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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.

picture of newton
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.

F(gravity)=G(M x m/d^2)=(mv^2)/r=F(centripital)

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 :

G((Mm/r^2))= (mv^2)/r

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:

M=(v^2r)/G

Info Click here for more information and a quiz on Newton's Law of Universal Gravitation.
Info Click here for more information and a quiz on uniform circular motion.
Quiz Click here to try using this technique to find the Sun's mass
Experiment Click here to launch a probe to Cyg X-1

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This page last updated: Monday, 30-Jan-2006 12:11:59 EST