Two-body problem
In physics, called the two-body problem, the task is to calculate the motion of two bodies that interact only with each other without external influences. Specifically, it is referred to as two-body problem, the task of classical mechanics to calculate the motion of two bodies mutually attract or repel with a force that decreases proportional to the square of the inverse mutual distance.
Well-known examples of such two-body problems are the motion of two spherical astronomical objects ( binary stars, planet and sun, planet and moon, etc.) that move around each other by mutual gravitational field, and the motion of two charged particles ( proton and electron in a hydrogen atom, but also elementary scattering problems ) that attract or repel each other by mutual electrostatic field.
In the astronomical context, the problem is also called the Kepler problem. It was originally thought the two-body problem would be sufficient to describe a heliocentric cosmos. In fact, however, been shown that the Kepler problem is an idealization that is only found in nature approximation. Regardless of the solution of the question is the basis of modern celestial mechanics. Moreover, many of astronomical systems, the influence of other body is relatively low and the solution of the two-body problem provides a good approximation of the exact path.
The applicability of the solution of the classical two-body problem in the electrostatic context, however, is severely limited by the quantum nature of atomic particles. A satisfactory approximate solution of such problems therefore only provides the quantum mechanical two-body problem.
- 4.1 Several body
- 4.2 deviation from the spherical shape
- 4.3 Perturbation Theory
- 4.4 Two -body systems in general relativity theory
The Kepler solution
The solution of the sky mechanical problem has historically been closely interwoven with the findings of Johannes Kepler, who described already before the modern formulation of mechanics, the orbits of the planets of the solar system in the form in which they were later found by Isaac Newton as solutions of the exact two-body problem. More precisely, the solutions of the resulting from the two-body problem Einzentrenproblems in which a gravity source in the center of the system is ( at Kepler the sun ), which attracts a single body (planet ) without a reaction of the body to the source takes place, so the remains in the center.
The solution of the problem is divided into the following parts:
Possible paths ( Kepler orbits ) are circles, ellipses, parabolas and hyperbolas in question. For circles and ellipses the bodies are bound to each other like the planets to the sun. Is the web form parabolic or hyperbolic, so only an encounter takes place, as for example some comets is the case.
The drawing shows the different trajectories for a Einzentrenproblem represents the different trajectories are characterized by a positive real number, called the eccentricity. Bound orbits (circles and ellipses ) have a numerical eccentricity of ε <1, the circle of eccentricity ε = 0. Larger eccentricities lead to open paths ( parabolas with ε = 1 and hyperbolas with ε > 1). These open courses have not yet been described by Kepler, but arrange themselves useful in its solution scheme.
The classic problem
The classic astronomical two-body problem is concerned with the movement of two single ( spherical or point-like ) celestial bodies in mutual gravitational field, as formulated by the Newtonian theory of gravitation. The gravitational force is in this theory a long-range force (ie, it acts instantaneously over any distance ). The gravitational force of each body acts radially symmetrical and decreases with the distance from the center of the body in proportion to the square of the inverse distance ( ~ 1/r2 ) and displays at any point in space in the direction of the center of the force-exerting member. In addition, the force is proportional to both the mass of the attracting body, as well as to the mass of the attracted body.
Formulated to the power law mathematically by introducing vectorial coordinates ( see adjacent figure) is obtained for the force of the second body on the first body
As well as the force of the first body to the second body
Both forces are, according to the third Newtonian axiom, equal but opposite from the amount in the direction.
After the second Newtonian axiom both forces cause each of the dressed body acceleration. Substituting this into the above force equation, one obtains a system of coupled differential equations:
In each equation, the mass of the attracted body was cut on both sides. In order to solve the system of equations, the equations must be decoupled. These are new coordinates, the Jacobi coordinates introduced (see figure):
The first vector describes the relative distance of the body, the second vector of the center of gravity ( bary center ) of the system. Through clever addition of suitable multiples of the two equations is now obtained the equations
The total mass respectively. The first equation describes the motion of the center of gravity and can be understood as a consequence of the broad priorities set: the center of mass moves in a straight line and uniform. This fact also applies to more complicated systems of several bodies.
The Einzentrenproblem
The second equation of the above system describes the relative movement of each of the two masses. It is the equation of motion of a single, smaller mass ( reduced mass ) moving in the gravitational field of a much larger mass. The solution to this so-called Einzentrenproblems goes back to Newton, and explains the previously defined by Kepler solution. You will be outlined here.
The first step to solving this problem is the fact that move the individual body in a central force field. This means that the force emanating from the source at the center, always points in the direction of the radius vector. A direct consequence of this is that the source exerts a force but no torque on the orbiting body, because this is the vector product of force and radius vector:
However, since the torque indicative of the temporal change of angular momentum, it follows that these have to be constant during the movement. The angular momentum is a vector which is perpendicular to both the radius vector and to the velocity vector of the body since it is given by the vector product of the two vectors:
This is only possible, if the whole motion occurs in a plane which is perpendicular to the angular momentum vector. Movements that take place in a plane that are easiest to describe in polar coordinates. The vectorial equation of motion can then be reformulated Einzentrenproblems in two coupled ordinary differential equations:
Wherein the second of these equations is the time derivative of the angular momentum is:
This size is called a first integral of motion ( conserved quantity ) and provides us with a relationship between the distance of the considered body from the center of gravity and its angular velocity. Furthermore, it corresponds to the second Kepler 's law, the surface sentence: The term is in fact just double the acreage of the area which sweeps the radius vector in an infinitesimal time interval. Since the time derivative of disappears, this surface is always the same.
Energy conservation
We use the above definition of the angular momentum in order to eliminate from the remaining differential equation the angle, we obtain a law for the distance R, the radial equation:
The multiplication by and in the form of
Can be written. The three terms in this equation correspond to the radial and angular part of the kinetic energy and the potential energy of the body. Together they form its total energy
According to the above equation and thus also is constant in time is an integral of motion. The total energy must of course be obtained if only because it is a conservative field in a gravitational field.
Trajectory
Priority before the values of the two integrals of movement E, and L, as can solve the equation of motion, by first radial motion r (t) from the mold of the energy integral ( last equation in the section above ), and then the angular movement of the angular momentum integral calculated. However, this path leads to equations that can be described as non-descriptive, because you can view them directly in the form of the web does not.
Therefore, it is common to either the radial or the energy integral equation φ in a first differential equation with respect to the angle in place of the time transform. Here is the second way is presented: [A 1]
The trajectory that solves this equation is, when considering the arbitrariness in the choice of the angle φ so exploited that the closest approach to the center ( periapsis ) at φ = 0 is of the form
Which you can verify by substituting the need for the two parameters and apply. This is the equation of a conic section with a numerical eccentricity ε.
Is the total energy is negative, the motion is connected, which means there is a maximum distance ( apoapsis ) with ε from the center less than one. Is it is in the path in this case is an ellipse, whose one focus is the center, the semi-major axis. This was exactly the statement of the first Kepler 's law ( ellipse set). The fact that the trajectory is also always closed with negative total energy, is a special case, the other radially symmetrical force fields only occurs for the harmonic oscillator, the force field increases in proportion to the distance from the center.
If the total energy is positive, the path is a hyperbola with the smallest distance from the center. The limiting case with energy E = 0 is a parabola whose minimum distance from the center is just equal to p / 2.
Time parameters
To obtain the time for a known movement path, it is now possible to use the integral of angular momentum, in order to obtain the function. By integrating this path leads to a function which then has to be inverted. A descriptive method to obtain the function is found by Kepler Kepler 's equation. This method is based on the Kepler face set, ie their physical basis also forms the angular momentum integral. However, the time dependence of the trajectory results except in special cases, ε = 0, and ε = 1 for the solution of a transcendental equation, so that the solution can not be represented in a closed form using standard functions. Specifically, the solution of this equation is therefore determined by numerical methods.
The orbital period T of the body on an elliptical orbit can be certain, however, directly from the angular momentum integral. Since the area of the ellipse and is also applicable, is
This is exactly the statement of the third Kepler 's law.
Joint motion
After the Einzentrenproblem is dissolved and movement of the center of gravity is also known, one can convert the Jacobian coordinates back into the original inertial coordinates:
The two bodies thus move around their common center of gravity in ellipses, the ratio of the distances is determined to focus by the mass ratio. The focus itself moves straight and uniform, so that the orbits of the two bodies describe a type of " snake curve" to the path of the center of gravity. This so-called wobbling motion is of great importance in the indirect observation of stars invisible companion (eg exoplanets ).
Inverse problem: orbit determination
With the solution of the two- body problem, it is possible to calculate when given a sufficient number of initial values of the trajectory of two celestial bodies, which can be considered sufficiently accurate as a two- body system. In celestial mechanics, one is, however, usually before the inverse problem: from the observed path to the model parameters ( initial values ) are calculated. Using the methods described above, it is then the position of celestial bodies for the (further ) calculate future when the disturbances are sufficiently small.
The number of initial values to be determined is always added by the original system of equations. Since it is a second order equation for the motion of two bodies in three-dimensional space, these are 2 × 2 × 3 = 12 parameters. In what form emerge these twelve values , however, depends on the specific situation and the chosen method. In the "brute - force" method of direct numerical integration of the initial system are given three values for the starting position and three values for the starting speed for example for each of the two bodies. If you select the above presented analytical way, so first three starting position values and speed values for the three starting center of mass motion are sought. The remaining Einzentrenproblem then requires six parameters which are conventionally specified by the web elements, two angles which determine the position of the plane of movement in the space ( and thus the position of the angular momentum vector ), an angle which describes the position of the web within this plane ( and hence the zero point of the polar angle φ ), as well as semi-major axis and eccentricity of the conic -shaped path (which together determine the energy and the amount of angular momentum ). In addition, the initial position of the orbiting body must be specified as an angle or as a time reference by specifying the Periapsiszeit. An alternative elegant method for specifying these six initial values is the specification of two vectors is constant over time: the angular momentum vector and the Laplace -Runge -Lenz vector. However, these two three-dimensional vectors determine not six, but only five of the orbital elements, since the vectors are necessarily perpendicular. So, again the time reference by specifying the Periapsiszeit must be established or a start angle can be determined.
The main methods for the determination of the orbital elements from the observational data goes back to Isaac Newton, Pierre- Simon Laplace and Carl Friedrich Gauss.
Boundaries of the two-body solution
The two-body problem is an idealization that rarely adequately reflected in concrete situations exactly the facts. The only exceptions are genuine double stars without planets or other dark companion, whose components are far enough apart so that tidal effects are negligible. As a classical two-body problem ( non- quantum mechanical ) models of the hydrogen atom can be considered as well as radially symmetric single-site scattering problems.
Several body
In almost all real situations, however, there are more than two bodies interact. The movement of multiple body problem can not be solved in a similar manner as presented here for two bodies. Even the three-body problem, ie the task of path computation when the interaction of a third body is taken into account, is not strictly solvable in general and can be solved in general only numerically. [A 2] This difficulty is made naturally with in solving many-body problems other components continued. Exceptions are only highly symmetric constellations, in which for example the body form a regular polygon lie on a line or extended shell around a center. An important application find such arrangements in the study of the motion of small bodies that are located in one of the five Lagrangian points of a two-body system.
Deviation from the spherical shape
Another problem is the deviation of one or both body from the spherical shape dar. Many astronomical bodies are described inaccurately by a radially symmetric mass distribution. In some cases, the objects can be modeled much more accurately, when considered as oblate spheroids. This applies to many planets, stars, but also for spiral galaxies, which can be well modeled as flat disks. Is one of the two bodies is substantially smaller than the other, such a system can be described as axisymmetric Einzentrenproblem, which is more general than the above-described, but is still accessible to a general solution. If both flattened body of comparable size and not in the same direction, however, this pathway is closed. In addition, tidal forces between the bodies can lead to dynamic deformations, as in close binary stars is often the case. These lead to a complex dynamic between rotation of the individual body and the movement of bodies around each other.
Perturbation theory
Nevertheless, the Kepler solution is the basis of all modern planetary theories (as well as the moon theories and theories of the motion of all other celestial bodies ). The orbits of almost all natural objects in our solar system, the most multiple stars and galaxies, are such that they can be in first approximation entirely described by the Kepler solution. The orbital elements of the Keplerian orbits, which are determined from the initial conditions, but are then no longer be assumed to be constant, but will be treated by perturbation theory. The orbital elements that are valid at a certain time, are then described as oskulierend, since they determine the Keplerian orbit, the real path as accurately as possible currently clings.
Furthermore, the influences of the bluff body to the two-body system can often over long periods making process, thus the description of the problem is gaining symmetry. Such influences lead z.T. on time-constant or periodic changes in the orbital elements. Examples of such phenomena are, for example, the uniform rotation of the line of apses, ie the position of the Keplerian orbit in the orbital plane, and the uniform displacement of the rail junction at an invariant plane ( the Laplace plane ). In the lunar theory are further examples of such periodic disturbances the evection and variation.
Two-body systems in general relativity theory
The modern theory of gravity finds its description in the General Relativity Theory ( ART). When the masses of the two bodies is sufficiently small, the distances between them is relatively large and the speeds of the body far below the speed of light, the system can be described by the Newtonian limiting case of the theory. In other words, the above sketched solution within the Newtonian gravity provides a very good approximate solution. If the conditions for the validity of the limiting case not satisfied or the requirements for accuracy are very high, but the problem must be resolved within the full ART - a task which proves to be much more complicated.
In the simplest case, which fortunately has very many applications, one of the two bodies is much larger than the other. It is then justified to consider the small object as a test body in the large object field, ie the small body caused no noticeable effect on the large. One can describe the problem then analogous to the Newtonian theory as allgemeinrelativistisches Einzentrenproblem. Also in the ART proves this problem due to the radial symmetry as well analyzed. In a similar way as it was described above, the integrals of the movement can be found [A 3] However, the analysis results to a radial equation, which includes an additional term compared to the Newtonian theory, the effect in the consequence that the sheets also at negative energy are not closed. Instead, the tracks, as is also true for two-body systems with different power laws as the Newtonian, rosette orbits. This effect has become famous because it makes it possible to explain the additional perihelion of Mercury.
The general relativistic two-body problem in all generality, ie with two bodies interact with each other is much more complicated. Since the presence of the two masses changed the space-time structure itself, concepts such as gravity, energy, angular momentum are no longer applicable. [A 4] Thus, no reduction of the problem to a Einzentrenproblem is possible. In addition, the influence of the space-time is thus anchored in the mathematical structure that the problem is not described by ordinary differential equations, but by partial differential equations. The non-linear structure of these equations give the solution of the equations themselves using numerical methods problematic. In heuristic approach can in the general case, try to take over the classical concepts approximated. This description leads to effects such as the emission of gravitational waves and an associated " angular momentum loss ." The orbits of the body then described spiral orbits around a common "center of gravity ", which are always close at ever shorter turnaround time. The exact description of these phenomena in the context of a post -Newtonian approximation is controversial due to unexplained convergence properties of the approximations.