Find the Masses of Stars in a Binary System *
Problem: Calculate the masses of Sirius A and Sirius B, given the data below. (Based on a problem by Balogh (n.d.) with additional data by Liebert et al. (2005).)
Hints:
Convert the angular extent of the observed semimajor axis, a_{o,} to radians and calculate the true semimajor axis as a function of cos(i).
Use this value to convert a_{o} to metres.
Calculate a_{B} (distance of Sirius B from the barycentre) and find the ratio of the masses of the two stars.
Solve for m_{B} (mass of Sirius B) in terms of cos(i).
Convert to solar masses.
The resultant masses are lower limits on the masses, absent the value of cos(i).
With the value of i given, calculate the actual masses of the two stars.
Data:
From visual observations of the Sirius A and B system:
 (1) 
 (2) 
 (3) 
 (4) 
Useful Equations:
Solution: Convert a_{o} to radians and calculate the true semimajor axis in parsecs.
 (5) 
An actual figure cannot be given unless the inclination angle, i, is known. Convert a to metres.
 (6) 
Calculate a_{B} and find the respective masses of the two stars.
 (7) 
 (8) 
Divide by the mass of the Sun to get the mass of star B in solar masses:
 (9) 
0.4 of a solar mass is a lower limit on the mass of star B without knowing the value of the inclination angle, i.
 (10) 
0.86 of a solar mass is a lower limit on the mass of star A without knowing the value of the inclination angle, i.
Measurements of the inclination yield a value of i = 43.5 degrees.
 (11) 
= 0.38. Divide by this number to find the true values of the masses of the two stars:
 (12) 
 (13) 
Star A has about 2.3 solar masses. Star B has about 1.1 solar masses. The currently accepted values are A: 2.02 and B:1.00 (Liebert et al. 2005).

Reference
Balogh, M. (n.d.) Lecture 5: Binary Stars. http://quixote.uwaterloo.ca/~mbalogh/teaching/.../PPT/Lecture5.ppt (Accessed 20151028).
Liebert, J. et al. (2005). The Age and Progenitor Mass of Sirius B. ApJ, 630(1), L69L72.
