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Figuring the Mass for Each Star in a Binary Star System

If you read last month's feature, you learned how easy it was to compute the (total) mass of a binary star system. As long as you know the average distance (d) between the two component stars in astronomical units (au), and the orbital period in Earth-years, you have all the information you need to figure the total mass of the binary star in solar masses:

mass = d3/t2, whereby d = distance and t = orbital period

Let's take the binary star Procyon, given that the distance (d) between the two component stars = 15 astronomical units and the orbital period (t) = 40.8 Earth-years. I use t to represent orbital period, following the suggestion given by a Golden Book guide. If you can't remember which unit to cube and which to square, remember Times Square!
mass = d3/t2
mass = 153/40.82
mass = 15 x 15 x 15/ 40.8 x 40.8
mass = 3375/1664.64
mass = 2.0274654 solar masses
Binary Star AnimationThe above answer gives the total mass of the Procyon star system. To figure out the mass for each component star, we need to know the average distance of each star from the center of mass.
Animation on right: Binary star of two equally massive components

The Solution!

Procyon's two components are called Procyon A (the more massive star) and Procyon B (the less massive star). According to solstation, Procyon A's distance (d1) from the center of mass = 4.3 astronomical units whereas the Procyon B's distance (d1) from the center of mass = 10.7 astronomical units.
Now that we know each star's distance from the center of mass, we can compute the mass of Procyon A (m1) from the following equation, given that m1 = the unknown = mass of Procyon A, d1 = Procyon A's distance from center of mass = 4.3, d = distance between Procyon A & Procyon B = 15, M = total mass = 2.0275

Formula for Finding the Mass of Procyon A

m1 = M(d-d1)/d
m1 = 2.0275(15-4.3)/15
m1 = 2.0275 x 10.7/15
m1 = 2.0275 x 0.71333...
m1 = 1.4462833 solar masses
See if you can solve for Procyon B's mass with the below equation, whereby m2 = the unknown = Procyon B's mass, and d2 = Procyon B's distance from center of mass = 10.7. M = 2.0275 and d = 15.

Formula for Finding the Mass of Procyon B

m2 = M(d-d2)/d
Up for more mental gymnastics? Then how about computing the mass of each component in the Sirius binary star system? Let's say M = total mass = 3.1 solar masses, d = total distance = 19.83 au, d1 = distance of Sirius A from center of gravity = 6.43 au, d2 = distance of Sirius B from center of mass = 13.4 au. What is the mass for Sirius A (m1), the more massive star, and Sirius B (m2), the less massive star?
Good Luck! Till next month . . .

copyright 2011 by Bruce McClure

October 2011 Feature * December 2011 Feature