Figure 1: Illustration of Kepler's three laws with two planetary orbits. (1) The orbits are ellipses, with focal points ƒ1 and ƒ2 for the first planet and ƒ1 and ƒ3 for the second planet. The Sun is placed in focal point ƒ1.
(2) The two shaded sectors A1 and A2 have the same surface area and the time for planet 1 to cover segment A1 is equal to the time to cover segment A2.
(3) The total orbit times for planet 1 and planet 2 have a ratio a13/2 : a23/2.
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.
Figure 2: Kepler's first law placing the Sun at the focus of an elliptical orbit
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 = rmin and for θ = 180°, r = rmax.
Mathematically, an ellipse can be represented by the formula:
The special case of a circle is ε = 0, resulting in r = p = rmin = rmax = a = b and A = π r2.
"A line joining a planet and the Sun sweeps out equal areas during equal intervals of time."
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
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  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. So it used to be known as the harmonic law.
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 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 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.
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.)
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 two-body 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.
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
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
4. Compute the heliocentric distancer from the first law:
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
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.
^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, 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.) 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, page 554.
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 Meriam, J. L. (1966, 1971). Dynamics, 2nd ed. New York: John Wiley. ISBN0-471-59601-9Check date values in: |date= (help).
Murray and Dermott, Solar System Dynamics, Cambridge University Press 1999, ISBN 0-521-57597-4
V.I. Arnold, Mathematical Methods of Classical Mechanics, Chapter 2. Springer 1989, ISBN 0-387-96890-3
B.Surendranath Reddy; animation of Kepler's laws: applet