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Kepler's Laws Of Planetary Motion

Kepler's laws of planetary motion

Johannes Kepler's primary contributions to astronomy/astrophysics were his three laws of planetary motion. Kepler, a nearly blind though brilliant German mathematician, derived these laws, in part, by studying the observations of the keen-sighted Danish astronomer Tycho Brahe. The article on Johannes Kepler gives a less mathematical description of the laws, as well as a treatment of their historical and intellectual context. Sir Isaac Newton's later discovery of the laws of motion and universal gravitation depended strongly on Kepler's work. Although from the modern point of view, Kepler's laws can be seen as a consequence of Newton's laws, historically, it was the other way around: Kepler provided a mathematically distilled description of the empirical observations, which Newton then interpreted.

Kepler's first law

The orbit of a planet about a star is an ellipse with the star at one focus. There is no object at the other focus of a planet's orbit. The semimajor axis, a, is half the major axis of the ellipse. In some sense it can be regarded as the average distance between the planet and its star, but it is not the time average in a strict sense, as more time is spent near apocentre than near pericentre.

Kepler's second law

A line joining a planet and its star sweeps out equal areas during equal intervals of time. This is also known as the law of equal areas. Suppose a planet takes 1 day to travel from points A to B. During this time, an imaginary line, from the Sun to the planet, will sweep out a roughly triangular area. This same amount of area will be swept every day. As a planet travels in its elliptical orbit, its distance from the Sun will vary. As an equal area is swept during any period of time and since the distance from a planet to its orbiting star varies, one can conclude that in order for the area being swept to remain constant, a planet must vary in velocity. The physical meaning of the law is that the planet moves faster when it is closer to the sun. This is because the sun's gravity accelerates the planet as it falls toward the sun, and decelerates it on the way back out.

Kepler's third law

The square of the sidereal period of an orbiting planet is directly proportional to the cube of the orbit's semimajor axis. :

P^2 \propto a^3

:P = object's sidereal period :a = orbit's semimajor axis Thus, not only does the length of the orbit increase with distance, also the orbital speed decreases, so that the increase of the sidereal period is more than proportional. See the actual figures: attributes of major planets. This law is also known as the harmonic law.

Accuracy and limitations

As discussed below, Kepler's third law needs to be modified when the orbiting body's mass is not negligible compared to the mass of the body being orbited. However, the correction is fairly small in the case of the planets orbiting the Sun. A more serious limitation of Kepler's laws is that they assume a two-body system. This is a particularly bad approximation in the case of the Earth-Sun-Moon system, and for calculations of the Moon's orbit, Kepler's laws are far less accurate than the empirical method invented by Ptolemy hundreds of years before. Newton generalized Kepler's first law with the realization that an object moving at greater than escape velocity (e.g., some comets) would have an open parabolic or hyperbolic orbit rather than a closed elliptical one. Thus, all of the conic sections are possible orbits. The second law is still valid for open orbits (since angular momentum is still conserved), but the third law is inapplicable because the motion is not periodic. Because electrical forces also obey an inverse square law, Kepler's laws also apply to bodies interacting electrically. If the interaction is repulsive, as in Rutherford scattering, then the orbit is hyperbolic, with the center of repulsion outside the hyperbola. In the planetary model of the atom, the electrons were imagined to orbit the nucleus exactly like the planets orbiting the Sun, and Niels Bohr even went so far as to attempt to assign specific atomic energy levels to specific elliptical orbits. The Keplerian approximation is successful when the quantum-mechanical wavelengths are small compared to the size of the orbit (as in Rutherford scattering, and in some unusual artificially populated electron orbitals with very high values of the principal quantum number). Kepler's laws do not consider the emission of radiation. The emission of gravitational radiation is negligible in our solar system, but important in some stellar systems containing black holes or neutron stars (any system involving large masses combined with large accelerations). In the planetary model of the atom, the emission of electromagnetic radiation should have led to the collapse of all atoms, and this was a hint that classical physics was in need of modification. Kepler's laws don't incorporate relativity, so, for example, they don't correctly predict the precession of Mercury's orbit. The successful explanation of this non-Keplerian behavior was one of the most dramatic early successes of general relativity. Kepler's laws fail even more dramatically in the case of an object orbiting a black hole; here, general relativity predicts that many orbits —those that pass through the event horizon— are one-way streets to a dead-end region of spacetime.

Connection to Newton's laws and conservation laws

Kepler did not understand why his laws were correct; it was Isaac Newton who discovered the answer to this more than fifty years later. The second law can also be seen as a statement of conservation of angular momentum, which is a logical consequence of Newton's laws in the special case of a force that acts along the line connecting two objects.

Kepler's first law

Newton proposed that "every object in the universe attracts every other object along a line of the centres of the objects, proportional to each object's mass, and inversely proportional to the square of the distance between the objects." The following assumes that acceleration is of the form :

\frac = f(r)\hat

. Recall that in polar coordinates: :

\frac = \dot r\hat + r\dot\theta\hat

\frac = (\ddot r - r\dot\theta^2)\hat + (r\ddot\theta + 2\dot r \dot\theta)\hat

In component form, we have: :

\ddot r - r\dot\theta^2 = f(r)

r\ddot\theta + 2\dot r\dot\theta = 0

Since

\dot r = dr/dt

and

\ddot\theta=/

, the latter equation is equivalent to :

\frac = -2\frac

. When integrated, this yields :

\log\dot\theta = -2\log r + \log\ell

, :

\ell = r^2\dot\theta

, for some constant

\ell

, which can be shown to be the specific angular momentum. Now we substitute. Let: :

r = \frac

\dot r = -\frac\dot u = -\frac\frac\frac= -\ell\frac

\ddot r = -\ell\frac\frac = -\ell\dot\theta\frac= -\ell^2u^2\frac

The equation of motion in the

\hat

direction becomes: :

\frac + u = - \fracf\left(\frac\right)

f(r)=-k/r^2

, as Newton's law of gravitation claims, then: :

\frac + u = \frac

where -k is our proportionality constant. This differential equation has the general solution: :

u = A\cos(\theta-\theta_0) + \frac.

Replacing u with r and letting θ₀=0: :

r = \frac

. This is indeed the equation of a conic section with the origin at one focus. Thus, Kepler's first law is a direct result of Newton's law of gravitation and Newton's second law of motion.

Kepler's second law

Assuming Newton's laws of motion, we can show that Kepler's second law is consistent. By definition, the angular momentum

\mathbf

of a point mass with mass

m

and velocity

\mathbf

is : :

\mathbf \equiv m \mathbf \times \mathbf

. where

\mathbf

is the position vector of the particle. Since

\mathbf = \frac

, we have: :

\mathbf = \mathbf \times m\frac

taking the time derivative of both sides: :

\frac = \mathbf \times \mathbf = 0

since the cross product of parallel vectors is 0. We can now say that

|\mathbf|

is constant. The area swept out by the line joining the planet and the sun is half the area of the parallelogram formed by

\mathbf

and

d\mathbf

. :

dA = \begin\frac\end |\mathbf \times d\mathbf| = \begin\frac\end \left|\mathbf \times \fracdt\right| = \fracdt

Since

|\mathbf|

is constant, the area swept out by is also constant. Q.E.D.

Kepler's third law

Newton used the third law as one of the pieces of evidence used to build the conceptual and mathematical framework supporting his law of gravity. If we take Newton's laws of motion as given, and consider a hypothetical planet that happens to be in a perfectly circular orbit of radius r, then we have

F=mv^2/r

for the sun's force on the planet. The velocity is proportional to r/T, which by Kepler's third law varies as one over the square root of r. Substituting this into the equation for the force, we find that the gravitational force is proportional to one over r squared. (Newton's actual historical chain of reasoning is not known with certainty, because in his writing he tended to erase any traces of how he had reached his conclusions.) Reversing the direction of reasoning, we can consider this as a proof of Kepler's third law based on Newton's of gravity, and taking care of the proportionality factors that were neglected in the argument above, we have: :

T^2 = \frac \cdot r^3

where:
- T = planet's sidereal period
- r = radius of the planet's circular orbit
- G = the gravitational constant
- M = mass of the sun The same arguments can be applied to any object orbiting any other object. This discussion implicitly assumed that the planet orbits around the stationary sun, although in reality both the planet and the sun revolve around their common center of mass. Newton recognized this, and modified this third law, noting that the period is also affected by the orbiting body's mass. However typically the central body is so much more massive that the orbiting body's mass may be ignored. Newton also proved that in the case of an elliptical orbit, the semimajor axis could be substituted for the radius. The most general result is: :

T^2 = \frac \cdot a^3

where:
- T = object's sidereal period
- a = object's semimajor axis
- G = 6.67 × 10⁻¹¹ N · m²/kg² = the gravitational constant
- M = mass of one object
- m = mass of the other object For objects orbiting the sun, it can be convenient to use units of years, AU, and solar masses, so that G, 4π² and the various conversion factors cancel out. Also with m<<M we can set m+M = M, so we have simply

T^2=a^3

Proving Kepler's third law

Define point A to be the periapsis, and point B as the apoapsis of the planet when orbiting the sun. Kepler's second law states that the orbiting body will sweep out equal areas in equal quantities of time. If we now look at a very small periods of time at the moments when the planet is at points A and B, then we can approximate the area swept out as a triangle with an altitude equal to the distance between the planet and the sun, and the base equal to the time times the speed of the planet. :

\frac\cdot(1-\epsilon)a\cdot V_A\,dt=\frac\cdot(1+\epsilon)a\cdot V_B\,dt

(1-\epsilon)\cdot V_A=(1+\epsilon)\cdot V_B

V_A=V_B\cdot\frac

Using the law of conservation of energy for the total energy of the planet at points A and B, :

\frac-\frac=\frac-\frac

\frac-\frac=\frac-\frac

\frac=\frac\cdot \left ( \frac-\frac \right )

\frac=\frac\cdot \left ( \frac \right )

V_B^2 \cdot \left ( \frac\right ) ^2-V_B^2=\frac\cdot \left ( \frac \right )

V_B^2 \cdot \left ( \frac\right )=\frac

V_B^2 \cdot \left ( \frac\right )=\frac

V_B^2 \cdot 4\epsilon=\frac

V_B=\sqrt.

Now that we have

V_B

, we can find the rate at which the planet is sweeping out area in the ellipse. This rate remains constant, so we can derive it from any point we want, specifically from point B. :

\frac=\frac=\frac\cdot(1+\epsilon)a\cdot V_B=\frac\cdot(1+\epsilon)a\cdot\sqrt=
\frac\cdot\sqrt

However, the total area of the ellipse is equal to

\pi a \sqrta

. (That's the same as

\pi a b

, because

b=\sqrta

). The time the planet take out to sweep out the entire area of the ellipse equals the ellipse's area, so, :

T\cdot \frac=\pi a \sqrta

T\cdot \frac\cdot\sqrt=\pi \sqrta^2

T=\frac=\frac=
\frac\sqrt

T^2=\fraca^3.

However, if the mass m is not negligible in relation to M, then the planet will orbit the sun with the exact same velocity and position as a very small body orbiting an object of mass

M+m

(see reduced_mass). To integrate that in the above formula, M must be replaced with

M+m

, to give

T^2=\fraca^3

. Q.E.D.

Solution for the motion as a function of time

The Keplerian problem assumes an orbit with semimajor axis a, semiminor axis b, and eccentricity ε. To convert the laws into predictions, Kepler began by adding the orbit's auxiliary circle (that with the major axis as a diameter) and defined these points:
- c center of auxiliary circle and ellipse
- s sun (at one focus of ellipse);

\mboxcs=a\varepsilon

- p the planet
- z perihelion
- x is the projection of the planet to the auxiliary circle; then

\mboxsxz=\frac ba\mboxspz

- y is a point on the circle such that

\mboxcyz=\mboxsxz=\frac ba\mboxspz

and three angles measured from perihelion:
- true anomaly

T=\angle zsp

, the planet as seen from the sun
- eccentric anomaly

E=\angle zcx

, x as seen from the centre
- mean anomaly

M=\angle zcy

, y as seen from the centre image:kepler's-equation-scheme.png Then :

\mboxcxz=\mboxcxs+\mboxsxz

=\mboxcxs+\mboxcyz

\frac2E=a\varepsilon\frac a2\sin E+\frac2M

giving Kepler's equation :

M=E-\varepsilon\sin E

. To connect E and T, assume

r=\mboxsp

then :

a\varepsilon+r\cos T=a\cos E

and

r\sin T=b\sin E

r=\frac=\frac

\tan T=\frac=\frac ba\frac=\frac

which is ambiguous but useable. A better form follows by some trickery with trigonometric identities: :

\tan\frac T2=\sqrt\frac\tan\frac E2

(So far only laws of geometry have been used.) Note that

\mboxspz

is the area swept since perihelion; by the second law, that is proportional to time since perihelion. But we defined

\mboxspz=\frac ab\mboxcyz=\frac ab\frac2M

and so M is also proportional to time since perihelion—this is why it was introduced. We now have a connection between time and position in the orbit. The catch is that Kepler's equation cannot be rearranged to isolate E; going in the time-to-position direction requires an iteration (such as Newton's method) or an approximate expression, such as :

E\approx M+\left(\varepsilon-\frac18\varepsilon^3\right)\sin M+\frac12\varepsilon^2\sin 2M+\frac38\varepsilon^3\sin 3M

via the Lagrange reversion theorem. For the small ε typical of the planets (except Pluto) such series are quite accurate with only a few terms; one could even develop a series computing T directly from M.[http://info.ifpan.edu.pl/firststep/aw-works/fsII/mul/mueller.html]

External links

- Crowell, Benjamin, Conservation Laws, [http://www.lightandmatter.com/area1book2.html http://www.lightandmatter.com/area1book2.html], an online book that gives a proof of the first law without the use of calculus. Category:Celestial mechanics Category:Equations Category:Eponymous laws ko:케플러 법칙 ja:ケプラーの法則

Johannes Kepler

Johannes Kepler (December 27, 1571 – November 15, 1630), a key figure in the scientific revolution, was a German mathematician, astronomer and astrologer of famed brilliance. He is best known for his laws of planetary motion, expounded in the two books Astronomia nova and Harmonice Mundi. Kepler was a professor of mathematics at the University of Graz, court mathematician to Emperor Rudolf II, and court astrologer to General Wallenstein. Early in his career, Kepler was an assistant to Tycho Brahe. Kepler's career also coincided with that of Galileo Galilei. He is sometimes referred to as "the first theoretical astrophysicist", although Carl Sagan also referred to him as the last scientific astrologer.

Life

Kepler was born on December 27, 1571 at the Imperial Free City of Weil der Stadt (now part of the Stuttgart Region in the German state of Baden-Württemberg, 30 km west of Stuttgart's center). His grandfather had been Lord Mayor of that town, but by the time Johannes was born, the Kepler family fortunes were in decline. His father earned a precarious living as a mercenary, and abandoned the family when Johannes was 17. His mother, an inn-keeper's daughter, had a reputation for involvement in witchcraft. Born prematurely, Johannes is said to have been a weak and sickly child, but despite his ill health, he was precociously brilliant - he often impressed travelers at the inn [aforementioned] with his phenomenal mathematical faculty as a child. Though he excelled in his schooling, Kepler was frequently bullied, and was plagued by a belief that he was physically repulsive, thoroughly unlikable and, compared to the other pupils, an outsider. This ostracizing probably led him to turn to the world of ideas, as well as an abiding religious conviction, for solace. He was introduced to astronomy/astrology at an early age, and developed a love for that discipline that would span his entire life. At age five, he observed the Comet of 1577, writing that he "...was taken by [his] mother to a high place to look at it." At age nine, he observed another astronomical event, the Lunar eclipse of 1580, recording that he remembered being "called outdoors" to see it and that the moon "appeared quite red." In 1587, Kepler began attending the University of Tübingen, where he proved himself to be a superb mathematician. Upon his graduation from that school in 1591, he went on to pursue study in theology, becoming a part of the Tübingen faculty. However, before he took his final exams he was recommended for the vacant post of teacher of mathematics and astronomy at the Protestant school in Graz, Austria. He accepted the position in April of 1594, at the age of 23. In April 1597, Kepler married Barbara Müller. She died in 1611 and was outlived by two children. In December 1599, Tycho Brahe wrote to Kepler, inviting Kepler to assist him at Benátky nad Jizerou outside Prague. After Tycho's death, Kepler was appointed Imperial Mathematician (from November 1601 to 1630) to the Habsburg Emperors. In October 1604, Kepler observed the supernova which was subsequently named Kepler's Star. In January 1612 the Emperor died, and Kepler took the post of provincial mathematician in Linz. In 1611, Kepler published a monograph on the origins of snowflakes, the first known work on the subject. He correctly theorized that their hexagonal nature was due to cold, but did not ascertain a physical cause for this. The question of snowflakes was not resolved until the 20th century. On March 8, 1618 Kepler discovered the third law of planetary motion: distance cubed over time squared. He initially rejected this idea, but later confirmed it on May 15 of the same year. In August of 1620, Katherine, Kepler's mother, was arrested in Leonberg as a witch; she was imprisoned for 14 months. She was released in October 1621 after attempts to convict her failed. Even though she was subjected to torture, she refused to confess to the charges. However, only the courageous personal intervention of Kepler (despite the risk to be arrested as well) and his reputation as the famous Imperial Mathematician rescued her.

Death

On November 15, 1630 Kepler died of a fever in Regensburg. In 1632, only two years after his death, his grave was demolished by the Swedish army in the Thirty Years' War.

Work

Kepler lived in an era when there was no clear distinction between astronomy and astrology, and no consensus on the scientific method as the correct way to decide what was correct or incorrect in science. His ideas are therefore a fascinating mixture of what would today be considered mathematical physics and nonsensical mysticism. Although the subsections below separate the two, Kepler did not see them as separate. Kepler was a Pythagorean numbers mystic. That is, he considered mathematical relationships to be at the base of all nature. Thus all creation was an integrated whole. This may be contrasted with the Platonic and Aristotelian notion that the Earth was fundamentally different from the rest of the universe, being composed of different substances and with different natural laws applying. Thus Kepler was the first to attempt to discover universal laws. This quest led him to apply terrestrial physics to celestial bodies, which produced the three Laws of Planetary Motion. Conversely, he was convinced that celestial bodies influence terrestrial events. One result of this belief was his correct assessment of the Moon's role in generating the tides, years before Galileo's incorrect formulation. Another was his belief that someday it would be possible to develop a "scientific astrology," while not believing that any astrologic rules currently propounded were at all valid.

Scientific work

Kepler's laws

Kepler inherited from Tycho Brahe a wealth of the most accurate raw data ever collected on the positions of the planets. The difficulty was to make sense of it. The orbital motions of the other planets are viewed from the vantage point of the Earth, which is itself orbiting the sun. As shown in the example below, this can cause the other planets to appear to move in strange loops. Kepler concentrated on the orbit of Mars, but he had to know the orbit of the Earth accurately first. In order to do this, he needed a surveyor's baseline. In a stroke of pure genius, he used Mars and the Sun as his baseline, since without knowing the actual orbit of Mars, he knew that it would be in the same place in its orbit at times separated by its orbital period. Thus the orbital positions of the Earth could be computed, and from them the orbit of Mars. He was able to deduce his planetary laws without knowing the exact distances of the planets from the sun, since his geometrical analysis needed only the ratios of their solar distances. Image:Retrograde-motion-of-mars.png Kepler, unlike Brahe, held to the heliocentric model of the solar system, and starting from that framework, he made twenty years of painstaking trial-and-error attempts at making some sense out of the data. He finally arrived at his three laws of planetary motion: three laws of planetary motion 1. Kepler's elliptical orbit law: The planets orbit the sun in elliptical orbits with the sun at one focus. 2. Kepler's equal-area law: The line connecting a planet to the sun sweeps out equal areas in equal amounts of time. 3. Kepler's law of periods: The time required for a planet to orbit the sun, called its period, is proportional to the long axis of the ellipse raised to the 3/2 power. The constant of proportionality is the same for all the planets. Using these laws, he was the first astronomer to successfully predict a transit of Venus (for the year 1631). Kepler's laws were the first clear evidence in favor of the heliocentric model of the solar system, because they only came out to be so simple under the heliocentric assumption. Kepler, however, never discovered the deeper reasons for the laws, despite many years of what would now be considered non-scientific mystical speculation. Isaac Newton eventually showed that the laws were a consequence of his laws of motion and law of universal gravitation. (From the modern vantage point, the equal-area law is more easily understood as arising from conservation of angular momentum.)

1604 supernova

angular momentum On October 17, 1604, Kepler observed that an exceptionally bright star had suddenly appeared in the constellation Ophiuchus. (It was first observed by several others on October 9.) The appearance of the star, which Kepler described in his book De Stella nova in pede Serpentarii ('On the New Star in Ophiuchus's Foot'), provided further evidence that the cosmos was not changeless; this was to influence Galileo in his argument. It has since been determined that the star was a supernova, the second in a generation, later called Kepler's Star or Supernova 1604. No further supernovae have been observed in the Milky Way, though others outside our galaxy have been seen.

Other scientific and mathematical work

Kepler also made fundamental investigations into combinatorics, geometrical optimization, and natural phenomena such as snowflakes, always with an emphasis on form and design. He was also one of the founders of modern optics, defining e.g. antiprisms and the Kepler telescope (see Kepler's books Astronomiae Pars Optica — i.a. theoretical explanation of the camera obscura — and Dioptrice). In addition, since he was the first to recognize the non-convex regular solids (such as the stellated dodecahedra), they are named Kepler solids in his honor.

Mysticism and astrology

Mysticism

Kepler discovered the laws of planetary motion while trying to achieve the Pythagorean purpose of finding the harmony of the celestial spheres. In his cosmologic vision, it was not a coincidence that the number of perfect polyhedra was one less than the number of known planets. Having embraced the Copernican system, he set out to prove that the distances from the planets to the sun were given by spheres inside perfect polyhedra, all of which were nested inside each other. The smallest orbit, that of Mercury, was the innermost sphere. He thereby identified the five Platonic solids with the five intervals between the six known planets — Mercury, Venus, Earth, Mars, Jupiter, Saturn; and the five classical elements. In 1596 Kepler published Mysterium Cosmographicum, or The Cosmic Mystery. Here is a selection explaining the relation between the planets and the Platonic solids: :… Before the universe was created, there were no numbers except the Trinity, which is God himself… For, the line and the plane imply no numbers: here infinitude itself reigns. Let us consider, therefore, the solids. We must first eliminate the irregular solids, because we are only concerned with orderly creation. There remain six bodies, the sphere and the five regular polyhedra. To the sphere corresponds the heaven. On the other hand, the dynamic world is represented by the flat-faces solids. Of these there are five: when viewed as boundaries, however, these five determine six distinct things: hence the six planets that revolve about the sun. This is also the reason why there are but six planets… sphere from Mysterium Cosmographicum (1596)]] sphere :… I have further shown that the regular solids fall into two groups: three in one, and two in the other. To the larger group belongs, first of all, the Cube, then the Pyramid, and finally the Dodecahedron. To the second group belongs, first, the Octahedron, and second, the Icosahedron. That is why the most important portion of the universe, the Earth—where God's image is reflected in man—separates the two groups. For, as I have proved next, the solids of the first group must lie beyond the earth's orbit, and those of the second group within… Thus I was led to assign the Cube to Saturn, the Tetrahedron to Jupiter, the Dodecahedron to Mars, the Icosahedron to Venus, and the Octahedron to Mercury… To emphasize his theory, Kepler envisaged an impressive model of the universe which shows a cube, inside a sphere, with a tetrahedron inscribed in it; another sphere inside it with a dodecahedron inscribed; a sphere with an icosahedron inscribed inside; and finally a sphere with an octahedron inscribed. Each of these celestial spheres had a planet embedded within them, and thus defined the planet's orbit. In his 1619 book, Harmonice Mundi or Harmony of the Worlds, as well as the aforementioned Mysterium Cosmographicum, he also made an association between the Platonic solids with the classical conception of the elements: the tetrahedron was the form of fire, the octahedron was that of air, the cube was earth, the icosahedron was water, and the dodecahedron was the cosmos as a whole or ether. There is some evidence this association was of ancient origin, as Plato tells of one Timaeus of Locri who thought of the Universe as being enveloped by a gigantic dodecahedron while the other four solids represent the "elements" of fire, air, earth, and water. In 1975, nine years after its founding, the College for Social and Economic Sciences Linz (Austria) was renamed Johannes Kepler University Linz in honor of Johannes Kepler, since he wrote his magnum opus harmonice mundi in Linz. To his disappointment, Kepler's attempts to fix the orbits of the planets within a set of polyhedrons never worked out, but it is a testimony to his integrity as a scientist that when the evidence mounted against the cherished theory he worked so hard to prove, he abandoned it. His most significant achievements came from the realization that the planets moved in elliptical, not circular, orbits. This realization was a direct consequence of his failed attempt to fit the planetary orbits within polyhedra. Kepler's willingness to abandon his most cherished theory in the face of precise observational evidence also indicates that he had a very modern attitude to scientific research. Kepler also made great steps in trying to describe the motion of the planets by appealing to a force which resembled magnetism, which he believed emanated from the sun. Although he did not discover gravity, he seems to have attempted to invoke the first empirical example of a universal law to explain the behaviour of both earthly and heavenly bodies.

Astrology

Kepler disdained astrologers who pandered to the tastes of the common man without knowledge of the abstract and general rules, but he saw compiling prognostications as a justified means of supplementing his meagre income. Yet, it would be a mistake to take Kepler's astrological interests as merely pecuniary. As one historian, John North, put it, 'had he not been an astrologer he would very probably have failed to produce his planetary astronomy in the form we have it.' Kepler believed in astrology in the sense that he was convinced that astrological aspects physically and really affected humans as well as the weather on earth. He strove to unravel how and why that was the case and tried to put astrology on a surer footing, which resulted in the On the more certain foundations of astrology (1601), in which, among other technical innovations, he was the first to propose the quincunx aspect. In The Intervening Third Man, or a warning to theologians, physicians and philosophers (1610), posing as a third man between the two extreme positions for and against astrology, Kepler advocated that a definite relationship between heavenly phenomena and earthly events could be established. At least 800 horoscopes and natal charts drawn up by Kepler are still extant, several of himself and his family, accompanied by some unflattering remarks. As part of his duties as district mathematician to Graz, Kepler issued a prognostication for 1595 in which he forecast a peasant uprising, Turkish invasion and bitter cold, all of which happened and brought him renown. Kepler is known to have compiled prognostications for 1595 to 1606, and from 1617 to 1624. As court mathematician, he explained to Rudolf II the horoscopes of the Emperor Augustus and Muhammad, and gave astrological prognosis for the outcome of a war between the Republic of Venice and Paul V. In the On the new star (1606) Kepler explicated the meaning of the new star of 1604 as the conversion of America, downfall of Islam and return of Christ. The De cometis libelli tres (1619) is also replete with astrological predictions.

Kepler on God

"I was merely thinking God's thoughts after him. Since we astronomers are priests of the highest God in regard to the book of nature," wrote Kepler, "it benefits us to be thoughtful, not of the glory of our minds, but rather, above all else, of the glory of God."

Writings by Kepler

astrological
- Mysterium cosmographicum (The Cosmic Mystery) (1596)
- Astronomiae Pars Optica (The Optical Part of Astronomy) (1604)
- De Stella nova in pede Serpentarii (On the New Star in Ophiuchus's Foot) (1604)
- Astronomia nova (New Astronomy) (1609)
- Dioptrice (Dioptre) (1611)
- Nova stereometria doliorum vinariorum (New Stereometry of wine barrels) (1615)
- Epitome astronomiae Copernicanae (published in three parts from 1618-1621)
- Harmonice Mundi (Harmony of the Worlds) (1619)
- Tabulae Rudolphinae (1627)
- Somnium (The Dream) (1634) - considered the first precursor of science fiction.

Kepler in fiction

- John Banville: Kepler: a novel. London: Secker & Warburg, 1981 ISBN 0-436-03264-3 (and later eds.). Also published: Boston, MA:Godine, 1983 ISBN 0-87923-438-5. Arthur Koestler : The Sleepwalkers; A History of Man's Changing Vision of the Universe. Publihed by Hutchinson. Synopsis; a profound outline of astrology from Pythagoras to Newton with weight given on Kepler.

Machines named in Kepler's honor

Kepler Space Observatory, a solar-orbiting, planet-hunting telescope due to be launched by NASA in 2008.

External links

- [http://posner.library.cmu.edu/Posner/books/annotation.cgi?call=520_K38PN Annotation: Posner Family Collection in Electronic Format] Harmonices mvndi The Harmony of the Worlds in fulltext facsimile in Latin
- Full text of [http://www.gutenberg.net/etext/12406 Kepler] by Walter W. Bryant, from Project Gutenberg
- [http://www.skyscript.co.uk/kepler.html Kepler and the "Music of the Spheres"]
- [http://www.dmoz.org/Science/Astronomy/History/People/Kepler,_Johannes/ Johannes Kepler Directory]
- [http://www.depauw.edu/sfs/backissues/8/christianson8art.htm Gale E. Christianson- Kepler's Somnium: Science Fiction and the Renaissance Scientist]
- Kepler, Johannes Kepler, Johannes Kepler, Johannes Kepler, Johannes Kepler, Johannes Kepler, Johannes Kepler, Johannes Kepler, Johannes Kepler, Johannes als:Johannes Kepler ko:요하네스 케플러 ja:ヨハネス・ケプラー

Astrophysics

] Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties (luminosity, density, temperature and chemical composition) of astronomical objects such as stars, galaxies, and the interstellar medium, as well as their interactions. The study of cosmology is theoretical astrophysics at the largest scales. Because it is a very broad subject, astrophysicists typically apply many disciplines of physics including, but not limited to, mechanics, electromagnetism, statistical mechanics, thermodynamics, quantum mechanics, relativity, nuclear and particle physics, and atomic and molecular physics. In practice, modern astronomical research involves a substantial amount of physics. The name of a university's department ("astrophysics" or "astronomy") often has to do more with the department's history than with the contents of the programs.

History

Although astronomy is as old as recorded history, it was long separated from the study of physics. In the Aristotelian worldview, the celestial pertained to perfection—bodies in the sky being perfect spheres moving in perfectly circular orbits—while the earthly pertained to imperfection; these two realms were seen as unrelated. For centuries, the apparently common-sense view that the Sun and other planets went round the Earth went basically unquestioned, until Nicolaus Copernicus suggested in the 16th century that the Earth and all the other planets in the Solar System orbited the Sun. This idea had been around, though, for nearly 2000 years when Aristarchus first suggested it, but not in such a nice mathematical model. Galileo Galilei made quantitative measurements central to physics, but in astronomy his observation did not have astrophysical significance. The availability of accurate observational data led to research into theoretical explanations for the observed behavior. At first, only ad-hoc rules were discovered, such as Kepler's laws of planetary motion, discovered at the start of the 17th century. Later that century, Isaac Newton, bridged the gap between Kepler's laws and Galileo's dynamics, discovering that the same laws that rule the dynamics of objects on earth rules the motion of planets and the moon. Celestial mechanics, the application of Newtonian gravity and Newton's laws to explain Kepler's laws of planetary motion, was the first unification of astronomy and physics. After Isaac Newton published his Principia, maritime navigation was transformed. Starting around 1670, the entire world was measured using essentially modern latitude instruments and the best available clocks. The needs of navigation provided a drive for progressively more accurate astronomical observations and instruments, providing a background for ever more available data for scientists. At the end of the 19th century it was discovered that, when decomposing the light from the Sun, a multitude of spectral lines were observed (regions where there was less or no light). Experiments with hot gases showed that the same lines could be observed in the spectra of gases, specific lines corresponding to unique chemical elements. In this way it was proved that the chemical elements found in the Sun (chiefly hydrogen) were also found on Earth. Indeed, the element helium was first discovered in the spectrum of the sun and only later on earth, hence its name. During the 20th century, spectroscopy (the study of these spectral lines) advanced, particularly as a result of the advent of quantum physics that was necessary to understand the astronomical and experimental observations. See also:
- Timeline of knowledge about galaxies, clusters of galaxies, and large-scale structure
- Timeline of white dwarfs, neutron stars, and supernovae
- Timeline of black hole physics
- Timeline of gravitational physics and relativity

Observational astrophysics

Most astrophysical processes cannot be reproduced in laboratories on Earth. However, there is a huge variety of astronomical objects visible all over the electromagnetic spectrum. The study of these objects through passive collection of data is the goal of observational astrophysics. The equipment and techniques required to study an astrophysical phenomenon can vary widely. Many astrophysical phenomena that are of current interest can only be studied by using very advanced technology and were simply not known until very recently. The majority of astrophysical observations are made using the electromagnetic spectrum.
- Radio astronomy studies radiation with a wavelength greater than a few millimeters. Radio waves are usually emitted by cold objects, including interstellar gas and dust clouds. The cosmic microwave background radiation is the redshifted light from the Big Bang. Pulsars were first detected at microwave frequencies. The study of these waves requires very large radio telescopes.
- Infrared astronomy studies radiation with a wavelength that is too long to be visible but shorter than radio waves. Infrared observations are usually made with telescopes similar to the usual optical telescopes. Objects colder than stars (such as planets) are normally studied at infrared frequencies.
- Optical astronomy is the oldest kind of astronomy. Telescopes paired with a charge-coupled device or a spectroscope are the most common instruments used. The Earth's atmosphere interferes somewhat with optical observations, so adaptive optics and space telescopes are used to obtain the highest possible image quality. In this range, stars are highly visible, and many chemical spectra can be observed to study the chemical composition of stars, galaxies and nebulae.
- Ultraviolet, X-ray and gamma ray astronomy study very energetic processes such as binary pulsars, black holes, magnetars, and many others. These kinds of radiation do not penetrate the Earth's atmosphere well, so they are studied with space-based telescopes such as RXTE, the Chandra X-ray Observatory and the Compton Gamma Ray Observatory. Other than electromagnetic radiation, few things may be observed from the Earth that originate from great distances. A few gravitational wave observatories have been constructed, but gravitational waves are extremely difficult to detect. Neutrino observatories have also been built, primarily to study our Sun. Cosmic rays consisting of very high energy particles can be observed hitting the Earth's atmosphere. Observations can also vary in their time scale. Most optical observations take minutes to hours, so phenomena that change faster than this cannot readily be observed. However, historical data on some objects is available spanning centuries or millennia. On the other hand, radio observations may look at events on a millisecond timescale (millisecond pulsars) or combine years of data (pulsar deceleration studies). The information obtained from these different timescales is very different. The study of our own Sun has a special place in observational astrophysics. Due to the tremendous distance of all other stars, the Sun can be observed in a kind of detail unparalleled by any other star. Our understanding of our own sun serves as a guide to our understanding of other stars. The topic of how stars change, or stellar evolution, is often modelled by placing the varieties of star types in their respective positions on the Hertzsprung-Russell diagram, which can be viewed as representing the state of a stellar object, from birth to destruction. The material composition of the astronomical objects can often be examined using:
- Spectroscopy
- Radio astronomy

Theoretical astrophysics

Theoretical astrophysicists create and evaluate models to reproduce and properly predict observations. They use a wide variety of tools which include analytical models (for example, polytropes to approximate the behaviors of a star) and computational numerical simulations. A few examples of this process: Dark matter and dark energy are the current leading topics in astrophysics, as their discovery and controversy originated during the study of the galaxies.

Astrodynamics

Main article: Astrodynamics Astrodynamics is the branch of celestial mechanics concerned with the motion of rockets, satellites and missiles. It is based upon Newton's laws of motion, and law of universal gravitation. The formula for escape velocity is defined in astrodynamics as:

v\geq\sqrt

Astrodynamics is also used to compute the position of a satellite at a given time, a problem first solved by Johannes Kepler, who computed the formula:

MT = E - e \sin E \;

This formula is commonly referred to as Kepler's equation, and can compute the time required for a satellite to travel from periapsis P to a given point S. Modern techniques for computing time-of-flight include the patched conic approximation, where one must choose the one dominant gravitating body in each region of space through which the trajectory will pass, and to model only that body's effects in that region, or the universal variable formulation.

Astrophysicists

Main article: List of astrophysicists

References

Herman Roth, "A Slowly Contracting or Expanding Fluid Sphere and its Stability" Phys. Rev. 39, 525–529 (1932) [Issue 3 – 1 February 1932 ]

External links

- [http://home.slac.stanford.edu/ppap.html Stanford Linear Accelerator Center, Stanford, California] Category:Astronomy Category:Applied and interdisciplinary physics ms:Astrofizik ja:天体物理学 simple:Astrophysics

Observation

:For the railroad use of the term observation, see observation car. ---- Observation in its most basic version means watching something and taking note of anything it does - although vision is the most often used, all the senses can be used to observe. For instance, you might observe a bird flying by watching it closely with binoculars. Direct observations form the foundations of all natural sciences. Some disciplines, such as biology and astronomy, have their historical basis in observations by amateurs - the participation of hobbyists is explained by the fact that there is pleasure in observation.

The role of Observation in the Scientific Method

The scientific method includes the following steps: # 'observe' a phenomenon, #'Hypothesize' an explanation for the phenomenon, #'predict' a logical consequence of the guess, #'test' the prediction, and #'review' for any mistakes. Observation plays a role in the first and fourth steps in the above list. Reliance is placed upon the five physical senses: visual perception, hearing (sense), taste, feeling, and olfaction, and upon measurement techniques. It is therefore understood that there are always certain limitations in making observations.

Example - The Big Bang

In cosmology, the original observation was that we seem to live in a firmament. The sun seemed to rise and set, travelling on a huge transparent bowl which was set around our world. Various paradigms which explained our world, came and went, but the universe seemed static. Even Einstein believed this.

Observation: Hubble's redshift

In the 1920s Edwin Hubble of Mount Wilson observatory [http://www.mtwilson.edu/his/art/g1a4.htm], observed that the galaxies, on the whole, were moving away from each other. Thus we live in an 'expanding universe'. The speed of expansion was apparently constant (Hubble's 'constant'), as evidenced by light from the galaxies, which was doppler-shifted in color toward the red side of the spectrum. Einstein correspondingly modified his field equation. See Cosmological constant

Hypothesis about the abundance of the elements

If the universe is expanding, then it must have been much smaller and therefore hotter and denser in the past. George Gamow hypothesized that the abundance of the elements in the Periodic Table of the Elements, might be accounted for by nuclear reactions in a hot dense universe. He was disputed by Fred Hoyle, who invented the term 'Big Bang' to disparage it. Fermi and others noted that this process would have stopped after only the light elements were created, and thus did not account for the abundance of heavier elements. Gamow's prediction: One consequence of this hypothesis was a 5–10 kelvin black body radiation temperature for the universe, after it cooled during the expansion.

Observation: the microwave background

In 1965, Arno A. Penzias and Robert W. Wilson announced that microwave radiation was surrounding us in all directions, at a level which was of the order of magnitude predicted by Gamow. Penzias and Wilson got the Nobel Prize for this discovery.

Big Bang Hypothesis now corroborated

After this piece of evidence, Gamow's hypothesis was now more likely. The age of the universe is currently estimated to be 13.7 billion years after the Big Bang.

Current observations

More refined measurements, such as those from the COBE satellite, are best fit by radiation from a pure 2.7 kelvin black body.

Future observations

It is, of course, entirely possible that observations made in the future may enable a different understanding. People of the future, looking back on the Big Bang theory may, perhaps, regard it with as much derision as the people of today regard the apparent geocentric universe of previous observations. All that is possible is to keep looking at the evidence as it comes in. Reference: J.A. Peacock, A.F. Heavens, A.T. Davies (eds.), 1989. Physics of the Early Universe. Proceedings of the 36th Scottish Universities Summer School in Physics (SUSSP). ISBN 0905945190.

The role of Observation in Philosophy

"Observe always that everything is the result of a change, and get used to thinking that there is nothing Nature loves so well as to change existing forms and to make new ones like them." Meditations. iv. 36. -Marcus Aurelius Observation in philosophical terms is the process of filtering sensory information through the thought process. Input is received via hearing, sight, smell, taste, or touch and then analyzed through either rational or irrational thought. You see a man beat his wife; you observe that such an action is either good or bad. Deductions about what behaviors are good or bad may be based on preferences about building relationships, or study of the consequences resulting from the observed behavior. With the passage of time, impressions stored in the consciousness about many related observations, together with the resulting relationships and consequences, permit the individual to build a construct about the moral implications of behavior. The defining characteristic of observation is that it involves drawing conclusions, as well as building personal views about how to handle similar situations in the future, rather than simply registering that something has happened. Observing is part of the process of developing a morality.

Hobbies that involve observation

Hobbies that involve observation depend for their interest on items being observed. A knowledge of these items and their habitats will develop over time in the observer, who may draw upon the experiences of others as conveyed in books or websites or by word of mouth. Most such hobbies involve classification of the items seen, with the precision and reliability of such classifications generally increasing over time. Depending on the geographic dispersal of the creatures or things being observed, pursuit of the hobby might well require or entice travel. When spotting natural creatures, an understanding of their migration patterns may be essential. Specific creatures may only be visible in particular places at certain times of the year. The creatures that can be observed include humans, e.g. from a sidewalk café. This may be especially interesting in an exotic country, or at a place where exotic people pass. Also one may like to look at sexually attractive people. There are parallels in those hobbies relating to man-made items. International political events may sometimes generate a gathering of VIP aircraft, and an international football match may cause a sudden influx of charter airliners to the region where the match is played. There is likely to be a social aspect to such hobbies, since fellow enthusiasts will normally alert a hobbyist to forthcoming (or even current) opportunities to witness unusual items within the scope of the shared pastime. New technologies such as mobile telephones and the Internet have clearly increased the opportunities for passing such information between fellow enthusiasts when it is timely.

Tycho Brahe

Tycho Brahe (born Tyge Ottesen Brahe) (December 14, 1546 – October 24 1601) was a Danish nobleman known primarily for his work as an astronomer and an astrologer (the two were highly related in his day), as well as an alchemist. He was granted an estate on the island of Hven and the funding to build the Uraniborg, an early research institute, where he built large astronomical instruments and took many careful measurements. As an astronomer, Tycho worked to combine what he saw as the geometrical benefits of the Copernican system with the philosophical benefits of the Ptolemaic system into his own model of the universe, the Tychonian system. His best known assistant was Johannes Kepler, who would later use Tycho's astronomical information to develop his own theories of astronomy. He is universally referred to as "Tycho" rather than by his surname "Brahe." Apparently his contemporaries did so and the usage has persisted. His name is pronounced , according to transcription in the International Phonetic Alphabet (in English, it is thought that Teeko is a close approximation of its probable pronunciation). While credited with the most accurate astronomical observations of his time, he was unable to carry the implications of the voluminous data he collected to its logical consequences; it was his assistant, Johannes Kepler, whose superior mathematical faculty would enable the full interpretation of his master's acute observations and allow for the discovery of the laws of planetary motion (it should also be noted that none before Tycho had attempted to make so many redundant observations, and the mathematical tools to take advantage of them had not yet been developed). Nonetheless, his mark in science should not be understated, for he had done what others before him were unable or unwilling to do — to catalogue the planets and stars with enough accuracy so as to determine whether the Ptolemaic or Copernican system was more valid in describing the heavens.

Early years

Tycho Brahe was born Tyge Ottesen Brahe (de Knudstrup), adopting the Latinised form Tycho at around age fifteen (sometimes written Tÿcho). He is often misnamed Tycho de Brahe. He was born at his family's ancestral seat of Knudstrup Castle, Denmark to Otte Brahe and Beate Bille. His twin brother was stillborn (Tycho wrote a Latin ode (Wittendorf 1994, p. 68) to his dead twin which was printed as his first publication in 1572). He also had two sisters, one older (Kirstine Brahe) and one younger (Sophie Brahe). Otte Brahe, Tycho's father, a nobleman, was an important figure in the Danish King's court. Beate Bille, Tycho's mother, also came from an important family which had produced leading churchmen and politicians. Tycho later wrote that when he was around two, his uncle, Danish nobleman Jørgen Brahe, ... without the knowledge of my parents took me away with him while I was in my earliest youth. Apparently this did not lead to any disputes nor did his parents attempt to get him back. Tycho lived with his childless uncle and aunt, Jørgen Brahe and Inger Oxe, in the Tostrup Castle until he was six years old. Around 1552 his uncle was given the command of Vordingborg Castle to which they moved, and where Tycho began a Latin education until he was 12 years old. On April 19 1559, Tycho began his studies at the University of Copenhagen. There, following the wishes of his uncle, he studied law but also studied a variety of other subjects and became interested in astronomy. It was, however, the eclipse which occurred on August 21 1560, particularly the fact that it had been predicted, that so impressed him that he began to make his own studies of astronomy helped by some of the professors. He purchased an ephemeris and books such as Sacrobosco's Tractatus de Sphaera, Apianus's Cosmographia seu descriptio totius orbis and Regiomontanus's De triangulis omnimodis. I've studied all available charts of the planets and stars and none of them match the others. There are just as many measurements and methods as there are astronomers and all of them disagree. What's needed is a long term project with the aim of mapping the heavens conducted from a single location over a period of several years. — Tycho Brahe, 1563 (aged 17). Tycho realized that progress in the science of astronomy could be achieved not by occasional haphazard observations, but only by systematic and rigorous observation, night after night, and by using instruments of the highest accuracy obtainable. He was able to improve and enlarge the existing instruments, and construct entirely new ones. Tycho's naked eye measurements of planetary parallax were accurate to the arcminute. (These measurements became the possessions of Kepler following Tycho's death.) While a student, Tycho lost part of his nose in a duel with broadswords with Manderup Parsbjerg, a fellow Danish nobleman. This occurred in the Christmas season of 1566, after a fair amount of drinking, while the just turned 20-year-old Tycho was studying at the University of Rostock in Germany. Attending a dance at a professor's house, he quarrelled with Parsbjerg. A subsequent duel (in the dark) resulted in Tycho losing the bridge of his nose. A consequence of this was that Tycho developed an interest in medicine and alchemy. For the rest of his life, he was said to have worn a replacement made of silver and gold blended into a flesh tone, and used an adhesive balm to keep it attached. In 1901, though, Tycho's tomb was reopened and his remains were examined by medical experts. The nasal opening of the skull was rimmed with green, a sign of exposure to copper, not silver or gold. Some historians have speculated that he wore a number of different prosthetics for different occasions, noting that a copper nose would have been more comfortable and less heavy than one of precious metals.

Death of his father

His foster father, uncle Jørgen Brahe, had already died in 1565 of pneumonia after rescuing Frederick II of Denmark from drowning. In April 1567, Tycho returned home from his travels, where his father wanted him to take up law, but was allowed to make trips to Rostock, then on to Augsburg (where he built a great quadrant), Basel, and Freiburg. He was informed about his father's illness at the end of 1570, so he returned to Knudstrup, where his father died in May 1571. Soon after, his other uncle Steen Bille helped him build an observatory and alchemical laboratory at Herrevad Abbey.

Family life

In 1572, in Knudstrup, Tycho fell in love with Kirsten Jørgensdatter, a commoner whose father, Pastor Jorgen Hansen, was the Lutheran clergyman of Knudstrup's village church. Under Danish law, when a nobleman and a common woman lived together openly as husband and wife, and she wore the keys to the household at her belt like any true wife, their alliance became a binding morganatic marriage after three years. The husband retained his noble status and privileges; the wife remained a commoner. Their children were legitimate in the eyes of the law, but they were commoners like their mother and could not inherit their father's name, coat of arms, or land property. (Skautrup 1941, pp. 24-5) Kirsten Jørgensdatter gave birth to their daughter, Kirstine (named after Tycho's late sister who died at 13) on October 12, 1573.

Nova

On November 11, 1572, Tycho observed (from Herrevad Abbey) a very bright star which unexpectedly appeared in the constellation Cassiopeia, now named SN 1572. Since it had been maintained since antiquity that the world of the fixed stars was eternal and unchangeable (a fundamental axiom of the Aristotelian world view: celestial immutability), other observers held that the phenomenon was something in the Earth's atmosphere. Tycho, however, observed that the parallax of the object did not change from night to night, suggesting that the object was far away. Tycho argued that a nearby object should appear to shift its position with respect to the background. He published a small book, De Stella Nova (1573), thereby coining the term nova for a "new" star (we now know that Tycho's star was a supernova). This discovery was decisive for his choice of astronomy as a profession. Tycho was strongly critical of those who dismissed the implications of the astronomical appearance, writing in the preface to De Stella Nova: "O crassa ingenia. O caecos coeli spectatores" ("Oh thick wits. Oh blind watchers of the sky"). Tycho's discovery was the inspiration for Edgar Allan Poe's poem, Al Aaraaf.

Heliocentrism

Kepler tried, but was unable, to persuade Tycho to adopt the heliocentric model of the solar system. Tycho believed in a modified geocentric model known as the Tychonian system, for the same reasons that he argued that the supernova of 1572 was not near the Earth. He argued that if the Earth were in motion, then nearby stars should appear to shift their positions with respect to background stars. In fact, this effect of parallax does exist; it could not be observed with the naked eye, or even with the telescopes of the next two hundred years, because even the nearest stars are much more distant than most astronomers of the time believed possible.

Uraniborg, Stjerneborg and Benátky nad Jizerou

telescopes King Frederick II of Denmark and Norway, impressed with Tycho's 1572 observations, financed the construction of two observatories for Tycho on the island of Hven in Copenhagen Sound. These were Uraniborg and Stjerneborg. Uraniborg also had a laboratory for his alchemical experiments. Because Tycho disagreed with Christian IV, the new king of his country, he moved to Prague in 1599. Sponsored by Rudolf II, the Holy Roman Emperor, he built a new observatory (in a castle in Benátky nad Jizerou, 50 km from Prague) and worked there until his death. In return for their support, Tycho's duties included preparing astrological charts and predictions for his patrons on events such as births, weather forecasting and astrological interpretations of significant astronomical events such as the comet of 1577 and the supernova of 1572.

Tycho and astronomy

1572 Tycho was the preeminent observational astronomer of the pre-telescopic period, and his observations of stellar and planetary positions achieved unparalleled accuracy for their time. For example, Tycho measured Earth's axial tilt as 23 degrees and 31.5 minutes, which he claimed to be more acurate than Copernicus by 3.5 minutes. After his death, his records of the motion of the planet Mars enabled Kepler to discover the laws of planetary motion, which provided powerful support for the Copernican heliocentric theory of the solar system. Tycho himself was not a Copernican, but proposed a system in which the planets other than Earth orbited the Sun while the Sun orbited the Earth. His system provided a safe position for astronomers who were dissatisfied with older models but were reluctant to accept the Earth's motion. It gained a considerable following after 1616 when Rome decided officially that the heliocentric model was contrary to both philosophy and Scripture, and could be discussed only as a computational convenience that had no connection to fact. His system also offered a major innovation: while both the geocentric model and the heliocentric model as set forth by Copernicus relied on the idea of transparent rotating crystalline spheres to carry the planets in their orbits, Tycho eliminated the spheres entirely. He was aware that a star observed near the horizon appears with a greater altitude than the real one, due to atmospheric refraction, and he worked out tables for the correction of this source of error. To perform the huge number of products needed to produce much of his astronomical data, Tycho relied heavily on the then-new technique of prosthaphaeresis, an algorithm for approximating products based on trigonometric identities that predated logarithms.

Tycho and Astrology

Like the fifteenth century astronomer Regiomontanus, Tycho Brahe appears to have accepted astrological prognostications on the principle that the heavenly bodies undoubtedly influenced (yet did not determine) terrestrial events, but expressed skepticism about the multiplicity of interpretative schemes, and increasingly preferred to work on establishing a sound mathematical astronomy. Two early tracts, one entitled Against Astrologers for Astrology, and one on a new method of dividing the sky into astrological houses, were never published and are unfortunately now lost. Tycho also worked in the area of weather prediction, produced astrological interpretations of the supernova of 1572 and the comet of 1577, and furnished his patrons Frederick II and Rudolph II with nativities and other predictions (thereby strengthening the ties between patron and client by demonstrating value). An astrological worldview was fundamental to Tycho's entire philosophy of nature. His interest in alchemy, particularly the medical alchemy associated with Paracelsus, was almost as long-standing as his study of astrology and astronomy simultaneously, and Uraniborg was constructed as both observatory and laboratory. In an introductory oration to the course of lectures he gave in Copenhagen in 1574, Tycho defended astrology on the grounds of correspondences between the heavenly bodies, terrestrial substances (metals, stones etc.) and bodily organs. He was later to emphasise the importance of studying alchemy and astrology together with a pair of emblems bearing the mottoes: Despiciendo suspicio ("By looking down I see upward") and Suspiciendo despicio ("By looking up I see downward"). As several scholars have now argued, Tycho's commitment to a relationship between macrocosm and microcosm even played a role in his rejection of Copernicanism and his construction of a third world-system.

Tycho's elk

Tycho often held large social gatherings in his castle, as he was a member of the nobility. He was said to own 1% of the entire wealth of Denmark at one point in the 1580s. Pierre Gassendi wrote that Tycho also had a tame elk, and that his mentor the Landgraf Wilhelm of Hesse-Kassel asked about an animal faster than a deer. Tycho replied writing there were none, but he could send his tame elk. When Wilhelm replied he would accept one in exchange for a horse, Tycho replied with the sad news that the elk just died on a visit to entertain a nobleman at Landskrona. Apparently during dinner the elk had drunk a lot of beer and fell down the stairs, and died.

Tycho's death

Tycho died on October 24 1601, several days after straining his bladder during a banquet. It has been said that to leave the banquet before it concluded, would be the height of bad manners and so he remained. His weakened state allowed an infection to invade his body and lead ultimately to his death. He was succeeded as Imperial Mathematicus by Kepler, two days later. However, recent investigations have suggested that Tycho did not die from urinary problems but most likely from mercury poisoning: toxic levels of it have been found in his hair and hair-roots. Tycho may have poisoned himself unintentionally by imbibing some mercury-containing medicine. Some have even speculated that Tycho may have been murdered, though there is no solid evidence for this. Tycho Brahe's body is currently interred in a tomb in the Church of Our Lady of Tyn on Old Town Square near Prague Astronomical Clock in Prague.

References

- Skautrup, Peter, 1941 Den jyske lov: Text med oversattelse og ordbog. Aarhus: Universitets-forlag.
- Wittendorff, ALex. 1994. Tyge Brahe. Copenhagen: G. E. C. Gad.
- from a translation from Gassendi
- page 210 refers to Tycho's elk as cited by:
-
-

External links

- [http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Brahe.html Brahe, Tycho] MacTutor History of Mathemtics
- [http://digital.lib.lehigh.edu/planets/brahe.php?num=F&exp=false&lang=lat&CISOPTR=404&limit=brahe&view=full Astronomiae instauratae mechanica, 1602 edition] - Full digital facsimile, Lehigh University.
- [http://www.sil.si.edu/DigitalCollections/HST/Brahe/brahe.htm Astronomiae instauratae mechanica, 1602 edition] - Full digital facsimile, Smithsonian Institution.
- [http://www.kb.dk/elib/lit/dan/brahe/index-en.htm Astronomiae instauratae mechanica, 1598 edition] - Full digital facsimile, the Royal Library, Denmark. Includes Danish and English translations.

Named after Tycho

- Tycho crater on the Moon.
- Tycho Brahe crater on Mars.
- The name of American electronic musician Tycho: [http://www.tychomusic.com Tycho] (http://www.tychomusic.com/)
- The name of an Australian powersynth band: [http://www.tycho.com.au Tycho Brahe] (http://www.tycho.com.au/)
- An old name of an Irish synthpop band, now called [http://www.tychonaut.com Tychonaut] (http://www.tychonaut.com).
- Tycho Brahe, pseudonym of Jerry Holkins and a character from the popular webcomic Penny Arcade.
- The AI Tycho from Bungie's computer game Marathon.
- Brother-Captain Tycho of the Blood Angels Chapter of Space Marines in Games Workshop's sci-fi tabletop wargame, Warhammer 40,000.
- a Scandlines ferry connecting Helsingør in Denmark and Helsingborg in Sweden. Tycho Brahe Tycho Brahe Category:Brahe Tycho Brahe Tycho Brahe Tycho Brahe Tycho Brahe ko:튀코 브라헤 ja:ティコ・ブラーエ

Johannes Kepler

Life

Death

On November 15, 1630 Kepler died of a fever in Regensburg. In 1632, only two years after his death, his grave was demolished by the Swedish army in the Thirty Years' War.

Kepler's laws of planetary motion

Kepler's first law

Kepler's second law

Kepler's third law

Accuracy and limitations

Connection to Newton's laws and conservation laws

Kepler's first law

Kepler's second law

Kepler's third law

Proving Kepler's third law

Solution for the motion as a function of time

See also

External links

Johannes Kepler

Life

Death

Work

Scientific work

Kepler's laws

1604 supernova

Other scientific and mathematical work

Mysticism and astrology

Mysticism

Astrology

Kepler on God

Writings by Kepler

See also

Kepler in fiction

Machines named in Kepler's honor

External links

Astrophysics

History

Observational astrophysics

Theoretical astrophysics

Astrodynamics

Astrophysicists

References

See also

External links

Observation

The role of Observation in the Scientific Method

Example - The Big Bang

Observation: Hubble's redshift

Hypothesis about the abundance of the elements

Observation: the microwave background

Big Bang Hypothesis now corroborated

Current observations

Future observations

The role of Observation in Philosophy

Hobbies that involve observation

See also

Tycho Brahe

Early years

Death of his father

Family life

Nova

Heliocentrism

Uraniborg, Stjerneborg and Benátky nad Jizerou

Tycho and astronomy

Tycho and Astrology

Tycho's elk

Tycho's death

Further reading

References

External links

Named after Tycho

Johannes Kepler

Life

Death

Work

Scientific work

Kepler's laws

1604 supernova

Other scientific and mathematical work

Mysticism and astrology

Mysticism