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.
Required:
Calculate the distance between the stars.

Answer 1

Answer:

The value is $d = 3.66 *10^{10} \ m$

Explanation:

From the question we are told that

The orbital period is $T = 31.3 \ days = 31.3 * 86400 = 2704320 \ s$

Generally for the centripetal force acting one of the stars will be equal to the gravitational force between the stars and this can be mathematically represented as

$F_g = F_c$

=> $\frac{Gm*m}{d^2} = \frac{mv^2}{r}$

Here m represents the mass of each star given that they have same mass and this equal to the mass of the sun whose value is

$m = 1.989*10^{30} \ kg$

d(2 * r) is the diameter of the orbit which is distance between each star

r is the radius of the orbit

G is the gravitational constant with value $G= 6.67*0^{-11} \ m^2/kg \cdot s^2$

v is the velocity of the stars which can be mathematically represented as

$v = \frac{2 \pi r}{T}$

So

$\frac{Gmm }{(2 * r)^2} = \frac{m (\frac{2\pi r}{T} )^2}{r}$

=> $r =[ \frac{Gm T^2}{16 \pi ^2} ]^{\frac{1}{3} }$

=> $r = [\frac{6.67 *10^{-11} * (1.989 *10^{30}) * 2704320^2}{16 * (3.142)^2} ]^{\frac{1}{3} }$

=> $r = 1.83 *10^{10} \ m$

Generally the distance between the star is

$d = 2 * r$

=> $d = 2 * 1.83 *10^{10}$

=> $d = 3.66 *10^{10} \ m$