Laws of Kepler - SpeedyLook encyclopedia

Statement of the three laws of Kepler

First law - Law of the orbits

The planets of the solar system describe elliptic trajectories whose Sun is a hearth.

In the heliocentric reference frame, the Sun always occupies one of both hearth S of the elliptic trajectory of the planets which revolve around him. With strictly speaking, it is the center of Masse which occupies this hearth; the greatest difference is reached with Jupiter which, because of its important mass, shifts this center of mass of: 743075 km; that is to say 1,07 solar rays - more important displacements can be obtained by cumulating the effects of planets on their Orbite. Except for Mercury, the ellipses which the centres of gravity of planets describe have a very weak orbital Excentricité, and their trajectory is quasi-circular.

From this first law, one deduces that the sun exerts on a planet a centripetal Force.

Second law - Law of the surfaces

If S is the Sun and M an unspecified position of a planet, the surface swept by the segment between two positions C and D is equal to the surface swept by this segment between two positions E and F if the duration which separates the positions C and D is equal to the duration which separates the positions E and F . The speed of a planet thus becomes higher when the planet approaches the sun. It is maximum in the vicinity of the shortest ray (Périhélie), and minimal in the vicinity of the largest ray (Aphélie).

From this second law, one deduces that the force exerted on planet is constantly directed towards the sun.

Third law - Law of the periods

The square of the sidereal Period T of an object (time between two successive passages in front of a remote star) is directly proportional to the cube of the Equatorial radius has elliptic trajectory of the object:

$\ frac {T^2} {a^3} =k$ , with K constant.

From this third law, one deduces that there exists a constant factor between the exerted force and planet considered, which is the constant of universal gravitation, or gravitational Constante masses it.
This formula with those of the ellipse make it possible to calculate the various parameters of an elliptic trajectory from very little information. Indeed, Johann Lambert (1728 - 1777) showed that the knowledge of three dated positions made it possible to find the parameters of the movement (for more deepened discussion, to see Lois of Kepler, demonstration; then Satellite, Orbitographie).

Newtonian form of the third law of Kepler

Newton included/understood the bond between the laws of the traditional Mécanique and the third law of Kepler. He deduced the following formula from it:

$T^2 = \ frac {4 \ pi^2} {\ mathrm {G} M} a^3$ ,

where

$T$ is the period of the object,
$a$ is the half main roads of the elliptic trajectory,
$\ mathrm G$ is the constant of the universal gravitation,
$M$ is the Masse of the object in the center.

The laws of Kepler are not only applicable to planets but to each time a mass is in orbit around an other mass. It is the case, for example, of the the Moon and the Ground or a satellite in orbit around this one.

This law is however applicable only for sufficiently distant important masses. Thus, for the displacement of a electron around the core of a Atom, one enters the field of the Quantum physics, which does not obey the same laws (this one is much more influenced by the electrostatic attraction than by the gravitational forces which play a negligible part).

Discovered new celestial bodies

Johannes Kepler discovered its laws thanks to a considerable work of analysis of the astronomical tables drawn up by Tycho Brahe. In particular the study of Mars enabled him to show that the movement was not epicyclic but elliptic.

Its laws allowed, themselves, to refine astronomical research and to highlight irregularities of movements of known bodies, by an astonishing progression of the analysis.

The most spectacular example was that of the irregularities of Uranus which allowed to the “discovery” of Neptune by the Glassmaker (1811 - 1877), by calculation: discovered confirmed by the observation of Galle (1812 - 1910) in 1846.

External bonds

Definition and demonstration of the laws of Kepler, on the Astrophy site (in French)
Demonstration of the three laws of Kepler and properties of an ellipse

Laws of Kepler, Kinematic demonstration
> elliptic Movement
Problem with two bodies
Revolution copernician
Equation of time. Give the difference between the hour indicated by the Sun and that indicated by a watch.
Keplerian Movement

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