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# Newton's Triumph

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ r / Tycho Brahe made his name as an astronomer by showing that the bright new star, today called a supernova, that appeared in the skies in 1572 was far beyond the Earth's atmosphere. This, along with Galileo's discovery of sunspots, showed that contrary to Aristotle, the heavens were not perfect and unchanging. Brahe's fame as an astronomer brought him patronage from King Frederick II, allowing him to carry out his historic high-precision measurements of the planets' motions. A contradictory character, Brahe enjoyed lecturing other nobles about the evils of dueling, but had lost his own nose in a youthful duel and had it replaced with a prosthesis made of an alloy of gold and silver. Willing to endure scandal in order to marry a peasant, he nevertheless used the feudal powers given to him by the king to impose harsh forced labor on the inhabitants of his parishes. The result of their work, an Italian-style palace with an observatory on top, surely ranks as one of the most luxurious science labs ever built. He died of a ruptured bladder after falling from a wagon on the way home from a party --- in those days, it was considered rude to leave the dinner table to relieve oneself. s / An ellipse is a circle that has been distorted by shrinking and stretching along perpendicular axes. t / An ellipse can be constructed by tying a string to two pins and drawing like this with the pencil stretching the string taut. Each pin constitutes one focus of the ellipse. u / If the time interval taken by the planet to move from P to Q is equal to the time interval from R to S, then according to Kepler's equal-area law, the two shaded areas are equal. The planet is moving faster during interval RS than it did during PQ, which Newton later determined was due to the sun's gravitational force accelerating it. The equal-area law predicts exactly how much it will speed up.

Isaac Newton's greatest triumph was his explanation of the motion of the planets in terms of universal physical laws. It was a tremendous psychological revolution: for the first time, both heaven and earth were seen as operating automatically according to the same rules.

Newton wouldn't have been able to figure out why the planets move the way they do if it hadn't been for the astronomer Tycho Brahe (1546-1601) and his protege Johannes Kepler (1571-1630), who together came up with the first simple and accurate description of how the planets actually do move. The difficulty of their task is suggested by figure q, which shows how the relatively simple orbital motions of the earth and Mars combine so that as seen from earth Mars appears to be staggering in loops like a drunken sailor. q / As the Earth and Mars revolve around the sun at different rates, the combined effect of their motions makes Mars appear to trace a strange, looped path across the background of the distant stars.

Brahe, the last of the great naked-eye astronomers, collected extensive data on the motions of the planets over a period of many years, taking the giant step from the previous observations' accuracy of about 10 minutes of arc (10/60 of a degree) to an unprecedented 1 minute. The quality of his work is all the more remarkable considering that his observatory consisted of four giant brass protractors mounted upright in his castle in Denmark. Four different observers would simultaneously measure the position of a planet in order to check for mistakes and reduce random errors.

With Brahe's death, it fell to his former assistant Kepler to try to make some sense out of the volumes of data. Kepler, in contradiction to his late boss, had formed a prejudice, a correct one as it turned out, in favor of the theory that the earth and planets revolved around the sun, rather than the earth staying fixed and everything rotating about it. Although motion is relative, it is not just a matter of opinion what circles what. The earth's rotation and revolution about the sun make it a noninertial reference frame, which causes detectable violations of Newton's laws when one attempts to describe sufficiently precise experiments in the earth-fixed frame. Although such direct experiments were not carried out until the 19th century, what convinced everyone of the sun-centered system in the 17th century was that Kepler was able to come up with a surprisingly simple set of mathematical and geometrical rules for describing the planets' motion using the sun-centered assumption. After 900 pages of calculations and many false starts and dead-end ideas, Kepler finally synthesized the data into the following three laws:

## Kepler's elliptical orbit law

The planets orbit the sun in elliptical orbits with the sun at one focus.

## Kepler's equal-area law

The line connecting a planet to the sun sweeps out equal areas in equal amounts of time.

## Kepler's law of periods

Let T, called the planet's period, be the time required for a planet to orbit the sun, and let a be the long axis of the ellipse. Then T2 is proportional to a3.

Although the planets' orbits are ellipses rather than circles, most are very close to being circular. The earth's orbit, for instance, is only flattened by 1.7% relative to a circle. In the special case of a planet in a circular orbit, the two foci (plural of “focus”) coincide at the center of the circle, and Kepler's elliptical orbit law thus says that the circle is centered on the sun. The equal-area law implies that a planet in a circular orbit moves around the sun with constant speed. For a circular orbit, the law of periods then amounts to a statement that T2 is proportional to r3, where r is the radius. If all the planets were moving in their orbits at the same speed, then the time for one orbit would only increase with the circumference of the circle, so we would have a simple proportionality between T and r. Since this is not the case, we can interpret the law of periods to mean that different planets orbit the sun at different speeds. In fact, the outer planets move more slowly than the inner ones.

## Example 13: Jupiter and Uranus

• The planets Jupiter and Uranus have very nearly circular orbits, and the radius of Uranus's orbit is about four times grater than that of Jupiter's orbit. Compare their orbital periods.
• If all the planets moved at the same speed, then it would take Uranus four times longer to complete the four-times-greater circumference of its orbit. However, the law of periods tells us that this isn't the case. We expect Uranus to take more than four times as long to orbit the sun.

The law of periods is stated as a proportionality, and proportionalities are statements about quantities in proportion to one another, i.e.. about division. We're given information about Uranus's orbital radius divided by Jupiter's, and what we should expect to get out is information about Uranus's period divided by Jupiter's. Let's call the latter ratio y. Then we're looking for a number y such that

equations

The law of periods predicts that Uranus's period will be eight times greater than Jupiter's, which is indeed what is observed (to within the precision to be expected since the given figure of 4 was just stated roughly as a whole number, for convenience in calculation).

What Newton discovered was the reasons why Kepler's laws were true: he showed that they followed from his laws of motion. From a modern point of view, conservation laws are more fundamental than Newton's laws, so rather than following Newton's approach, it makes more sense to look for the reasons why Kepler's laws follow from conservation laws. The equal-area law is most easily understood as a consequence of conservation of angular momentum, which is a new conserved quantity to be discussed in chapter 3. The proof of the elliptical orbit law is a little too mathematical to be appropriate for this book, but the interested reader can find the proof in chapter 5 of my online book Conservation Laws. v / Connecting Kepler's law of periods to the laws of physics.

The law of periods follows directly from the physics we've already covered. Consider the example of Jupiter and Uranus. We want to show that the result of example 13 is the only one that's consistent with conservation of energy and momentum, and Newton's law of gravity. Since Uranus takes eight times longer to cover four times the distance, it's evidently moving at half Jupiter's speed. In figure v, the distance Jupiter covers from A to B is therefore twice the distance Uranus covers, over the same time, from D to E. If there hadn't been any gravitational force from the sun, Jupiter would have ended up at C, and Uranus at F. The distance from B to C is a measure of how much force acted on Jupiter, and likewise for the very small distance from E to F. We find that BC is 16 millimeters on this scale drawing, and EF is 1 mm, but this is exactly what we expect from Newton's law of gravity: quadrupling the distance should give 1/16 the force.