Roche Potential 1
Binary Stars and the Roche Potential
Binary star
Lagrange points
The Roche potential function for a close binary star is plotted above (the larger star is twice the mass of the
smaller star) using an in-house Windows forms-based application written in C#. Lines connecting regions of the
same colour are gravitational equipotentials along which the gravitational field is constant - in other words, the
plots show the gravitational field around a close binary star. Surfaces of equal pressure, such as the surfaces of
stars, coincide with these equipotentials and so the plots show us the possible shapes of the stars. The plots are
colour coded from red, through green to blue. If each star fills the red spherical regions, then although the two
stars are very close to one-another, they are not in contact. However, if the two stars fill the hour-glass shape,
then the two stars are in physical contact (a contact binary) and joined to one-another by a narrow neck region,
as illustrated in the 3D Pov-Ray model of this equipotential shown below:
Above: a 3D model of a contact binary. The
hourglass shape, formed when the two stars are
barely touching at a point, forms the pair of Roche
lobes. Both the stars here fill slightly beyond their
Roche lobes. Stars that fill their Roche lobe are
elongated toward their companion.

Left: two stars of the same mass that are not in
contact, but form a close binary. Neither star is
filling its Roche lobe.
For the plots of the contact binary shown above, the larger star was twice the mass of the smaller star and a = 1
and the 2D plots represent slices through the equatorial plane of the binary (z = 0). The plot below is labelled with
the centre-of-mass of the system (cross) and the five Lagrange points (indicated approximately) where
equipotential lines meet. In the centre of the blue areas are the L4 and L5 Lagrange points, which form 60 degree
angles with the centre of the more massive star and the axis connecting the two stellar centres and so are angled
back from the main star. These two points create regions of gravitational stability, created by Coriolis forces, where
objects may reside without falling into either star (assuming the radiation and temperature would not be too
severe). These points also exist around planets and moons and asteroids may reside here, as in the Trojan
asteroids that orbit Jupiter's Lagrange points. Such asteroids are called lagrangian asteroids, or Trojan asteroids
(after the case of Jupiter).

L1 is a particularly important point. It is a saddle point, which makes it unstable - it is like a narrow mountain pass
between the two potential valleys where the star centres reside and matter will tend to fall down one side or the
other. If one star fills its Roche lobe whilst its companion is still well within its Roche lobe then matter will begin to fall
from the Roche-lobe filling star, through L1, toward the companion. As the binary system is rotating, this stream of
matter will spiral and form a disc of matter slowly spiralling in to the companion star - a so-called
accretion disc.
Mass ratio 1.0
Mass ratio 2
Mass ratio 1.898
Mass ratio 1.85
Mass ratio 1.75
Mass ratio = 1:1 = 1
Mass ratio = 1.75
Mass ratio = 1.85
Mass ratio = 1.898, showing the Roche
lobes - should both stars fill these lobes
then they will just be touching at L1.
Mass ratio = 2
Mass ratio = 3
Mass ratio = 3, scale halved
Mass ratio = 3. Both stars could become
enclosed by a common envelope.