ASTRONOMY 9: HISTORY OF COSMOLOGY

Handout #14

J. E. Baker
UC Berkeley, Spring 2000

Tycho Brahe and Johannes Kepler

1.

Tycho Brahe (1546-1601, Denmark)

• The ``Astronomer Prince'', a very eccentric character, pompous and arrogant nobleman
• 1557: Studies law in Copenhagen (age 13)
• 1558: Witnesses partial solar eclipse, amazed that astronomical events are predictable, devotes self to astronomy
• Dispute with relative over who was best at math $\Rightarrow$ duel, nose sliced off and replaced with silver!
• Important recognition: astronomy needs continuing observations of greater precision and accuracy
• Builds improved instruments, becomes greatest pre-telescopic observational astronomer
• Positional data accurate to $2^\prime$ (1/30th of a degree)!
• Important astronomical events in Tycho's life

  (a)

  1558: Partial solar eclipse

  (b)

  1572: Supernova in Cassiopeia

  • No measured parallax $\Rightarrow$ well beyond the moon in Aristotle's celestial realm
  • Heavenly change, contradicts Aristotelian cosmos

  (c)

  1577: Comet

  • Limits on parallax $\Rightarrow$ at least 6$\times$ farther than Moon
  • No mere ``atmospheric phenomenon''!

• King Frederick II gives Tycho the entire island of Hveen!
• Tycho builds grandiose observatory Uraniborg, ``Castle of the Heavens''
• Rules the island in grand dictatorial style
• Tycho's cosmology
  • Compromise between geocentric and Copernican ideas
  • Earth remains motionless at cosmic center (plausible, since fixed stars did not have measurable parallax)
  • Sun goes round earth as in geocentric model
  • But other planets go round Sun, as in Copernican model!
  • Note: Tycho's model breaks the ``celestial spheres''
• Becomes ``Imperial Mathematicus'' to Rudolph II in Prague (1596)
• 1601: Dies of uremia after dinner with the Rosenbergs, ``let me not seem to have lived in vain''

2.

Johannes Kepler (1571-1630, b. Germany)

• Founded ``astrophysics''
• Search for mathematical laws to describe natural phenomena (planetary motion)
• Nightmarish childhood
• Attended Lutheran seminary from age 13-17
• Prof of math and astronomy at Gratz by age 23!
• Kepler deeply influenced by Pythagorean thought
  • Beauty of geometry and mathematics
  • Harmony of the spheres
  • Convinced that cosmos must be geometrically beautiful
  • Mystical inspiration for work
• Kepler's ``day job'': astrology!
  • 1601: writes treatise that breaks with principles of Ptolemaic astrology
  • Tries to make astrology more ``certain'', based on harmonies
  • Importance at imperial court of Holy Roman Emp. Rudolph II
  • Thought souls interacted with planets, ``the hidden cause''
• Kepler's main obsession: significance of the fact that there were six known planets
  • Today we know that planets formed out of gravitational instability in a disk-shaped nebula 5 billion years ago
  • Fact that there are 9 is an ``accident'' of the initial conditions of the nebula (rarity of such a configuration is unknown--so far other solar systems look very different, but we can't detect similar ones to our own for another decade or two)
  • For Kepler, six planets must be related to the five regular polyhedra (Platonic solids)
  • Solids with regular polygons as faces and all vertices look the same
  • Can prove that only five exist:

    (a)

    Tetrahedron: 4 triangular faces, 3 edges at each vertex

    (b)

    Octahedron: 8 triangular faces, 4 edges at each vertex

    (c)

    Icosahedron: 20 triangular faces, 5 edges at each vertex

    (d)

    Cube: 4 square faces, 3 edges at each vertex

    (e)

    Dodecahedron: 12 pentagonal faces, 3 edges at each vertex

  • Kepler thought that planets moved on six spheres circumscribed and inscribed around the five solids
  • Finally failed to reconcile with observations, though fit to within about 5% error
  • Very modern scientific innovation: if theory doesn't fit the observations, must discard it!
• ``Under a calamitous sky'', Kepler marries a rich widow, ``simple of mind and fat of body'' with a ``stupid, sulking, lonely, melancholy complexion''
• Very personal and amusing (though obscure) writings, unusual record of the blind alleys and frustrations in scientific thought
• 1597: Mysterium Cosmographicum (Secret of the Universe)
  • Presents Kepler's cosmology involving Platonic solids
  • First outspoken defense of Copernicus (50 yrs after death!)
  • Search for the physical causes of planetary motion
• Kepler recognizes that he needs good data $\Rightarrow$ Tycho
• 1600: Tycho invites Kepler to join him; rocky relationship, constant arguing
• Tycho knows Kepler has theoretical brilliance to interpret his data, but jealously guards his data
• After Tycho's death, Kepler gets Tycho's title and data
• Struggles long and hard trying to understand the motion of Mars
• Heliocentric model of orbit of Mars using circles leads to a stubborn error of $8^\prime$, even with use of equant
• ``Because these $8^\prime$ could not be ignored, they alone have led to a total reformation of astronomy.''
• 1609: Astronomia Nova (New Astronomy)
  • Problem of orbit shape; spends years trying all kinds of ovals and epicycles, finally stumbles across answer
  • First 2 of Kepler's three laws:

    1.

    Planetary orbits are ellipses with Sun at one focus

    2.

    Orbits sweep out equal areas in equal times

  • Finally smashes the dogma of uniform circular motion!
  • No more epicycles
  • First precise, verifiable natural laws
  • Work on magnetism at this time by William Gilbert (1544-1603, physician to Elizabeth I)
  • Kepler thinks the planets are held in their orbits by ``magnetic'' force from Sun which ``diminishes in ratio to distance as does the force of light''
  • Connection of Earthly with celestial
  • Tantalizingly close to idea of universal gravity, but doesn't quite get it
  • Kepler's cosmos and the Holy Trinity: Father--Sun at center, Son--Sphere of stars at circumference, Holy Ghost--force from Sun in intervening space
• 1619: Harmonice Mundi (Harmony of the Worlds)
  • Importance of Pythagorean harmony and geometry
  • Compares ratio of min/max distances of each planet from Sun with musical harmonies; music of the heavens!
  • After much trial and error, stumbles onto Third law: $P^2 \propto a^3$
    • P= period of planet (how long to go around Sun)
    • a= semi-major axis of ellipse (for special case of circle, radius)
    • If P in years, a in A.U., then constant is 1:
      
      $$\left(\frac{P}{1\ \mathrm{year}}\right)^2 = \left(\frac{a}{1\ \mathrm{AU}}\right)^3$$
      
      

    • A.U. = astronomical unit = Earth-Sun distance = $1.5\times 10^8$ km
• 1621: Epitome of Copernican Astronomy
  • Generalization of 3 laws to all planets
  • First textbook of Copernican astronomy
  • Placed on Index of forbidden books
• 1627: Rudolphine Tables
  • Use of 3 laws to predict planetary motions
• Kepler still does not understand physical reasons why his laws exist...
• Somnium: last published work
  • First real modern science fiction story
  • Dream of a journey to the moon
• Dies sickly and poor, unable to collect monies owed to him

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