Roche limit

The Roche limit [ ʀɔʃ - ] is a criterion for the assessment of internal stability, ie, the cohesion of a celestial body that orbits another. Here, the gravitational forces that hold together the celestial body internally, compared with the tidal forces that pull apart him. The Roche limit is named after Édouard Albert Roche, who discovered them in 1850.

Cause of the tidal forces is the fact that the attraction force is greater by the partner on the side facing it the celestial body than on the side facing away. Since the celestial bodies orbiting the partners as a whole, it comes to internal stresses or deformations that may lead to the dissolution of the celestial body.

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

• As a limit for its orbit (English Roche limit). Moves the heavenly bodies outside this orbit, so the stabilizing inner gravitational forces dominate the tidal forces. This meaning is used in particular when the stability of a moon is seen orbiting a planet.
• As a limit for its geometric shape (English Roche lobe ). The celestial body is inside this form, so it is stable. This importance is particularly used when two stars orbit each other and deforming.

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 is due to the tidal forces acting on it, torn. It is assumed here that the body is held together only by the own gravitational forces, and the mechanical strength is negligible. For real solids, this assumption is better fulfilled, the greater the body. Therefore, artificial satellites can easily circulate within the Roche limit, while large objects such as moons and planets can not exist there. If there is material that has not yet aggregated into a single body, in an orbit around the main body, this material is distributed within the Roche limit ring on the orbit, while it forms a lump outside the border.

In fact, all known planetary rings are within the Roche limit of their planet. They could therefore have formed either directly from the protoplanetary accretion disk, since the tidal forces have prevented moons form of this material, or fragments of destroyed moons that have moved from the outside of the Roche limit. All the major moons of the solar system are, however, far outside the Roche limit, but smaller moons are able to stay well within the Roche limit. Thus the orbits of Jupiter's moon Metis and Saturn's moon Pan are inside the Roche limit for so-called liquid body. The mechanical strength of this body acts on the one directly opposite the tidal forces acting on the body, and on the other hand, the strength also causes these bodies remain rigid, ie not change their shape - an effect which is described below will and the tidal forces reinforced. This effect is described particularly clearly by the fact that an object that would " set " on the surface of such a moon would not hold on the moon, but would pulled away by tidal forces from the surface. A body with lower mechanical strength, such as a comet would be destroyed in these regions, as was to be seen in the example of the comet Shoemaker- Levy 9, whose orbit in July 1992 broke through Jupiter's Roche limit, after which the nucleus of the comet into numerous fragments disintegrated. At the closest approach to the planet in 1994, these fragments then collided with the planet.

Determination of the Roche limit

The Roche limit depends on the deformability of the satellite approaches the main body. To calculate this limit, therefore, two extreme cases are considered. In the first case, it is believed that the body remains absolutely rigid, until the body is torn apart by tidal forces. The opposite case is a so-called " fluid body ", that is, a satellite, the deformation did not oppose and will therefore initially deformed elongated in the approach to the Roche limit and then tears. The second case provides, as expected, the greater distance to the planet as Roche limit.

Rigid body

When rigid satellite is assumed that the internal forces keep the shape of the body stable and the body but is still held together only by its own gravity. More idealizations are neglecting any deformation of the main body by tidal forces or its own rotation, and the self-rotation of the satellite. The Roche limit is in this case

Wherein said radius and the density of the main body, and describes the density of the satellite.

It will be noted in the above formula that the Roche limit of a rigid body for satellites, the density of which is more than twice as high as the density of the main body located within the main body. This case occurs in many rocky moons of the gas giants in our solar system, for example. Such satellites are therefore not disrupted by the tidal forces also at the next approximation to the main body.

To derive the above formula, we assume that a small mass on the surface of the satellites lie on the point which is closest to the main body. The satellite itself is considered in this approach as spherical and has a radius r and mass m. In the small mass u, which lies on the surface, now two forces:

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

• The tidal force u acting on the mass, as it is attracted to the main body, but that is not the focus of the satellite in free fall ( Orbit ) moves around the main body. In the rotating frame of reference of the satellite, this can be tidal and the difference between the gravitational force u applied from the main body to the ground, and the centrifugal force to view. For it results in first approximation

The Roche limit is reached when the small test body starts on the surface of the satellite to hover, that is, if the gravitational force and the tidal power assume the same amount. For this case is obtained from the above equations, the relationship

U no longer contains the test mass. Pressing the masses of the two bodies by their densities and their radii and in and out, we obtain the above mass and radius of the satellite independent relationship.

Liquid body

A model of a liquid satellite orbiting the main body, forms the opposite as compared to a fixed satellite limit. Liquid means that the satellite is the deformation caused by tidal forces do not opposes. (Surface tension and other is negligible. ) The tidal forces lead to an elongated deformation of the satellite in the direction of the line connecting the satellite and the main body. Indeed, this is exactly the effect that we know on Earth as tides, in which deform the liquid oceans on the Earth's surface in the direction of the connecting line to the moon and form two flood 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, however, ensures even greater tidal power. Therefore, the Roche limit is the orbital radius of a liquid Satellite substantially larger than we calculated in the rigid model, namely:

So about twice as large as in the rigid model. Roche has calculated this threshold distance already around the year 1850 ( see bibliography ), where he somewhat ansetzte the numerical factor in the formula with 2.44 too high, but this is no surprise especially since this value has to be calculated numerically, and to the 19th century, there was no computer available.

The Roche limit real satellite will be expected somewhere between the limits in the two extreme models, and depends on the degree of rigidity of the corresponding satellite.

To derive the above formula, is much more effort than necessary in the case of the rigid body. First we need to specify the concept of the liquid body. This refers to a body which consists of an incompressible fluid, the thus independent of the external and internal forces ρm a predetermined density, and has a predetermined volume V. Furthermore, we assume that the satellite moves in bound rotation on a circular path, that is, his focus rests in a fixed angular velocity ω rotating reference system with origin at the center of the whole system. The angular velocity is given by the third Kepler's law:

In this reference system is called synchronous rotation of the satellite that the fluid from which the satellite is not moving, the problem can therefore be regarded as static. Therefore, the viscosity and friction of the fluid play no role in this model, as these variables would be included in the calculation only on motion of the fluid.

Now applied to the liquid of the satellite following forces act in the rotating frame:

• The force of gravity of the main body.
• The centrifugal force as apparent force in a rotating frame of reference.
• The gravitational force of the satellite itself

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

We already know that the gravitational force of the main body and the centrifugal force in the satellite center of mass cancel, as it moves on a ( free-falling ) circular path. The external force acting on the fluid particles, therefore depends on the distance to the center of gravity and the tidal power used ones in the rigid model. For small body is the distance from the centroid of the liquid particles small compared to the distance d from the main body, and the tidal force can be linearized, thus it yields the formula given above for FT. As distance from the center, only the radius r of the satellite was considered in the rigid model, but is now considered an arbitrary point on the surface of the satellite, so that there effective tidal power depends on the distance of the point? D to focus in the radial direction (ie parallel to the connecting line from the satellite to the main body ). Since the tidal force is linear in the radial distance? D, their potential in this variable is square, and that results in ( for ):

So now we are looking for a form for the satellite, so that its self-gravity potential just so overlaid this tidal potential that the total potential on the surface is constant. One such problem is in general very difficult to solve, due to the simple quadratic dependence of the tidal potential on the distance from the centroid, the solution this problem can be found but fortunately through skillful rates.

Since the tidal potential in only one direction, namely in the direction of the main body changed, it is obvious that the satellite maintains its axisymmetric deformation around this connecting line, thus forming a body of revolution. The self-potential of such a rotary body on the surface can depend only on the radial distance from the center of gravity, since the cutting surface of such body at a fixed radial distance is precisely a circular disc whose edge has certainly constant potential. Are now the sum of the self potential and the potential at any point of tidal surface may be equal to, the self- potential of a quadratic dependence must be exactly as the tidal potential hold on the radial distance. It turns out that you then have to choose as a form prolates ( cigar-shaped ) spheroid. At a given density and the volume of such a self-potential depends on the numerical ellipsoid eccentricity ε of the ellipsoid:

Wherein the self-potential is constant on the circular edge of the central plane of symmetry at? d = 0. The dimensionless function f is to be determined from the exact solution of the potential of an ellipsoid of revolution and is given by

And depends surprisingly not on the volume of the satellite.

Thus the dependence of the function f complicated by the eccentricity is, we still only need now the appropriate value for the eccentricity defined so is constant in the single local variable? D. This is exactly the case when

Is an equation that can solve any computer numerically slightly. As you can see in the adjacent diagram f the course of the function, this equation has in general two solutions, the smaller solution, ie the lower eccentricity, the stable equilibrium position represents. Therefore, this solution of the equation indicates the eccentricity of the Gezeitenellipsoids that sets in at a specified distance from the main body.

The Roche limit is created now by the fact that the function f that can be regard as the strength of the force that wants to shape the ellipsoid in the spherical shape, can not be arbitrarily large. There is a certain eccentricity at which this power is maximized. But since the tidal power can increase as it approaches the main body beyond all limits, it is clear that there is a limit distance in which the ellipsoid is torn.

The maximum eccentricity of the Gezeitenellipsoids is calculated numerically from the zero of the derivative of the function f, which is shown in the graph. Is obtained:

Which corresponds to an aspect ratio of about 1:1.95. Substituting this value into the function f a, we can calculate the minimum distance at which there is such a Gezeitenellipsoid - the Roche limit:

Roche limits of selected examples

The following table lists the densities and radii of selected objects in our solar system.

The above values ​​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 500 kg/m3. True Roche limit depends on the flexibility of the respective satellite, but also by many other parameters such as the deformation of the main body and the accurate density distribution within the satellite, and is usually from between the two values ​​given.

You can see on the chart above that for particularly dense satellites orbiting a far less dense main body, the Roche limit are within the main body (eg the Sun-Earth System). The next table shows a few more examples will be presented, with the actual distance of the satellite is given in percent of the Roche limit. One sees, for example, that the moon Neptune Naiad is particularly close to the Roche limit of the rigid model and therefore probably its actual physical Roche limit is already quite close.

Roche limit as geometric boundary shape

Encircled a star for a partner, so it is deformed by tidal forces. If the star is large and close enough so he assumes a teardrop shape with a tip that is facing the partner. If it is in a phase of expansion, such as in the transition to a red giant, so he can not continue to grow, but it flows material on this tip to the partner. These drops form is also referred to as Roche limit. Because of this mass loss, the Roche limit reduced ( for the shape of the radius ends ), the whole system may become unstable and completely flow over the star to his partner.

Is it where the partner is a compact object such as a white dwarf, a neutron star or a black hole so dramatic processes take place during material transfer. See novae and X-ray binaries.

The Roche limit of the overall system is composed of the two teardrop- shaped equipotential surfaces that are touching at the tips and a form as the form of eight. This tip is known as the Lagrange point L1 of the system. This potential energy surface has to be calculated for a co-rotating coordinate system. Is an effective potential which takes into account in addition to the gravitational forces and the centrifugal forces. Once moving material in this system, it undergoes additional Coriolis forces that can only be described by a velocity-dependent potential, however.