# Two Black Holes orbiting as a binary system

1. Mar 22, 2017

### FelaKuti

1. The problem statement, all variables and given/known data
In 2009, astronomers found convincing evidence of two such black holes orbiting as a binary system. From data collected, they estimated that the separation of the black holes was 3.2 × 1015 m and that their masses were 1.6 × 1039 kg and 4.0 × 1037 kg.

The black holes orbit about a point 7.7 × 1013 m from the larger mass black hole.

Show that the orbital time of the binary system is about 100 years.

2. Relevant equations

F = GMM / r2

F = Mw2r

2pi/w = T

3. The attempt at a solution

I know I have to equate centripetal force to the gravitational force, then substitute the 2pi/w = T to get an expression involving the time period. However I'm just stumped on how to consider the radius in this question.

2. Mar 22, 2017

### Staff: Mentor

Hint: The radius of the orbit circle for a given object is not the same as the distance between the objects.

3. Mar 22, 2017

### FelaKuti

So towards the centre point, but I'm guessing I should add the radiuses together perhaps? I'm also confused about the masses here, should I add them together too?

4. Mar 22, 2017

### Staff: Mentor

Adding the radii gives you the total distance between the objects, which is a given value already.
No. How does Newton's law of gravitation work? How about centripetal force for a rotating body? Does it ever depend on anything other than the rotating body itself?

Make a quick sketch showing the circle that one of the masses describes as it orbits. What's its center and radius? How does that radius compare to the distance between the two objects? What distance matters as far as Newton's law of gravity is concerned? What distance matters as far as circular motion (centripetal force) is concerned?

5. Mar 22, 2017

### FelaKuti

For centripetal force it's the distance between the mass and the centre.
For Newton's law of gravitation it's the distance between the two objects.

so it should be this?

So F = Mw2R = GMm / r2

6. Mar 22, 2017

### Staff: Mentor

Yes, provided that you define r and R appropriately.

7. Mar 22, 2017

### FelaKuti

Got it, thanks~