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Johannes Kepler (`There was an error working with the wiki: Code[6]`

`There was an error working with the wiki: Code[7]`

, based on his works `There was an error working with the wiki: Code[66]`

, `There was an error working with the wiki: Code[67]`

and the textbook `There was an error working with the wiki: Code[68]`

.

Through his career Kepler was a mathematics teacher at a Graz seminary school (later the `There was an error working with the wiki: Code[8]`

, mathematics teacher in `There was an error working with the wiki: Code[9]`

. He also did fundamental work in the field of `There was an error working with the wiki: Code[69]`

and helped to legitimize the telescopic discoveries of his contemporary `There was an error working with the wiki: Code[70]`

.

He is sometimes referred to as "the first `There was an error working with the wiki: Code[10]`

", although `There was an error working with the wiki: Code[71]`

also referred to him as the last scientific `There was an error working with the wiki: Code[72]`

.

Kepler was born on `There was an error working with the wiki: Code[73]`

`There was an error working with the wiki: Code[74]`

at the `There was an error working with the wiki: Code[75]`

of `There was an error working with the wiki: Code[76]`

(now part of the `There was an error working with the wiki: Code[77]`

in the German state of `There was an error working with the wiki: Code[78]`

, 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 `There was an error working with the wiki: Code[79]`

, and he left the family when Johannes was five years old. He was believed to have died in the war in the Netherlands. His mother, an inn-keeper's daughter, was a healer and herbalist who was later tried for `There was an error working with the wiki: Code[80]`

. Born prematurely, Johannes claimed to have been a weak and sickly child. Despite his ill health, he was precociously brilliant. As a child, he often impressed travelers at his grandfather's inn with his phenomenal mathematical faculty.

He was introduced to astronomy/astrology at an early age, and he developed a love for it that would span his entire life. At age five, he observed the `There was an error working with the wiki: Code[81]`

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 `There was an error working with the wiki: Code[82]`

of 1580, recording that he remembered being "called outdoors" to see it and that the moon "appeared quite red". However, childhood `There was an error working with the wiki: Code[83]`

left him with weak vision, limiting him to the mathematical rather than observational aspects of astronomy.

In 1589, after moving through grammar school, Latin school, and lower and higher seminary in the Lutheran education system, Kepler began attending the `There was an error working with the wiki: Code[11]`

. Under the instruction of `There was an error working with the wiki: Code[12]`

he became a Copernican at that time, defending `There was an error working with the wiki: Code[84]`

from both a theoretical and theological perspective in student debates. Despite his desire to become a minister, near the end of his studies, Kepler was recommended for a position as teacher of mathematics and astronomy at the Protestant school in `There was an error working with the wiki: Code[85]`

, `There was an error working with the wiki: Code[86]`

. He accepted the position in April 1594, at the age of 23.

In Graz, Kepler began developing an original theory of cosmology based on the Copernican system, which was published in 1596 as Mysterium Cosmographicum&mdash`There was an error working with the wiki: Code[87]`

. In April 1597, Kepler married Barbara Müller. She died in 1611 and was outlived by two of Johannes's children and one by an earlier marriage. In December 1599, `There was an error working with the wiki: Code[88]`

wrote to Kepler, inviting Kepler to assist him at `There was an error working with the wiki: Code[89]`

outside `There was an error working with the wiki: Code[90]`

. Pressured to leave Graz by increasingly strict `There was an error working with the wiki: Code[91]`

policies restricting the religious practices and political rights of `There was an error working with the wiki: Code[92]`

, Kepler joined Tycho in 1600. After Tycho's death in 1601, Kepler was appointed Imperial Mathematician in his place, a post he would retain through the reigns of three `There was an error working with the wiki: Code[93]`

Emperors (from November 1601 to 1630).

As Imperial Mathematician, Kepler inherited Tycho's responsibility for the Emperor's `There was an error working with the wiki: Code[13]`

&mdashNew Astronomy. Astronomia Nova contained what would become the first and second `There was an error working with the wiki: Code[14]`

.

In October 1604, Kepler observed the `There was an error working with the wiki: Code[94]`

which was subsequently named `There was an error working with the wiki: Code[95]`

(a term which may also refer to the `There was an error working with the wiki: Code[96]`

). In 1611, Kepler published (as a letter to a friend) a monograph on the origins of `There was an error working with the wiki: Code[97]`

flakes, 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. In January 1612, the Emperor died. To escape the growing religious tension in Prague, Kepler took the post of Provincial Mathematician in `There was an error working with the wiki: Code[98]`

.

In 1615, Kepler married Susanna Ruettinger, with whom he would have several children.

In 1617, Kepler's mother Katharina was accused of being a witch in Leonberg. Beginning in August 1620 she was imprisoned for fourteen months. Thanks in part to the extensive legal defense drawn up by Kepler, she was released in October 1621 after failed attempts to convict her. However, she was subjected to territio verbalis, a graphic description of the `There was an error working with the wiki: Code[99]`

awaiting her as a witch, in a final attempt to make her confess. Throughout the trial, Kepler postponed his other work (on the `There was an error working with the wiki: Code[100]`

and a multi-volume astronomy textbook) to focus on his "harmonic theory". The result, published in 1619 as `There was an error working with the wiki: Code[101]`

("Harmony of the Worlds") contained the third law of planetary motion.

Kepler completed the last of seven volumes of his textbook Epitome of Copernican Astronomy in 1621, which brought together and extended his previous work and would become very influential in the acceptance of the Copernican system over the next century. In 1627 he completed the Rudolphine Tables, which provided accurately calculated future positions of the planets and allowed the prediction of rare astronomical events.

On `There was an error working with the wiki: Code[15]`

army in the `There was an error working with the wiki: Code[102]`

. Kepler had incidentally composed the epitaph for his own tombstone, which read:

:I measured the skies, now the shadows I measure,Sky-bound was the mind, earth-bound the body rests

Kepler lived in an era when there was no clear distinction between `There was an error working with the wiki: Code[16]`

within the `There was an error working with the wiki: Code[17]`

). He also incorporated religious arguments and reasoning into his work, such that the basis for many of his most important contributions was essentially theological (Barker & Goldstein, 2001).

Kepler was a Pythagoras mystic. He considered mathematical relationships to be at the base of all nature, and all creation to be an integrated whole. This was in contrast to the `There was an error working with the wiki: Code[18]`

notion that the Earth was fundamentally different from the rest of the universe, being composed of different substances and with different natural laws applying. In his attempt to discover universal laws, Kepler applied terrestrial physics to celestial bodies famously, his effort produced the three Laws of Planetary Motion. Kepler was also convinced that celestial bodies influence terrestrial events. One result of this belief was his correct assessment of the moon's role in generating the `There was an error working with the wiki: Code[103]`

, years before Galileo's incorrect formulation. Another was his belief that someday it would be possible to develop a "scientific astrology", despite his general disdain for most of the astrology of his time.

Kepler inherited from `There was an error working with the wiki: Code[104]`

a wealth of the most accurate raw data ever collected on the positions of the `There was an error working with the wiki: Code[105]`

s. 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 trying to understand the orbit of `There was an error working with the wiki: Code[106]`

, 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.

Unlike Brahe, Kepler held to the heliocentric model of the solar system. Starting from that framework, Kepler made twenty years of painstaking trial-and-error attempts at making some sense out of the data. He finally arrived at his `There was an error working with the wiki: Code[19]`

:

#Kepler's elliptical orbit law: The planets orbit the sun in elliptical orbits with the sun at one focus.

#Kepler's equal-area law: The line connecting a planet to the sun sweeps out equal areas in equal amounts of time.

#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 `There was an error working with the wiki: Code[20]`

and `There was an error working with the wiki: Code[107]`

.

Kepler first discovered his distance-cubed-over-time-squared (or 'third') law of planetary motion on March 8, 1618 but rejected the idea until May 15, 1618, when he verified his result. This result was published in his Harmonices Mundi (1619).

Kepler's primary contributions to `There was an error working with the wiki: Code[108]`

/`There was an error working with the wiki: Code[109]`

were his three laws of planetary motion. Kepler, a nearly blind though brilliant German mathematician, derived these laws, in part, by studying the `There was an error working with the wiki: Code[110]`

s of the keen-sighted Danish astronomer `There was an error working with the wiki: Code[111]`

. 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 `There was an error working with the wiki: Code[21]`

and `There was an error working with the wiki: Code[22]`

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 `There was an error working with the wiki: Code[23]`

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.

The `There was an error working with the wiki: Code[25]`

. There is no object at the other focus of a planet's orbit. The `There was an error working with the wiki: Code[26]`

than near `There was an error working with the wiki: Code[27]`

. Strictly speaking the paths of both objects form ellipses around their common `There was an error working with the wiki: Code[112]`

. As the mass of the star is so much greater than that of the planet, the centre of mass of the system is very close to that of the star and the orbit of the star will be very small.

Newton said 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 a is of the form

: {a} = \frac{d^2{r}}{dt^2} = f(r)\hat`There was an error working with the wiki: Code[1]`

.

Recall that in `There was an error working with the wiki: Code[113]`

s

:\frac{d{r}}{dt} = \dot r\hat`There was an error working with the wiki: Code[2]`

+ r\dot\theta\hat{\theta},

:\frac{d^2{r}}{dt^2} = (\ddot r - r\dot\theta^2)\hat`There was an error working with the wiki: Code[3]`

+ (r\ddot\theta + 2\dot r \dot\theta)\hat{\theta}.

In component form, we have

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

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

Substituting for \ddot \theta and \dot r in the second equation, we have

:r { d \dot\theta \over dt } + 2 {dr \over dt} \dot\theta = 0

which simplifies to

:\frac{d\dot\theta}{\dot\theta} = -2\frac{dr}{r}.

When integrated, this yields

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

: \log\ell = \log r^2 + \log\dot\theta,

:\ell = r^2\dot\theta,

for some constant \ell, which can be shown to be the `There was an error working with the wiki: Code[114]`

. Now we substitute. Let

:r = \frac{1}{u},

:\dot r = -\frac{1}{u^2}\dot u = -\frac{1}{u^2}\frac{d\theta}{dt}\frac{du}{d\theta}= -\ell\frac{du}{d\theta},

:\ddot r = -\ell\frac{d}{dt}\frac{du}{d\theta} = -\ell\dot\theta\frac{d^2u}{d\theta^2}= -\ell^2u^2\frac{d^2u}{d\theta^2}.

The equation of motion in the \hat`There was an error working with the wiki: Code[4]`

direction becomes

:\frac{d^2u}{d\theta^2} + u = - \frac{1}{\ell^2u^2}f\left(\frac{1}{u}\right).

Newton's law of gravitation relates the force per unit mass to the radial distance as

: f \left( {1 \over u} \right) = f(r)= - \, { GM \over r^2 } = - GM u^2

where G is the constant of universal gravitation and M is the mass of the star.

As a result,

:\frac{d^2u}{d\theta^2} + u = \frac{GM}{\ell^2}

This differential equation has the general solution:

:u = \frac{GM}{\ell^2} \bigg[ 1 + e\cos(\theta-\theta_0) \bigg] .

for arbitrary constants of integration e and ?0.

Replacing u with 1/r and letting ?0 = 0:

:r = { 1 \over u } = \frac{ \ell^2 / GM }{ 1+ e\cos\theta}

This is indeed the equation of a `There was an error working with the wiki: Code[28]`

e and 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.

A `There was an error working with the wiki: Code[29]`

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 one day to travel from `There was an error working with the wiki: Code[30]`

s A to B''. During this time, an imaginary line, from the Sun to the planet, will sweep out a roughly `There was an error working with the wiki: Code[31]`

area. This same amount of area will be swept every day regardless of where in its orbit the planet is. As a planet travels in its elliptical orbit, its `There was an error working with the wiki: Code[115]`

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 `There was an error working with the wiki: Code[116]`

, a planet must vary in `There was an error working with the wiki: Code[117]`

. 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.

Assuming Newton's laws of motion, we can show that Kepler's second law is consistent. By definition, the `There was an error working with the wiki: Code[118]`

{L} of a point mass with mass m and velocity {v} is :

:{L} \equiv {r} \times {p} = {r} \times ( m {v} ).

where {r} is the position vector of the particle and {p} = m {v} is the momentum of the particle.

By definition,

:{v} = \frac{d{r}}{dt} .

As a result, we have

:{L} = {r} \times m\frac{d{r}}{dt}.

Taking the derivative of both sides with respect to time, we arrive at a solution that is dependent on the cross product of parallel `There was an error working with the wiki: Code[32]`

at 0. Note that F is always parallel to r, since the force is entirely radial, and p is always parallel to v by definition. Therefore, we can now say that |{L}| is constant.

The area swept out by the line joining the `There was an error working with the wiki: Code[119]`

and the `There was an error working with the wiki: Code[120]`

is half the `There was an error working with the wiki: Code[121]`

of the `There was an error working with the wiki: Code[122]`

formed by {r} and d{r}.

:dA = \frac{1}{2} |{r} \times d{r} = \frac`There was an error working with the wiki: Code[5]`

{2m}dt

Since |{L}| is constant, the area swept out by the planet in a time interval is also a constant. Q.E.D.

The `There was an error working with the wiki: Code[33]`

s of the `There was an error working with the wiki: Code[34]`

to the `There was an error working with the wiki: Code[35]`

s of the `There was an error working with the wiki: Code[123]`

of the orbits.

:T^2 \propto a^3

:T = orbital period of planet

:a = semimajor axis of orbit

So the expression T2a–3 has the same value for all planets in the solar system as it has for `There was an error working with the wiki: Code[124]`

. That value is (with T in seconds, a in meters) 3.00\times 10^{-19} \frac{s^{2}}{m^{3}} \pm \ 0.7%\, . Thus, not only does the length of the orbit increase with distance, the `There was an error working with the wiki: Code[125]`

decreases, so that the increase of the `There was an error working with the wiki: Code[126]`

is more than proportional. This law is also known as the harmonic 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 law of gravity, and taking care of the proportionality factors that were neglected in the argument above, we have:

:T^2 = \frac{4\pi^2}{GM} \cdot r^3

where:

T = planet's `There was an error working with the wiki: Code[36]`

r = radius of the planet's circular orbit

G = the `There was an error working with the wiki: Code[127]`

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 `There was an error working with the wiki: Code[37]`

axis could be substituted for the radius. The most general result is:

:T^2 = \frac{4\pi^2}{G(M + m)} \cdot a^3

where:

T = object's `There was an error working with the wiki: Code[38]`

a = object's `There was an error working with the wiki: Code[39]`

G = 6.67 × 10&minus11 N · m²/kg2 = the `There was an error working with the wiki: Code[128]`

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, 4?2 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. Note that the values of G and planetary masses are not known with good accuracy however, the products GM (the Keplerian attraction) are known to extremely high precision.

Define point A to be the `There was an error working with the wiki: Code[40]`

, and point B as the `There was an error working with the wiki: Code[41]`

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{1}{2} \cdot(1-\epsilon)a\cdot V_A\,dt= \frac{1}{2} \cdot(1+\epsilon)a\cdot V_B\,dt

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

:V_A=V_B\cdot\frac{1+\epsilon}{1-\epsilon}

Using the `There was an error working with the wiki: Code[42]`

for the total energy of the planet at points A and B,

:\frac{mV_A^2}{2}-\frac{GmM}{(1-\epsilon)a} =\frac{mV_B^2}{2}-\frac{GmM}{(1+\epsilon)a}

:\frac{V_A^2}{2}-\frac{V_B^2}{2} =\frac{GM}{(1-\epsilon)a}-\frac{GM}{(1+\epsilon)a}

:\frac{V_A^2-V_B^2}{2}=\frac{GM}{a}\cdot \left ( \frac{1}{(1-\epsilon)}-\frac{1}{(1+\epsilon)} \right )

:\frac{\left ( V_B\cdot\frac{1+\epsilon}{1-\epsilon}\right ) ^2-V_B^2}{2}=\frac{GM}{a}\cdot \left ( \frac{1+\epsilon-1+\epsilon}{(1-\epsilon)(1+\epsilon)} \right )

:V_B^2 \cdot \left ( \frac{1+\epsilon}{1-\epsilon}\right ) ^2-V_B^2=\frac{2GM}{a}\cdot \left ( \frac{2\epsilon}{(1-\epsilon)(1+\epsilon)} \right )

:V_B^2 \cdot \left ( \frac{(1+\epsilon)^2-(1-\epsilon)^2}{(1-\epsilon)^2}\right )=\frac{4GM\epsilon}{a\cdot(1-\epsilon)(1+\epsilon)}

:V_B^2 \cdot \left ( \frac{1+2\epsilon+\epsilon^2-1+2\epsilon-\epsilon^2}{(1-\epsilon)^2} \right) =\frac{4GM\epsilon}{a\cdot(1-\epsilon)(1+\epsilon)}

:V_B^2 \cdot 4\epsilon =\frac{4GM\epsilon\cdot (1-\epsilon)^2}{a\cdot(1-\epsilon)(1+\epsilon)}

:V_B =\sqrt{\frac{GM\cdot(1-\epsilon)}{a\cdot(1+\epsilon)}}.

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. However, the total area of the ellipse is equal to \pi a \sqrt{(1-\epsilon^2)}a. (That's the same as \pi a b, because b=\sqrt{(1-\epsilon^2)}a). The time the planet take out to sweep out the entire area of the ellipse equals the ellipse's area, so,

:T\cdot \frac{dA}{dt}=\pi a \sqrt{(1-\epsilon^2)}a

:T\cdot \frac{1}{2} \cdot\sqrt{GMa\cdot(1-\epsilon)(1+\epsilon)}=\pi \sqrt{(1-\epsilon^2)}a^2

:T=\frac{2\pi \sqrt{(1-\epsilon^2)}a^2}{\sqrt{GMa\cdot(1-\epsilon)(1+\epsilon)}} =\frac{2\pi a^2}{\sqrt{GMa}}=

\frac{2\pi}{\sqrt{GM}}\sqrt{a^3}

:T^2=\frac{4\pi^2}{GM}a^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 `There was an error working with the wiki: Code[129]`

). To integrate that in the above formula, M must be replaced with M+m, to give

:T^2=\frac{4\pi^2}{G(M+m)}a^3.

Q.E.D.

Kepler did not understand why his laws were correct it was `There was an error working with the wiki: Code[130]`

who discovered the answer to this more than fifty years later. The second law can also be seen as a statement of conservation of `There was an error working with the wiki: Code[131]`

, which is a logical consequence of Newton's laws in the special case of a force that acts along the line connecting two objects.

The `There was an error working with the wiki: Code[43]`

?. 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) \mbox{length }cs=a\varepsilon

p the planet

z perihelion

x is the projection of the planet to the auxiliary circle then \mbox{area }sxz=\frac ab\mbox{area }spz

y is a point on tnot truehe circle such that \mbox{area }cyz=\mbox{area }sxz=\frac ab\mbox{area }spz

and three angles measured from perihelion:

`There was an error working with the wiki: Code[132]`

T= angle zsp\,, the planet as seen from the sun

`There was an error working with the wiki: Code[133]`

E= angle zcx\,, x as seen from the centre

`There was an error working with the wiki: Code[134]`

M= angle zcy\,, y as seen from the centre

Then

:\mbox{area }cxz=\mbox{area }cxs+\mbox{area }sxz=\mbox{area }cxs+\mbox{area }cyz

:\frac{a^2}2E=a\varepsilon\frac a2\sin E+\frac{a^2}2M

giving `There was an error working with the wiki: Code[135]`

:M=E-\varepsilon\sin E.

To connect E and T, assume r=\mbox{length }sp then

:a\varepsilon+r\cos T=a\cos E and r\sin T=b\sin E

:r=\frac{a\cos E-a\varepsilon}{\cos T}=\frac{b\sin E}{\sin T}

:\tan T=\frac{\sin T}{\cos T}=\frac ba\frac{\sin E}{\cos E-\varepsilon}=\frac{\sqrt{1-\varepsilon^2}\sin E}{\cos E-\varepsilon}

which is ambiguous but useable. A better form follows by some trickery with `There was an error working with the wiki: Code[44]`

:

:\tan\frac T2=\sqrt\frac{1+\varepsilon}{1-\varepsilon}\tan\frac E2

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

Note that \mbox{area }spz is the area swept since perihelion by the second law, that is proportional to time since perihelion. But we defined \mbox{area }spz=\frac ba\mbox{area }cyz=\frac ba\frac{a^2}2M and so M is also proportional to time since perihelion&mdashthis 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 `There was an error working with the wiki: Code[136]`

) 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 `There was an error working with the wiki: Code[45]`

) 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

On `There was an error working with the wiki: Code[137]`

, `There was an error working with the wiki: Code[138]`

, Kepler observed that an exceptionally bright star had suddenly appeared in the constellation `There was an error working with the wiki: Code[139]`

. (It was first observed by several others on `There was an error working with the wiki: Code[140]`

.) 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 were not changeless this was to influence Galileo Galilei in his argument. It has since been determined that the star was a `There was an error working with the wiki: Code[141]`

, the second in a generation, later called `There was an error working with the wiki: Code[142]`

or `There was an error working with the wiki: Code[143]`

. No further supernovae have been observed in the `There was an error working with the wiki: Code[144]`

, though others outside our galaxy have been seen.

Kepler also made fundamental investigations into `There was an error working with the wiki: Code[145]`

, geometrical optimization, and natural phenomena such as `There was an error working with the wiki: Code[146]`

flakes, always with an emphasis on form and design. He was also one of the founders of modern `There was an error working with the wiki: Code[147]`

, defining for example `There was an error working with the wiki: Code[148]`

s and the Kepler `There was an error working with the wiki: Code[149]`

(see Kepler's books Astronomiae Pars Optica&mdashi.a. theoretical explanation of the `There was an error working with the wiki: Code[150]`

&mdashand Dioptrice). In addition, since he was the first to recognize the non-convex regular solids (such as the stellated dodecahedra), they are named `There was an error working with the wiki: Code[151]`

s in his honor.

Kepler also was in contact with `There was an error working with the wiki: Code[152]`

, inventor of the first automatic `There was an error working with the wiki: Code[153]`

, whose

letters to Kepler show how to use the machine for calculating astronomical tables.

Kepler discovered the laws of planetary motion while trying to achieve the Pythagorean purpose of finding the harmony of the `There was an error working with the wiki: Code[46]`

was one less than the number of known `There was an error working with the wiki: Code[47]`

, `There was an error working with the wiki: Code[48]`

, `There was an error working with the wiki: Code[49]`

, `There was an error working with the wiki: Code[50]`

, `There was an error working with the wiki: Code[51]`

, `There was an error working with the wiki: Code[52]`

) and the five classical elements.

In 1596 Kepler published Mysterium Cosmographicum, or The Sacred Mystery of the Cosmos. 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 `There was an error working with the wiki: Code[154]`

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…

: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 `There was an error working with the wiki: Code[53]`

, then the `There was an error working with the wiki: Code[54]`

, and finally the `There was an error working with the wiki: Code[155]`

. To the second group belongs, first, the `There was an error working with the wiki: Code[156]`

, and second, the `There was an error working with the wiki: Code[157]`

. That is why the most important portion of the universe, the Earth&mdashwhere God's image is reflected in man&mdashseparates 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, `There was an error working with the wiki: Code[55]`

was earth, the `There was an error working with the wiki: Code[158]`

was water, and the `There was an error working with the wiki: Code[159]`

was the cosmos as a whole or ether. There is some evidence this association was of ancient origin, as `There was an error working with the wiki: Code[160]`

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.

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 `There was an error working with the wiki: Code[161]`

to explain the behaviour of both earthly and heavenly bodies.

Kepler disdained astrologers who pandered to the tastes of the common man without knowledge of the `There was an error working with the wiki: Code[56]`

and general rules, but he saw compiling prognostications as a justified means of supplementing his meager 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." However, Kepler's views on astrology were quite unconventional for his time he argued for a system of astrology based largely on `There was an error working with the wiki: Code[57]`

." Information relating to his theories can be found in his book `There was an error working with the wiki: Code[162]`

.

Kepler believed in astrology in the sense that he was convinced that `There was an error working with the wiki: Code[58]`

, `There was an error working with the wiki: Code[163]`

and `There was an error working with the wiki: Code[164]`

(1610), posing as a third man between the two extreme positions for and against `There was an error working with the wiki: Code[165]`

, Kepler advocated that a definite relationship between heavenly phenomena and earthly events could be established.

At least 800 horoscopes and `There was an error working with the wiki: Code[59]`

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, Kepler explained to `There was an error working with the wiki: Code[60]`

, downfall of `There was an error working with the wiki: Code[166]`

and return of `There was an error working with the wiki: Code[167]`

. The De cometis libelli tres (1619) is also replete with `There was an error working with the wiki: Code[168]`

predictions.

`There was an error working with the wiki: Code[169]`

(The Sacred Mystery of the Cosmos) (1596)

De Fundamentis Astrologiae Certioribus (On The More Certain Fundamentals of Astrology) (1601)

Astronomiae Pars Optica (The Optical Part of Astronomy) (1604)

De Stella nova in pede Serpentarii (On the New Star in Ophiuchus's Foot) (1604)

`There was an error working with the wiki: Code[170]`

(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)

`There was an error working with the wiki: Code[171]`

(Harmony of the Worlds) (1619)

Tabulae Rudolphinae (1627)

`There was an error working with the wiki: Code[61]`

(The Dream) (1634) - considered the first precursor of `There was an error working with the wiki: Code[172]`

.

`There was an error working with the wiki: Code[173]`

, a solar-orbiting, planet-hunting telescope due to be launched by `There was an error working with the wiki: Code[174]`

in 2008.

The `There was an error working with the wiki: Code[62]`

, a set of geometrical constructions, two of which were described by him.

`There was an error working with the wiki: Code[175]`

, Supernova 1604, which he observed and described.

`There was an error working with the wiki: Code[176]`

about `There was an error working with the wiki: Code[177]`

, proved true 400 years later.

`There was an error working with the wiki: Code[63]`

, a crater on the moon

`There was an error working with the wiki: Code[64]`

, a crater on Mars

`There was an error working with the wiki: Code[178]`

is an `There was an error working with the wiki: Code[179]`

.

In 1975, nine years after its founding, the College for Social and Economic Sciences Linz (Austria) was renamed `There was an error working with the wiki: Code[180]`

in honor of Johannes Kepler, since he wrote his `There was an error working with the wiki: Code[181]`

Harmonice Mundi in Linz.

Johannes Kepler's Gymnasium in Prague

Keplerstraße in Hanau near Frankfurt am Main

Kepler-Gymnasium in `There was an error working with the wiki: Code[182]`

, Germany.

`There was an error working with the wiki: Code[183]`

: 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. Draws heavily on Koestler's account of Kepler in The Sleepwalkers.

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.

Main

Peter Barker and Bernard R. Goldstein: "Theological Foundations of Kepler's Astronomy". Osiris, Volume 16: Science in Theistic Contexts. University of Chicago Press, 2001.

Max Caspar: Kepler transl. and ed. by C. Doris Hellman with a new introduction and references by Owen Gingerich bibliographic citations by Owen Gingerich and Alain Segonds. New York: Dover, 1993 ISBN 0-486-67605-6

James A. Connor: Kepler's Witch: An Astronomer's Discovery of Cosmic Order Amid Religious War, Political Intrigue, and the Heresy Trial of His Mother. HarperSanFrancisco, 2004 ISBN 0-06-052255-0

J.V. Field: Kepler's geometrical cosmology. Chicago: Chicago University Press, 1988 ISBN 0-226-24823-2

Owen Gingerich: The eye of heaven: Ptolemy, Copernicus, Kepler. New York: American Institute of Physics, 1993 ISBN 0-88318-863-5 (Masters of modern physics v. 7)

Kitty Ferguson: The nobleman and his housedog: Tycho Brahe and Johannes Kepler: the strange partnership that revolutionized science. London : Review, 2002 ISBN 0-747270-22-8 (published in the US as: Tycho & Kepler: the unlikely partnership that forever changed our understanding of the heavens. New York: Walker, 2002 ISBN 0-8027-1390-4)

`There was an error working with the wiki: Code[187]`

: `There was an error working with the wiki: Code[188]`

: A History of Man's Changing Vision of the Universe. (1959). ISBN 0-140-19246-8

John Lear: Kepler's Dream. Berkeley: University of California Press, 1965.

Bruce Stephenson: Kepler's physical astronomy. New York: Springer, 1987 ISBN 0-387-96541-6 (Studies in the history of mathematics and physical sciences 13)

Crowell, Benjamin, Conservation Laws, http://www.lightandmatter.com/area1book2.html, an online book that gives a proof of the first law without the use of calculus.

Gerneal

Annotation: Posner Family Collection in Electronic Format Harmonices mundi ("The Harmony of the Worlds") in fulltext facsimile in `There was an error working with the wiki: Code[189]`

Full text of Kepler by Walter W. Bryant, from `There was an error working with the wiki: Code[190]`

Gale E. Christianson- Kepler's Somnium: Science Fiction and the Renaissance Scientist

Kepler's Belief in Astrology by Nick Kollerstrom

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, Wikipedia: The Free Encyclopedia. Wikimedia Foundation.

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