In other words, in the inertial frame, the moon and the planet orbit in a fixed plane about their common center of mass at the angular velocity . It is convenient to transform to a non-inertial reference frame that rotates (with respect to the inertial frame), about an axis passing through , at the angular velocity

Suppose that the moon does not rotate (about an axis passing through its center of mass) in the non-inertial reference frame. This implies that, in the inertial
frame, the moon appears to rotate about an axis parallel to the
-axis, and passing through
, at the same angular velocity as
it orbits about
. This type of rotation is termed *synchronous*, and ensures that the same
hemisphere of the moon is always directed toward the planet. Such rotation is fairly common in the solar system. For instance,
the Moon rotates synchronously in such a manner that the same hemisphere is always visible from the Earth. Synchronous
rotation in the solar system is a consequence of process known as *tidal locking* (Murray and Dermott 1999).

Because a synchronously rotating moon is completely stationary in the aforementioned non-inertial frame, its internal pressure, , is governed by a force balance equation of the form [cf., Equation (2.83)]

where is the uniform internal mass density, the gravitational potential due to the moon, the gravitational potential due to the planet, and

the centrifugal potential due to the fact that the non-inertial frame is rotating (about an axis parallel to the -axis and passing through point ) at the angular velocity [cf., Equation (2.84)]. Suppose that the moon is much less massive that the planet (i.e., ). In this limit, the centrifugal potential (2.148) reduces to

where use has been made of Equation (2.146).

Suppose that the planet is spherical. It follows that the potential is the same as that which would be generated by a point mass located at . In other words,

where we have expanded up to second order in , et cetera.

The previous two equations can be combined to give

where

(2.152) |

and any constant terms have been neglected. Thus, the net force field experienced by the moon due to the combined action of the fictitious centrifugal force and the gravitational force field of the planet is

(2.153) |

The previous type of force field is known as a

Suppose that the bounding surface of the moon is the ellipsoid

(2.154) |

where . It follows, from Appendix D, that the gravitational potential of the moon at an interior point can be written

(2.155) |

where the integrals , for , are defined in Equations (D.30) and (D.31). Hence, from Equations (2.147) and (2.151), the pressure distribution within the moon is given by

(2.156) |

where is the central pressure. The pressure must be zero on the moon's bounding surface, otherwise this surface would not be in equilibrium. Thus, in order to achieve equilibrium, we require

(2.157) |

whenever

(2.158) |

The previous two equations can only be simultaneously satisfied if

Let and , where . It is also helpful to define . With the help of some of the analysis presented in the previous section, the integrals , for , can be shown to take the form

(2.160) | ||

(2.161) | ||

(2.162) |

where the incomplete elliptic integrals and are defined in Equations (2.141) and (2.142), respectively. Thus, Equation (2.159) yields

subject to the constraint

where

(2.165) |

and is the mean radius of the moon. The dimensionless parameter measures the strength of the tidal distortion field, generated by the planet, that acts on the moon. There is an analogous parameter,

(2.166) |

where is the mean radius of the planet, which measures the tidal distortion field, generated by the moon, that acts on the planet. We previously assumed that the former distortion field is much stronger than the latter, allowing us to neglect the tidal distortion of the planet altogether, and so to treat it as a sphere. This assumption is only justified if , which implies that

(2.167) |

where and are the mean densities of the moon and the planet, respectively. Assuming that these densities are similar, the previous condition reduces to , or, equivalently, . In other words, neglecting the tidal distortion of the planet, while retaining that of the moon, is generally only reasonable when the mass of the moon is much less than that of the planet, as was previously assumed to be the case.

Equations (2.163) and (2.164), which describe the ellipsoidal equilibria of
a synchronously rotating, relatively low mass, liquid moon due to the tidal force field of the planet about which it orbits,
were first obtained by Édouard Roche (1820-1883) in 1850. The properties of the so-called *Roche ellipsoids* are
set out in Table 2.3, and Figures 2.8 and 2.9.

It can be seen, from Table 2.3 and Figure 2.8, that the eccentricity of a Roche ellipsoid in the - plane is almost equal to its eccentricity in the - plane. In other words, Roche ellipsoids are almost spheroidal in shape, being elongated along the -axis (i.e., the axis joining the centers of the moon and the planet), and compressed by almost equal amounts along the - and -axes. In the limit , in which the tidal distortion field due to the planet is weak, it is easily shown that

(2.168) |

For the case of the tidal distortion field generated by the Earth, and acting on the Moon, which is characterized by and , we obtain (Yoder 1995). It follows that , and . In other words, were the Moon a homogeneous liquid body, the elongation generated by the tidal field of the Earth would be about .

It can be seen, from Table 2.3 and Figure 2.9, that the parameter attains a maximum value as the eccentricity of a Roche ellipsoid varies from 0 to 1. In fact, this maximum value, , occurs when and . It follows that there is a maximum strength of the tidal distortion field, generated by a planet, that is consistent with an ellipsoidal equilibrium of a synchronously rotating, homogeneous, liquid moon in a circular orbit about the planet. It is plausible that if this maximum strength is exceeded then the moon is tidally disrupted by the planet. The equilibrium condition is equivalent to

(2.169) |

where and are the mean densities of the moon and the planet, respectively. According to the previous expression, there is a minimum orbital radius of a moon circling a planet. Below this radius, which is called the