Binary Star Analysis: Mass Calculation from 10pc, 33yr Period, i=90°

[CaptainEvil]

Homework Statement

We observe a binary system in which the two stars are 1 and 2 arc sec, respectively, from the center of mass. The system is 10pc from us. The period is 33 yr. What are the masses of the two stars, assuming that i=90 degrees?

Homework Equations

The Attempt at a Solution

I want to derive an equation for the masses based on what we know, so here goes.

Based on a simple sketch of the system, we know m_ar_a = m_br_b.

We can label the distance between the two stars r = r_a + r_b

re-arranging and substituting yields m_ar_a = m_b(r-r_a)

Then r_a = m_br/m_a + m_b = m_br/M where M = m_a + m_b

Now, force of Gravity must equal centripetal force in this binary system, so Gm_am_b/r² = m_av_a²/r_a

since v = d/t, v_a = 2$$\pi$$r_a/T where T is the period.

re-arranging once again yields r_a = Gm_bT²/4r²$$\pi$$²

subbing in from our equation for r_a yields M = 4r³$$\pi$$²/GT²

Now we have an equation for the total mass of the system, but I need individual masses.

Can anyone confirm my equation? Am I on the right track? Where does the inclination of the system come in? Is it just for an assumption? Likewise, where does the system distance d away from us come into play? Am I missing something here? Any help would be appreciated thanks.

 

[Jhero]

I would first like to commend you on your attempt at deriving an equation for the masses of the two stars in the binary system. Your approach seems to be on the right track, but there are a few things that need to be clarified.

Firstly, the formula you have derived for the total mass of the system is correct, assuming that the inclination angle (i) is indeed 90 degrees. This is because when i=90 degrees, the distance between the two stars (r) is equal to the distance between the center of mass and one of the stars (ra or rb). However, if the inclination angle is different, the formula would need to be modified accordingly.

Secondly, the distance of the system from us (d) does not come into play in this equation. This is because the distance between the two stars (r) is already taken into account in the formula. The distance from us only affects the apparent separation between the stars, but not their actual masses.

Lastly, in order to solve for the individual masses, you would need to have another equation that relates the period of the binary system to the individual masses of the stars. This equation is known as Kepler's Third Law and states that the square of the period of a binary system is proportional to the cube of the semi-major axis (a) of the orbit, which is related to the individual masses of the stars.

Hope this helps clarify your approach and good luck with your calculations!