Kepler's laws of planetary motion

The three laws of Kepler are named after the astronomer and natural philosopher Johannes Kepler. He found these fundamental laws for the orbits of the planets around the sun, when he took them to a desired harmonic with respect and the deviations from a circular path of Mars analyzed mathematically. The sentences describe the motion of an ideal celestial bodies.

The conditions for the three Kepler's laws are different. The second law applies to all central forces (which do not even need to be conservative ). The third law applies to forces, which decrease with the square of distance. Deviations from the first law we can see that already in the solar system: comets often move in parabola-like webs, the axis of Mercury's orbit slowly rotates due to the influences of the other planets around the sun.

• 2.1 Kepler's First Law ( ellipse set)
• 2.2 Kepler's Second Law ( face set )
• 2.3 Kepler's Third Law

Fundamental importance in astronomy

The first two Kepler laws represent the exact solution of a two-body problem in the framework of Newtonian mechanics, which is described by the Kepler equation. They apply to all two systems when

• Body mass points,
• No gravitational forces of other bodies out,
• Nichtgravitative forces are negligible and
• Relativistic effects can be neglected.

The third law is a very good approximation for the solution of a many-body problem, in which the mass of a body is much larger than the other. Is the solution of one of the body have been found, to obtain also information on the movement of the other body.

Although the three laws of motion only in a system of two celestial body exactly describe ( equation of motion of idealized two-body system in the potential theory ), they are a good approximation for the actual traversed in a solar system planetary orbits. The to be removed from Kepler's laws six orbital elements (ie the semi-major axis length, eccentricity, the inclination, the node length, Periapsiswinkel and Periapsiszeit ) are the basis of every orbit determination. The small deviations from the Keplerian orbits by the gravitational interaction of the lighter body with each other are called perturbations.

History

Kepler formulated the geometry and kinematics of the planetary orbits in three laws, of which he was the first two relatively quickly ( Astronomia Nova, "New Astronomy ", 1609). The search for the third lasted, however - including several wrong turns over compound curves - a decade, he found it in mid 1618 ( Harmonice mundi, Harmony of the World ', published 1619). Here, considerations for help, who are now called anthropic principle came to him, and probably also to be found in music harmony. An important basis for Kepler were the observations of Tycho Brahe and his own as Tycho's assistant, who formed the data base as Rudolfinian boards, his model tested at the Kepler. The exquisite, created in Prague observational data of Brahe and Kepler from the planet Mars was particularly significant for the first two laws ( ellipses and face set ).

Another -positioned in this context of Kepler law on the acting force, the Anima motrix has proved to be not true. Kepler's laws of planetary motion were placed in the general context of his law of gravitation of Newton later.

Problem of multi-body systems

Even if two bodies orbiting each other, also acts as a gravity from small to larger body: therefore also moves this and is not in the focus of the ellipse, but both revolve around the barycenter ( center of mass ) of the system ( inadequacy of the heliocentric world view: The sun is not in the "center " of the solar system ).

When three or more bodies orbit each other, there are further perturbations of the gravitational influences among themselves, but for which Kepler's laws of orbital elements and a reference system used to represent today ( osculating ellipses, a temporary approximate solution ). Are numerous body gravitationally bound together, reign almost always chaotic, so long-term highly unstable states. Thus, for example, giving way to the movement of the stars around the galactic center markedly from the second and third Kepler law from, and also the solar system is not stable to all eternity Kepler system (such as the intersection of Neptune and Pluto shows ).

Heliocentric and fundamental formulation of laws

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

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

In a cosmic scale, but the relativistic effects are increasingly starting to impact, and the differences from the Keplerian model primarily serve as a criterion for modern concepts of astrophysics. The formation mechanisms in spiral galaxies about can no longer be traced consistent with a purely based on the Kepler laws model.

Derivation and modern representation

The Kepler laws can be derived directly from the elegant Newton's theory of motion. The second set is a geometric interpretation of the angular momentum theorem, the first sentence follows from the Clairaut equation that describes a complete solution of a movement in rotationally symmetrical force fields, and by means of integration of the Kepler equation and the Gaussian constant is followed by the third sentence of the second, or by means of the hodograph directly from Newton's laws.

Kepler tried to describe the movements of the planets with its laws. From the observed values ​​, in particular the orbit of Mars, he knows that he must deviate from the ideal of circular orbits. Unlike later, Newton 's laws are therefore empirically derived and not from theoretical considerations. From today's perspective, however, we can start from the knowledge of Newtonian gravitation and thereby to justify the validity of Kepler's laws.

Kepler's First Law ( ellipse set)

This law is derived from Newton's law of gravitation, provided the mass of the central body is much larger than that of the satellites and the interaction of the satellites can be neglected on the central body.

The energy for satellites with a mass in the Newtonian gravitational field of the sun with mass in cylindrical coordinates

With the help of and

Allows the energy equation to

Reshape. This differential equation is the polar coordinate representation

A conic compared. For this purpose, the derivative

Formed and all phrases that contain the insertion to be carried,

Transformed equation of the trajectory eliminated:

By comparing the coefficients of powers of r.

This solution depends solely on the specific energy and the specific angular momentum. The parameters and the eccentricity, the shape elements of the web. In case the following applies:

This is the solution that offers the first Kepler set. The general solution of the equation of motion are conic sections, Kepler orbits. These are in the case of closed orbits ellipses. A body that is not gravitationally bound to the center of gravity, ie, has too high a speed to form a closed path, the field goes through on a parabolic or hyperbolic path ( → parabola, hyperbola → ). As a special case still exists a solution in which the mass zustürzt on a straight line to the center.

For forces, there is a conserved quantity, which is crucial for the direction of the elliptical orbit, the Runge- Lenz vector pointing along the major axis. Small changes in the force field (usually by the influences of the other planets) leave this vector slowly change its direction, whereby, for example, the perihelion of Mercury's orbit can be explained.

Although Kepler's laws were originally formulated only for the gravitational force, so the solution above applies also for the Coulomb force. For each repulsive charges the effective potential is then always positive and you only obtain hyperbolic.

The web describes an ellipse with foci and the numerical eccentricity and a the semi-major axis of the main peak are in ( pericentre, Perizentrumsdistanz ) and ( Apozentrum, Apozentrumsdistanz ). Thus, the law makes a statement about the geometric form of a sheet and is used to determine their shape elements ( semi-major axis / eccentricity).

If you put (unlike Kepler ) is not centrally symmetric force field based, but alternating direction of gravity, so also form elliptical orbits. However, both move body toward the center of the orbits is the common focus of " central body " and Trabant, as fictitious central mass, the total mass of the system is to be assumed. 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 resonates a little under the influence of the orbiting planet ( longitude of the sun ≠ 0). The Earth-Moon system, however, shows greater fluctuations in terms of the path geometry, here the system focus is still within the earth. Satellite even respond to fluctuations in irregular by the earth's force field.

Kepler's Second Law ( face set )

Under the driving beam is defined as the line connecting the center of gravity of a celestial body, such as a planet or moon, and the Gravizentrum, such as a first approximation of the sun or of the planet to which he is moving.

A simple derivation result, when considering the surface, by inserting the driving beam in a small period of time. In the graph on the right Z is the center of power. The Trabant first moves from A to B. If his speed does not change, he would move the next time step from B to C. It is readily apparent that the two triangles and ZAB ZBC include the same area. Now acts a force in Z direction, the velocity v is deflected to one that is parallel to the common basic example of the two triangles. Instead of the Trabant at C so ends up at C '. Since the two triangles and ZBC ZBC ' have the same base and the same height, and their surface is the same. Thus, the area rate applies to the two small time intervals and. Integrating such small time steps ( with infinitesimal time steps ), we obtain the surface sentence.

The swept area is for an infinitesimal time step

Since, due to a central force, the angular momentum

Is constant, the surface integral is thus just

For equal time differences, therefore, results in the same swept area.

So the second Kepler law defines both the geometrical basis of astrometric path ( as a path in a plane ), as well as their web dynamics ( temporal behavior). Kepler formulated the law only for circulation of the planets around the sun, but it also applies to non- closed orbits. The second law of Kepler is not limited to the force of gravity, in contrast to the other two laws ( Kepler actually went with his anima motrix also from a power off), but applies generally to all central forces. Kepler was only interested in a description of the planetary orbits, but the second law already the first formulation of the law, which we now know as the conservation of angular momentum. The second Kepler set can be seen as a special formulation of the angular momentum theorem.

The second Kepler law has two basic consequences for the movement conditions in multi-body systems, both for solar systems, as well as for space: The constancy of the normal vector web stating that elementary celestial mechanics is a plane problem. In fact, here deviations by the volumes of the heavenly bodies, so that mass is outside the orbital plane, and the orbital planes precess ( their position in space change ). Therefore, the orbits of the planets are not all in one plane ( the ideal solar system plane of the ecliptic ), rather they show an inclination and perihelion, also varies the ecliptic latitude of the sun. Conversely, it is relatively easy to move a spacecraft in the solar system level, but enormously costly to place such a probe over the north pole of the sun.

The constancy of the speed of surface states that an imaginary line connecting the central body, or more precisely the focus of the two celestial bodies, and a moon in equal times, the same area is always covered. A body thus moves faster when it is close to its center of gravity and the slower the further he is away from it. This applies for example to the drive of the earth around the sun as well as for the run of the moon, or a satellite to the Earth. A web arises as a constant free fall, close Gone swinging around the center of gravity, and remount the farthest point of culmination of the web is: The body is always faster in the pericentre ( center next point ) has the highest speed and is from then on more and more slowly until Apozentrum ( zentrumsfernsten point) from which it is re-accelerated. Seen in the Kepler ellipse is a special case of the oblique cast, which closes in its path. This consideration plays a central role, where it comes with an initial pulse suitably chosen to produce ( by the start ) a suitable orbit in space physics: the more circular the web, the more uniform the velocity of circulation.

Third Kepler Law

Or

Kepler used for the path axes a is the average distance from the sun (in the sense of the composition of perihelion and Apheldistanz ).

In combination with the law of gravitation, Kepler's third law obtained for the motion of two masses M and m have the form:

Wherein the approximation is valid if the mass m is negligibly small compared with M ( as in the solar system ). Through this form you can determine approximately the total mass of binary systems from measurement of the orbital period and distance.

Taking into account the different masses of two celestial bodies and the above formula, so is a more precise formulation of Kepler's third law:

Obviously, the deviation is gaining in importance only when both satellites are very different in their masses and the central object has a mass M, which does not deviate greatly from the one of the two satellites.

The third law of Kepler is paid for all forces, which decrease with the square of distance, as can be easily derived from the viewing scale. In the equation

R appeared in the cube and square of t. Under a scale transformation is therefore the same equation is replaced if it is. On the other hand, doing so will quickly be seen that the analogue of Kepler's third law for closed orbits in a force field for arbitrary k is even.