Roche limit

The rigid-body Roche limit is a simplified calculation for a spherical satellite. Irregular shapes such as those of tidal deformation on the body or the primary it orbits are neglected. It is assumed to be in hydrostatic equilibrium. These assumptions, although unrealistic, greatly simplify calculations.

The Roche limit for a rigid spherical satellite is the distance, ${\displaystyle d}$, from the primary at which the gravitational force on a test mass at the surface of the object is exactly equal to the tidal force pulling the mass away from the object:[3][4]

${\displaystyle d=R_{M}\left(2{\frac {\rho _{M}}{\rho _{m}}}\right)^{\frac {1}{3}}}$

where ${\displaystyle R_{M}}$ is the radius of the primary, ${\displaystyle \rho _{M}}$ is the density of the primary, and ${\displaystyle \rho _{m}}$ is the density of the satellite. This can be equivalently written as

${\displaystyle d=R_{m}\left(2{\frac {M_{M}}{M_{m}}}\right)^{\frac {1}{3}}}$

where ${\displaystyle R_{m}}$ is the radius of the secondary, ${\displaystyle M_{M}}$ is the mass of the primary, and ${\displaystyle M_{m}}$ is the mass of the secondary.

This does not depend on the size of the objects, but on the ratio of densities. This is the orbital distance inside of which loose material (e.g. regolith) on the surface of the satellite closest to the primary would be pulled away, and likewise material on the side opposite the primary will also go away from, rather than toward, the satellite.

Note that this is an approximate result as inertia force and rigid structure are ignored in its derivation.

The orbital period then depends only on the density of the secondary:

${\displaystyle P=2\pi \left({\frac {d^{3}}{GM_{M}}}\right)^{1/2}=2\pi \left({\frac {d^{3}}{(4/3)\pi GR_{M}^{3}\rho _{M}}}\right)^{1/2}={\sqrt {\frac {6\pi }{G\rho _{m}}}}}$

where G is the gravitational constant. For example, a density of 3.346 g/cc (the density of our moon) corresponds to an orbital period of 2.552 hours.

Derivation of the Roche limit

In order to determine the Roche limit, consider a small mass ${\displaystyle u}$ on the surface of the satellite closest to the primary. There are two forces on this mass ${\displaystyle u}$: the gravitational pull towards the satellite and the gravitational pull towards the primary. Assume that the satellite is in free fall around the primary and that the tidal force is the only relevant term of the gravitational attraction of the primary. This assumption is a simplification as free-fall only truly applies to the planetary center, but will suffice for this derivation.[5]

The gravitational pull ${\displaystyle F_{\text{G}}}$ on the mass ${\displaystyle u}$ towards the satellite with mass ${\displaystyle m}$ and radius ${\displaystyle r}$ can be expressed according to Newton's law of gravitation.

${\displaystyle F_{\text{G}}={\frac {Gmu}{r^{2}}}}$

the tidal force ${\displaystyle F_{\text{T}}}$ on the mass ${\displaystyle u}$ towards the primary with radius ${\displaystyle R}$ and mass ${\displaystyle M}$, at a distance ${\displaystyle d}$ between the centers of the two bodies, can be expressed approximately as

${\displaystyle F_{\text{T}}={\frac {2GMur}{d^{3}}}}$.

To obtain this approximation, find the difference in the primary's gravitational pull on the center of the satellite and on the edge of the satellite closest to the primary:

${\displaystyle F_{\text{T}}={\frac {GMu}{(d-r)^{2}}}-{\frac {GMu}{d^{2}}}}$

${\displaystyle F_{\text{T}}=GMu{\frac {d^{2}-(d-r)^{2}}{d^{2}(d-r)^{2}}}}$

${\displaystyle F_{\text{T}}=GMu{\frac {2dr-r^{2}}{d^{4}-2d^{3}r+r^{2}d^{2}}}}$

In the approximation where ${\displaystyle r\ll R}$ and ${\displaystyle R<d}$, it can be said that the ${\displaystyle r^{2}}$ in the numerator and every term with ${\displaystyle r}$ in the denominator goes to zero, which gives us:

${\displaystyle F_{\text{T}}=GMu{\frac {2dr}{d^{4}}}}$

${\displaystyle F_{\text{T}}={\frac {2GMur}{d^{3}}}}$

The Roche limit is reached when the gravitational force and the tidal force balance each other out.

${\displaystyle F_{\text{G}}=F_{\text{T}}\;}$

or

${\displaystyle {\frac {Gmu}{r^{2}}}={\frac {2GMur}{d^{3}}}}$,

which gives the Roche limit, ${\displaystyle d}$, as

${\displaystyle d=r\left(2\,{\frac {M}{m}}\right)^{\frac {1}{3}}}$

The radius of the satellite should not appear in the expression for the limit, so it is re-written in terms of densities.

For a sphere the mass ${\displaystyle M}$ can be written as

${\displaystyle M={\frac {4\pi \rho _{M}R^{3}}{3}}}$ where ${\displaystyle R}$ is the radius of the primary.

And likewise

${\displaystyle m={\frac {4\pi \rho _{m}r^{3}}{3}}}$ where ${\displaystyle r}$ is the radius of the satellite.

Substituting for the masses in the equation for the Roche limit, and cancelling out ${\displaystyle 4\pi /3}$ gives

${\displaystyle d=r\left({\frac {2\rho _{M}R^{3}}{\rho _{m}r^{3}}}\right)^{1/3}}$,

which can be simplified to the following Roche limit:

${\displaystyle d=R\left(2\,{\frac {\rho _{M}}{\rho _{m}}}\right)^{\frac {1}{3}}\approx 1.26R\left({\frac {\rho _{M}}{\rho _{m}}}\right)^{\frac {1}{3}}}$.

Comparison of the Hill spheres and Roche limits of the Sun-Earth-Moon system (not to scale) with shaded regions denoting stable orbits of satellites of each body

Consider a planet with a density of ${\displaystyle \rho _{m}}$ and a radius of ${\displaystyle r}$, orbiting a star withis is the physical meaning ? of Roche limit, Roche lobe and Hill sphere.

Formula(2) (link?) can be described as: ${\displaystyle R_{\text{Roche}}=R_{\text{Hill}}{\sqrt[{3}]{\frac {3M}{m}}}=R_{\text{secondary}}{\sqrt[{3}]{\frac {3M}{m}}}}$, a perfect mathematical symmetry.
This is the astronomical significance of Roche limit and Hill sphere.

Note : Roche limit and Hill sphere are completely different from each other but are both work of Édouard Roche.

Hill sphere of an astronomical body is the region in which it dominates the attraction of satellites whereas Roche limit is the minimum distance to which a satellite can approach its primary body without tidal force overcoming the internal gravity holding the satellite together.

A more accurate approach for calculating the Roche limit takes the deformation of the satellite into account. An extreme example would be a tidally locked liquid satellite orbiting a planet, where any force acting upon the satellite would deform it into a prolate spheroid.

The calculation is complex and its result cannot be represented in an exact algebraic formula. Roche himself derived the following approximate solution for the Roche limit:

${\displaystyle d\approx 2.44R\left({\frac {\rho _{M}}{\rho _{m}}}\right)^{1/3}}$

However, a better approximation that takes into account the primary's oblateness and the satellite's mass is:

${\displaystyle d\approx 2.423R\left({\frac {\rho _{M}}{\rho _{m}}}\right)^{1/3}\left({\frac {(1+{\frac {m}{3M}})+{\frac {c}{3R}}(1+{\frac {m}{M}})}{1-c/R}}\right)^{1/3}}$

where ${\displaystyle c/R}$ is the oblateness of the primary. The numerical factor is calculated with the aid of a computer.

The fluid solution is appropriate for bodies that are only loosely held together, such as a comet. For instance, comet Shoemaker–Levy 9's decaying orbit around Jupiter passed within its Roche limit in July 1992, causing it to fragment into a number of smaller pieces. On its next approach in 1994 the fragments crashed into the planet. Shoemaker–Levy 9 was first observed in 1993, but its orbit indicated that it had been captured by Jupiter a few decades prior.[6]

As the fluid satellite case is more delicate than the rigid one, the satellite is described with some simplifying assumptions. First, assume the object consists of incompressible fluid that has constant density ${\displaystyle \rho _{m}}$ and volume ${\displaystyle V}$ that do not depend on external or internal forces.

Second, assume the satellite moves in a circular orbit and it remains in synchronous rotation. This means that the angular speed ${\displaystyle \omega }$ at which it rotates around its center of mass is the same as the angular speed at which it moves around the overall system barycenter.

The angular speed ${\displaystyle \omega }$ is given by Kepler's third law:

${\displaystyle \omega ^{2}=G\,{\frac {M+m}{d^{3}}}.}$

When M is very much bigger than m, this will be close to

${\displaystyle \omega ^{2}=G\,{\frac {M}{d^{3}}}.}$

The synchronous rotation implies that the liquid does not move and the problem can be regarded as a static one. Therefore, the viscosity and friction of the liquid in this model do not play a role, since these quantities would play a role only for a moving fluid.

Given these assumptions, the following forces should be taken into account:

• The force of gravitation due to the main body;
• the centrifugal force in the rotary reference system; and
• the self-gravitation field of the satellite.

Since all of these forces are conservative, they can be expressed by means of a potential. Moreover, the surface of the satellite is an equipotential one. Otherwise, the differences of potential would give rise to forces and movement of some parts of the liquid at the surface, which contradicts the static model assumption. Given the distance from the main body, the form of the surface that satisfies the equipotential condition must be determined.

Radial distance of one point on the surface of the ellipsoid to the center of mass

As the orbit has been assumed circular, the total gravitational force and orbital centrifugal force acting on the main body cancel. That leaves two forces: the tidal force and the rotational centrifugal force. The tidal force depends on the position with respect to the center of mass, already considered in the rigid model. For small bodies, the distance of the liquid particles from the center of the body is small in relation to the distance d to the main body. Thus the tidal force can be linearized, resulting in the same formula for F_T as given above.

While this force in the rigid model depends only on the radius r of the satellite, in the fluid case, all the points on the surface must be considered, and the tidal force depends on the distance Δd from the center of mass to a given particle projected on the line joining the satellite and the main body. We call Δd the radial distance. Since the tidal force is linear in Δd, the related potential is proportional to the square of the variable and for ${\displaystyle m\ll M}$ we have

${\displaystyle V_{T}=-{\frac {3GM}{2d^{3}}}\Delta d^{2}\,}$

Likewise, the centrifugal force has a potential

${\displaystyle V_{C}=-{\frac {1}{2}}\omega ^{2}\Delta d^{2}=-{\frac {GM}{2d^{3}}}\Delta d^{2}\,}$

for rotational angular velocity ${\displaystyle \omega }$.

We want to determine the shape of the satellite for which the sum of the self-gravitation potential and V_T + V_C is constant on the surface of the body. In general, such a problem is very difficult to solve, but in this particular case, it can be solved by a skillful guess due to the square dependence of the tidal potential on the radial distance Δd To a first approximation, we can ignore the centrifugal potential V_C and consider only the tidal potential V_T.

Since the potential V_T changes only in one direction, i.e. the direction toward the main body, the satellite can be expected to take an axially symmetric form. More precisely, we may assume that it takes a form of a solid of revolution. The self-potential on the surface of such a solid of revolution can only depend on the radial distance to the center of mass. Indeed, the intersection of the satellite and a plane perpendicular to the line joining the bodies is a disc whose boundary by our assumptions is a circle of constant potential. Should the difference between the self-gravitation potential and V_T be constant, both potentials must depend in the same way on Δd. In other words, the self-potential has to be proportional to the square of Δd. Then it can be shown that the equipotential solution is an ellipsoid of revolution. Given a constant density and volume the self-potential of such body depends only on the eccentricity ε of the ellipsoid:

${\displaystyle V_{s}=V_{s_{0}}+G\pi \rho _{m}\cdot f(\varepsilon )\cdot \Delta d^{2},}$

where ${\displaystyle V_{s_{0}}}$ is the constant self-potential on the intersection of the circular edge of the body and the central symmetry plane given by the equation Δd=0.

The dimensionless function f is to be determined from the accurate solution for the potential of the ellipsoid

${\displaystyle f(\varepsilon )={\frac {1-\varepsilon ^{2}}{\varepsilon ^{3}}}\cdot \left[\left(3-\varepsilon ^{2}\right)\cdot \operatorname {artanh} \varepsilon -3\varepsilon \right]}$

and, surprisingly enough, does not depend on the volume of the satellite.

The graph of the dimensionless function f which indicates how the strength of the tidal potential depends on the eccentricity ε of the ellipsoid.

Although the explicit form of the function f looks complicated, it is clear that we may and do choose the value of ε so that the potential V_T is equal to V_S plus a constant independent of the variable Δd. By inspection, this occurs when

${\displaystyle {\frac {2G\pi \rho _{M}R^{3}}{d^{3}}}=G\pi \rho _{m}f(\varepsilon )}$

This equation can be solved numerically. The graph indicates that there are two solutions and thus the smaller one represents the stable equilibrium form (the ellipsoid with the smaller eccentricity). This solution determines the eccentricity of the tidal ellipsoid as a function of the distance to the main body. The derivative of the function f has a zero where the maximal eccentricity is attained. This corresponds to the Roche limit.

The derivative of f determines the maximal eccentricity. This gives the Roche limit.

More precisely, the Roche limit is determined by the fact that the function f, which can be regarded as a nonlinear measure of the force squeezing the ellipsoid towards a spherical shape, is bounded so that there is an eccentricity at which this contracting force becomes maximal. Since the tidal force increases when the satellite approaches the main body, it is clear that there is a critical distance at which the ellipsoid is torn up.

The maximal eccentricity can be calculated numerically as the zero of the derivative of f'. One obtains

${\displaystyle \varepsilon _{\text{max}}\approx 0{.}86}$

which corresponds to the ratio of the ellipsoid axes 1:1.95. Inserting this into the formula for the function f one can determine the minimal distance at which the ellipsoid exists. This is the Roche limit,

${\displaystyle d\approx 2{.}423\cdot R\cdot {\sqrt[{3}]{\frac {\rho _{M}}{\rho _{m}}}}\,.}$

Surprisingly, including the centrifugal potential makes remarkably little difference, though the object becomes a Roche ellipsoid, a general triaxial ellipsoid with all axes having different lengths. The potential becomes a much more complicated function of the axis lengths, requiring elliptic functions. However, the solution proceeds much as in the tidal-only case, and we find

${\displaystyle d\approx 2{.}455\cdot R\cdot {\sqrt[{3}]{\frac {\rho _{M}}{\rho _{m}}}}\,.}$

The ratios of polar to orbit-direction to primary-direction axes are 1:1.06:2.07.