While helping an astronomy professor, you discover a binary star system in which

Question

While helping an astronomy professor, you discover a binary star system in which the two stars are in circular orbits about the system's center of mass. From their color and brightness, you determine that each star has the same mass as our Sun. The orbital period of the pair is 31.3 days , based on the oscillation of brightness observed as one star occludes (hides) the other. From this information you are able to ascertain the distance between the stars.

Calculate the distance between the stars.

Explanation / Answer

givn time period, T = 29.3 days

= 29.3*24*60*60

= 2531520 s

mass of sun, m = 1.99*10^30 kg

let the distance between the two stars is d.

so radius of orbit of stars motion, r = d/2

from the given data, linear speed of stars, v = 2*pi*r/T

now use Newton's universal law of gravitation.

G*m^2/d^2 = m*v^2/r

G*m^2/d^2 = m*(2*pi*r/T)^2/r

G*m^2/d^2 = m*(pi*d/T)^2/(d/2)

G*m^2/d^2 = m*pi^2*d^2/(T^2*d/2)

G*m^2/d^2 = 2*m*pi^2*d/T^2

G*m/d^2 = 2*pi^2*d/T^2

d^3 = G*m*T^2/(2*pi^2)

d = (G*m*T^2/(2*pi^2))^(1/3)

= (6.67*10^-11*1.99*10^30*2531520^2/(2*pi^2))^(1/3)

= 3.51*10^10 m <<<<<<<<------------Answer

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