chart linked here.

A. TYCHO (d. 1601)

• Observer--greatest before invention of telescopes

• His data for the "supernova" (an exploding star) of 1572 demolishes the Aristotlean doctrine of heavenly perfection & permanence. T. uses a parallax/triangulation technique to show that the supernova is more distant than Saturn and therefore must reside in the region the Greeks believed to be unchanging.

• Collected massive set of unprecedentedly accurate (1 arc-minute) data on planetary motions, later analyzed by Kepler.
  Although T. died before he was able to analyze his data, he favored a geocentric universe (though one in which the Earth rotated and all the other planets orbited the Sun) because he did not believe the stars could be so distant that he could not measure their parallaxes.

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

• Physicist & astronomer

• Experimentally demonstrates that acceleration of objects under gravity is independent of their mass. This, and many other of G's experimental results, flatly contradict the principles of Aristotlean physics, which, owing to Greek disdain for experiment, had never been subjected to empirical tests.

• 1610: first astronomical use of the telescope.
  
  One of Galileo's telescopes. (Click for info.)

  
  

  • G's telescopes were small (above), with relatively crude lenses only a few inches in diameter...but they yielded the first fundamentally new astronomical data in over 1500 years and utterly transformed astronomy.

  • Discovers mountains/craters on the Moon; and spots on the Sun.

  • Discovers the existence of thousands of stars fainter than the eye can see.

  • Shows that stars and planets are different kinds of objects: planets show perceptible disks, but stars are points of light.

  • Discovers the 4 brightest satellites of Jupiter---the first new planetary bodies discovered in recorded history.

  • Discovers the phases of Venus

  • These discoveries strongly support the Copernican interpretation, though don't actually prove it:
    • The craters on the Moon and spots on the Sun contradict the claims of the Greeks (and also the Catholic Church) that the heavenly bodies are perfect.
      
    • The phases show that Venus orbits the Sun, not the Earth.
      
    • Jupiter's satellites show the existence of a second "center of motion" in the solar system and that planets can move without "losing" their satellites. (It had been argued that if the Earth moved around the Sun, the Moon could not be bound to it.)

  • The telescope is sufficiently cheap and easy to build that anyone can test claims made about the nature of astronomical objects. Encourages questioning of traditional authorities.

• Mathematician

• Insisted that models agree with observations within observational error ====> the basis of modern empirical science

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

• Observations for Mars could not be fit by models based on circular motions
  K. works years to resolve an 8 arc-min discrepancy & discovers Mars' orbit is an ellipse, not a circle

• Reinterprets data for all planets and condenses his conclusions to three "Laws of Planetary Motion." Note that these are derived empirically from Tycho's data.

• Kepler's Laws:
  1. Planetary orbits are ellipses with the Sun at one focus
     
     Notes: The Sun is not in the center of the ellipse. The Sun is in the same plane as the ellipse, but the orbits of different planets lie in different planes. The planetary orbits are not very elliptical, which is why circles are fair approximations, as in Copernicus' model. There is nothing at the second focus of the orbital ellipses.

  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's Second Law -- Click for animation.
     This implies a given planet moves faster when it is nearer the Sun.
     This is also the first hint of a universal physical principle: 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).
     In equation form, P² = K a³, where P is the period, a is the semi-major axis, and K is a constant.
     The time taken to complete one orbit grows more than in direct proportion to orbital size. The orbital period is the circumference of th orbit divided by the planet's mean velocity. Since the circumference of an orbit increases in direct proportion to its semi-major axis, the Third Law implies that the velocities of planets in larger orbits are slower than for planets nearer the Sun.
     A planet with an orbital diameter 5 times the Earth's will require 11 Earth years to complete an orbit.

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

• Net result: tens of thousands of individual observations have been reduced to a small set of simple geometric and arithmetic relationships. All the arbitrary complexity of Ptolemy has vanished.

• The concept of force:
  • In the early Greek cosmologies, there was no need for the concept of a special "force" to control the planets in their orbits, since they were thought to be affixed to crystalline spheres, and the tendency of things to fall toward Earth was thought to be the product of our special location at the center of the universe.

  • But in the real world revealed by Kepler's work, the Sun is the key to the planetary motions. It is not just the center of the solar system but is also directly linked to the geometry of planetary orbits and to motions within those orbits. Kepler therefore postulated that the Sun exerts a force on the planets. Although he was unable to formulate this idea properly, Newton did so with phenomenal success.

• Mathematician & physicist

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

• Drawing on Galileo's experiments, Newton's formulation of dynamics directly contradicts Aristotle, who believed that objects only move if subjected to a force. Newton shows that objects only accelerate (i.e. change velocity) if subjected to a force.
  
• Newton's Second Law of Motion:
  (Force) = (Mass) x (Acceleration)

• To explain the motions of falling objects near the Earth as well as the movement of the Moon and planets in their orbits, Newton postulated the existence of a universal gravitational force
  • Every object with mass exerts an attractive force on every other object with mass.

  • F_grav = G M₁ x M₂ / R²
    where G is a universal constant, the M's are the masses, and R is the distance between the two masses

  • The same expression applies to the Earth's influence on the apple, to the Earth's influence on the Moon, to the Sun's influence on the Earth, and so on.

• To predict motion under gravity, substitute F_grav in the Second Law and solve for acceleration ==> a differential equation for a planet's motion. To solve it, Newton invents calculus. More details on gravitational orbits are given in Guide 9.

• For two gravitating objects (e.g. one planet and the Sun) the resulting orbit satisfies ALL THREE of Kepler's Laws

• Newton's theory explained not only all the known properties of the planetary orbits but also the motion of objects (like bullets and pendulums) moving in Earth's gravity, the tides, the Earth's precession, the Earth's oblate shape, and many other previously mysterious phenomena. In the 1800's it correctly predicted the location of a new planet, Neptune. Newton's theory was complete, quantitative, and predictive.

• Newton's legacy: first generalized physical laws; modern physics & astronomy; calculus; modern engineering. Newton's Laws became the cornerstone of the "Scientific Revolution."

Reading: FMW: Sec. 1.4, 2.1, 2.2
Further optional reading: Arthur Koestler, The Sleepwalkers; Timothy Ferris, Coming of Age in the Milky Way; J. L. E. Dreyer, A History of Astronomy from Thales to Kepler; Richard S. Westfall, Never at Rest: A Biography of Isaac Newton.

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Last modified September 2003 by rwo

Text copyright © 1998-2003 Robert W. O'Connell. All rights reserved. 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 121 at the University of Virginia.