock? Take a minute to think about it.

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The solution lies in the fact that the extreme force imparted by the tether on each satellite is actually not perfectly orthogonal to gravity. The tether is orthogonal to gravity at the tether’s center point, but not at its endpoints; it forms a chord with respect to the orbital plane connecting the two satellites. So each satellite experiences a massive tug sideways, but also a very slight (relatively speaking) tug downwards!

When they connect to the tether, the satellites will start spinning around each other at roughly 246Hz, or nearly 15,000rpm. Using a = v² / r, the inward pull on each satellite is about 11.95 million m/s², or about 1.22 million G’s. (Interpreted as surface gravity, this would lie somewhere between a white dwarf and a neutron star.) But since the tether forms a 10-meter chord, in a circle of roughly 6700km radius, the angle at each satellite deviates from horizontal (from the satellite’s perspective) by roughly 5 / 6700000 radians, or 0.0000428 degrees. The value 5 / 67000000 is equal to 1 / 1340000; this fraction of the tether force will be pulling downwards, and 1.22 million G’s divided by 1340000 is equal to about 0.91 G’s, which is (lo and behold) the same as gravity at that altitude. Thus, the satellites will be pulled downward by the tether at 0.91 G’s, precisely counteracting the upward centrifugal force, which leaves only actual gravity to act on them as the only net force. This accounts for the combined system’s fall to Earth.

What do you think? Is this the correct explanation that resolves the apparent paradox? Leave your thoughts in the comments!