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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 12.1 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.
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