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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 semi-major axis, ao, to radians and calculate the true semimajor axis as a function of cos(i).
Use this value to convert ao to metres.
Calculate aB (distance of Sirius B from the barycentre) and find the ratio of the masses of the two stars.
Solve for mB (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:
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| (1) |
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| (2) |
![a[o] := 5.52*Unit('arcsec')](/view.aspx?SI=154338/f7a267d7de7065a19ba89b81a459d800.gif)
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| (3) |
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| (4) |
Useful Equations:
Solution: Convert ao to radians and calculate the true semimajor axis in parsecs.
![a = a[o]/(206265*Unit('arcsec')*p*cos(i))](/view.aspx?SI=154338/b57759122264ed94621c471385d8a0c2.gif)
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| (5) |
An actual figure cannot be given unless the inclination angle, i, is known. Convert a to metres.
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| (6) |
Calculate aB and find the respective masses of the two stars.
![a[B] := a/(1+.466)](/view.aspx?SI=154338/7ef395a06f63873aa9c75f5b8aa12cff.gif)
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| (7) |
![evalf(m[B] = 4*Pi^2*(2.190847000*10^12/cos(i))^3/((6.673*10^(-11)*(60*(24*(49.94*365)*60))^2)*(1+1/.466)))](/view.aspx?SI=154338/ce71e6d137d00849919b51e112135897.gif)
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![m[B] = 0.7972901819e30/cos(i)^3](/view.aspx?SI=154338/c59cbaa00c93ed7947bc66e87c2ac0cf.gif)
| (8) |
Divide by the mass of the Sun to get the mass of star B in solar masses:
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| (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.
![solve(.4/(cos(i)^3*m[A]) = .466, m[A])](/view.aspx?SI=154338/6d33a72d65529e53737b4436378d967f.gif)
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| (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.
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| (11) |
 = 0.38. Divide by this number to find the true values of the masses of the two stars:
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| (12) |
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| (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).
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Reference
Balogh, M. (n.d.) Lecture 5: Binary Stars. http://quixote.uwaterloo.ca/~mbalogh/teaching/.../PPT/Lecture5.ppt (Accessed 2015-10-28).
Liebert, J. et al. (2005). The Age and Progenitor Mass of Sirius B. ApJ, 630(1), L69-L72.
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