Lindblad resonance
Lindblad resonances (named after its discoverer Bertil Lindblad ) is a resonance phenomenon of the galaxies theory. It involves resonances of the Railways of individual stars within the galaxy with large-scale galactic structures such as spiral arms, galactic bars or even close companions of the galaxy. These resonances could play a crucial role for the existence of long-lived spiral and bar structures in galaxies.
Other applications of the theory of Lindblad resonances can be found in the Declaration of structures in planetary rings and in protoplanetary disks.
Explanation
A spiral galaxy can be considered in first approximation as an axisymmetric cluster of stars. The axis of symmetry runs vertically to the disc through the center of the galaxy. The plurality of star then generate a common gravitational field that can be assumed to be continuous large scale and also axially symmetric. The individual stars move in this common gravitational field on tracks within the meaning of the total angular momentum run almost exclusively around the center of the galaxy, while periodically change the radial distance and the vertical distance to the galactic plane. Close encounters with other stars who generally involves. lead chaotic changes in these pathways are left in this consideration in mind.
The periodic approximation and distance of a star from the galactic center occurs with a certain angular frequency κ ( Epizykelfrequenz ), which depends on the distance range of the star to the center and on the specific radial profile of the common gravitational field of the stars in the galaxy. When Kepler problem, for example, in which the total mass is concentrated in a spherically symmetric central body, this angular frequency coincides exactly with the angular frequency of the circulation, resulting in the well-known elliptical orbits. Since galaxies in the matter is united not only in the center, but is distributed over the entire galaxy, the gravitational field falls to the outside less. In general, the angular frequency is then κ not in integer ratio to the angular frequency of the orbit, and the orbits have the form of a rosette, which does not close again. This phenomenon is also known from the perturbation theory of the Kepler problem, where it leads to the so-called Apsidendrehung ( perihelion ).
The density wave theory suggests that the spiral arms of a rotating galaxy can be stabilized by a density wave, which rotates in the gravitational field of the galaxy with a constant angular frequency ΩS. The orbits of the stars in the galactic plane being disturbed by the spiral arms or a galactic bar. The disturbance while rotating at a constant angular frequency, which is not generally coincident with the rotational angular frequency of the individual stars. The fault is modeled as an additional well on the angle in the disc -dependent gravitational potential.
If the difference between the angular frequency ΩS of the disturbance and the angular frequency of the circulation of a star Ω ( R), the center R depends on the mean distance, just an integral multiple m of the Epizykelfrequenz κ (R ), which also depends on the average distance from the center is, resonance occurs between the web and disorder:
Wherein the natural number m for the multiplicity of the symmetry of the disorder, for example, the number of spiral arms (usually two ) is available. The periodic oscillation distance of the star is then affected in any approach to the problem to the same extent.
The response types and their radii
The resonance occurs in certain path radii R, the resonant radius that can be estimated for a given model. Particularly relevant is the case m = 2, since the resonance then for specific potential models, the stabilization of the helical structure and so hard to support the observation findings that most spiral galaxies have two arms, explained. In typical characteristic of the potential three resonance radii that are highlighted in yellow in the adjacent animation result:
- The inner Lindblad resonance (ILR ) near the center of galaxies, wherein the helical structure begins. The orbits of the stars on these orbits are approximately elliptical around the center with two approximations to the center per orbit in the reference system of the disorder. The fault runs slower than the stars.
- The co-rotating resonance ( CR) in the middle distance from the galaxy center. The orbits of the stars on these orbits are also approximately elliptical, but not at the center but at a fixed position in the reference system of the disorder. There is always an approximation to the center per orbit in the reference system of the disorder.
- The outer Lindblad resonance ( ORF ) in the " visible edge " galaxy, wherein the spiral structure is closing. The orbits of the stars on these orbits are again approximately elliptical around the center with two approximations to the center per orbit in the reference system of the disorder. The fault runs faster than the stars.
All other orbits are rosettes in the reference system of the disorder.
Density waves, which occur in the spiral galaxy can survive only between the inner and outer Lindblad resonance. Occur on the spiral arms in this area only. This density waves can not penetrate through the ILC into the core. They are absorbed in this limit, like waves on a beach, and only form of so-called evanescent waves. The bar of a barred spiral galaxy extends no further than to the CR. Star Rings that are found in spiral galaxies form at the CR and the OLR. The gas of a galaxy collects at the ILR. There also is a ring of gas and newly formed stars can then form.
° values for the Milky Way with m = 2 and ΩS ≈ 15 km / s / kpc
^ Relative frequency
Mathematical derivation
Rosette tracks
The primary gravitational potential U (r, φ, z, t) is a spiral galaxy is to be stationary, axially symmetrical and mirror-symmetrically to the galactic plane (U (r, φ, z, t ) = U (r, z) = U (r, z ) ) is assumed, in which first the compressed spiral arms, bars, or other disturbances can thereby unnoticed. In the following we restrict ourselves to the galactic plane, ie there she always z = 0 and call the potential there simply U ( r). Orbits near the Galactic plane cause vibrations in the z- direction at z = 0, which should not be considered further here. The general equations of motion in the galactic plane
Then be usefully in plane polar coordinates (r, φ ) formulated:
In such a potential exist for any distance R to the center of the galaxy in the galactic plane stable circular orbits. The stars on the orbits rotate at a constant angular velocity Ω, the results from the central force for such paths:
Most of the stars of a spiral galaxy have tracks that are in a very small radial distance range around a circular path. It is then justified to linearly approximate the general equations of motion in the plane of this circular path. Therefore one is the deviations from the circular path as follows:
Linearization of the force, the second derivative of the potential:
The second derivative of U can be expressed by the derivative of the angular velocity Ω according to the orbit radius:
The derivative of the angular velocity after the orbit radius we call in the following Ω '. The quantities Ω and Ω ' are observables, which can be determined from the rotation curve of a galaxy. In particular, they can be determined from the Oort constants.
The linearized equations of motion now read:
The second equation can be, as well as the corresponding non- linearized equation, integrate directly. The constant of the movement
Is related to the difference between the angular momentum and orbital impaired sheet. It can be set equal to zero without loss of generality, since, for every angular momentum of a suitable undisturbed orbit. Substituting the equation resulting in radial movement equation is obtained:
This is a homogeneous wave equation with angular frequency
With solution
The equation for the angular displacement:
Then provides a phase-shifted by 90 ° oscillation
With amplitude b / R = 2aΩ / ( κR ). The perturbed orbit leads to relatively undisturbed circular orbit with the same angular momentum of an elliptical orbit with semi-axes a and b, the ratio is just b / a = 2Ω / κ. The ellipse is called epicycles ( even if it is not a circle). The angular frequency is κ thus called Epizykelfrequenz. The railway, which results from the superposition of circular motion and Epizykelbewegung, rosette web is called. The picture to some examples are shown.
To potentials which are either proportional to the logarithm or a power function of the pure R, Ω (R) is proportional to a purely exponential function of r. From the above formula we see that the Epizykelfrequenz then is proportional to the angular velocity of the circular path and the ratio of the two so that a constant is obtained (this is also the case in the adjacent animation). For the potential of a spherically symmetric central mass U (r) ~ 1 / r arises, for example, so that closed orbits with a pericentre and a Apozentrum per revolution yield, as also specify the Kepler's laws. For a logarithmic potential, that is a realistic approximation of a typical galaxy potential, calculated as the ratio. Measurements of the Oort constants in the solar neighborhood provide value to our galactic neighborhood. For a rigidly rotating disk (a model for the galactic nucleus applies quite well ), so that the stars are there on almost elliptical orbits with the galaxy center in the center (not the focal point as the Kepler problem) move.
Movement relative to the interference
Bars, spiral arms or close companion of a galaxy can be treated as a fault of the axisymmetric primary potential U, which rotates at a constant angular velocity ΩS. If a change in a reference system that rotates along with the disorder, then the trajectories of the stars transform in a way that the angular velocities of circular orbits on Ω '= Ω - ΩS be reduced while the Epizykelbewegung of the transformation remains unaffected. The star therefore remains at a rosette orbits, but with a different frequency ratio κ / Ω '. In the special case of a co-rotating path is Ω '= 0 and only the Epizykelbewegung visible, ie the star moves on a relatively stationary for fault ellipse. Since Ω ' is usually the amount is less than Ω, the most rosette orbits of the stars in the comoving system run much more Epizykeldurchläufe per cycle than in the non- co-moving system. In addition, the sign of the relative angular velocity for stars within the co-rotating orbit is positive, negative outside.
If the frequency ratio κ / Ω ' integer, so the rosette orbits are closed with κ / Ω ' Epizykelumläufen per orbit around the center of the galaxy. While the fault affects the path of the star particularly strong if the amount of the ratio κ / Ω ' just the multiplicity M of the symmetry of the error:
The path points with maximum distance to the center be rotated from the disorder over time in the disturbance and the semi-axes of epicycles increase. A detailed description of this resonance phenomenon in the context of perturbation theory possible that exceeds the scope of this article. In the following paragraph, a self-consistent approach is presented how the Lindblad resonance effect on the stabilization and expansion of the spiral structure of the galaxy.
Effect of resonance
You can see the effect of the resonances to model mathematically, if one chooses a continuum mechanical approach to the description of the galaxy including disorder. When it is assumed that the density distribution area Σ is not stationary in a system can be considered the time evolution. To this end, consider first the Euler equation
The functions (area density ), ( flow field ) and ( potential), and the " speed of sound " includes. The latter is determined by a heuristic state equation which a " pressure " p assigns the areal density by Σ. All variables are then considered as the sum of an unperturbed time-independent size and a disturbance. This means that for all functions a location-and time-dependent perturbation approach is made. Thus, for example, the surface density in accordance with
Disturbed.
You then eliminated in the following from the disturbance equations approach the undisturbed portions, one obtains a Poisson and three disturbance equations. This system of equations is, for example, through an approach the density of the form
Solved. This approach is consistent spiral density waves with poor shape and function f (r), which rotate rigidly with frequency. At rest, it follows for the density maxima of the pattern
What a spiral:
If one finds for the other perturbation equations self-consistent solutions, we obtain an algebraic equation system that results in further then to a dispersion relation which expresses the condition of a spiral density wave. This in turn is followed by the dispersion equation
In the stands for the Different from the angular velocity of a circular path with a radius r with the angular velocity of the disturbance, the circular radial wave number of the spiral structure and the Epizykelfrequenz that is as above:
By rearranging the dispersion equation to
It can be seen that the solution for the circular wave number generally involves. has two branches:
Short-wave ( ) and long wave (-) can be mentioned. The Lindblad resonances it is now recognized at the places where the long waves disappear, as their circular wave number is zero:
Outside the OLR and ILR can only exist within the shortwave. The region around the co-rotating orbits in this model shows a behavior out fine, because there the expression under the square root is negative, since ω is very small, ie,
The waves here have a complex wave number and therefore disappear exponentially in penetrating this region ( evanescent waves). In this region are formed in some galaxies so -called star rings.