Calculating Angular Diameter of an Orbit Using Kepler's Law

[Kynsuo]

Homework Statement

A star orbits a black hole at the centre of a galaxy. Assuming these orbits are circular and that the distance to the centre of the galaxy is $d$, find the angular diameter of an orbit who's period is $T$.

Relevant Equations

I have the radius of the black hole $R$, the mass of the black hole $M$.

I am confused because the question implies that I need to do some sort of calculation with Kepler's law. I got
$r+d = \sqrt[3]{\frac{T^2 GM}{4 \pi^2} }$

But don't understand why I need this, since I already have the distance and the angular diameter should be $\arctan (2R/d)$ I think I am missing something.

 

[Doc Al]

But don't understand why I need this, since I already have the distance and the angular diameter should be $\arctan (2R/d)$ I think I am missing something.

You need the radius of the orbit, not the radius of the black hole.

 

[Kynsuo]

You need the radius of the orbit, not the radius of the black hole.

Thanks DocAI. I am given the distance to the centre of the galaxy which is $d$, I'm assuming that this is the radius of the orbit. I'm confused what this has to do with the period of orbit. Also, don't I need the radius of the black hole to take the ratio of the radius of the black hole to the radius of orbit in order to find the angular diameter?

 

[Doc Al]

I am given the distance to the centre of the galaxy which is d, I'm assuming that this is the radius of the orbit.

No. d is the distance that the star & black hole system is from you, not the radius of the orbit of the star around the black hole.

I'm confused what this has to do with the period of orbit.

The distance d has nothing to do with the period of the orbit.

Also, don't I need the radius of the black hole to take the ratio of the radius of the black hole to the radius of orbit in order to find the angular diameter?

Think of it this way. You are observing, from a distance "d", a star orbiting a black hole. How big the orbit appears to you -- its angular diameter -- depends upon the size of the orbit (the diameter of the orbit) and how far away it is (the distance "d"). The first step is to calculate, using the given information, the size of the orbit. Only then can you worry about the angular diameter.

 

[Kynsuo]

Thanks! This is what I was missing. Using the $T$, $G$ and $M$, I can find and expression for $R$, the radius of orbit. Then once I have the orbit I can find the angular diameter. Thanks.