Dynamics of Galaxies and Clusters of Galaxies Luis A. Aguilar Instituto de Astronom´ıa, UNAM. Apdo. Postal 877, Ensenada, Baja California, M´exico. [email protected] Submitted to Lecture Notes in Physics, Springer Lecture notes of the 2005 Guillermo Haro Advances School: A Panchromatic View of Clusters of Galaxies and the LSS. Ed. O. L´opez–Cruz, M. Plionis and David Hughes 1 Introduction The purpose of these lectures is to introduce students to some of the dynam- ical phenomena that are important in the evolution of galaxies and clusters of galaxies. Galactic Dynamics is usually perceived as an arid discipline, full of math and hard to apply to real astronomical problems. This is very unfor- tunate, since gravity is behind all astronomical phenomena and its resulting dynamics has a role to play. This misconception of Galactic Dynamics stems from the fact that often students are introduced to the subject using a formal approach, full of mathematical rigor, that leads through a lengthy path be- fore reaching astronomical applications. Although mathematical correctness is gained, a physical grasp of the dynamics behind is sometimes lost. It is my intention to bypass the usual formal introduction and to use an eclectic list of topics related to the dynamics of galaxies and clusters of galax- ies, to illustrate the use of Galactic Dynamics. Our main goal is understanding, rather than rigor. I will try to show the student how, learning to use some basic equations, we can extract useful dynamical information that can be used to aid in our understanding of the realm of galaxies. In section 2, Poisson’s and Boltzmann’s equations are presented as the ba- sic equations of Galactic Dynamics. Then, after briefly reviewing what makes a dynamical system collisionless, in section 3 one of the Jeans equations in spherical coordinates is introduced as a quick tool to derive dynamical prop- erties of a model built from a density profile, without having to build a self- consistent dynamical model. The Navarro, Frenk and White profile is used as an example. A Mathematica Notebook version of this section is provided as 4 Luis A. Aguilar well. Students with access to this program can interact with this version and change parameters in the examples provided. The full Notebook is available from the Guillermo Haro and the author’s web pages. In section 4 we mention two additional density profiles and suggest them as exercises for the reader. We also note that simulations of interacting galaxies 4 tend to give density profiles that go as ρ r− at large radii when simulated in isolation, as opposed to those of systems∝ immersed in an expanding back- 3 ground, which tend to give the gentler ρ r− of the NFW profile. We show why the former dependence is expected for∝an isolated finite mass system that has been perturbed. In section 5 we talk about the orbital structure of spherical potentials. Although idealized, spherical potentials allow us to introduce basic concepts about orbits. We describe the use of the Lindblad diagram as a tool to classify orbits in spherical potentials. In section 6 we tackle a sticky issue: Dynamical friction, one of the main culprits responsible for robbing center of mass motion to satellite systems, causing them to spiral toward the center of the host system. As we will see, this force is a case of bipolar personality, depending on the speed of the object being slowed, its behavior can change quite a bit. In section 7 we examine the tidal force, responsible for truncating small systems when they move within the gravitational influence of a larger one. We will see that, contrary to common opinion, tides do not necessarily stretch along the radial direction. It all depends on the mass distribution of the tide- producing object. We then discuss the concept of tidal radius at some length, going from a very simplified static model of two point particles, to the inclusion of effects due to the motion of the satellite and host systems, non-circular orbits and extended mass distributions. In section 8 we talk about what happens when two galaxies collide. The rapid variation of the perturbing force pumps orbital energy into the internal degrees of motion of the interacting galaxies and heats them up, promoting mass loss above the rate due to tidal radius truncation. We discuss the role of two very important timescales: the interaction time and the internal dynamical time. As we will see, the amount of damage the interacting systems suffer, depends to a great extent on the ratio of these two numbers. Section 9 presents a simple exercise, where the combined action of dynam- ical friction and tidal truncation, acting on a Plummer sphere that spirals following a path close to circular within a flat rotation curve halo, is modeled. This wraps up much of what has been discussed in the three previous sections. Finally, in section 10 we suggest some references for the interested student to pursue in more detail the topics covered in these lectures. Section 11 pro- vides some concluding remarks and section 12, is the homework! After all, it wouldn’t be a school if there is no homework. Dynamics of Galaxies and Clusters of Galaxies 5 2 Basic Galactic Dynamics The two most important equations of Stellar and Galactic Dynamics are Pois- son’s equation, that relates the gravitational potential φ with its source, the mass density function ρ, 2φ = 4πGρ, (1) ∇ and Boltzmann’s equation, which is a transport equation that describes the evolution of a dynamical system in phase space, ∂f ∂f ∂f ∂f + v φ = (2) ∂t · ∂x − ∇ · ∂v ∂t ! "coll The phase space distribution function, f(x, v, t), contains all the dynam- ical information of the system. Its projection in configuration space gives the spatial density of the system, ρ(x, t) = f(x, v, t)d3v. # The right hand side of Boltzmann’s equation contains the so-called colli- sional term. Its effect differs from the term in the left hand side, mainly by the timescale on which the collisions it describes operates. The left hand side describes the flow of particles in a given parcel of the system as it moves in phase-space, on a timescale determined essentially by its potential, whereas the right hand side describes the flow of particles in and out of this parcel due to 2-body collisions, which usually are so fast with respect to the for- mer timescale, that the collisional time derivative appears to the rest of the equation as an instantaneous source and sink term. The natural timescale of the left hand side of Boltzmann’s equation is the so-called dynamical timescale, which is essentially the orbital time for particles within the system. As a very rough approximation to a system-wide average of this timescale, we can use the ratio of the system size and the rms velocity, t R/v dyn ≈ rms The effect of the right hand side of Boltzmann’s equation operates on the so-called collisional timescale. It can be shown that this time is of the form (e.g. see section 3.2 of [49]), t (R/v )N/ log(N), coll ≈ rms where N is the number of particles within the system. The ratio of tcoll to tdyn is a measure of the degree of collisionality of a dynamical system: 6 Luis A. Aguilar t /t N/ log(N) (3) coll dyn ≈ Notice that the parameters that characterize the properties of the system in physical units have cancelled out, and the only dependence left is on the number of particles within the system: the more particles, the larger this ratio is, and so the less important collisions are. This is a bit counterintuitive and must be explained further. One would think that the more particles there are in a system, the more collisions will be, and indeed this is the case, but what matters is not the number of collisions, as the effect they have in the system. Let us take a system with N particles. If we split each particle in two keeping everything else in the system, like total mass, size and rms velocity, constant, we double the number of particles and thus the number of collisions. But since the strength of the gravitational interaction scales as the product of the masses, and each mass is now one half of what it was, the result is a reduction in the effect of collisions. In reality, doubling the number of particles 1/3 also reduces the mean interparticle distance (l N − ), which has the effect of increasing the collisional effect, but in 3D ∝space this effect is dominated by the mass effect, even when taking into account the increased number of collisions per unit time. So, the more particles the more collisions, but the less effect they will have. When the effect of collisions is negligible over the period of time we are interested in, we are lead to the collisionless Boltzmann equation: ∂f ∂f ∂f + v φ = 0 (4) ∂t · ∂x − ∇ · ∂v This is a partial differential equation that, together with Poisson’s equa- tion and proper boundary conditions, presumably set by observations, can in principle be solved. In practice, there is not enough observational information and furthermore, the mathematical complexity of the task of solving it, makes the direct solution an impossibility for realistic cases. Other alternative approaches have been used. In particular, a fruitful ap- proach is obtained rewriting the previous equation as a total time derivative, using the fact that minus the gradient of the potential is just the acceleration, ∂f ∂f ∂f Df + x˙ + v˙ = = 0 (5) ∂t · ∂x · ∂v Dt This means that, as we move along dynamical trajectories, the value of the distribution function does not change.