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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.
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
centrepital 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:
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