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where R is the primary's Radius , ho_M is the primary's Density and ho_m is the satellite's density.

For a fluid satellite, tidal forces cause the satellite to elongate, further compounding the tidal forces and causing it to break apart more readily. The calculation is complex and cannot be solved exactly, but a close approximation is the following:

: d \approx 2.423R\left( rac { ho_M} { ho_m} ight)^{ rac{1}{3}}

which indicates that a fluid satellite will disintegrate at almost twice the distance of a rigid sphere of similar density.

Most real satellites are somewhere between these two extremes, with internal Friction , Viscosity , and chemical bonds rendering the satellite neither perfectly rigid nor perfectly fluid.


Rigid satellites

As stated above, the formula for calculating the Roche limit, d, for a rigid Spherical satellite orbiting a spherical primary is:

: d = R\left( 2\; rac { ho_M} { ho_m} ight)^{ rac{1}{3}}

where R is the Radius of the primary, ho_M is the Density of the primary, and ho_m is the density of the satellite. As described below, this rigid-body approximation does not take into account the deformation of the satellite's spherical shape due to tidal effects and is only an approximation of what a real satellite's Roche limit would be.

Notice that if the satellite is more than twice as dense as the primary (as can easily be the case for a rocky moon orbiting a gas giant) then the Roche limit will be inside the primary and hence not relevant.


Derivation of the formula

In order to determine the Roche limit, we consider a small mass u on the surface of the satellite closest to the primary. There are two forces on this mass u: the gravitational pull towards the satellite and the gravitational pull towards the primary. Since the satellite is already in orbital Free Fall around the primary, the Tidal Force is the only relevant term of the gravitational attraction of the primary.

The gravitational pull F_G on the mass u towards the satellite with mass m and radius r can be expressed according to Newton's Law Of Gravitation .

: F_G = rac{Gmu}{r^2}

The Tidal Force F_T on the mass u towards the primary with radius R and a distance d between the centers of the two bodies can be expressed as:

: F_T = rac{2GMur}{d^3}

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

: F_G = F_T

or

: rac{Gmu}{r^2} = rac{2GMur}{d^3}

Which quickly gives the Roche limit, d, as:

: d = r \left( 2 M / m ight)^{ rac{1}{3}}

However, we don't really want the radius of the satellite to appear in the expression for the limit, so we re-write this in terms of densities.

For a sphere the mass M can be written as:

: M = rac{4\pi ho_M R^3}{3} where R is the radius of the primary.

And likewise:

: m = rac{4\pi ho_m r^3}{3} where r is the radius of the satellite.

Substituting for the masses in the equation for the Roche limit, and cancelling out 4\pi/3 gives:

: d = r \left( rac{ 2 ho_M R^3 }{ ho_m r^3 } ight)^{1/3}

which can be simplified to the Roche limit:

: d = R\left( 2\; rac { ho_M} { ho_m} ight)^{ rac{1}{3}}


Fluid satellites

A more correct 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 the satellite. In this case, the satellite is deformed into a Prolate Spheroid .

The calculation is complex and cannot be solved exactly. Historically, Roche himself derived the following numerical solution for the Roche Limit:

: d \approx 2.44R\left( rac { ho_M} { ho_m} ight)^{1/3}

However, with the aid of a computer a better numerical solution is:

: d \approx 2.423 R\left( rac { ho_M} { ho_m} ight)^{1/3} \left( rac{(1+ rac{m}{3M})+ rac{c}{3R}(1+ rac{m}{M})}{1-c/R} ight)^{1/3}

where c/R is the Oblateness of the primary.



ROCHE LIMITS FOR SELECTED EXAMPLES


The table below shows the mean density and the equatorial radius for selected objects in our Solar System .

Using these data, the Roche Limits for rigid and fluid satellites can easily be calculated. The average density of Comet s is around 500 kg/m3.

The table below gives the Roche limits expressed in metres and in primary radii. The true Roche Limit for a satellite depends on its flexibility, and will be somewhere between the rigid and fluid Roche Limits given below.

If the primary is less than half as dense as the satellite, the rigid-body Roche Limit is less than the primary's radius, and the two bodies may collide before the Roche limit is reached. For example, the Sun-Earth Roche Limit indicates that the Earth would collide with the Sun before disintegrating due to tidal forces.

How close are the solar system's moons to their Roche limits? The table below gives each inner satellite's orbital radius divided by its own Roche radius, for both the rigid and fluid cases. (Note Naiad in particular, which may in fact be quite close to its actual Roche limit.)


SEE ALSO



REFERENCES


  • Édouard Roche: ''La figure d'une masse fluide soumise à l'attraction d'un point éloigné'', Acad. des sciences de Montpellier, Vol. 1 (1847-50) p. 243



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