Galactic Dynamics Pdf

[Ilona Brownson]

Inthe present article we focus on topics of galactic dynamics which are most relevant to the general theory and methods of dynamical astronomy (see also scholarpedia section "Extragalactic Astronomy").

The main components of the Universe are the galaxies, which arecomposed of billions of stars, but also of gas, dust and darkmatter. The most well known classification of galaxies is due toHubble. The main types are elliptical galaxies (E), normal spiralgalaxies (S), barred spiral galaxies (SB) and irregular galaxies(I)( Figure 1 ).

The elliptical galaxies have various degrees of ellipticity fromzero (E0) to \(0.7\) (E7), and they are slowly rotating. On the other hand the spiral and barred spiral galaxies are rotating fast and they contain spiral arms. There are tight spirals (Sa, SBa),intermediate (Sb, SBb) and open spirals (Sc, SBc). They are flatsystems, containing also gas and dust, out of which new stars areformed continuously. The spiral and barred spiral galaxies are also called `disc galaxies', since the spiral arms are embedded in a thin disc (the thickness is of order less than 10%). Finally, the irregular galaxies are relatively small systems that accompany large spiral galaxies.

The shapes of the galaxies are governed mainly by the gravitational interactions between their individual mass components, i.e. the stars, dark matter, and gas. The stars form the main body of the luminous part of a galaxy. The gas forms a relatively small proportion of matter (up to 10% in disc galaxies). The stars and the gas together are called `baryonic matter'. The dark matter, on the other hand, contains more mass than the stars and gas, and it extends to large distances (one order of magnitude larger than the baryonic matter).

The motions of the stars and of the dark matter elements are governed purely by their gravitational forces. The study of these motions and their combinations to form self-consistent statistical mechanical configurations, is called stellar dynamics. This constitutes the central approach to galactic dynamics. Gas dynamics, on the other hand, is governed also by dissipative forces due to pressure, radiation, magnetic fields etc. The latter also influence, to some extent, particular morphological and kinematical features of the galaxies.

The gravitational potential of a galaxy is composed of a mean field, due to the average distribution of the galactic matter, and fluctuationsdue to the approaches (encounters) between individual stars. Anestimate of the effects of these encounters is provided by the"relaxation time" of the system (Chandrasakhar), which can be estimated by the time required by a star to change its average direction of motion by 90 due to the cumulative effects of encounters. This time is very long, of the order of 1012 years, while the periods of the motions of the stars around the center of the galaxy are of the order of 108 years ("dynamical time"). Thus, in a first approximation, we may consider the orbits of the stars as due to the general distribution of the galactic matter. The same applies to the orbits of dark matter elements. In a better approximation, however, various details of the relaxation process have to be taken into account. This issue is discussed in section 6 (N-body systems).

In the treatment of dark matter we may consider i) rigid halo models with a fixed dark matter density \(\rho_d\ ,\) where \(f\) and \(\rho\) refer to the stellar distribution function and density while (4) is replaced by \(\nabla^2V=4\pi G(\rho+\rho_d)\ ,\) or ii) live halo models in which the dark matter is responsive to, and affects the collective motions of stars. Rigid halo models have been used extensively in orbital studies of stars (or clusters). However, the rigid halo approach is not convenient in circumstances where the halo interacts collectively with a particular stellar sub-structure like a bar (see section 6). N-body simulations have shown that in such cases only a `live' halo model can capture correctly the effects of such interactions.

By solving the equations of motion in a given potential\(V(\textbfx,t)\) we define smooth orbits of the stars or dark matter elements. The superposition of many orbits with appropriate weights yields a response density. The system must be self -consistent (self-gravitating) i.e. the response density must be equal to the imposed density.

The self-consistency condition is satisfied accurately in N-bodysimulations (section 6). However in many cases we assume a fixedpotential that represents a model galaxy. In this case theself-consistency condition can be checked a posteriori, i.e. afterthe orbits are calculated in the fixed potential.

We frequently consider stationary models, i.e. \(V=V(\textbfx)\) is considered independent of time. In N-body simulations, we often consider successive snapshots and study the forms of the orbits at these times.

In stationary galactic systems of \(n\) degrees of freedom there can be at most \(n\) independent integrals which appear as arguments of \(f\ .\) The energy \(E\) is an obvious integral in all stationary systems, while the angular momentum \(L\) along the axis of symmetry is a second integral in axisymmetric systems. Additional `third integrals', of exact or approximative form, can be found in various cases and they play a key role in dynamics.

where \(K\) and \(W\) are the total kinetic and potential energies of the stellar system in equilibrium. Depending on the shape of a system, different states of virial equilibrium may exist with different degrees of kinetic energy that go to rotation or to velocity anisotropy. Accordingly, the elliptical galaxies as well as the spheroidal bulges of disc galaxies are divided into those being rotationally supported or pressure supported.

The study of individual orbits in fixed potential models with different degrees of symmetry is a central subject of galactic dynamics, since such models offer idealizations of the true gravitational potential of galaxies. A generic feature of such models is the co-existence of ordered and chaotic orbits. In fact, the properties of ordered orbits can be unraveled using various forms of the canonical perturbation theory. Such is the theory of the third integral as well as the Kolmogorov-Arnold-Moser (KAM) theory. On the other hand, the properties of chaotic orbits can be explored mainly by numerical means such as the Poincar surface of section or quantities such as the Lyapunov characteristic number. Readers are deferred to (Contopoulos 2004) for an instructive introduction to these topics. Below, we review only the most basic facts relevant to the orbits in galaxies. We examine first the orbits and forms of integrals in simple (non-rotating) systems like elliptical galaxies. Then we examine the orbits and integrals in fast rotating disc galaxies. We return to the issue of self-consistency in sections 5 (spiral structure), and in section 6 (N-body simulations).

Within the approximation of collisionless stellar dynamics, the orbits in an axisymmetric galaxy are governed by a smooth time-independent axisymmetric gravitational potential \(V(R,z)\ ,\) which corresponds to the solution of Poisson's equation for an axisymmetric matter distribution \(\rho(R,z)\ .\) Then, the orbit of a star (point particle) becomes independent of its mass, and it is given by a Hamiltonian of the form\[\tag8H\equivp_R^2\over 2+p_\vartheta^2\over 2R^2+p_z^2\over2+V(R,z)=E\]

Ordered orbits are found by calculating a `third integral' of motionin the form of a series (Contopoulos). Expanding the Hamiltonian withrespect to the radius \(R_0\) of a circular orbit, the Hamiltonian describing the motion in a meridian plane takes the form\[\tag12H=\frac12(\dotx^2+\doty^2+\omega^2_1x^2+\omega^2_2y^2)+\varepsilon xy^2+\varepsilon'x^3+\mboxhigher order terms\]

provided that \(\Phi_2\) is a properly chosen function. The successive terms in the power series are of increasing order in a properly chosen small parameter. The latter gives the order of the distance of orbits in phase space from the equatorial circular orbit. More generally, in `third integral' expansions the small parameter can be chosen so as to reflect the deviations of the model considered from an integrable model.

The ordered orbits obeying a non-resonant third integral are called`tube' orbits if \(J_0\neq 0\ ,\) and `box' orbits if \(J_0=0\ .\) Box orbits pass arbitrarily close to the center, while tube orbits leave a hole around the axis of symmetry. On the other hand, the orbits obeying a resonant third integral are either periodic or they form thin tubes around their corresponding periodic orbits.

In the case of a galaxy with a smooth core the orbits near thecenter are ordered. The boxes are deformed Lissajous figures( Figure 3 a ), while the orbits around particular resonant periodic orbits form elongated tubes ( Figure 3 b ). But there also many chaotic orbits ( Figure 3 c ), mainly in the region separating the box orbits from the main tube orbits.

If now the core of a galaxy is cuspy (e.g. the density rises as a power law as we approach the center), or if we add a central mass (e.g. a black hole) at the center of the galaxy, all the orbits near the center become chaotic. At the same time the number of tube orbits increases. In order to distinguish between ordered and chaotic orbits we use a Poincar surface of section, i.e. a surface in phase space that intersects all the orbits, and find the distribution of the successive intersections (iterates) of each orbit by this section. In the case of orbits in the meridian plane \((R,z)\) of a galaxy with a plane of symmetry \(z=0\ ,\) the plane \(z=0\) is a Poincar surface of section ( Figure 4 ). The energy\[\tag15E=\frac12(\dotR^2+\dotz^2)+V(R,z)\]

In a generic dynamical system the ordered and chaotic orbitscoexist. In fact, if an integrable system is perturbed slightly,there is a large set of integral surfaces containing regular orbits.This result is based on the famous Kolmogorov, Arnold, Moser (KAM)theorem that can be stated as follows. If an autonomous Hamiltonian system of N degrees of freedom is close to an integrable systemexpressed in action-angle variables, there are N-dimensionalinvariant tori containing quasi-periodic motions with frequencies \(\omega_j\) satisfying a Diophantine condition\[\tag19\mathbf\omega\cdot k=\omega_1k_1+\omega_2k_2+...+\omega_Nk_N>\frac\gamma+...\]

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