Postalo Vleeptron / Commemorative: Kepler's Laws of Planetary Motion / be a sentient! learn these laws now! / sister driveby / send all smart kids to St. John's College, Annapolis Maryland USA

 

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Commemorative Sheet
Postalo Vleeptron &
...Vleeptron
...Advanced
...Mathematics
...Research
...Institute

Kepler's Laws 
of Planetary Motion
Epicycle / Conic Sections

Quiz Tomorrow, worth 40 percent of semester grade.

If you've memorized and understand the Laws, the Quiz will be a walk in the park, you can leave early and have Latte with Brainy Marissa.

If you don't know the Laws, expect 50 minutes of torment, guilt, shame, squirming, frustration, self-anger, agonizing doubt, worry, followed by a D or an F. You'll need all 50 minutes to pull a D.

Every sentient on Vleeptron knows these Laws inside and out. Instead of Spelling Bees, Vleeptron has Kepler's Laws Bees. Last year, the Planet-Wide Bee was held in the Ramada Ciudad Vleeptron, sponsored by VAMRI and Agence-Vleeptron Presse. Wz9+z0, a home-schooler, won. She knew what an epicycle was.

Do you?

Do you know the Conic Sections? I'm sorry I'm such an incompetent draftsman or I would have sliced the ice-cream cones with the razor blade to make a parabola and a hyperbola. I'm sure it's a snap with CAM-CAD, which can bite me. I'll keep working on it in MS_Paint.



 
Okay, I got all the Conic Sections now, including the parabola and hyperbola.
 

The background blue is a Custom Color, selected for how pretty I thought it was. Usually I'm content with MS_Paint's garish, lurid, blunt, short Crayola pallet. But this is Important, it needs a Custom Color.

My sister did a driveby for the 4th of July, and she and her husband very kindly assembled my Christmas Celestron Refractor Telescope.

Now I can see my beautiful Night Skies, and forest creatures cavorting, gamboling and predating (but all upside-down, such is the Way of the Refractor).

She told me about some work she did with NASA's Project Kepler.

When it was all over, and she'd learned 1000 Amazing Cutting-Edge Muy Starwarz Science Space Things, she still had no idea who Kepler was.

So I told her who Kepler was, in e-mails she read as she flew home to the Eastern Shore of Maryland USA. With pictures.

(Some filched, the rest me own invention.)

* * *

Not far from you is St. John's College, the Antidote to your woes. Every other college/university has a Math and Physics Department which crams math and physics into your head -- but none of the history that led up to K's revolutionary discovery -- and in the History department, there's almost never any History of Science, it's all Mary Queen of Scots and the Constitutional Convention.

St. John's is all "Great Books" seminars, but no segregation of subjects -- you get the math, the science, the history in the same room from the same teacher -- or, rather, you get it directly from Galileo's Dialogue of the Two World Systems and from Newton's Principia.

The history is super-ancient, and begins with

The Night Sky Puzzle

To the unaided (pre-telescopic) eye, the night sky (and it used to be spectacularly clearer) features the Moon, the "Fixed Stars," and then 5 star-like "planets" -- Greek for Wanderers.

The fixed stars are locked in a never-changing pattern. The pattern rotates around the North Star every night, but the Bear is always the exact same Bear, the Dragon is always the Dragon, etc. They may stand on their heads or lie on their sides in different seasons, but their stars always keep their fixed patterns.

But the Planets wander all over the place from night to night and season to season in crazy ways. Mars is particularly nutty -- for months it goes in one direction, and then suddenly it stops and goes backwards for a month or two -- retrograde motion.

The Babylonians and ancient Egyptians made amazing discoveries in astronomy. (The Babylonians could predict solar and lunar eclipses -- a nifty trick modern astronomers couldn't duplicate for 37 centuries.)

The Greeks -- sailors, travellers, shameless shoplifters and plagiarists -- were the first to try to find deeper, more profound answers to The Puzzle of the Planets, a scientific explanation of their screwy sky wanderings.

Celestial navigation was a huge motivator for the Greek merchant sailors. (Can you use a sextant?) A better bag of tricks about the night skies made the difference between commercial success and ruin, between life and death. So both kinds of Greeks -- head-in-the-clouds scientific dreamers, and practical, hard-headed merchants -- evolved better and better guesses about The Puzzle.

One barrier were some arbitrary philosophical, mystical and religious beliefs about Nature, due to Pythagoras and his disciple Plato.

The worst hurdle was the belief that God, being Perfect, would move every object through space along "perfect" mathematical paths, and the circle was deemed the perfect mathematical object for planet paths.

But when the Greeks observed the Planets and crunched the numbers, it was obvious the planets did not move in circle orbits.

Finally, the ancient world's most mature system, Ptolemy's Almagest (Arabic for "the best," circa 100 AD), was a scheme in which the planets moved in small circles whose centers moved around bigger circles. Adding enough of these epicycles could crudely explain the planets' odd motions. The Almagest epicycle scheme was the best explanation for 1500 years.

Tycho Brahe suspected the big problem was the lousy, unreliable, imprecise Ptolemy-era observations of the planets' motions.

He built an observatory on a Baltic island -- Uraniborg -- and because telescopes hadn't been invented, observed the planets through very long sighting tubes which revolved (azimuth) and aimed high or low (elevation). He made nightly observations for about 30 years, toward the end with young Kepler as his assistant.

Kepler inherited the observations, easily 10 times more precise, thorough and reliable than Ptolemy's. Kepler first talked Prince Rudolph into bankrolling the publication of the Rudolphine Tables.

Then he set about trying to pry the solution to The Puzzle out of them.

This is a huge numerical database -- date and time of observation, elevation, azimuth -- and would require a crushing volume of arithmetic, all done by hand.

But Burghi (a Swiss clockmaker) and Napier (an Edinburgh nobleman and mega-eccentric) had just discovered the computational shortcut of logarithms, and the first practical log tables began circulating in Europe. Logs cut Kepler's computational effort by a third, and made the analysis possible.

Today we'd call his problem "curve fitting" -- finding (from hundreds of candidates) a geometrical curve, like a circle, or straight line, or spiral, which fit Tycho's data.

The ancient Greeks had studied a set of curves called the Conic Sections -- the 2D plane curves you get when you slice up-and-down ice-cream cones with a razor blade. They found them mathematically interesting, but because they lacked Platonic Perfection, dismissed them as candidates for heavenly orbits; God would never use such curves.

Kepler -- who hovered between the Middle Ages and the modern scientic age -- returned to the Conic Sections, and realized that the ellipse perfectly fit Tycho's observations.

(He first studied the ellipse because all beer barrels were ellipses -- so a toppled barrel wouldn't roll down the street.)

From that Eureka moment he soon found Law 2: Equal Times anywhere in a planet's orbit sweep out Equal Areas.

This implies that planets move fastest near the Sun, slowest when they're far from the Sun. Obviously one hard part of The Puzzle was that planets change speed at different points in their orbit. Law 2 now described and predicted precisely how and where the speed changed.

Law 3 took Kepler another 10 years to discover. It's true, but certainly not obvious or common-sensical. I can't imagine how he stumbled on the relationship between

* the square of a planet's Orbital Period

Earth: (365.25 days)^2 = 133407.5625
 
and

* the cube of half the longer axis of a planet's ellipse

Earth: (149,597,890 km)^3 = 3.34793027158529 x 10^24

but for every massive object that orbits the Sun, the ratio of these quantities is a constant, the same for every planet (and every comet).

This is what Nature looks like at its deepest core -- pure mathematics. (Richard Feynman said: We have no clue why Nature is entirely mathematical, but we should be grateful that it is.)

Newton proved Kepler's Laws in the Principia (1687). (He'd proved them decades earler, but was a secretive, paranoid guy who dreaded controversy and being misunderstood and criticized by idiots.)

All of this -- Brahe, Kepler (both astrologers), Burghi and Napier (Edinburgh feared him as a dangerous wizard), Newton (a student of astrology, but dropped it to become an obsessive alchemist), Liebniz (co-discoverer of the calculus) -- were the revolutions that ended the Middle Ages and gave birth to modern Science.

I'd throw in Copernicus too, but many ancient Greek astronomers believed in the Sun-center Solar System (rather than the Judaeo-Christian-friendly Earth-center scheme).

Galileo died the year Newton was born. The telescope you very kindly assembled for me, the refractor, is the telescope Galileo improved after reading a letter about the anonymous Dutch invention.

But his had no Warning Tag about looking at the Sun; he made the first observations of sun spots, and eventually went blind.

Newton was annoyed by the color blur [chromatic aberration] at the fringe of the refractor view, and knew nothing could fix it. So he invented the reflector, whose parabaloid mirror reflects starlight to one point, the eyepiece, and doesn't blur the colors. You can see the model of the reflector young Newton presented to the Royal Society; I think they made him a member that evening.

B