Expert: Philip A. Stahl - 6/4/2008

Question
Can you help me with this problem? Find the mass of the stars Sirius A and Sirius B which form a binary system with a true relative orbit of 7.5" dia. and at a distance of 2.67 parsecs. The period of the system is 50 years, and B (the fainter star) is twice as far from the center of mass as A.

Answer
Well, what I will do is set this problem up so you can work it out. First, note that for a “true relative orbit” this means the binary orbit is seen “face on”, e.g. not distorted by any tilts. (Which would make it appear more elliptical than it is). Also note here that the question wording makes no sense if 7."5 is the "diameter"! It has to be the semi-major axis, or the masses make no sense and are not consistent with our data. You may wish to re-read that question again, if from a text.

Basically, to solve the problem, one needs to make use of Kepler’s third law:

(m1 + m2) P^2   =    (a)^3

Where m1, m2 are the binary masses (in this case for Sirius A, B) and P is the period in years, and a is the semi-major axis in astronomical units. Thus, the law is posed in solar units.

Bear in mind here that for a linear arc length, s, at some radius r, which is subtended by an angle theta, one has:

s = r (theta)

in other words, the arc length is the product of the radius r, by the angle theta.

Analogous to the example, s would be the distance between the two stars in astronomical units.

Given this, we may rewrite the Kepler equation as:

(m1 +  m2) P^2   =  (r x a”)^3

where r is the radius (2.67 pc) and a” = 7.”5 = theta

from here you should be able to work this out.

Just remember that in determining each separate component mass you need to be aware that the distance of each star from the barycenter (center of mass) of the system is inversely proportional to its own mass.

Thus, if d1, and d2 are the respective distances from the barycenter, for mass m1 and m2  and mass m2 >  m1, then:

m2/ m1  =   d1/ d2

Thus,  if m2 = 3 m1  then d2 = (d1) /3

Hope this helps! If not, let me know and we can proceed further with the solution!