Binary system of stars (##\alpha## - centauri)

[Aleolomorfo]

Homework Statement

$\alpha$-Centuary is in a binary visual system with another star. Their separation, from their CM, is 8.0'' and 9.7''. The distance from the Earth is 1.31pc. Their revolution period around the CM is 80.1 years. I have to find masses and luminosities for each star.

Homework Equations

Third Newton's law: $\omega^2 = \frac{G(M_1+M_2)}{a^3}$

The Attempt at a Solution

[/B]
With the help of this picture I can calculate $a=1.7''\times1.31pc$. Then I can solve this system of equations:
$$\omega^2 = \frac{G(M_1+M_2)}{a^3}$$
$$\omega = \frac{2\pi}{T}$$
In this way I find only $M_1+M_2$. I need another equation but I do not find it.
For the luminosity I can use the scaling relation that $L\propto M^4$. However, firstly I need to find $M_1$ and $M_2$ separately.

 

[Aleolomorfo]

Sorry but I have noticed that I had switched the separations from the CM. I put 9,7'' on the shorter segment and viceversa, but it is not a problem since I have used the sum.

 

[Aleolomorfo]

Maybe I have an idea: I can use the definition of CM.
$$M_1\times r_1=M_2\times r_2$$ with $r1=1,31pc\times 9.7''$ and $r_2=1.31pc\times 8''$
I think it is ok, isn't it?

 

[tms]

You can't just multiply arcseconds by distance to get the angular distance; you have to convert to radians first.

 

[Aleolomorfo]

You can't just multiply arcseconds by distance to get the angular distance; you have to convert to radians first.

Yes, I have implied it, I should have written explicitly, sorry. I have done it and I have found reasonable results, so I think it is ok.

 

[gneill]

Yes, I have implied it, I should have written explicitly, sorry. I have done it and I have found reasonable results, so I think it is ok.

You should show your work and the values that you obtained.