# How would a retrograde satellite experiencing tidal deceleration affect the rotation of the primary?

There are two common scenarios like this one, where an orbiting body orbits its primary slower than the primary rotates, resulting in the orbiting body moving away and the primary experiencing a slowdown in rotation. Whereas in the opposite, an orbiting body that orbits faster than the primary's rotation spirals in while the primary's rotation spins faster.

I know that a retrograde satellite will experience tidal deceleration and spiral in toward its primary body, but how will it affect the rotation of the primary?

For a retrograde satellite, you are right that the satellite will migrate inwards towards the planet. Contrary to a prograde orbit, the rotation of the primary will slow down.

Think about it in terms of angular momentum. Let the primary have a positive angular momentum and the satellite a negative one (since they rotate/orbit in opposite directions). Since the satellite is being pulled inwards, its angular momentum is being lowered in magnitude (becoming less negative). In that case the primary's spin angular momentum needs to become less positive to conserve the total angular momentum. This means the primary's rotation will slow down.

An example with numbers (unitless): Primary(initial) = 10 // Satellite(initial) = -5 /// Primary(final) = 7 // Satellite(final) = -2 /// Thus a transfer of 3 angular momentum units has occurred.

Eventually the satellite will slow down the primary until the primary is no longer rotating (assuming the satellite is not lost by this point). Then it will cause the primary to start rotating in the same direction as the satellite is orbiting.

Hope this helps!

EDIT:

In this paper, see equation (7), the torque on the primary due to the satellite. If we only care about the sign, we can note that $$N_m \textit{~}\, (n_m -\Omega_p)$$ where $$n_m$$ is the orbital frequency of the satellite and $$\Omega_p$$ is the rotational frequency of the primary. Let's take our previous example of the $$\Omega_p$$ positive and $$n_m$$ negative. This would make the torque $$N_m$$ negative, no matter what the magnitudes of the frequencies are. Consequently, the positive $$\Omega_p$$ would be reduced, signifying the slowing of the primary's rotation.

$$N_m = 3 k_2 \tau (n_m - \Omega_p) \frac{GM_m^2 R_p^2}{a_m^6} \tag{7}$$

where the subscripts $$m$$ and $$p$$ refer to the moon and planet respectively, $$k2$$ is the Love number of the planet that depends on its rigidity and $$\tau$$ is the tidal time lag of the planet.

• It is possible to either add some math that demonstrates that this is true, or to cite a supporting reference our source for that? If the total angular momentum is conserved and the angular momentum vectors point in opposite directions, then the magnitudes of both could increase or decrease. I don't think that one conservation law alone is enough to conclude definitively without using any math, or perhaps also including conservation of energy or a bit of thermodynamics. – uhoh Sep 7 '20 at 5:11
• Sure, I'll edit my post to include a reference and math – Armen Sep 7 '20 at 17:50
• I've added equation (7) here using MathJax and explained the terms, please feel free to edit further. Can you double check if the convention for the definition of $\tau$ and see if it does or does not also change sign in the case of retrograde motion? The word "retrograde" does not appear anywhere explicitly and it's possible that prograde motion was implicitly assumed in some earlier work on which this is based but not noted again here. cf. Figure 2 of Piro 2018. Thanks! – uhoh Sep 8 '20 at 0:56