Leo Suresh, Leoba Media

### HTML Lessons

KEPLER'S LAWS OF PLANETARY MOTION
```orbits are not circles
Aristotle's cosmology required circular orbits
sun is not the center

orbits are not at constant speed
Aristotle's cosmology required constant speed

* called The Harmonic Law
* period is the time required for one orbit
* distance refers to the average distance from the sun
* planets further from the sun move slower
* Aristotle's cosmology required all planets to move at
the same speed
* the relationship (harmonic ratio) is the same for earth
and planets except the moon

Ellipse (Conic Sections)
Appearance is that of squashed circle
Ellipse - locus of points for which sum of distances from
foci is a constant
Foci - two points symmetrically located on major axis either
side of center
Major axis is longest axis of figure
(semi-major axis is half of major axis)
Minor axis is perpendicular to major axis and is
shortest distance
Eccentricity - ratio of distance of either foci from
center to length of semi-major axis```

Kepler's Three Laws of Planetary Motion In the 16th century, the Polish astronomer Nicolaus Copernicus replaced the traditional Earth-centered view of planetary motion with one in which the Sun is at the center and the planets move around it in circles. Although the Copernican model came quite close to correctly predicting planetary motion, discrepancies existed. This became particularly evident in the case of the planet Mars, whose orbit was very accurately measured by the Danish astronomer Tycho Brahe. The problem was solved by the German mathematician Johannes Kepler, who found that planetary orbits are not circles, but ellipses. Kepler described planetary motion according to three laws. Each of these laws is illustrated by an applet. Law I: Each planet revolves around the Sun in an elliptical path, with the Sun occupying one of the foci of the ellipse. Each planet moves in an elliptic orbit around the Sun with the Sun occupying one foci (the other is empty) Appearance of an ellipse - looks like a squashed circle; one long axis, called major axis; perpendicular to major axis is minor, or shorter, axis Foci - two points symmetrically located on major axis either side of center of ellipse Ellipse - locus of points for which the sum of their distances from the foci is a constant Eccentricity - ratio of distance of either foci from center to the length of semi-major axis (one-half of major axis) Law II: The straight line joining the Sun and a planet sweeps out equal areas in equal intervals of time. Imaginary line connecting any planet to Sun sweeps over equal areas in equal intervals of time Law III: The squares of the planets' orbital periods are proportional to the cubes of the semimajor axes of their orbits. Kepler's third law is illustrated for circular orbits, although the law also applies to elliptical orbits. Click on either of the buttons on the right. The satellite's motion will simulate either the orbit of NASA's space shuttle or that of a satellite in a geosynchronous orbit. A geosynchronous orbit is one with an orbital period equal to the period of the rotation of the Earth. This means that for circular orbits above the Earth's equator, the satellite will always remain above the same point on the Earth. You can also click on the satellite and drag it to other orbits. The satellite's altitude above the Earth's surface and orbital period are given on the right. Notice how much slower the satellite moves in a large orbit, and how much longer it takes to complete an orbit than when it is closer to the Earth. A mathematical relationship exists between the orbital period and the size of the orbit, i.e., the distance between the center of the Earth and the satellite. The relationship states that the square of the orbital period is proportional to the cube of the size of the orbit Square of any planet's orbital period (sidereal) is proportional to cube of its mean distance (semi-major axis) from Sun Mathematical statement: P = ka3/2 , where P = sideral period, and a = semi-major axis Example - If a is measured in astronomical units (AU = semi-major axis of Earth's orbit) and sidereal period in years (Earth's sidereal period), then the constant k in mathematical expression for Kepler's third law is equal to 1, and the mathematical relation becomes P2 = a3 Kepler's laws apply not just to planets orbiting the Sun, but to all cases in which one celestial body orbits another under the influence of gravitation -- moons orbiting planets, artificial satellites orbiting the Earth and other solar system bodies, and stars orbiting each other
Mathematical mysteries: Kepler's conjecture Sir Walter Raleigh is perhaps best known for playing bowls while the Spanish Armada sailed into the English Channel. However, he also started a mathematical quest which to this day remains unsolved. Sir Walter posed a simple question to his mathematical assistant, Thomas Harriot. How can I calculate the number of cannon balls in a stack? Harriot solved this problem without difficulty but started to wonder about a more general problem. What arrangement of the balls takes up the least space? Harriot wrote about the problem to his colleague Johannes Kepler, best known for his work on planetary orbits. Kepler experimented with the problem and concluded that an arrangement known as the face centred cubic packing, a pattern favoured by fruit sellers, could not be bettered. This statement has become known as "Kepler's conjecture" or simply the sphere packing problem To measure how good an arrangement is, mathematicians imagine that all of space is filled with spheres. They calculate the proportion of this infinite space, or volume, occupied by the spheres themselves. The remainder of the volume or space is formed by the gaps between the spheres. The sphere packing problem has various applications. Firstly, it extends quite simply into other dimensions (in two dimensions it becomes the circle packing problem). It is also analogous to the problem of constructing optimal codes (see "Coding theory: the first 50 years" elsewhere in this issue). A proof of the conjecture may also help physicists to understand the structure of crystals, something that Harriot himself urged Kepler to consider in his work on optics. Although the sphere packing problem has frustrated mathematicians for nearly four centuries there is light at the end of the tunnel. In 1953 the Hungarian, László Fejes-Tóth showed that given sufficient computing power the problem could be solved. The American mathematician, Thomas Hales is currently trying to do just that and is hopeful that within "a year or two" he will complete the calculations The trouble with Hales' solution is that it cuts right to the heart of what many believe a mathematical proof should be. If Hales is successful there may be a feeling of anti-climax with no new insight immediately apparent. It is therefore not surprising that some people are convinced that a more traditional proof is possible. Wu-Yi Hsiang, from California published just such a proof in 1990 but after much deliberation, the majority of mathematicians in the field remain sceptical claiming that there are holes in his argument. Whether the holes in Hsiang's proof will be filled before Hales finishes his computation remains to be seen but it seems likely that Kepler's conjecture will, at least for the time being, remain a mathematical mystery
Johannes Kepler was born on December 27, 1571 in Weil der Stadt, Germany. Kepler's grandfather was supposedly from a noble background, and once Mayor of Weil. However, Kepler's father became a mercenary who narrowly avoided the gallows. Kepler's mother, Katherine, was raised by an aunt who was eventually burned as a witch. In later years, Katherine herself was accused of Devil worship, and barely escaped from being burned at the stake. Kepler had six brothers and sisters, three of which, died in infancy. In his youth, Johannes was described as: "...a sickly child, with thin limbs and a large, pasty face surrounded by dark curly hair. He was born with defective eyesight-myopia plus anocular polyopy (multiple vision). His stomach and gall bladder gave constant trouble; he suffered from boils, rashes, and possibly from piles, for he tells us that he could never sit still for any length of time..." (Koestler, p 24) From this inauspicious start, Johannes Kepler began his fascinating journey as a pioneer in astronomy.
Johannes Kepler graduated from the Faculty of Arts at the University of Tuebingen at the age of twenty, intending to matriculate into the Theological Faculty. It was here that Kepler learned and became an adherent of the heliocentric theory of planetary motion, first developed by the Polish astronomer Nicolaus Copernicus. In 1594, Kepler left Tuebingen for the University of Graz to become a professor of astronomy. It was here that Kepler realized that figures of the type shown here determine a definite fixed ratio between the sizes of the two circles, provided the triangle has all sides equal, and a different ratio of sizes will occur for a square between the two circles, another for a regular pentagon, and so on. Kepler believed that this could be used to determine the orbits of planets in the solar system. Unfortunately, Kepler proceeded from a false assumption: namely, that the orbits of the planet were circular. Despite the fact that his calculations did not match known planetary data, Kepler presumed that Copernicus's data was in error, and produced this diagram of orbits, where the outer ring represents the orbit of Saturn. Kepler published these ideas in a work entitled Mysterium Cosmographicum in 1596 Tycho Brahe (1546-1601) By all accounts, the relationship between the two was strained. We have previously discussed Kepler's upbringing. In contrast, Brahe was from an aristocratic background who shared Kepler's less than scintillating personality. As a result, the two continuously quarreled, and usually failed to resolve their academic and personal differences. However, the two realized that they needed each other. As a result, both learned from each other's writings. Brahe died in 1601, and Kepler assumed his post as imperial mathematicus. In addition, Rudolph II requested his service as court astronomer, which Kepler preformed until Rudolph's death in 1612, During his tenure as court astronomer, Johannes Kepler labored over one of his most impressive works: Astronomia Nova. His primary motivation was to attempt to calculate the orbit of Mars. One offshot of this work was the formulation of the concepts that were eventually known as the first two of Kepler's Laws. In 1612, Kepler became provincial mathematician to Linz, in upper Austria. Over the next fourteen years, Kepler published Harmonice Mundi, in which Kepler outlined his third law. Furthermore, he published the Epitome Astronomiae Copernicanae, which combined all of his discoveries together. However, Kepler's personal life was far less successful. His first wife, Barbara, and their two sons died from the fever and small pox in 1612. In 1615, Kepler was excommunicated from the church, and his mother was placed on trial for being a witch.. Despite these tribulations, Kepler completed the Tabulae Rudolfinae in 1625. These tables reduced the mean errors in tables of planetary motion significantly. However, political unrest led in the destruction of his home during a peasant revolt, leaving Kepler without a permanent residence. Johannes Kepler was named the private mathematicus in the newly acquired Duchy of Sagan in 1628. Unfortunately, neither this position nor his previous one was a lucrative profession due to the Thirty Years War. As such, Kepler was left having to borrow money to travel to collect an old debt leaving his second wife and children behind, penniless. Sadly, he died en route on November 15, 1630 in the village of Ratisbon. Kepler, to this day, remains one of the greatest figures in astronomy. However, his endeavors were not just limited to this field. He is often called the founder of modern optics for his first use of eyeglasses designed for nearsightedness and farsightedness, his explantions of vision by refraction within the eyes, and his explantion of the use of both eyes for depth perception. Furthermore, he explanied the principles of the telescope. Additional Firsts (from N.A.S.A. Kepler Mission Home Page) His book Stereometrica Doliorum formed the basis of integral calculus. First to explain that the tides are caused by the Moon (Galileo rebuked him for this). First to use stellar parallax caused by the Earth's orbit to try to measure the distance to the stars; the same principle as depth perception. First to suggest that the Sun rotates about its axis in Astronomia Nova First to derive the birth year of Christ, that is now universally accepted.

1. Dates
Born: Weil der Stadt, Germany, 27 Dec. 1571
Died: Regensburg, 15 Nov. 1630
Dateinfo: Dates Certain
Lifespan: 59
2. Father
Occupation: Soldier
Common soldier of fortune.
Poor.
3. Nationality
Career: Germany
Death: Regensburg, Germany
4. Education
Schooling: Tübingen, M.A.
1579, German and Latin Schreibschule, Leonberg.
1584, Adelberg monastery school (lower seminary).
1586, Maulbronn, a prepatory school for the university of Tuebingen (higher seminary).
1587, matriculated Univ. of Tuebingen, but the Stift, the seminary for scholarship students was full, so he stayed at Maulbronn for another two years. 1589, taken into the Stift.
1588, passed Baccalaureat exam.
1591, M.A., Tuebingen. He began the theology course, but was called away to Graz in second year.
5. Religion
Affiliation: Lutheran
6. Scientific Disciplines
Primary: Astronomy, Optics, Mathematics
Subordinate: Astrology
7. Means of Support
Primary: Schoolmastering, Patronage
Secondary: Per (From Wife), Astrology, Calendars
Education: After passing competitive Wuerttemberg state examinations he entered the school at Maulbronn. In 1589, he entered the Stift as a scholarship student, recieving 6 gulden per year; as a scholarship student he was forever bound into ducal service. In addition, his maternal grandfather granted him the income from a meadow.
Graz: 1594, teacher of mathematics at the Lutheran Stiftschule at Graz. He earned a salary of 150 gulden (his predecessor had received 200), which was raised to 200 after his marriage, and was granted 60 gulden moving expendses. The job of district mathematician and calendar maker was later added to his duties. He made 5 calendars in all; for the first he received an honorarium of 20 gulden (no information on the others). He also received income doing astrological nativities and prognostications for lords.
1597, married the reasonably wealthy two-time widow, daughter of a wealthy mill owner, Barbara Mueller. Her wealth was tied up in land, and was sufficient, Kepler reckoned, to support him after a few years. Her holdings included 10,000 gulden from which Kepler received 70 gulden for maintenence, the yield of a vineyard, and a house. Unfortunately. it was difficult to liquidate her assets when Kepler was forced to leave Catholic Graz summarily in 1598, especially because it became illegal for Protestants to lease to Catholics. He was allowed to return, but sought other employment, working for a few months with Tycho Brahe, but then returning shortly to Graz before being banished altogether in 1600. Upon his ultimate expulsion, he was dismissed from his job, paid a half-year's salary, and provided with a letter of recommendation. His exit tax was lowered to 5% from 10%.
1600, went to work for Tycho Brahe in Prague, until Tycho died and Kepler assumed his position.
1601, Imperial Mathematician to Rudolf (500 gulden salary). Barbara inherited about 3000 gulden of landed property from her father in 1603. She died in 1611, leaving no will and consequently he got nothing, except for 2000 gulden for her children which he invested (1615) in the treasury of the Upper Austrian representatives. Rudolf abdicated in 1611. In 1612, he died, freeing Kepler to leave Prague.
1612-1628, district mathematician and teacher in Linz (to which was added the task of completing a map of Upper Austria, which he later had removed). He received a salary of 400 gulden (plus travel expenses for the map). (In late 1616, after a political battle over whether he ought to be retained, he was granted a small hononrarium as a consolation for the insult.) 1613, he married an orphan, Susanna Reuttinger, a ward of Baroness Elizabeth von Starhemberg. While in Linz, Kepler again supplemented his income with proceeds from calendars. For instance, he paid for the publication of the ephemerides of 1617 with a calendar for 1616. In all, he made six popluar calendars between 1617 and 1624. Kepler was allowed to stay on in Linz after the expulsion of the Protestants in 1626, and even retained his position despite a prolonged absence from 1626-1628. 1612, Emperor Matthais continued the post of Imperial Mathematician at reduced salary of 300 fl/yr plus 60 gulden for dwellings and wood costs. and gave his consent for Kepler to move to Linz. Ferdinand continued Kepler's appointment as imperial mathematician until Kepler's death. 1628, moved to Sagan to work for Albrecht von Wallenstein, at a salary of 1000 gulden, and a press, for which Wallenstein promised to provide twenty bales of paper and 1,040 gulden priting costs annually.
In 1630, the congress of electors dismissed Wallenstein. Kepler tried to return to Linz, to collect on two 6% bonds (2000 and 1500 gulden), but died en route.
8. Patronage
Types: Scientist, Court Official, Aristrocrat, Government Official Education: After Landesexamen in 1583, Kepler got a scholarship from the Duke of Wuerttemberg; this scholarship bound him forever into ducal service. In 1590, the magistrate of Weil proposed him to the senate of the university of Tuebingen for a stipend of 20 gulden per year, which was granted. The senate renewed it in 1591. The senate then recommended him for the post of teacher of mathematics to the protestant school at Graz. Graz: For moving to Graz, Kepler received a loan of 50 gulden from Prof. Gerlach, the superintendent of the school; Kepler was later granted 60 gulden moving expenses. The commissioners of the school upon the occasion of his wedding granted him as a "veneration" a silver cup worth 27 gulden. Kepler requested and received from the commissioners a raise to a 200 gulden salary after his wedding. He was sought out by Lords to do nativities and prognostications with which he supplemented his income. (I include this under Miscellaneous) The city granted him a honorarium of 20 gulden for his first calendar. The city was generally kind to him; after the expulsion of Protestants he was given a half-year's salary and a letter or recommendation.
In 1596, he visted the Duke of Wuerttemberg and presented him with the idea of an artistic representation of his system of nested spheres from the Mysterium which he wanted to dedicate to the Duke. He later applied to Duke Johann Friedrich of Wuerttemberg for a job, but was rejected because he was suspected of being "a sly Calvinist." The Duke of Wuerttemberg later arranged that the court documents of Kepler's mother's trial be sent to Tuebingen for the decision of the legal faculty (in which Kepler had a contact, Christoph Besold), and he was responsible for the order that she be absolved and the charges dismissed. The Mysterium was dedicated to the estates of Styria, from whom Kepler received a 250 gulden honararium in 1600. After the general expulsion order for Protestant teachers in 1598, archduke Ferdinand made an exception in the case of Kepler, allowing him to continue as district mathematician. Kepler attributed his favor at court to the regimental counsellor Manechio. Ferdinand also rewarded Kepler for an astronomical essay Kepler addressed to him in 1600. After the expulsion of all Protestants, Ferdinand ordered Kepler be reimbursed his 5% exit tax, but this was never carried out. As emperor, Ferdinand confirmed him as imperial mathematician, and again excepted him when expelling non- Catholic teachers. Ferdinand also finally wrote drafts to pay him some of his back pay to finance the publication of the Rudolfine Tables, which was eventually dedicated to him. Ferdinand approved Kepler's move to Ulm in 1626.
He recieved Kepler graciously in Prague in 1627, awarding him 4000 gulden (in drafts) for dedication of the Rudolfine Tables. He could have stayed in service, but would have had to become Catholic. Even as Kepler lay dying the Emperor sent a gentleman with 25 or 30 Hungarian ducats to his aid. The Emperor owed 12,694 gulden to his family in 1633. Though claims continued to be made up until 1717, this money was never paid. At the end of his life Kepler called himself Duke of Friedland. I do not know why. Presumaby he received the title from Wallenstein or Ferdinand (perhaps given the title after Wallenstein's downfall?). The Bavarian chancellor Herwart von Hohenburg was a major patron to Kepler. They corresponded and H.v.H. lent Kepler books which he did not have. H.v.H. was probably partly responsible for Kepler's permission to stay in Graz (see directly above). Kepler appealed to Michael Maestlin often as though to a patron, for instance asking around for a job in 1600, but Maestlin seldom was able to oblige him. 1600, Tycho supported him in Prague. Kepler moved very much in the orbit of Tycho, including living in the same house from the beginning of 1601. Tycho assigned Kepler the task of refuting Ursus. 1601, Kepler became Imperial mathematician to Rudolf II. He had tremendous trouble getting paid his 500 gulden salary. At one point, the Emperor paid 400 gulden in printing costs, which Kepler spent on household expenses, he then granted an additional 500 gulden for which he reserved the entire edition of Astronomia nova for himself (Kepler eventually had to sell the entire edition to the printer). He often gave extra compensation, such as a draft for 2000 talers in 1610, on which Kepler could not collect.
Kepler dedicated numerous works to the Emperor, including the Astronomia nova, Astronomiae pars optica, and De stella nova. He also prepared numerous reports for the Emperor on astrological or scientific matters. After abdicating, Rudolf asked Kepler to stay on, which he did, until the Emperor's death. By March 1611, he was 3000 gulden in arrears. Casting about for another job, Kepler presented a copy of the Astronomiae pars optica to Duke Maximilian of Bavaria. The Duke's gift to Kepler was so small that Herwart von Hohenburg increased it out of his own pocket. Elector Ernst of Cologne sometimes occupied days of his time at court, and lent him a telescope. Kepler presented him with the manuscript of Dioptrice. Baron Johann Friedrich Hoffman von Gruenbueehel und Strechau, an imperial advisor, housed Kepler when he first visited Prague, and later provided Kepler with two instruments built on Tychonic designs.