Detailed Partial Solution for Mass of Cyg X-1

First, re-arrange the mass function

to

Then expand the quadratic and rearrange into a cubic for m₂ in the form x³ + ax² + bx + c = 0:

We can now identify the coefficients as

We use the following values for the measured quantities, and convert to mks units:

• P = 5.6 days = 4.838 x 10⁵ seconds
• v₁ = 75 km/s = 7.5 x 10⁴ m/s
• i = 30 degrees
• m₁ = 30 solar masses = 30 x (2.0 x 10³⁰) = 60 x 10³⁰ kg

Noting that

We're now ready to compute the quantities necessary to find the real cubic root (see Solving a Cubic Equation). We first calculate the quantities

We next compute

and compute S and T:

Then the real root is then given by

Putting in the values for S, T and a, we obtain

Finally, convert to solar masses by remembering that the sun's mass is 2.0 x 10³⁰ kg. This gives the mass of the black hole in Cygnus X-1 to be:

Click here return to solve for the mass of Cygnus X-1.