Partial Solution for Mass of Cyg X1
Working carefully, the mass function can be solved to find the
mass of the black hole in Cygnus X1.
First, rearrange the mass function
to
Then expand the quadratic and rearrange into a cubic
for m_{2} in the form x^{3} + ax^{2} + 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 10^{5} seconds
 v_{1} = 75 km/s = 7.5 x 10^{4} m/s
 i = 30 degrees
 m_{1} = 30 solar masses = 30 x (2.0 x 10^{30}) =
60 x 10^{30} kg
Noting that
the values for the coefficients are
We're now ready to compute the quantities necessary to find the
real cubic root (see Solving a Cubic Equation).
