A binary holds two stars

PHYSICS · Multi-part · Fermi University Physics · Astrophysics & Cosmology · Celestial mechanics and planetary systems · Stellar death and compact objects

Stars from the Inside, Ch Accretion power: mass transfer, discs and the weighing of black holes

Problem

A binary holds two stars, treated as point masses, in a circular orbit. Mass moves from one star to the other by Roche-lobe overflow, and the transfer is conservative: no mass and no orbital angular momentum leaves the system. ((a)) The orbital angular momentum of a circular binary of separation aa is J=m1m2Gam1+m2J = m_1 m_2\sqrt{\dfrac{Ga}{m_1+m_2}}. Using J˙=0\dot J = 0 and m˙2=−m˙1\dot m_2 = -\dot m_1 (conservative transfer), show by logarithmic differentiation that a˙a=2m˙1(m1−m2)m1m2.\frac{\dot a}{a} = \frac{2\dot m_1(m_1-m_2)}{m_1 m_2}. ((b)) A donor of mass m1=2.0 M⊙m_1 = 2.0\,M_\odot transfers mass steadily onto a companion of mass m2=0.8 M⊙m_2 = 0.8\,M_\odot at a rate ∣m˙1∣=2.0×10−7 M⊙ yr−1|\dot m_1| = 2.0\times10^{-7}\,M_\odot\,\mathrm{yr^{-1}}. Evaluate a˙/a\dot a/a and state whether the orbit is presently shrinking or widening.

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