Roche-Model for binary stars

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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

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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

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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

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Roche Lobes

• Lagrange points L1, L2, L3, and L4, L5

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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

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Roche potential wells
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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 gtgt a

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Binaries in Roche-Lobes

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Inner Lagrange point

• to find L1
• for which a solution for x1 can be found
  numerically for a given mass ratio q

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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