Roche lobe

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 tear-drop-shaped region bounded by a critical gravitational equipotential, with the apex of the tear drop 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 another heavier body around which it orbits. 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 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.

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.

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

Potential array

In the gravity potential graphics, L1, L2, L3, L4, L5 are 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 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 the any of Hill sphere (inner circles of blue and light blue) and the communal gravity region (8's rings of yellow and green in the inner side).

Hill sphere and horseshoe orbit

L2 and L3 are gravitational perturbation equilibria points. Passing through these two equilibrium points, debris can commute between the external region (8's rings of yellow and green in the outer side) and the communal gravity region of the binary system.

L4, 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.

Geometry

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 is

$\frac{r_1}{A} = 0.38+0.2\log\frac{M_1}{M_2}$

for $0.3<\frac{M_1}{M_2}<20$

and

$\frac{r_1}{A} = 0.46224\left(\frac{M_1}{M_1+M_2}\right)^{1/3}$

for $\frac{M_1}{M_2}<0.8$

where A is the orbital separation of the system and $r_1$ is the radius of the Roche lobe around mass $M_1$. These formulas are accurate to within about 2%.

Another approximate formula by Eggleton is as follows:

$\frac{r_1}{A} = \frac{0.49q^{2/3}}{0.6q^{2/3} + \ln(1 + q^{1/3})}$

where $q=M_1/M_2$. This formula gives results up to 1% accuracy over the entire range of $q$.

Sources

• Morris, S.L., "Two Mathematical Expansions of the Roche Equipotentials", PASP 106: 154-155, 1994 February
• Morris, S.L., "The Limits of Inclination for Binary Star Partial Eclipses", ApJ 520: 797-804, 1999 August 1