Gravitational Force

Kepler's Laws

Galileo's premonition: "to understand motion is to understand nature" is seen most clearly with the attempt to understand the kinematics and dynamics of planetary motion. The attempts to understand the motions of the planets had been undertaken since at least the time of the ancient Greeks, yet, by the end of the 16th century there was still no satisfactory theory for the motions of the planets. Johannes Kepler, using data taken by Tycho Brahe, published, over a period of 20 years of research, three "laws" of planetary motion. Though they provide some descriptive and prescriptive power, they cannot be thought of as kinematics since they are purely empirical. The laws are:

1. The planets move around the sun in elliptical orbits with the sun at one focus.
2. The line joining the sun to a planet sweeps out equal areas in equal times.
3. The square of the period of a planet's orbit around the sun is proportional to the cube of its mean orbital radius from the sun.

Although we need a more sophisticated mathematical approach than we have undertaken thus far to understand the first of these laws, Newton was able to show that all of them are a direct consequence of his own laws of motion and the isotropy of three dimensional space. The reason it took so long to see this relatively simple set of laws is based on the fact that the orbital motion we see has to take into account the motion of the earth around its own axis and around the Sun. Even for a simple system like the earth-moon system, the fact that both objects actually orbit a common center-of-mass rather than just the earth around the moon complicates the description of the observed motion of the moon.

f apples and moons

Sometime during the period of 1665-1666, Newton returned home to his mother's farm since the university was closed due to the plague. While there, Newton, according to his own words, considered the problem of planetary motion and focused on coming up with a theory to explain the motion of the moon around the earth. Newton's view, for which there is considerable doubt, is that he had at hand all the information to consider a thought experiment in which the same gravitational force that pulls a falling apple to the ground could be the same force that holds the moon in orbit around the earth. In fact, most of the information Newton needed was not clearly dilineated until after 1675, but the thought experiment is impressive and informative nonetheless. It begins with Newton's statement
"as the power [of gravity] is not found sensibly diminished at the remotest distance from the center of the Earth to which we can rise, neither at the tops of the loftiest buildings, not even the summits of the highest mountains; it appeared reasonable to conclude that this power must extend much further than is usually thought; why not as high as the Moon; and if so, her motion must be influenced by it; perhaps she is retained in her orbit thereby."

The thought experiment begins with Kepler's third law applied to circular orbits. It continues with Christian Huygen's publication, in 1673, of the formula for the centripetal acceleration of a body in a circular orbit, ar = 4*Pi2r/T2. By 1679, Robert Hooke, Edmund Hillary, and Christopher Wren had shown that, for a circular orbit, the force delivering the centripetal acceleration must have a strength which is proportional to 1/r2 to fit in with Kepler's Third Law for planetary orbits. This is easy to see. Consider a point object with a mass m, a non-zero, constant velocity and a centripetal force acting on it toward a central body as shown below.

We have a_r = v²/r. The period of the circular orbit must be T = 2*Pi*r/v, therefore,

ar	= v2/r
	= 4*Pi2r2/(r*T2)
ar	= 4*Pi2r/T2

This result from Huygen can be combined with Kepler's 3rd Law, T² = kr³, to show that the force providing a_r must vary as 1/r²

Fcentral	= mar
	= m[4*Pi2r/(kr3)]
Fcentral	= 4*Pi2m/(kr2)

So the central force is proportional to 1/r². The 1/r² form is also reasonable from the isotropy of space argument presented in the text.

With the knowledge that, at least for circular orbits, the central force must vary as 1/r² and knowledge that the distance from the center of the earth to the center of the moon is about 60 earth radii, Newton could perform the quick calculation that the acceleration toward the center of the earth of an apple near the surface of the earth is consistent with the moon being attracted to the center of the earth by the same force, as long as that force varies as 1/r². Say that the magnitude of the acceleration a body experiences due to the gravitational attraction of the earth is a_r = k_earth/r² where r is the distance from the center of the earth. Then k_earth = a_rr². For the apple, r = R_earth = 6.37 x 10⁶ m. For the moon, r = R_e-m = 60R_earth. Therefore, if the same earth is responsible for the motion of a falling apple and the curved orbit of the moon, then a_appleR_earth² = a_moonR_e-m(60R_earth)², so, using our established connection between centripetal acceleration and period of orbit, we find

aappleRearth2	= amoonRe-m(re-m)2 ==>
gRearth2	= [4*Pi2re-m/Tmoon2] (re-m)2
gRearth2	= [4*Pi2/Tmoon2] (60Rearth)3 ==>
T2	= [4*Pi2/g](60)3Rearth ==>
T	= sqrt[4*Pi2/(10 m/s2)] (60)3(6.4 x 106 m)]
	= 2.34 x 106 s (1 day/86400 s)
T	= 27.1 days

The actual sidereal time for one complete lunar orbit is 27.3 days so Newton's guess about gravity being the same force that causes the motion of a falling apple and the orbiting moon looks to be correct or, in Newton's words,
"I deduced that the forces which keep the planets in their orbs must be reciprocally as the squares of their distances from the centers about which they revolve; and thereby compared the force requisite to keep the Moon in her orb with the force of gravity at the surface of the Earth; and found them answer pretty nearly."

This is a remarkable advance! Newton has shown that the same laws of motion that we study on earth are applicable to the motions of bodies far beyond the environs of earth. Laboratory experiments here can be used to deduce theories of nature which are universally applicable.

The remaining things for Newton to do to come up with a complete theory of gravity were non-trivial. First, he had to be clear in his notions about his second and third laws of motion (he had already a clear concept of inertia), he had to reconcile his results with the fact that the orbits of moons and planets are not circular, but elliptical as Kepler had shown, he had to deal with the "action-at-a-distance" issue, and he had to correctly deal with mass as a generator of gravitational force. For all but the last two of these, Newton came up with definitive descriptions that were not substantially improved upon for more than 300 years. The last two he acknowledged as puzzles that would be left to later generations to fully resolve. We have not fully resolved them yet.