Lindblad response

Lindblad resonances (named after their discoverer Bertil Lindblad ) are a resonance phenomenon from galaxy theory . These are resonances of the orbits of individual stars within the galaxy with large-scale galactic structures, such as spiral arms , galactic bars or close companions of the galaxy. These resonances could play a crucial role in the existence of long-lived spiral and bar structures in galaxies.

Other applications of the theory of Lindblad resonances can be found in the explanation of structures in planetary rings and in protoplanetary disks .

Explanation

A spiral galaxy can be used as a first approximation axisymmetric be regarded collection of stars. The axis of symmetry runs perpendicular to the disk through the center of the galaxy. The multitude of stars then generate a common gravitational field , which can be assumed to be continuous and also axially symmetrical over a large area. The individual stars move in this common gravitational field on orbits which almost without exception run around the center of the galaxy in the sense of the total angular momentum and periodically change the radial distance and the perpendicular distance to the galactic plane. Close encounters with other stars leading to i. General chaotic changes in these orbits are disregarded in this consideration.

The periodic approach and distance of a star from the galactic center takes place with a certain angular frequency κ (epicyclic frequency), which depends on the distance range of the star to the center and on the concrete radial course of the common gravitational field of all stars in the galaxy. In the case of the Kepler problem, for example, in which the total mass is combined in a spherically symmetrical central body, this circular frequency corresponds exactly to the circular frequency of the revolution, which results in the known elliptical orbits. Since in galaxies the matter is not only united in the center, but is distributed over the entire galaxy, the gravitational field decreases less strongly towards the outside. In general, the angular frequency κ is then not in an integer ratio to the angular frequency of the revolution, and the paths have the shape of a rosette that does not close again. This phenomenon is also known from the perturbation theory of the Kepler problem, where it leads to the so-called rotation of the apses (perihelion).

The density wave theory states that the spiral arms of a rotating Galaxy be stabilized by a density wave, which in the gravitational field of the galaxy with a constant angular frequency Ω _S rotates. The orbits of the stars in the galactic plane are disturbed by the spiral arms or a galactic bar . The disturbance rotates at a constant angular frequency, which generally does not match the orbital frequency of the individual stars. The disturbance is modeled as an additional gravitational potential that is also dependent on the angle in the disk .

If the difference between the angular frequency Ω _{S of} the disturbance and the angular frequency of the orbit of a star Ω (R) , which depends on the mean distance from the center R , is just an integral multiple m of the epicyclic frequency κ (R) , which is also the mean distance from the center depends, there is a resonance between the path and the disturbance:

${\displaystyle m\left[\Omega _{\mathrm {S} }-\Omega (R)\right]=\pm \kappa (R),}$

where the natural number m stands for the number of symmetry of the disturbance, for example the number of spiral arms (usually two). The periodic distance oscillation of the star is then influenced to the same extent with every approach to the disturbance.

The types of resonance and their radii

Animation of the Lindblad resonances. There are three resonance radii in the underlying model potential. The resonant paths are marked in yellow.

Resonance occurs at certain orbital radii R , the resonance radii , which can be estimated for a given model. The case m = 2 is particularly relevant , since the resonance then supports the stabilization of the spiral structure particularly strongly for concrete potential models and thus explains the observation that most spiral galaxies have two arms. With a typical course of the potential, there are three resonance radii, which are marked in yellow in the animation below:

• The Lindblad Inner Resonance (ILR) near the galaxy center, where the spiral structure begins. The orbits of the stars on these orbits are approximately elliptical around the center with two approaches to the center per revolution in the reference system of the disturbance. The disturbance orbits more slowly than the stars.
• The corotating resonance (CR) at mean distance from the center of the galaxy. The orbits of the stars on these orbits are also approximately elliptical, but not around the center, but around a fixed position in the reference system of the disturbance. There is one approach to the center per revolution in the reference system of the disturbance.
• The outer Lindblad resonance (OLR) at the “visible edge” of the galaxy, where the spiral structure ends. The orbits of the stars on these orbits are again approximately elliptical around the center with two approaches to the center per revolution in the reference system of the disturbance. The disturbance orbits faster than the stars.

All other orbits are rosette-shaped in the reference system of the disturbance.

Density waves that occur in the spiral galaxy can only survive between the inner and outer Lindblad resonances. The spiral arms only appear in this area. These density waves cannot penetrate the nucleus through the ILR. They are absorbed at this limit, like waves on a beach, and only form so-called evanescent waves . The bar of a barred spiral galaxy does not extend further than to CR. Star rings found in spiral galaxies form on the CR and the OLR. The gas of a galaxy gathers at the ILR. A ring of gas and newly formed stars can then also form there.

Lindblad resonances in the Milky Way

Type of resonance (abbreviation)	Resonance radius °	Relative frequency ^	description
external Lindblad resonance (OLR)	20 kpc	+1	Disturbance moves faster than the stars
corotating resonance (CR)	14 kpc	0	Stars and disturbance rotate at the same speed
Inner Lindblad Resonance (s) (ILR) (Depending on the parameters of the system, there can be zero to multiple ILRs.)	3 kpc	−1	Disturbance orbits more slowly than the stars

° Values ​​for the Milky Way with m = 2 and Ω _S ≈ 15 km / s / kpc

^ Relative frequency ${\displaystyle \nu ={\frac {m}{\kappa }}\left[\Omega _{\mathrm {S} }-\Omega (R)\right]}$

Mathematical derivation

Rosette strips

The primary gravitational potential U (r, φ, z, t) of a spiral galaxy is said to be stationary, axially symmetric and mirror symmetric to the galactic plane ( U (r, φ, z, t) = U (r, z) = U (r, -z ) ), whereby the compressed spiral arms, bars or other disturbances are initially ignored. In the following we restrict ourselves to the galactic plane, i.e. H. let z = 0 always apply and simply call the potential U (r) there . Orbits in the vicinity of the galactic plane carry out oscillations in the z-direction around z = 0 , which should not be considered further here. The general equations of motion in the galactic plane

${\displaystyle {\ddot {\mathbf {x} }}=-\nabla U(r)}$

are then formulated in plane polar coordinates (r, φ) :

${\displaystyle {\ddot {r}}-r{\dot {\phi }}^{2}={\frac {\partial U(r)}{\partial r}};\quad r{\ddot {\phi }}+2{\dot {r}}{\dot {\phi }}=0}$

In such a potential stable circular orbits exist for every distance R to the center of the galaxy in the galactic plane. The stars on the circular orbits revolve with a constant angular velocity Ω, which results from the central force for such orbits:

${\displaystyle \left.{\frac {\partial U}{\partial r}}\right|_{r=R}=\Omega ^{2}R.}$

Most of the stars in a spiral galaxy have orbits that are located in a very small radial range around a circular orbit. It is then justified to approximate the general equations of motion in the plane linearly around this circular path. To do this, the deviations from the circular path are defined as follows:

${\displaystyle r=R+\Delta r\quad \phi =\Omega t+\Delta \phi .}$

The linearization of the force contains the second derivative of the potential:

${\displaystyle {\frac {\partial U(r)}{\partial r}}=\left.{\frac {\partial U(r)}{\partial r}}\right|_{r=R}+\left.{\frac {\partial ^{2}U(r)}{\partial r^{2}}}\right|_{r=R}\cdot \Delta r.}$

The second derivative of U can be expressed by the derivative of the angular velocity Ω with respect to the circular path radius:

${\displaystyle \left.{\frac {\partial ^{2}U(r)}{\partial r^{2}}}\right|_{r=R}=2R\Omega {\frac {d\Omega }{dR}}+\Omega ^{2}.}$

In the following we call the derivation of the angular velocity according to the circular path radius Ω '. The quantities Ω and Ω 'are observable quantities that can be determined from the rotation curve of a galaxy. In particular, they can be determined from the Ort constants .

The linearized equations of motion are now:

${\displaystyle \Delta {\ddot {r}}-2R\Omega \Delta {\dot {\phi }}-\Omega ^{2}\Delta r=(2R\Omega \Omega '+\Omega ^{2})\Delta r;\quad R\Delta {\ddot {\phi }}+2\Omega \Delta {\dot {r}}=0}$

The second equation, like the corresponding non-linearized equation, can be integrated directly. The constant of movement

${\displaystyle \Delta l=R\Delta {\dot {\phi }}+2\Omega \Delta r}$

is related to the angular momentum difference between the circular path and the disturbed path. It can be set equal to zero without restricting the general validity, since there is a suitable undisturbed circular path for every angular momentum. If the following equation is inserted into the radial equation of motion, one obtains:

Different rosette orbits in the model gravitational field of a galaxy.

${\displaystyle \Delta {\ddot {r}}=(2R\Omega \Omega '+4\Omega ^{2})\Delta r.}$

This is a homogeneous oscillation equation with a circular frequency

${\displaystyle \kappa ={\sqrt {2R\Omega \Omega '+4\Omega ^{2}}}}$

with solution

${\displaystyle \Delta r(t)=a\sin(\kappa t).}$

The equation for the angular offset:

${\displaystyle \quad \Delta {\dot {\phi }}=-{\frac {2\Omega }{R}}\Delta r}$

then delivers an oscillation out of phase by 90 °

${\displaystyle \Delta \phi (t)=-{\frac {b}{R}}\cos(\kappa t)}$

with amplitude b / R = 2aΩ / (κR). Relative to the undisturbed circular orbit with the same angular momentum, the disturbed path executes an elliptical path with semiaxes a and b , the ratio of which is just b / a = 2Ω / κ . The ellipse is called an epicyclic (even if it is not a circle). The angular frequency κ is therefore called the epicyclic frequency . The path that results from the superimposition of circular and epicyclic movements is called the rosette path . Some examples can be seen in the adjacent picture.

For potentials that are either proportional to the logarithm or a pure power function of r , Ω ( R ) is proportional to a pure power function of r . From the above formula you can see that the epicyclic frequency is then proportional to the angular velocity of the circular path and the ratio of the two results in a constant (this is also the case in the adjacent animation). For the potential of a spherically symmetrical central mass U (r) ~ 1 / r it results, for example , that closed orbits with one pericenter and one apocenter per revolution result, as also dictated by Kepler's laws. For a logarithmic potential, which is a realistic approximation of a typical galaxy potential, the ratio is . Measurements of the Oort constants in the solar environment provide the value for our galactic neighborhood . For a rigidly rotating disk (a model that applies very well to the galaxy nucleus) , the stars move there on almost elliptical orbits with the center of the galaxy in the center (not in the focal point as in the Kepler problem). ${\displaystyle \kappa /\Omega =1}$ ${\displaystyle \kappa /\Omega ={\sqrt {2}}}$ ${\displaystyle \kappa /\Omega =1{,}3}$ ${\displaystyle \kappa /\Omega =2}$

Movement relative to the disturbance

Rosette tracks in the reference system moving with the disturbance.

Bars, spiral arms or close companions of a galaxy can be seen as a disturbance of the axially symmetric primary potential U , which rotates with a constant angular velocity Ω _S. If you change to a reference system that rotates with the disturbance, the trajectories of the stars are transformed in such a way that the angular velocities of the circular orbits are reduced to Ω '= Ω-Ω _S , while the epicyclic movement remains unaffected by the transformation. The stars continue to move on rosette orbits, but with a different frequency ratio κ / Ω '. In the special case of a corotating orbit, Ω '= 0 and only the epicyclic movement is visible, i.e. H. the star moves on an ellipse that is stationary relative to the fault. Since Ω 'is mostly smaller than Ω in magnitude, most of the rosette orbits of the stars in the moving frame of reference carry out many more epicycles per revolution than in the system that is not moving with them. In addition, the sign of the relative angular velocity for stars inside the corotating orbit is positive, outside it is negative.

If the frequency ratio κ / Ω 'is an integer, the rosette orbits are closed with κ / Ω' epicyclic revolutions per revolution around the center of the galaxy. The perturbation has a particularly strong influence on the orbit of a star if the magnitude of the ratio κ / Ω 'is precisely the number m of the symmetry of the perturbation:

${\displaystyle \kappa =\pm m\Omega '}$

The points of the path with the maximum distance from the center are rotated by the disturbance in the course of time into the disturbance and the semiaxes of the epicycles enlarge. A precise description of this resonance phenomenon is possible within the framework of perturbation theory, which is beyond the scope of this article. The following paragraph presents a self-consistent approach to how the Lindblad resonance affects the stabilization and expansion of the spiral structure of the galaxy.

Effect of the resonance

The effect of the resonances can be modeled mathematically if one chooses a continuum mechanical approach to describe the galaxy including the disturbance. If one assumes that the area density distribution Σ in a system is not stationary, one can consider its development over time. To do this, first consider the Euler equation

${\displaystyle {\frac {\partial \Sigma \mathbf {u} }{\partial t}}+\operatorname {div} \left(\Sigma \mathbf {u} \otimes \mathbf {u} \right)=-\nabla c_{s}-\Sigma \nabla U,}$

which contains the functions (surface density), (flow field) and (potential), as well as the "speed of sound" . The latter is determined by means of a heuristic equation of state which assigns a “pressure” p to the surface density Σ . All variables are then considered as the sum of an undisturbed time-independent variable and a disturbance. This means that a location- and time-dependent fault approach is made for all functions . So z. B. the areal density according to ${\displaystyle \Sigma }$ ${\displaystyle \mathbf {u} }$ ${\displaystyle U}$ ${\displaystyle c_{S}}$ ${\displaystyle c_{S}=dp/d\Sigma }$ ${\displaystyle \Sigma }$

${\displaystyle \Sigma (r,\phi ,t)=\Sigma _{0}(r)+{\tilde {\Sigma }}(r,\phi ,t)}$

disturbed.

If one then eliminates the undisturbed components in the equations following from the perturbation approach, one obtains a Poisson and three perturbation equations. This system of equations is given by an approach z. B. for the density of the shape

${\displaystyle {\tilde {\Sigma }}(r,\phi ,t)={\tilde {\Sigma }}^{*}(r)\cos \left(m\Omega _{S}t-m\phi +f(r)\right)}$

solved. This approach corresponds to spiral density waves with arms and a shape function f (r) that rotate rigidly with frequency . The pattern follows from this for the density maxima ${\displaystyle m}$ ${\displaystyle \Omega _{S}}$

${\displaystyle m\Omega _{S}t_{0}-m\phi +f(r)=0\!\,}$ ,

what a spiral is: ${\displaystyle m\neq 0}$

${\displaystyle \phi =\phi (r)={\frac {1}{m}}f(r).}$

Galaxy ( ESO 269-57 ) with a star ring and two clearly separated spiral arms

If one finds self-consistent solutions for the other perturbation equations as well, one obtains an algebraic system of equations which then leads to a dispersion relation which expresses the condition for a spiral density wave. From this in turn the dispersion equation follows

${\displaystyle 1-{\frac {m^{2}\omega ^{2}}{\kappa ^{2}}}+{\frac {k^{2}c_{s}^{2}}{\kappa ^{2}}}={\frac {2\pi G\Sigma _{0}|k|}{\kappa ^{2}}}}$,

in which stands for the difference from the angular velocity of a circular path with radius r with the angular velocity of the disturbance, is the radial circular wave number of the spiral structure and the epicyclic frequency, which results as above: ${\displaystyle \omega =\Omega (r)-\Omega _{S}}$ ${\displaystyle k={\frac {df}{dr}}}$ ${\displaystyle \kappa }$

${\displaystyle \kappa ^{2}=2R\Omega {\frac {d\Omega }{dr}}+4\Omega ^{2}.}$

If one reshapes the dispersion equation to

${\displaystyle m^{2}\omega ^{2}=\kappa ^{2}+k^{2}c_{s}^{2}-2\pi G\Sigma _{0}|k|}$,

so one recognizes that the solution for the circular wave number i. General has two branches:

${\displaystyle c_{s}^{2}|k|=G\pi \Sigma \pm {\sqrt {G^{2}\pi ^{2}\Sigma ^{2}+c_{s}^{2}m^{2}\omega ^{2}-c_{s}^{2}\kappa ^{2}}},}$

the shortwave (+) and longwave (-) are called. The Lindblad resonances can now be recognized at the points where the long waves disappear because their circular wave number becomes zero:

${\displaystyle m\omega =\pm \kappa .}$

Outside the OLR and within the ILR, only short waves can exist. The region around the corotating orbits shows a prominent behavior in this model, since there the expression under the root becomes negative, since ω is very small, i.e. H.

${\displaystyle G^{2}\pi ^{2}\Sigma ^{2}+c_{s}^{2}m^{2}\omega ^{2}<c_{s}^{2}\kappa ^{2}.}$

The waves there have a complex wave number and therefore disappear exponentially when entering this region (evanescent waves). In this region, so-called star rings form in some galaxies .

literature

• J. Binney, S. Tremaine: Galactic dynamics (=  Princeton series in astrophysics ). Princeton University Press, 1988, ISBN 0-691-08445-9 ( Online Google Books).

Web links

• Vereinigung der Sternfreunde eV: Circular: density waves and Lindblad resonances
• Kley, Wilhelm: Planet formation, Chapter 7: Development of planetary systems
• Pattern velocity and Lindblad resonances ; In: Astron. Astrophys. 333: 79-91 (1998).
• The Origins of Spiral Arms
• Christian Lerrahn: "Superhumps behind bars - what Euler says about superhumps" ( Memento from September 29, 2007 in the Internet Archive ) Diploma thesis with references.

Individual evidence

1. ↑ Binney, Tremaine: Galactic dynamics . 1988, p. 149 ff .
2. ↑ Tayler, RJ: Galaxies - Structure and Development . Vieweg, 1986.
3. ↑ Kley, Wilhelm: Planetary Origin , Chapter 7: Development of Planetary Systems ( Memento of the original from March 4, 2016 in the Internet Archive ) Info: The archive link was automatically inserted and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
4. ↑ ^{a b c d} Combes, F. et al .: Galaxies and Cosmology (=  A&A Library ). Springer, 1995.
5. ↑ Binney, Tremaine: Galactic dynamics . 1988, p. 399 ff .
6. ^ ^{A b c d} Whittle, Marc: Extragalactic Astronomy, Lecture Notes, University of Virginia
7. ^ Frank H. Shu: The Physics of Astrophysics: Gas dynamics . University Science Books, 1992, ISBN 0-935702-65-2 , pp. 147 ff . ( Online ).
8. ↑ Binney, Tremaine: Galactic dynamics . 1988, p. 365 ff .