7. THE DISCOVERY OF GRAVITY

• 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.

• Similarly, T. showed that the bright comet of 1577 lay beyond the Moon. Previously, it was assumed that comets (obviously changing from day to day) were phenomena inside Earth's atmosphere. This poked another hole in the cozy medieval picture of a tranquil, eternal universe beyond earth..

• The idea that there were changes in the cosmos other than the serene circular motions favored by astronomers up to this time was startling. This 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 undoubtedly 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.

• 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 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) about the perfection of the heavenly bodies.
    
  • Venus' 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.

C. KEPLER (d. 1630)

This is the conceptual foundation of modern empirical science

• K. works eight years to resolve an 8 arc-min discrepancy. (This is 8 times the observational accuracy of Tycho's data.)

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

• The methods Kepler used to achieve this breakthrough are re-created in the ASTR 121 optional laboratory on the Orbit of Mars.

1. Planetary orbits are ellipses with the Sun at one focus
   
   • Note that the Sun is not at the center of the ellipse. 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: 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 proportional to a x a^1/2 and therefore 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.

• 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---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 drew directly from Galileo's experiments.

• The presence of a force implies 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 occurs in the absence of a force.

• The quantitative relationship between applied force and resulting acceleration is
  (Force) = (Mass) x (Acceleration)

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

• Every object with mass exerts an attractive force on every other object with mass. Quantitatively, the force is an "inverse square law":
  • 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

• 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 the apple, to the Earth's influence on the Moon, to the Sun's influence on the Earth, and so on.