# Kepler's laws

Graphical summary of the three Kepler laws:
1. Two elliptical orbits with the sun (sun) at the focal point F 1 .
F 2 and a 1 are the other focal point or the major semi-axis for Planet1, F 3 and a 2 for Planet2.
2. The two gray sectors A 1 and A 2 , which have the same area, are swept over in the same time.
3. The squares of the orbital times of Planet1 and Planet2 behave like a 1 3  : a 2 3 .

The three Kepler laws are the fundamental laws of the orbit of the planets around the sun. Johannes Kepler found it at the beginning of the 17th century when he tried to adapt the heliocentric system according to Copernicus to the precise astronomical observations of Tycho Brahe . At the end of the 17th century, Isaac Newton was able to derive Kepler's laws in the classical mechanics he founded as an exact solution to the two-body problem , if there is an attraction between the two bodies that decreases with the square of the distance. Kepler's laws are:

1. Kepler's law

The planets move on elliptical orbits. The sun is in one of its focal points.

2. Kepler's law

A beam drawn from the sun to the planet covers areas of the same size at the same time.

3. Kepler's law

The squares of the orbital times of two planets behave like the cubes (third powers ) of the major semiaxes of their orbital ellipses .

Kepler's laws apply to the planets in the solar system to a good approximation. The deviations in the positions in the sky are usually less than 1 '( arc minute ), i.e. about 1/30 full moon diameter. They are known as orbital disruptions and are mainly based on the fact that the planets are not only attracted by the sun, but also attract one another. Further, much smaller corrections can be calculated according to general relativity .

Kepler's Laws were an essential step in overcoming medieval science and establishing modern science. They are of fundamental importance in astronomy to this day .

## history

### Kepler's starting point

Kepler was convinced of the heliocentric system of Copernicus (1543) because it was conceptually simpler and got along with less assumed circles and parameters than the geocentric system of Ptolemy, which had prevailed since about 150 AD. The Copernican system also made it possible to ask further questions, because for the first time the size of all planetary orbits in relation to the size of the earth's orbit was clearly defined here, without trying further hypotheses. Kepler spent his life looking for a deeper explanation for these proportions. It also became clear at that time that the planets could not be moved by fixed rotating crystal spheres in a given way along their deferents and epicycles , because, according to Tycho Brahe's observations on the comet of 1577, it should have penetrated several such shells. Apparently the planets found their way through space on their own. Their speeds, which could be determined from the size of their orbit and their orbital time, were also in contradiction to the philosophically based assumptions in the Ptolemaic system. It was well known that it did not remain constant along the path, but now, like the shape of the path, demanded a new explanation. All of this motivated Kepler to take the decisive step in astronomy, to assume "physical" causes for the planetary motion, that is, those that were already revealed when studying earthly motions. In doing so, he contradicted the previously sacrosanct Aristotelian doctrine of a fundamental opposition between heaven and earth and made an important contribution to the Copernican revolution .

In order to investigate this more precisely, it was first necessary to determine the actual planetary orbits. For this purpose, Kepler had access to data from Tycho's decades of sky observations, which were not only much more accurate for the first time since ancient times (maximum uncertainty approx. 2 ′), but also extended over large parts of the planetary orbits. When evaluating these data, Kepler for the first time consistently followed the guiding principle that the physical cause of the planetary movements lies in the sun, and consequently not in the fictitious point called the " central sun " (introduced by Ptolemy and placed in the empty center of that circle by Copernicus that he had assigned to the earth), but in the true physical sun. He imagined that the sun was acting on the planets like a magnet, and he also executed this picture in detail.

In his work, Kepler broke new ground in other ways. As a starting point for the analysis of the orbits, unlike all earlier astronomers, he did not take the uniform circular motion prescribed by the philosophers since Plato and Aristotle, to which further uniform circular motions were then added to improve the correspondence with the planetary positions observed in the sky ( epicyclic theory ). Rather, he tried to reconstruct the actual orbits and the variable speed with which the planets move on them directly from the sky observations.

Thirdly, Kepler broke new ground in the way his work was presented. Until then, it was common for astronomers to describe their worldview in a fully developed state. They explained how to build it up piece by piece, citing philosophical or theological justifications for each of the necessary individual assumptions. Kepler, on the other hand, described step by step the actual progress of his many years of work, including his intermittent failures due to unsuitable approaches. In 1609 he published the first part of his results as Astronomia Nova with the significant addition in the title (translated) "New astronomy, causally founded, or physics of the sky, [...] according to the observations of the nobleman Tycho Brahe". The work culminates in the first two Kepler laws, which each apply to a single planetary orbit. Kepler's deeper explanation of the entire system and the relationships between the planetary orbits appeared in 1619 under the title Harmonices mundi ("Harmonies of the World"). There is a proposition in it that later became known as the third Kepler law.

### Kepler's approach

Kepler's first result at work was that neither the Ptolemaic nor the Copernican system could reproduce the planetary positions with sufficient accuracy, even after improving individual parameters, e.g. B. the eccentricities . However, he continued to use these models as an approximation in order to select those from Tycho's observations that would be most suitable for a more precise characterization of the orbits. So he found that the eccentric orbits of Mars and Earth with respect to the fixed stars remain fixed (with sufficient accuracy), that each runs in a plane in which the sun is, and that the two planes are slightly inclined towards each other.

So Kepler could assume that Mars, although its exact orbit was still unknown, would take up the same position in space after each of its orbits around the sun, even if it appears at different heavenly positions when viewed from the earth, because then the earth each Sometimes it is at a different point in its path. From this he initially determined the earth's orbit with approx. 4-digit accuracy. On this basis, he evaluated the other observations of Mars, in which the deviations from a circular path are clearer than in the case of Earth. When, after many failures and long trials, he could not reduce the maximum error in the position of Mars in the sky below 8 ′ (about 1/4 full moon diameter), he took another attempt and found - half by chance - that the Martian orbit was best through a Ellipse is to be reproduced with the sun in one of its focal points. This result was also confirmed for the earth's orbit, and it also matched all other planets observed by Tycho. Kepler knew that an elliptical path can also be composed exactly of two circular movements, but he did not consider this possibility any further. For an exact representation of the movement, these circular movements would have to run around their respective center points with variable speed, for which no physical reason is apparent:

"Kepler did not make use of the epicyclic generation of the ellipse because it does not agree with the natural causes which produce the ellipse [...]. "

In the subsequent search for the law of the entire structure of the solar system, which in turn lasted about a decade, Kepler pursued the idea of ​​a harmony underlying the creation plan, which - as in the case of harmony in music - should be found in simple numerical relationships . He published his result in 1619 as Harmonice mundi ('Harmonies of the World'). For later astronomy only the short message (in the 5th book of the work) is of lasting value, according to which the squares of the orbital times of all planets are in the same ratio as (in modern words) the third powers of the major semiaxes of their orbital ellipses.

Kepler also looked for a physical explanation of how the sun could act on the planets to cause the observed movements. His considerations about a magnetic action at a distance or an anima motrix inherent in the planets remained fruitless. Isaac Newton was later able to prove that the three Kepler laws represent the exact solution of the motion of a body under the action of a force according to Newton's law of gravitation . This is considered a significant step in the development of classical mechanics and modern science as a whole.

### Heliocentric and fundamental formulation of the laws

Kepler formulated the law for the planets known to him. But the cosmological principle applies to the laws , since they are valid everywhere in the universe .

However, the heliocentric case of the solar system is by far the most important, which is why they are often formulated restrictively in the literature only for planets. Of course, they also apply to moons , the asteroid belt and the Oort cloud , or the rings of Jupiter and Saturn , to star clusters as well as to objects in orbit around the center of a galaxy , and to all other objects in space . They also form the basis of space travel and the orbits of satellites .

On a cosmic scale, however, the relativistic effects are beginning to have an increasing effect, and the differences to the Kepler model serve primarily as a test criterion for more modern concepts about astrophysics. The mechanisms of formation in spiral galaxies, for example, can no longer be consistently reproduced with a model based purely on Kepler's laws.

## Derivation and modern representation

Kepler tried to describe the planetary movements with his laws. From the observed values, especially the orbit of Mars, he knew that he had to deviate from the ideal of circular orbits. Unlike Newton's later theoretical derivations, his laws are therefore empirical. From today's point of view, however, we can start from the knowledge of Newton's gravity and thus justify the validity of Kepler's laws.

Kepler's laws can elegantly be derived directly from Newton's theory of motion.

The first law follows from Clairaut's equation , which describes a complete solution of a movement in rotationally symmetrical force fields.

The second law is a geometric interpretation of the conservation of angular momentum .

Using integration, the Kepler equation and the Gaussian constant , the third law follows from the second or, using the hodograph, directly from Newton's laws. Furthermore, according to the principle of mechanical similarity , it follows directly from the inverse-quadratic dependence of the gravitational force on the distance.

### 1. Kepler's law (theorem of ellipses)

1. Kepler's law
The orbit of a satellite is an ellipse . One of their focal points lies in the center of gravity of the system.

This law results from Newton's law of gravitation , provided that the mass of the central body is significantly greater than that of the satellite and the effect of the satellite on the central body can be neglected.

The energy for a satellite with mass in the Newtonian gravitational field of the sun with mass is in cylindrical coordinates${\ displaystyle m}$${\ displaystyle M}$

${\ displaystyle E = E _ {\ mathrm {kin}} + E _ {\ mathrm {pot}} = {\ frac {1} {2}} m ({\ dot {r}} ^ {2} + r ^ { 2} {\ dot {\ phi}} ^ {2}) - {\ frac {GMm} {r}}.}$

With the help of angular momentum and ${\ displaystyle L = mr ^ {2} d \ phi / dt}$

${\ displaystyle {\ frac {dr} {dt}} = {\ frac {dr} {d \ phi}} {\ frac {d \ phi} {dt}} \ \ Rightarrow \ {\ dot {r}} = {\ frac {dr} {d \ phi}} {\ dot {\ phi}}}$

allows the energy equation

${\ displaystyle \ left ({\ frac {dr} {d \ phi}} \ right) ^ {2} = 2m {\ frac {r ^ {4}} {L ^ {2}}} \ left [E + { \ frac {GMm} {r}} - {\ frac {L ^ {2}} {2mr ^ {2}}} \ right]}$

reshape. This differential equation is used with the polar coordinate representation

${\ displaystyle r (\ phi) = {\ frac {p} {1+ \ varepsilon \ cdot \ cos \ phi}}}$

of a conic section. The derivative

${\ displaystyle {\ frac {dr} {d \ phi}} = {\ frac {p \ varepsilon \ cdot \ sin \ phi} {(1+ \ varepsilon \ cdot \ cos \ phi) ^ {2}}}}$

and all expressions that contain are formed by substituting the to ${\ displaystyle \ phi}$

${\ displaystyle \ varepsilon \ cdot \ cos \ phi = {\ frac {p} {r}} - 1}$

The transformed equation of the trajectory is eliminated:

${\ displaystyle {\ frac {dr} {d \ phi}} = {\ frac {r ^ {2} \ varepsilon \ cdot \ sin \ phi} {p}} \ Rightarrow}$
${\ displaystyle \ left ({\ frac {dr} {d \ phi}} \ right) ^ {2} = {\ frac {r ^ {4} \ varepsilon ^ {2} (1- \ cos ^ {2} \ phi)} {p ^ {2}}} = {\ frac {r ^ {4}} {p ^ {2}}} \ left (\ varepsilon ^ {2} - \ left ({\ frac {p} {r}} - 1 \ right) ^ {2} \ right) = {\ frac {r ^ {4}} {p ^ {2}}} \ left (\ varepsilon ^ {2} -1 + {\ frac {2p} {r}} - {\ frac {p ^ {2}} {r ^ {2}}} \ right) \ Rightarrow}$
${\ displaystyle p = {\ frac {({\ frac {L} {m}}) ^ {2}} {GM}} {\ text {and}} \ varepsilon = {\ sqrt {1 + {\ frac { 2 {\ frac {E} {m}} ({\ frac {L} {m}}) ^ {2}} {G ^ {2} M ^ {2}}}}}}$

by comparing the coefficients of the powers of ${\ displaystyle r.}$

This solution depends only on the specific energy and the specific orbital angular momentum . The parameter and the numerical eccentricity are the design elements of the path . The following applies in the event : ${\ displaystyle E / m}$${\ displaystyle L / m}$${\ displaystyle p}$ ${\ displaystyle \ varepsilon}$${\ displaystyle 0 <\ varepsilon <1 \, (E <0)}$

${\ displaystyle {\ text {The conic section described by}} r (\ phi) {\ text {is an ellipse.}}}$1. Kepler's law
 Major semi-axis ${\ displaystyle a = {\ frac {p} {1- \ varepsilon ^ {2}}}}$ Small semi-axis ${\ displaystyle b = {\ frac {p} {\ sqrt {1- \ varepsilon ^ {2}}}}}$ Foci ${\ displaystyle F_ {1} = (0,0)}$${\ displaystyle F_ {2} = \ left (- {\ frac {2 \ varepsilon p} {1- \ varepsilon ^ {2}}}, 0 \ right)}$ Pericenter ${\ displaystyle P = \ left (r _ {\ mathrm {min}} = {\ frac {p} {1+ \ varepsilon}}, 0 \ right)}$ Apocenter ${\ displaystyle A = \ left (-r _ {\ mathrm {max}} = - {\ frac {p} {1- \ varepsilon}}, 0 \ right)}$
Two bodies revolve around their common center of gravity -
here idealized circular paths as a special shape of the ellipse

If (unlike Kepler) one does not use a centrically symmetrical force field as a basis, but reciprocally acting gravitation, then elliptical orbits are also formed. Both bodies move, however, the center of the orbits is the common center of gravity of the “central body” and the satellite, the total mass of the system is to be assumed as the fictitious central mass. However, the common center of gravity of the solar system planets and the sun (the barycenter of the solar system) is still within the sun: The sun does not rest relative to it, but swings a little under the influence of the orbiting planets ( length of the sun  ≠ 0). The earth-moon system , on the other hand, shows greater fluctuations in terms of the orbit geometry, here too the center of gravity is still within the earth. Satellites even react to fluctuations in the force field, which is irregular due to the shape of the earth .

Although Kepler's laws were originally formulated only for the gravitational force, the above solution also applies to the Coulomb force . For charges that repel each other, the effective potential is then always positive and only hyperbolic orbits are obtained.

For forces there is also a conserved quantity that is decisive for the direction of the elliptical orbit, the Runge-Lenz vector , which points along the main axis. Small changes in the force field (usually due to the influences of the other planets) let this vector slowly change its direction. B. the perihelion of Mercury's orbit can be explained. ${\ displaystyle 1 / r ^ {2}}$

### 2. Kepler's law (area theorem)

2. Kepler's law
At the same time, the driving beam sweeps the object – center of gravity over the same areas.

The driving beam is the line connecting the center of gravity of a celestial body , e.g. B. a planet or moon, and the center of gravity , z. B. in a first approximation of the sun or the planet around which it moves.

Illustration for deriving the area set from a small time step

A simple derivation is obtained if one considers the areas that the driving beam covers in a small period of time. In the graphic on the right, Z is the center of force. The Trabant initially moves from A to B. If its speed did not change, it would move from B to C in the next time step . It can quickly be seen that the two triangles ZAB and ZBC contain the same area. If a force is now acting in direction Z, the speed is deflected by an amount that is parallel to the common base ZB of the two triangles. Instead of C the Trabant lands at C '. Since the two triangles ZBC and ZBC 'have the same base and the same height, their area is also the same. This means that the area law applies to the two small time segments and . If one integrates such small time steps (with infinitesimal time steps ), one obtains the area theorem. ${\ displaystyle v}$${\ displaystyle \ Delta v}$${\ displaystyle [- \ Delta t, 0]}$${\ displaystyle [0, \ Delta t]}$${\ displaystyle \ Delta t}$

The swept area is for an infinitesimal time step

${\ displaystyle F (t, t + \ operatorname {d} t) = 1/2 | \ mathbf {r} (t) \ times \ mathbf {\ dot {r}} (t) \ operatorname {d} t | = {\ frac {L} {2m}} \ operatorname {d} t}$.

Since the angular momentum is due to a central force

${\ displaystyle \ mathbf {\ dot {L}} = m (\ mathbf {\ dot {r}} \ times \ mathbf {\ dot {r}} + \ mathbf {r} \ times \ mathbf {\ ddot {r }}) = m \ mathbf {r} \ times \ mathbf {\ ddot {r}} = m \ mathbf {r} \ times f (r) \ mathbf {r} = 0}$

is constant, the area integral is straight

${\ displaystyle F (t_ {0}, t) = {\ frac {1} {2}} \ int _ {t_ {0}} ^ {t} | \ mathbf {r} (\ tau) \ times \ mathbf {\ dot {r}} (\ tau) | \ operatorname {d} \ tau = {\ frac {1} {2}} {\ frac {L} {m}} \ int _ {t_ {0}} ^ {t} \ operatorname {d} \ tau = {\ frac {1} {2}} {\ frac {L} {m}} (t-t_ {0})}$.

The same swept area results for the same time differences . ${\ displaystyle t-t_ {0}}$

Kepler's 2nd law defines both the geometric basis of an astrometric orbit (as a path in a plane) and its orbit dynamics (the behavior over time). Kepler formulated the law only for the orbit of the planets around the sun, but it also applies to non-closed orbits . In contrast to the other two laws, Kepler's second law is not restricted to the force of gravity (in fact, Kepler's anima motrix also assumed a force), but applies in general to all central forces and movements with constant angular momentum. Kepler was only interested in a description of the planetary orbits, but the second law is already the first formulation of the law we know today as conservation of angular momentum . The 2nd Kepler's law can be seen as a special formulation of the angular momentum law, see also the law of swirl # surface theorem . ${\ displaystyle 1 / r ^ {2}}$${\ displaystyle 1 / r}$

Kepler's 2nd law also has two fundamental consequences for the movement relationships in multi-body systems, both for solar systems and for space travel: The constancy of the orbital normal vector means that elementary celestial mechanics is a flat problem. In fact, there are also deviations here due to the volumes of the celestial bodies, so that the mass lies outside the plane of the orbit and the planes of the orbit precess (change their position in space). Therefore, the orbits of the planets do not all lie in one plane (the ideal solar system plane , the ecliptic ), they rather show an inclination and perihelion rotation , and the ecliptical latitude of the sun also fluctuates . Conversely, it is relatively easy to move a spacecraft in the plane of the solar system, but it is extremely time-consuming to position a probe over the north pole of the sun.

The constancy of the surface velocity means that an imaginary connecting line between the central body, more precisely the center of gravity of the two celestial bodies, and a satellite always sweeps over the same area at the same time. So a body moves faster when it is close to its center of gravity and the slower the further it is away from it. This applies for example to the course of the earth around the sun as well as to the course of the moon or a satellite around the earth. A path presents itself as a constant free fall , swinging close to the center of gravity, and ascent again to the furthest culmination point of the path: The body becomes faster and faster, has the highest speed in the pericenter (point closest to the center) and then becomes slower and slower to the apocenter ( most distant point), from which it accelerates again. Seen in this way, the Keplerellipse is a special case of the crooked throw that closes in its path. This consideration plays a central role in space physics, where it is a matter of generating a suitable orbit with a suitably selected initial impulse (through the start): the more circular the orbit, the more uniform the orbital speed.

### 3. Kepler's law

The squares of the orbital times and two satellites around a common center are proportional to the third powers of the major semiaxes and their elliptical orbits.${\ displaystyle T_ {1}}$${\ displaystyle T_ {2}}$ ${\ displaystyle a_ {1}}$${\ displaystyle a_ {2}}$

or

The squares of the orbital times are in the same ratio as the cubes of the major semiaxes:
${\ displaystyle \ left ({\ frac {T_ {1}} {T_ {2}}} \ right) ^ {2} = \ left ({\ frac {a_ {1}} {a_ {2}}} \ right) ^ {3}}$3. Kepler's law

Kepler used the mean distances from the sun for half the orbit axes (in the sense of the mean of perihelion distance and apheld distance ). ${\ displaystyle a_ {1,2}}$

${\ displaystyle C = {\ frac {T ^ {2}} {a ^ {3}}}}$... 3. Kepler's law, mass-independent formulation with Kepler's constant of the central mass ( Gaussian gravitational constant of the solar system)

In combination with the law of gravitation , the 3rd law of Kepler for the motion of two masses and the form ${\ displaystyle M}$${\ displaystyle m}$

${\ displaystyle T ^ {2} = {\ frac {4 \ pi ^ {2}} {G (M + m)}} \ cdot a ^ {3} \ approx {\ frac {4 \ pi ^ {2} } {GM}} \ cdot a ^ {3}}$3. Kepler's law, formulation with two masses

The approximation applies if the mass is negligibly small compared to (e.g. in the solar system). With this form one can determine the total mass of binary star systems from the measurement of the period of revolution and the distance. ${\ displaystyle m}$${\ displaystyle M}$

Taking into account the different masses of two celestial bodies and the above formula, a more exact formulation of Kepler's 3rd law is:

${\ displaystyle \ left ({\ frac {T_ {1}} {T_ {2}}} \ right) ^ {2} = \ left ({\ frac {a_ {1}} {a_ {2}}} \ right) ^ {3} {\ frac {M + m_ {2}} {M + m_ {1}}}}$3. Kepler's law, formulation with three masses

Obviously, the deviation only becomes more important if both satellites differ greatly in their masses and the central object has a mass that does not deviate significantly from that of one of the two satellites. ${\ displaystyle M}$

The 3rd Kepler law applies to all forces that decrease quadratically with the distance, as one can easily deduce from the scale consideration. In the equation

${\ displaystyle m {\ frac {d ^ {2} r} {(dt) ^ {2}}} = {\ frac {GMm} {r ^ {2}}}}$

appears in the third power and square. A scale transformation thus gives the same equation if is. On the other hand, is characterized quickly be seen that the analogue of 3. Kepler's law for closed paths in a force field for any currently is. ${\ displaystyle r}$${\ displaystyle t}$${\ displaystyle r '= \ alpha r, t' = \ beta t}$${\ displaystyle \ alpha ^ {3} = \ beta ^ {2}}$${\ displaystyle 1 / r ^ {k}}$${\ displaystyle k}$${\ displaystyle (a_ {1} / a_ {2}) ^ {k + 1} = (T_ {1} / T_ {2}) ^ {2}}$

## literature

• Johannes Kepler: Astronomia nova aitiologetos seu Physica coelestis . In: Max Caspar (Ed.): Collected works . tape 3 . C. H. Beck, Munich 1938.
• Johannes Kepler: Harmonices Mundi Libri V . In: Max Caspar (Ed.): Collected works . tape 6 . C. H. Beck, Munich 1990, ISBN 3-406-01648-0 .
• Andreas Guthmann: Introduction to celestial mechanics and ephemeris calculus . BI-Wiss.-Verlag, Mannheim 1994, ISBN 3-411-17051-4 .

Commons : Kepler's Laws  - collection of images, videos and audio files

## Individual evidence

1. Thomas S. Kuhn: The Copernican Revolution. Vieweg (Braunschweig 1980), ISBN 3-528-08433-2 .
2. ^ Carl B. Boyer: Note on Epicycles & the Ellipse from Copernicus to Lahire . In: Isis . tape 38 , 1947, pp. 54-56 . The Kepler sentence quoted here in italics is given there as a direct translation.
3. Arthur Koestler: The night walkers: The history of our world knowledge . Suhrkamp, ​​1980.
4. Bruce Stephenson: Kepler's physical astronomy . Springer Science & Business Media Vol. 13, 2012.
5. Martin Holder: The Kepler Ellipse . universi, Siegen 2015 ( online [PDF; accessed November 1, 2017]).
6. ^ Curtis Wilson: How Did Kepler Discover His First Two Laws? In: Scientific American . tape 226 , no. 3 , 1972, p. 92-107 , JSTOR : 24927297 .
7. Guthmann, § II.2.37 Solution of Clairot's equation: The case e <1. P. 81 f.
8. Guthmann, § II.1 One- and two-body problem. Introduction, p. 64 f. and 30. Clairot's equation. P. 71 ff.
9. Guthmann, § II.1.26 The area set. P. 66 f.
10. Guthmann, § II.5 orbital dynamics of the Kepler problem. P. 108 ff.
11. David L. Goodstein, Judith R. Goodstein: Feynman's lost lecture: The movement of planets around the sun . Piper Verlag GmbH, Munich 1998.
12. LD Landau and EM Lifshitz: Mechanics . 3. Edition. Butterworth-Heinemann, Oxford 1976, ISBN 978-0-7506-2896-9 , pp. 22-24 (English).
13. J. Wess: Theoretical Mechanics. Jumper. Chapter on the two-body problem.