## Its Not Just a Good Idea

In subsequent chapters discussing the planets, we will look more precisely at how mathematics describes planetary motion. For now, consider the following general points.

Astro Byte

Newton's laws of motion were the result of brilliant insight and considerable hard work. When asked later how he had unraveled the mathematics behind the mysteries of planetary motion, he responded, "By thinking of them without ceasing."

### Astro Byte

Newton's laws of motion were the result of brilliant insight and considerable hard work. When asked later how he had unraveled the mathematics behind the mysteries of planetary motion, he responded, "By thinking of them without ceasing." In Principia, Newton proposed that the force of gravity exerted by objects upon one another is proportional to the mass of the two objects, and weakens as the square of the distance between those objects.

Specifically, he postulated that the gravitational force between two objects is directly proportional to the product of their masses (mass of object A times mass of object B). So two objects, one very massive and the other with very little mass, will "feel" the same mutual attraction. In addition, he claimed that the force between two objects will decrease in proportion to the square of the distance. This "inverse-square law" means that the force of gravity mutually exerted by two objects, say 10 units of distance apart, is 100 times (102) weaker than that exerted by objects only 1 unit apart— yet this force never reaches zero. The most distant galaxies in the universe exert a gravitational pull on one another. These relations between mass, distance, and force comprise what we call Newton's Law of Universal Gravitation.

Consider the solar system, with the planets moving in elliptical orbits around the sun. Newton's Principia explained not only what holds the planets in their elliptical orbits (an "inverse-square" force called gravity), but also predicted that the planets themselves (massive Jupiter in particular) would have a small but measurable effect on each other's orbits.

Newton's mathematical prowess was legendary, and he had both the brilliance and the determination to solve problems that had stymied others for decades. His friend, the great astronomer Edmund Halley, and others had been thinking for some time about a way to account for the elliptical orbits of planets, and thought that if the force of gravity decreased with the square of the distance, that the elliptical orbits of Kepler could be explained. But Halley was unable to prove this hunch mathematically. One day, Halley asked Newton what would be the trajectory of a planet moving in a gravitational force field that decreased in strength with the square of the distance from the sun. Newton instantly responded: "An ellipse."

Halley asked to see the mathematical proof, and a few months later, Newton sent him a letter in which he derived Kepler's three laws of planetary motion from the basis of a gravitational attraction that decreased with the square of the distance between the sun and the planets. With Halley's encouragement, Newton began his masterwork, Principia, in which he put forth his famous Laws of Motion and the Law of Universal Gravitation.

Close Encounter

### Close Encounter

Like any good scientific theory, Newton's laws not only explained what was already observed (the motion of the planets), but was able to make testable predictions. The orbit of Saturn, for example, was known to deviate slightly from what one would expect if it were simply in orbit around the sun (with no other planets present). The mass of Jupiter has a small, but measurable, effect on its orbital path. Newton noted with a sense of humor that the effect of Jupiter on Saturn's orbit made so much sense (according to his theory) that "astronomers are puzzled with it."

For the first time, a scientist had claimed that the rules of motion on the earth were no different from the rules of motion in the heavens. The moon was just a big apple, much farther away, falling to the earth in its own way. The planets orbit the sun following the same rules as a baseball thrown up into the air, and the pocket watch of the earth is held in its orbit by a chain called gravity.

Did Newton bring the celestial sphere down to Earth, or elevate us all to the status of planets? Whatever you think, we have never looked at the solar system or the universe in the same way since.

The Least You Need to Know

V While Europe labored through the Middle Ages, Arabian astronomers, inspired by a translation of Ptolemy, were busy making remarkable observations.

V Seeking an explanation of the motion of the planets accurate enough to enable him to revise inaccurate European calendars, Copernicus (following the lead of Aristarchus before him) put the sun at the center of the solar system, with the planets, including Earth, in orbit around it.

V Tycho Brahe made extraordinary astronomical observations, but it was his student, Johannes Kepler, who reduced these data to three basic laws of planetary motion. These laws, based on elliptical planetary orbits, brought the heliocentric model of the solar system into close agreement with observations.

V Galileo used the newly invented telescope to provide experimental proofs of the Copernican idea.

V With the sun-centered universe in place, Isaac Newton put forth in Principia the three laws of motion and the law of universal gravitation, which accounted for the forces behind planetary motion. For the first time, someone proposed that the laws governing the motion of objects here on Earth and in the heavens were one and the same. ## Telescopes Mastery

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