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Roche-Model for binary stars

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

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Roche-Model for binary stars

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  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. if we normalise to a =1 • then we can define • 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 • 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 • radius of Roche-lobe RL • find by numerical integration of potential • Effectively, it is a tidal radius where • densities in lobes are equal

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