# Doing Correct Calculations with Binary Star Systems

I've been experimenting with Binary Star System calculations but came across some issues.

I can correctly calculate the Semi-Major Axis of the Earth's orbit around the Sun by doing the equation:

Using data from Wikipedia.

a = (1.521e11 + 1.47095e11) / 2.   <- average radius of Earth's orbit (meters)
μ = G * M = 6.67408e(-11) * 1.98847e30 gravitational constant * Mass of Sun (kg)


I got ≈ 31557884 seconds which equates to ≈ a year

Now my issue is that I've been searching around for a formula that calculates the period and etc of a Binary System where two masses (such as stars) orbit around each other.

So then I try essentially the same formula, this:

I used data from the Procyon A and B (binary system)

a = Semi-Major Axis = 4.3 AU
M1 = Procyon A Mass = 1.499 Solar Masses
M2 = Procyon B Mass = 0.602 Solar Masses


these give me a T (period) of 4731227. It says 40.82 years on Wiki.

What am I doing wrong? Is it the data? is it the formula? My comprehension of the question? Thank you for the help!

• Welcome to Astronomy! For starters, in case you are in a hurry: en.wikipedia.org/wiki/Reduced_mass but I think someone will write a full answer fairly soon. – uhoh Nov 11 '18 at 9:52
• – uhoh Nov 11 '18 at 10:47