After years of vastly improved technology, our image of how planets move is still the one Kepler created. How was it discovered?
For 2, years, astronomers placed the earth at the center of the universe and assumed that all heavenly bodies moved in perfect circles around it. But predictions using this system never matched actual measurements. Scientists invented epi-circles — small circles that the planets actually rolled around that, themselves, rolled around the great circular orbits of each planet. Still, there were errors, so scientists created epi-circles on the epi-circles. Kepler suffered through a troubled upbringing.
His aunt was burned at the stake as a witch. His mother almost suffered the same fate. The boy was often sick and had bad eyesight that glasses could not correct.
Still, Kepler enjoyed a brilliant— but again troubled—university career. In , he took a position as an assistant to Tycho Brahe, famed German astronomer. For decades Tycho had been measuring the position of the planets especially Mars with far greater precision than any other European astronomer. When Tycho died in he left all his notes and tables of planetary readings to Kepler. Gleick, J. Isaac Newton. New York: Vintage Books. Gribbon, J. New York: Random House. Hawking, S. The Illustrated on the Shoulders of Giants.
Philadelphia: Running Press. Iannotta, B. GPS Satellite Orbits. Serway, R. Physics for Scientists and Engineers, 3rd ed. Wolfe, J. Einstein Light. In his writings, Kepler is given to laying his opinions on the line - which is very convenient for historians. Religious intolerance sharpened in the following years. Kepler was excommunicated in This caused him much pain, but despite his by then relatively high social standing, as Imperial Mathematician, he never succeeded in getting the ban lifted.
Kepler's first cosmological model Instead of the seven planets in standard geocentric astronomy the Copernican system had only six, the Moon having become a body of kind previously unknown to astronomy, which Kepler was later to call a 'satellite' a name he coined in to describe the moons that Galileo had discovered were orbiting Jupiter, literally meaning 'attendant'.
Why six planets? Moreover, in geocentric astronomy there was no way of using observations to find the relative sizes of the planetary orbs; they were simply assumed to be in contact. This seemed to require no explanation, since it fitted nicely with natural philosophers' belief that the whole system was turned from the movement of the outermost sphere, one or maybe two beyond the sphere of the 'fixed' stars the ones whose pattern made the constellations , beyond the sphere of Saturn.
In the Copernican system, the fact that the annual component of each planetary motion was a reflection of the annual motion of the Earth allowed one to use observations to calculate the size of each planet's path, and it turned out that there were huge spaces between the planets. Why these particular spaces? He suggested that if a sphere were drawn to touch the inside of the path of Saturn, and a cube were inscribed in the sphere, then the sphere inscribed in that cube would be the sphere circumscribing the path of Jupiter.
Then if a regular tetrahedron were drawn in the sphere inscribing the path of Jupiter, the insphere of the tetrahedron would be the sphere circumscribing the path of Mars, and so inwards, putting the regular dodecahedron between Mars and Earth, the regular icosahedron between Earth and Venus, and the regular octahedron between Venus and Mercury.
This explains the number of planets perfectly: there are only five convex regular solids as is proved in Euclid 's Elements , Book Kepler did not express himself in terms of percentage errors, and his is in fact the first mathematical cosmological model, but it is easy to see why he believed that the observational evidence supported his theory. Kepler saw his cosmological theory as providing evidence for the Copernican theory. Before presenting his own theory he gave arguments to establish the plausibility of the Copernican theory itself.
Kepler asserts that its advantages over the geocentric theory are in its greater explanatory power. For instance, the Copernican theory can explain why Venus and Mercury are never seen very far from the Sun they lie between Earth and the Sun whereas in the geocentric theory there is no explanation of this fact.
The agreement with values deduced from observation was not exact, and Kepler hoped that better observations would improve the agreement, so he sent a copy of the Mysterium cosmographicum to one of the foremost observational astronomers of the time, Tycho Brahe - Kepler got the job.
The 'War with Mars' Naturally enough, Tycho 's priorities were not the same as Kepler's, and Kepler soon found himself working on the intractable problem of the orbit of Mars [ See the History Topic on Kepler's planetary laws ]. He continued to work on this after Tycho died in and Kepler succeeded him as Imperial Mathematician.
Conventionally, orbits were compounded of circles, and rather few observational values were required to fix the relative radii and positions of the circles. Tycho had made a huge number of observations and Kepler determined to make the best possible use of them. Essentially, he had so many observations available that once he had constructed a possible orbit he was able to check it against further observations until satisfactory agreement was reached. Kepler concluded that the orbit of Mars was an ellipse with the Sun in one of its foci a result which when extended to all the planets is now called "Kepler's First Law" , and that a line joining the planet to the Sun swept out equal areas in equal times as the planet described its orbit "Kepler's Second Law" , that is the area is used as a measure of time.
After this work was published in Astronomia nova, Both laws relate the motion of the planet to the Sun; Kepler's Copernicanism was crucial to his reasoning and to his deductions.
The actual process of calculation for Mars was immensely laborious - there are nearly a thousand surviving folio sheets of arithmetic - and Kepler himself refers to this work as 'my war with Mars', but the result was an orbit which agrees with modern results so exactly that the comparison has to make allowance for secular changes in the orbit since Kepler's time. Observational error It was crucial to Kepler's method of checking possible orbits against observations that he have an idea of what should be accepted as adequate agreement.
From this arises the first explicit use of the concept of observational error. Kepler may have owed this notion at least partly to Tycho , who made detailed checks on the performance of his instruments see the biography of Brahe. Optics, and the New Star of The work on Mars was essentially completed by , but there were delays in getting the book published. Meanwhile, in response to concerns about the different apparent diameter of the Moon when observed directly and when observed using a camera obscura , Kepler did some work on optics, and came up with the first correct mathematical theory of the camera obscura and the first correct explanation of the working of the human eye, with an upside-down picture formed on the retina.
Following Galileo 's use of the telescope in discovering the moons of Jupiter, published in his Sidereal Messenger Venice, , to which Kepler had written an enthusiastic reply , Kepler wrote a study of the properties of lenses the first such work on optics in which he presented a new design of telescope, using two convex lenses Dioptrice , Prague, This design, in which the final image is inverted, was so successful that it is now usually known not as a Keplerian telescope but simply as the astronomical telescope.
Leaving Prague for Linz Kepler's years in Prague were relatively peaceful, and scientifically extremely productive. In fact, even when things went badly, he seems never to have allowed external circumstances to prevent him from getting on with his work.
Things began to go very badly in late First, his seven year old son died. Kepler wrote to a friend that this death was particularly hard to bear because the child reminded him so much of himself at that age.
Then Kepler's wife died. Then the Emperor Rudolf, whose health was failing, was forced to abdicate in favour of his brother Matthias, who, like Rudolf, was a Catholic but unlike Rudolf did not believe in tolerance of Protestants.
Kepler had to leave Prague. Before he departed he had his wife's body moved into the son's grave, and wrote a Latin epitaph for them. He and his remaining children moved to Linz now in Austria. The length of the string remains the same, so that the sum of the distances from any point on the ellipse to the foci is always constant.
The distance 2a is called the major axis of the ellipse. The shape roundness of an ellipse depends on how close together the two foci are, compared with the major axis. The ratio of the distance between the foci to the length of the major axis is called the eccentricity of the ellipse. If the foci or tacks are moved to the same location, then the distance between the foci would be zero. This means that the eccentricity is zero and the ellipse is just a circle; thus, a circle can be called an ellipse of zero eccentricity.
In a circle, the semimajor axis would be the radius. Next, we can make ellipses of various elongations or extended lengths by varying the spacing of the tacks as long as they are not farther apart than the length of the string.
The greater the eccentricity, the more elongated is the ellipse, up to a maximum eccentricity of 1. The size and shape of an ellipse are completely specified by its semimajor axis and its eccentricity. The eccentricity of the orbit of Mars is only about 0. Kepler generalized this result in his first law and said that the orbits of all the planets are ellipses. Here was a decisive moment in the history of human thought: it was not necessary to have only circles in order to have an accepTable cosmos.
The universe could be a bit more complex than the Greek philosophers had wanted it to be. He expressed the precise form of this relationship by imagining that the Sun and Mars are connected by a straight, elastic line. When Mars is closer to the Sun positions 1 and 2 in Figure 4 , the elastic line is not stretched as much, and the planet moves rapidly. Farther from the Sun, as in positions 3 and 4, the line is stretched a lot, and the planet does not move so fast.
As Mars travels in its elliptical orbit around the Sun, the elastic line sweeps out areas of the ellipse as it moves the colored regions in our figure. Kepler found that in equal intervals of time t , the areas swept out in space by this imaginary line are always equal; that is, the area of the region B from 1 to 2 is the same as that of region A from 3 to 4.
If a planet moves in a circular orbit, the elastic line is always stretched the same amount and the planet moves at a constant speed around its orbit. But, as Kepler discovered, in most orbits that speed of a planet orbiting its star or moon orbiting its planet tends to vary because the orbit is elliptical. The orbital speed of a planet traveling around the Sun the circular object inside the ellipse varies in such a way that in equal intervals of time t , a line between the Sun and a planet sweeps out equal areas A and B.
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