Newton’s Law of Gravity

Kepler’s laws were a beautifully simple explanation of what the planets did, but they did not address why they moved as they did. Did the sun exert a force that pulled a planet toward the center of its orbit, or, as suggested by Descartes, were the planets circulating in a whirlpool of some unknown liquid? Kepler, working in the Aristotelian tradition, hypothesized not just an inward force exerted by the sun on the planet, but also a second force in the direction of motion to keep the planet from slowing down. Some speculated that the sun attracted the planets magnetically.

Once Newton had formulated his laws of motion and taught them to some of his friends, they began trying to connect them to Kepler’s laws. It was clear now that an inward force would be needed to bend the planets’ paths. This force was presumably an attraction between the sun and each planet (although the sun does accelerate in response to the attractions of the planets, its mass is so great that the effect had never been detected by the prenewtonian astronomers). Since the outer planets were moving slowly along more gently curving paths than the inner planets, their accelerations were apparently less. This could be explained if the sun’s force was determined by distance, becoming weaker for the farther planets. Physicists were also familiar with the noncontact forces of electricity and magnetism, and knew that they fell off rapidly with distance, so this made sense.

In the approximation of a circular orbit, the magnitude of the sun’s force on the planet would have to be:

F=ma=mv2/r.

Now although this equation has the magnitude, vv, of the velocity vector in it, what Newton expected was that there would be a more fundamental underlying equation for the force of the sun on a planet, and that that equation would involve the distance, rr, from the sun to the object, but not the object’s speed, vv—motion doesn’t make objects lighter or heavier.

Equation [1] was thus a useful piece of information which could be related to the data on the planets simply because the planets happened to be going in nearly circular orbits, but Newton wanted to combine it with other equations and eliminate vv algebraically in order to find a deeper truth.

To eliminate vv, Newton used the equation:

v=circumferenceT=2πrT.

This equation would also only be valid for planets in nearly circular orbits. Plugging this into equation [1] to eliminate vv gives:

F=4π2mrT2.

This unfortunately has the side-effect of bringing in the period, TT, which we expect on similar physical grounds will not occur in the final answer. That is where the circular-orbit case, T∝r3/2T∝r3/2, of Kepler’s law of periods comes in. Using it to eliminate TT gives a result that depends only on the mass of the planet and its distance from the sun:

\begin{multline*}
F\propto m/r^2 . \shoveright{\text{[force of the sun on a planet of mass}}\\
\shoveright{\text{$m$ at a distance $r$ from the sun; same}}\\
\text{proportionality constant for all the planets]}
\end{multline*}
\begin{multline*} F\propto m/r^2 . \shoveright{\text{[force of the sun on a planet of mass}}\\ \shoveright{\text{$m$ at a distance $r$ from the sun; same}}\\ \text{proportionality constant for all the planets]}\end{multline*}

(Since Kepler’s law of periods is only a proportionality, the final result is a proportionality rather than an equation, so there is no point in hanging on to the factor of 4π24π2.)

As an example, the “twin planets” Uranus and Neptune have nearly the same mass, but Neptune is about twice as far from the sun as Uranus, so the sun’s gravitational force on Neptune is about four times smaller.

The forces between heavenly bodies are the same type of force as terrestrial gravity.

Okay, but what kind of force was it? It probably was not magnetic, since magnetic forces have nothing to do with mass. Then came Newton’s great insight. Lying under an apple tree and looking up at the moon in the sky, he saw an apple fall. Might not Earth also attract the moon with the same kind of gravitational force? The moon orbits Earth in the same way that the planets orbit the sun, so maybe Earth’s force on the falling apple, Earth’s force on the moon, and the sun’s force on a planet were all the same type of force.

There was an easy way to test this hypothesis numerically. If it was true, then we would expect the gravitational forces exerted by Earth to follow the same F∝m/r2F∝m/r2 rule as the forces exerted by the sun, but with a different constant of proportionality appropriate to Earth’s gravitational strength. The issue arises now of how to define the distance, rr, between Earth and the apple. An apple in England is closer to some parts of Earth than to others, but suppose we take rr to be the distance from the center of Earth to the apple, i.e., the radius of Earth (the issue of how to measure rr did not arise in the analysis of the planets’ motions because the sun and planets are so small compared to the distances separating them). Calling the proportionality constant kk, we have:

Fearth on apple Fearth on moon=kmapple/r2earth=kmmoon/d2earth-moon.

Newton’s second law says a=F/ma=F/m, so:

aappleamoon=k/r2earth=k/d2earth-moon.

The Greek astronomer Hipparchus had already found 2000 years before that the distance from Earth to the moon was about 60 times the radius of Earth, so if Newton’s hypothesis was right, the acceleration of the moon would have to be 602=3600602=3600 times less than the acceleration of the falling apple.

Applying a=v2/ra=v2/r to the acceleration of the moon yielded an acceleration that was indeed 3600 times smaller than 9.8 m/s29.8 m/s2, and Newton was convinced he had unlocked the secret of the mysterious force that kept the moon and planets in their orbits.

Newton’s law of gravity

The proportionality F∝m/r2F∝m/r2 for the gravitational force on an object of mass mm only has a consistent proportionality constant for various objects if they are being acted on by the gravity of the same object. Clearly the sun’s gravitational strength is far greater than Earth’s, since the planets all orbit the sun and do not exhibit any very large accelerations caused by Earth (or by one another). What property of the sun gives it its great gravitational strength? Its great volume? Its great mass? Its great temperature? Newton reasoned that if the force was proportional to the mass of the object being acted on, then it would also make sense if the determining factor in the gravitational strength of the object exerting the force was its own mass. Assuming there were no other factors affecting the gravitational force, then the only other thing needed to make quantitative predictions of gravitational forces would be a proportionality constant. Newton called that proportionality constant GG, so here is the complete form of the law of gravity he hypothesized.

Newton’s law of gravity

\begin{multline*}
F = \frac{Gm_1m_2}{r^2} \shoveright{\text{[gravitational force between objects of mass}}\\
\shoveright{\text{ $m_1$ and $m_2$, separated by a distance $r$; $r$ is not}}\\
\text{the radius of anything ]}
\end{multline*}
\begin{multline*} F = \frac{Gm_1m_2}{r^2} \shoveright{\text{[gravitational force between objects of mass}}\\ \shoveright{\text{ $m_1$ and $m_2$, separated by a distance $r$; $r$ is not}}\\ \text{the radius of anything ]}\end{multline*}

Newton conceived of gravity as an attraction between any two masses in the universe. The constant GG tells us how many newtons the attractive force is for two 1-kg masses separated by a distance of 1 m. The experimental determination of GG in ordinary units was not accomplished until long after Newton’s death.

The proportionality to 1/r21/r2 in Newton’s law of gravity was not entirely unexpected. Proportionalities to 1/r21/r2 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/r21/r2, because at a distance rr from the candle, the light has to be spread out over the surface of an imaginary sphere of area 4πr24πr2. The same is true for the intensity of sound from a firecracker, or the intensity of gamma radiation emitted by the Chernobyl reactor. It is 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/r21/r2, 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/r21/r2 force, however.

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 Earth, or it can be launched at a higher speed, giving a more gently curved ellipse that reaches farther from 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 momentum.

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