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• The comet of 1577 aroused Kepler’s interest in the skies while he was a boy
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Kepler’s ambition was different from Tycho’s
He was a mystic who wanted to find out the secret of the universe.
Copernicus’ sun centered universe appealed to his mystic model.
Planet Modern Copernicus
Mercury 0.387 0.376
Venus 0.723 0.719
Earth 1.000 1.000
Mars 1.523 1.520
Jupiter 5.202 5.219
Saturn 9.554 9.174
He began to search for a harmonic relationship between sizes
The tunes of the cosmic music box.
Like Pythagoras, Kepler searched for mathematical patterns in nature Copernicus Table:
Orbital Radii were not in simple ratios like : 4:3, 5:3
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Sudden Insight
Saturn
Jupiter
The ratio of the diameters of the two circles is very nearly the same as 9.6/5.2 = 1.9 nearly 2, the same as the ratio of the orbits of the outermost two planets, Saturn and Jupiter.
SaturnJupiter
Mars
Earth
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Kepler’s thinking was a revival of classical thought.
With their emphasis on geometric order and symmetry.
God must have made this world in a goodly pattern.
But the universe must be constructed according to the geometry of three dimensional solids !
…..
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Kepler’s Architecture of the Heavens Based on Perfect Solids and Spheres
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Classical Geometry influenced
Michelangelo
Sistine Chapel
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Sistine Ceiling: Story of the World in Geometrical Frames
Influence of classical geometry on art and science
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Sistine Lunette
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Leonardo da Vinci - The Last Supper 1498
Convent of Santa Maria delle Grazie (Refectory), Milan
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ANDREA DEL CASTAGNO Last Supper 1447
Sant'Apollonia, Florence
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It was beautiful!
But it did not fit the data!Where have we heard this before? “Let all keep silence and hark to Tycho, who has devoted thirty-five years to his observations…he shall explain to me the order and arrangement of the orbits…”
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Kepler Joins Tycho in PragueAfter years of wrangling Tycho released the data for Mars
Kepler boasted he could solve the orbit in 8 days.
Finally Tyco near death:
“The work, the work! Not to have lived in vainInto whose hands can I entrust it all?I thought to find him standing by the way,Waiting to seize the splendor from my hand,The swift, the young-eyed runner with the torch.Let me not live in vain, let me not fallBefore I yield it to the appointed soul.”
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After 2 years of trial and error and 70 separate attempts, he thought he finally found a combination of circles that would describe the path.
Except for a slight misfit, about 8 minutes of an arc.
Remember that a circle has 360 degrees
and each degree has 60 minutes.
8 minutes is the miniscule angle that the second hand moves in 0.02 sec!
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The years in which he watched that planet Mars’His patient notes and records, all were mine;And, mark you, had he clipped or trimmed one factBy even a hair’s breadth, so that his resultsMade a pure circle of that planet’s pathIt might have baffled us for an age and drownedAll our new light in darkness. But he held to what he saw. He might so easily,So comfortably have said,”My instruments Are crude and fallible. In so fine a pointEyes may have erred, too. Why not acquiesce?Why mar the tune, why dislocate a world,For one slight class of seeming fact with faith?”But no, though stars may swerve, he held his course,Recording only what his eyes could seeUntil death closed them.
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Kepler knew that Tycho’s measurements were accurate to 1-2 arc-minutes. “Since divine goodness has granted us a most diligent observer, Tycho Brahe,
from whose observations the error in this calculation of 8 minutes in Mars is revealed, it is fitting that we recognize and make use of this good gift of God with a graceful mind.”
It was the dawn of a new era in physics: the era of rigor. Kepler was willing to go wherever the observations led him.
The theory should fit the data. Otherwise the theory is worthless. Provided there is sufficient reason to trust the data, which Tycho supplied in
ample measure.
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Back to the start…
Kepler was an accomplished mathematician.
He was aware of new techniques, such as logarithms to reduce the drudgery of numerical complications.
Parabolas and ellipses worked out by Archimedes and Apollonius in Alexandria.
Once again, revival of classical ideas proved fruitful
Q: Choose the ellipse (s)?A)B) C) D) A and B
A
C
B
C
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Centuries of efforts using perfect cycles for planetary paths were doomed.
He was prepared to try a new mathematical figure. He considered the ellipse.
It worked for the orbit of Mars, better than any combination of circles. The sun had to be positioned at one of the focal points of the ellipse.
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Definition of Ellipse:
X
all the points X on the ellipse obey the rule
XA +XB = constant
xx
A B
Focii
axisminor
major axis
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Kepler's First LawKepler's first law of undisturbed planetary motion:
The orbit of each planet is an ellipse and the Sun is at one focus. The sun lies in the plane of all the planetary orbits
In one decisive stroke, Kepler demolished the antique universe on wheels !
A friend of his complained:How can you replace the sacred perfect circle with the imperfect ellipse?
Kepler replied:“The circle is a voluptuous whore, enticing astronomers away from the honest maiden of nature.”
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Planet Eccentricity
Mercury .0206
Venus 0.007
Earth 0.017
Mars 0.093
Jupiter 0.048
Saturn 0.056
a
ae
Focus
If e = 0, the ellipse will become a circle
Nearly all of the planets have eccentricities which are small, only a few percent, and their orbits are close to being circular.
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Kepler's First LawKepler's first law of undisturbed planetary motion:
The orbit of each planet is an ellipse and the Sun is at one focus. The sun lies in the plane of all the planetary orbits
In one decisive stroke, Kepler demolished the antique universe of wheels upon wheels !
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Like the Baroque artists, Kepler swept away the classical rules
‘This music leads us far From all our guides, except that faith in law.Your quest for knowledge how it rests on that!
How sure the soul is that if truth destroyThe temple, in three days the truth will buildA nobler temple; and that order reignsIn all things.
Noyes
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Baroque: Fall of the Giants by Guilio Romano, 1534
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ANDREA DEL CASTAGNO Last Supper 1447
Sant'Apollonia, FlorenceClassical rules
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Ghirlandaio,1480
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Last Supper by Tintoretto 1592-94
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Lecture Quiz Grading
• All right = A
• One wrong = A-
• Two wrong = B
• Three wrong = B-
• Four wrong = C
• Five wrong = C-
• Six (rare) wrong = D
If the biggest bar agrees with the right answser, the Question will count
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Annular Total Solar Eclipse
Total solar eclipse What is the reason for the occasional annular shape of the sun during a total solar eclipse?(a) The moon is slightly smaller than the sun, so the moon blocks out most but not all of the sun(b) The orbit of the moon around the earth is elliptical, so the moon is sometimes closer to the earth.(c) The solar flares from the sun make the entire rim of the sun visible.(d) all of the above.
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Paradise, TintorettoLike a Baroque artist,Kepler was interested in motion.What was the relationship between the speeds of the planets?
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Kepler was bothered by the ellipse.
The distance from the planet to the sun is not constant,
The speed at which a planet moves is not constant.
fast
slow
Sun
Planet
In which part of the orbit of a planet is the speed the highest(a) when it is farthest from the sun(b) when it is nearest to the sun(c) the speed is the same in all parts of the orbit(d) none of the above
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2nd Law:In its orbit, a planet sweeps out equal areas in equal times.
(The three shaded sections all have the same area)
He continued to calculate with different quantities…Suppose a spoke connects the sun with the planet. The spoke sweeps out equal intervals of time. Kepler was delighted. IT was his second law.
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But there was still no connection between the orbital size or the period of the different planets.
Kepler continued his search for 10 years.
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“…at last, at last, the true relationship…overcame by storm the shadows of my mind, with such fullness of agreement…that I at first believed that I was dreaming…”
PlanetRelative
Distance from Sun
Period in Year
Mercury 0.387 0.241
Venus 0.723 0.615
Earth 1 1
Mars 1.524 1.881
Jupiter 5.203 11.862
Saturn 9.539 29.458
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Planet Relative Period in Years Square of Period Cube of
Distance
Distance From
sun
Mercury 0.387 0.241 0.058 0.058
Venus 0.723 0.615 0.378 0.378
Earth 1 1 1 1
Mars 1.524 1.881 3.538 3.54
J upiter 5.203 11.862 140.707 140.851
Saturn 9.539 29.458 867.774 867.977
T2 R3
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The law he found was now called Kepler’s Third law:
R3/T2 is a constant = 1
R: average radius of the orbit in AU.
T: time taken to make one revolution around sun in years.
The most important aspect of this law is that the value of the constant is the same for every planet!
He had finally hit upon the “numerical harmonies” that Pythagoras had foreseen.
It was celestial music indeed, not to his ears, but to his mind.
Admittedly, the relationship was not elegant as Pythagoras and Kepler himself had dreamed about.
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And so in music men might find the road
To truth, at many a point where sages grope.
One day, a greater Plato would arise
To write a new philosophy, he said,
Showing how music is the golden clue
To all the windings of the world’s dark maze
Himself had used it, partly proved it, too,
In his own book, - the Harmonies of the World.
“All that the years discover points one way
To the great ordered harmony,” he said,
“Revealed on earth by music. Planets move
In subtle accord like notes of one great song
Audible only to the Artificer,
The Eternal Artist...
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Two new planets, Uranus and Neptune were discovered after Newton. The semi-major axis of Uranus' orbit (relative to the earth's orbit) is: 19.19. Using Kepler's 3rd Law, calculate the periods (in years) of Uranus. Chose the closest answer.
A) 4.3 yearsB) 19 yearsC) 84 yearsD) 361 years
R
3
= 19.19
3
= 7066.8
T
2
= 7066.8
T = 7066.8 = 84 years
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What did Tycho and Kepler have in common?
(a) They learned much from the scientists of the classical period.
(b) They advanced precision in astronomy.
(c) They completely accepted Copernicus' heliocentric system.
(d) all of the above
(e) (a) and (b).
