Roche lobe
Encyclopedia
The Roche lobe is the region of space
around a star
in a binary system within which orbiting material is gravitationally bound to that star. If the star expands past its Roche lobe, then the material can escape the gravitational pull of the star. If the star is in a binary system
then the material will fall in through the inner Lagrangian point
. It is an approximately tear-drop shaped region bounded by a critical gravitational equipotential
, with the apex of the tear-drop pointing towards the other star (and the apex is at the Lagrangian point
of the system). It is 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 force
s. It is different from the Roche sphere which approximates the gravitational sphere of influence of one astronomical body in the face of perturbations from another heavier body around which it orbit
s. The Roche lobe, Roche limit and Roche sphere are named after the French
astronomer
Édouard Roche
.
, 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 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.
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).
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 star
s consisting of a red giant
and a white dwarf
that are sufficiently close enough together that material from the red giant dribbles down onto the white dwarf), X-ray binaries
and millisecond pulsar
s.
for
and
for
where A is the semi-major axis
of the system and
is the radius of the Roche lobe around mass 
. These formulas are accurate to within about 2%.
Another approximate formula by Eggleton is as follows:
where
. This formula gives results up to 1% accuracy over the entire range of 
.
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...
around a star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
in a binary system within which orbiting material is gravitationally bound to that star. If the star expands past its Roche lobe, then the material can escape the gravitational pull of the star. If the star is in a binary system
Binary system
Binary system may refer to:* Binary numeral system, the base-2 internal "machine language" of computers* Binary opposition, a bipolar distinction in philosophy, structuralism and critical theory...
then the material will fall in through the inner Lagrangian point
Lagrangian point
The Lagrangian points are the five positions in an orbital configuration where a small object affected only by gravity can theoretically be stationary relative to two larger objects...
. It is an approximately tear-drop shaped region bounded by a critical gravitational equipotential
Equipotential
Equipotential or isopotential in mathematics and physics refers to a region in space where every point in it is at the same potential. This usually refers to a scalar potential , although it can also be applied to vector potentials...
, with the apex of the tear-drop pointing towards the other star (and the apex is at the Lagrangian point
Lagrangian point
The Lagrangian points are the five positions in an orbital configuration where a small object affected only by gravity can theoretically be stationary relative to two larger objects...
of the system). It is different from the Roche limit
Roche limit
The Roche limit , sometimes referred to as the Roche radius, is the distance within which a celestial body, held together only by its own gravity, will disintegrate due to a second celestial body's tidal forces exceeding the first body's gravitational self-attraction...
which is the distance at which an object held together only by gravity begins to break up due to tidal force
Tidal force
The tidal force is a secondary effect of the force of gravity and is responsible for the tides. It arises because the gravitational force per unit mass exerted on one body by a second body is not constant across its diameter, the side nearest to the second being more attracted by it than the side...
s. It is different from the Roche sphere which approximates the gravitational sphere of influence of one astronomical body in the face of perturbations from another heavier body around which it orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
s. The Roche lobe, Roche limit and Roche sphere are named after the French
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
astronomer
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
Édouard Roche
Édouard Roche
Édouard Albert Roche was a French astronomer and mathematician, who is best known for his work in the field of celestial mechanics...
.
Definition of the Roche lobes
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 potentialScalar potential
A scalar potential is a fundamental concept in vector analysis and physics . The scalar potential is an example of a scalar field...
, so that, for example, the stellar surfaces lie along equipotential surfaces.
Close to each star, surfaces of equal gravitational potential are approximately spherical
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
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 Lagrangian point
Lagrangian point
The Lagrangian points are the five positions in an orbital configuration where a small object affected only by gravity can theoretically be stationary relative to two larger objects...
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.
Where matter moves
Mass transfer
Mass transfer is the net movement of mass from one location, usually meaning a stream, phase, fraction or component, to another. Mass transfer occurs in many processes, such as absorption, evaporation, adsorption, drying, precipitation, membrane filtration, and distillation. Mass transfer is used...
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
Conservative force
A conservative force is a force with the property that the work done in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the net work done by a conservative force is zero.It is possible to define a numerical value of...
(i.e. not representable by a scalar potential).
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
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
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
Nova
A nova is a cataclysmic nuclear explosion in a star caused by the accretion of hydrogen on to the surface of a white dwarf star, which ignites and starts nuclear fusion in a runaway manner...
(binary star
Binary star
A binary star is a star system consisting of two stars orbiting around their common center of mass. The brighter star is called the primary and the other is its companion star, comes, or secondary...
s consisting of a red giant
Red giant
A red giant is a luminous giant star of low or intermediate mass in a late phase of stellar evolution. The outer atmosphere is inflated and tenuous, making the radius immense and the surface temperature low, somewhere from 5,000 K and lower...
and a white dwarf
White dwarf
A white dwarf, also called a degenerate dwarf, is a small star composed mostly of electron-degenerate matter. They are very dense; a white dwarf's mass is comparable to that of the Sun and its volume is comparable to that of the Earth. Its faint luminosity comes from the emission of stored...
that are sufficiently close enough together that material from the red giant dribbles down onto the white dwarf), X-ray binaries
X-ray binary
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 compact: a white dwarf, neutron star, or black hole.The infalling matter releases...
and millisecond pulsar
Millisecond pulsar
A millisecond pulsar is a pulsar with a rotational period in the range of about 1-10 milliseconds. Millisecond pulsars have been detected in the radio, X-ray, and gamma ray portions of the electromagnetic spectrum. The origin of millisecond pulsars is still unknown...
s.
Geometry of the Roche lobe
The precise shape of the Roche lobe depends on the mass ratio, 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 isfor
and
for
where A is the semi-major axis
Semi-major axis
The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...
of the system and
is the radius of the Roche lobe around mass . These formulas are accurate to within about 2%.Another approximate formula by Eggleton is as follows:
where
. This formula gives results up to 1% accuracy over the entire range of .See also
- Binary starBinary starA binary star is a star system consisting of two stars orbiting around their common center of mass. The brighter star is called the primary and the other is its companion star, comes, or secondary...
- Hill sphereHill sphereAn astronomical body's Hill sphere is the region in which it dominates the attraction of satellites. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere. That moon would, in turn, have a Hill sphere of its own...
- Roche limitRoche limitThe Roche limit , sometimes referred to as the Roche radius, is the distance within which a celestial body, held together only by its own gravity, will disintegrate due to a second celestial body's tidal forces exceeding the first body's gravitational self-attraction...
- RocheworldRocheworldRocheworld is a science fiction novel by Robert Forward in which he uses a light sail propulsion system to set the crew on an interstellar mission. The spaceship and crew of 20 have to travel 5.9 light-years Rocheworld (first published in serial form in 1982; first book publication, under the...
for a hard science fiction novel based on this concept.