Calculating Mass of Stars in a Binary System

[chazgurl4life]

Q:
Suppose that a binary star system consists of two stars of equal mass. They are observed to be separated by 340 million kilometers and take 5.0 Earth years to orbit about a point midway between them. What is the mass of each?
I figured out that:
mass=4pi^2(radius)^2/Gravitational Force(#of years)*(distance)2
m= [4(3.14)^2(3.3x10^29)^3]/[(6.67x10^-11){(8.0 years)(3.4x10^7}^2] =3.33x10^29 then (3.33x10^29)/2 = 1.7x10^29

I don't know what I'm doing wrong here. Any ideas?

 

[tony873004]

Q:
mass=4pi^2(radius)^2/Gravitational Force(#of years)*(distance)2

Maybe an algebra mistake in getting to this point. Should be:
mass=4pi^2(radius)^3/(Gravitational Force(#of years*seconds per year)^2)

How did you know your answer was wrong? It's very close to correct. Does the back of the book give the answer?

 

[chazgurl4life]

so if i reply this equation it comes out as:
m=4pi^2(3.3x10^29)^3/6.67e-11(5yearsx 3.155815296E7 sec per yr)^2

is that right? or am i using the wrong radius? isn't the radius half the distance between the two stars? if that's true than the radius id 170 million, isn't it? I am so confused!

 

[tony873004]

Where did you get 3.3 x 10^29. They give you the distance of 340,000,000 million kilometers. This becomes your a or radius (3.4 x 10^11 meters)

It's not half the distance between the 2 stars since. Pretend 1 star is still, and the other orbits it. It will trace an orbit whose diameter is twice the distance between the 2 stars. Therefore, the distance between the 2 stars becomes the radius, or semi-major axis in this problem.