# National Aeronautics and Space Administration

## Newton's Law

### Solving for the Mass of Cyg X-1 Using Newton's Law of Universal Gravitation

The mass of Cygnus X-1 can be determined if a probe is launched toward it and programmed to orbit at a fixed distance from Cygnus X-1. The probe will be kept in orbit around Cyg X-1 by the force of gravity, whose strength is dependent only upon Cyg X-1's mass. After arriving and successfully orbiting Cygnus X-1, the probe will radio back information about its orbital speed. It will then be possible to determine the mass of Cygnus X-1 using the orbital radius and speed of the probe around Cygnus X-1, Newton's Law of Universal Gravitation, and circular motion equations.

http://www-groups.dcs.st-and.ac.uk/~history/Posters2/Newton.html

Realizing that the gravitational attraction of Cygnus X-1 provides the centripetal force that is responsible for the circular motion of an orbiting satellite allows us to equate the two forces, as follows.

Look again at the equations equating the force of gravity and the force of centripetal motion above. Because the orbital radius is the same as the distance between Cygnus X-1 and the satellite, the equations can now be rewritten :

Because m, the mass of the satellite, appears on both sides of the equation, it can be eliminated. Rearranging to solve for M, the mass of Cygnus X-1, we get an exprssion with the measurable values of the probe's speed and distance from Cyg X-1 and the known value of Newton's constant of universal gravitation, G:

 A service of the High Energy Astrophysics Science Archive Research Center (HEASARC), Dr. Alan Smale (Director), within the Astrophysics Science Division (ASD) at NASA/GSFC