Explanation A certain quaternary star system consist of three stars, each of mass m , moving in a same circular orbit of radius r about a central star of mass M . The stars orbit in the same sense and the positioned one-third of a revolution apart from one another. The diagram for the given condition is shown below. Figure (1) As the stars are moving in the same circular orbit of radius r and positioned one-third of a revolution apart from one another, their positions forms an equilateral triangle. Formula to calculate the period of the each of three stars is, T = 2 π r v (1) Here, v is the velocity of the each star and r is the radius of the circle. The net force act on the each star is, F = F 2 + 2 F 1 (2) Here, F 1 is the force exerted by the one star on the other star and F 2 is the force exerted by the star on the star placed at the center. Formula to calculate the force exerted by the one star on the other star is, F 1 = G m 2 ( d ) 2 cos θ Here, G is the universal gravitational constant, m is the mass of the star at placed at the circumference of the circle, d is the distance between the position of two stars and θ is the angle made by the force with the horizontal. Substitute 2 r cos 30.0 ° for d and 30.0 ° for θ in above expression. F 1 = G m 2 ( 2 r cos 30.0 ° ) 2 cos 30.0 ° Formula to calculate the force exerted by the star on the star placed at the center is, F 2 = G M m r 2 Here, M is the mass of the star at placed at the center of the circle and r is the radius of the circular orbit. Substitute G m 2 ( 2 r cos 30 ° ) 2 cos 30 ° for F 1 and G M m r 2 for F 2 in equation (2)

9th Edition

ISBN: 8220100461262

Author: SERWAY

Publisher: Cengage Learning US

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Chapter 13, Problem 13.66AP

To determine

The time period of each of three stars is T=2πr3G(M+m3).

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The time period of each of three stars is T = 2 π r 3 G ( M + m 3 ) .

Solution Summary: The author describes a certain quaternary star system consisting of three stars, each of mass m, moving in the same circular orbit, positioned one-third of the revolution apart from one another.

Concept explainers

The time period of each of three stars is T=2πr3G(M+m3).

Chapter 13 Solutions