
Finding the Mass of a Star in a binary system
Kepler's Laws of planetary motion apply to any bodies orbiting about one
another, including binary stars. Newton realized that it was the force of
gravity that governed the motions of such orbiting bodies and caused their
characteristic motions (as stated by Kepler). Newton, who formulated the
Universal Law of Gravity, was able to generalize Kepler's Laws to apply to
any two bodies orbiting each other. He found that Kepler's Laws can be
derived from first principles, specifically Newton's three laws of motion
and his law of Universal Gravitation.
 First Law  Orbits are conic sections with the centerofmass of the
two bodies at the focus.
 Second Law  angular momentum conservation.
 Third Law  Generalized to depend on the masses of the two bodies.
Using these principles, he derived the following expression relating the
masses of a binary pair:
where m_{1} and m_{2} are the two masses, P is the period
of revolution, G is the gravitational constant, and v_{1} is the radial
component of the velocity of one of the stars (m_{1}). If both of
the stars' radial velocities are measured, as with visual binaries, the
equation can be manipulated so that both masses can be determined. In the
case of Cygnus X1, however, only one of the stars can be seen (Cygnus
X1's visual companion), so in order to determine the mass of the unseen
object, it is necessary to know, or to estimate, the mass of the companion
star. In this case, m_{1} and v_{1} refer to the companion
star and m_{2} refers to Cugnus X1, the unknown mass for which we
want to solve.
This equation indicates that m_{2} must increase as sin(i)
decreases. It will be necessary for this calculation to make an educated
guess at the value for i, the inclination angle.
(See
http://www.astronomy.ohiostate.edu/~pogge/Ast161/Unit4/orbits.html for a
more detailed description of this derivation.)
Once the observables are measured or estimated, it remains to rearrange
the equation above in order to end up with a cubic equation of the form
x^{3} + ax^{2} + bx + c = 0, which can then be solved.
