ASTR 1210 (O'Connell) Study Guide



7. THE DISCOVERY OF GRAVITY


Thomas Digges' version of the
Copernican Universe (1576)

A. Expanding Horizons

Copernicus' heliocentric universe sparked immediate controversy because it contradicted both the scientific and religious conventions of the times. It was hotly debated in the 150 years following publication of De revolutionibus, the most famous episode being the recantation of heliocentrism forced upon Galileo by the Catholic Church (1633).

But a number of thinkers quickly and enthusiastically embraced the Copernican model, including Thomas Digges (d. 1595) in England and Giordano Bruno (d. 1600) in Italy. Even though Copernicus himself had pictured the stars as lying in a shell at a fixed distance from Earth, both Digges and Bruno realized that the model allowed the stars to be arbitrarily far away. In the first published description of heliocentrism in English, Digges drew the stars as stretching away to infinity (see the picture at the top of this page), and his inscription ( "This orb of stars fixed infinitely up extendeth itself in altitude spherically...with perpetual shining glorious lights innumerable...replenished with perfect endless joy...") shows that he was thrilled by this prospect. His concrete depiction of an infinite universe had lasting influence on later English scientists. Bruno emphasized the possibility that the stars were other Suns and that an unbounded universe was filled with inhabited planets orbiting other stars (a "plurality of worlds").

After 1600, scientific discoveries about the natural world progressed rapidly, at least by earlier standards. Scientific information and the standards of the "scientific method" were quickly disseminated by printed books. The next key development for physics & astronomy was the discovery, understanding, and quantification of gravity.

Here, we describe the work of four astronomers/physicists whose work was pivotal in understanding gravity and founding modern science. The timeline chart below will help you keep track of who's who and when. Click for a larger version.

B. Tycho (d. 1601)

  • Observer -- the greatest before the invention of telescopes. See his picture here.

  • His observations of the "supernova" of 1572 (an exploding star) demolish the Aristotlean doctrine of heavenly perfection & permanence.

  • At his magnificent, state-sponsored observatory (see picture at right), Tycho compiled a massive set of unprecedentedly accurate (uncertainties less than about 1 arc-minute) data on planetary motions, later analyzed by his assistant, Kepler. The accuracy of Tycho's data was the best possible without optical instruments.

  • Although Tycho died before he was able to analyze his data, he favored a geocentric universe (albeit one in which the Earth spun on its axis and all the other planets orbited the Sun).


    Galileo's notes on the discovery of the satellites of Jupiter.

    C. Galileo (d. 1642)

  • Galileo (picture here) played a pivotal role in the transition from medieval to modern science. He made fundamental contributions in three separate areas: experimental physics, astronomy, and popularizing science. Ironically, it was his success as a popularizer, more than as a scientist, that embroiled him in political difficulties with Church authorities.

    "You must read the book of Nature... In other words, observe and do experiments. This is against the medieval idea of scholasticism--that all wisdom and knowledge are best found in ancient authorities."

    "Truth cannot be found in the book of Aristotle but in the book of Nature; and the book of Nature is written in the language of mathematics."

  • As a physicist:

  • As an astronomer:


    D. Kepler (d. 1630)

  • Mathematician (see his picture here)

  • Analyzes Tycho's data, all obtained without telescopes but more accurate than any previous.

  • Without any deliberate intent, Kepler introduces the conceptual foundation of modern empirical science:

  • Kepler quickly discovers that models based on pure circular motions could not fit Tycho's data for Mars.

  • Kepler reinterprets the data for all planets and condenses his conclusions to three "Laws of Planetary Motion."

    Kepler's Laws

    1. Planetary orbits are ellipses with the Sun at one focus

      Ellipse Geom

      • Note that the Sun is not at the center of the ellipse and that there is nothing at the second focus of the orbital ellipse. The distance between any planet and the Sun will vary as it moves around its orbit.

      • The Sun is in the same plane as the ellipse for a given planet, but the orbits of different planets can lie in different planes.

      • The planetary orbits are not very elliptical, which is why circles are fair approximations, as in Copernicus' model.

    2. For a given planet, a line joining the planet as it moves and the Sun sweeps out equal areas in the orbital plane in equal times.

      Kepler 2nd Law

      Kepler's Second Law -- Click for animation.

        This implies a given planet moves faster when it is nearer the Sun, with a specific relationship between its sideways motion and its distance.

        [This behavior is also the first hint of a universal physical principle not recognized until after Newton: the conservation of angular momentum.]

       
    3. The squares of the orbital periods of different planets are proportional to the cubes of the orbital sizes (semi-major axes).

      Orbital Velocities
        In equation form, P2 = K a3, where P is the period, a is the semi-major axis, and K is a constant.

        The time P taken to complete one orbit is therefore proportional to (a x a1/2) and grows more than in direct proportion to orbital size.

          A planet with an orbital diameter 5 times the Earth's will require 11 Earth years to complete an orbit.

        The easiest way to think about the Third Law is that it implies that the velocities of planets in larger orbits are slower than for planets nearer the Sun:

          A planet's mean velocity in its orbit is equal to the circumference of the orbit divided by its orbital period. Since the circumference of an orbit increases in direct proportion to its semi-major axis, but the period increases more than in direct proportion, the mean velocity of planets in larger orbits is slower. See graph above right.

  • Java illustrations of Kepler's three laws are available at this web site.

  • Net result: A tremendous simplification. Tens of thousands of individual observations have been reduced to a small set of simple geometric and arithmetic relationships. All the arbitrary complexity ("wheels within wheels") of Ptolemy has vanished. So, too, however, has the perfection of uniform, circular motion.

  • Note that Kepler's Laws were derived empirically from Tycho's data. They are not "theoretical." They simply summarize the central observational facts.

  • Kepler was a very smart person, but his breakthrough was entirely dependent on the large body of highly accurate data compiled by Tycho.

    The concept of force

    E. Newton (d. 1727)

  • Isaac Newton was a mathematician and physicist and ranks as one of the two or three most important people in human history because of the profound influence of his work on all later science and technology. His picture on the British Pound note is shown here.

  • Attempting to understand Kepler's Laws, Newton develops the basic principles of dynamics---i.e. the methods needed to predict how objects move in response to forces.

    Newton's First Law of Motion

    Newton's Second Law of Motion

    Gravity

    AppleInspiration Predictions

    Newton's Legacy


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    Last modified July 2014 by rwo

    Text copyright © 1998-2014 Robert W. O'Connell. All rights reserved. Timeline chart copyright © by Cengage Learning, Inc. . Illustrations of Kepler's laws by Nick Strobel. Falling apple animation from ASTR 161 UTenn at Knoxville. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 1210 at the University of Virginia.