How to Calculate Mass in a Binary System with Given Orbital Period and Radius?

[Lengalicious]

Homework Statement

See attachment.

Homework Equations

The Attempt at a Solution

Ok, so for Qd) my attempt was:
d = vt = 75x10^3 * (2*2.8*86400*365) = 1.32x10^13 m, (circumference of orbit)
because the star periodically shifts from red to blue shifted frequencies every 2.8 years so I guess the full orbit to be 2*2.8 years.

Then 2∏r = 1.32x10^13, -> r = 2.1x10^12m?

Now for Qe) All I could figure is that the relevant eqn is (2∏/T)² = G*(m_a + m_b)/(r_a + r_b)³

But with the information I have, 1stly from the assumpt that m_a >> m_b I can say that:

(2∏/T)² = G*(m_a)/(r_a + r_b)³ right?

Considering X-major is m_a. Now I can rearrange for the mass but I still have missing information, I just have radius of the orbit for x-minor, but not x-major. I even have the orbital period. Just not r_a.

Can someone give me a helping hand here, how do I find r_a? Also have I done everything thus far correctly?

 

[TSny]

Your work looks correct to me.

Under the assumption that m_a >> m_b, you have r_a << r_b. So, I think you can safely neglect r_a relative to r_b.

 

[Lengalicious]

Lol that does make sense, silly me, thanks. Hopefully that's correct. :)