Formula for finding the Mass of a Star in a binary system
Finding the Mass of a Star in a binary system
- 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.
|Click here for more information on Newton's Law of Universal Gravitation.|
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.
|Click here to estimate the mass of the companion star.|
|Click here to estimate the inclination of the Cygnus X-1 system.|
|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.
|Click here to see how to solve cubic equations.|
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