Lecture 4

Early Discoveries about the Earth and Sky (cont.)

Examples of deductive reasoning

• Is the Earth flat or round?
  • early sea-faring peoples noticed that ships disappeared over the horizon, but most ships returned

  • Aristotle concluded the Earth was spherical for several other good reasons...

    

    

• Which is closer to the Earth, the Sun or the Moon? Aristarchus of Samos figured it out in about 280 B.C. How can you tell? Hint:

  

  Go here for an animation showing the phases of the Moon.

  • Moon reflects sunlight

  • how does Moon look to us for different Sun, Moon, Earth arrangements?

  • at 90 degrees, if Moon was further than Sun, it would look fully illuminated

  • at 90 degrees, if Moon was closer than Sun, it would look 1/2 illuminated ---> thus, Moon closer than Sun

• Which is larger, the Earth or the Moon? How can you tell? Hint:

  

  Go here for an animation showing an eclipse.

  • lunar eclipse -- how many Moon's fit in Earth's shadow? about 3 Moons

• What is the distance to the Moon?
  • size on the sky is about 1/2 degree (1800 arcseconds)

  • small angle equation

    

    

    All measurements on the sky are in angular units, not in linear units.

    

  • distance to Moon (D) = 206,265 x size of Moon (d) / angle on sky in arcseconds (alpha)

  • thus, D = 2.1 x 10⁵ d / 1.8 x 10³ = 120 d

  • from lunar eclipses, the diameter of the Moon (d) is 1/3 diameter of Earth

  • so, D = 40 times Earth's diameter

• What is Earth's size? Eratosthenes, 200 B.C., Alexandria

  

  • compared length of shadows at same time of day at two different locations

  • difference in length of shadows related to the angle at which you are looking at Sun, to Earth's curvature from Syene and Alexandria (1/50 of Earth's circumference)

  • relating angular difference to linear distance gives you the circumference of Earth

  • calculation was only off true value by 2%!

• How do we know the Earth's tilt? Eratosthenes again...
  • measured difference between noontime elevation of Sun in winter and summer

  • divide the difference by two to get the angle of the tilt

  • deduced that Earth's equator is tilted by 23.5 degrees

  • difference in height of Sun at different times gives latitude

• What causes seasons?

  Go here to relate the composer Vivaldi's ``The Four Seasons'' to the motion of the Earth.

  

  • light from a flashlight is more intense when focused over a smaller area

  • light from a flashlight is less intense when it is spread out by tilting the flashlight

    

  • similarly, the tilt of Earth's rotational axis and the Earth's orbit around the Sun work together to create the seasons

  • because the Earth's axis tilts toward Sun in summer, the Northern Hemisphere receives concentrated light

  • because the Earth's axis tilts away from the Sun in summer, the Northern Hemisphere receives light that has been spread out ---> less of the Sun's energy heats a given area of the surface ---> cooler

The Copernican Revolution

A brief history of cosmology

• in 350 B.C., Aristotle proposes that Sun and all other heavenly bodies orbit Earth (geocentric model)

  

• in 250 B.C., Aristarchus proposes Sun-centered, or heliocentric, model, but not accepted because
  • contradicted the great Aristotle

  • and predicted parallax, the change in the position of the stars as the Earth moves, which no one observed at that time

    

    

    To try an experiment using parallax, click here.

  • but Aristarchus said -- correctly! -- that stellar parallax would not be easy to observe because the stars are very far away

  • this idea of very distant stars also explained why stars don't get brighter or fainter as the Earth moves toward or away from them

• in 140 A.D., Ptolemy revives Aristotlean cosmology

• but planets show retrograde motion! Ptolemy was forced to explain such motion with a complicated system of epicycles

  Retrograde Motion
  Inferior Planets
  Superior Planets

  

• in 1500, Copernicus adopts heliocentric model
  • beautiful and elegant in its explanation of retrograde motion (click on figure for retrograde animation)

    

    The Copernican explanation of retrograde motion. As Earth overtakes Mars (a-c), it appears to slow its eastward motion. As Earth passes Mars(d), it appears to move westward (retrograde). As Earth draws ahead of Mars (e-g), it resumes its eastward motion against the background stars. The positions of Earth and Mars are shown at equal intervals of 1 month.

  • not totally correct, as it predicted that the planets all move in circular orbits around the Sun

Birth of modern astronomy

• Galileo's observations

  

  • tests of Copernican theory

    • sunspots and Sun's rotation (click image to see the Sun rotate)

      

    • Jupiter's four large (Galilean) moons

      

      

    • phases of Venus (full when smallest ---> full on far side of Sun ---> orbiting Sun)

      

  • mechanics
    • acceleration

      • rate of change of velocity

      • independent of weight or composition

      • example: feather and hammer fall to ground at same time on Moon (astronaut movie)

    • inertia
      • resistance of any object to a change in its motion

      • example: ball dropped from ship's mast

    • frame of reference
      • atmosphere carried with Earth, so we don't feel our motion

      • cabin of smoothly sailing ship

      • astronauts on Space Shuttle

• Kepler's Laws
  • Johannes Kepler used Tycho Brahe's data

    

  • three laws of planetary motion (click here to experiment with them)
    • the orbits of the planets are ellipses -- not circles -- with the Sun at one focus of the ellipse

      

    • a line from the planet to the Sun sweeps over equal areas in equal intervals of time ---> planet moves faster when closer to the Sun

      

      Click the figure above for another applet.

    • a planet's orbital period squared is proportional to its average distance from the Sun cubed; or, P²(in years) is proportional to a³(in AU), where P is the orbital period and a is the semimajor axis of the ellipse