THE COPERNICAN REVOLUTION AND
THE HELIOCENTRIC SOLAR SYSTEM

The finale of the Greek astronomical contributions:

Hypatia (370--415 CE) of Alexandria, built better instruments and made more accurate
positional measurements. She was murdered by monks who objected to
her paganism and her astrology.

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After the burning of the Alexandria library and the fall of Rome,
Astronomy in Europe withered, with only parts of Greek and Roman knowledge retained.

Rise of Islam led to large observatories in Samarkand in Central Asia and Persia,
with more careful observations and improved instruments.
Greek and Indian knowledge (e.g., zero) were combined and preserved. -->

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THE COPERNICAN REVOLUTION

Nicholas COPERNICUS (1473--1543), a Polish cleric, argued in favor of
a HELIOCENTRIC COSMOLOGY ---
THE EARTH IS NOT THE CENTER OF THE UNIVERSE.

  • The center of the Earth is the center of gravity for objects
    near it and for the MOON.
  • The other PLANETS REVOLVE around the SUN.
  • The STARS are MUCH FARTHER from the EARTH than is the SUN.
  • STELLAR (and SOLAR) ``MOTIONS'' ARE APPARENT:
    ALL ARE DUE TO MOTION OF THE EARTH.
  • PLANETARY ``MOTIONS'' are also substantially DUE TO THE EARTH'S MOTIONS.

    Copernicus' model still assumed perfectly circular orbits and did not
    dispense with epicycles --- but now the main orbits went
    around the Sun and the epicycles could be smaller.

  • While this model fit the data available then, it was only
    slightly better and certainly not proven.

    SO WHY IS IT ``BETTER''?

    THE HELIOCENTRIC MODEL is BOTH SIMPLER and more BEAUTIFUL
    than the Ptolemaic geocentric model.

  • "Occam's razor" suggests that if two hypotheses both describe the
    same data equally well, choose the simpler one.
  • At the time of Copernicus, each gave pretty equivalent explanations
    of the current data, but the Heliocentric model was a little simpler.

    Still, it gained few adherents: it went against "common sense" and was written in Latin.

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    BETTER OBSERVATIONS

    TYCHO Brahe (1546--1601), a Danish noble, designed and had built HUGE instruments
    at Uraniborg on Hveen, including:

  • sextants (for measuring angles)
  • astrolabes (for locating positions on the sky)

    These allowed for PRECISION MEASUREMENTS, particularly of planetary positions.

  • Accuracies of about 1 arc minute in planetary and stellar positions were achieved.
  • Tycho was the first to quote errors along with his measurements.

    After the peasants protested that his taxes were too high, he lost favor
    with the new Danish king and moved to Prague in 1597.

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    EMPIRICAL DISCOVERY

    Johannes KEPLER (1571--1630) an Austrian, was an accomplished
    amateur observer, who worked under Tycho from 1600 and inherited his data.

    He had already accepted the basic Copernican picture and tried to use this
    superior data to see if it could fit better than the Ptolemaic theory.

  • Tycho assigned Kepler to examine the data on Mars first,
    since it required the biggest epicycles.

    After many trials, he was able to make sense of the relative
    positions of Mars and Earth if

  • MARS MOVED IN AN ELLIPSE, WITH THE SUN AT ONE FOCUS.
    This broke with the longstanding assumption that combinations of circles
    were the only way to explain things; after all, a circle is just a special ellipse,
    with ECCENTRICITY = 0.

    By 1609 he had also realized that Mars' orbit swept out equal areas in equal times.

    KEPLER'S LAWS of PLANETARY MOTION

    Ten more years of work led to the THREE EMPIRICAL LAWS:

    1. All planets follow elliptical orbits, with the Sun located at one focus.

    2. Every planet sweeps out equal areas in equal times as it orbits the Sun.

  • In other words, planets move fastest when closest to the Sun (near PERIHELION)
    and slowest when furthest away (APHELION).

    3. The cube of the semi-major axis of a planet's orbit is
    proportional to the square of its period.

  • a^3 = P^2
  • if a is in units of AU and P is in years (in OUR solar system).
    Examples: a_Mars = 1.524 AU so P_Mars = (1.524)^3/2 = 1.881 years;
    a_Jupiter = 5.20 AU so P_Jupiter = (5.20)^3/2 = 11.86 years.
  • More generally, a propto P^{2/3} OR
  • P propto a^{3/2}
  • Later Newton showed that this general proportionality (but not equality)
    was always true for systems with a single dominating mass, not just our solar system.

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    SIDEBAR ON ELLIPSES:

  • An ellipse is a conic section: slice a cone with a plane not parallel
    to its base but not as steep as its side and you'll get an ellipse.
  • The easiest way to draw one is to loop a string around two tacks,
    holding it taught with a pencil point.
  • A definition is that an ellipse is the locus of points the
    sum of whose distances from two other points (the foci) is constant.
  • The longest axis through an ellipse is the MAJOR AXIS.
  • One half of that is the SEMI-MAJOR AXIS, a
  • One half of the shortest axis (perpendicular to the major axis) is the SEMI-MINOR AXIS, b
  • The distance from the Center to each Focus is the semi-major axis times the ECCENTRICITY
    or FC = ae
  • In equations: e = [1 - (b/a)^2]^{1/2}
    OR: b^2 = a^2 (1 - e^2)
  • If e = 0 we have a circle (b=a; foci and center coincide);
    if e = 1 we have a line-segment