ASTR 1210 (O'Connell) Study Guide


Thomas Digges' version of the
Copernican Universe (1576)

A. Expanding Horizons

Copernicus' conception of a 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"):

"There are countless suns and countless earths all rotating round their suns in exactly the same way as the seven planets of our system. We see only the suns because they are the largest bodies and are luminous, but their planets remain invisible to us because they are smaller and non-luminous. The countless worlds in the universe are no worse and no less inhabited than our earth...The unnumbered worlds in the universe are all similar in form and rank and subject to the same forces and the same laws."

                                        --- Giordano Bruno (1584)

The parallax technique introduced by the heliocentric model provided, for the first time, a practical method to estimate the distances to stars by triangulation and hence their intrinsic brightnesses. Well before the first actual measurement of a stellar parallax, the improving lower limits on stellar distances made by successive attempts to measure parallax with telescopes demonstrated that stars might well be, as Bruno and others believed, as bright intrinsically as the Sun. The case that the Sun is a star gradually grew stronger.

After 1600, scientific discoveries about the natural world progressed rapidly, at least by earlier standards. Scientific discoveries and the protocols of the "scientific method" were quickly disseminated by printed books.

The next key development for physics and astronomy was the discovery that what we call "gravity" is really only the local manifestation of a universal force between all forms of matter and that it is this same force that controls the motions of the planets and moons in the Solar System.

Here, we describe the work of four astronomers/physicists whose work was pivotal in understanding and quantifying gravity -- a watershed for all of science. This period includes the first use of telescopes in astronomy (1609). The telescope was the first of the major pieces of instrumentation that would become the foundation of modern scientific research. The timeline chart below will help you keep track of who's who and when. Click for a larger version.

B. Tycho (d. 1601)

  • A Danish astronomer, Tycho Brahe was an observer -- the greatest before the invention of telescopes -- with a flamboyant personality and lifestyle. 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).

  • Literary footnote: The Rosencrantz and Guildenstern families were relatives of Tycho. William Shakepeare evidently became aware of that during Tycho's 1592 visit to England, and he named the two famous incidental characters in Hamlet (which is set in Denmark) after them.

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

    C. Galileo (d. 1642)

  • The Italian scientist Galileo Galilei (pictured at the right) 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."

    "...without [mathematics] we are wandering in vain through a dark labyrinth."

  • As a physicist:

  • As an astronomer:

    D. Kepler (d. 1630)

  • Johannes Kepler was a German mathematician. His picture is at the right.

  • He analyzes Tycho's data, all obtained without telescopes but much more accurate than any previous. He adopts the Copernican interpretation (that the Earth is a planet in orbit around the Sun) as the basis of his model for the planetary system.

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

  • In analyzing Tycho's observations of Mars, Kepler quickly discovers that models based on pure circular motions could not fit the data.

  • 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 there or 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 fact that the planes of the orbits of the other planets always include the Sun but do not include the Earth is a simple but important objection to geocentric interpretations of the Solar System.

      • 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 (inverse) 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 -- see the illustration of Kepler's First Law above).

      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 and the beautiful symmetry so admired by the Greeks.

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

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

    Kepler's laws and the concept of force

    Kepler's laws and the speed of light

    E. Newton (d. 1727)

  • Isaac Newton was an English mathematician and physicist and ranks among a handful of the most consequential 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 at the right.

  • 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 applied forces. He first published his ideas in The Mathematical Principles of Natural Philosophy in 1687.

  • He starts his formulation of dynamics with several basic postulates ("laws of motion") that are drawn directly from Galileo's experiments.



    Newton's Legacy

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    Last modified November 2023 by rwo

    Text copyright © 1998-2023 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.