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12.3 Dynamical Friction 12.3 Dynamical Friction When an object of mass MS (hereafter the subject mass) moves through a large collisionless system whose constituent particles (the field particles) have mass m ≪ MS, it experiences a drag force, called dynamical friction, which transfers energy and momentum from the subject mass to the field particles. Intuitively, this can be understood from the fact that two-body encounters cause particles to exchange energies in such a way that the system evolves towards thermodynamic equilibrium. Thus, in a system with multiple populations, each with a different particle mass mi, two-body encounters drive the system towards equipartition, in which the mean kinetic 2 2 energy per particle is locally the same for each population: m1⟨v1 ⟩ = m2⟨v2⟩ = mi⟨vi ⟩. Since MS ≫ m and particles at the same radius in inhomogeneous self gravitating systems tend to have similar orbital velocities, the subject mass usually has a much larger kinetic energy than the typical field particles it encounters, producing a net tendency for it to lose energy and momentum. 1 12.3 Dynamical Friction An alternative but equivalent way to think about dynamical friction, is that the moving subject mass perturbs the distribution of field particles causing a trailing enhancement (or “wake”) in their density. The gravitational force of this wake on the subject mass MS then slows it down (see Fig. 12.3). 2 12.3 Dynamical Friction The time it takes for a satellite to merge into a central galaxy due to the dynamical friction force can be calculated from the Chandrasekhar’s formula: where msat represents the satellite mass, ln(Λ) is the Coulomb logarithm (assumed to be ln (1 + M/msat)), ρ is the local density and vrel represents the relative velocity of the satellite. B(x) is given by where x = |vrel|/ 2σ. 3 12.3 Dynamical Friction The work done on the satellite by this force (F.v) will produce a change on its total energy over time: where r represent the radius of the satellite’s orbit and M the mass of the central halo. Considering this orbit to be circular, v2 = GM/r, and the equation can be rewritten as from which follows 4 12.3 Dynamical Friction Assuming the distribution of mass in the host halo to be an isothermal sphere (only for (2.27) this calculation), M = 2σ2r/ G and equation 2.27 can be rewritten as where σ represents the velocity dispersion of the central halo. Finally which integrated from the centre of the central galaxy to the initial position of the satellite, and for B(x=1), gives where rsat is the satellite position at the time it lost the dark matter halo and became a type 2. 5 6 12.2 Tidal Stripping We now examine how tidal forces impact a collisionless system in the more general case. As we will see, even in a static configuration tidal forces can strip material from the outer parts of a collisionless system, which is generally known as tidal stripping. Assuming a slowly varying system (a satellite in a circular orbit) with a spherically symmetric mass distribution, material outside the tidal radius will be stripped from the satellite. This radius can be identified as the distance from the satellite centre at which the radial forces acting on it cancel. These forces are the gravitational binding force of the satellite, the tidal force from the central halo and the centrifugal force and following King (1962) the disruption radius Rt will be given by: where msat is the mass of the satellite, ω is its orbital angular velocity and φ represents the potential of the central object. 7 12.2 Tidal Stripping The second derivative of the potential from the central object, d2φ/dr2, is given by where r represents the radial distance from the central galaxy to the satellite and M(< r) the mass distribution of the central object within that radius. Equation 5.1 can then be rewritten as: where ρ and M represent respectively the density and mass of the central halo. 8 12.2 Tidal Stripping 2 Taking the isothermal sphere approximation (M = 2σh alor/G), the disruption radius becomes: where σhalo represents the velocity dispersion of the central halo. Using the same approx- imation for the distribution of mass in the satellite galaxy, Rt becomes, where σsat is the velocity dispersion of the satellite galaxy, which we approximate by the velocity dispersion of the satellite halo just before it becomes stripped. 9 12.2 Tidal Stripping From the assumption that satellites follow circular orbits we get ω2 sat = GM/r3 and The final expression for the disruption radius will therefore be: The material outside this radius is assumed to disrupted and becomes part of the intra-cluster medium component. 10 11 12.4 Galaxy Merging If the orbital energy is sufficiently low, close encounters between two systems can lead to a merger. In the hierarchical scenario of structure formation, mergers play an extremely important role in the assembly of galaxies and dark matter halos. Mergers between systems of comparable mass typically cannot be treated analytically. When two systems merge, their orbital energy is transferred to the internal energy of the merger product. In addition, some of the orbital energy can be carried away by material ejected from the progenitors (for instance in the form of tidal tails). In the case of galaxies embedded in extended dark matter halos, a significant fraction of the orbital energy of the stellar components can be transferred to the dark matter by dynamical friction. Due to the strong tidal perturbations and the exchange of energy between components, the system needs to settle to a new (virial) equilibrium after merging, which it does via violent relaxation, phase mixing and Landau damping of global modes. The outcome of such relaxation is difficult to predict theoretically, and one has to resort to numerical simulations. 12 12.4 Galaxy Merging the structure of the remnant of a merger between two galaxies depends primarily on four properties: • The progenitor mass ratio q≡M1/M2, where M1 ≥M2. If q<∼4 (q>∼4) one speaks of a major (minor) merger. In major mergers violent relaxation plays an important role during the relaxation of the merger remnant, and the remnant typically has little resemblance to its progenitors. In minor mergers, on the other hand, phase mixing and/or Landau damping dominate, and the merger is less destructive. Consequently, the remnant of a minor merger often resembles its most massive progenitor. • The morphologies of the progenitors (disks or spheroids). Galactic disks are fragile, and are therefore relatively easy to destroy, especially when q is small. Disks that accrete small satellites (i.e., in the minor merger regime with q >∼ 10) typically survive the merger event but can undergo considerable thickening. Mergers that involve one or more disk galaxies tend to create tidal tails, which are absent in mergers between two spheroids. 13 12.4 Galaxy Merging • The gas mass fractions of the progenitors. Unlike stars and dark matter particles, gas responds to pressure forces as well as gravity and can lose energy through radiative cooling. Moreover, gas flows develop shocks whereas streams of stars can freely interpenetrate. Consequently, mergers between gas-rich progenitors (often called ‘wet’ mergers) can have a very different outcome from mergers between gas-poor progenitors (‘dry’ mergers). • The orbital properties. The orbital energy and angular momentum not only determine the probability for a merger to occur, but also have an impact on the merger outcome. For example, the relative orientation of the orbital spin with respect to the intrinsic spins of the progenitors (prograde or retrograde) is an important factor determining the prominence of tidal tails. 14 15 12.4.3 The Connection between Mergers, Starbursts and AGN An important aspect of (wet) mergers, and of interactions in general, is that they may be responsible for triggering (nuclear) starbursts and AGN activity. Numerical simulations of mergers and encounters between gas-rich disks show that the tidal perturbations can cause the disks to become globally unstable and to develop pronounced bars. Since the gas and stars do not have the same response to the tidal force, the gaseous and stellar bars generally have different phases. This phase difference gives rise to torques that can effectively remove angular momentum from the gas. As a result, the gas flows towards the central region and eventually forms a dense gas concentration at the center of the merger remnant. 16 12.4.3 The Connection between Mergers, Starbursts and AGN There is strong observational support that some starbursts and active galaxies are triggered by mergers and/or encounters between gas-rich galaxies. Using integrated colors, Larson & Tinsley (1978) showed that many interacting and merging galaxies have undergone short bursts (<∼ 108 yr) of star formation involving up to ∼ 5 percent of their total luminous mass. 12 All ULIRGs, which are starbursting galaxies with far-infrared luminosities exceeding 10 L⊙ (see §2.3.7), show clear signs of recent or continuing interactions, such as shells, tidal tails and complex velocity fields. In addition, most ULIRGs have dominant non-thermal optical emission lines, indicative of an embedded active galactic nucleus. Many bright radio galaxies also reveal structural peculiarities indicative of a recent interaction. Together with the absence of nearby companions, this suggests that some radio galaxies may have resulted from mergers of disk galaxies. Although many details are still poorly understood, it is clear from these and other observations that the presence of gas in a merger between two galaxies can result in enhanced star formation and/or AGN activity. Important outstanding questions are what fraction of the present-day stars formed in merger-induced starbursts and what is the role of feedback in regulating and terminating the starburst and AGN activity.
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