In astronomy, Kepler's laws of planetary motion are three scientific laws describing the motion of planets around the Sun, published by Johannes Kepler between 1609 and 1619. These improved the heliocentric theory of Nicolaus Copernicus, replacing its circular orbits with epicycles with elliptical trajectories, and explaining how planetary velocities vary.^[2] The laws state that:

The elliptical orbits of planets were indicated by calculations of the orbit of Mars.^[4] From this, Kepler inferred that other bodies in the Solar System, including those farther away from the Sun, also have elliptical orbits. The second law helps to establish that when a planet is closer to the Sun, it travels faster. The third law expresses that the further a planet is from the Sun, the longer its orbit, and vice versa.

Isaac Newton showed in 1687 that relationships like Kepler's would apply in the Solar System to a good approximation, as a consequence of his own laws of motion and law of universal gravitation.

Kepler's laws improved the model of Copernicus. If the eccentricities of the planetary orbits are taken as zero, then Kepler basically agreed with Copernicus:

The eccentricities of the orbits of those planets known to Copernicus and Kepler are small, so the foregoing rules give fair approximations of planetary motion, but Kepler's laws fit the observations better than does the model proposed by Copernicus.

Kepler's corrections are not at all obvious:

The 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

which is close to the correct value (0.016710219) (see Earth's orbit).

The calculation is correct when perihelion, the date the Earth is closest to the Sun, falls on a solstice. The current perihelion, near January 3, is fairly close to the solstice of December 21 or 22.

It took nearly two centuries for current formulation of Kepler's work to take on its settled form. Voltaire's Eléments de la philosophie de Newton (Elements of Newton's Philosophy) of 1738 was the first publication to use the terminology of "laws".^[5][6] The Biographical Encyclopedia of Astronomers in its article on Kepler (p. 620) states that the terminology of scientific laws for these discoveries was current at least from the time of Joseph de Lalande.^[7] It was the exposition of Robert Small, in An account of the astronomical discoveries of Kepler (1814) that made up the set of three laws, by adding in the third.^[8] Small also claimed, against the history, that these were empirical laws, based on inductive reasoning.^[6][9]

Further, the current usage of "Kepler's Second Law" is something of a misnomer. Kepler had two versions, related in a qualitative sense: the "distance law" and the "area law". The "area law" is what became the Second Law in the set of three; but Kepler did himself not privilege it in that way.^[10]

Johannes Kepler published his first two laws about planetary motion in 1609, having found them by analyzing the astronomical observations of Tycho Brahe.^[11][2][12] Kepler's third law was published in 1619.^[13][2] Kepler had believed in the Copernican model of the solar system, which called for circular orbits, but he could not reconcile Brahe's highly precise observations with a circular fit to Mars' orbit – Mars coincidentally having the highest eccentricity of all planets except Mercury.^[14] His first law reflected this discovery.

Kepler in 1621 and Godefroy Wendelin in 1643 noted that Kepler's third law applies to the four brightest moons of Jupiter.^[1] The second law, in the "area law" form, was contested by Nicolaus Mercator in a book from 1664, but by 1670 his Philosophical Transactions were in its favour. As the century proceeded it became more widely accepted.^[15] The reception in Germany changed noticeably between 1688, the year in which Newton's Principia was published and was taken to be basically Copernican, and 1690, by which time work of Gottfried Leibniz on Kepler had been published.^[16]

Newton was credited with understanding that the second law is not special to the inverse square law of gravitation, being a consequence just of the radial nature of that law; while the other laws do depend on the inverse square form of the attraction. Carl Runge and Wilhelm Lenz much later identified a symmetry principle in the phase space of planetary motion (the orthogonal group O(4) acting) which accounts for the first and third laws in the case of Newtonian gravitation, as conservation of angular momentum does via rotational symmetry for the second law.^[17]

The mathematical model of the kinematics of a planet subject to the laws allows a large range of further calculations.

Mathematically, an ellipse can be represented by the formula:

For an ellipse 0 < ε < 1 ; in the limiting case ε = 0, the orbit is a circle with the Sun at the centre (i.e. where there is zero eccentricity).

At θ = 0°, perihelion, the distance is minimum

At θ = 180°, aphelion, the distance is maximum (by definition, aphelion is – invariably – perihelion plus 180°)

The semi-major axis a is the arithmetic mean between rmin and rmax:

The semi-minor axis b is the geometric mean between rmin and rmax:

The semi-latus rectum p is the harmonic mean between rmin and rmax:

The eccentricity ε is the coefficient of variation between rmin and rmax:

The area of the ellipse is

The special case of a circle is ε = 0, resulting in r = p = rmin = rmax = a = b and A = πr2.

The orbital radius and angular velocity of the planet in the elliptical orbit will vary. This is shown in the animation: the planet travels faster when closer to the Sun, then slower when farther from the Sun. Kepler's second law states that the blue sector has constant area.

and the mean motion of the planet around the Sun

satisfies

This captures the relationship between the distance of planets from the Sun, and their orbital periods.

Kepler enunciated in 1619^[13] 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.^[18] So it was known as the harmonic law.^[19]

Using Newton's Law of gravitation (published 1687), this relation can be found in the case of a circular orbit by setting the centripetal force equal to the gravitational force:

Then, expressing the angular velocity in terms of the orbital period and then rearranging, we find Kepler's Third Law:

The following table shows the data used by Kepler to empirically derive his law:

Upon finding this pattern Kepler wrote:^[20]

For comparison, here are modern estimates:

Isaac Newton computed in his Philosophiæ Naturalis Principia Mathematica the acceleration of a planet moving according to Kepler's first and second law.

This implies that the Sun may be the physical cause of the acceleration of planets. However, Newton states in his Principia that he considers forces from a mathematical point of view, not a physical, thereby taking an instrumentalist view.^[21] Moreover, he does not assign a cause to gravity.^[22]

Newton defined the force acting on a planet to be the product of its mass and the acceleration (see Newton's laws of motion). So:

The Sun plays an unsymmetrical part, which is unjustified. So he assumed, in Newton's law of universal gravitation:

As the planets have small masses compared to that of the Sun, the orbits conform approximately to Kepler's laws. Newton's model improves upon Kepler's model, and fits actual observations more accurately (see two-body problem).

Below comes the detailed calculation of the acceleration of a planet moving according to Kepler's first and second laws.

Differentiate the position vector twice to obtain the velocity vector and the acceleration vector:

So

where the radial acceleration is

and the transversal acceleration is

Kepler's second law says that

is constant.

So the acceleration of a planet obeying Kepler's second law is directed towards the Sun.

Kepler's first law states that the orbit is described by the equation:

Differentiating with respect to time

or

Differentiating once more

Substituting the equation of the ellipse gives

where

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:

The acceleration of solar system body number i is, according to Newton's laws:

In the special case where there are only two bodies in the Solar System, Earth and Sun, the acceleration becomes

which is the acceleration of the Kepler motion. So this Earth moves around the Sun according to Kepler's laws.

If the two bodies in the Solar System are Moon and Earth the acceleration of the Moon becomes

So in this approximation, the Moon moves around the Earth according to Kepler's laws.

In the three-body case the accelerations are

These accelerations are not those of Kepler orbits, and the three-body problem is complicated. But Keplerian approximation is the basis for perturbation calculations. See Lunar theory.

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 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 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.^[24] 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 a2/2 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.

But note: Cartesian position coordinates reference the center of ellipse are (a cos E, b sin E)

Reference the Sun (with coordinates (c,0) = (ae,0) ), r = (a cos E – ae, b sin E)

True anomaly would be arctan(r**y/r**x), magnitude of r would be √r · r.

Note from the figure that

so that

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 gives the result

This is 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:

Using the relation above between θ and E the final equation for the distance r is: