1. Roche-Model for binary stars
Stars deform in close binary systems
due to mutual gravitational potential
tides
rotation
Observations
show aspherical distortions in close systems
e.g. from light curves in eclipsing systems
Small perturbations
Use Legendre polynomials
When strongly deformed, need description for ellipsoidal shape of star
Use potential in system
effective surfaces
Important for binary evolution

2. Potential in close binaries
P(x,y,z)
C: centre of mass
reference frame centred on more massive star m1
rotating with angular velocity w, same as binary system
circular orbit
Potential at P(x,y,z) is then r1 r2 y m1 m2 C x z

3. then we can define if we normalise to a =1
the normalised gravitational potential,
and the mass ratio

4. Equipotential surfaces
The total potential may then be calculated at any point P with respect to the binary system.
Surfaces of constant potential may be found
shape of stars is given by these equipotential surfaces
Deformation from spherical depends on size relative to semi major axis, a, and mass ratio q

5. Roche Lobes
Lagrange points L1, L2, L3, and L4, L5

6. Lagrange points Points where L1 - Inner Lagrange Point
in between two stars
matter can flow freely from one star to other
mass exhange
L2 - on opposite side of secondary
matter can most easily leave system
L3 - on opposite side of primary
L4, L5 - in lobes perpendicular to line joining binary
form equilateral triangles with centres of two stars
Roche-lobes:: surfaces which just touch at L1
maximum size of non-contact systems

7. Roche potential wells

8. Types of Binaries Detached systems Semi-detached systems
Inside Roche-lobes
Semi-detached systems
at least one star filling its Roche-lobe
Contact systems
two stars touching at inner lagrange point L1
Over-Contact systems
two stars overfilling Roche-lobes
neck of material joining them
Common-envelope systems
Two stars have one near-spherical envelope
R >> a

9. Binaries in Roche-Lobes

10. Inner Lagrange point to find L1:
for which a solution for x1 can be found numerically for a given mass ratio q

11. Roche-Lobe Effective size Effectively, it is a tidal radius where
radius of Roche-lobe RL
find by numerical integration of potential
Effectively, it is a tidal radius where
densities in lobes are equal