7. THE DISCOVERY OF GRAVITY

• Tycho uses a triangulation, or "parallax", technique to show that the supernova is more distant than Saturn and therefore must reside in the cosmic region the Greeks believed to be unchanging.

• Similarly, Tycho showed that the bright comet of 1577 lay beyond the Moon. Previously, it was assumed that comets (obviously changing from day to day) had to be phenomena inside Earth's atmosphere.

• The idea that there were changes in the distant cosmos other than the serene circular motions favored by astronomers up to this time was startling. Not only did this fatally damage the cozy medieval picture of a tranquil, eternal universe beyond Earth, but it obviously made the universe of much greater intellectual interest than it may have been before.

• Primary reason? Because he did not believe the stars could be so distant that he could not measure their parallaxes. This is a sound scientific argument and a major obstacle to earlier acceptance of the Copernican model.

• If Tycho could have measured stellar parallaxes, he would have instantly become a Copernican. But the distances between stars would have had to be about 100 times smaller than they are for that to be possible without telescopes. (Interestingly, that would have been the case had the Earth been situated in the central core of our Galaxy rather than in its outskirts.)

• Here is an animation of the stellar parallax effect as it would be observed with a modern telescope.

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

B. Galileo (d. 1642)

• Experimentally demonstrates that objects falling in response to gravity accelerate---that is, increase their velocity---in a highly systematic way and that their acceleration 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.

• 1609: 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. They 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) about the perfection of the heavenly bodies.

  • Venus' phases prove 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. Somewhat like the Internet in our time, this encourages the questioning of traditional authorities.

C. Kepler (d. 1630)

• Because of his great respect for Tycho's precise observational technique, Kepler insisted that interpretive models agree with the observations within observational uncertainty or "error".

• This is the basis of modern empiricism. Today, this requirement for acceptable interpretations is taken to extend to all relevant data. It is a demand for consistency with all interpreted phenomena and established physical principles. This is the "cumulative" aspect of science described in Study Guide 1.

• The term "error" here does not imply some kind of mistake. Instead, it refers to the uncertainties that are inherent in any measuring process, no matter how careful. An essential part of a scientist's job is always to make good estimates of the uncertainty in any published data.

• He works eight years to resolve an 8 arc-min discrepancy between the models and data. (This is 8 times the observational precision of Tycho's data.)

• He finally realizes that Mars' orbit is an ellipse, not a circle

• The methods Kepler used to achieve this breakthrough are re-created in the ASTR 1210 optional laboratory on the Orbit of Mars.

Kepler's Laws

1. Planetary orbits are ellipses with the Sun at one focus
   
   • 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 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'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).
   
   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 a^1/2) 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.

The complex epicycles in Ptolemy's model were necessitated by two factors: the fact that the Earth moves, whereas it was assumed to be stationary, and the fact that actual orbits are elliptical, whereas they were assumed to be circular.

The concept of force

• In the early Greek cosmologies, there was no need for a special "force" to control the planets in their orbits or for interactions between the planets or with the Sun, since all the cosmic bodies were thought to be affixed to crystalline spheres, and the tendency of things to fall downward was thought to be the product of Earth's special location at the center of the universe, not something caused by the Earth itself.

• 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---e.g. one which either pulls or pushes planets along their orbits. Although he was unable to formulate this idea properly, Newton did so with phenomenal success.

D. Newton (d. 1727)

• His formulation of dynamics draws directly from Galileo's experiments.

• In the absence of a (net) force, an object will remain at rest or in straight-line motion with no change in speed.

• The presence of a net force produces a change in velocity (i.e. an acceleration).

• This law directly contradicts Aristotle, who believed that objects cannot move at all unless subjected to a continuous force. Newton argues that continuous, uniform motion can occur in the absence of a force.

• The quantitative relationship between applied force and resulting acceleration is
  (Force) = (Mass) x (Acceleration)
  Or, solving for the acceleration: (Acceleration) = (Force)/(Mass)
  There is a directionality inherent in this formulation. Mass has no "direction," but forces do. The Second Law implies that the acceleration will occur in the same direction as the applied force.

• The second law is covered in more detail in Guide 8 and in the Supplements.

• Kepler had introduced the idea of a special force, exerted by the Sun on the planets to keep them moving in their elliptical orbits. Newton brilliantly adapted this concept, realizing that all bodies could exert a force on one another. In particular, the Earth could exert a force on the Moon to keep it in its orbit around the Earth. But in that case the Earth would also exert a force on objects near the Earth's surface.

• Newton argued that our everday experience of "gravity" (our sense of a downward pull or the downward acceleration of falling objects) was a manifestation of exactly the same kind of force, exerted by the Earth, that kept the Moon in its orbit. He sought a formulation that would simultaneously explain local gravity and planetary orbits.

• So Newton postulated the existence of a universal gravitational force in the following form: Every object with mass exerts an attractive force on every other object with mass. The force declines with distance. Quantitatively, the force is an "inverse square law":
  F_grav = G M₁ M₂ / R²
  where G is a universal constant, the M's are the masses of the two objects, and R is the distance between the two masses
  The gravitational force diminishes rapidly with distance. The Sun's gravitational force on two identical planets, one of which is 1 AU from the Sun and the other of which is 5 AU distant would differ by a factor of 25.
  Importantly, however, the force only becomes very small with distance, it never actually goes to zero. So, all the stars in our Galaxy, for example, despite being on average 1000's of light years away, combine to affect the motion of the Sun around the Galaxy.

• Gravity acts along the (radial) line connecting the two bodies. It therefore does not (as Kepler thought) necessarily act to "push" or "pull" objects along their current directions of motion. In fact, for a circular orbit, the gravitational force acts perpendicular to the direction of motion.

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

• Newton's formulation of gravity would have been conceptually impossible without the elimination by Copernicus of the assumption of the special, central location of the Earth.

• [Modern footnote: Gravity was only the first universal force to be recognized. Physics now recognizes four types of force, the other three all much stronger than gravity but having much shorter ranges in practice. See Supplements 2 & 3.]

• Newton combined his gravitational force law with his Second Law of motion in order to predict the motion of an object in an external gravitational field. To solve the equations, Newton had to invent calculus.

• For two gravitating objects (e.g. one planet and the Sun) the resulting orbit satisfies all three of Kepler's Laws. Details are given in Study Guide 8.

• Newton's theory explained all the known properties of the planetary orbits and also the motion of objects (like apples, bullets, and pendulums) moving in Earth's gravity. Remarkably, it was quickly extended to determine the orbit of Halley's Comet (which was correctly predicted to return in 1759) and to explain the tides, the Earth's precession, the Earth's oblate shape, and many other formerly mysterious phenomena.

• In 1846 Newton's theory was used to correctly predict the location of a new planet, Neptune based on unexpected residuals in the orbit of the planet Uranus. This was regarded as the "greatest triumph" of Newtonian mechanics.

Newton's Laws were complete, quantitative, and predictive. They were amazingly clarifying and their importance extended vastly beyond their original frame of reference. They represent the first generalized physical laws. From them emerged modern physics, mathematics, and engineering. They became the cornerstone of the "Scientific Revolution."

Bennett textbook: Ch. 3.3 (Copernicus, Tycho, Kepler, Galileo)
Study Guide 7
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.
Puzzlah for Wednesday, Feb. 13:
You have two objects, A and B, both of which are the same shape. B weighs twice as much as A. You drop both simultaneously from a height of 3 feet. What happens?
1. A (the lighter object) hits the ground first.
2. B (the heavier object) hits the ground first.
3. They hit at the same time.

You should attempt your own experiments to determine the answer...don't just take the word of the readings!

Bennett textbook: Ch. 4.1, 4.2, 4.3, 4.4 (Newtonian dynamics & gravitational orbits)
Study Guide 8

More information on Tycho
A Tycho home page
Images of Tycho (Oxford University exhibition)
The Galileo Project. (elaborate site devoted to Galileo and his times)
More information on telescopes (Study Guide 14)
Biography of Kepler (Mike Fowler, PHYS 109)
Biography of Kepler (MacTutor Archive)
More information on Kepler's Laws (U.Tenn)
Kepler's Discovery (background on Kepler and his science)
Flashlet Interactive Simulation of Kepler's Laws (Mike Fowler)
Java Simulation of Solar System Orbits Shows shapes and orbital speeds. (UMd)
Java Simulation of Parallax (UBC)
Directory of links for Isaac Newton
Biography and Contributions of Newton, with many supporting links (Weisstein)
Voltaire's Comments on Newton (at a distance of 100 years)
More information on Newton's Laws (Study Guide 10/Supplements II and III)
More information on Newton's Laws (U.Tenn)
JPL Notes on Gravitation & Mechanics (Kepler's & Newton's laws & applications to spaceflight)

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Last modified February 2013 by rwo

Text copyright © 1998-2013 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 1210 at the University of Virginia.