In a binary each star moves on its own ellipse about the centre of mass

PHYSICS · Subjective · Fermi University Physics · Scientific Practice & Experimental Methods · Astrophysics & Cosmology · Celestial mechanics and planetary systems

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

Problem

In a binary each star moves on its own ellipse about the centre of mass, with semi-major axes a1a_1 and a2a_2 and a=a1+a2a=a_1+a_2, and each shows a velocity amplitude Kj=2πajsin⁡iP1−e2,j=1,2.K_j = \frac{2\pi a_j\sin i}{P\sqrt{1-e^2}},\qquad j=1,2. (a) Show that K1/K2=m2/m1K_1/K_2 = m_2/m_1 and that asin⁡i=P1−e2 (K1+K2)/2πa\sin i = P\sqrt{1-e^2}\,(K_1+K_2)/2\pi. (b) Using Kepler's third law (2π/P)2a3=G(m1+m2)(2\pi/P)^2a^3 = G(m_1+m_2), derive m1sin⁡3i=P(1−e2)3/22πG(K1+K2)2K2m_1\sin^3 i = \frac{P(1-e^2)^{3/2}}{2\pi G}(K_1+K_2)^2K_2 and the matching expression for m2sin⁡3im_2\sin^3 i. (c) For a pair with K1=K2K_1 = K_2, each amplitude measured independently to 22 per cent, find the fractional uncertainty in m1sin⁡3im_1\sin^3 i. Why is it larger than the uncertainty in either amplitude?

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