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Asynchronous Case Edit

Let us consider a deformable body of mass $ M_b $. Under its own gravitational attraction its shape is a sphere of radius $ R $. Now suppose we introduce another point mass $ M_d $ at a distance $ d > R $. Gravity due to $ M_d $ deforms $ M_b $. The new shape of $ M_b $ is bounded by an equipotential surface. We denote by x the maximum deviation from a sphere of radius $ R $. The difference in the tidal potential due to $ M_d $ is $ \frac{G M_d R^2}{d^3} $, while the difference due the displacement by $ x $ is $ \frac{G M_b x}{R^2} $, hence

$ \frac{x}{R} \approx \frac{M_d}{M_b} \left(\frac{R}{d} \right)^3 $

The mass protruding from the unperturbed radius $ R $ is

$ \Delta M = M_d \left(R/d \right)^3 $

This mass manifests as two lobes, one on the point closest to $ M_d $, and another on the farthest, antipodal point. If $ M_d $ is spinning faster than $ M_b $, then there is a misalignment between the line connecting the lobes and the line connecting $ M_d $ and $ M_b $. This misalignment gives rise to torques. For simplicity, we will assume that the angle between the two lines is of order unity. The torque is therefore given by the difference between the tidal forces acting on the two lobes

$ T \approx \frac{G M_d}{d^3} R \cdot R \cdot \Delta M \approx \frac{G M_d^2 R^5}{d^6} $

This process will exert a torque on $ M_b $ until it rotates at the same angular velocity as that of the orbital motion of $ M_d $, which we denote by $ \Omega $. At point we say that the system is synchronised. The time for synchronisation is

$ \tau_s \approx \frac{M_b R^2 \Omega}{T} \approx \frac{M_b}{M_d} \frac{d^3}{R^3} \frac{\Omega d^3}{G M_d} $

Synchronous Case Edit

Let us consider a satellite with mass $ M $ and radius $ R $ that is moving around an object with mass $ m $. The semi major axis is $ a $ and a small eccentricity $ e $. We assume that the the satellite is synchronised, so that dissipation is only due to the change in distance between the objects, by a length of about $ l \approx e a $.

The tidal acceleration close to the surface of the satellite is $ \frac{G m}{d^3} R $. However, this acceleration is constant, and the satellite adjusts itself such that the pressure gradient will cancel this acceleration. The amplitude of the time changing acceleration is $ g \approx \frac{G m R}{d^3} \frac{l}{d} $. We assume that the dynamical time of the satellite $ T \approx \sqrt{R^3/GM} $ is shorter than the orbital period $ P \approx \sqrt{a^3/G m} $, otherwise the satellite will be tidally disrupted. Therefore, the induced velocity is

$ \delta v \approx g T $

The vibrational energy is

$ \delta U \approx M \delta v^2 $

Maximum power is attained when this energy is dissipated over a single orbital period

$ L \approx \frac{\delta U}{P} \approx \frac{G m^2}{R P} \left(\frac{R}{a}\right)^6 \left(\frac{l}{d}\right)^2 $

The dissipation power is more commonly expressed in the following way

$ L \approx \frac{n^5 R^5 e^2}{G} $

where $ n \approx 1/P $ is the mean motion.

When satellite first form or are captured, they will usually have a spin period that is different from their orbital period. Tidal forces will tend to synchronise the two periods. For close enough satellites, this synchronisation is attained within the lifetime of the system. At this point the satellite is said to be tidally locked. Later on, tidal dissipation continues at a slower rate, due to the eccentricity of the orbit.