Newtonian Gravitation and the

Laws of Kepler

We now come to the great synthesis of dynamics and astronomy accomplished by Newton: the Laws of Kepler for planetary motion may be derived from Newton's Law of Gravitation. Furthermore, Newton's Laws provide corrections to Kepler's Laws that turn out to be observable, and Newton's Law of Gravitation will be found to describe the motions of all objects in the heavens, not just the planets.

Notice that (because
of Kepler's 2nd Law) the velocity vector is constantly changing both its
magnitude and its direction as it moves around the elliptical
orbit (if the orbit were circular, the magnitude of the velocity would remain
constant but the direction would change continuously). Since either a
change in the magnitude or the direction of the velocity vector constitutes an
acceleration, there is a continuous acceleration as the
planet moves about its orbit (whether circular or elliptical),
and therefore by Newton's 2nd Law there is a force that acts at every
point on the orbit. Furthermore, the force is not constant in
magnitude, since the change
in velocity (acceleration) is larger when the planet is near the Sun on the
elliptical orbit.

- Since the planets move on ellipses
(Kepler's 1st Law), they are continually accelerating, as
we have noted above. As we have also noted above, this implies a force acting
continuously on the planets.
- Because the planet-Sun line sweeps out equal areas in equal times
(Kepler's 2nd Law), it is possible to show that the force must be directed
toward the Sun from the planet.
- From Kepler's 1st Law the orbit is an ellipse with the Sun at one focus;
from Newton's laws
it can be shown that this means that the magnitude of the force must vary as
one over the square of the distance between the planet and the Sun.
- Kepler's 3rd Law and Newton's 3rd Law imply that the force must be proportional to the product of the masses for the planet and the Sun.

For the ellipse (and its special case, the circle), the plane intersects opposite "edges" of the cone. For the parabola the plane is parallel to one edge of the cone; for the hyperbola the plane is not parallel to an edge but it does not intersect opposite "edges" of the cone. (Remember that these cones extend forever downward; we have shown them with bottoms because we are only displaying a portion of the cone.)

- The orbits of some of the planets (e.g., Venus) are ellipses of such small
eccentricity that they are essentially circles, and we can put artificial
satellites into orbit around the Earth with circular orbits if we choose.
- The orbits of the planets generally are ellipses.
- Some comets have parabolic orbits; this means that they pass the Sun once
and then leave the Solar System, never to return. Other comets have elliptical
orbits and thus orbit the Sun with specific periods.
- The gravitational interaction between two passing stars generally results in hyperbolic trajectories for the two stars.

- Java applet illustrating properties of a circle
- Java applet illustrating properties of an ellipse
- Java applet illustrating properties of an hyperbola
- Java applet illustrating properties of a parabola

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