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.
