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 center-of-mass 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 m1 and m2 are the two masses, P is the period
of revolution, G is the gravitational constant, and v1 is the radial
component of the velocity of one of the stars (m1). 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 X-1, however, only one of the stars can be seen (Cygnus
X-1'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, m1 and v1 refer to the companion
star and m2 refers to Cugnus X-1, the unknown mass for which we
want to solve.
This equation indicates that m2 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.
http://www.astronomy.ohio-state.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
x3 + ax2 + bx + c = 0, which can then be solved.