Roche limit

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View from above of the orbital plane of the satellite: Simulation of a liquid satellite that is only held together by its own gravity. Far from the Roche limit, the mass of the satellite (red / blue) practically forms a sphere.

Closer to the Roche limit, the tidal forces deform the satellite into an ellipsoid.

Within the Roche limit, the body can no longer withstand the tidal forces and dissolves.

Particles that are closer to the main body move faster than those that are further from the main body (see the red arrows of different lengths).

After a while, this differential rotation creates a ring.

The Roche limit [ ʀɔʃ- ] is a criterion for assessing the internal stability, that is, the cohesion of a celestial body orbiting another. The gravitational forces that hold the celestial body together internally are compared with the tidal forces that pull it apart. The Roche border is named after Édouard Albert Roche , who discovered it in 1850.

The cause of the tidal forces is the fact that the force of attraction by the partner is greater on the side of the celestial body facing him than on the opposite. This leads to internal stresses or deformations that can lead to the dissolution of the celestial body.

The term Roche limit of a celestial body is used in two different meanings:

• As a limit for its orbit ( Roche limit ): If the celestial body moves outside this orbit, the stabilizing internal gravitational forces dominate the tidal forces. This meaning is especially used when considering the stability of a moon orbiting a planet .
• As a limit for its geometric shape (English Roche lobe ): If the celestial body is within this shape, it is stable. This meaning is especially used when two stars orbit each other and deform in the process.

Roche limit as the limit for the orbit

The Roche limit of a celestial body orbiting a main body is the distance at which the body will be torn apart due to the tidal forces acting on it. It is assumed that the body is only held together by its own gravitational forces and that its mechanical strength is negligible. For real solids, this assumption is the better the larger the body is. Therefore, artificial satellites can easily orbit within the Roche boundary, while large objects such as moons and planets cannot exist there. If there is material that has not yet aggregated into a single body in an orbit around the main body, then this material will distribute itself in a ring around the orbit within the Roche boundary , while outside the boundary it will form a lump.

In fact, all known planetary rings are within the Roche boundary of their planets. They could therefore either have formed directly from the protoplanetary accretion disk , as tidal forces prevented moons from forming from this material, or they could be fragments of destroyed moons that moved from outside across the Roche boundary. All larger moons of the solar system , however, are located far outside the Roche limit, but smaller moons are also able to be within the Roche limit. The orbits of Jupiter's moon Metis and Saturn's moon Pan are within the Roche limit for so-called liquid bodies . The mechanical strength of these bodies acts on the one hand directly against the tidal forces that act on the body, and on the other hand, the strength also ensures that these bodies remain rigid, i.e. H. do not change their shape - an effect which is described below and which additionally intensifies the tidal forces. This effect is particularly vividly described by the fact that an object that would be "placed" on the surface of such a moon would not stay on the moon, but would be pulled away from the surface by the tidal forces. A body with less mechanical strength, such as a comet, would be destroyed in these regions, as was shown by the example of the comet Shoemaker-Levy 9 , whose orbit penetrated Jupiter's Roche boundary in July 1992, whereupon the comet's core was broken into numerous fragments disintegrated. During the next approach to the planet in 1994, these fragments collided with the planet.

Determination of the Roche limit

The Roche limit depends on the deformability of the satellite approaching the main body. Two extreme cases are therefore considered to calculate this limit. In the first case, the body is assumed to remain absolutely rigid until the body is torn apart by the tidal forces. The opposite case is a so-called "liquid body", i. H. a satellite that does not resist the deformation at all and is therefore initially deformed in an elongated manner as it approaches the Roche limit and then tears. As expected, the second case provides the greater distance to the planet than the Roche limit.

Rigid bodies

In the case of the rigid satellite it is assumed that the internal forces keep the shape of the body stable, but the body is still only held together by its own gravity. Further idealizations are the neglect of possible deformations of the main body by tidal forces or its own rotation, as well as the own rotation of the satellite. The Roche limit is in this case ${\displaystyle d}$

${\displaystyle d=R\cdot {\sqrt[{3}]{\frac {2\,\rho _{M}}{\rho _{m}}}}\approx 1{,}25992\cdot R\cdot {\sqrt[{3}]{\frac {\rho _{M}}{\rho _{m}}}}}$

wherein the radius and the density is the main body, and describes the density of the satellite. ${\displaystyle R}$ ${\displaystyle \rho _{M}}$ ${\displaystyle \rho _{m}}$

One notices from the above formula that the Roche limit of a rigid body for satellites whose density is more than twice the density of the main body lies within the main body. This case occurs e.g. B. in many rocky moons of the gas giants of our solar system. Such satellites are therefore not torn apart by the tidal forces even when the main body comes close.

Derivation of the formula

Sketch for deriving the Roche limit

To derive the above formula, let's assume that there is a small mass on the surface of the satellite at the point closest to the main body. With this approach, the satellite itself is considered to be spherical and has a radius  r and a mass  m . Two forces now act on the small mass  u lying on the surface: ${\displaystyle u}$

• the gravitational force with which the satellite attracts the mass u lying on its surface  :

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

• the tidal force acting on the mass  u as it is attracted to the main body but is not in the center of gravity of the satellite, which is in free fall (orbit) around the main body. In the reference system rotating with the satellite, this tidal force can also be seen as the difference between the gravitational force exerted by the main body on the mass  u and the centrifugal force. For them, the first approximation results

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

The Roche limit is reached when the small test body begins to hover on the surface of the satellite, i.e. i.e. when the gravitational force and the tidal force assume the same amount. In this case, the relationship is obtained from the above equations ${\displaystyle F_{G}}$ ${\displaystyle F_{T}}$

${\displaystyle d=r\,{\sqrt[{3}]{2\,M/m}}}$

which no longer contains the test mass u . If one expresses the masses of the two celestial bodies by their average densities and and their radii and , one obtains the above relationship which is independent of the mass and radius of the satellite. ${\displaystyle \rho _{M}}$ ${\displaystyle \rho _{m}}$ ${\displaystyle R}$ ${\displaystyle r}$

Liquid body

The model of a liquid satellite orbiting the main body is the opposite limit case compared to a rigid satellite. Liquid means that the satellite does not oppose the deformation caused by tidal forces at all. ( Surface tension and other things are negligible.) The tidal forces then lead to an elongated deformation of the satellite in the direction of the connecting line between the satellite and the main body. In fact, this is exactly the effect we on Earth know as tides , in which the liquid oceans on the Earth's surface deform in the direction of the line connecting the Moon and form two tidal mountains. Since the strength of the tidal force increases with the expansion of the body in the direction of the connecting line, a strong deformation of the satellite causes an even greater tidal force. Therefore, the Roche limit for the orbit radius of a liquid satellite is much larger than we calculated in the rigid model, namely:

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

d. H. about twice the size of the rigid model. Roche calculated this limit distance as early as 1850 (see literature), but set the numerical factor in the formula at 2.44 a little too high. The Roche limit of real satellites lies between the two limit models and depends on the size and rigidity of the satellite.

Derivation of the formula

In order to derive the formula given above, much more effort is required than in the case of the rigid body. First we have to specify the concept of the liquid body . What is meant by this is a body that consists of an incompressible liquid that has a specified density ρ _m and a specified volume V independent of the external and internal forces . Furthermore, we assume that the satellite moves in a bound rotation on a circular orbit, i.e. That is, its center of gravity rests in a reference system rotating at a fixed angular velocity ω with its origin in the center of gravity of the overall system. The angular velocity is given by Kepler's third law :

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

In this frame of reference, bound rotation of the satellite means that the fluid that the satellite is made of does not move, so the problem can be viewed as static. Therefore, the viscosity and friction of the fluid do not play a role in this model, since these variables would only be included in the calculation if the fluid were to move.

The following forces now act on the liquid of the satellite in the rotating reference system:

• the gravitational force of the main body
• the centrifugal force as an apparent force in the rotating frame of reference
• the gravitational force of the satellite itself

Since all occurring forces are conservative , they can all be represented by a potential. The surface of the satellite takes on the form of an equipotential surface of the total potential, since otherwise there would be a horizontal component of the force on the surface that parts of the liquid would follow. What shape the satellite has to assume at a given distance from the main body so that this requirement is met will now be discussed.

Radial distance of a point on the surface of the ellipsoid to the center of gravity

We already know that the gravitational force of the main body and the centrifugal force cancel each other out in the center of gravity of the satellite, since it moves on a (freely falling) circular path. The external force that acts on the liquid particles is therefore dependent on the distance to the center of gravity and is the tidal force already used in the rigid model . For small bodies, the distance of the liquid particles from the center of gravity is small compared to the distance  d to the main body, and the tidal force can be linearized, giving the formula for F _T given above . In the rigid model, only the radius of the satellite r was considered as the distance from the center of gravity,  but if you consider any point on the surface of the satellite, the tidal force effective there depends on the distance Δd between the point and the center of gravity in the radial direction (i.e. parallel to the connecting line from the satellite to the main body). Since the tidal force is linear in the radial distance Δd, its potential in this variable is quadratic, namely (for ): ${\displaystyle m\ll M}$

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

So we are now looking for a shape for the satellite so that its own gravitational potential is superimposed on this tidal potential in such a way that the total potential on the surface becomes constant. Such a problem is generally very difficult to solve, but because of the simple quadratic dependence of the tidal potential on the distance from the center of gravity, the solution to this problem can fortunately be found by clever guesswork .

The dimensionless function  f , which gives the strength of the tidal potential as a function of the eccentricity of an ellipsoid of revolution.

Since the tidal potential only changes in one direction, namely in the direction of the main body, it is obvious that the satellite remains axially symmetrical around this connecting line when it is deformed, i.e. forms a body of revolution. The intrinsic potential of such a body of revolution on the surface can then only depend on the radial distance to the center of gravity, since the cut surface of such a body is a circular disk with a fixed radial distance, the edge of which certainly has constant potential. If the sum of the intrinsic potential and the tidal potential should be the same at every point on the surface, the intrinsic potential, just like the tidal potential, must have a quadratic dependence on the radial distance. It turns out that one then has to choose a prolate (cigar-shaped) ellipsoid of revolution as the shape . With a given density and volume, the intrinsic potential of such an ellipsoid depends on the numerical eccentricity ε of the ellipsoid:

${\displaystyle V_{S}=V_{S0}+G\pi \rho _{m}f(\epsilon )\Delta d^{2}}$

where the constant self potential on the circular edge of the central plane of symmetry is at Δd = 0. The dimensionless function  f is to be determined from the exact solution of the potential of an ellipsoid of revolution and results from ${\displaystyle V_{S0}}$

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

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

The derivative of the function  f has a zero at the maximum eccentricity of the tidal ellipsoid. This zero marks the Roche limit.

As complicated as the dependence of the function  f on the eccentricity is, we now only need to determine the appropriate value for the eccentricity so that Δd is constant in the single position variable. This is the case if and only if ${\displaystyle V_{T}+V_{S}}$

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

is, an equation that any computer can easily solve numerically. As can be seen from the curve of the function  f in the graphic opposite, this equation generally has two solutions, whereby the smaller solution, i.e. the lower eccentricity, represents the stable equilibrium position. This solution of the equation therefore gives the eccentricity of the tidal ellipsoid, which occurs at a fixed distance from the main body.

The Roche limit arises from the fact that the function  f , which can be seen as the strength of the force that wants to reshape the ellipsoid into a spherical shape, cannot be arbitrarily large. There is a certain eccentricity at which this force becomes maximum. Since the tidal force can rise above all limits when approaching the main body, it is clear that there is a limit distance at which the ellipsoid is torn apart.

The maximum eccentricity of the tidal ellipsoid is calculated numerically from the zero of the derivative of the function  f , which is shown in the graphic. You get:

${\displaystyle \epsilon _{\text{max}}\approx 0{,}86}$

which corresponds to an aspect ratio of about 1: 1.95. If you insert this value into the function  f , you can calculate the minimum distance in which such a tidal ellipsoid exists - the Roche limit :

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

Roche limits of selected examples

Densities and radii of selected objects in our solar system

object	Density in g / cm³	Radius in km
Sun	1,400	695,000
Jupiter	1.330	071,500
earth	5.515	006,376.5
moon	3.340	001,737.4

The values ​​given above are now used to calculate the Roche limits for the rigid model and the liquid model. The mean density of a comet is assumed to be 500 kg / m³. The true Roche limit depends on the flexibility of the satellite in question, but also on numerous other parameters, such as the deformation of the main body and the exact density distribution within the satellite, and is usually between the two specified values.

Main body	satellite	Roche limit (rigid)		Roche limit (liquid)
		Orbit radius in km	Radius quotient	Orbit radius in km	Radius quotient
earth	moon	9,496	1.49	18,261	2.86
earth	comet	17,883	2.80	34,392	5.39
Sun	earth	554.441	0.80	1,066,266	1.53
Sun	Jupiter	890.745	1.28	1,713,025	2.46
Sun	moon	655.323	0.94	1,260,275	1.81
Sun	comet	1,234,186	1.78	2,373,509	3.42

You can see from the table above that for particularly dense satellites orbiting a much less dense main body, the Roche limit can lie within the main body (e.g. in the sun-earth system). A few more examples are presented in the next table, where the actual distance of the satellite is given as a percentage of the Roche limit. One sees z. B. that the Neptune moon Naiad is particularly close to the Roche limit of the rigid model and is therefore probably already quite close to its actual physical Roche limit.

Main body	satellite	Orbit radius vs. Roche limit		Main body	satellite	Orbit radius vs. Roche limit
		rigid	liquid			rigid	liquid
Sun	Mercury	10400%	5400%	Uranus	Cordelia	155%	79%
earth	moon	4100%	2100%		Ophelia	167%	86%
Mars	Phobos	172%	89%		Bianca	184%	94%
	Deimos	451%	233%		Cressida	192%	99%
Jupiter	Metis	186%	93%	Neptune	Naiad	140%	72%
	Adrastea	220%	110%		Thalassa	149%	77%
	Amalthea	228%	114%		Despina	153%	78%
	Thebe	260%	129%		Galatea	184%	95%
Saturn	Pan	174%	85%		Larissa	220%	113%
	Atlas	182%	89%	Pluto	Charon	1400%	720%
	Prometheus	185%	90%
	Pandora	185%	90%
	Epimetheus	198%	97%

Roche limit as a geometric limit shape

3D representation of the Roche potential of a binary star system with a mass ratio of 2: 1, including the 2D projection

If a star orbits a partner, it is deformed by the tidal forces. If the star is big and close enough, it takes on a teardrop shape with a point facing the partner. If it is in an expansion phase, such as in the transition phase to a red giant , it cannot grow any further, but material flows over this tip to the partner. This drop shape is also known as the Roche limit. Since this loss of mass reduces the Roche limit (for the shape of the end of the circle), the whole system can become unstable and the star can flow over completely to its partner.

If the partner is a compact object such as a white dwarf , a neutron star or a black hole , dramatic processes take place during the transfer of material. See also Novae and X-ray binary stars .

The Roche limit of the overall system consists of the two drop-shaped equipotential surfaces that touch at the tips and thus form the shape of an eight. This point is the so-called Lagrange point L _{1 of} the system. This potential surface must be calculated for a co-rotating coordinate system . It is an effective potential that takes into account centrifugal forces as well as gravitational forces . As soon as material moves in this system, it also experiences Coriolis forces , which can only be described by a speed-dependent potential.

See also

• Hill sphere

literature

• 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

Web links

• Detailed derivation of the Roche limit Scienceworld Wolfram (however very confusing)