Tidal torques

Defining a spherical coordinate system, , , , whose origin is the center of the Earth, and which is orientated such that the Earth-Moon axis always corresponds to (see Figure 6.7), we find that the Earth's external gravitational potential is [cf., Equation (3.65)]

where is the ellipticity induced by the tidal field of the Moon. Note that the second term on the right-hand side of this expression is the contribution of the Earth's tidal bulge, which attains its maximum amplitude at , rather than , because of the aforementioned misalignment between the bulge and the Earth-Moon axis. Equations (6.76), (6.79), and (6.88) can be combined to giveFrom Equation (3.7), the torque about the Earth's center that the terrestrial gravitational field exerts on the Moon is

where use has been made of Equation (6.89), as well as the fact that is a small angle. There is zero torque in the absence of a misalignment between the Earth's tidal bulge and the Earth-Moon axis. The torque acts to increase the Moon's orbital angular momentum. By conservation of angular momentum, an equal and opposite torque, , is applied to the Earth; it acts to decrease the terrestrial rotational angular momentum. Incidentally, if the Moon were sufficiently close to the Earth that its orbital angular velocity exceeded the Earth's rotational angular velocity (i.e., if ) then the phase lag between the Earth's tidal elongation and the Moon's tidal field would cause the tidal bulge to fall slightly behind the Earth-Moon axis (i.e., ). In this case, the gravitational torque would act to reduce the Moon's orbital angular momentum, and to increase the Earth's rotational angular momentum.The Earth's rotational equation of motion is

(6.91) |

(6.92) |

(6.94) |

Up to now, we have concentrated on the effect of the tidal torque on the rotation of the Earth. Let us now examine its effect on the orbit of the Moon. The total angular momentum of the Earth-Moon system is

(6.95) |

(6.96) |

(6.97) |

The net rate at which the tidal torques acting on the Moon and the Earth do work is

(6.98) |

Of course, we would expect spatial gradients in the gravitational field of the Earth to generate a tidal bulge in the Moon. We would also expect dissipative effects to produce a phase lag between this bulge and the Earth. This would allow the Earth to exert a gravitational torque that acts to drive the Moon toward a synchronous state in which its rotational angular velocity matches its orbital angular velocity. By analogy with the previous analysis, the de-spinning rate of the Moon is estimated to be

(6.99) |

(6.100) |

(6.101) |