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

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

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

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

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

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

Roche potential wells

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

Binaries in Roche-Lobes

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

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