# Kepler's laws

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## Kepler's laws,

three mathematical statements formulated by the German astronomer Johannes Kepler that accurately describe the revolutions of the planets around the sun. Kepler's laws opened the way for the development of celestial mechanics, i.e., the application of the laws of physics to the motions of heavenly bodies. His work shows the hallmarks of great scientific theories: simplicity and universality.### Summary of Kepler's Laws

The first law states that the shape of each planet's orbit is an ellipse**ellipse,**

closed plane curve consisting of all points for which the sum of the distances between a point on the curve and two fixed points (foci) is the same. It is the conic section formed by a plane cutting all the elements of the cone in the same nappe.**.....** Click the link for more information. with the sun at one focus. The sun is thus off-center in the ellipse and the planet's distance from the sun varies as the planet moves through one orbit. The second law specifies quantitatively how the speed of a planet increases as its distance from the sun decreases. If an imaginary line is drawn from the sun to the planet, the line will sweep out areas in space that are shaped like pie slices. The second law states that the area swept out in equal periods of time is the same at all points in the orbit. When the planet is far from the sun and moving slowly, the pie slice will be long and narrow; when the planet is near the sun and moving fast, the pie slice will be short and fat. The third law establishes a relation between the average distance of the planet from the sun (the semimajor axis of the ellipse) and the time to complete one revolution around the sun (the period): the ratio of the cube of the semimajor axis to the square of the period is the same for all the planets including the earth.

### Development of Kepler's Laws

Earlier theories of planetary motion, such as the geocentric Ptolemaic system**Ptolemaic system**

, historically the most influential of the geocentric cosmological theories, i.e., theories that placed the earth motionless at the center of the universe with all celestial bodies revolving around it (see cosmology).**.....** Click the link for more information. and the heliocentric Copernican system**Copernican system,**

first modern European theory of planetary motion that was heliocentric, i.e., that placed the sun motionless at the center of the solar system with all the planets, including the earth, revolving around it.**.....** Click the link for more information. , had allowed only perfect circles as orbits and were therefore compelled to combine many circular motions to reproduce the variations in the planets' motions. Kepler eliminated the epicycles and deferents that had made each planet a special case. His laws apply generally to all orbiting bodies.

Kepler's first and second laws were published in 1609 in *Commentaries on the Motions of Mars.* Because Mars was the planet whose motions were in greatest disagreement with existing theories, its orbit provided the critical test for his hypotheses. To do this Kepler was able to rely on the astronomical observations of his mentor, Tycho Brahe, which were much more accurate than any earlier work. The third law appeared in 1619 in *Harmony of the Worlds.*

### Kepler's Foretelling of the Law of Gravity

Kepler believed that the sun did not sit passively at the center of the solar system but that through some mysterious power or "virtue" actually compelled the planets to hold to their orbits. Because the planets moved slower when they were farther from the sun, this power must diminish with increasing distance. The idea that the planets were controlled by the sun was developed by Isaac Newton in his laws of motion**motion,**

the change of position of one body with respect to another. The rate of change is the speed of the body. If the direction of motion is also given, then the velocity of the body is determined; velocity is a vector quantity, having both magnitude and direction, while speed**.....** Click the link for more information. and law of gravitation**gravitation,**

the attractive force existing between any two particles of matter. **The Law of Universal Gravitation**

Since the gravitational force is experienced by all matter in the universe, from the largest galaxies down to the smallest particles, it is often called**.....** Click the link for more information. . Newton assumed that the sun continuously exerts a force on each planet that pulls the planet toward the sun. He calculated that elliptical orbits would result if the force varied inversely as the square of the distance from the sun (i.e., when the distance doubles, the force becomes four times weaker). His law of universal gravitation predicts that the planets exert small forces on each other although subject to the dominant force of the sun. These small additional forces explain most of the small departures from Kepler's laws revealed by later, more accurate observations.

## Kepler's laws

(**kep**-lerz) The three fundamental laws of planetary motion that were formulated by Johannes Kepler and were based on the detailed observations of the planets made by Tycho Brahe, with whom Kepler had worked. The laws state that

**1.**The orbit of each planet is an ellipse with the Sun at one focus of the ellipse.

**2.**Each planet revolves around the Sun so that the line connecting planet and Sun (the radius vector) sweeps out equal areas in equal times (see illustration). Thus a planet's velocity decreases as it moves farther from the Sun.

**3.**The squares of the sidereal periods of any two planets are proportional to the cubes of their mean distances from the Sun. If the period,

*P*, is measured in years and the mean distance,

*a*, in astronomical units, then

*P*

^{2}≊

*a*

^{3}for any planet.

The first two laws were published in 1609 in *Astronomia Nova * and the third law in 1619 in *Harmonices Mundi *. The third law, sometimes called the *harmonic law*, allowed the relative distances of the planets from the Sun to be calculated from measurements of the planetary orbital periods. Kepler's laws gave a correct description of planetary motion. The physical nature of the motion was not explained until Newton proposed his laws of motion and gravitation. From these laws can be obtained Newton's form of Kepler's third law:

*P*

^{2}= 4π

^{2}

*a*

^{3}/

*G*(

*m*

_{1}+

*m*

_{2})

where *G * is the gravitational constant, *m *_{1} and *m *_{2} are the masses of the Sun and a planet, *a * is the semimajor axis of the planet, and *P * is its sidereal period; all quantities are in SI units. If *P *, *a *, and *m * are expressed in years, astronomical units, and solar masses, then

*P*

^{2}=

*a*

^{3}/(

*m*

_{1}+

*m*

_{2})

## Kepler’s Laws

three laws of planetary motion discovered by J. Kepler at the beginning of the 17th century. Kepler’s principal work, *Astronomia nova,* published in 1609, contained the first two laws. The third law was discovered later; in the third chapter of his fifth book *De harmonice mundi* (1619), Kepler noted that the idea of a new law flashed into his mind suddenly on Mar. 8, 1618, and by May 15 he had completed all the necessary calculations proving that the law was valid. Subsequently, Kepler’s laws were refined and received the following final formulation.

(1) First law. In unperturbed motion (that is, in the two-body problem) the orbit of a moving point is a second-degree curve, at one of whose foci is located the center of the force of attraction. Thus the orbit of a mass point in unperturbed motion is a certain conic section, that is, a circle, ellipse, parabola, or hyperbola.

(2) Second law. In unperturbed motion the area described by the radius vector of the moving point varies proportionally with time. Kepler’s first two laws are valid only for unperturbed motion arising from the action of a force of attraction that is inversely proportional to the square of the distance from the center of the force.

(3) Third law. In the unperturbed elliptical motion of two mass points the products of the squares of the periods of revolution times the sums of the masses of the central and the moving points are to each other as the cubes of the semimajor axes of their orbits, that is,

where T_{1} and T_{2} are the periods of revolution of the two points, m_{1}*a* and m_{2} are their masses, mo is the mass of the central point, and *a*_{1} and *a*_{2} are the semimajor axes of the orbits of the points. Neglecting the masses of the planets compared to the mass of the sun, we obtain Kepler’s third law in its original form: the squares of the periods of revolution of two planets around the sun are to each other as the cubes of the semimajor axes of their elliptical orbits. Kepler’s third law may be applied only in the case of elliptical orbits and therefore is not of such general importance as the first two laws. However, when applied to planets, satellites of planets, and components of binary stars, all of which move in elliptical orbits, it permits several characteristics of celestial bodies to be determined. Thus, on the basis of the Kepler’s third law, it is possible to compute the masses of the planets, taking the mass of the sun *m*_{0} = 1. Knowing from observation the period of revolution of one component of a binary star relative to the other and having measured its parallax, one may find the sum of their masses. If the parallaxes of stars are not known, then on the basis of the assumption that the masses of the components correspond to their physical properties, it is possible to compute the distances to the stars by Kepler’s third law (these are the dynamic parallaxes of stars).

Having discovered the first two laws, Kepler compiled tables of planetary motion on their basis, which were published in 1627 under the name *Tabulae Rudolphinae.* These tables far exceeded in their precision all previous ones, and they were used in practical astronomy in the 17th and 18th centuries. Kepler’s success in explaining planetary motion depended on a new methodological approach to the solution of the problem: for the first time in the history of astronomy an attempt was made to determine planetary orbits directly from observations.

It was clear even to Kepler that the laws discovered by him were not entirely exact. If they are fulfilled with great precision for the planets, then in order to represent the motion of the moon it was necessary to introduce an ellipse with a rotating line of apsides and to add inequalities called evection and variation. These inequalities had already been discovered empirically by Ptolemy in the second century (evection) and Tycho Brahe in the 16th century (variation) and were explained only after the discovery of the law of universal gravitation by I. Newton in the 17th century. Kepler’s laws, found by observation, were derived by Newton as a rigorous solution to the two-body problem.

### REFERENCES

Duboshin, G. N.*Nebesnaia mekhanika. Osnovnye zadachi i metody,*2nd ed. Moscow, 1968.

Subbotin, M. F.

*Vvedenie ν teoreticheskuiu astronomiiu.*Moscow, 1968.

Riabov, Iu. A. “K 350-letiiu otkrytiia pervykh dvukh zakonov Keplera.” In the book

*Astronomicheskii kalendar’ na 1959.*Moscow, 1958.

G. A. CHEBOTAREV

## Kepler's laws

[′kep·lərz ′lȯz]## Kepler's laws

**i**. The path of a planet around the sun is an ellipse, and the sun is at one of the foci of the ellipse.

**ii**. The line joining the planet to the sun sweeps out equal areas in equal times.

**iii**. The square of the time of an orbital revolution is proportional to the cube of the major semi-axis.