Chapter 13, Problem 054 Your answer is partially correct. Try again. Hunting a b

Question

Chapter 13, Problem 054 Your answer is partially correct. Try again. 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 = 9.02 days, and approximate mass m1 = 5.3Ms, where Ms 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 Ms

Explanation / Answer

w = 2pi / T = 2 pi / (9.02 x 24 x 3600s )

w = 7.905 x 10^-6 rad/s

v = w r1

r1 = (270 x 10^3) / w = 3.416 x 10^10 m

Fg = G m1 m2 / (r1 + r2)^2 = m1 v1^2 / r1

m1 r1 = m2 r2

r2 = (m1 / m2) r1

G m2 / (r1 + (m1/m2)r1)^2 = v1^2 / r1

G m2 / r1 (1 + (m1/m2))^2 = v1^2

m2 = 3.733 x 10^31 (m2 + m1)^2 / m2^2

m2^3 = 18.8M (m2^2 + m1^2 + 2m1 m2)

m2^3 - (18.8 M) m2^2 - (199 M^2) m2 + 528 M = 0

m2 = 25.7 Ms

m2 / Ms = 25.7 ........Ans

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