# Would tidal forces be strong enough to stabilize a configuration where one satellite is being used to tow another satellite?

The scenario that I'm contemplating is one in which there is a first "depleted satellite" that has run out of propellant and a second "rescue satellite" that is designed to rescue the depleted satellite.

The rescue satellite is equipped with:

1. An ion thruster,
2. Reaction wheels for adjusting its orientation,
3. At least one grappling gun that can shoot a magnet on a string.

The rescue satellite would maneuver close to the depleted satellite and then use its grappling gun(s) to shoot at least one magnet on a string at the depleted satellite. (We'll assume here that the depleted satellite has enough ferromagnetic material in its hull plating for magnets to stick.)

Let's assume that the satellites are in a 550 km LEO orbit and the effective range of the grappling gun(s) is 10 m.

Would the tidal forces acting on the coupled pair of satellites stabilize them enough to permit the rescue satellite to maintain the depleted satellite's orbit by thrusting with its ion drive? Or, would any attempt to use the ion drive for this purpose just cause the pair to start spinning out of control?

Note: related to connecting two satellites.

### Tidal Force

Over such a short distance you can determine the tidal force between two satellites connected by a tether using

$$F = \mu m(1/r_1^2-1/r_2^2)$$

where

$$F$$ is the force between the satellites and the minimum strength for your bond,

$$\mu$$ is the standard gravitational parameter (3.986004418E14 $$m^3/s^2$$ for Earth),

$$m$$ is the mass of each satellite (let's assume 1000 kg),

$$r_1$$ is the distance from the center of the planet to the lower satellite (assume 6,378,100+550,000-5 m), and

$$r_2$$ is the distance from the center of the planet to the upper satellite (assume 6,378,100+550,000+5 m)

(ref - thanks Woody!)

The force between them will be 24 milliNewtons.

This is pretty small. From Wikipedia...

Ion thrusters in operation typically consume 1–7 kW of power, have exhaust velocities around 20–50 km/s (Isp 2000–5000 s), and possess thrusts of 25–250 mN.

### Aerodynamic Drag

To determine how much thrust we need, let's calculate the drag force on the satellites using the drag equation

$$F=0.5 \rho v^2 C_d A$$

We can estimate the air density, $$\rho$$, as 2.34E-14$$kg/m^3$$ using data from the MSISE-90 Model of Earth's Upper Atmosphere.

Using $$v=\sqrt{\mu/r}$$ we can determine that the satellite's velocity will be 7585 m/s.

Let's assume the coefficient of drag, $$C_d$$ is 1 and the total area of both satellites, $$A$$, is 10 $$m^2$$.

The drag force on the satellites will then be 0.0067 mN.

### Conclusion

While tiny, the tidal forces are 3560 times stronger than the thrust needed to counter aerodynamic drag.

Therefore, if the rescue satellite was outfitted with an ion drive that could throttle down to a sufficiently low thrust level, it is conceivable that the tidal forces would be strong enough to stabilize the conjoined pair.

The tether should be designed to not exert a force that would pull the satellites together. Something like a fine jewelry chain might work.

Verify that the satellites do not generate magnetic or electrostatic forces that are strong enough to attract the satellites together.

You might be able to dispense with the grappler and fine string/chain altogether and instead use a solar-powered electromagnet controlled by a distance sensor to couple the satellites.

• Wouldn't the orientation of the two objects get messed up as they orbit? 180 degrees around the orbit, the trailing object is going to wind up in front of the leader. If they're attached by a string, then even if the "lead" satellite is rotating to stay pointed prograde, the trailing satellite is going to constantly try to wrap the tow cable around it. (or more accurately, the string and trailer are retaining orientation relative to the fixed stars and the leader is rotating to keep its nose pointed prograde, so the leader is the one turning and wrapping the tow cable around itself...) Jan 10 at 15:51
• Assume the satellites start out attached, one above the another, with the string direction normal to the Earth's surface. If you cut the string, what would happen? Answer: The lower sat will be at the apogee of its elliptical orbit and the upper sat will be at the perigee of its elliptical orbit. Therefore, orbital mechanics will not let them get closer to each other. So the tow cable can't get wrapped around a sat. Jan 10 at 21:27
• Forget the rope. Imagine you're standing on one satellite, and the other is above your head and the earth below your feet, facing the direction of travel. Neither satellite is rotating with respect to the stars. As you orbit the earth, the satellite above you will stay put, but you'll see the earth rise up behind you, go over your head, and come down in front of you until it's back below your feet when you get back to your original point in the orbit. Over the course of that orbit, you started out facing prograde, halfway around you were facing retrograde, and you end up prograde. Jan 10 at 21:42
• But if you want to fire a thruster the whole time, you can't just face one way with respect to the stars. You need to rotate the satellite so that it always faces prograde. And if you do that, then the earth stays put from your perspective, and satellite "above" is going to orbit around you as you move. Put the string in, and now it's going to slowly wrap around you. Jan 10 at 21:44
• Ok, this rescue idea probably works best for nadir-pointing satellites. For a star-stationary satellite, like the Hubble space telescope, it isn't a good solution as proposed. Keep in mind that even when the thruster is firing, the string is still more-or-less normal to the surface of the Earth. I'm not following your logic in the second comment though. Don't think "orbit around you as you move" is correct. Jan 10 at 22:02

Answer: No. Tidal tension on a 10m tether would not even keep the tether taut.

Tidal force (due to the difference in orbital radius) creates tension on a long tether.

Tension on the tether is the difference in gravitational attraction due to difference in orbital altitude. See http://www.physicsbootcamp.org/gravity-Tidal-Forces.html to plug in values for your model. The tension will be proportional to the masses and proportional to the mass's distance from the common center of gravity. See p104 https://ntrs.nasa.gov/api/citations/19920010006/downloads/19920010006.pdf

A 10 m tether would be inadequate to provide stability. In 1966 Gemini 11 released the Agena target vehicle on a 30m tether, but this length was insufficient to even keep the tether taut.

The TSS-1R mission managed to deploy 19.7km of tether before the tether broke. The tension was 65N at tether failure.