A double-lined eclipsing binary has a circular orbit ( )

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 double-lined eclipsing binary has a circular orbit (

e=0

), period

P = 4.00\,\mathrm{d}

, and radial-velocity semi-amplitudes

K_1 = 150\,\mathrm{km\,s^{-1}}

and

K_2 = 50\,\mathrm{km\,s^{-1}}

. Its light curve shows total eclipses consistent only with an inclination indistinguishable from

i=90^\circ

(edge-on). The light-curve model gives fractional radii corresponding to actual radii

R_1 = 4.60\,R_\odot

(the star whose own velocity amplitude is

K_1

) and

R_2 = 1.60\,R_\odot

(the star whose own velocity amplitude is

K_2

). You may use the double-lined mass relations

\begin{aligned} m_{1}\sin^3 i &= \frac{P(1-e^2)^{3/2}(K_1+K_2)^2K_2}{2\pi G}, \\ m_{2}\sin^3 i &= \frac{P(1-e^2)^{3/2}(K_1+K_2)^2K_1}{2\pi G}, \end{aligned}

and the Eggleton formula for the volume-equivalent Roche-lobe radius of a star of mass ratio

q

(its own mass over its companion's),

\frac{r_L}{a} = \frac{0.49\,q^{2/3}}{0.6\,q^{2/3}+\ln(1+q^{1/3})}.

((a)) Find the masses

m_1

and

m_2

in solar masses. ((b)) Find the orbital separation

a

in solar radii. ((c)) Find the Roche-lobe radius of each star, and decide whether each star lies inside, or fills, its lobe. ((d)) Classify the system as detached, semi-detached or contact, and state which star (if any) is transferring mass. ((e)) Given that, all else equal, a more massive star exhausts its core hydrogen and evolves off the main sequence sooner than a less massive star born at the same time, explain the puzzle this system's masses and radii pose.

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