Roche lobe

Last updated December 26, 2023

In astronomy, the Roche lobe is the region around a star in a binary system within which orbiting material is gravitationally bound to that star. It is an approximately teardrop-shaped region bounded by a critical gravitational equipotential, with the apex of the teardrop pointing towards the other star (the apex is at the L₁ Lagrangian point of the system).

The Roche lobe is different from the Roche sphere, which approximates the gravitational sphere of influence of one astronomical body in the face of perturbations from a more massive body around which it orbits. It is also different from the Roche limit, which is the distance at which an object held together only by gravity begins to break up due to tidal forces. The Roche lobe, Roche limit, and Roche sphere are named after the French astronomer Édouard Roche.

Definition

A three-dimensional representation of the Roche potential in a binary star with a mass ratio of 2, in the co-rotating frame. The droplet-shaped figures in the equipotential plot at the bottom of the figure define what are considered the Roche lobes of the stars. L1, L2 and L3 are the Lagrangian points where forces (considered in the rotating frame) cancel out. Mass can flow through the saddle point L1 from one star to its companion, if the star fills its Roche lobe. RochePotential.jpg — A three-dimensional representation of the Roche potential in a binary star with a mass ratio of 2, in the co-rotating frame. The droplet-shaped figures in the equipotential plot at the bottom of the figure define what are considered the Roche lobes of the stars. L₁, L₂ and L₃ are the Lagrangian points where forces (considered in the rotating frame) cancel out. Mass can flow through the saddle point L₁ from one star to its companion, if the star fills its Roche lobe.

In a binary system with a circular orbit, it is often useful to describe the system in a coordinate system that rotates along with the objects. In this non-inertial frame, one must consider centrifugal force in addition to gravity. The two together can be described by a potential, so that, for example, the stellar surfaces lie along equipotential surfaces.

Close to each star, surfaces of equal gravitational potential are approximately spherical and concentric with the nearer star. Far from the stellar system, the equipotentials are approximately ellipsoidal and elongated parallel to the axis joining the stellar centers. A critical equipotential intersects itself at the L₁ Lagrangian point of the system, forming a two-lobed figure-of-eight with one of the two stars at the center of each lobe. This critical equipotential defines the Roche lobes.^[2]

Where matter moves relative to the co-rotating frame it will seem to be acted upon by a Coriolis force. This is not derivable from the Roche lobe model as the Coriolis force is a non-conservative force (i.e. not representable by a scalar potential).

Further analysis

In the gravity potential graphics, L₁, L₂, L₃, L₄, L₅ are in synchronous rotation with the system. Regions of red, orange, yellow, green, light blue and blue are potential arrays from high to low. Red arrows are rotation of the system and black arrows are relative motions of the debris.

Debris goes faster in the lower potential region and slower in the higher potential region. So, relative motions of the debris in the lower orbit are in the same direction with the system revolution while opposite in the higher orbit.

L₁ is the gravitational capture equilibrium point. It is a gravity cut-off point of the binary star system. It is the minimum potential equilibrium among L₁, L₂, L₃, L₄ and L₅. It is the easiest way for the debris to commute between a Hill sphere (an inner circle of blue and light blue) and communal gravity regions (figure-eights of yellow and green in the inner side).

L₂ and L₃ are gravitational perturbation equilibria points. Passing through these two equilibrium points, debris can commute between the external region (figure-eights of yellow and green in the outer side) and the communal gravity region of the binary system.

L₄ and L₅ are the maximum potential points in the system. They are unstable equilibria. If the mass ratio of the two stars becomes larger, then the orange, yellow and green regions will become a horseshoe orbit.

The red region will become the tadpole orbit.

Mass transfer

When a star "exceeds its Roche lobe", its surface extends out beyond its Roche lobe and the material which lies outside the Roche lobe can "fall off" into the other object's Roche lobe via the first Lagrangian point. In binary evolution this is referred to as mass transfer via Roche-lobe overflow.

In principle, mass transfer could lead to the total disintegration of the object, since a reduction of the object's mass causes its Roche lobe to shrink. However, there are several reasons why this does not happen in general. First, a reduction of the mass of the donor star may cause the donor star to shrink as well, possibly preventing such an outcome. Second, with the transfer of mass between the two binary components, angular momentum is transferred as well. While mass transfer from a more massive donor to a less massive accretor generally leads to a shrinking orbit, the reverse causes the orbit to expand (under the assumption of mass and angular-momentum conservation). The expansion of the binary orbit will lead to a less dramatic shrinkage or even expansion of the donor's Roche lobe, often preventing the destruction of the donor.

To determine the stability of the mass transfer and hence exact fate of the donor star, one needs to take into account how the radius of the donor star and that of its Roche lobe react to the mass loss from the donor; if the star expands faster than its Roche lobe or shrinks less rapidly than its Roche lobe for a prolonged time, mass transfer will be unstable and the donor star may disintegrate. If the donor star expands less rapidly or shrinks faster than its Roche lobe, mass transfer will generally be stable and may continue for a long time.

Mass transfer due to Roche-lobe overflow is responsible for a number of astronomical phenomena, including Algol systems, recurring novae (binary stars consisting of a red giant and a white dwarf that are sufficiently close that material from the red giant dribbles down onto the white dwarf), X-ray binaries and millisecond pulsars. Such mass transfer by Roche lobe overflow (RLOF) is further broken down into three distinct cases:

Case A

Case A RLOF occurs when the donor star is hydrogen burning. According to Nelson and Eggleton, there are a number of subclasses^[3] which are reproduced here:

AD dynamic

when RLOF happens to a star with a deep convection zone. Mass transfer happens rapidly on the dynamical time scale of the star and may end with a complete merger.

AR rapid contact

similar to AD, but as the star onto which matter is rapidly accreting gains mass, it gains physical size enough for it to reach its own Roche-lobe. As such times, the system manifests as a contact binary such as a W Ursae Majoris variable.

AS slow contact

similar to AR, but only a short period of fast mass transfer happens followed by a much longer period of slow mass transfer. Eventually the stars will come into contact, but they have changed substantially by the point this happens. Algol variables are the result of such situations.

AE early overtaking

similar to AS, but the star gaining mass overtakes the star donating mass to evolve past the main sequence. The donor star can shrink so small to stop mass transfer, but eventually mass transfer will start again as stellar evolution continues leading to the cases

AL late overtaking

the case when the star that initially was the donor undergoes a supernova after the other star has undergone its own round of RLOF.

AB binary

the case where the stars switch back and forth between which one is undergoing RLOF at least three times (technically a subclass of the above).

AN no overtaking

the case when the star that initially was the donor undergoes a supernova before the other star reaches a RLOF phase.

AG giant

Mass transfer does not begin until the star reaches the red giant branch but before it has exhausted its hydrogen core (after which the system is described as Case B).

Case B

Case B happens when RLOF starts while the donor is a post-core hydrogen burning/hydrogen shell burning star. This case can be further subdivided into classes Br and Bc^[4] according to whether the mass transfer occurs from a star dominated by a radiation zone (Br) and therefore evolves as the situation with most Case A RLOF or a convective zone (Bc) after which a common envelope phase may occur (similar to Case C).^[5] An alternative division of cases is Ba, Bb, and Bc which are roughly corresponding to RLOF phases that happen during helium fusion, after helium fusion but before carbon fusion, or after carbon fusion in the highly evolved star.^[6]

Case C

Case C happens when RLOF starts when the donor is at or beyond the helium shell burning phase. These systems are the rarest observed, but this may be due to selection bias.^[7]

Geometry

The precise shape of the Roche lobe depends on the mass ratio $q=M_{1}/M_{2}$ , and must be evaluated numerically. However, for many purposes it is useful to approximate the Roche lobe as a sphere of the same volume. An approximate formula for the radius of this sphere is

{\frac {r_{1}}{A}}=\max {[f_{1},f_{2}]}

, for

0<q<20

where $f_{1}=0.38+0.2\log {q}$ and $f_{2}=0.46224\left({\frac {q}{1+q}}\right)^{1/3}$ . Function $f_{1}$ is greater than $f_{2}$ for $q\gtrsim 0.5228$ . The length A is the orbital separation of the system and r₁ is the radius of the sphere whose volume approximates the Roche lobe of mass M₁. This formula is accurate to within about 2%.^[2] Another approximate formula was proposed by Eggleton and reads as follows:

{\frac {r_{1}}{A}}={\frac {0.49q^{2/3}}{0.6q^{2/3}+\ln(1+q^{1/3})}}

.

This formula gives results up to 1% accuracy over the entire range of the mass ratio $q$ .^[8]

Related Research Articles

In celestial mechanics, the Lagrange points are points of equilibrium for small-mass objects under the gravitational influence of two massive orbiting bodies. Mathematically, this involves the solution of the restricted three-body problem.

In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.

The tidal force or tide-generating force is a gravitational effect that stretches a body along the line towards and away from the center of mass of another body due to spatial variations in strength in gravitational field from the other body. It is responsible for the tides and related phenomena, including solid-earth tides, tidal locking, breaking apart of celestial bodies and formation of ring systems within the Roche limit, and in extreme cases, spaghettification of objects. It arises because the gravitational field exerted on one body by another is not constant across its parts: the nearer side is attracted more strongly than the farther side. The difference is positive in the near side and negative in the far side, which causes a body to get stretched. Thus, the tidal force is also known as the differential force, residual force, or secondary effect of the gravitational field.

A binary star or binary star system is a system of two stars that are gravitationally bound to and in orbit around each other. Binary stars in the night sky that are seen as a single object to the naked eye are often resolved using a telescope as separate stars, in which case they are called visual binaries. Many visual binaries have long orbital periods of several centuries or millennia and therefore have orbits which are uncertain or poorly known. They may also be detected by indirect techniques, such as spectroscopy or astrometry. If a binary star happens to orbit in a plane along our line of sight, its components will eclipse and transit each other; these pairs are called eclipsing binaries, or, together with other binaries that change brightness as they orbit, photometric binaries.

Tidal locking between a pair of co-orbiting astronomical bodies occurs when one of the objects reaches a state where there is no longer any net change in its rotation rate over the course of a complete orbit. In the case where a tidally locked body possesses synchronous rotation, the object takes just as long to rotate around its own axis as it does to revolve around its partner. For example, the same side of the Moon always faces the Earth, although there is some variability because the Moon's orbit is not perfectly circular. Usually, only the satellite is tidally locked to the larger body. However, if both the difference in mass between the two bodies and the distance between them are relatively small, each may be tidally locked to the other; this is the case for Pluto and Charon, as well as for Eris and Dysnomia. Alternative names for the tidal locking process are gravitational locking, captured rotation, and spin–orbit locking.

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. Inside the Roche limit, orbiting material disperses and forms rings, whereas outside the limit, material tends to coalesce. The Roche radius depends on the radius of the first body and on the ratio of the bodies' densities.

X-ray binaries are a class of binary stars that are luminous in X-rays. The X-rays are produced by matter falling from one component, called the donor, to the other component, called the accretor, which is either a neutron star or black hole. The infalling matter releases gravitational potential energy, up to 30 percent of its rest mass, as X-rays. The lifetime and the mass-transfer rate in an X-ray binary depends on the evolutionary status of the donor star, the mass ratio between the stellar components, and their orbital separation.

X-ray bursters are one class of X-ray binary stars exhibiting X-ray bursts, periodic and rapid increases in luminosity that peak in the X-ray region of the electromagnetic spectrum. These astrophysical systems are composed of an accreting neutron star and a main sequence companion 'donor' star. There are two types of X-ray bursts, designated I and II. Type I bursts are caused by thermonuclear runaway, while type II arise from the release of gravitational (potential) energy liberated through accretion. For type I (thermonuclear) bursts, the mass transferred from the donor star accumulates on the surface of the neutron star until it ignites and fuses in a burst, producing X-rays. The behaviour of X-ray bursters is similar to the behaviour of recurrent novae. In the latter case the compact object is a white dwarf that accretes hydrogen that finally undergoes explosive burning.

Geopotential is the potential of the Earth's gravity field. For convenience it is often defined as the negative of the potential energy per unit mass, so that the gravity vector is obtained as the gradient of the geopotential, without the negation. In addition to the actual potential, a hypothetical normal potential and their difference, the disturbing potential, can also be defined.

Newton's law of universal gravitation says every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Separated objects attract and are attracted as if all their mass were concentrated at their centers. The publication of the law has become known as the "first great unification", as it marked the unification of the previously described phenomena of gravity on Earth with known astronomical behaviors.

<span class="mw-page-title-main">Hill sphere</span> Region in which an astronomical body dominates the attraction of satellites

The Hill sphere is a common model for the calculation of a gravitational sphere of influence. It is the most commonly used model to calculate the spatial extent of gravitational influence of an astronomical body (m) in which it dominates over the gravitational influence of other bodies, particularly a primary (M). It is sometimes confused with other models of gravitational influence, such as the Laplace sphere or being called the Roche sphere, the latter causing confusion with the Roche limit. It was defined by the American astronomer George William Hill, based on the work of the French astronomer Édouard Roche.

Algol variables or Algol-type binaries are a class of eclipsing binary stars that are similar to the prototype member of this class, β Persei. An Algol binary is a system where both stars are near-spherical such that the timing of the start and end of the eclipses is well-defined. The primary is generally a main sequence star well within its Roche lobe. The secondary may also be a main sequence star, referred to as a detached binary or it may an evolved star filling its Roche lobe, referred to as a semidetached binary.

In astronomy, a polar is a highly magnetic type of cataclysmic variable (CV) binary star system, originally known as an AM Herculis star after the prototype member AM Herculis. Like other CVs, polars contain two stars: an accreting white dwarf (WD), and a low-mass donor star which is transferring mass to the WD as a result of the WD's gravitational pull, overflowing its Roche lobe. Polars are distinguished from other CVs by the presence of a very strong magnetic field in the WD. Typical magnetic field strengths of polar systems are 10 million to 80 million gauss. The WD in the polar AN Ursae Majoris has the strongest known magnetic field among cataclysmic variables, with a field strength of 230 million gauss.

The two-body problem in general relativity is the determination of the motion and gravitational field of two bodies as described by the field equations of general relativity. Solving the Kepler problem is essential to calculate the bending of light by gravity and the motion of a planet orbiting its sun. Solutions are also used to describe the motion of binary stars around each other, and estimate their gradual loss of energy through gravitational radiation.

An AM Canum Venaticorum star, is a rare type of cataclysmic variable star named after their type star, AM Canum Venaticorum. In these hot blue binary variables, a white dwarf accretes hydrogen-poor matter from a compact companion star.

The sphere of influence is a region around a supermassive black hole in which the gravitational potential of the black hole dominates the gravitational potential of the host galaxy. The radius of the sphere of influence is called the "(gravitational) influence radius".

In astronomy, a common envelope (CE) is gas that contains a binary star system. The gas does not rotate at the same rate as the embedded binary system. A system with such a configuration is said to be in a common envelope phase or undergoing common envelope evolution.

A binary black hole (BBH), or black hole binary, is a system consisting of two black holes in close orbit around each other. Like black holes themselves, binary black holes are often divided into stellar binary black holes, formed either as remnants of high-mass binary star systems or by dynamic processes and mutual capture; and binary supermassive black holes, believed to be a result of galactic mergers.

In astrophysics the chirp mass of a compact binary system determines the leading-order orbital evolution of the system as a result of energy loss from emitting gravitational waves. Because the gravitational wave frequency is determined by orbital frequency, the chirp mass also determines the frequency evolution of the gravitational wave signal emitted during a binary's inspiral phase. In gravitational wave data analysis it is easier to measure the chirp mass than the two component masses alone.

An accretion disk is a structure formed by diffuse material in orbital motion around a massive central body. The central body is most frequently a star. Friction, uneven irradiance, magnetohydrodynamic effects, and other forces induce instabilities causing orbiting material in the disk to spiral inward toward the central body. Gravitational and frictional forces compress and raise the temperature of the material, causing the emission of electromagnetic radiation. The frequency range of that radiation depends on the central object's mass. Accretion disks of young stars and protostars radiate in the infrared; those around neutron stars and black holes in the X-ray part of the spectrum. The study of oscillation modes in accretion disks is referred to as diskoseismology.

References

↑ Source
1 2 Paczynski, B. (1971). "Evolutionary Processes in Close Binary Systems". Annual Review of Astronomy and Astrophysics. 9: 183–208. Bibcode:1971ARA&A...9..183P. doi:10.1146/annurev.aa.09.090171.001151.
↑ Nelson, C. A.; Eggleton, P. P. (2001). "A Complete Survey of Case A Binary Evolution with Comparison to Observed Algol-type Systems". The Astrophysical Journal. 552 (2): 664–678. arXiv: astro-ph/0009258 . Bibcode:2001ApJ...552..664N. doi:10.1086/320560. S2CID 119505485.
↑ Vanbeveren, D.; Mennekens, N. (2014-04-01). "Massive double compact object mergers: gravitational wave sources and r-process element production sites". Astronomy & Astrophysics. 564: A134. arXiv: 1307.0959 . Bibcode:2014A&A...564A.134M. doi: 10.1051/0004-6361/201322198 . ISSN 0004-6361.
↑ Vanbeveren, D.; Rensbergen, W. van; Loore, C. de (2001-11-30). The Brightest Binaries. Springer Science & Business Media. ISBN 9781402003769.
↑ Bhattacharya, D; van den Heuvel, E. P. J (1991-05-01). "Formation and evolution of binary and millisecond radio pulsars". Physics Reports. 203 (1): 1–124. Bibcode:1991PhR...203....1B. doi:10.1016/0370-1573(91)90064-S. ISSN 0370-1573.
↑ Podsiadlowski, Philipp (February 2014). "The evolution of binary systems". Accretion Processes in Astrophysics. pp. 45–88. doi:10.1017/CBO9781139343268.003. ISBN 9781139343268 . Retrieved 2019-08-12.{{cite book}}: |website= ignored (help)
↑ Eggleton, P. P. (1 May 1983). "Approximations to the radii of Roche lobes". The Astrophysical Journal. 268: 368. Bibcode:1983ApJ...268..368E. doi:10.1086/160960.

Sources

Morris, S.L. (Feb 1994). "Two Mathematical Expansions of the Roche Equipotentials". Publications of the Astronomical Society of the Pacific. 106 (696): 154–155. Bibcode:1994PASP..106..154M. doi:10.1086/133361. JSTOR 40680260. S2CID 121386366.
Morris, S.L. (Aug 1, 1999). "The Limits of Inclination for Binary Star Partial Eclipses". Astrophysical Journal. 520 (2): 797–804. Bibcode:1999ApJ...520..797M. doi: 10.1086/307488 .

External links

From Hot Jupiters to Super-Earths via Roche Lobe Overflow - Cornell University

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Source

[pacz71-2] 1 2 Paczynski, B. (1971). "Evolutionary Processes in Close Binary Systems". Annual Review of Astronomy and Astrophysics. 9: 183–208. Bibcode:1971ARA&A...9..183P. doi:10.1146/annurev.aa.09.090171.001151.

[3] Nelson, C. A.; Eggleton, P. P. (2001). "A Complete Survey of Case A Binary Evolution with Comparison to Observed Algol-type Systems". The Astrophysical Journal. 552 (2): 664–678. arXiv: astro-ph/0009258 . Bibcode:2001ApJ...552..664N. doi:10.1086/320560. S2CID 119505485.

[4] Vanbeveren, D.; Mennekens, N. (2014-04-01). "Massive double compact object mergers: gravitational wave sources and r-process element production sites". Astronomy & Astrophysics. 564: A134. arXiv: 1307.0959 . Bibcode:2014A&A...564A.134M. doi: 10.1051/0004-6361/201322198 . ISSN 0004-6361.

[5] Vanbeveren, D.; Rensbergen, W. van; Loore, C. de (2001-11-30). The Brightest Binaries. Springer Science & Business Media. ISBN 9781402003769.

[6] Bhattacharya, D; van den Heuvel, E. P. J (1991-05-01). "Formation and evolution of binary and millisecond radio pulsars". Physics Reports. 203 (1): 1–124. Bibcode:1991PhR...203....1B. doi:10.1016/0370-1573(91)90064-S. ISSN 0370-1573.

[7] Podsiadlowski, Philipp (February 2014). "The evolution of binary systems". Accretion Processes in Astrophysics. pp. 45–88. doi:10.1017/CBO9781139343268.003. ISBN 9781139343268 . Retrieved 2019-08-12.{{cite book}}: |website= ignored (help)

[eggleton-8] Eggleton, P. P. (1 May 1983). "Approximations to the radii of Roche lobes". The Astrophysical Journal. 268: 368. Bibcode:1983ApJ...268..368E. doi:10.1086/160960.

[2]

[3]

[4]

[5]

[6]

[7]

[8]