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| In [[astronomy]], '''Kepler's laws of planetary motion''' are three [[scientific law]]s describing [[motion (physics)|motion]] of [[planet]]s around the [[Sun]].
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| [[Image:Kepler laws diagram.svg|thumb|300px|Figure 1: Illustration of [[Johannes Kepler|Kepler's]] three laws with two planetary orbits.<br> (1) The orbits are ellipses, with focal points ''ƒ''<sub>1</sub> and ''ƒ''<sub>2</sub> for the first planet and ''ƒ''<sub>1</sub> and ''ƒ''<sub>3</sub> for the second planet. The Sun is placed in focal point ''ƒ''<sub>1</sub>. <br><br> (2) The two shaded sectors ''A''<sub>1</sub> and ''A''<sub>2</sub> have the same surface area and the time for planet 1 to cover segment ''A''<sub>1</sub> is equal to the time to cover segment ''A''<sub>2</sub>. <br><br> (3) The total orbit times for planet 1 and planet 2 have a ratio ''a''<sub>1</sub><sup>3/2</sup> : ''a''<sub>2</sub><sup>3/2</sup>.]]
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| {{Astrodynamics |Equations}}
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| Kepler's laws are:
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| #The [[orbit]] of every [[planet]] is an [[ellipse]] with the Sun at one of the two [[Focus (geometry)|foci]].
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| #A [[line (geometry)|line]] joining a planet and the Sun sweeps out equal [[area]]s during equal intervals of time.<ref name="Wolfram2nd">Bryant, Jeff; Pavlyk, Oleksandr. "[http://demonstrations.wolfram.com/KeplersSecondLaw/ Kepler's Second Law]", ''[[Wolfram Demonstrations Project]]''. Retrieved December 27, 2009.</ref>
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| #The [[square (algebra)|square]] of the [[orbital period]] of a planet is [[Proportionality (mathematics)|proportional]] to the [[cube (arithmetic)|cube]] of the [[semi-major axis]] of its orbit.
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| ==History==
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| [[Johannes Kepler]] published his first two laws about planetary motion in 1609, having found them by analyzing the astronomical observations of [[Tycho Brahe]].<ref name=Holton>{{cite book
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| |title=Physics, the Human Adventure: From Copernicus to Einstein and Beyond
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| |author=Holton, Gerald James |coauthor=Brush, Stephen G. |pages=40–41
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| |url=http://books.google.com/?id=czaGZzR0XOUC&pg=PA40
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| |edition=3rd paperback |isbn=0-8135-2908-5 |publisher=Rutgers University Press
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| |location=Piscataway, NJ |accessdate=December 27, 2009 |year=2001}}</ref> Kepler's third law was published in 1619.<ref name=Holton />
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| Kepler's laws challenged the long-accepted [[geocentric model]]s of [[Aristotle]] and [[Ptolemy]], and followed the [[Copernican heliocentrism|heliocentric theory]] of [[Nicolaus Copernicus]] by asserting that the Earth orbited the Sun, proving that the planets' speeds varied, and using elliptical orbits rather than circular orbits with [[epicycle]]s.<ref name=Holton/>
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| Most planetary orbits are almost circles, so it is not obvious that they are actually ellipses. Detailed calculations for the orbit of the planet Mars first indicated to Kepler its elliptical shape, and he inferred that other heavenly bodies, including those farther away from the Sun, have elliptical orbits too.
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| Kepler in 1622 and [[Godefroy Wendelin]] in 1643 noted that Kepler's third law applies to the four brightest moons of [[Jupiter]].<ref group="Nb">Godefroy Wendelin wrote a letter to Giovanni Battista Riccioli about the relationship between the distances of the Jovian moons from Jupiter and the periods of their orbits, showing that the periods and distances conformed to Kepler's third law. See: Joanne Baptista Riccioli, ''Almagestum novum'' … (Bologna (Bononiae), (Italy): Victor Benati, 1651), volume 1, [http://books.google.com/books?id=_mJDAAAAcAAJ&pg=PA492#v=onepage&q&f=false page 492 Scholia III.] In the margin beside the relevant paragraph is printed: ''Vendelini ingeniosa speculatio circa motus & intervalla satellitum Jovis''. (The clever Wendelin's speculation about the movement and distances of Jupiter's satellites.)<br>In 1622, Johannes Kepler had noted that Jupiter's moons obey (approximately) his third law in his ''Epitome Astronomiae Copernicanae'' [Epitome of Copernican Astronomy] (Linz (“Lentiis ad Danubium“), (Austria): Johann Planck, 1622), book 4, part 2, [http://books.google.com/books?id=wa2SE_6ZL7YC&pg=PA554#v=onepage&q&f=false page 554].</ref>
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| [[Isaac Newton]] proved in 1687 that relationships like Kepler's would apply in the [[solar system]] to a good approximation, as consequences of Newton's own [[Newton's laws of motion|laws of motion]] and [[Newton's law of universal gravitation|law of universal gravitation]].
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| [[Voltaire]]'s ''Eléments de la philosophie de Newton'' (''Elements of Newton's Philosophy'') was in 1738 the first publication to call Kepler's Laws "laws".<ref name="Wilson 1994">{{cite journal
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| |last=Wilson |first=Curtis |authorlink= |date=May 1994 |title=Kepler's Laws, So-Called |journal=HAD News |volume=
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| |issue=31 |pages=1–2 |publisher=Historical Astronomy Division, [[American Astronomical Society]]
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| |location=Washington, DC|pmid= |pmc= |doi= |bibcode= |oclc= |id= |url=http://had.aas.org/hadnews/HADN31.pdf
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| |accessdate=December 27, 2009 |laysummary= |laysource= |laydate= |quote=
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| }}</ref>
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| Together with Newton's theories, they are part of the foundation of modern [[astronomy]] and [[physics]].<ref name=smith-sep>See also G E Smith, [http://plato.stanford.edu/archives/win2008/entries/newton-principia/ "Newton's Philosophiae Naturalis Principia Mathematica"], especially the section [http://plato.stanford.edu/archives/win2008/entries/newton-principia/#HisConPri ''Historical context ...''] in ''The Stanford Encyclopedia of Philosophy'' (Winter 2008 Edition), Edward N. Zalta (ed.).</ref>
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| ==First Law==
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| :"The [[orbit]] of every [[planet]] is an [[ellipse]] with the Sun at one of the two [[Focus (geometry) | foci]]."
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| [[Image:kepler-first-law.svg|thumb|Figure 2: Kepler's first law placing the Sun at the focus of an elliptical orbit]]
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| [[Image:Ellipse latus rectum.PNG|thumb|Figure 4: Heliocentric coordinate system ''(r, θ)'' for ellipse. Also shown are: semi-major axis ''a'', semi-minor axis ''b'' and semi-latus rectum ''p''; center of ellipse and its two foci marked by large dots. For θ = 0°, ''r = r<sub>min</sub>'' and for θ = 180°, ''r = r<sub>max</sub>''.]]
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| Mathematically, an ellipse can be represented by the formula:
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| :<math>r=\frac{p}{1+\varepsilon\, \cos\theta},</math>
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| where (''r'', ''θ'') are [[polar coordinates]], ''p'' is the [[semi-latus rectum]], and ''ε'' is the [[Eccentricity (mathematics)|eccentricity]] of the ellipse.
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| Note that 0 < ''ε'' < 1 for an ellipse; in the limiting case ''ε'' = 0, the orbit is a circle with the sun at the centre (see section [[#Zero eccentricity|Zero eccentricity]] below).
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| For a planet ''r'' is the distance from the Sun to the planet, and ''θ'' is the angle to the planet's current position from its closest approach, as seen from the Sun. | |
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| At ''θ'' = 0°, [[perihelion]], the distance is minimum
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| :<math>r_\mathrm{min}=\frac{p}{1+\varepsilon}.</math>
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| At ''θ'' = 90° and at ''θ'' = 270°, the distance is <math>\, p.</math>
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| At ''θ'' = 180°, [[aphelion]], the distance is maximum
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| :<math>r_\mathrm{max}=\frac{p}{1-\varepsilon}.</math>
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|
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| The [[semi-major axis]] ''a'' is the [[arithmetic mean]] between ''r''<sub>min</sub> and ''r''<sub>max</sub>:
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| :<math>\,r_\max - a=a-r_\min</math>
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|
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| :<math>a=\frac{p}{1-\varepsilon^2}.</math>
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| The [[semi-minor axis]] ''b'' is the [[geometric mean]] between ''r''<sub>min</sub> and ''r''<sub>max</sub>:
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| :<math>\frac{r_\max} b =\frac b{r_\min}</math>
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| :<math>b=\frac p{\sqrt{1-\varepsilon^2}}.</math>
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| The [[semi-latus rectum]] ''p'' is the [[harmonic mean]] between ''r''<sub>min</sub> and ''r''<sub>max</sub>:
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| :<math>\frac{1}{r_\min}-\frac{1}{p}=\frac{1}{p}-\frac{1}{r_\max}</math>
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| :<math>pa=r_\max r_\min=b^2\,.</math>
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| The [[eccentricity (mathematics)|eccentricity]] ''ε'' is the [[coefficient of variation]] between ''r''<sub>min</sub> and ''r''<sub>max</sub>:
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| :<math>\varepsilon=\frac{r_\mathrm{max}-r_\mathrm{min}}{r_\mathrm{max}+r_\mathrm{min}}.</math>
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| The [[area]] of the ellipse is | |
| :<math>A=\pi a b\,.</math>
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| The special case of a circle is ''ε'' = 0, resulting in ''r'' = ''p'' = ''r''<sub>min</sub> = ''r''<sub>max</sub> = ''a'' = ''b'' and ''A'' = π ''r''<sup>2</sup>.
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| ==Second law==
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| :"A [[line (geometry)|line]] joining a planet and the Sun sweeps out equal areas during equal intervals of time."<ref name="Wolfram2nd"/>
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| [[Image:kepler-second-law.gif|thumb|The same blue area is swept out in a given time. The green arrow is velocity. The purple arrow directed towards the Sun is the acceleration. The other two purple arrows are acceleration components parallel and perpendicular to the velocity]]
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| In a small time <math>dt\,</math> the planet sweeps out a small triangle having base line <math>r\,</math> and height <math>r d\theta\,</math> and area
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| <math>dA=\tfrac 1 2\cdot r\cdot r d\theta</math> and so the constant [[areal velocity]] is <br>
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| <math>\frac{dA}{dt}=\tfrac{1}{2}r^2 \frac{d\theta}{dt}.</math>
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| The planet moves faster when it is closer to the Sun.
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| The area enclosed by the elliptical orbit is
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| <math>\pi ab.\,</math>
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| So the period
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| <math>P\,</math>
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| satisfies
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| :<math>P\cdot \tfrac 12r^2 \frac{d\theta}{dt}=\pi a b</math>
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| and the [[mean motion]] of the planet around the Sun
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| <math>n = {2\pi}/P </math>
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| satisfies
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| :<math>r^2{d\theta} = a b n dt .</math>
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| ==Third law==
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| <!--- The article Gravitation links to this subsection. Please do not change the title of this subsection without making the appropriate amendments to all articles that link to it --->
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| :"The [[square (algebra)|square]] of the [[orbital period]] of a planet is directly [[Proportionality (mathematics)|proportional]] to the [[cube (arithmetic)|cube]] of the [[semi-major axis]] of its orbit."
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| The third law, published by Kepler in 1619 [http://www-istp.gsfc.nasa.gov/stargaze/Skeplaws.htm] captures the relationship between the distance of planets from the Sun, and their orbital periods.
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| Kepler enunciated this third law in a laborious attempt to determine what he viewed as the "[[Musica universalis|music of the spheres]]" according to precise laws, and express it in terms of musical notation.<ref>[[Edwin Arthur Burtt|Burtt, Edwin]]. ''The Metaphysical Foundations of Modern Physical Science''. p. 52.</ref>
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| So it used to be known as the ''harmonic law''.<ref name=Holton3>{{cite book |title=Physics, the Human Adventure |author=Gerald James Holton, Stephen G. Brush |page=45 |url=http://books.google.com/?id=czaGZzR0XOUC&pg=PA45&dq=Kepler+%22harmonic+law%22 |isbn=0-8135-2908-5 |publisher=Rutgers University Press |year=2001}}</ref>
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| Mathematically, the law says that the expression <math> P^2/a^3</math> has the same value for all the planets in the solar system.
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| ==Zero eccentricity==
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| Kepler's laws refine the model of Copernicus, which assumed circular orbits. If the eccentricity of a planetary [[orbit]] is zero, then Kepler's laws state:
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| #The planetary orbit is a circle
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| #The Sun is in the center
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| #The speed of the planet in the orbit is constant
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| #The square of the [[orbital period|sidereal period]] is proportionate to the [[cube]] of the distance from the Sun.
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| Actually, the eccentricities of the orbits of the six planets known to Copernicus and Kepler are quite small, so the rules above give excellent approximations of planetary motion, but Kepler's laws fit observations even better.
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| Kepler's corrections to the Copernican model are not at all obvious:
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| #The planetary orbit is ''not'' a circle, but an ''ellipse''
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| #The Sun is ''not'' at the center but at a ''focal point''
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| #Neither the linear speed nor the angular speed of the planet in the orbit is constant, but the ''area speed'' is constant.
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| #The square of the [[orbital period|sidereal period]] is proportionate to the cube of the ''mean between the maximum and minimum'' distances from the Sun.
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| The nonzero eccentricity of the orbit of the earth makes the time from the [[March equinox]] to the [[September equinox]], around 186 days, unequal to the time from the September equinox to the March equinox, around 179 days. A diameter would cut the orbit into equal parts, but the plane through the sun parallel to the [[equator]] of the earth cuts the orbit into two parts with areas in a 186 to 179 ratio, so the eccentricity of the orbit of the Earth is approximately
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| :<math>\varepsilon\approx\frac \pi 4 \frac {186-179}{186+179}\approx 0.015,</math>
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| which is close to the correct value (0.016710219). (See [[Earth's orbit]]).
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| The calculation is correct when the [[perihelion]], the date that the Earth is closest to the Sun, is on a [[solstice]]. The current perihelion, near January 4, is fairly close to the solstice on December 21 or 22.
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| ==Planetary acceleration==
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| [[Image:Newtons proof of Keplers second law.gif|right|thumb|A sudden sunward velocity change is applied to a planet. Then the areas of the triangles defined by the path of the planet for fixed time intervals will be equal. (Click on image for a detailed description.)]]
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| [[Isaac Newton]] computed in his [[Philosophiæ Naturalis Principia Mathematica]] the [[acceleration]] of a planet moving according to Kepler's first and second law.
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| #The ''direction'' of the acceleration is towards the Sun.
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| #The ''magnitude'' of the acceleration is in inverse proportion to the square of the distance from the Sun.
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| This suggests that the Sun may be the physical cause of the acceleration of planets.
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| Newton defined the [[force]] on a planet to be the product of its [[mass]] and the acceleration. (See [[Newton's laws of motion]]). So:
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| #Every planet is attracted towards the Sun.
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| #The force on a planet is in direct proportion to the mass of the planet and in inverse proportion to the square of the distance from the Sun.
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| Here the Sun plays an unsymmetrical part, which is unjustified. So he assumed [[Newton's law of universal gravitation]]:
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| #All bodies in the solar system attract one another.
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| #The force between two bodies is in direct proportion to the product of their masses and in inverse proportion to the square of the distance between them.
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| As the planets have small masses compared to that of the Sun, the orbits conform to Kepler's laws approximately. Newton's model improves upon Kepler's model and fits actual observations more accurately. (See [[two-body problem]]).
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| A deviation in the motion of a planet from Kepler's laws due to the gravity of other planets is called a [[perturbation (astronomy)|perturbation]].
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| Below comes the detailed calculation of the acceleration of a planet moving according to Kepler's first and second laws.
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| === Acceleration vector ===
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| {{See also| Polar coordinate#Vector calculus| Mechanics of planar particle motion}}
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| From the [[heliocentric]] point of view consider the vector to the planet <math>\mathbf{r} = r \hat{\mathbf{r}} </math> where <math> r</math> is the distance to the planet and the direction <math> \hat {\mathbf{r}} </math> is a [[unit vector]]. When the planet moves the direction vector <math> \hat {\mathbf{r}} </math> changes:
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| :<math> \frac{d\hat{\mathbf{r}}}{dt}=\dot{\hat{\mathbf{r}}} = \dot\theta \hat{\boldsymbol\theta},\qquad \dot{\hat{\boldsymbol\theta}} = -\dot\theta \hat{\mathbf{r}}</math>
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| where <math>\scriptstyle \hat{\boldsymbol\theta}</math> is the unit vector orthogonal to <math>\scriptstyle \hat{\mathbf{r}}</math> and pointing in the direction of rotation, and <math>\scriptstyle \theta</math> is the polar angle, and where a dot on top of the variable signifies differentiation with respect to time.
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| So differentiating the position vector twice to obtain the velocity and the acceleration vectors:
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| :<math>\dot{\mathbf{r}} =\dot{r} \hat{\mathbf{r}} + r \dot{\hat{\mathbf{r}}}
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| =\dot{r} \hat{\mathbf{r}} + r \dot{\theta} \hat{\boldsymbol{\theta}},</math>
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| :<math>\ddot{\mathbf{r}}
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| = (\ddot{r} \hat{\mathbf{r}} +\dot{r} \dot{\hat{\mathbf{r}}} )
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| + (\dot{r}\dot{\theta} \hat{\boldsymbol{\theta}} + r\ddot{\theta} \hat{\boldsymbol{\theta}}
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| + r\dot{\theta} \dot{\hat{\boldsymbol{\theta}}})
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| = (\ddot{r} - r\dot{\theta}^2) \hat{\mathbf{r}} + (r\ddot{\theta} + 2\dot{r} \dot{\theta}) \hat{\boldsymbol{\theta}}.</math>
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| So
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| :<math>\ddot{\mathbf{r}} = a_r \hat{\boldsymbol{r}}+a_\theta\hat{\boldsymbol{\theta}}</math>
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| where the '''radial acceleration''' is
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| :<math>a_r=\ddot{r} - r\dot{\theta}^2</math>
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| and the '''transversal acceleration''' is
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| :<math>a_\theta=r\ddot{\theta} + 2\dot{r} \dot{\theta}.</math>
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| ===The inverse square law===
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| Kepler's laws say that
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| :<math>r^2\dot \theta = nab </math>
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| is constant.
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| The transversal acceleration <math>a_\theta</math> is zero:
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| :<math>\frac{d (r^2 \dot \theta)}{dt} = r (2 \dot r \dot \theta + r \ddot \theta ) = r a_\theta = 0. </math>
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| So the acceleration of a planet obeying Kepler's laws is directed towards the sun.
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| The radial acceleration <math>a_r </math> is
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| :<math>a_r = \ddot r - r \dot \theta^2= \ddot r - r \left(\frac{nab}{r^2}
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| \right)^2= \ddot r -\frac{n^2a^2b^2}{r^3}. </math>
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| Kepler's first law states that the orbit is described by the equation:
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| :<math>\frac{p}{r} = 1+ \varepsilon \cos\theta.</math>
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| Differentiating with respect to time
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| :<math>-\frac{p\dot r}{r^2} = -\varepsilon \sin \theta \,\dot \theta </math>
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| or
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| :<math>p\dot r = nab\,\varepsilon\sin \theta. </math>
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| Differentiating once more
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| :<math>p\ddot r =nab \varepsilon \cos \theta \,\dot \theta
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| =nab \varepsilon \cos \theta \,\frac{nab}{r^2}
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| =\frac{n^2a^2b^2}{r^2}\varepsilon \cos \theta . </math>
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| The radial acceleration <math>a_r </math> satisfies
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| :<math>p a_r = \frac{n^2 a^2b^2}{r^2}\varepsilon \cos \theta - p\frac{n^2 a^2b^2}{r^3}
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| = \frac{n^2a^2b^2}{r^2}\left(\varepsilon \cos \theta - \frac{p}{r}\right). </math>
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| Substituting the equation of the ellipse gives
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| :<math>p a_r = \frac{n^2a^2b^2}{r^2}\left(\frac p r - 1 - \frac p r\right)= -\frac{n^2a^2}{r^2}b^2. </math>
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| The relation <math>b^2=pa</math> gives the simple final result
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| :<math>a_r=-\frac{n^2a^3}{r^2}. </math>
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| This means that the acceleration vector <math>\mathbf{\ddot r}</math> of any planet obeying Kepler's first and second law satisfies the '''inverse square law''' | |
| :<math>\mathbf{\ddot r} = - \frac{\alpha}{r^2}\hat{\mathbf{r}}</math>
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| where
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| :<math>\alpha = n^2 a^3\,</math>
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| is a constant, and <math>\hat{\mathbf r}</math> is the unit vector pointing from the Sun towards the planet, and <math>r\,</math> is the distance between the planet and the Sun. | |
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| According to Kepler's third law, <math>\alpha</math> has the same value for all the planets. So the inverse square law for planetary accelerations applies throughout the entire solar system.
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| The inverse square law is a [[differential equation]]. The solutions to this differential equation include the Keplerian motions, as shown, but they also include motions where the orbit is a [[hyperbola]] or [[parabola]] or a [[straight line]]. See [[Kepler orbit]].
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| ===Newton's law of gravitation===
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| By [[Newton's second law]], the gravitational force that acts on the planet is:
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| :<math>\mathbf{F} = m_{Planet} \mathbf{\ddot r} = - {m_{Planet} \alpha}{r^{-2}}\hat{\mathbf{r}}</math>
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| where <math>m_{Planet}</math> is the mass of the planet and <math>\alpha</math> has the same value for all planets in the solar system. According to [[Newton's Third Law|Newton's third Law]], the Sun is attracted to the planet by a force of the same magnitude. Since the force is proportional to the mass of the planet, under the symmetric consideration, it should also be proportional to the mass of the Sun, <math>m_{Sun}</math>. So
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| :<math>\alpha = Gm_{Sun}</math>
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| where <math>G</math> is the [[gravitational constant]].
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| The acceleration of solar system body number ''i'' is, according to Newton's laws:
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| :<math>\mathbf{\ddot r_i} = G\sum_{j\ne i} {m_j}{r_{ij}^{-2}}\hat{\mathbf{r}}_{ij} </math>
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| where <math>m_j </math> is the mass of body ''j'', <math>r_{ij} </math> is the distance between body ''i'' and body ''j'', <math>\hat{\mathbf{r}}_{ij} </math> is the unit vector from body ''i'' towards body ''j'', and the vector summation is over all bodies in the world, besides ''i'' itself.
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| In the special case where there are only two bodies in the world, Earth and Sun, the acceleration becomes | |
| :<math>\mathbf{\ddot r}_{Earth} = G{m_{Sun}}{r_{{Earth},{Sun}}^{-2}}\hat{\mathbf{r}}_{{Earth},{Sun}}</math>
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| which is the acceleration of the Kepler motion. So this Earth moves around the Sun according to Kepler's laws.
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| If the two bodies in the world are Moon and Earth the acceleration of the Moon becomes | |
| :<math>\mathbf{\ddot r}_{Moon} = G{m_{Earth}}{r_{{Moon},{Earth}}^{-2}}\hat{\mathbf{r}}_{{Moon},{Earth}}</math>
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| So in this approximation the Moon moves around the Earth according to Kepler's laws.
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| In the three-body case the accelerations are
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| :<math>\mathbf{\ddot r}_{Sun} = G m_{Earth}{r_{{Sun},{Earth}}^{-2}}\hat\mathbf{r}_{{Sun},{Earth}} + G{m_{Moon}}{r_{{Sun},{Moon}}^{-2}}\hat{\mathbf{r}}_{{Sun},{Moon}}</math>
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| :<math>\mathbf{\ddot r}_{Earth} = G{m_{Sun}}{r_{{Earth},{Sun}}^{-2}}\hat{\mathbf{r}}_{{Earth},{Sun}} + G{m_{Moon}}{r_{{Earth},{Moon}}^{-2}}\hat{\mathbf{r}}_{{Earth},{Moon}} </math>
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| :<math>\mathbf{\ddot r}_{Moon} = G{m_{Sun}}{r_{{Moon},{Sun}}^{-2}}\hat{\mathbf{r}}_{{Moon},{Sun}}+G{m_{Earth}}{r_{{Moon},{Earth}}^{-2}}\hat{\mathbf{r}}_{{Moon},{Earth}}</math>
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| These accelerations are not those of Kepler orbits, and the [[three-body problem]] is complicated. But Keplerian approximation is basis for [[perturbation (astronomy)|perturbation]] calculations. See [[Lunar theory]].
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| ==Position as a function of time {{anchor|position_function_time}}==
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| Kepler used his two first laws to compute the position of a planet as a function of time. His method involves the solution of a [[Transcendental function|transcendental equation]] called [[Kepler's equation]].
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| The procedure for calculating the heliocentric polar coordinates (''r'',''θ'') of a planet as a function of the time ''t'' since [[perihelion]], is the following four steps:
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| :1. Compute the '''[[mean anomaly]]'''
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| ::<math>M=nt</math>
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| :2. Compute the '''[[eccentric anomaly]]''' ''E'' by solving Kepler's equation:
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| ::<math>\ M=E-\varepsilon\cdot\sin E</math>
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| :3. Compute the '''[[true anomaly]]''' ''θ'' by the equation:
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| ::<math>\tan\frac \theta 2 = \sqrt{\frac{1+\varepsilon}{1-\varepsilon}}\cdot\tan\frac E 2</math>
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| :4. Compute the '''heliocentric distance''' ''r'' from the first law:
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| ::<math>r=\frac p {1+\varepsilon\cdot\cos\theta}</math>
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| The important special case of circular orbit, ε = 0, gives simply ''θ'' = ''E'' = ''M''. Because the uniform circular motion was considered to be ''normal'', a deviation from this motion was considered an '''anomaly'''.
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| The proof of this procedure is shown below.
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| ===Mean anomaly, ''M''===
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| [[Image:Anomalies.PNG|thumb |200px| FIgure 5: Geometric construction for Kepler's calculation of θ. The Sun (located at the focus) is labeled ''S'' and the planet ''P''. The auxiliary circle is an aid to calculation. Line ''xd'' is perpendicular to the base and through the planet ''P''. The shaded sectors are arranged to have equal areas by positioning of point ''y''.]]
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| The Keplerian problem assumes an [[Elliptic orbit|elliptical orbit]] and the four points:
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| :''s'' the Sun (at one focus of ellipse);
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| :''z'' the perihelion
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| :''c'' the center of the ellipse
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| :''p'' the planet
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| and
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| :<math>\ a=|cz|,</math> distance between center and perihelion, the '''semimajor axis''',
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| :<math>\ \varepsilon={|cs|\over a},</math> the '''eccentricity''',
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| :<math>\ b=a\sqrt{1-\varepsilon^2},</math> the '''semiminor axis''',
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| :<math>\ r=|sp| ,</math> the distance between Sun and planet.
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| :<math>\theta=\angle zsp,</math> the direction to the planet as seen from the Sun, the '''[[true anomaly]]'''.
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| The problem is to compute the [[polar coordinates]] (''r'',''θ'') of the planet from the '''time since perihelion''', ''t''.
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| It is solved in steps. Kepler considered the circle with the major axis as a diameter, and
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| :<math>\ x,</math> the projection of the planet to the auxiliary circle
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| :<math>\ y,</math> the point on the circle such that the sector areas ''|zcy|'' and ''|zsx|'' are equal,
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| :<math>M=\angle zcy,</math> the '''[[mean anomaly]]'''.
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| The sector areas are related by <math>|zsp|=\frac b a \cdot|zsx|.</math>
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| The [[circular sector]] area <math>\ |zcy| = \frac{a^2 M}2.</math>
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| The area swept since perihelion,
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| :<math>|zsp|=\frac b a \cdot|zsx|=\frac b a \cdot|zcy|=\frac b a\cdot\frac{a^2 M}2 = \frac {a b M}{2}, </math>
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| is by Kepler's second law proportional to time since perihelion. So the mean anomaly, ''M'', is proportional to time since perihelion, ''t''.
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| :<math>M=n t,</math>
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| where ''n'' is the [[mean motion]].
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| ===Eccentric anomaly, ''E''===
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| When the mean anomaly ''M'' is computed, the goal is to compute the true anomaly ''θ''. The function ''θ''=''f''(''M'') is, however, not elementary.<ref>{{cite web|url=http://info.ifpan.edu.pl/firststep/aw-works/fsII/mul/mueller.html|title=EQUATION OF TIME -- PROBLEM IN ASTRONOMY|last=MÜLLER|first=M|year=1995|publisher=Acta Physica Polonica A|accessdate=23 February 2013}}</ref> Kepler's solution is to use | |
| :<math>E=\angle zcx</math>, ''x'' as seen from the centre, the '''[[eccentric anomaly]]'''
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| as an intermediate variable, and first compute ''E'' as a function of ''M'' by solving Kepler's equation below, and then compute the true anomaly ''θ'' from the eccentric anomaly ''E''. Here are the details.
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| :<math>\ |zcy|=|zsx|=|zcx|-|scx|</math>
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| :<math>\frac{a^2 M}2=\frac{a^2 E}2-\frac {a\varepsilon\cdot a\sin E}2</math>
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| Division by ''a''<sup>2</sup>/2 gives '''[[Kepler's equation]]'''
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| :<math>M=E-\varepsilon\cdot\sin E.</math>
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| This equation gives ''M'' as a function of ''E''. Determining ''E'' for a given ''M'' is the inverse problem. Iterative numerical algorithms are commonly used.
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| Having computed the eccentric anomaly ''E'', the next step is to calculate the true anomaly ''θ''.
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| ===True anomaly, θ===
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| Note from the figure that
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| :<math>\overrightarrow{cd}=\overrightarrow{cs}+\overrightarrow{sd}</math>
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| so that | |
| :<math>a\cdot\cos E=a\cdot\varepsilon+r\cdot\cos \theta.</math>
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| Dividing by <math>a</math> and inserting from Kepler's first law
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| :<math>\ \frac r a =\frac{1-\varepsilon^2}{1+\varepsilon\cdot\cos \theta} </math>
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| to get | |
| :<math>\cos E
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| =\varepsilon+\frac{1-\varepsilon^2}{1+\varepsilon\cdot\cos \theta}\cdot\cos \theta
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| </math> <math>=\frac{\varepsilon\cdot(1+\varepsilon\cdot\cos \theta)+(1-\varepsilon^2)\cdot\cos \theta}{1+\varepsilon\cdot\cos \theta}
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| </math> <math>=\frac{\varepsilon +\cos \theta}{1+\varepsilon\cdot\cos \theta}.</math>
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| The result is a usable relationship between the eccentric anomaly ''E'' and the true anomaly ''θ''.
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| A computationally more convenient form follows by substituting into the [[trigonometric identity]]:
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| :<math>\tan^2\frac{x}{2}=\frac{1-\cos x}{1+\cos x}.</math>
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| Get
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| :<math>\tan^2\frac{E}{2}
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| =\frac{1-\cos E}{1+\cos E}
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| </math> <math>=\frac{1-\frac{\varepsilon+\cos \theta}{1+\varepsilon\cdot\cos \theta}}{1+\frac{\varepsilon+\cos \theta}{1+\varepsilon\cdot\cos \theta}}
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| </math> <math>=\frac{(1+\varepsilon\cdot\cos \theta)-(\varepsilon+\cos \theta)}{(1+\varepsilon\cdot\cos \theta)+(\varepsilon+\cos \theta)}
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| </math> <math>=\frac{1-\varepsilon}{1+\varepsilon}\cdot\frac{1-\cos \theta}{1+\cos \theta}=\frac{1-\varepsilon}{1+\varepsilon}\cdot\tan^2\frac{\theta}{2}.</math>
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| Multiplying by (1+ε)/(1−ε) and taking the square root gives the result
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| :<math>\tan\frac \theta2=\sqrt\frac{1+\varepsilon}{1-\varepsilon}\cdot\tan\frac E2.</math>
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| We have now completed the third step in the connection between time and position in the orbit.
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| === Distance, ''r'' ===
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| The fourth step is to compute the heliocentric distance ''r'' from the true anomaly ''θ'' by Kepler's first law:
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| :<math>\ r=a\cdot\frac{1-\varepsilon^2}{1+\varepsilon\cdot\cos \theta}.</math>
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| ==See also==
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| *[[Circular motion]]
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| *[[Free-fall time]]
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| *[[Gravity]]
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| *[[Kepler orbit]]
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| *[[Kepler problem]]
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| *[[Kepler's equation]]
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| *[[Laplace–Runge–Lenz vector]]
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| ==Notes==
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| <references group="Nb"/>
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| ==References==
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| {{reflist}}
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| ==Bibliography==
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| *Kepler's life is summarized on pages 523–627 and Book Five of his ''magnum opus'', ''[[Harmonice Mundi]]'' (''harmonies of the world''), is reprinted on pages 635–732 of ''On the Shoulders of Giants'': The Great Works of Physics and Astronomy (works by Copernicus, [[Johannes Kepler|Kepler]], [[Galileo Galilei|Galileo]], [[Isaac Newton|Newton]], and [[Albert Einstein|Einstein]]). [[Stephen Hawking]], ed. 2002 ISBN 0-7624-1348-4
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| *A derivation of Kepler's third law of planetary motion is a standard topic in engineering mechanics classes. See, for example, pages 161–164 of {{Cite document|first=J. L. |last=Meriam|date=1966, 1971|title=Dynamics, 2nd ed |location=New York|publisher=John Wiley|isbn=0-471-59601-9|postscript=<!--None-->}}.
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| *Murray and Dermott, Solar System Dynamics, Cambridge University Press 1999, ISBN 0-521-57597-4
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| *V.I. Arnold, Mathematical Methods of Classical Mechanics, Chapter 2. Springer 1989, ISBN 0-387-96890-3
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| ==External links==
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| * B.Surendranath Reddy; animation of Kepler's laws: [http://www.surendranath.org/Applets/Dynamics/Kepler/Kepler1.html applet]
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| * Crowell, Benjamin, ''Conservation Laws'', [http://www.lightandmatter.com/area1book2.html http://www.lightandmatter.com/area1book2.html], an [[On-line book|online book]] that gives a proof of the first law without the use of calculus. (see section 5.2, p. 112)
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| * David McNamara and Gianfranco Vidali, ''Kepler's Second Law - Java Interactive Tutorial'', [http://www.phy.syr.edu/courses/java/mc_html/kepler.html http://www.phy.syr.edu/courses/java/mc_html/kepler.html], an interactive Java applet that aids in the understanding of Kepler's Second Law.
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| * Audio - Cain/Gay (2010) [http://www.astronomycast.com/history/ep-189-johannes-kepler-and-his-laws-of-planetary-motion/ Astronomy Cast] Johannes Kepler and His Laws of Planetary Motion
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| * University of Tennessee's Dept. Physics & Astronomy: Astronomy 161 page on Johannes Kepler: The Laws of Planetary Motion [http://csep10.phys.utk.edu/astr161/lect/history/kepler.html]
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| * Equant compared to Kepler: interactive model [http://people.scs.fsu.edu/~dduke/kepler.html]
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| * Kepler's Third Law:interactive model [http://people.scs.fsu.edu/~dduke/kepler3.html]
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| * Solar System Simulator ([http://user.uni-frankfurt.de/~jenders/NPM/NPM.html Interactive Applet])
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| * [http://www.phy6.org/stargaze/Skeplaws.htm Kepler and His Laws], educational web pages by David P. Stern
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| {{orbits}}
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| {{Johannes Kepler}}
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| {{DEFAULTSORT:Kepler's Laws Of Planetary Motion}}
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| [[Category:Johannes Kepler]]
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| [[Category:Orbits]]
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| [[Category:Equations]]
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| [[Category:Copernican Revolution]]
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| {{Link FA|he}}
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