The radial velocity of a star in an eccentric spectroscopic orbit is v_r(t) = + K [ (…

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

The radial velocity of a star in an eccentric spectroscopic orbit is vr(t)=γ+K[cos⁡(ν(t)+ω)+ecos⁡ω],v_r(t) = \gamma + K\big[\cos(\nu(t)+\omega) + e\cos\omega\big], where the true anomaly ν(t)\nu(t) is linked to time through Kepler's equation and the anomaly relation M=E−esin⁡E,tan⁡ν2=1+e1−etan⁡E2,M=2πP(t−t0),M = E - e\sin E, \qquad \tan\frac{\nu}{2} = \sqrt{\frac{1+e}{1-e}}\tan\frac{E}{2}, \qquad M = \frac{2\pi}{P}(t-t_0), with t0t_0 the time of periastron passage. The general mass function of the same orbit is f(m)=PK132πG(1−e2)3/2.f(m) = \frac{PK_1^3}{2\pi G}(1-e^2)^{3/2}. ((a)) Show that at e=0e=0 the eccentric and mean anomalies and the true anomaly are all equal, E=M=νE=M=\nu, for every value of tt. ((b)) Hence show that the radial-velocity curve above reduces exactly to the single sinusoid vr(t)=γ+Kcos⁡ ⁣[2πP(t−t0)+ω],v_r(t) = \gamma + K\cos\!\left[\frac{2\pi}{P}(t-t_0)+\omega\right], and note which general relation for KK this is consistent with once e=0e=0 is substituted in it. ((c)) Show that the general mass function reduces, in the same limit, to the circular-orbit mass function f(m)=PK13/(2πG)f(m) = PK_1^3/(2\pi G). ((d)) Now keep terms of first order in a small ee. Show that ν≈M+2esin⁡M\nu \approx M + 2e\sin M, and hence that vr≈γ+K[cos⁡(M+ω)+ecos⁡(2M+ω)].v_r \approx \gamma + K\big[\cos(M+\omega) + e\cos(2M+\omega)\big]. Explain why a fit to a nearly circular orbit determines ecos⁡ωe\cos\omega and esin⁡ωe\sin\omega far better than it determines ω\omega itself.

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