The proportionality to 1/r
in Newton’s law of gravity was not entirely
unexpected. Proportionalities to 1/r
are found in many other phenomena
in which some effect spreads out from a point. For instance, the intensity of
the light from a candle is proportional to 1/r
, because at a distance r from
the candle, the light has to be spread out over the surface of an imaginary
sphere of area 4
. The same is true for the intensity of sound from a
firecracker, or the intensity of gamma radiation emitted by the Chernobyl
reactor. It’s important, however, to realize that this is only an analogy. Force
does not travel through space as sound or light does, and force is not a
substance that can be spread thicker or thinner like butter on toast.
Although several of Newton’s contemporaries had speculated that the
force of gravity might be proportional to 1/r
, none of them, even the ones
who had learned Newton’s laws of motion, had had any luck proving that
the resulting orbits would be ellipses, as Kepler had found empirically.
Newton did succeed in proving that elliptical orbits would result from a 1/r
force, but we postpone the proof until the end of the next volume of the
textbook because it can be accomplished much more easily using the
concepts of energy and angular momentum.
Newton also predicted that orbits in the shape of hyperbolas should be
possible, and he was right. Some comets, for instance, orbit the sun in very
elongated ellipses, but others pass through the solar system on hyperbolic
paths, never to return. Just as the trajectory of a faster baseball pitch is
flatter than that of a more slowly thrown ball, so the curvature of a planet’s
orbit depends on its speed. A spacecraft can be launched at relatively low
speed, resulting in a circular orbit about the earth, or it can be launched at a
higher speed, giving a more gently curved ellipse that reaches farther from
the earth, or it can be launched at a very high speed which puts it in an
even less curved hyperbolic orbit. As you go very far out on a hyperbola, it
approaches a straight line, i.e. its curvature eventually becomes nearly zero.
Newton also was able to prove that Kepler’s second law (sweeping out
equal areas in equal time intervals) was a logical consequence of his law of
gravity. Newton’s version of the proof is moderately complicated, but the
proof becomes trivial once you understand the concept of angular momen-
tum, which will be covered later in the course. The proof will therefore be
deferred until section 5.7 of book 2.
The conic sections are the curves
made by cutting the surface of an infi-
nite cone with a plane.
An imaginary cannon able to shoot
cannonballs at very high speeds is
placed on top of an imaginary, very
tall mountain that reaches up above
the atmosphere. Depending on the
speed at which the ball is fired, it may
end up in a tightly curved elliptical or-
bit, a, a circular orbit, b, a bigger ellip-
tical orbit, c, or a nearly straight hy-
perbolic orbit, d.
Section 10.2Newton’s Law of Gravity