Johannes Kepler – A Life for Science and Religion (Dedicated to Walter Thirring for his 80th Birthday
In western science Johannes Kepler was the turning point from a magical-alchemistic to a rational-mathematical conception of the laws of nature. In his life he worked on both sides, but what earned him eternal fame are his three laws of planetary motion. Johannes Kepler was both a scientist and a religious man. In his time a career in these two fields was not that astonishing as it seems for us nowadays â€“ because the view of the world was not yet separated into a religious and a scientific one. Therefore religious as well as scientific thoughts can be found in one and the same work of Johannes Kepler (e. g. in â€œAd Vitellionem Paralipomenaâ€ about optics). Although his scientific discoveries are very important, he never fulfilled his biggest wish namely to become a Protestant pastor.
As Stephen Hawking (born 1942) mentioned: If there had been a prize for the man who struggled for the highest exactness in his work, it should have been awarded to Kepler. By his passion for accurate measurements he got so involved in his astronomical research that he worked out the most precise astronomical indexes of his time. Eventually this resulted in the acceptance of the heliocentric theory of the planetary system.
Like Nikolaus Kopernikus (1473-1543), whose work inspired him, Kepler was a very religious man. He saw his studies about universal features as his Christian duty and its fulfillment to understand the Universe that God had created. But contrary to Kopernikus, Keplerâ€™s life was far from being peaceful and poor in events.
fig. 1: Johannes Kepler.
II. YOUTH AND EDUCATION OF JOHANNES KEPLER
Johannes Kepler was born as the first child of Heinrich and Katharina Kepler on December 27, 1571 in Weil der Stadt near Stuttgart (in WÃ¼rttemberg, Germany). Before he was one year old, his father left the family to fight for the Dutch Duke. Finally, Keplerâ€™s mother Katharina followed her husband and the grandparents took care of the child. Kepler was a sickly child and at the age of six years he almost died because of smallpox, which left a detriment of his eyes behind. In 1575 his parents returned from the Netherlands and the family moved to Leonberg (because Katharina did not get along with her parents-in-law). With the money earned in the war they bought a house there. But his father did not stay longtime at Leonberg â€“ after one year he went away to the Netherlands again. By the way, the erratic way of living of Keplerâ€™s father was a reason why his mother strongly influenced him when he was a boy.
So in 1577 Keplerâ€™s mother showed him the yearâ€™s comet. This experience had an important impact on his life because it started his passion for astronomy.
Once again Keplerâ€™s father returned to Leonberg. However, because of an incautious bail he had lost his house and estate. So in 1579 the family leased the tavern â€œZur Sonneâ€ (â€œThe Sunâ€) in Elmerdingen near Pforzheim.
fig. 2: The places where Johannes Kepler lived.
Beside the comet from 1577 the observation of two lunar eclipses (1582 and 1588) motivated Keplerâ€™s interest in astronomy. The years from the comet until the second lunar eclipse were the most important ones for his education. Only six years old, Kepler learned to write and read. Then he learned German at Leonberg, but later acquired Latin at school and he wrote his works in the Latin writing-style. In 1583 it was clear that the boy was not able to help with the farm work, because he was too weak. Furthermore, he passed the difficult entrance examination for the convent school in Adelberg very successfully. This school was a pre-stage for higher education at the grammar school of Maulbronn, where Kepler was accepted as a fifteen-year-old. There Greek and Latin philosophers were read in the original language. But also rhetorics, dialectics, music, astronomy, and arithmetics were taught. Despite Kepler came to blows with his schoolmates, his scientific achievements always came to the fore.
Although his family was poor and he was often ill, Kepler started to study Lutheran theology and philosophy in TÃ¼bingen in 1589. There he also got to know Kopernikusâ€™s heliocentric theory. But it was forbidden to advocate these ideas in public because of their alleged contradiction to certain parts of the Bible. Michael MÃ¤stlin was Keplerâ€™s most important teacher in mathematics and astronomy at this time. Kepler attended his lectures about Euclideâ€™s â€œElementaâ€ and the astronomy of Ptolemaios and Regiomontan.
After his masterâ€™s degree at the school of artists Kepler was allowed to participate actively in academic disputations. For example, he defended Kopernikusâ€™ teachings and praised the Sun as regent of the motion of the planets, inter alia by the means of newplatonic thoughts.
For the main subject of his studies, in which Kepler wanted to achieve the doctorate of theology, Kepler had to get to know the different doctrines. His most important teacher in theology was Matthias Hafenreffer (1561-1619), who did not always agree with his opinions concerning theology, but nevertheless the two men were friends for a long time. Despite being of Lutheran confession, Kepler was more inclined to follow the Calvinists as far as the doctrine of the Last Supper was concerned. Through the comments of the Holy Bible by the Wittenbergian theologian Aegidius Hunnius (1550-1603) he came to more insight as through the lectures in TÃ¼bingen. In a humanistic spirit Kepler saw the aim of church policy of the different confessions in the reunification of the split Church â€“ because in his opinion all Christian Confessions participated in the indivisible truth. Kepler wanted to become a priest, but because of his opposition to the common doctrines and his adoration of the heliocentric theory he came into conflict with the Protestant theologians of WÃ¼rttemberg. The praising of the heliocentric system was a problem because it opened a disruption between astronomy and theology, which led to a change of the theological idea of mankind â€“ because the geocentric view of the Bible stated that the human beings were the most important ones in the cosmos and should therefore be in its center. (By the way, Tycho Braheâ€™s view of the world was still geocentric and at his deathbed Brahe asked Kepler to use his observations of Mars to prove his geocentric Tychonic system. But for Kepler it was always evident that Kopernikusâ€™ method was the right one.)
Finally, in 1594 Kepler abandoned his studies of theology in TÃ¼bingen, however, this decision was not completely voluntary.
Later Kepler wrote to MÃ¤stlin that he had wanted to become a theologian, but by his efforts people should now see how God was honoured in astronomy too (KGW XIII, 40).
III. THE YEARS IN GRAZ
Because of his mathematical ability and the necessity to earn some money Kepler started to teach mathematics in a Protestant school at Graz at the age of 22. However, the main focus of the school were Protestant theology and apologetic so that Keplerâ€™s instruction of mathematics in the highest class did not meet with much response. In order to obtain more students he had to offer further courses in rhetorics, and later also in history and ethics. Kepler also became a mathematician for land surveying in Graz. At that time astronomy and astrology were not yet distinctly separated from each other – and therefore one of Keplerâ€™s duties as mathematician for land surveying was to create a calendar including predictions about weather, political events, etc. Being financially poor this was an important source of income for Kepler and furthermore he became a local celebrity because his predictions wondrously turned out to be true. However, he never took astrology seriously, but called it the silly little daughter of astronomy.
During a lecture about geometry Kepler suddenly had a revelation, which turned out to change his whole life. He considered this revelation as the secret clue to understand the world. Kepler drew an equilateral triangle in a circle and another circle in the triangle on the blackboard in front of the class. Immediately he noticed that the proportions of the circles accorded to the proportion of the orbits of Saturn and Jupiter. Later Kepler proposed this theory about the motions of the planets: The planets are arranged in such a way that the five Platonic bodies of the ancient Greeks can fit between them. This was an explanation in pyhtagoreic and platonic tradition for the orbits of the planets as well as their mutual distances. Kepler sent this work to his former WÃ¼rttembergian teacher Michael MÃ¤stlin (1550-1631) to have it printed in TÃ¼bingen. But the senate of the University of TÃ¼bingen complained about Keplerâ€™s theory of the motion of the earth, which could derogate the reputation of the Bible. Finally, those problems were solved (that is to say Kepler had to discard a passage in which he had explained the compatibility of Kopernikusâ€™ theory with the Holy Scripture) and in 1596 Kepler published this theory in his first work â€œMysterium cosmographicumâ€. Although there is no scientific foundation for this theory, it highly influenced the development of physics. Kepler also sent this work to Galileo Galilei (1564-1642) and Tycho Brahe. Galilei did not think much of it, but Brahe was enthusiastic. Brahe lauded Keplerâ€™s basic concept and advised him to improve his astronomy by the means of observations. And Kepler followed this advise to ensure the cosmology, which was drafted by a speculative way, by data from observations. This meant an approach by an epistemology which was more based on empiricism.
Because of the publication of â€œMysterium cosmographicumâ€ Keplerâ€™s self-confidence was established and the first verse of Persiusâ€™ satires, which he quoted in an entry in a register, remained his motto: â€œO curas hominum, o quantum est in rebus inane.â€ (Oh, the troubles of the human beings, oh, how many void things reside in these matters.)
fig. 3: an illustration to Keplerâ€™s theory about the orbits of the planets
In 1597 another important thing happened in Keplerâ€™s life: He married Barbara MÃ¼ller, who was already widowed twice. But the marriage was not very happy and only two of their five children survived.
Kepler was also busy with new methods of observation in Graz. So he upgraded the ecliptical instrument for the observation of solar eclipses. The common principle of this instrument was based on the method of projecting a picture on a screen, but the axis of Keplerâ€™s instrument was also traversable in the horizon-system.
IV. KEPLERâ€™S WORK IN PRAGUE
At that time Ferdinand II. started to rule Styria and strictly carried out the counter-reformation. Therefore the Protestant Kepler had to leave Graz in 1597. He went to Prague, where he became Braheâ€™s assistant. The two men complemented each other excellently because Brahe made observations, but he was not very talented in mathematics. On the other hand the mathematician Kepler was not very skilled in making exact observations. As his first great work in Prague Kepler had to analyze the highly excentric motion of Mars. Because of this project Kepler later developed his theory of planetary motion, which revolutionized theoretical astronomy. At Braheâ€™s institution Kepler also wrote the work â€œDe fundamentis astrologiae certioribusâ€.
Only some days after Braheâ€™s death on October 24, 1601 Kepler became his successor as mathematician and astronomer at the court. Mourning about Braheâ€™s death and hoping to have the chance to continue his work, Kepler composed an elegy about Braheâ€™s passing. Kepler explained Braheâ€™s observations and the result of this work were the â€œTabulae Rudolfinaeâ€. (Kepler had promised Brahe to finish his work.) However, Kepler did not publish this work until 1627 in Ulm because he was busy with a lot of interesting aspects in Braheâ€™s results â€“ Kepler even calculated the date of a Venus-transit.
With an enormous effort of computation Kepler wrote the â€œTabulae Rudolfinaeâ€, his penultimate work. These studies contain information about the position of Sun and Moon, which was also useful for metering solar and lunar eclipses. Particularly the positions of planets at any time B.C. and A.D. can be found in this work. In the appendix Kepler provided instructions for using the tables and a list of 530 small towns which were arranged in the geographic graticule so that the â€œTabulae Rudofinaeâ€ represented a perfect orientation also for the mariners. (The zero meridian [which has been metered from Greenwich since 1883] went through Tycho Braheâ€™s former observatory at the Danish island Hven.) Another important part of this work is a catalogue of 1000 fixed stars, which hearkens back to Tycho Brahe. Kepler already included the constellations of the southern heaven, which had emerged because of recent expeditions. Later a world map was added to Keplerâ€™s Tabulae. (This map was exact except for two longitudes and nowadays it is still unknown how such an accuracy could be obtained at that time.) In short, the â€œTabulae Rudofinaeâ€ remained the astronomical standard work for the next half century and they carried Keplerâ€™s fame around the world.
fig. 4: Keplerâ€™s â€žTabulae Rudolfinaeâ€œ.
In 1604 Kepler wrote the work â€œAstronomiae pars opticaâ€, in which he analyzes the nature of light and colour. The decrease of the light-intensity with the square of the distance as well as the amount of refraction of starlight because of the atmosphere are discussed. Furthermore, this work contains analyses about the shadows of Earth and Moon during eclipses and the process of seeing. Finally Kepler names the focal point with the Latin word â€œfocusâ€ as the first scientist.
On October 11, 1604 Kepler observed a nova and he wrote the book â€œGrÃ¼ndlicher Bericht von einem ungewÃ¶hnlichen neuen Sternâ€ about this discovery in German. Two years later he published the work â€žDe stella nova in pede Serpentariiâ€œ about the same nova. It represents a scientific analysis which Emperor Rudolf II. commissioned. Kepler describes colour, luminosity and speculates about the distance as well as the matter of the â€œnewâ€ star. On the other hand he criticizes the common astrological â€œgossipâ€ about that event, but he includes the human beings in cosmic events. In the fourth chapter Kepler assumes that the star of the three wise men at the time of Jesus Christâ€™s birth coincided with the great conjunction of the bright planets â€“ and because of this fact a new date for Jesus Christâ€™s year of birth could be calculated.
At that time Kepler was also busy with geodesics and proposed a measuring of the Earth in a letter to Herwart in 1607. He proposed to make observations from two towers to determine the zenith-distances and the space between the two points of observation. So the extent of the Earth and its possible deviations from the spheric figure could be specified.
Keplerâ€™s astronomical chief work was the â€œAstronomia novaâ€ of 1609, which contains his first and second law (see later). With the help of Tycho Braheâ€™s data from measures of Mars Kepler calculated the orbits of the planets. For the first time in history of astronomy a new theory of planetary motion is proposed which is equally valid for all planets. By describing unequal elliptical motion of the planets, Kepler invalidates the ancient axiom of the circular steady motion of the planets, which Kopernikus had also followed. As far as the scientific approach is concerned, Kepler states a new method: A theory is only applicable, when it is mathematically correctly affirmed by observations. In other words, assumptions and beliefs are not enough. In this work Kepler also creates the basis for the infinitesimal calculus, which Newton and Leibniz advanced later. The â€œAstronomia Novaâ€ is the first work in history, in which the motion of celestial bodies is demonstrated as licit necessity. This work was written in a difficult style and therefore remained misunderstood to a large extent. However, Keplerâ€™s fame increased because of his exact calculations, which were based on the new planetary astronomy.
In 1610 Kepler observed Jupiterâ€™s moons to affirm Galileo Galileiâ€™s discoveries. Kepler made his observations with a Galileian telescope, which he had received by a benefactor. (Galilei did not want or could not supply Kepler with a telescope.) Kepler published his analyses concerning this field in the brief scripture â€œNarratio de Jovis satellitibusâ€. As far as Jupiterâ€™s moons are concerned Kepler proposed Galilei the Latin word â€œsatellesâ€, which Galilei adopted immediately. But Galilei never admitted that this term had been invented by Kepler.
Furthermore he dealt with optics once more and the book â€žDioptriceâ€œ, which he published in 1611, was the basis for the foundation of optics as a science. Inter alia the theory of the refracting telescope, analyses of optical path and refraction in three sided prisms, and a draft of Keplerâ€™s later astronomical telescope with two convex lenses can be found in this work.
fig. 5: Galileiâ€™s telescope (top) contrary to Keplerâ€™s telescope.
In Prague Kepler reached the acme of his scientific work. Beside the important books he wrote at that time, he was respected very much as imperatorial mathematician and his opinion about various scientific problems was highly estimated at court. However, Kepler was never appointed professor at a university. Therefore he did not have the chance to establish a scientific school and to present his doctrines to his pupils. Furthermore his financial means were not enough to afford assistants.
IV. KEPLER IN LINZ
After the death of Rudolf II. Kepler went to Linz (Upper Austria), where he worked as a mathematician for land surveying and as a teacher. He had to give his pupils an understanding of mathematical as well as philosophical questions and he should also create a map of Upper Austria. But this map was never completed.
Keplerâ€™s employment in Linz was less demanding than that in Prague, but he could keep the freedom of his mind there. His life in Linz was quieter than in Prague and soon after his arrival in Upper Austria he married a second time (his first wife had already died in 1611). From eleven candidates he finally chose Susanna Reuttinger, who later gave birth to seven children, but only one of them survived.
Although Kepler was very happy in Linz, he had a conflict of faith at the beginning of his stay there: When Kepler criticised certain articles of faith of the preacher Daniel Hitzler, Hitzler required Keplerâ€™s signature of the doctrines of the â€œConcordian Formulaâ€. Because of conscientious reasons Kepler could not agree and preferred his personal freedom instead.
Kepler continued his research in Linz by the help of various splendid libraries. Some aristocratic patrons protected him from conflicts, but Kepler did not have a chance for an exchange of ideas with other friendly academic people.
As far as theology is concerned, Keplerâ€™s studies about Jesus Christâ€™s year of birth must be mentioned. He published his analyses about this topic in the work â€œBericht vom Geburtsjahr Christiâ€ in 1613. Kepler claimed that Jesus Christ was born five years before the common date. Kepler had already dealt with this chronological problem in 1606 and took up his former thoughts at that time. (By the way the Christian monk Dionysius Exiguus [ca. 470 – ca. 540] is well known as initiator of such calculations.) Keplerâ€™s book was a polemic paper against the doctor and astrologer HelisÃ¤us RÃ¶slin, who had doubted all of Keplerâ€™s thoughts about the â€œnew starâ€, the comets, and the date of Jesus Christâ€™s year of birth. So Kepler defended his point of view in this book, in which he proved to be an excellent historian.
In Linz Kepler was engaged with mathematical problems and in 1615 he published the book â€žStereometria Doliorum Vinariorumâ€œ according to which â€œKeplerâ€™s barrel ruleâ€ was named later. This work contains novel calculations of volume. The new method was to calculate the volume of barrels of vine by dipping a measuring-rood into it. One year later a much simpler version of those ideas followed with the work â€œAuszug aus der uralten MeÃŸkunst Archimedisâ€. The intention of this book was to emphasize the practical applications. Moreover at the end of the book there was an index of technical terms, where Kepler provided German translations for the Latin termini.
In 1615 Keplerâ€™s mother was accused of sorcery. He tried hard to defend her and five years later he eventually managed that she was acquitted and released. But one year later she died because of the consequences of the torture. Because of those events Kepler was not very motivated to produce further scientific work in Linz, nevertheless he began to write the studies â€œHarmonices mundiâ€, which can be seen as the most profound of Keplerâ€™s works. In 1619 he finished these five books, which deal with an application of the theory of harmony in music, astrology, geometry and astronomy. As far as music is concerned Kepler believed in a musical consonance as help for the arrangement of the spaces between the planets corresponding to sorts of tone. In connection with this theory he dealt with the question why human beings regard certain intervals as euphonic, but others as cacophonous. Kepler thought that he had understood Godâ€™s logic at the creation of the world with the theses contained in this work, which is fraught with platonic-pythagoreic spirit and especially influenced by Proklos. Accordingly he was in high spirits.
Great is our god and great is his force and of his wisdom is not a number.
Praise him, heavens, praise him, Sun, Moon and planets,
Which sense you have to cognize, which tongue to glorify your creator.
Praise him, you celestial harmonies, praise him, you all,
Who you are witnesses of the now discovered harmonies!
You must praise too, my soul, the God, your creator, as long as I will be.
(loose translation of a part of the prayer)
Kepler also wanted to manifest the wonderfulness of the divine creation in the â€œHarmonices mundiâ€. But in this work there is an antithesis of Keplerâ€™s mystic and his scientific claim at certain parts â€“ for example on the tides. Formerly he had seen the tide as a result of the attraction of the moon, but now he gives a mystical-poetical answer to this question by referring to the breath of the impressed body of the Earth. It honours Kepler that he had both perceptions and that there was no conflict between them. However, the â€œHarmonices mundiâ€ were difficult to read, so there was no feedback. Keplerâ€™s ideas of a universal harmony formed an antithesis to the religious conflicts and the horror of the Thirty Yearsâ€™ War. Therefore Kepler passionately argued for peace in his letters and the dedications of his books.
Kepler also disagreed with the harsh proceeding of the Papal Censorship of Publications Board. In 1616 after the admonition of Galilei the scriptures of Kopernikus and his followers had been put on the index. Kepler feared that his works would be forbidden at least in Italy and wrote to the booksellers that they should be cautious and not sell his â€œHarmonices mundiâ€ to everybody.
So Kepler maintained the high aim of confessional and therefore political peace as well as scientific truth at the acme of his producing.
Kepler was also a passionate follower of Kopernikusâ€™ heliocentric theory and in 1618 he wrote the â€œEpitome Astronomiae Copernicaeâ€, in which his discoveries are described and which is the first textbook about the heliocentric theory. Keplerâ€™s intention was to provide plain information in these seven books. He overcomes Kopernikusâ€™ mistakes and imprecision without undermining the importance of Kopernikusâ€™ theories. In the first part the doctrine of his antecessor is gloried and that was the reason why this part was put on the index in 1619 before further chapters were published. In Italy only some respectable scientists were allowed to own this book. The major topics of the â€œEpitome Astronomiae Copernicaeâ€ are celestial mechanics, celestial dynamics, and celestial physics â€“ all illustrated by geometrical figures and plain examples. But also corresponding questions about natural philosophy are an important part of the contents. Despite the accusation of sorcery of his mother and the occupation of Linz by Maximilianâ€™s troops Kepler continued this work. As a â€œpriest of God on the book of natureâ€ he devoted the â€œEpitome Astronomiae Copernicaeâ€ to the praise of the creator.
Kepler was also a genius in calculating and dealt with logarithms. He recalculated the logarithms which Neper had invented and he included particularly the simplified mode of calculation for astronomical arithmetic problems. In 1624 Kepler published all this in his work â€œChilias Logarithmorumâ€, which contained for example the 1000 logarithms from 100 to 100.000 each with 100 proceeding.
V. KEPLERâ€™S LAST YEARS
In 1626 Kepler was forced to leave Linz because of the Thirty Yearsâ€™ War. The farmers revolted against the occupation by the Bavarians and the procedures of the counter-reformation. Linz was occupied and because of the acts of violence Kepler and his family left the city at the first chance.
Kepler went to Ulm to complete the print of his â€œTabulae Rudolfinaeâ€. After having taken the finished work to the Frankfurtian Autumn Fair in September 1627, he handed it over to the emperor in Prague the following year. The emperor made a splendid offer to Kepler, but we do not know what it was. Anyway, Kepler should become a Catholic. However, Kepler could not accept that and so he refused the emperorâ€™s offer.
The finishing of the â€œTabulae Rudolfinaeâ€ was the reason why Albrecht Wallenstein, the Duke of Freidland and Sagan and the commander-in-chief of the imperial troops, offered Kepler a new job. Without doubt Wallenstein hoped that Kepler would support him in further astrological questions, because in 1608 Kepler had already interpreted the horoscope of nativity of a Bohemian man called Waltstein. (To keep the promised discretion Kepler had written that name in a cryptograph which he rarely used.) For Wallensteinâ€™s horoscope Kepler should overwork his rectification based on the tables and his ephemerises. Furthermore the imperial party thought to improve the predefinition of the outgoing of the war by predictions from Keplerâ€™s calculations of the orbits of the planets.
In the small town Sagan Kepler was cut off from intellectual life, but he began to print his â€žSomnium sive Astronomia Lunarisâ€œ, which could nearly be called a science-fiction novel because it describes the observation of astronomical phenomena by inhabitants of the Moon. With the â€œSomniumâ€ Kepler wanted to popularize the Copernican theory, but it was published after Keplerâ€™s death by his son Ludwig in 1634. This work can be seen as a forerunner of science-fiction novels like Voltaireâ€™s â€œMicromÃ©gasâ€œ or Jules Verneâ€™s â€œDe la terre Ã la luneâ€.
Finally Kepler finished the â€œEphemeridesâ€ in exhausting work which lasted for months. This represented the end of Keplerâ€™s scientific activity.
Later Kepler was hard-pressed for money again and therefore he went to Regensburg via Leipzig and NÃ¼rnberg to collect some money. But his health was already weak. Few days after his arrival he fell mortally ill and died on November 15th 1630. Kepler was buried outside the city gates, however, three years later the Thirty Yearsâ€™ War left no trace of his grave behind. We only have a transcription of the inscription on his grave from Ludwig Kepler, which he composed himself. It says:
“Mensus eram coelos, nunc terrae metior umbras.
Mens coelestis erat, corporis umbra jacet.â€
I measured the sky, now I measure the shadow of the earth. The mind was celestial, the bodyâ€™s shadow here is at rest.
VI. KEPLERâ€™S MOST IMPORTANT SCIENTIFIC ACHIEVEMENT
Although Kepler was never as famous as Galilei, he left behind an enormous work, which was very useful for later astronomers too. However, Keplerâ€™s most important findings were his â€œthree lawsâ€, which pupils have been learning at school up to this very day. In the original wording they are formulated as follows:
- “… ut sequenti capite patecet: ubi simul etiam demonstrabitur, nullam Planetae relinqui figuram Orbitae, praeterquam perfecte ellipticam; conspirantibus rationibus, a principiis Physicis, derivatis, cum experientia observationum et hypotheseos vicariae hoc capite allegata”
- “Quare ex superioribus, sicut se habet CDE area ad dimidium temporis restitutorii, quod dicatur nobis 180Â° gradus: sic CAG, CAH areae ad morarum in CG et CH diuturnitatem. Itaque CGA area fiet mensura temporis seu anomaliae mediae, quae arcui eccentrici CG respondet, cum anomalia media tempus metiatur”
- “Sed res est certissima exactissimaque, quod proportio quae est inter binorum quorumcunque Planetarum tempora periodica, sit praecise sesquialtera proportionis mediarum distantiarum, id est Orbium ipsorum”
As far as his first law is concerned, Kepler was inspired by Gilbertâ€™s work â€žDe magnete, magneticisque corporibus et de magno magnete tellureâ€œ as well as Braheâ€™s question about the orbit of Mars. In 1609 he published it together with the second law in â€œAstronomia novaâ€ (see above). (Kepler had discovered his second law before the third.) However, the way to those findings was not easy: Kepler had difficulties to get at least an approximately coinciding result between calculus and observation of four observations of Mars-oppositions. He complained about his clumsiness in calculating and only achieved a difference of eight angular minutes between calculus and observation, which got in the way of a final solution for some time.
Keplerâ€™s train of thoughts to get to the elliptical planetary motion was the following: He acted on the assumption of the Copernican theory of planetary motion, but contrary to that model (with the benchmark in the center of the terrestrial orbit) he put the benchmark into the true Sun. Kepler extended his model of planetary motion in the length by making use of an additional point (â€œpoint of balanceâ€) on the line of apsis (the linear connection between the center of the circle and the center of the Sun). From this point planetary motion seemed to be steady. This model leads to the setting-up of the â€œhypothesis vicariaâ€, which allows a satisfying calculation of ecliptical lengths. Kepler developed this extended model with the help of Braheâ€™s exact observations of oppositions. Furthermore for the analysis of the different distances of the planet from the Sun (i. e. the planetâ€™s radial motion) Kepler calculated the triangles which are formed by the Sun and the positions of the Earth and the planet. Either the position of the Earth or of the planet is assumed as held on, whereas the other celestial body has different positions at the same times. This approach affirms the finding of the observation of the latitudes that the hypothesis vicaria does not reproduce the ratios of the orbits according to reality. Additionally, through triangulations like that the relative radial distances of Mars and the Earth from the Sun can be defined and so the planetâ€™s ratios of the orbits can be effectively fathomed.
The verification that the velocity of the planet in the extreme points of the orbit is inversely proportional to the distance Sun-Earth is inductively extended on all points of the orbit. This so-called proposition of the radii is the basis of the proposition of the areas.
Finally, after investigating the mechanism of motion on the basis of the terrestrial orbit Kepler developed the ellipse for the orbit of Mars by means of geometry. He used oval orbits as approach to the true form of the orbit.
fig. 6: an illustration to the deduction of the hypothesis vicaria.
In the fifth book of â€œHarmonices mundiâ€ Kepler published his third law in 1619, which describes the correlation of the orbits of two planets with their average distances from the Sun. Contrary to the two first strictly valid laws, the third law is only valid if the sun has much more mass than the planet. However, the purpose of this law in the â€œHarmonices mundiâ€ is not an astronomical one, but it enables a new calculation of the distances of the planets through harmonic ratios of the extreme motions of the planets. The more particular function in this work is the fine adjustment of the cosmos which is introduced as harmonically structured. As far as the preparatory work for this â€œaurea regula Kepleriâ€ is concerned, nothing is known about it nowadays, because these sheets were removed from Keplerâ€™s estate.
And not an apple, as sometimes told, but Keplerâ€™s third law took Isaak Newton (1643-1727) to the discovery of his gravitation law about 60 years later (F = ,
G = 6.672.10-11 m3kg-1s-2, M â€¦ mass of the sun). (Instead of gravitation Kepler had introduced a magnetic force of the Sun, the so-called â€œanima motrixâ€:
intensity 1 / distance2.)
Nowadays Keplerâ€™s laws are formed as abstract theorems:
1. The orbit of all planets is an ellipse with the Sun in one of its foci (ellipse-proposition).
fig. 7: the elliptical planetary motion.
2. The radius vector (the line between the Sun and a planet) sweeps out equal areas in equal times (area-proposition).
fig. 8: the equal areas of the radius vector.
3. The squares of the periods of revolution of two planets are proportional to the cubes of their semimajor axes.
fig. 9: planetary motion and elliptical axes.
By formulas Keplerâ€™s three laws can be expressed in the following way:
pe . pa = (a â€“ e)( a + e) = a2 â€“ e2.
a â€¦ arithmetic mean distance sun-planet;
e â€¦ excentricity;
pe â€¦ perihelion â€¦ a â€“ e;
pa â€¦ aphelion â€¦ a + e;
With the theorem of Pythagoras the result for the semi-minor axis b is: b = and we can easily prove the equation for the ellipse:
b2 x2 + a2 y2 = a2 b2 cartesian coordinates,
r = 1 + e cos E; r cos v = e + cos E polar coordinates.
fig. 10: an illustration of the parameters of elliptical planetary motion.
Formulas for the computation of the velocities in various points of the ellipse:
Mean velocity: V= 10-3 km/s
Perihel- velocity: V= 10-3 km/s
Aphel- velocity: V= 10-3 km/s
Velocity in a various point with distance r of the sun: V= 10-3 km/s
The planet moves fastest when it is near perihelion and most slowly when it is near aphelion.
fig. 11: an illustration to Keplerâ€™s
Examples for Keplerâ€™s third Law:
The constant of proportionality is the same quantity for all planets. It depends only on G, the constant of universal gravitation, and M, the mass of the Sun.
Since Newton we have been able to prove easily that = .
As far as mathematics is concerned, especially the third law is remarkable and was further evolved.
If the different masses of two planets in the three-body-problem are considered, the exact formulation of Keplerâ€™s third law is the following:
In combination with the law of gravitation we get a planetâ€™s time of circulation T around the Sun:
With his three laws Kepler effected the transition from the qualitative natural philosophy to the quantitative natural science and he was one of the founders of modern physics.
Johannes Kepler was an ingenious man, whose perception of science as priestly service on nature was unique. All of his findings originated from his enthusiasm for nature â€“ he really exulted when he discovered the principles of order and harmony. He believed in the progress of science and wrote to the emperor in the dedication of his work about astronomical optics: Inexhaustible is the treasure of the secrets of nature, it is an indescribable richness, and who brings new things to light in this field, achieves nothing else than disclosing other people the way to new research (KGW II, 8).
Later academic people appreciated Kepler too:
Firstly, Pierre Simon de Laplace (1749-1827) said that Kepler was one of those rare people who are sometimes given to science by nature â€“ and they formulated the important theories which were prepared by the work of some centuries. Secondly, Albert Einstein (1879-1955) thought that Kepler was one man among few who could not help speaking his mind in all respects. Last but not least Walter Thirring (born 1927) wrote that because of the rations of the irrational radii Keplerâ€™s theory seemed to lead to a form of truth which could not be understood when they are only cursorily examined.
Additionally further honors are accorded to Kepler nowadays too: The University of Linz, the observatories of Weil der Stadt, Graz, and Linz are named after Kepler. Furthermore the nova which Kepler discovered, a lunar crater, and an asteroid are called â€œKeplerâ€. Additionally, a spacecraft which will search for extrasolar planets and will be launched by the NASA in November 2008 is named after Kepler. Finally the way of constructing telescopes which Kepler developed is still used nowadays. â€“ That is why Kepler had an enormous impact on the progress of science. He was a man who adored harmony and order, and everything he discovered was associated with his perception of God. Divinely ordered numerical proportions and mental harmonies were important for him. Especially his work â€œHarmonices mundiâ€ can absolutely be called modern â€“ with the search for a â€œtheory of everythingâ€ and the claim to draw the cultural and social conclusions from the knowledge with the ethic aim to improve human life and to increase the desire for harmony. So as a theologian and astronomer Kepler wanted to understand how and why God had created the world.
fig. 12: the lunar crater â€œKeplerâ€œ.
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