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Partial Solution for Mass of Cyg X-1

Working carefully, the mass function can be solved to find the mass of the black hole in Cygnus X-1.

First, re-arrange the mass function

((m2)^3(sin i)^3)/(m1+m2)^2 = (Pv^3)/(2(pi)(G))

to

(m2)^3(sin i)^3 = (Pv^3/(2(pi)(G)))(m1+m2)^2

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

(m2)^3-(Pv^3/(2(pi)(G)(sin i)^3))(m2)^2-2(Pv^3/(2(pi)(G)(sin i)^3))(m1)(m2)-(Pv^3/(2(pi)(G)(sin i)^3))(m1)^2=0

We can now identify the coefficients as

a=(Pv^3/(2(pi)(G)(sin i)^3)),b=-2(Pv^3/(2(pi)(G)(sin i)^3))(m1),c=-(Pv^3/(2(pi)(G)(sin i)^3))(m1)^2

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

Pv^3/(2(pi)(G)(sin i)^3)=(4.838x10^5s)(7.50x10^4m/s)^3/(2(pi)(sin 30)^3(6.67x10^-11m^3/kg-s^2))=3.90x10^30 kg

the values for the coefficients are

a=-3.90x10^30 kg,b=-4.68x10^62 kg^2,c=-1.40x10^94 kg^3

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

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

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