Determining the masses of 2 stars in a binary system.

[knowlewj01]

Homework Statement

A binary star is resolved on the sky, the orbit is circular, it has a period of 30 years and the semi-minor and semi-major axes are observed to be 0.5 and 4 arcseconds respectively. If the distance to the system is 10pc, what are the masses of the stars? give an answer in solar masses.

Homework Equations

i guess this one has to be used:

M₁+M₂=$$\frac{4\pi^2a^3}{GP^2}$$

also this one to eliminate one of the M's

M₁r₁=M₂r₂

The Attempt at a Solution

got as far as drawing a diagram but I don't know what the semi major and minor axes correspond to, are they the radii of the circular orbits of the separate stars about the centre of mass? any help would be good. Cheers.

 

[Unto]

I don't know what they are playing at putting the minor axes in this questions...

Badly worded question, but I guess you should assume that the 2 stars are of similar mass and orbit in a circular fashion around a common center of mass. This means the stars share exactly the same orbit.

You are told that the maximum separation between the 2 stars is 4 arcseconds, from our point of veiw to the center of mass which which take as a point, this is 2 arcseconds either side of us for each star.

Given that the distance to the center of mass is 10pc, use simple trigonometry to find the distance from the center of mass to 1 of the stars - this distance will be shared by the other star.

2M = $$\frac{4\pi^2a^3}{GP^2}$$

Since you would have to change into units of AU and years, you can forget the constant and go ahead with:

2M = $$\frac{a^3}{P^2}$$

Total separation (which a is) between the stars is double of the orbital radii. Plug in and divide by 2, you should get nice values for the stars in solar masses.

 

[knowlewj01]

Ok, thanks i'll give this a go and see what comes out of it. It just threw me off when it said the orbit was circular but still had semi major and semi minor axes. Think it means that the distance from the first and second star to the CM are 0.5" and 4". Thanks, ill see what happens.

 

[knowlewj01]

M₁R₁=M₂R₂

R₁ and R₂ are 0.5" and 4" respectively, I worked these out to be 5AU and 40AU at a distance of 10pc by geometry.

using the definition of centre of mass at the beggining of this post i found M₂ in terms of M₁, R₁ and R₂, this was

$$\frac{5M1}{40}$$

Sub this value for M₂ into:

M₁+M₂=$$\frac{a^3}{P^2}$$

M₁ turns out to be 90 solar masses
M₂ turns out as 11.25 solar masses

can anyone see any problems with this?

Thanks