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Unformatted text preview: These lecture notes were prepared for Rutgers Physics 341/342: Principles of Astrophysics by Prof. Chuck Keeton, and modified by Profs. Saurabh Jha and Eric Gawiser. All rights reserved. c 2011 Lecture 9: Binary Stars (cont’d) I. Double-Lined Spectroscopic Binary Suppose we do not see the two stars individually, but in the spectrum we see two sets of absorption lines. We can use the lines (with the Doppler effect) to measure the line-of-sight velocities of the two stars. What can we do then? For illustration, let’s go back to circular orbits. From before we can write the true velocities as ~v 1 ( t ) = d~ r 1 dt =- μ m 1 aω (- sin ωt ˆ x + cos ωt ˆ y ) ~v 2 ( t ) = d~ r 2 dt = + μ m 2 aω (- sin ωt ˆ x + cos ωt ˆ y ) But we don’t see the true velocities, only the radial velocities , that is, the components along the line-of-sight, v 1 ,obs = v 1 ,y sin i =- μ m 1 a ω cos( ωt ) sin i v 2 ,obs = v 2 ,y sin i = + μ m 2 a ω cos( ωt ) sin i Notice that if we take the ratio most of the factors drop out: | v 1 ,obs | | v 2 ,obs | = m 2 m 1 So we can get the mass ratio very directly from the observations. What about the total mass? Let the maximum radial velocities be K 1 and K 2 (these are called the radial velocity amplitudes) K 1 = μ m 1 a ω sin i K 2 = μ m 2 a ω sin i = ⇒ K 1 K 2 = m 2 m 1 Notice that K 1 + K 2 = μ 1 m 1 + 1 m 2 a ω sin i ⇒ a = P 2 π K 1 + K 2 sin i 1 So now with Kepler III we have:...
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