Question

Hunting a black hole. Observations of the light from a certain star indicate that it is part of a binary (two-star) system. This visible star has orbital speed v = 270 km/s, orbital period T = 21.5 days, and approximate mass m₁ = 5.3M_s, where M_s is the Sun's mass, 1.99 x 10³⁰ kg. Assume that the visible star and its companion star, which is dark and unseen, are both in circular orbits (see the figure). Find the ratio of the approximate mass m₂ of the dark star to M_s.

mo 0 19 m

Answer 1

Using newtons second law, F-Gm,m, _ m,v? FR Using center of mass relative to my m (0)+m,r 1 = m, + m2 mr m; + m2 m2 + m2; m2 Using the orbital speed equation, 2777 т Then ma Using the force equation,

m, VT (m+m,)? 27G (270x10°) (21.5)(86400) 21(6.67x10-1) = 8.72x10 kg m," ,-43.84 M (m, + m2) _m 2-43.84 M (5.3M. +m,) m² = 43.84 m2 +10.6m2 +28.09M.) m2 = 36.91m3 + 464.704m2 +1219.106M, m, = 47.28M = 47M, m2 = 47 M.