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 approximation 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 12.1 High-Speed Encounters

In general, an encounter between two collisionless systems is extremely complicated, and one typically has to resort to numerical simulations to investigate its outcome.

However, in the limiting case where the encounter velocity is much larger than the internal velocity dispersion of the perturbed system the change in the internal energy can be approximated analytically.

Such high-speed encounters play an important role in galaxy clusters, where the velocity dispersion of the member −1 galaxies (σcluster ∼ 1000 km s ) is significantly larger than that of the individual galaxies.

12 12.1 High-Speed Encounters

Consider the encounter between S and P. Let v∞ be the initial velocity of P with respect to S when their separation is large.

In the large-v∞ limit the tidal forces due to P act on a time scale that is much shorter than the dynamical time of S, and the encounter is said to be impulsive. This means that we may consider any test particle, q, inside S to be stationary with respect to the center of S during the encounter, only experiencing a change ∆v in its velocity. In this impulse approximation, the potential energy of q before and after the encounter is the same (i.e. the density distribution of S remains unchanged during the encounter), so that the change in the total energy per unit mass of a particle of S is given by

We are interested in computing ∆ES, obtained by integrating ∆E over the entire system S. Because of symmetry, the integral of the first term on the right-hand side of Eq. (12.3) is typically equal to zero, so that

13 12.1 High-Speed Encounters

In the impulse approximation, the encounter only changes the kinetic energy of a system, but leaves its potential energy intact. Consequently, after the encounter the system is no longer in virial equilibrium, and has to undergo a relaxation process in order to settle to a new virial equilibrium.

Let the initial kinetic and total energies of S be KS and ES, respectively. According to the virial theorem we have that ES =

−KS .

Due to the encounter, ES → ES + ∆ES and, since all this energy is invested in the internal kinematics of S, we also have that

KS → KS + ∆ES . After S has relaxed to a new virial equilibrium, KS = −(ES + ∆ES). Thus, the relaxation process decreases the kinetic energy by 2∆ES. This energy is transferred to potential energy, which becomes less negative, implying that tidal shocks cause systems to expand.

14 12.1 High-Speed Encounters

The net effect of pumping energy into the system is therefore a decrease of its kinetic energy (i.e. the system gets ‘colder’). This is a consequence of the negative specific heat of self- gravitating systems. By analogy with the particles in an ideal gas, the kinetic energy in an N-body system of equal point masses can be assigned to a mean ‘temperature’:

2 Here kB is Boltzmann’s constant, and ⟨v ⟩ and ⟨T⟩ are the mean velocity dispersion and mean temperature, respectively. According to the virial theorem we have that

which allows us to define the heat capacity of the system as

C is always negative, so that a system becomes hotter when it loses energy. This is a characteristic, and somewhat counter-intuitive, property of all systems in which the dominant forces are gravitational. This includes the Sun, where the stability of nuclear burning is a consequence of C < 0: If the reaction rates become too high, the excess energy input into the core makes the core expand and cool. This makes the reaction rates drop, bringing the system back to equilibrium.

15 16 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.

Contrary to the cases of high-speed encounters and dynamical friction, for which reasonably accurate analytical descriptions are available, 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.

17 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.

18 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.

19 20 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.

21 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.

22 23 12.5 Transformation of Galaxies in Clusters

Denser environments host larger fractions of galaxies morphologically classified as early-types. In addition, galaxies in denser environments are on average redder, less gas-rich, and have lower specific star formation rates. This strong environment dependence is often interpreted as indicating that galaxies undergo transformations (e.g. late- type → early type, star forming → passive) once they enter or become part of a denser environment.

14 15 Clusters of galaxies are the largest virialized structures in the Universe, with masses of about 10 − 10 M⊙, and velocity dispersions of about 1000 km s−1, and the environments with the highest number densities of galaxies.

Hence, galaxy interactions are frequent, making clusters the ideal environments to look for possible transformation processes. Roughly speaking, cluster galaxies can be effected by the cluster environment in three different ways: (i) tidal interactions with other cluster members and with the cluster potential, (ii) dynamical friction, which causes the galaxy to slowly make its way to the cluster center, and

(iii) interactions with the hot, X-ray emitting intra-cluster medium (ICM) that is known to permeate clusters.

24 12.5.1 Galaxy Harassment

In a cluster of galaxies, the typical velocity of a galaxy is of the order of the velocity dispersion of the cluster, which is much larger than the internal velocity dispersion of the galaxy. Encounters are all high-speed in nature - colliding galaxies are impulsively heated (i.e. its internal energy is increased). As a result, the perturbed galaxy becomes less bound, and more vulnerable to disruptions by further encounters and by tidal interactions with the global cluster potential - harassment.

Early studies of this process focused on how it modifies the structure of elliptical galaxies in clusters and whether it can account for the extended outer envelopes of central cluster galaxies. A particularly interesting result by Aguilar & White (1986) was that the de Vaucouleurs surface brightness profile appears invariant to harrassment, i.e. rapid encounters between galaxies with this structure can cause substantial mass loss but the profile of the final object is still well fit by the r1/4-law; there is no tendency for harrassment to produce a tidal cut-off in the density profiles of cluster galaxies.

25 12.5.1 Galaxy Harassment

Hydro sims show that disks can be almost entirely destroyed by one or two passages through the cluster.

This is particular true for the fragile disks of late-type (Sc-Sd) spiral galaxies. If such galaxies experience several close encounters with relatively massive cluster members, they may lose very substantial amounts of mass as impulsive heating pushes stars onto unbound orbits. The disk stars that remain bound to the galaxy are also heated, causing a transformation of the (dynamically cold) disk into a spheroidal component closely resembling a dwarf elliptical. Since dwarf ellipticals are ubiquitous in clusters, it may well be that they are the remnants of the disk galaxies that have experienced such harassment.

This is consistent with the fact that the galaxy population of clusters is observed to have evolved rapidly over the past few billion years: at redshifts z ∼ 0.4, clusters contain a large population of star-forming galaxies, many of which are disturbed and show evidence for multiple bursts of star formation. This population of “Butcher-Oemler” galaxies is almost entirely absent from clusters at z ∼ 0 (Butcher & Oemler, 1978).

However, this scenario is made less plausible by the fact that many dwarf ellipticals show little or no rotation, while the objects formed in numerical simulations by harassing disk galaxies always seem to preserve significant amounts of rotation.

26 12.5.1 Galaxy Harassment

Although harassment may have a strong impact on the morphology of late-type (Sc-Sd) disk galaxies, which typically have relatively low surface density disks, numerical simulations indicate that it has little impact on the more compact early-type (Sa-Sb) disk galaxies. This is easy to understand from the fact that the dynamical time in denser, more compact disks is shorter. In Sa and Sb galaxies the orbital time within a couple of disk scale lengths is short enough for the disk to respond adiabatically to the high-speed encounters experienced in clusters. In addition, since denser systems have a smaller fraction of their mass located beyond the tidal radius, they are also less susceptible to tidal stripping. Tidal shocks therefore can neither remove large amounts of material from early-type disk galaxies, nor transform them into spheroids.

Nevertheless harassment can still significantly heat the disks and drive disk instabilities that funnel gas into the central regions. Combined with ram-pressure stripping, it is thus plausible that harassment can transform Sa-Sb galaxies into S0 galaxies.

27 12.5.2 Galactic Cannibalism

Galaxies in clusters are unlikely to merge since their encounter speed is typically much larger than their internal velocity dispersion.

One important exception: because of dynamical friction, galaxies lose energy and momentum which causes them to ‘sink’ towards the center of the potential well. If the dynamical friction time is sufficiently short, the galaxy will ultimately reach the cluster center, where it will merge with the central galaxy already residing there. Hence, a central cluster galaxy may accrete satellite galaxies, a process called galactic cannibalism.

Roughly speaking, a satellite galaxy will be cannibalized by its central galaxy when dynamical friction can bring it to the center of the potential well within a Hubble time 1/H(z). From Eq. (12.45) we see that the critical radius for this to occur to a satellite galaxy of mass MS in an isothermal host halo of mass Mh is

where we have used Vh/rh = 10H(z). Thus, although it is preferentially the massive satellites that will be cannibalized, in the central regions of a cluster even low mass galaxies can be accreted by the central galaxy within a Hubble time.

28 12.5.2 Galactic Cannibalism

Galactic cannibalism has two important effects: it causes the central galaxy to increase in mass, and it causes a depletion of massive satellite galaxies, for which the dynamical friction time is the shortest.

Consequently, cannibalism causes an increase of the magnitude difference, ∆M12, between the brightest and second brightest member of a cluster.

If galactic cannibalism is the main mechanism regulating the luminosity of the central galaxy, the magnitude gap ∆M12 can thus be used as a measure for the dynamical age of the cluster: older systems will have a larger magnitude gap.

29 12.5.2 Galactic Cannibalism

The central cluster galaxy is typically also the brightest cluster galaxy and often has an extraordinarily diffuse and extended outer envelope, in which case it is called a cD galaxy. Numerous studies have suggested that these cD galaxies are the product of galactic cannibalism. This would explain not only their large masses, but also their diffuse envelopes, which, in this picture, consist of material tidally stripped from the cannibalized galaxies as they spiral into the cluster center.

Some of the assumptions made in these early models were poorly chosen and led to overestimates of the efficiency of the process. For example, most studies ignored mass loss from the satellite galaxies due to tidal stripping [as does Eq. (12.77)]. This can increase dynamical friction times by a factor of several, a correction that results in current cD growth rates too low to explain the observed luminosities but in excellent agreement with the rates inferred from the fraction of cD galaxies harboring multiple nuclei.

This is not fatal, since clusters have been growing at the same time as their central galaxies, and merging onto the central object should have occurred more rapidly in lower mass progenitors than in today’s cluster. As noted by, this implies that cannibalism must be followed throughout the hierarchical growth of the cluster, rather than just in the present-day system. The merger tree techniques discussed in §7.3 make it possible to solve this problem in detail in a CDM cosmology. Current results show that the cannabalism model provides a detailed explanation for most properties of the cluster/cD population.

30 12.5.3 Ram-Pressure Stripping

When a galaxy moves through the intracluster medium (ICM), its gas component experiences a ram pressure, just like one feels wind drag when cycling. As first discussed by Gunn & Gott (1972), if the ram pressure is sufficiently strong, it may strip the gas initially associated with the galaxy.

Consider a disk galaxy of radius Rd moving through an ICM of density ρICM with velocity V . For simplicity, assume that the velocity vector of the disk galaxy is pointing perpendicular to the disk, so that it experiences a face-on wind. The 2 amount of ICM material swept per unit time by the disk is πR dρICMV. If we assume that the wind is stopped by the interstellar medium (ISM) of the disk, then the momentum transferred to the 2 2 2 disk per unit time is πR dρICMV , corresponding to a ram pressure Pram = ρICMV . If this pressure exceeds the force per unit area that binds the ISM to the disk, the gas will be stripped.

To estimate the binding force, assume that the mean surface density of interstellar gas is ΣISM and that the mean mass density (usually dominated by stars) of the disk is Σ⋆. The gravitational field of the disk is approximately 2πGΣ⋆ in the disk and so the gravitational force per unit area on the interstellar gas is 2πGΣ⋆ΣISM. Hence, we expect that stripping occurs if

31 12.5.3 Ram-Pressure Stripping

10 To put this in perspective, consider a disk similar to that of the Milky Way, with a stellar mass of 5 × 10 M⊙ and an ISM 9 mass of 5 × 10 M⊙, both spread over a disk of radius 10 kpc.

Suppose that this disk is moving at a speed V = 1000 km s−1 (the typical velocity dispersion of galaxies in rich clusters) −27 −3 with respect to an ICM. Eq. (12.78) then gives ρICM > 4.6 × 10 g cm as the condition for ram-pressure stripping to occur.

The typical density of the ICM in clusters is ∼ 10−3 particles per cubic centimeter, or ∼ 10−27gcm−3, of the same order as required for ram-pressure stripping to be effective.

In general, a disk galaxy will be on an eccentric orbit along which its velocity and the ICM density change as function of time. Hence, whether condition (12.78) is satisfied or not depends on time. In general, though, since the surface densities of stars and gas decline as a function of galactocentric distance, one can typically identify a radius in the disk beyond which ram-pressure stripping is efficient.

32 12.5.3 Ram-Pressure Stripping

When a spiral galaxy loses most of its interstellar gas, its potential for future star formation is greatly reduced. Ram pressure stripping is therefore often invoked to explain why dense environments, such as clusters, reveal a clear deficit of gas-rich, star-forming galaxies. With (most of) its interstellar medium removed, and with star formation quenched, the resulting disk galaxy may look like a S0 galaxy. Thus, ram-pressure stripping may explain why clusters contain a larger fraction of S0 galaxies than the field.

However, the importance of ram-pressure stripping for transforming spirals into S0s, and for quenching star formation, is still a matter of debate. Although there is ample observational evidence that ram-pressure stripping is occurring, in almost all cases only the gas at relatively large galactocentric radii is being stripped. This is consistent with numerical, hydrodynamical simulations, which show that if a Milky-Way like galaxy falls through the center of the Coma cluster, only about 80 percent of its gas mass is stripped; the inner 20 percent survives a plunge through the densest region of the ICM. Furthermore, it is not even clear that ram-pressure stripping necessarily results in a reduction of the galaxy’s star formation rate. The remaining, non-stripped gas may actually be compressed by the ram-pressure, giving rise to enhanced star-formation in the disk. Somewhat surprisingly, the effect of ram pressure stripping seems to be enhanced if the gas disk is porous, so that part of the wind can stream through the holes. Although this results in less direct momentum transfer to the gas disk, streaming through the holes can ablate their edges through turbulent viscous stripping, and can prevent the stripped gas from falling back. However, it should be noted that the apparent HI holes in real disk galaxies may not be empty, but rather filled with molecular gas. In this case, the flow cannot pass through the disk, and ram-pressure stripping remains inefficient in removing the inner material. 33 12.5.4 Strangulation

The ram-pressure stripping discussed above may strip a galaxy of its entire cold gas reservoir, causing an abrupt quenching of its star formation. However, if only the outer parts of a galaxy’s gas disk are stripped, star formation may continue until all fuel is exhausted. In normal field spirals, the gas consumption timescale is typically only a few Gyrs. Their lifetimes can be significantly extended only if they continue to accrete fresh gas. Galaxies are believed to be surrounded not only by halos of dark matter, but also by substantial amounts of hot/warm gas. This consists: in part of gas that is just falling onto the system for the first time, in part of gas which has already fallen in and been shocked to high temperature, but has not yet cooled, and in part of gas that has been reheated and expelled from the galaxy by feedback processes. The infall and cooling of this material can replenish the ISM as it is consumed by star formation, allowing the galaxy to continue forming stars over long timescales. Hence, this extended gas component can be regarded as a reservoir of fuel for future star formation. Since this gas reservoir is only relatively loosely bound to the galaxy, it is easily removed, either by tides or by ram- pressure when the galaxy falls into a larger system such as a cluster. Thereafter the galaxy loses its supply of new gas and gradually exhausts the remaining fuel in its disk. This process is called strangulation, and produces a gradual decline of a galaxy’s star formation rate on the gas consumption timescale.

34 12.5.4 Strangulation

It is well established that galaxies in the field form stars at rates several times higher than systems of similar luminosity in the denser environments associated with groups and clusters.

Although this is partly a reflection of the well- known morphology-density relation, in that ellipticals and S0 galaxies, which have lower specific star formation rates than spirals, are more abundant in denser environments, even galaxies of given stellar mass and given internal structure show a strong correlation between star formation rate and environment density. Late-type disk galaxies in clusters clearly have less gas and form stars at lower rates than those in the field. As argued by several studies, strangulation seems to be the main mechanism responsible for this environment dependence of the specific star formation rates.

Most current models for the evolution of the galaxy population and its environment dependence include strangulation by assuming that a galaxy’s gas reservoir is completely and instantaneously stripped when it is accreted onto a bigger host system. In combination with merging, strangulation can reproduce most of the observed trends of star formation activity and morphology with stellar mass and environment. Nevertheless, a detailed comparison shows that current implementations predict too many faint galaxies in clusters, and, in particular, much too high a fraction of red satellite galaxies. Extending the timescale on which the gas reservoir is stripped to ∼ 2 Gyrs reproduces the color distribution of satellite galaxies considerably better. Such delayed stripping is also suggested by hydrodynamical simulations (McCarthy et al., 2008), which show that up to ∼ 30 per cent of the initial hot gas can remain bound to a satellite galaxy even 10 Gyr after it has been accreted.