Galactic Dynamics: Second Edition

“One of these becomes the vertical energy Ez, which in turn goes as z2, having manifest even parity, as one approaches the Galactic plane (Binney & Tremaine 2008).”

“Relaxed systems correspond to stationary solutions of these equations, and much of the field of galactic dynamics is concerned with understanding different aspects of these solutions (see Binney & Tremaine 2008, for an excellent exposition).”

“…which read as follows: r p = - + + - dP r dr r P r M r r P r r r M r 4 2 , 13 3( ) ( ( ) ( )) ( ) ( ) ( ( )) ( ) p = + - dN r dr M r r P r r r M r 2 4 2 , 14 3( ) ( ) ( ) ( ( )) ( ) where P(r) is the DM pseudo-pressure that can also be defined for collisionless particles (Binney & Tremaine 2008).”

“In general,vc differs from the stellar rotation velocityvϕ (often referred to as streaming motion) which is usually lower due to the effect of the stellar velocity dispersion, which acts as a pressure term (see e.g. Binney & Tremaine 2008).”

“Here we briefly recall the relevant equations for a local stability analysis of a self-gravitating gaseous disc (e.g. Binney & Tremaine 2008).”

“…for the luminous King component and the dark halo is obtained assuming isotropy and using the Ed- c© 2016 RAS, MNRAS 000, 1–14 dington’s formula (Binney & Tremaine 2008), namely fi( ) = 1√ 8π2 ∫ 0 d2ρi dΨ2 dΨ√ −Ψ (4) Here, is the binding energy and Ψ the potential, which is taken as that…”

“Note that this expression accounts for the progressive tidal stripping of the secondary galaxy along the evolution (Binney & Tremaine 2008, at least as long as both galaxies are modeled simplistically as isothermal spheres).”

“Their high stellar densities, sometimes exceeding 105M⊙ pc −3, lead to a variety of complex dynamical phenomena (Spitzer 1987; Heggie & Hut 2003; Binney & Tremaine 2008).”

“The typical scales in debris (which characterize the dispersions σE and σL) can be computed following previous work (e.g. Johnston 1998; Binney & Tremaine 2008).”

“Following Binney & Tremaine (2008), we first consider the collisional relaxation time of such a system, neglecting the gas component.”

“Dynamical friction refers to an effective drag force induced on a massive object via gravitational interaction with nearby matter (Chandrasekhar 1949; Binney & Tremaine 2008).”

“Assuming that the distribution of DM particles in the Galaxy is spherically symmetric and isotropic, and in the limit of spherical symmetry for the underlying gravitational potential for the Galaxy Φ(r), Eddington’s formula [28] gives an one-to-one correspondence between the DM halo density profile ρh(r) and its phase…”

“…fragmentation scale, with characteristic fragmentation scale given by the most unstable wavelength, situated at one-half of the Toomre wavelength: λ(most unstable) ≡ pλT, (4) where p is a constant equal to 0.5 or 0.55 for zero-thickness fluid or stellar discs, respectively (Binney & Tremaine 2008).”

“When the system is in dynamical equilibrium and collisionless under the gravitational smooth potential, the distribution function obeys the steady-state collisionless Boltzmann equation (Binney & Tremaine 2008).”

“σ2 PBHrh/G − 1, otherwise the assumption of PBHs travelling in an homogeneous star field will not be valid [62].”

“As a self-gravitating system, the inner halo has negative heat capacity, the more heat is extracted by dark matter collisions, the further it collapses to convert its gravitational energy to kinetic energy [60].”

“…using the dispersion relation of the density wave from which they develop that is given by[ m(Ω−Ωp) ]2 = κ2 + c2s k 2, (12) (equation (6.55) of Binney & Tremaine (2008) without considering self-gravity) where Ω and Ωp are the gas and pattern angular frequencies while κ2 = R−3 d(R4Ω2)/dR…”

“The integrals of motion, J , completely describe the orbit, which determines the orbital energy (for more information see Binney & Tremaine 2008).”

“…cold exponential disc as in SB09), and to convolve these with the Green’s function G( J, J ′, τ ) that solves the action-space diffusion equation (e.g. Binney & Tremaine 2008, section 7.4.2c) ∂f ∂t = ∂ ∂ J · ( −D(1)f + D(2) · ∂f ∂ J ) , (13) for a delta-function source of stars at J ′ a time τ in…”

“We then impose the equilibrium of the first and second moments of velocities by solving Jeans’ equations (see e.g., Binney & Tremaine 2008) with a constantQT .”

“A general orbit in these triaxial potentials does not necessarily have conserved integrals of motion (e.g., actions Binney & Tremaine 2008).”

“…the assumption that the smallest gap length that is visible is the width of stream, on the basis that the random motions of the stars in the stream are well described by epicycles which have an extent perpendicular to the steam nearly equal to the extent along the stream (Binney & Tremaine 2008).”

“…the volume number density and radial component of the velocity dispersion at radius r for a given tracer population, and βr is the anisotropy parameter defined as 1−σ2t /2σ2r , with σt being the tangential components (σ 2 t = σ 2 θ + σ 2 φ) of the velocity dispersion (Binney & Tremaine 2008).”

“At the same tii s velocity dispersion will increase (e.g. Binney & Tremaine 2008), so that, all other parameters and properties remaining constant, its spirals should become considerably more open.”

“A key element of the eccentric instability is thatf(e) in Equation (2) is an increasing function of eccentricity (e.g. Binney & Tremaine 2008), so that an orbit that is slightly more eccentric than average will lag behind in precession.”

“…friction is only important in a massive galaxy, such as the eventual remnant of the merging Antennae, for a few very massive clusters located very near the galactic center and can safely be neglected for all the other clusters in a first approximation (Fall & Zhang 2001; Binney & Tremaine 2008).”

“We follow the general relativistic approach in SFW, which extended the Newtonian treatment in [48–50] (see also [51]).”

“One remarkable property realized early on is that the cluster evolution proceeds asymptotically in a self-similar fashion, that is, the cluster density profiles differ only in scale and normalization at late times (e.g., Cohn 1980; Binney & Tremaine 2008).”

“All orbits in a smooth spherically symmetric potential are rosettes confined to a single plane (Binney & Tremaine 2008) and are regular.”

“In the solar neighborhood, Dehnen & Binney (1998a) found that there is a linear relation between lag and the square of the radial velocity dispersion, which is in good agreement with theory (e.g., Section 4.8.2a of Binney & Tremaine 2008).”

“For a logarithmic potential, at distances much larger than the core radius, the relation between qφ and qρ can be approximated as (Binney & Tremaine 2008): 1− qφ ≈ 1 3 (1− qρ) .”

“Assuming dynamical equilibrium (steady state) and axisymmetry, the z-Jeans equation in cylindrical coordinates becomes (Binney & Tremaine 2008; Silverwood et al. 2016) 1 ν ∂ ∂z (νσ 2z ) + 1 Rν ∂ ∂R (RνσRz)︸ ︷︷ ︸ tilt term:T = −∂ ∂z , (5) where z is the perpendicular distance to the Galactic plane,…”

“…computed an upper limit to the escape velocity, vesc , using the potential of a thin exponential disk, with the form presented in equation 2.165 of Binney & Tremaine (2008): Φ(r, 0) = −πGΣ0r [I0(y)K1(y) − I1(y)K0(y)] (6) where y = r/2rd , In,Kn are modified Bessel functions, rd the disk scale…”

“For example, assuming a spherical dark matter halo, in the disk’s outer regions where its mass can be regarded as negligible, the circular, Ω, and vertical, ν, frequencies are expected to be equal (Binney & Tremaine 2008).”

“Classically, a simple characterization of the filtering scale is given by (Binney & Tremaine 2008) λ2F,Jeans(t) = c2s(t)a(t) 4πGρ̄m0 , (2) in comoving units, where ρ̄m0 is the average comoving matter density, a is the cosmological scale factor, and cs is the sound speed.”

“Here, we assume C = 2.25, which is the average value of known galactic mass distribution models (Binney & Tremaine 2008).”

“The interested reader can refer to excellent reviews in Binney & Tremaine (2008) and Sellwood (2010) for more information.”

“(8-72) of Binney & Tremaine (2008): Trh = 6.5× 108 yr ln(0.4N) ( M 105M )1/2( M 〈m〉 )( Rh pc )3/2 .”

“For simplicity, we set μ = 1 throughout.3 The vertical scale height is estimated by noting that H = cs,z/ν in a vertically isothermal gas with effective vertical speed of sound cs,z and vertical frequency ν; in the solar neighborhood, ν 2Ω (Binney & Tremaine 2008).”

“(e.g. Binney & Tremaine 2008, section 6.1.5), so the capacity of a wave to force stars does not extend further than 1/k from the plane.”

“The gravitational acceleration in the plane of a thin, exponential disc is (Binney & Tremaine 2008, Ch. 2.6) g(R̃) = −dΦ dR = −πGΣs R̃×[ I0 ( R̃ 2 ) K0 ( R̃ 2 ) − I1 ( R̃ 2 ) K1 ( R̃ 2 )] , (32) where Iν and Kν are the modified Bessel functions of the 4 In order to gauge the stability of the disc…”

“Support for the bar/spiral connection is provided by the examples where prominent spirals extend from the ends of the bar (see e.g. NGC 1300, p. 525 in Binney & Tremaine 2008, or NGC 986 in Buta et al. 2010).”

“In particular, assuming spherical symmetry and isotropy of the distribution function [40], σ(2) χ,r(r) = 1 ρχ(r) ∫ ∞”

“The dark matter halo is modelled as a Plummer sphere A47, page 2 of 14 (Binney & Tremaine 2008) with mass MH = 1.6×1011 M , characteristic radius rH = 10 kpc, and nhalo = 5 × 105 particles, each with mass 3.2 × 105 M .”

“…radii shows an increased spread due to scattering by molecular clouds, which increases their epicyclic energy (“blurring”, Spitzer & Schwarzschild 1953; Binney & Tremaine 2008); – the orbit of stars is moved inward or outward either by resonant scattering by the transient spiral structure…”

“…same order of magnitude as the true physical separations ri, we expect their survival times to obey the dynamical friction timescale from equations 8.2 and 8.12 of Binney & Tremaine (2008), applicable for small satellites of massive galaxies: tfric = 19 Gyr lnΛ ( ri 5 kpc )2 σ 200 kms−1 108 M⊙ M .”

“This quantity gathers the first velocity moments so that a decrease in the dispersion should be partly compensated by an increase in velocity if the difference in dispersion were fully due to a variation in the enclosed mass (Binney & Tremaine 2008).”

“…this effect, we follow the method used in Barausse (2012) and approximate the sinking timescale for each wandering SMBH using the formula derived in Binney & Tremaine (2008). tdf ∼ ( 19 Gyr ln Λ ) ( ri 5 kpc ) ( σ 200 km/s ) ( 108 M mbh ) (1) We assume Λ ∼ bmax/bmin with the maximum impact…”

“The potential of a thin Mestel disk is r V r r ln , 90 2 max f = - ⎛ ⎝⎜ ⎞ ⎠⎟( ) ( ) where rmax is the maximum extent of the disk (Binney & Tremaine 2008).”

“For those two disruption regimes (Binney & Tremaine 2008, Equation8.”

“Three actions Ji and three conjugate angles coordinates θi provide canonical coordinates for six-dimensional phase space (e.g. Binney & Tremaine 2008, §3.5).”

“The normalisationA of the dynamical friction time is often chosen to be independent of the shape of the satellite’s orbit (e.g. 1.17 according to Binney & Tremaine 2008).”

“…7/4 cusp stellar densities are extremely high at the center and the fraction 4 Direct N-body simulations compute the gravitational accelerations between particles using the exact Newton’s law, without introducing any approximations to compute the gravitational potential (Binney and Tremaine 2008).”

“…the resonances that we will explore in forthcoming work is to pass, in their vicinity, to another system of angle-action variables (fast and slow variables), that allows us to focus on the librations around the resonant orbits, neglecting all the high-frequency motions (see Binney & Tremaine 2008).”

“For a pressure supported system, one can use the Collisionless Boltzmann Equation (CBE) to related the six-dimensional (6D) phase-space distribution function, f (®r, ®v), of a tracer particle, to the underlying gravitational potential (Binney & Tremaine 2008).”

“The evaporation timescale for soft binaries is (Binney & Tremaine 2008): TEV = Mbσ G16 √ πMa1ρ lnΛ (1) ≈3×108 ( 2 lnΛ )( Mb 2M⊙ )( M⊙ M )( σ 100km/s ) × ( 1AU a1 )( ρ0 ρ ) yrs, wherer is the distance from the MBH,ρ the local density of stars, lnΛ the Coulomb logarithm,M the mass of the field…”

“It follows that this DF unambiguously specifies the singular isothermal sphere, in contrast to the DF (23), from which one can derive both cored and singular isothermal spheres (e.g. Binney & Tremaine 2008, §4.3.3b).”

“The evaporation timescale is directly linked to the relaxation time th which scales with r 3/2 h (Binney & Tremaine 2008, Eq.”

“One way to gain insight into this relation is to interpret it in terms of the scalar virial theorem (e.g. Binney & Tremaine 2008).”

“Galaxies and star clusters are distinctively different stellar environments, in the sense that they have very different relaxation time-scales (Binney & Tremaine 2008).”

“The gas particles are sampled in order to reproduce the mass distribution in the Galactic disc given by Binney & Tremaine (2008): Mg(R, z) = 2π ∫ z −z ∫ R 0 R ρg(R, z) dR dz , (11) with ρg(R, z) = g 2zg exp ( − R Rg − |z| zg ) , (12) where g = 131.8 M pc−2 is the central surface gas density, zg =…”

“We shall work with a simplified model of a planetary system based on the sheared-sheet or Hill’s approximation (Spitzer & Schwarzschild 1953; Hénon 1969, 1970; Petit & Hénon 1986; Binney & Tremaine 2008).”

“Similarly, Binney & Tremaine (2008) show that a factor of K = 3.35 is appropriate for a variety of geometries and mass-distributions.”

“(A7) Note that I, A1, and A3 in equations (A2) – (A4) are identical to the dimensionless coefficients for the gravitational potential of uniform spheroids tabulated in Table 2.1 of Binney & Tremaine (2008).”

“(48d) 1 The conventions used here differ slightly from those of Binney & Tremaine (2008) and others which define J = 1 2π ∮ p dq.”