Kepler's laws of planetary motion

From this, Kepler inferred that other bodies in the Solar System, including those farther away from the Sun, also have elliptical orbits.

The third law expresses that the farther a planet is from the Sun, the longer its orbital period.

It took nearly two centuries for the current formulation of Kepler's work to take on its settled form.

[6][7] 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.

[8] 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.

[9] Small also claimed, against the history, that these were empirical laws, based on inductive reasoning.

[11] Kepler published his first two laws about planetary motion in 1609,[12] having found them by analyzing the astronomical observations of Tycho Brahe.

[16][14] 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.

In 1621, Kepler noted that his third law applies to the four brightest moons of Jupiter.

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.

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

Kepler's second law states that the blue sector has constant area.

for a small piece of the orbit dx and time to cover it dt.

, Kepler's equal area law will hold for any system that conserves angular momentum.

Since any radial force will produce no torque on the planet's motion, angular momentum will be conserved.

[26] The original form of this law (referring to not the semi-major axis, but rather a "mean distance") holds true only for planets with small eccentricities near zero.

[27] 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 ω in terms of the orbital period

The following table shows the data used by Kepler to empirically derive his law: Kepler became aware of John Napier's recent invention of logarithms and log-log graphs before he discovered the pattern.

For comparison, here are modern estimates:[citation needed] Isaac Newton computed in his Philosophiæ Naturalis Principia Mathematica the acceleration of a planet moving according to Kepler's first and second laws.

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

is the polar angle, and where a dot on top of the variable signifies differentiation with respect to time.

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

So the inverse square law for planetary accelerations applies throughout the entire Solar System.

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

Kepler used his two first laws to compute the position of a planet as a function of time.

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 five steps: The position polar coordinates (r,θ) can now be written as a Cartesian vector

Kepler considered the circle with the major axis as a diameter, and The sector areas are related by

But note: Cartesian position coordinates with reference to the center of ellipse are (a cos E, b sin E) With reference to the Sun (with coordinates (c,0) = (ae,0) ), r = (a cos E – ae, b sin E) True anomaly would be arctan(ry/rx), magnitude of r would be √r · r. Note from the figure that

The fourth step is to compute the heliocentric distance r from the true anomaly θ by Kepler's first law:

Illustration of Kepler's laws with two planetary orbits.
  1. The orbits are ellipses, with foci F 1 and F 2 for Planet 1, and F 1 and F 3 for Planet 2. The Sun is at F 1 .
  2. The shaded areas A 1 and A 2 are equal, and are swept out in equal times by Planet 1's orbit.
  3. The ratio of Planet 1's orbit time to Planet 2's is .
Kepler's first law placing the Sun at one of the foci of an elliptical orbit
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 min and for θ = 180° , r = r max .
The same (blue) area is swept out in a fixed time period. 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.
Log-log plot of period T vs semi-major axis a (average of aphelion and perihelion) of some Solar System orbits (crosses denoting Kepler's values) showing that a ³/ T ² is constant (green line)
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 .