Two Black Holes orbiting as a binary system

[FelaKuti]

Homework Statement

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.

Homework Equations

F = GMM / r²

F = Mw²r

2pi/w = T

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.

 

[gneill]

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

 

[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?

 

[gneill]

So towards the centre point, but I'm guessing I should add the radiuses together perhaps?

Adding the radii gives you the total distance between the objects, which is a given value already.

I'm also confused about the masses here, should I add them together too?

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?

 

[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

 

[gneill]

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

Yes, provided that you define r and R appropriately.

 

[FelaKuti]

Got it, thanks~