7. THE DISCOVERY OF GRAVITY

• Tycho uses a version of the parallax technique, based on observations taken 6-8 hours apart, 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 distant 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 gave experiment and observation explicit precedence over authority. (Recall the discussion of the role of authority in modern science in Guide 1.) His many amazing discoveries made with small telescopes were a resounding demonstration of the superiority of empiricism in learning about nature. His disdain for a reliance on authority and his devotion to mathematical empiricism are often expressed in his writings:

• Experimentally, he demonstrates that objects falling in response to gravity accelerate---that is, increase their velocity---in a highly systematic way (directly proportional to time) and that their acceleration is independent of their mass.
  Among other things, this means that two objects if dropped simultaneously from any height will hit the ground at the same time (ignoring extraneous effects like atmospheric drag) no matter how much they differ in mass.

• He realizes that these results, and those of many other of his experiments, flatly contradict the principles of Aristotlean physics. Aristotle had naively assumed, as do most people, that heavier objects fall faster than lighter ones and that downward velocities are constant. Owing to the Greek distrust of experiment, Aristotle's assumptions had not been subjected to empirical tests, nor had most of the other details of his writings on physics.
  Isn't it amazing that you, if you did the "object drop" assignment in Guide 6, were able to make a very simple experiment in your dorm room that invalidates the claims of one of the most famous philosophers in history and that were assumed for be true for 1900 years? That's the power of empirical exploration of the real world.
  At least one Greek scientist, Strato, was aware of gravitational acceleration: he had observed the increasing separation between water droplets as they fall from a spout. But as in the case of Aristarchus, this important insight was submerged by the overwhelming influence of Aristotle on later thinking.

• Galileo made the first astronomical use of a telescope in 1609. He had learned of the invention of telescopes by Lippershey in Holland and quickly decided to make his own.

  One of Galileo's telescopes. (Click for details.)

  Galileo demonstrating telescope to his patrons

  • Galileo's telescopes were small (above), with relatively crude lenses only a few inches in diameter. But they yielded the first fundamentally new observational insights in over 1500 years. They utterly transformed astronomy. They allowed Galileo to discover:
    • Mountains and craters on the Moon.

    • That spots first observed by others on the Sun were features on the solar surface (not intervening material, for instance) and showed that the Sun rotated on an axis.

    • The existence of thousands of stars fainter than the eye can see. The "Milky Way" was revealed to be made up of a myriad of stars.

    • That stars and planets are different kinds of objects: planets show perceptible disks, but stars are points of light, which implies they are probably very far away.

    • Four satellites in orbit around Jupiter (now named the "Galilean satellites"). These were the first new planetary bodies discovered in recorded history. They are the four brightest of the (now known) 92 satellites of the planet.

    • The phases of Venus

• These discoveries strongly support the Copernican interpretation, though they don't actually prove it:
  • The craters on the Moon and spots on the Sun contradict the claims of the Greeks and the Catholic Church about the perfection of the heavenly bodies, which then cast doubt on their other pronouncements.

  • The rotation of the Sun demonstrates that celestial bodies can spin, which supports Copernicus' assumption that the Earth spins on its axis.

  • Copernicus' assumption that the Earth is a planet meant that the other planets should share some of Earth's characteristics: they should be round, they should shine by reflected Sunlight, and they should show surface features. All of these "predictions" were verified by Galileo's observations.

  • The fact that Venus displays an almost full range of phases from "crescent" to nearly "full" proves that Venus orbits the Sun, not the Earth. In Ptolemy's model, the epicycles of both Mercury and Venus lie between Earth and the Sun, so Venus can never appear in other than a crescent phase. See this illustration of the difference between the Copernican and Ptolemaic interpretation.

  • Jupiter's satellites show the existence of a second "center of motion" in the solar system. This ruins the symmetry of the geocentric model, in which the Earth lies at the only center of motion in the universe. Galileo's observations also showed that planets can move without "losing" their satellites. It had been argued that if the Earth moved around the Sun, the Moon would be left behind. Not so.

• In another blow to conventional wisdom, the discovery of vast numbers of previously invisible stars shows that the stars are not simply some kind of celestial decoration, "put there" for the benefit or instruction of human beings. This is the first evidence of a huge physical system of stars surrounding the Sun, confirming the speculations of Digges and Bruno, and the most important contribution to that time to the argument that the Sun itself is a star.

• The telescope is sufficiently cheap and easy to build that anyone can test claims made about the nature of astronomical objects. The availability of printed books and simple instruments like telescopes encouraged the questioning of traditional authorities, somewhat like the Internet in our time.
  • The subsequent history of astronomy was revolutionized at intervals by the advent of new telescope designs and auxiliary equipment.

• 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." Where there is a disagreement, it is the model, not the data, that must be revised. Empirical data takes precedence.

• 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" used in this context 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. (The discrepancy was 8 times the observational uncertainty of Tycho's data.)

• He finally realizes that Mars' orbit is an ellipse, not a circle
  Ellipses are perfectly well-defined geometrical figures, first studied by the ancient Greek mathematicians. But because they are less symmetrical than circles, Ptolemy did not attempt to incorporate them in his cosmological model.
  Other than Mercury, Mars has the most eccentric (non-circular) orbit of the five planets then known -- but Mercury is harder to observe and had less extensive coverage in Tycho's data sets. That is why the data for Mars was critical to Kepler's discoveries.

• 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 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'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).
   
   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 complexities such as epicycles in Ptolemy's model were needed because of three principal factors: the fact that the Earth moves, whereas Ptolemy assumed it to be stationary; the fact that actual orbits are elliptical, whereas he assumed them to be circular; and the fact that the velocity of a planet in its orbit varies, whereas he assumed constant circular velocities.

Kepler's laws and the concept of force

• In the early Greek cosmologies, there was no need for a special "force" to control the motions of planets in their orbits, 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, there are no crystalline spheres; rather, 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 by precise but non-intuitive mathematical relations. Kepler therefore postulated that the Sun exerts a force on the planets---e.g. one which either pulls or pushes planets along their orbits. Another kind of force was thought to be required in order to keep the Moon and the satellites of Jupiter bound to their parent planets as they moved around the Sun. Although Kepler was unable to formulate this idea properly, Newton did so with phenomenal success.

Kepler's laws and the speed of light

• Within 100 years, an avalanche of observations had shown that planetary motions obey Kepler's laws with remarkable precision. The laws were even extended to the motion of the four Galilean moons in orbit around Jupiter. The laws allowed astronomers to predict the eclipses of the moons by the body of Jupiter.
  However, measurements over about a decade showed that there were worrisome systematic discrepancies from the predicted Keplerian timing of eclipses. These were not understood, but it did emerge that they depended on the Earth's distance from Jupiter (which, due to their relative orbital sizes, varies from 4 to 6.5 AU's).

• In 1676 Danish astronomer Ole Romer, shown observing at the right, interpreted the discrepancies as the consequence of a finite value for the speed of light: light took longer to reach the Earth from Jupiter when the two planets were farther apart and delayed the times of the expected eclipses. Allowing for this delay confirmed the predictions from Kepler's Laws. This was the first evidence that light has a finite propagation speed. Romer's estimated value for the speed of light was about 25% smaller than modern determinations, but this was an extraordinary achievement for his time.

E. Newton (d. 1727)

Newton's First Law of Motion
• In the absence of a (net) force, an object will remain at rest or in straight-line motion with no change in velocity.

• 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 only occurs in the absence of a force.

Newton's Second Law of Motion
• 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 should 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, and the force declines with distance. Quantitatively:
  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 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.
  The gravitational force diminishes rapidly with distance. Quantitatively, it is an "inverse square law," with the force declining as the square of the distance between the two masses. 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 5² or 25.
  Importantly, however, the force only becomes very small with distance, it never actually vanishes (or goes to zero). So, all the stars in our Galaxy, for example, combine gravitationally to determine the motion of the Sun around the Galaxy, despite their being, on average, tens of thousands of light years away.
  Also, by its formulation, the strength of the gravitational force is the same on both masses and differs only in direction. Both masses will move in response to the force -- but the motion is inversely proportional to mass, so the smaller mass moves faster (see Guide 8).

• 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. Although this sounds paradoxical, it has a simple explanation in the context of Newtonian dynamics, which we explain in Study Guide 8.

• 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.
  And conversely, the geocentric, circular-motion universe becomes impossible if one accepts the reality of Newtonian gravity.

• [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 predicted 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 ocean 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 and theory of gravity were complete, quantitative, and predictive. They were profoundly 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.

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

Slides shown in lecture
The Path to Newton (Summarizes in text and graphical form the key conceptual
developments leading to Newton's laws of motion and gravity)
More information on Tycho
Images of Tycho (Oxford University exhibition)
The Galileo Project. (elaborate site devoted to Galileo and his times)
The Trial of Galileo (the most famous collision between science and religion)
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 (Wikipedia)
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)
The Newton Project (Oxford University)
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)
JPL Notes on Gravitation & Mechanics (Kepler's & Newton's laws & applications to spaceflight)

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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.