# I.              Kepler's  Laws and their proof from principals of Newtonian physics

A.   The square of the period of a planet in a near circular orbit is proportional to the cube of its distance from the sun, or T2 = A R3.  Proof:  the centripetal force, F = mv2/R,  required to move a mass m  a circular orbit of radius R about another mass M is the gravitational force.  Equate the centripetal force to the gravitational force, GMm/R2,  substitute v = R w for the magnitude of linear velocity tangent to the circle in the term for centripetal force and remember that the angular velocity w = 2 p  / T.
Note: Equating  centripetal force to  the form of an unknown gravitational force  was probably the inspiration that lead Newton to  conclude that gravity force between two masses must be inversely proportional to the squared distance between two masses.

B.   The motion of any body orbiting the sun sweeps out equal areas in equal time intervals of its orbit.  Proof:                     Sketch an ellipse and show that areas of triangles dA in time intervals dt are such that dA/dt = (R Rdq/dt )/2, which is a                 constant proportional to the constant (conserved) angular momentum , L= I w  = mR2 w, where w = dq/dt.

C.   Orbits of planets about the sun are ellipses.  Proof: using conservation of energy write K.E. + P.E. = E, where E                     is constant, write velocity as the derivative of path length ds/dt.  Then manipulate the energy conservation              equation into the form of a differential equation for dr/dq,  which can be integrated over q into the form of the equation r(q) for an ellipse in polar coordinates.

II.            The center of mass system and its importance.

A.   An accurate periodic law must be derived in a center of mass system.

B.   All objects in an N body system orbit about a common center of mass.

C.   Other solar systems may be detected from oscillations of a star toward and away from an Earth observer caused by the star orbiting the center of mass of its solar system. These oscillations may be measured from Doppler shifts of its spectrum. For an example solar system discovered by this technique click here.

III.         N body problem stable/unstable points

A.   Stable points of a 3 body celestial mechanic problem are called Lagrangian points.  Objects tend to be stable in these points. Example: Trojan asteroids near Jupiter.

B.   Unstable orbits related to resonances between orbital periods of two bodies.  The ratios of the two periods are simple integer mulitples.  Material tends to be removed from orbits having these period ratios. Example: Kirkwood gaps in the asteroid belt between Mars and Jupiter.