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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.

Info Click here for more information on Newton's Law of Universal Gravitation.
Using these principles, he derived the following expression relating the masses of a binary pair:

(m2^3)(sin^3i)/(m1+m2)^2=(v1^3)P/(2(pi)(G))

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.

(See http://www.astronomy.ohio-state.edu/~pogge/Ast161/Unit4/orbits.html for a more detailed description of this derivation.)

Data Click here to estimate the mass of the companion star.
Data Click here to estimate the inclination of the Cygnus X-1 system.
Data Click here to measure the velocity of Cygnus X-1's companion.

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.

Tools Click here to see how to solve cubic equations.

times the mass of the Sun.

Experiment Return to the beginning and try another approach

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This page last updated: Wednesday, 04-Oct-2006 11:33:23 EDT