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# Find an expression for the smallest inclination angle

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Unformatted text preview: Earth 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 EG And Z And T CrB BF Cyg V1329 Cyg CI Cyg AG Dra V443 Her BX Mon RS Oph V2116 Oph AG Peg AX Per FG Ser V343 Ser Porb (days) 482.6 759.0 227.6 757.2 956.5 853.8 548.7 599.4 1259 455.7 1042 818.2 682.1 633.5 450.5 K1 (km s−1 ) 7.3 6.7 23.9 6.7 7.8 6.7 5.9 2.5 4.6 16.7 16.0 5.4 7.8 6.9 2.7 (a) Recall from class the deﬁnition of the binary mass function, f1 ≡ (M2 sin i)3 4π 2 (a1 sin i)3 = . 2 (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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## This note was uploaded on 01/24/2014 for the course PHYS 8.901 taught by Professor Deeptochakrabarty during the Spring '06 term at MIT.

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