In astronomy, Kepler's laws of planetary motion are three scientific laws describing orbital motion, originally formulated to describe the motion of planets around the Sun.
Kepler's laws are:
 The orbit of every planet is an ellipse with the Sun at one of the two foci.
 A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.^{[1]}
 The square of the orbital period of a planet is proportional to the cube of the semimajor axis of its orbit.
History
Johannes Kepler published his first two laws in 1609, having found them by analyzing the astronomical observations of Tycho Brahe.^{[2]} Kepler discovered his third law many years later, and it was published in 1619.^{[2]} At the time, Kepler's laws were radical claims; the prevailing belief (particularly in epicyclebased theories) was that orbits were perfect circles. Most of the planetary orbits can be rather closely approximated as circles, so it is not immediately evident that the orbits are 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. Kepler's laws and his analysis of the observations on which they were based challenged the longaccepted geocentric models of Aristotle and Ptolemy, and generally supported the heliocentric theory of Nicolaus Copernicus (although Kepler's ellipses likewise did away with Copernicus's circular orbits and epicycles), by asserting that the Earth orbited the Sun, proving that the planets' speeds varied, and using elliptical orbits rather than circular orbits with epicycles.^{[2]}
Some eight decades later, Isaac Newton proved that relationships like Kepler's would apply exactly under certain ideal conditions that are to a good approximation fulfilled in the solar system, as consequences of Newton's own laws of motion and law of universal gravitation.^{[Nb 1]}^{[3]} Because of the nonzero planetary masses and resulting perturbations, Kepler's laws apply only approximately and not exactly to the motions in the solar system.^{[Nb 2]}^{[3]} 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".^{[4]} Together with Newton's mathematical theories, they are part of the foundation of modern astronomy and physics.^{[3]}
First Law
 "The orbit of every planet is an ellipse with the Sun at one of the two foci."
An ellipse is a closed plane curve that resembles a stretched out circle (see the figure to the right). Note that the Sun is not at the center of the ellipse, but at one of its foci. This focal point is sometimes called the occupied focus. The other focal point, known as the empty or vacant focus, marked with a lighter dot, has no physical significance for the orbit. The center of an ellipse is the midpoint of the line segment joining its focal points. A circle is a special case of an ellipse where both focal points coincide.
The relative distance between a focal point and the center is known as the eccentricity. It can take any value greater than or equal to 0 (a circle) and less than 1 (as the eccentricity tends to 1, the ellipse tends to a parabola). The eccentricities of the planets known to Kepler varied from 0.007 (Venus) to 0.2 (Mercury). (See List of planetary objects in the Solar System for more detail).
After Kepler's death, however, bodies with highly eccentric orbits were identified, among them many comets and asteroids. Heavenly bodies such as comets with parabolic or even hyperbolic orbits are possible under the Newtonian theory and have been observed.^{[5]}
In mathematical equations of the ellipse, using Cartesian coordinates, the ellipse has its center of symmetry as its center. This however is not helpful in studying planetary orbits, since the sun is at one focus of the ellipse, not at the center. Therefore polar coordinates rather than Cartesian coordinates are used here.
Symbolically, an ellipse can be represented in polar coordinates as:
 $r=\backslash frac\{p\}\{1+\backslash varepsilon\backslash ,\; \backslash cos\backslash theta\},$
where (r, θ) are the polar coordinates (from the focus) for the ellipse, p is the semilatus rectum, and ε is the eccentricity of the ellipse. Note that 0 < ε < 1 for a proper ellipse; if ε = 0, the ellipse has no eccentricity and can be analyzed directly as a case of a simple circle with the sun at the centre (see section Zero eccentricity below). For a planet orbiting the Sun, r is the distance from the Sun to the planet and θ is the angle between the planet's current position and its closest approach, with the Sun as the vertex.
At θ = 0°, perihelion, the distance is minimum
 $r\_\backslash mathrm\{min\}=\backslash frac\{p\}\{1+\backslash varepsilon\}.$
At θ = 90° and at θ = 270°, the distance is $\backslash ,\; p.$
At θ = 180°, aphelion, the distance is maximum
 $r\_\backslash mathrm\{max\}=\backslash frac\{p\}\{1\backslash varepsilon\}.$
The semimajor axis a is the arithmetic mean between r_{min} and r_{max}:
 $\backslash ,r\_\backslash max\; \; a=ar\_\backslash min$
 $a=\backslash frac\{p\}\{1\backslash varepsilon^2\}.$
The semiminor axis b is the geometric mean between r_{min} and r_{max}:
 $\backslash frac\{r\_\backslash max\}\; b\; =\backslash frac\; b\{r\_\backslash min\}$
 $b=\backslash frac\; p\{\backslash sqrt\{1\backslash varepsilon^2\}\}.$
The semilatus rectum p is the harmonic mean between r_{min} and r_{max}:
 $\backslash frac\{1\}\{r\_\backslash min\}\backslash frac\{1\}\{p\}=\backslash frac\{1\}\{p\}\backslash frac\{1\}\{r\_\backslash max\}$
 $pa=r\_\backslash max\; r\_\backslash min=b^2\backslash ,.$
The eccentricity ε is the coefficient of variation between r_{min} and r_{max}:
 $\backslash varepsilon=\backslash frac\{r\_\backslash mathrm\{max\}r\_\backslash mathrm\{min\}\}\{r\_\backslash mathrm\{max\}+r\_\backslash mathrm\{min\}\}.$
The area of the ellipse is
 $A=\backslash pi\; a\; b\backslash ,.$
The special case of a circle is ε = 0, resulting in r = p = r_{min} = r_{max} = a = b and A = π r^{2}.
Second law
 "A line joining a planet and the Sun sweeps out equal areas during equal intervals of time."^{[1]}
In a small time $dt\backslash ,$ the planet sweeps out a small triangle having base line $r\backslash ,$ and height $r\; d\backslash theta\backslash ,$.
The area of this triangle is given by
$dA=\backslash tfrac\; 1\; 2\backslash cdot\; r\backslash cdot\; r\; d\backslash theta$
and so the constant areal velocity is
$\backslash frac\{dA\}\{dt\}=\backslash tfrac\{1\}\{2\}r^2\; \backslash frac\{d\backslash theta\}\{dt\}.$
Now as the first law states that the planet follows an ellipse, the planet is at different distances from the Sun at different parts in its orbit. So the planet has to move faster when it is closer to the Sun so that it sweeps equal areas in equal times.
The total area enclosed by the elliptical orbit is
 $A=\backslash pi\; ab.\backslash ,$
Therefore the period
 $P\backslash ,$
satisfies
 $\backslash pi\; ab=P\backslash cdot\; \backslash tfrac\; 12r^2\; \backslash dot\backslash theta$
or
 $r^2\backslash dot\; \backslash theta\; =\; nab$
where
 $\backslash dot\backslash theta=\backslash frac\{d\backslash theta\}\{dt\}$
is the angular velocity, (using Newton notation for differentiation), and
 $n\; =\; \backslash frac\{2\backslash pi\}\{P\}$
is the mean motion of the planet around the Sun.
Third law
 "The square of the orbital period of a planet is directly proportional to the cube of the semimajor axis of its orbit."
The third law, published by Kepler in 1619 [1] captures the relationship between the distance of planets from the Sun, and their orbital periods. Symbolically, the law can be expressed as
 $P^2\; \backslash propto\; a^3\; ,$
where $P$ is the orbital period of the planet and $a$ is the semimajor axis of the orbit.
The constant of proportionality is
 $\backslash frac\{P\_\{planet\}^2\}\{a\_\{planet\}^3\}\; =\; \backslash frac\{P\_\{earth\}^2\}\{a\_\{earth\}^3\}\; =\; 1\; \backslash frac\{\; \backslash rm\{yr^2\}\; \}\{\; \backslash rm\{AU^3\}\; \}$
for a sidereal year (yr), and astronomical unit (AU). (For numeric values see List of gravitationally rounded objects of the Solar System).
Kepler enunciated this third law in a laborious attempt to determine what he viewed as the "music of the spheres" according to precise laws, and express it in terms of musical notation.^{[6]}
So it used to be known as the harmonic law.^{[7]}
For circular orbits, Kepler's 3rd Law is also commonly represented as
 $\backslash frac\{4\; \backslash pi^2\}\{T^2\}\; =\; \backslash frac\{G\; M\}\{R^3\}$
Where $T$ is the period, $G$ is the Gravitational constant, $M$ is the mass of the larger body, and $R$ is the distance between the centers of mass of the two bodies.
Generality
Godefroy Wendelin, in 1643, noted that Kepler's third law applies to the four brightest moons of Jupiter.^{[Nb 3]}
In fact, these laws approximately describe the motion of any two bodies in orbit around each other. (The statement in the first law about the focus becomes closer to exactitude as one of the masses becomes closer to zero mass. Where there are more than two masses, all of the statements in the laws become closer to exactitude as all except one of the masses become closer to zero mass and as the perturbations then also tend towards zero).^{[Nb 4]} The masses of the two bodies can be nearly equal, e.g. Charon—Pluto (~1:10), in a small proportion, e.g. Moon—Earth (~1:100), or in a great proportion, e.g. Mercury—Sun (~1:10,000,000).
In all cases of twobody motion, rotation is about the barycenter of the two bodies, with neither one having its center of mass exactly at one focus of an ellipse. However, both orbits are ellipses with one focus at the barycenter. When the ratio of masses is large, the barycenter may be deep within the larger object, close to its center of mass. In such a case it may require sophisticated precision measurements to detect the separation of the barycenter from the center of mass of the larger object. But in the case of the planets orbiting the Sun, the largest of them mass as much as 1/1047.3486 (Jupiter) and 1/3497.898 (Saturn) of the solar mass,^{[8]} and so it has long been known that the solar system barycenter can sometimes be outside the body of the Sun, up to about a solar diameter from its center.^{[Nb 5]} Thus Kepler's first law, though not far off as an approximation, does not quite accurately describe the orbits of the planets around the Sun under classical physics.
Zero eccentricity
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:
 The planetary orbit is a circle
 The Sun is in the center
 The speed of the planet in the orbit is constant
 The square of the sidereal period is proportionate to the cube of the distance from the Sun.
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.
Kepler's corrections to the Copernican model are not at all obvious:
 The planetary orbit is not a circle, but an ellipse
 The Sun is not at the center but at a focal point
 Neither the linear speed nor the angular speed of the planet in the orbit is constant, but the area speed is constant.
 The square of the sidereal period is proportionate to the cube of the mean between the maximum and minimum distances from the Sun.
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 equator 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
 $\backslash varepsilon\backslash approx\backslash frac\; \backslash pi\; 4\; \backslash frac\; \{186179\}\{186+179\}\backslash approx\; 0.015,$
which is close to the correct value (0.016710219). (See Earth's orbit).
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.
Planetary acceleration
Relation to Newton's laws
Isaac Newton computed in his Philosophiæ Naturalis Principia Mathematica the acceleration of a planet moving according to Kepler's first and second law.
 The direction of the acceleration is towards the Sun.
 The magnitude of the acceleration is in inverse proportion to the square of the distance from the Sun.
This suggests that the Sun may be the physical cause of the acceleration of planets.
Newton defined the force on a planet to be the product of its mass and the acceleration. (See Newton's laws of motion). So:
 Every planet is attracted towards the Sun.
 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.
Here the Sun plays an unsymmetrical part, which is unjustified. So he assumed Newton's law of universal gravitation:
 All bodies in the solar system attract one another.
 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.
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 twobody problem).
A deviation in the motion of a planet from Kepler's laws due to the gravity of other planets is called a perturbation.
Acceleration vector
From the heliocentric point of view consider the vector to the planet $\backslash mathbf\{r\}\; =\; r\; \backslash hat\{\backslash mathbf\{r\}\}$ where $r$ is the distance to the planet and the direction $\backslash hat\; \{\backslash mathbf\{r\}\}$ is a unit vector. When the planet moves the direction vector $\backslash hat\; \{\backslash mathbf\{r\}\}$ changes:
 $\backslash frac\{d\backslash hat\{\backslash mathbf\{r\}\}\}\{dt\}=\backslash dot\{\backslash hat\{\backslash mathbf\{r\}\}\}\; =\; \backslash dot\backslash theta\; \backslash hat\{\backslash boldsymbol\backslash theta\},\backslash qquad\; \backslash dot\{\backslash hat\{\backslash boldsymbol\backslash theta\}\}\; =\; \backslash dot\backslash theta\; \backslash hat\{\backslash mathbf\{r\}\}$
where $\backslash scriptstyle\; \backslash hat\{\backslash boldsymbol\backslash theta\}$ is the unit vector orthogonal to $\backslash scriptstyle\; \backslash hat\{\backslash mathbf\{r\}\}$ and pointing in the direction of rotation, and $\backslash scriptstyle\; \backslash theta$ is the polar angle, and where a dot on top of the variable signifies differentiation with respect to time.
So differentiating the position vector twice to obtain the velocity and the acceleration vectors:
 $\backslash dot\{\backslash mathbf\{r\}\}\; =\backslash dot\{r\}\; \backslash hat\{\backslash mathbf\{r\}\}\; +\; r\; \backslash dot\{\backslash hat\{\backslash mathbf\{r\}\}\}$
=\dot{r} \hat{\mathbf{r}} + r \dot{\theta} \hat{\boldsymbol{\theta}},
 $\backslash ddot\{\backslash mathbf\{r\}\}$
= (\ddot{r} \hat{\mathbf{r}} +\dot{r} \dot{\hat{\mathbf{r}}} )
+ (\dot{r}\dot{\theta} \hat{\boldsymbol{\theta}} + r\ddot{\theta} \hat{\boldsymbol{\theta}}
+ r\dot{\theta} \dot{\hat{\boldsymbol{\theta}}})
= (\ddot{r}  r\dot{\theta}^2) \hat{\mathbf{r}} + (r\ddot{\theta} + 2\dot{r} \dot{\theta}) \hat{\boldsymbol{\theta}}.
So
 $\backslash ddot\{\backslash mathbf\{r\}\}\; =\; a\_r\; \backslash hat\{\backslash boldsymbol\{r\}\}+a\_\backslash theta\backslash hat\{\backslash boldsymbol\{\backslash theta\}\}$
where the radial acceleration is
 $a\_r=\backslash ddot\{r\}\; \; r\backslash dot\{\backslash theta\}^2$
and the tangential acceleration is
 $a\_\backslash theta=r\backslash ddot\{\backslash theta\}\; +\; 2\backslash dot\{r\}\; \backslash dot\{\backslash theta\}.$
The inverse square law
Kepler's second law implies that the areal velocity $\backslash tfrac\; 1\; 2\; r^2\; \backslash dot\; \backslash theta$ is a constant of motion.
The tangential acceleration $a\_\backslash theta$ is zero by Kepler's second law:
 $\backslash frac\{d\; (r^2\; \backslash dot\; \backslash theta)\}\{dt\}\; =\; r\; (2\; \backslash dot\; r\; \backslash dot\; \backslash theta\; +\; r\; \backslash ddot\; \backslash theta\; )\; =\; r\; a\_\backslash theta\; =\; 0.$
So the acceleration of a planet obeying Kepler's second law is directed exactly towards the sun.
Kepler's first law implies that the area enclosed by the orbit is $\backslash pi\; ab$, where $a$ is the semimajor axis and $b$ is the semiminor axis of the ellipse. Therefore the period $P$ satisfies $\backslash pi\; ab=\backslash tfrac\; 1\; 2\; r^2\backslash dot\; \backslash theta\; P$ or
 $r^2\backslash dot\; \backslash theta\; =\; nab$
where
 $n\; =\; \backslash frac\{2\backslash pi\}\{P\}$
is the mean motion of the planet around the sun.
The radial acceleration $a\_r$ is
 $a\_r\; =\; \backslash ddot\; r\; \; r\; \backslash dot\; \backslash theta^2=\; \backslash ddot\; r\; \; r\; \backslash left(\backslash frac\{nab\}\{r^2\}$
\right)^2= \ddot r \frac{n^2a^2b^2}{r^3}.
Kepler's first law states that the orbit is described by the equation:
 $\backslash frac\{p\}\{r\}\; =\; 1+\; \backslash varepsilon\; \backslash cos\backslash theta.$
Differentiating with respect to time
 $\backslash frac\{p\backslash dot\; r\}\{r^2\}\; =\; \backslash varepsilon\; \backslash sin\; \backslash theta\; \backslash ,\backslash dot\; \backslash theta$
or
 $p\backslash dot\; r\; =\; nab\backslash ,\backslash varepsilon\backslash sin\; \backslash theta.$
Differentiating once more
 $p\backslash ddot\; r\; =nab\; \backslash varepsilon\; \backslash cos\; \backslash theta\; \backslash ,\backslash dot\; \backslash theta$
=nab \varepsilon \cos \theta \,\frac{nab}{r^2}
=\frac{n^2a^2b^2}{r^2}\varepsilon \cos \theta .
The radial acceleration $a\_r$ satisfies
 $p\; a\_r\; =\; \backslash frac\{n^2\; a^2b^2\}\{r^2\}\backslash varepsilon\; \backslash cos\; \backslash theta\; \; p\backslash frac\{n^2\; a^2b^2\}\{r^3\}$
= \frac{n^2a^2b^2}{r^2}\left(\varepsilon \cos \theta  \frac{p}{r}\right).
Substituting the equation of the ellipse gives
 $p\; a\_r\; =\; \backslash frac\{n^2a^2b^2\}\{r^2\}\backslash left(\backslash frac\; p\; r\; \; 1\; \; \backslash frac\; p\; r\backslash right)=\; \backslash frac\{n^2a^2\}\{r^2\}b^2.$
The relation $b^2=pa$ gives the simple final result
 $a\_r=\backslash frac\{n^2a^3\}\{r^2\}.$
This means that the acceleration vector $\backslash mathbf\{\backslash ddot\; r\}$ of any planet obeying Kepler's first and second law satisfies the inverse square law
 $\backslash mathbf\{\backslash ddot\; r\}\; =\; \; \backslash frac\{\backslash alpha\}\{r^2\}\backslash hat\{\backslash mathbf\{r\}\}$
where
 $\backslash alpha\; =\; n^2\; a^3=\backslash frac\{4\backslash pi^2\; a^3\}\{P^2\}\backslash ,$
is a constant, and $\backslash hat\{\backslash mathbf\; r\}$ is the unit vector pointing from the Sun towards the planet, and $r\backslash ,$ is the distance between the planet and the Sun.
According to Kepler's third law, $\backslash alpha$ has the same value for all the planets. So the inverse square law for planetary accelerations applies throughout the entire solar system.
The inverse square law is a differential equation. The solutions to this differential equation includes 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.
Newton's law of gravitation
By Newton's second law, the gravitational force that acts on the planet is:
 $\backslash mathbf\{F\}\; =\; m\; \backslash mathbf\{\backslash ddot\; r\}\; =\; \; \backslash frac\{m\; \backslash alpha\}\{r^2\}\backslash hat\{\backslash mathbf\{r\}\}$
where $\backslash alpha$ only depends on properties of the Sun. According to Newton's third Law, the Sun is also pulled towards 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. So the equation for gravitational force should be
 $\backslash mathbf\{F\}\; =\; \; \backslash frac\{GMm\}\{r^2\}\backslash hat\{\backslash mathbf\{r\}\}$
where $G$ is a universal constant. This is Newton's law of universal gravitation.
The acceleration of solar system body i is, according to Newton's laws:
 $\backslash mathbf\{\backslash ddot\; r\_i\}\; =\; G\backslash sum\_\{j\backslash ne\; i\}\; \backslash frac\{m\_j\}\{r\_\{ij\}^2\}\backslash hat\{\backslash mathbf\{r\}\}\_\{ij\}$
where $m\_j$ is the mass of body j, $r\_\{ij\}$ is the distance between body i and body j, $\backslash hat\{\backslash mathbf\{r\}\}\_\{ij\}$ is the unit vector from body i pointing towards body j, and the vector summation is over all bodies in the world, besides i itself. In the special case where there are only two bodies in the world, Planet and Sun, the acceleration becomes
 $\backslash mathbf\{\backslash ddot\; r\}\_\{Planet\}\; =\; G\backslash frac\{m\_\{Sun\}\}\{r\_^2\}\backslash hat\{\backslash mathbf\{r\}\}\_$
which is the acceleration of the Kepler motion.
Position as a function of time
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 equation called Kepler's equation.
The procedure for calculating the heliocentric polar coordinates (r,θ) of a planet as a function of the time t since perihelion, and the mean motion n = 2π/P, is the following four steps:
 1. Compute the mean anomaly
 $M=nt$
 2. Compute the eccentric anomaly E by solving Kepler's equation:
 $\backslash \; M=E\backslash varepsilon\backslash cdot\backslash sin\; E$
 3. Compute the true anomaly θ by the equation:
 $\backslash tan\backslash frac\; \backslash theta\; 2\; =\; \backslash sqrt\{\backslash frac\{1+\backslash varepsilon\}\{1\backslash varepsilon\}\}\backslash cdot\backslash tan\backslash frac\; E\; 2$
 4. Compute the heliocentric distance r from the first law:
 $r=\backslash frac\; p\; \{1+\backslash varepsilon\backslash cdot\backslash cos\backslash theta\}$
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.
The proof of this procedure is shown below.
Mean anomaly, M
The Keplerian problem assumes an elliptical orbit and the four points:
 s the Sun (at one focus of ellipse);
 z the perihelion
 c the center of the ellipse
 p the planet
and
 $\backslash \; a=cz,$ distance between center and perihelion, the semimajor axis,
 $\backslash \; \backslash varepsilon=\{cs\backslash over\; a\},$ the eccentricity,
 $\backslash \; b=a\backslash sqrt\{1\backslash varepsilon^2\},$ the semiminor axis,
 $\backslash \; r=sp\; ,$ the distance between Sun and planet.
 $\backslash theta=\backslash angle\; zsp,$ the direction to the planet as seen from the Sun, the true anomaly.
The problem is to compute the polar coordinates (r,θ) of the planet from the time since perihelion, t.
It is solved in steps. Kepler considered the circle with the major axis as a diameter, and
 $\backslash \; x,$ the projection of the planet to the auxiliary circle
 $\backslash \; y,$ the point on the circle such that the sector areas zcy and zsx are equal,
 $M=\backslash angle\; zcy,$ the mean anomaly.
The sector areas are related by $zsp=\backslash frac\; b\; a\; \backslash cdotzsx.$
The circular sector area $\backslash \; zcy\; =\; \backslash frac\{a^2\; M\}2.$
The area swept since perihelion,
 $zsp=\backslash frac\; b\; a\; \backslash cdotzsx=\backslash frac\; b\; a\; \backslash cdotzcy=\backslash frac\; b\; a\backslash cdot\backslash frac\{a^2\; M\}2\; =\; \backslash frac\; \{a\; b\; M\}\{2\},$
is by Kepler's second law proportional to time since perihelion. So the mean anomaly, M, is proportional to time since perihelion, t.
 $M=n\; t,$
where n is the mean motion.
Eccentric anomaly, E
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.^{[9]} Kepler's solution is to use
 $E=\backslash angle\; zcx$, x as seen from the centre, the eccentric anomaly
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.
 $\backslash \; zcy=zsx=zcxscx$
 $\backslash frac\{a^2\; M\}2=\backslash frac\{a^2\; E\}2\backslash frac\; \{a\backslash varepsilon\backslash cdot\; a\backslash sin\; E\}2$
Division by a^{2}/2 gives Kepler's equation
 $M=E\backslash varepsilon\backslash cdot\backslash sin\; E.$
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.
Having computed the eccentric anomaly E, the next step is to calculate the true anomaly θ.
True anomaly, θ
Note from the figure that
 $\backslash overrightarrow\{cd\}=\backslash overrightarrow\{cs\}+\backslash overrightarrow\{sd\}$
so that
 $a\backslash cdot\backslash cos\; E=a\backslash cdot\backslash varepsilon+r\backslash cdot\backslash cos\; \backslash theta.$
Dividing by $a$ and inserting from Kepler's first law
 $\backslash \; \backslash frac\; r\; a\; =\backslash frac\{1\backslash varepsilon^2\}\{1+\backslash varepsilon\backslash cdot\backslash cos\; \backslash theta\}$
to get
 $\backslash cos\; E$
=\varepsilon+\frac{1\varepsilon^2}{1+\varepsilon\cdot\cos \theta}\cdot\cos \theta
$=\backslash frac\{\backslash varepsilon\backslash cdot(1+\backslash varepsilon\backslash cdot\backslash cos\; \backslash theta)+(1\backslash varepsilon^2)\backslash cdot\backslash cos\; \backslash theta\}\{1+\backslash varepsilon\backslash cdot\backslash cos\; \backslash theta\}$ $=\backslash frac\{\backslash varepsilon\; +\backslash cos\; \backslash theta\}\{1+\backslash varepsilon\backslash cdot\backslash cos\; \backslash theta\}.$
The result is a usable relationship between the eccentric anomaly E and the true anomaly θ.
A computationally more convenient form follows by substituting into the trigonometric identity:
 $\backslash tan^2\backslash frac\{x\}\{2\}=\backslash frac\{1\backslash cos\; x\}\{1+\backslash cos\; x\}.$
Get
 $\backslash tan^2\backslash frac\{E\}\{2\}$
=\frac{1\cos E}{1+\cos E}
$=\backslash frac\{1\backslash frac\{\backslash varepsilon+\backslash cos\; \backslash theta\}\{1+\backslash varepsilon\backslash cdot\backslash cos\; \backslash theta\}\}\{1+\backslash frac\{\backslash varepsilon+\backslash cos\; \backslash theta\}\{1+\backslash varepsilon\backslash cdot\backslash cos\; \backslash theta\}\}$ $=\backslash frac\{(1+\backslash varepsilon\backslash cdot\backslash cos\; \backslash theta)(\backslash varepsilon+\backslash cos\; \backslash theta)\}\{(1+\backslash varepsilon\backslash cdot\backslash cos\; \backslash theta)+(\backslash varepsilon+\backslash cos\; \backslash theta)\}$ $=\backslash frac\{1\backslash varepsilon\}\{1+\backslash varepsilon\}\backslash cdot\backslash frac\{1\backslash cos\; \backslash theta\}\{1+\backslash cos\; \backslash theta\}=\backslash frac\{1\backslash varepsilon\}\{1+\backslash varepsilon\}\backslash cdot\backslash tan^2\backslash frac\{\backslash theta\}\{2\}.$
Multiplying by (1+ε)/(1−ε) and taking the square root gives the result
 $\backslash tan\backslash frac\; \backslash theta2=\backslash sqrt\backslash frac\{1+\backslash varepsilon\}\{1\backslash varepsilon\}\backslash cdot\backslash tan\backslash frac\; E2.$
We have now completed the third step in the connection between time and position in the orbit.
Distance, r
The fourth step is to compute the heliocentric distance r from the true anomaly θ by Kepler's first law:
 $\backslash \; r=a\backslash cdot\backslash frac\{1\backslash varepsilon^2\}\{1+\backslash varepsilon\backslash cdot\backslash cos\; \backslash theta\}.$
See also
Notes
References
Bibliography
 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 .
 Murray and Dermott, Solar System Dynamics, Cambridge University Press 1999, ISBN 0521575974
 V.I. Arnold, Mathematical Methods of Classical Mechanics, Chapter 2. Springer 1989, ISBN 0387968903
External links
 B.Surendranath Reddy; animation of Kepler's laws: applet
 Crowell, Benjamin, Conservation Laws, online book that gives a proof of the first law without the use of calculus. (see section 5.2, p. 112)
 David McNamara and Gianfranco Vidali, Kepler's Second Law  Java Interactive Tutorial, 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.
 Audio  Cain/Gay (2010) Astronomy Cast Johannes Kepler and His Laws of Planetary Motion
 University of Tennessee's Dept. Physics & Astronomy: Astronomy 161 page on Johannes Kepler: The Laws of Planetary Motion [2]
 Equant compared to Kepler: interactive model [3]
 Kepler's Third Law:interactive model [4]
 Solar System Simulator (Interactive Applet)
 Kepler and His Laws, educational web pages by David P. Stern


   

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Template:Johannes Kepler
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