Editing Roche limit

{{short description|Orbital radius at which a satellite might break up due to gravitational force}}
{{About|the orbit within which particles might form rings or objects on a stable orbit might disintegrate into rings|the limits at which an orbiting object will be captured|Roche lobe|the gravitational sphere of influence of one astronomical body in the face of perturbations from another heavier body around which it orbits|Roche sphere}}
{{Multiple image
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| image1 = Roche limit (far away sphere).svg
| caption1 = A celestial body (yellow) is orbited by a mass of fluid (blue) held together by gravity, here viewed from above the orbital plane. Far from the Roche limit (white line), the mass is practically spherical.
| image2 = Roche limit (tidal sphere).svg
| caption2 = Closer to the Roche limit, the body is deformed by [[tidal force]]s.
| image3 = Roche limit (ripped sphere).svg
| caption3 = Within the Roche limit, the mass's own gravity can no longer withstand the tidal forces, and the body disintegrates.
| image4 = Roche limit (top view).svg
| caption4 = Particles closer to the primary move more quickly than particles farther away, as represented by the red arrows.
| image5 = Roche limit (ring).svg
| caption5 = The varying orbital speed of the material eventually causes it to form a ring.
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| alt1 = 
}}

In [[celestial mechanics]], the '''Roche limit''', also called '''Roche radius''', is the distance from a celestial body within which a second celestial body, held together only by its own force of [[gravity]], will disintegrate because the first body's tidal forces exceed the second body's [[self-gravitation]].<ref name="wolf">{{Cite web |author=Weisstein |first=Eric W. |date=2007 |title=Eric Weisstein's World of Physics&nbsp;– Roche Limit |url=http://scienceworld.wolfram.com/physics/RocheLimit.html |access-date=September 5, 2007 |publisher=scienceworld.wolfram.com}}</ref> Inside the Roche limit, [[Planetary orbit|orbiting]] material disperses and forms [[Ring system (astronomy)|rings]], whereas outside the limit, material tends to [[coalescence (physics)|coalesce]]. The Roche radius depends on the radius of the second body and on the ratio of the bodies' densities.

The term is named after [[Édouard Roche]] ({{IPA|fr|ʁɔʃ|lang}}, {{IPAc-en|lang|r|ɒ|ʃ}} {{respell|ROSH}}), the [[France|French]] [[astronomer]] who first calculated this theoretical limit in 1848.<ref name="two">{{Cite web |url=http://saturn.jpl.nasa.gov/faq/FAQSaturn/#q11 |archive-url=https://web.archive.org/web/20090423160619/http://saturn.jpl.nasa.gov/faq/FAQSaturn/#q11 |url-status=dead |archive-date=April 23, 2009 |title=What is the Roche limit? |access-date=September 5, 2007 |publisher=NASA&nbsp;– JPL |author=NASA}}</ref>

== Explanation ==
[[File:Shoemaker-Levy 9 on 1994-05-17.png|thumb|upright=1.2|Comet [[Shoemaker–Levy 9]] was disintegrated by the tidal forces of [[Jupiter]] into a string of smaller bodies in 1992, before colliding with the planet in 1994.]]
The Roche limit typically applies to a [[Natural satellite|satellite]]'s disintegrating due to [[tidal force]]s induced by its ''primary'', the body around which it [[orbit]]s. Parts of the satellite that are closer to the primary are attracted more strongly by gravity from the primary than parts that are farther away; this disparity effectively pulls the near and far parts of the satellite apart from each other, and if the disparity (combined with any centrifugal effects due to the object's spin) is larger than the force of gravity holding the satellite together, it can pull the satellite apart. Some real satellites, both [[natural satellite|natural]] and [[artificial satellite|artificial]], can orbit within their Roche limits because they are held together by forces other than gravitation. Objects resting on the surface of such a satellite would be lifted away by tidal forces. A weaker satellite, such as a [[comet]], could be broken up when it passes within its Roche limit.

Since, within the Roche limit, tidal forces overwhelm the gravitational forces that might otherwise hold the satellite together, no satellite can gravitationally coalesce out of smaller particles within that limit. Indeed, almost all known [[planetary ring]]s are located within their Roche limit. (Notable exceptions are Saturn's [[Rings of Saturn#E Ring|E-Ring]] and [[Rings of Saturn#Phoebe ring|Phoebe ring]]. These two rings could possibly be remnants from the planet's proto-planetary [[accretion disc]] that failed to coalesce into moonlets, or conversely have formed when a moon passed within its Roche limit and broke apart.)

The gravitational effect occurring below the Roche limit is not the only factor that causes comets to break apart. Splitting by [[thermal stress]], internal [[gas pressure]], and rotational splitting are other ways for a comet to split under stress.

== Determination ==
The limiting distance to which a [[Natural satellite|satellite]] can approach without breaking up depends on the rigidity of the satellite. At one extreme, a completely rigid satellite will maintain its shape until tidal forces break it apart. At the other extreme, a highly fluid satellite gradually deforms leading to increased tidal forces, causing the satellite to elongate, further compounding the tidal forces and causing it to break apart more readily.

Most real satellites would lie somewhere between these two extremes, with tensile strength rendering the satellite neither perfectly rigid nor perfectly fluid. For example, a [[Rubble pile|rubble-pile asteroid]] will behave more like a fluid than a solid rocky one; an icy body will behave quite rigidly at first but become more fluid as tidal heating accumulates and its ices begin to melt.

But note that, as defined above, the Roche limit refers to a body held together solely by the gravitational forces which cause otherwise unconnected particles to coalesce, thus forming the body in question. The Roche limit is also usually calculated for the case of a circular orbit, although it is straightforward to modify the calculation to apply to the case (for example) of a body passing the primary on a parabolic or hyperbolic trajectory.

=== Rigid satellites ===
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, <math>d</math>, 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:<ref>see calculation in Frank H. Shu, ''The Physical Universe: an Introduction to Astronomy,'' p. 431, University Science Books (1982), {{ISBN|0-935702-05-9}}.</ref><ref>{{cite web | url=http://www.asterism.org/tutorials/tut25-1.htm | title=Roche Limit: Why Do Comets Break Up? | access-date=2012-08-28 | archive-date=2013-05-15 | archive-url=https://web.archive.org/web/20130515200347/http://www.asterism.org/tutorials/tut25-1.htm | url-status=dead }}</ref>

:<math> d = R_M\left(2 \frac {\rho_M} {\rho_m} \right)^{\frac{1}{3}} </math>

where <math>R_M</math> is the [[radius]] of the primary, <math>\rho_M</math> is the [[density]] of the primary, and <math>\rho_m</math> is the density of the satellite. This can be equivalently written as

:<math> d = R_m\left(2 \frac {M_M} {M_m} \right)^{\frac{1}{3}} </math>

where <math>R_m</math> is the radius of the secondary, <math>M_M</math> is the [[mass]] of the primary, and <math>M_m</math> is the mass of the secondary. A third equivalent form uses only one property for each of the two bodies, the mass of the primary and the density of the secondary, is

:<math> d = 0.7816 \left( \frac {M_M} {\rho_m} \right)^{\frac{1}{3}} </math>

These all represent 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.

=== Fluid satellites ===
A more accurate approach for calculating the Roche limit takes the deformation of the satellite into account. An extreme example would be a [[tidal locking|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:

:<math> d \approx  2.44 \ R\left( \frac {\rho_M} {\rho_m} \right)^{1/3} </math>

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

:<math> d \approx 2.423 \ R\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} </math>

where <math>c/R</math> is the [[oblateness]] of the primary.  
<!--- Numerical solution obtained from http://scienceworld.wolfram.com/physics/RocheLimit.html,
equation (17) using the e value given at (20) (= 1.676 554) gives eqn. (21) (= 0.070 310 549),
extract cubic root and invert (=2.422 849 866 704...).  The result is insensitive to the
precision of (20) because it is a functional minimum: using e = 1.676600 or 1.676500 changes only the tenth digit --->

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.<ref>[http://www2.jpl.nasa.gov/sl9/hst1.html International Planetarium Society Conference, Astronaut Memorial Planetarium & Observatory, Cocoa, Florida] Rob Landis 10–16 July 1994 [https://web.archive.org/web/19961221154010/http://www.seds.org/sl9/landis.html archive 21/12/1996]</ref>

== See also ==
{{div col|colwidth=20em}}
* [[Roche lobe]]
* [[Chandrasekhar limit]]
* [[Spaghettification]] (the extreme case of tidal distortion)
* [[Hill sphere]]
* [[Sphere of influence (black hole)]]
* [[Black hole]]
* [[Triton (moon)]] (Neptune's satellite)
* [[Comet Shoemaker–Levy 9]]
{{div col end}}

== References ==
{{Reflist}}

== Sources ==
* {{cite book|first=Édouard|last=Roche|url=https://books.google.com/books?id=UmoVAAAAQAAJ|chapter=La figure d'une masse fluide soumise à l'attraction d'un point éloigné, part 1|publisher=Académie des sciences de Montpellier|title=Mémoires de la section des sciences, Volume 1|date=1849|pages=243–262}} 2.44 is mentioned on page 258.
* {{cite book|first=Édouard|last=Roche|url=https://books.google.com/books?id=UmoVAAAAQAAJ|chapter=La figure d'une masse fluide soumise à l'attraction d'un point éloigné, part 2|publisher=Académie des sciences de Montpellier|title=Mémoires de la section des sciences, Volume 1|date=1850|pages=333–348}}
* {{cite book|first=Édouard|last=Roche|url=https://books.google.com/books?id=x3gVAAAAQAAJ|chapter=La figure d'une masse fluide soumise à l'attraction d'un point éloigné, part 3|publisher=Académie des sciences de Montpellier|title=Mémoires de la section des sciences, Volume 2|date=1851|pages=21–32}}
* {{cite book|first=George|last=Howard Darwin|url=https://books.google.com/books?id=8dPPAAAAMAAJ|chapter=On the figure and stability of a liquid satellite|title=Scientific Papers, Volume 3|date=1910|pages=436–524}}
* {{cite book|first=James|last=Hopwood Jeans|url=https://books.google.com/books?id=coE-AAAAYAAJ|title=Problems of cosmogony and stellar dynamics|chapter=Chapter III: Ellipsoidal configurations of equilibrium|date=1919}}
* {{Cite book |last=Chandrasekhar |first=Subrahmanyan |url=https://books.google.com/books?id=amOfQgAACAAJ |title=Ellipsoidal Figures of Equilibrium |date=1969 |publisher=Yale University Press |isbn=978-0-300-01116-6 |language=en}}
* {{Cite journal |last=Chandrasekhar |first=S. |year=1963 |title=The Equilibrium and the Stability of the Roche Ellipsoids. |journal=The Astrophysical Journal |language=en |volume=138 |pages=1182 |bibcode=1963ApJ...138.1182C |doi=10.1086/147716 |issn=0004-637X}}

== External links ==
* [http://www.merlyn.demon.co.uk/gravity6.htm#Roche Discussion of the Roche Limit]; {{Webarchive|url=https://web.archive.org/web/20190916174931/http://www.merlyn.demon.co.uk/gravity6.htm#Roche |date=2019-09-16 }}
* [http://www.astronomycast.com/2007/08/episode-48-tidal-forces-across-the-universe/ Audio: Cain/Gay&nbsp;– Astronomy Cast] Tidal Forces Across the Universe&nbsp;– August 2007
* [https://ntrs.nasa.gov/citations/19740055720 Roche Limit Description from NASA]

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