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In astronomy, Kepler's laws of planetary motion are three scientific laws describing motion of planets around the Sun.
Astrodynamics 

Orbital mechanics 
Equations

Efficiency measures 
Kepler's laws are:
Johannes Kepler published his first two laws in 1609, having found them by analyzing the astronomical observations of Tycho Brahe.^{[2]} Kepler's third law was published in 1619.^{[2]}
Kepler's laws challenged the longaccepted geocentric models of Aristotle and Ptolemy, and followed the 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 epicycles.^{[2]}
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.
Kepler in 1622 and Godefroy Wendelin in 1643 noted that Kepler's third law applies to the four brightest moons of Jupiter.^{[Nb 1]}
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 laws of motion and law of universal gravitation.
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".^{[3]}
Together with Newton's theories, they are part of the foundation of modern astronomy and physics.^{[4]}
Mathematically, an ellipse can be represented by the formula:
where (r, θ) are polar coordinates, p is the semilatus rectum, and ε is the eccentricity of the ellipse.
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 below).
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.
At θ = 0°, perihelion, the distance is minimum
At θ = 90° and at θ = 270°, the distance is
At θ = 180°, aphelion, the distance is maximum
The semimajor axis a is the arithmetic mean between r_{min} and r_{max}:
The semiminor axis b is the geometric mean between r_{min} and r_{max}:
The semilatus rectum p is the harmonic mean between r_{min} and r_{max}:
The eccentricity ε is the coefficient of variation between r_{min} and r_{max}:
The area of the ellipse is
The special case of a circle is ε = 0, resulting in r = p = r_{min} = r_{max} = a = b and A = π r^{2}.
In a small time the planet sweeps out a small triangle having base line and height and area and so the constant areal velocity is
The planet moves faster when it is closer to the Sun.
The area enclosed by the elliptical orbit is So the period satisfies
and the mean motion of the planet around the Sun satisfies
The third law, published by Kepler in 1619 [1] captures the relationship between the distance of planets from the Sun, and their orbital periods.
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.^{[5]} So it used to be known as the harmonic law.^{[6]}
Mathematically, the law says that the expression has the same value for all the planets in the solar system.
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:
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 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 throught 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
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.
Isaac Newton computed in his Philosophiæ Naturalis Principia Mathematica the acceleration of a planet moving according to Kepler's first and second law.
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:
Here the Sun plays an unsymmetrical part, which is unjustified. So he assumed Newton's law of universal gravitation:
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.
Below comes the detailed calculation of the acceleration of a planet moving according to Kepler's first and second laws.
From the heliocentric point of view consider the vector to the planet where is the distance to the planet and the direction is a unit vector. When the planet moves the direction vector changes:
where is the unit vector orthogonal to and pointing in the direction of rotation, and 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:
So
where the radial acceleration is
and the tangential acceleration is
Kepler's laws say that
is constant.
The tangential acceleration is zero:
So the acceleration of a planet obeying Kepler's laws is directed towards the sun.
The radial acceleration is
Kepler's first law states that the orbit is described by the equation:
Differentiating with respect to time
or
Differentiating once more
The radial acceleration satisfies
Substituting the equation of the ellipse gives
The relation gives the simple final result
This means that the acceleration vector of any planet obeying Kepler's first and second law satisfies the inverse square law
where
is a constant, and is the unit vector pointing from the Sun towards the planet, and is the distance between the planet and the Sun.
According to Kepler's third law, 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 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.
By Newton's second law, the gravitational force that acts on the planet is:
where is the mass of the planet and has the same value for all the planets. According to 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, . So
where 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:
where is the mass of body j, is the distance between body i and body j, 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
which is the acceleration of the Kepler motion.
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, is the following four steps:
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.
The Keplerian problem assumes an elliptical orbit and the four points:
and
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
The sector areas are related by
The circular sector area
The area swept since perihelion,
is by Kepler's second law proportional to time since perihelion. So the mean anomaly, M, is proportional to time since perihelion, t.
where n is the mean motion.
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.^{[7]} Kepler's solution is to use
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.
Division by a^{2}/2 gives Kepler's equation
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.
Note from the figure that
so that
Dividing by and inserting from Kepler's first law
to get
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:
Get
Multiplying by (1+ε)/(1−ε) and taking the square root gives the result
We have now completed the third step in the connection between time and position in the orbit.
The fourth step is to compute the heliocentric distance r from the true anomaly θ by Kepler's first law:

