How Do You Convert Arcseconds to Meters for Calculating Celestial Masses?

[tomeatworld]

Homework Statement

By mapping the star S2 close to Galactic Centre, the orbit is found to have a semi-major axis of 0.119 arcsec , an inclination of 46$^{o}$, an orbital period of 15.2 years, an eccentricity of 0.87 and a pericentre separation of 0.0155.
a) Estimate the mass of object at the centre of the Galaxy.
b) Place a limit on the density.

Edit: For a) told to assume distance to galactic centre is 8kpc.

Homework Equations

T$^{2}$ = $\frac{4\pi^{2}}{GM} \alpha^{3}$

The Attempt at a Solution

The problem I have is using the semimajor axis in the calculation. It's given in arcsec but (I assume) it's needed in m. How do you convert? All attempts I made, using trig, gave me the wrong answer. Do I need to convert? How would I go about it?

 

[Filip Larsen]

Perhaps if you know the distance to the Galactic Center over which the specified angles are observed you could convert them to a distance.

 

[tomeatworld]

Just added an edit. We are told to assume the distance to galactic centre is 8kpc.

Ok. Very silly of me. Worked it out perfectly. Thanks for the push in the right direction!

I can't however work the second bit out. It's wanted in $M_{SUN}$ $pc^{-3}$ but I just don't know where to start.

 

[gneill]

Just added an edit. We are told to assume the distance to galactic centre is 8kpc.

Ok. Very silly of me. Worked it out perfectly. Thanks for the push in the right direction!

I can't however work the second bit out. It's wanted in $M_{SUN}$ $pc^{-3}$ but I just don't know where to start.

Well, let's see. You've worked out a mass for the central object in part (a), and the orbit of the star would seem to describe certain limits to the area enclosed by that orbit...

 

[tomeatworld]

The 8kpc is for the Suns orbit. How can you use that for S2's orbit?

 

[gneill]

The 8kpc is for the Suns orbit. How can you use that for S2's orbit?

I thought you'd solved part (a)? What did you use for "a" in your period equation? What would be the pericentre separation?

 

[tomeatworld]

So using the pericentre (is that the right point?) you can get the closest approach to the black hole. Then assuming a sphere get the volume etc etc.

I used the semimajor axis in a).

 

[gneill]

So using the pericentre (is that the right point?) you can get the closest approach to the black hole. Then assuming a sphere get the volume etc etc.

I used the semimajor axis in a).

Yes, the pericentre separation is the distance of closest approach of the star to the galactic centre. And yes, this pericentre distance will give you a upper bound on the volume of whatever it is that's inside that distance.

 

[tomeatworld]

Ok, thanks! Sorts everything out I think!