Roche Ellipsoids

Consider a homogeneous liquid moon of mass $M$ that is in a circular orbit of radius $R$ about a planet of mass $M'$ . Let $C$ , $C'$ , and $C''$ be the center of the moon, the center of the planet, and the center of mass of the moon-planet system, respectively. As is easily demonstrated, all three points lie on the same straight-line, and the distances between them take the constant values $CC'=R$ and $CC'' = [M'/(M+M')]\,R$ (Fitzpatrick 2012). Moreover, according to standard Newtonian dynamics, there exists an inertial frame of reference in which $C''$ is stationary, and the line $CC'$ rotates at the fixed angular velocity $\omega$ , where (Fitzpatrick 2012)

$$\omega^{\,2} = \frac{G\,(M+M')}{R^{\,3}}.$$ (2.146)

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 $\omega$ . It is convenient to transform to a non-inertial reference frame that rotates (with respect to the inertial frame), about an axis passing through $C''$ , at the angular velocity $\omega$ . It follows that points $C$ , $C'$ , and $C''$ appear stationary in this frame. It is also convenient to adopt the standard right-handed Cartesian coordinates, $x_1$ , $x_2$ , $x_3$ , and to choose the coordinate axes such that $\omega$ $=\omega\,{\bf e}_3$ , $C=(0,\,0,\,0)$ , $C'=(R,\,0,\,0)$ , and $C''=([M'/(M+M')]\,R,\,0,\,0)$ . Thus, in the non-inertial reference frame, the orbital rotation axis runs parallel to the $x_3$ -axis, and the centers of the moon and the planet both lie on the $x_1$ -axis.

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 $x_3$ -axis, and passing through $C$ , at the same angular velocity as it orbits about $C''$ . 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, $p$ , is governed by a force balance equation of the form [cf., Equation (2.83)]

$${\bf0 } = \nabla p + \rho\,\nabla({\mit\Psi}+{\mit\Psi}'+{\mit\Psi}'') ,$$ (2.147)

where $\rho$ is the uniform internal mass density, ${\mit\Psi}$ the gravitational potential due to the moon, ${\mit\Psi}'$ the gravitational potential due to the planet, and

$${\mit\Psi}'' = -\frac{1}{2}\,\omega^2\left[\left(x_1-\frac{M'}{M+M'}\,R\right)^2+x_2^{\,2}\right]$$ (2.148)

the centrifugal potential due to the fact that the non-inertial frame is rotating (about an axis parallel to the $x_3$ -axis and passing through point $C''$ ) at the angular velocity $\omega$ [cf., Equation (2.84)]. Suppose that the moon is much less massive that the planet (i.e., $M/M'\ll 1$ ). In this limit, the centrifugal potential (2.148) reduces to

$${\mit\Psi}'' \simeq -\left[\frac{1}{2}-\frac{x_1}{R}+\frac{(1/2)\,x_1^{\,2}+(1/2)\,x_2^{\,2}}{R^{\,2}}\right],$$ (2.149)

where use has been made of Equation (2.146).

Suppose that the planet is spherical. It follows that the potential ${\mit\Psi}'$ is the same as that which would be generated by a point mass $M'$ located at $C'$ . In other words,

$${\mit\Psi}' =-\frac{G\,M'}{R}\left(1-2\,\frac{x_1}{R} + \frac{x_1^{\,2}+x_2^{\,2}+x_3^{\,2}}{R^{\,2}}\right)^{-1/2}$$

$$\simeq -\frac{G\,M'}{R}\left[1+\frac{x_1}{R}+ \frac{x_1^{\,2}-(1/2)\,x_2^{\,2}-(1/2)\,x_3^{\,2}}{R^{\,2}}+\cdots\right],$$ (2.150)

where we have expanded up to second order in $x_1/R$ , et cetera.

The previous two equations can be combined to give

$${\mit\Psi}'+{\mit\Psi}'' \simeq -\lambda \left(\frac{3}{2}\,x_1^{\,2}-\frac{1}{2}\,x_3^{\,2}\right),$$ (2.151)

where

$$\lambda = \frac{G\,M'}{R^{\,3}},$$ (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

$$-\rho\,\nabla({\mit\Psi}'+{\mit\Psi}'') = \rho\,\lambda\,(3\,x_1,\,0,\,-x_3).$$ (2.153)

The previous type of force field is known as a tidal force field, and clearly acts to elongate the moon along the axis joining the centers of the moon and planet (i.e., the $x_1$ -axis), and to compress it along the orbital rotation axis (i.e., the $x_3$ -axis). Moreover, the magnitude of the tidal force increases linearly with distance from the center of the moon. The tidal force field is a consequence of the different spatial variation of the centrifugal force and the planet's gravitational force of attraction. This different variation causes these two forces, which balance one another at the center of the moon, to not balance away from the center (Fitzpatrick 2012). As a result of the tidal force field, we expect the shape of the moon to be distorted from a sphere. Of course, the moon also generates a tidal force field that acts to distort the shape of the planet. However, we are assuming that the tidal distortion of the planet is much smaller than that of the moon (which justifies our earlier statement that the planet is essentially spherical). As will be demonstrated later, this assumption is reasonable provided the mass of the moon is much less than that of the planet (assuming that the planet and moon have similar densities).

Suppose that the bounding surface of the moon is the ellipsoid

$$\frac{x_1^{\,2}}{a_1^{\,2}}+ \frac{x_2^{\,2}}{a_2^{\,2}}+\frac{x_3^{\,2}}{a_3^{\,2}}=1,$$ (2.154)

where $a_1\geq a_2\geq a_3$ . It follows, from Appendix D, that the gravitational potential of the moon at an interior point can be written

$${\mit\Psi} = - \frac{3}{4}\,G\,M\left(\alpha_0-\sum_{i=1,3}\alpha_i\,x_i^{\,2}\right),$$ (2.155)

where the integrals $\alpha_i$ , for $i=0,3$ , 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

$$p = p_0 -\frac{1}{2}\,\rho\left[\left(\frac{3}{2}\,G\,M\,\alpha_1... ..._2\,x_2^{\,2}+\left(\frac{3}{2}\,G\,M\,\alpha_3+\lambda\right)x_3^{\,2}\right],$$ (2.156)

where $p_0$ 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

$$\frac{1}{2}\,\rho\left[\left(\frac{3}{2}\,G\,M\,\alpha_1-3\,\lamb... ...2^{\,2}+\left(\frac{3}{2}\,G\,M\,\alpha_3+\lambda\right)x_3^{\,2}\right] = p_0,$$ (2.157)

whenever

$$\frac{x_1^{\,2}}{a_1^{\,2}}+ \frac{x_2^{\,2}}{a_2^{\,2}}+\frac{x_3^{\,2}}{a_3^{\,2}}=1.$$ (2.158)

The previous two equations can only be simultaneously satisfied if

$$\left[\alpha_1-\frac{3\,\lambda}{(3/2)\,G\,M}\right]a_1^{\,2} =\alpha_2\,a_2^{\,2} = \left[\alpha_3+\frac{\lambda}{(3/2)\,G\,M}\right] a_3^{\,2}.$$ (2.159)

Figure 2.8: Properties of the Roche ellipsoids.

Let $a_2=a_1\,\cos\beta$ and $a_3=a_1\,\cos\gamma$ , where $\gamma\geq \beta$ . It is also helpful to define $\alpha=\sin^{-1}(\sin\beta/\sin\gamma)$ . With the help of some of the analysis presented in the previous section, the integrals $\alpha_i$ , for $i=1,3$ , can be shown to take the form

$$\alpha_1 =\frac{2}{a_1^{\,3}\,\sin^3\gamma}\,\frac{F(\gamma,\alpha)-E(\gamma,\alpha)}{\sin^2\alpha},$$ (2.160)

$$\alpha_2 =\frac{2}{a_1^{\,2}\,\sin^3\gamma}\left[\frac{E(\gamma,\alpha)}{\... ...{\sin^2\alpha} - \frac{\cos\gamma\,\sin\gamma}{\cos^2\alpha\,\cos\beta}\right],$$ (2.161)

$$\alpha_3 = \frac{2}{a_1^{\,2}\,\sin^3\gamma}\left[-\frac{E(\gamma,\alpha)}{\cos^2\alpha} + \frac{\cos\beta\,\sin\gamma}{\cos^2\alpha\,\cos\gamma}\right],$$ (2.162)

where the incomplete elliptic integrals $E(\gamma,\alpha)$ and $F(\gamma,\alpha)$ are defined in Equations (2.141) and (2.142), respectively. Thus, Equation (2.159) yields

$$\mu = \frac{1}{\sin\beta\,\tan\beta\,\tan\gamma}\left[F(\gamma,\alpha... ...^2\beta)-E(\gamma,\alpha)\left(1+\frac{\cos^2\beta}{\cos^2\alpha}\right)\right.$$

$$\phantom{=}\left.+ \frac{\sin\alpha\,\sin\beta\,\cos\beta\,\cos\gamma}{\cos^2\alpha}\right],$$ (2.163)

subject to the constraint

$$=\cos^2\gamma\left[F(\gamma,\alpha)-2\,E(\gamma,\alpha) + E(\gamm... ...pha} - \frac{\sin\alpha\,\sin\beta\,\cos\beta\,\cos\gamma}{\cos^2\alpha}\right]$$ 0

$$\phantom{=}+ (3+\cos^2\gamma)\left[E(\gamma,\alpha)+[F(\gamma,\alpha)\,\cos^2\alpha-2\,E(\gamma,\alpha)]\,\frac{\cos^2\beta}{\cos^2\alpha}\right.$$

$$\phantom{=}\left. +\frac{2\,\sin\alpha\,\sin\beta\,\cos\beta\,\cos\gamma}{\cos^2\alpha}\right],$$ (2.164)

where

$$\mu = \frac{M'}{M} \,\frac{a_0^{\,3}}{R^{\,3}},$$ (2.165)

and $a_0=(a_1\,a_2\,a_3)^{1/3}$ is the mean radius of the moon. The dimensionless parameter $\mu$ measures the strength of the tidal distortion field, generated by the planet, that acts on the moon. There is an analogous parameter,

$$\mu'= \frac{M}{M'}\,\frac{{a_0'}^{\,3}}{R^{\,3}},$$ (2.166)

where $a_0'$ 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 $\mu\gg \mu'$ , which implies that

$$\frac{M}{M'} \ll \frac{\rho'}{\rho},$$ (2.167)

where $\rho=M/[(4/3)\,\pi\,a_0^{\,3}]$ and $\rho'=M'/[(4/3)\,\pi\,{a_0'}^{\,3}]$ are the mean densities of the moon and the planet, respectively. Assuming that these densities are similar, the previous condition reduces to $M\ll M'$ , or, equivalently, $a_0<a_0'$ . 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.

Table 2.3: Properties of the Roche ellipsoids.

$e_{12}$	$e_{13}$	$\mu$	$e_{12}$	$e_{13}$	$\mu$
0.00	0.00000	0.00000	0.52	0.56740	0.33440
0.04	0.04613	0.00213	0.56	0.60632	0.38204
0.08	0.09223	0.00852	0.60	0.64445	0.43094
0.12	0.13809	0.01913	0.64	0.68182	0.48027
0.16	0.18364	0.03392	0.68	0.71848	0.52890
0.20	0.22879	0.05282	0.72	0.75446	0.57532
0.24	0.27346	0.07573	0.76	0.78984	0.61729
0.28	0.31756	0.10253	0.80	0.82472	0.65150
0.32	0.36104	0.13308	0.84	0.85923	0.67265
0.36	0.40383	0.16721	0.88	0.89353	0.67151
0.40	0.44588	0.20470	0.92	0.92793	0.62978
0.44	0.48718	0.24528	0.96	0.96294	0.50135
0.48	0.52769	0.28865	1.00	1.00000	0.00000

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.

Figure 2.9: Properties of the Roche ellipsoids.

It can be seen, from Table 2.3 and Figure 2.8, that the eccentricity $e_{12}=\sin\beta$ of a Roche ellipsoid in the $x_1$ -$x_2$ plane is almost equal to its eccentricity $e_{13}=\sin\gamma$ in the $x_1$ -$x_3$ plane. In other words, Roche ellipsoids are almost spheroidal in shape, being elongated along the $x_1$ -axis (i.e., the axis joining the centers of the moon and the planet), and compressed by almost equal amounts along the $x_2$ - and $x_3$ -axes. In the limit $\mu\ll1$ , in which the tidal distortion field due to the planet is weak, it is easily shown that

$$e_{12}^{\,2}\simeq e_{13}^{\,2}\simeq \frac{15}{2}\,\mu.$$ (2.168)

For the case of the tidal distortion field generated by the Earth, and acting on the Moon, which is characterized by $M/M'=0.01230$ and $R/a_0=221.29$ , we obtain $\mu=7.50\times 10^{-6}$ (Yoder 1995). It follows that $e_{13}=7.50\times 10^{-3}$ , and $(a_1-a_3)/a_1 \simeq e_{13}^{\,2}/2 = 2.81\times 10^{-5}$ . In other words, were the Moon a homogeneous liquid body, the elongation generated by the tidal field of the Earth would be about $50\,{\rm m}$ .

It can be seen, from Table 2.3 and Figure 2.9, that the parameter $\mu$ attains a maximum value as the eccentricity of a Roche ellipsoid varies from 0 to 1. In fact, this maximum value, $\mu=0.06757$ , occurs when $e_{12}=0.8594$ and $e_{13}= 0.8759$ . 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 $\mu<0.06757$ is equivalent to

$$\frac{R}{a_0'} > 2.455\left(\frac{\rho'}{\rho}\right)^{1/3},$$ (2.169)

where $\rho=M/[(4/3)\,\pi\,a_0^{\,3}]$ and $\rho'=M'/[(4/3)\,\pi\,{a_0'}^{\,3}]$ 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 Roche radius, the moon is presumably torn apart by tidal effects. The Roche radius for a synchronously rotating, self-gravitating, liquid moon in a circular orbit about a spherical planet is about $2.5$ times the planet's radius (assuming that the moon and the planet have approximately the same mass density). Of course, relatively small objects, such as artificial satellites, which are held together by internal tensile strength, rather than gravity, can orbit inside the Roche radius without being disrupted.