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to come into rotational equilibrium with the Moon’s orbit, estimate the time (from the current
epoch) when this equilibrium conﬁguration will be reached.
8. Eclipsing binaries. Assume that two stars are in circular orbits about a mutual center of mass and
are separated by a distance a. Assume also that the binary inclination angle is i (deﬁned as the angle
between the line-of-sight and the orbital angular momentum vector, with 0◦ ≤ i ≤ 90◦ ) and that the
two stellar radii are R1 and R2 . Find an expression for the smallest inclination angle that will just
barely produce an eclipse.
9. Ensemble of binaries. The table below (from a paper by Hinkle et al. 2003) contains the measured
orbital parameters (binary period Porb and radial velocity semi-amplitude K1 ) for a group of “singleline” spectroscopic binaries (for which the Doppler radial velocity curve for M1 is measured, but M2
is not directly observed). You may assume that all the orbits are circular. Star
V343 Ser Porb
(km s−1 )
2.7 (a) Recall from class the deﬁnition of the binary mass function,
f1 ≡ (M2 sin i)3
4π 2 (a1 sin i)3
(M1 + M2 )2
GPorb Derive an expression for the mass function in terms of the observables Porb and K1 . What is the
physical signiﬁcance or interpretation of the mass function?
(b) The systems in the table are a group of symbiotic binaries, which consist of a red giant star with
a hot, degenerate white dwarf companion. The radial velocity measurements are for the red giant
component. Typical component masses are M1 = 1.5M for the red giant and M2 = 0.56M for
the white dwarf.
A random ensemble of binaries (i.e., one whose orbital angular momentum vectors are isotropicall...
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