Answer to Question #86970 in Astronomy | Astrophysics for Maria

Question

Two equal-mass stars maintain a constant distance apart of 9.0×1010 m and rotate about a point midway between them at a rate of one revolution every 12.6 yr.
PART A
Why don't the two stars crash into one another due to the gravitational force between them?
PART B
What must be the mass of each star?
Express your answer using two significant figures.

Answer 1

PART A

Gravity is not strong enough to get them any closer, and if it were to somehow get a little stronger, the stars would speed up as they move and they would reach another equilibrium point with the stronger gravity.

PART B

1 yr=3.156×10⁷ s

T=12.6 yr=12.6×3.156×10⁷ s

The planets are orbiting a point midway between them. Their radius of orbit is half their separation. Since they are 9.0x10¹⁰ m apart, they move in a circle 4.50x10¹⁰ m in radius. We have gravity providing the centripetal force for them to orbit each other, so we set the force of gravity equal to the centripetal force needed for them to orbit the center. The only hitch is that if "r" is the radius of orbit, the center to center distance is now 2r. So let's set up the orbital condition with the period in it:

$$\frac{G(m1m2)}{(2r)^2}=\frac{m1(4π^2r)}{T^2}$$

where m1=m2=m

We find:

$$m =\frac{16π^2r^3}{GT^2}$$

m=1.41×10²⁷ kg

Answer: 1.41×10²⁷ kg