- What were Tycho Brahe's main contributions to astronomy?
- What was Tycho's model of the solar system?
- What did Johannes Kepler discover about the motions of planets? What role did Tycho's observations play in these discoveries?
- What are Kepler's three laws of planetary motions?
- What role did physical and philosophical reasoning play in Kepler's discoveries?

Early work:

- Uses absence of parallax to demonstrate that "new star" of 1572 (now known to be a supernova -- exploding star) further than Moon.
- Explodes Aristotle's conception of immutable heavens.
- Parallax observations of comets show also beyond Moon.
- Shatters Aristotelian idea of "crystalline spheres" moving planets.

- King of Denmark, impressed, gives money to build two large observatories.
- With specially designed, large instruments, Tycho measures planetary and stellar motions with unprecedented accuracy.
- For best observations, accurate to 1-2 arc-minutes (1 arc-minute = 1/30 diameter of Moon).
- Typical observations accurate to 4 arc-minutes.
- Measurements only superseded after invention of telescope.

- Anti-Copernican, on philosophical grounds and because his observations do not reveal stellar parallax.
- Proposes mixed model (the
*Tychonic model*): planets orbit the Sun, but the Sun orbits the Earth. - Philosophically very different, but nearly impossible to distinguish from Copernican model on observational grounds. (Stellar parallax is the difference.)

An enthusiastic Copernican from early on.

Astronomical work motivated by two overarching principles:

- Discover mathematical harmonies governing the structure of the cosmos.
- Explain physical causes of celestial motions.

Second principle a modern one, breaking with long tradition (Aristotle through Copernicus) that laws of motion on Earth don't apply to heavens.

Early idea: orbital radii of planets (distances from Sun) determined
by a geometry of the five regular polyhedra.

(Modern note: This idea is completely wrong.)

Knew this idea described the data approximately, but committed to testing
it with best observations available.

Tycho had the best observations.

Kepler became his research assistant, in 1600. Tycho assigned
him problem of detemining orbit of Mars.

Kepler inherited Tycho's position and, more importantly, data,
after Tycho's death in 1601.

- Key idea: planets move under influence of force from Sun.
- Thought this force might be magnetism, or something like it.
- Had "Aristotelian" idea of how force works: force needed to keep something in motion, planets "pushed" around their orbits.
- Noted that more distant planets move more slowly, and individual planets move slower when further from Sun.
- Thought this reflected force getting weaker further away from Sun.

- Tried to construct orbits for Earth and Mars using circular motions.
- Best solution disagreed with several of Tycho's observations by 8 arc-minutes.
- Thought measurement could not possibly be this far off, decided to try other shapes.
- Needed rule for how speed of planet varies with distance from Sun.
- Motivated by idea of force weakening with distance, came up with "equal area rule": the line joining the Sun and the planet sweeps out equal areas in equal times.
- Tried various possible shapes for orbits.
- After eight years of work, success at last: the orbit of Mars is an ellipse, with the Sun at one focus.

Kepler's empirical results for motions of planets summarized by three "laws".

These *completely* replace the epicycles and other complications of
the Ptolemaic and Copernican models.

**First Law**:
The orbits of the planets are *ellipses * with the Sun at one focus.

An ellipse:

- Hammer two nails into board. Surround with longer loop of string.
- Draw oval with string stretched as far as it will go.
- This is an ellipse, and the positions of the nails are the
*foci*. - Characterized by
*semi-major axis*, half the length of the longest axis, and*eccentricity*, departure from circularity. - Limiting case is a circle: both foci at same position, semi-major axis is radius, eccentricity is zero.

**Second Law:** The line joining the Sun and the planet sweeps out equal
areas in equal times.

- Completely specifies change of speed with distance from Sun.
- Planets move fastest when closest to the Sun (
*perihelion*). - Slowest when furthest from the Sun (
*aphelion*).

**Third Law:**
The square of a planet's orbital period is proportional
to the cube of the semi-major axis of its orbit.

Define one *astronomical unit (AU)* to be the semi-major axis
of the Earth's orbit.

Third law can be expressed as an equation:

a = semi-major axis in AU.

Kepler discovered third law 10 years after first two.

Delighted by implied harmony of cosmos.

Results:

- Description of planetary orbits by Kepler's laws much simpler than Ptolemaic or Copernican system.
- Also much more accurate; first quantitative breakthrough in 1400 years.
- Greatly strengthens case for heliocentric cosmos: it allows a simpler and more accurate description of celestial motions.
- Three laws provide concise empirical summary of planetary motions, a clear target for Newton to aim at.

- Revolutionary idea: use physical reasoning to understand planetary motions.
- Kepler's understanding of physics seriously flawed.
- Nonetheless, physical approach crucial to his success:
- Referred planetary positions to true position of Sun, not center of Earth's orbit. Simplifies problem drastically.
- Reasoning about force from Sun led to equal areas law.
- Provided principle to replace uniform circular motions.

- Kepler's musings on mathematical harmony of solar system seem largely misplaced, but they gave him motivation for years of difficult work.

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Updated: 2005 April 3 [dhw]