A binary-star system contains a visible star and a black hole moving around their center of mass in circular orbits with radii r1 and r2 , respectively. The visible star has an orbital speed of v=5.36x105 ms-1 and a mass of m1 =5Ms ,where Ms= 1.98x1030kg is the mass of our Sun. Moreover, the orbital period of the visible star is T = 30 hours.(a) What is the radius r1 of the orbit of the visible star?(b) Calculate the mass m2 of the black hole in terms of MS . [Hint: One root of the equation x3 = 20a(5a+x)2  , where a is a constant, is x = 28a .]

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Asked Dec 16, 2019
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A binary-star system contains a visible star and a black hole moving around their center of mass in circular orbits with radii r1 and r2 , respectively. The visible star has an orbital speed of v=5.36x105 ms-1 and a mass of m1 =5Ms ,where Ms= 1.98x1030kg is the mass of our Sun. Moreover, the orbital period of the visible star is T = 30 hours.
(a) What is the radius r1 of the orbit of the visible star?
(b) Calculate the mass m2 of the black hole in terms of MS . [Hint: One root of the equation x3 = 20a(5a+x)2  , where a is a constant, is x = 28a .]

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Expert Answer

Step 1

A binary star system is represented by the figure given below.

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W2 72

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Step 2

a)

As the visible star is executing circular motion about the center of mass, its radius r1 is calculated as,

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2лк V = v,T 2л

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Step 3

Substitute 5.36(105) ms-1 for v1 and 30 hr for T and solve for ...

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60min 30hr 60s (5.36x10°ms ) lhr Imin = 9.21×10°m

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