How to Determine Masses in a Binary Star System?

[pierce15]

Hello,

I wasn't sure whether I should post this is the homework section since it's technically a textbook problem, but I figured I'd get better responses here. The problem is as follows:

Sirius is a visual binary with a period of 49.94 years. Its measured parallax is .37931"$\pm$.00158", and the angular extent of the semimajor axis of the reduced mass is 7.61". The ratio of the distances of Sirius A and B to the center of mass is $a_A / a_B = .466$. Find the masses of the two stars, assuming that the motion is in the plane of the sky.

First, you can use the ratio to get $m_A / m_B = 1/.466 = 2.146$. I'm pretty sure I next have to use the 7.61", but I don't know how. After that, I would have all the unknowns in Kepler's third except the masses, so I could solve the system. So how do I get the semimajor axis of the smaller star?

 

[mfb]

I wasn't sure whether I should post this is the homework section since it's technically a textbook problem, but I figured I'd get better responses here.

Textbook questions belong to the homework section. I moved it with a redirect in the original forum.

First, you can use the ratio to get $m_A / m_B = 1/.466 = 2.146$.

Okay.

I'm pretty sure I next have to use the 7.61", but I don't know how.

This is related to the true semi-major axis of the system, if you know the distance. There is another parameter given that allows to calculate the distance.

So how do I get the semimajor axis of the smaller star?

Find the semi-major axis of the reduced mass first.

 

[pierce15]

This is related to the true semi-major axis of the system, if you know the distance. There is another parameter given that allows to calculate the distance.

Using the parallactic angle yields $d [pc] = 1/p" = 1/.37921 = 2.6363 pc$. Now what?

By the way, the "reduced mass" just refers to the star with lower mass, right?

 

[mfb]

Using the parallactic angle yields $d [pc] = 1/p" = 1/.37921 = 2.6363 pc$. Now what?

You got an angle (as seen from earth) and a distance...

By the way, the "reduced mass" just refers to the star with lower mass, right?

No.

 

[pierce15]

Yeah, my bad... 7.61" = a / 2.636 pc ---> a = 3.00 E12 after converting 7.61" to rad and 2.636 to m. So is this the same a that goes in kepler's third equation? Or do I have to go back and use the semimajor axis ratio that I was given

 

[mfb]

So is this the same a that goes in kepler's third equation?

Should be. Check the link to the reduced mass.

Or do I have to go back and use the semimajor axis ratio that I was given

There was no given semi-major axis, you had to calculate it.

 

[pierce15]

Should be. Check the link to the reduced mass.
There was no given semi-major axis, you had to calculate it.

Got it. Thank you very much.