Distance between Binary Star Systems

[G01]

A binary star system has a period of T=90 days. Each star is 2X the mass of the sun. They rotate like a dumbell around the center of mass. what is the distance d between them?
OK first, Mass of sun = $$1.99 X 10^{30} kg$$
Lets concentrate on one of the stars rotating around the center of mass. The force of gravity on this star is the centripetal force keeping the star circling the center. So:
$$F_{g} = F_{c}// \frac{\G(2M)(4M)}{r^2} = (2M)r\omega^2// \frac{G(4M)}{\omega^2} = r^3// r = \sqrt [3] {\frac{\G(4M)}{\omega^2}}$$
If $$\omega = 2 \pi f$$ converting T to f and converting to the right units we get
$$\omega = 8.08 X 10^{-7} rad/s$$
Now the answer in the back of my book gives the answer as $$9.33 X 10^{10} m$$ (Knight, Chapt 12 #61) I'm getting double that value somehow. I can't see what I did wrong. I'm beginning to think that the answer in the back is giving the radius of rotation when it should be giving the diameter. Chances are I'm doing something stupid. Hopefully someone here may see what I'm doing wrong. Thanks for you help. Also if someone can tell me how to make something go to a new line in latex my formulas will be much more legible

 

[maverick6664]

I've got the same answer as your book.

You seem to make one or two mistakes in the distance of two stars or the mass of each star in $$F_g$$

My calculation is

Let $$r$$ be the radius (half the distance).

$$F_g = \frac {G(2M)^2}{(2r)^2}, F_c = (2M)r\omega^2$$

so

$$r = \sqrt[3]{\frac {GM}{2 \omega^2} } = 4.66*10^{10} m$$ where $$G= 6.672*10^{-11}$$

$$distance = 2r = 9.33*10^{10} m$$

 

[G01]

ahh i see thank you.

 

[salu azmat]

how did u do the second step? i.e.,r=(GM/2w square)*1/3

 

[rwisz]

Hi there,

Thank you for sharing your calculations and thought process. It seems like you have done the calculations correctly, and your answer is indeed double the value given in the book. This could be due to a couple of reasons:

1. The answer in the book may be incorrect. It is always possible for errors to occur in published materials, and it is important to double-check and verify information.

2. The answer in the book may be giving the radius of rotation instead of the diameter. As you mentioned, this is a possibility and it would explain the discrepancy in your calculations.

In terms of formatting in LaTeX, to go to a new line you can use two forward slashes (//) or use the "newline" command (\newline).

I hope this helps and keep up the good work in your calculations!