Project Jupiter

Attachment D: Weighing Jupiter, the Mathematics

Taken from Physics for Students of Science and Engineering, Combined Ed., John Wiley & Sons, D. Halliday and R. Resnick, 1965

Consider two spherical bodies of masses M and m moving in circular orbits under the influence of each other’s gravitational attraction. The center of mass of this system of two bodies lies along the line joining them at a point C such that mr =MR. The large body of mass m moves in an orbit of constant radius R and the small body of mass m in an orbit of constant radius r, both having the same angular velocity. The gravitational force acting on each body must provide the necessary centripetal acceleration. The centripetal force is mj 2 r, thus because of the equality mr=MR, we find

Balancing the centripetal and gravitational forces we get

If one body has a much greater mass than the other, the R is negligible compared to r. The above equation simplifies to

If the angular velocity is expressed in terms of the period of the revolution,

Then the equation becomes

Re-arranging we have the form of Kepler’s third law:

For Project Jupiter we now solve for M, the mass of Jupiter