A detached eclipsing binary in a circular orbit has a double-lined spectroscopic…

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

Stars from the Inside, Ch Binary stars: fitted orbits, modelled eclipses and the Roche geometry

Problem

A detached eclipsing binary in a circular orbit has a double-lined spectroscopic solution giving asin⁡i=20.0 R⊙a\sin i = 20.0\,R_\odot. A joint light-curve and radial-velocity fit gives the inclination i=87.0∘i = 87.0^\circ. The eclipse is total: first contact (external tangency, the geometric onset of eclipse) occurs at a phase angle with sin⁡ϕ1=0.230\sin\phi_1 = 0.230, and second contact (internal tangency, the onset of totality) occurs at sin⁡ϕ2=0.070\sin\phi_2 = 0.070, where sin⁡2ϕ=(r1±r2)2−cos⁡2isin⁡2i,\sin^2\phi = \frac{(r_1\pm r_2)^2 - \cos^2 i}{\sin^2 i}, the ++ sign applying at first/fourth contact (giving r1+r2r_1+r_2) and the −- sign at second/third contact (giving r1−r2r_1-r_2), with r1=R1/ar_1 = R_1/a and r2=R2/ar_2 = R_2/a the fractional radii. Label the larger star 1, so that r1≥r2r_1 \ge r_2. ((a)) Find the fractional radii r1r_1 and r2r_2. ((b)) Find aa, and hence R1R_1 and R2R_2 in solar radii.

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