Galactic Dynamics: Second Edition

“It is important to note here that the potential is intrinsically rounder than the density distribution, for example for a cored logarithmic potential 1−(c/a)ρ ≈ 3[1−(c/a)Φ] outside the core (Binney & Tremaine 2008).”

“We follow Brandt [28] and calculate the evolution of radial shells by using the virial theorem and the diffusion coefficient for weak scattering of stars off black holes (see also [46]).”

“…mass distribution is obtained either from the Jeans equations and from some assumption about the anisotropy of stellar orbits (Łokas & Mamon 2003; Binney & Tremaine 2008), or by finding the potentials in which a distribution of orbits reproduces the observed surface-brightness profile and…”

“Because in the solar neighborhood the rotation curve is almost flat (Binney & Tremaine 2008; Bovy et al. 2012a), ρeffDM ρDM(R ).”

“We compute the distribution function (DF) fi for each component i, using Eddington’s formula (Binney & Tremaine 2008), fi(E) = 1√ 8π2 ∫ Φ=E Φ=0 d2ρi dΦ2T dΦT√ E −ΦT , (3) where ρi is the density profile of component i, E is the relative (positive) energy and ΦT is the total gravitational potential…”

“In order to achieve a steady state, we need to derive the particle distribution function from integrals of motion based on Jeans Theorem (Chap 4. of Binney & Tremaine 2008).”

“We use the spherical Jeans equation (Binney & Tremaine 2008) to solve for the circular velocity Vc given the radial velocity dispersion σr of tracer particles: V 2c = GM(r) r = σ 2r ( −d ln ρtr d ln r − d ln σ 2 r d ln r − 2β ) , (1) where M(r) is the enclosed total mass within radius r, ρtr is the…”

“…Obtained by approximating the mass distribution as spherical and using Vc (r) = √ GM(< r)/r 5 κ(Rg) = √ (R dΩ2dR + 4Ω2) in the epicyclic approximation; see Binney & Tremaine (2008) 6 These are the locations of the resonances strictly only for mildly non-axisymmetric systems surface density images…”

“Binney & Tremaine (2008) suggest that C = 2.25 is an average value of known galactic mass distribution models.”

“This value ranges from 2.25 for an average value of known galactic mass distribution models (Binney & Tremaine 2008) to 6.7 for dispersiondominated systems (Förster Schreiber et al. 2009).”

“The gravitational potential of a spherically symmetric mass distribution, ρ(r), can be found through the expression (Binney & Tremaine 2008), (r) = 4πG [∫ r 0 dr ′ r ′2ρ(r ′) + ∫ ∞ r dr ′ r ′ρ(r ′) ] .”

“For example, globular cluster M 54 (NGC 6715) sits right at the heart of the Sagittarius (Sgr) dwarf spheroidal galaxy, though it shows clear kinematical differences from the coincident Sgr stars so it is believed to have formed in the halo of Sgr and sunk to the centre as a result of dynamical friction (Bellazzini et…”

“Assuming the cluster is in virial equilibrium, one may estimate that v∞ ≃ √ 0.4GMc/rh (Binney & Tremaine 2008), where Mc is the total cluster mass.”

“…passage is given by∆E ∝ M2hostMR2V−2rel , whereMhost is the mass of the host,M denotes the mass of the dwarf,R is a characteristic radius of the dwarf (related tormax), andVrel is the relative velocity between the two galaxies at the pericenter of the orbit (Spitzer 1958; Binney & Tremaine 2008).”

“…for stars to random walk in angular momentum by Jlc via the two-body relaxation process is tJ(ε) ∼ [ Jlc(ε) Jc(ε) ]2 tr (a(ε)) , (21) where two-body relaxation time is tr(r) = 5.68 × 10 19 ln Λ M M∗ 104 M pc−3 ρ∗(r) ( σ (r) 107 cm/s )3 s, (22) (Equations (7)–(107) in Binney & Tremaine 2008).”

“…& Spergel (2001) Drimmel & Spergel (2001) Drimmel & Spergel (2001) model +Marshall et al. (2006) +Marshall et al. (2006) ISM local Robin et al. (2003) Binney & Tremaine (2008) Binney & Tremaine (2008) density local stellar Wielen (1974) Wielen (1974) Jahreiß & Wielen (1997) mass density…”

“…this characteristic mass; the structural and kinematic properties of such ultra-massive and rare spheroids show unequivocal evidence for a dissipationless formation history (Bender et al. 1989; Carollo et al. 1993; Faber et al. 1997; Binney & Tremaine 2008; Cappellari 2016, and references therein).”

“The dark matter halos in which galaxies are embedded are satisfactorily modeled as smooth quasi-isothermal spheroids, such as Hernquist (1990) or Navarro-FrenkWhite (NFW, 1997) functions, as described in authoritative texts such as Binney & Merrifield (1998) and Binney & Tremaine (2008).”

“…disk stars due to the dynamical heating by clumps with mass Mc, whose number density is nc, is given by tg ≃ v3 πncG2(Mc +ms)2 ln Λ , (3) where v andms are the three-dimensional velocity dispersion of disk stars and mass of a star respectively, and lnΛ is Coulomb logarithm (Binney & Tremaine 2008).”

“…of the tracers’ orbits) to β = 1 (corresponding to completely radial orbits of the tracers), (2) a radially varying β of the Osipkov–Merritt (OM) form (see Binney & Tremaine 2008, pp. 297–298) given by β(r) = (1 + r2a /r2)−1, where ra is the “anisotropy radius,” and (3) a radial profile of β…”

“(16) Here Ri is the initial galactocentric radius of the star cluster, vc is the circular velocity of the (assumed isothermal) galaxy, and MGC is the mass of the star cluster (Binney & Tremaine 2008).”

“Assuming spherical symmetry, the Jeans equation for an nrotating system can be written (Binney & Tremaine 2008) 1 ν d dr (νv̄2r ) + 2 βv̄2r r = −GM(r) r2 , (2) whereν(r), v̄2r (r), andβ(r) ≡ 1− v̄2θ/v̄2r represent the 3-dimensional density, radial velocity dispersion, and orbital anisotropy,…”

“…central mass densities rb,0, rh,0, and the surface densities St , ST—which are calculated by integrating the mass density—the values are taken from Binney & Tremaine (2008) for the bulge component, BlandHawthorn & Gerhard (2016) for the disc component, and Kafle et al. (2014) for the halo…”

“To calculate the rate of change due to stars we use equation 8.20 from Binney & Tremaine (2008): d dt (1 a ) stars = −14.3Gρ σ (20) or the following which is in a more directly comparable form to equation 16(da dt ) stars = 14.3 Gρ σ a2 .”

“…≫ a/N where a is the disc radius and N is the number of stars) approximately satisfy the WKB dispersion relation for bending waves (Hunter & Toomre 1969; Binney & Tremaine 2008) (ω −mΩ)2 = ν2 + (2π)2GΣ λ (52) where ν is the vertical frequency arising from an external potential and m is the…”

“(24) See, e.g., Binney & Tremaine (2008) for a derivation of this result.”

“As a simple analytic alternative, we have investigated modelling the halos at infall as isothermal spheres (as applied to Binney & Tremaine 2008, Eq.”

“…the other hand, it is also possible that the BH sinking time scale estimated in our simulation in the no-boost case could overestimate the true sinking time, as the high-density stellar bulges are not fully resolved (e.g. Antonini & Merritt 2012; Dosopoulou & Antonini 2017; Biernacki et al. 2017).”

“The origin of spiral patterns in galaxies still has no fully satisfactory dynamical explanation (see reviews by Binney & Tremaine 2008; Sellwood 2013a).”

“The second term represents the energy- momentum term which can be written as (Binney & Tremaine 2008): ∫ ∂ ∂xi (ρuiuj)x kdV = − ∫ ρujukdV = −2Kjk. (19) For simplicity, we can split the kinetic tensor into the energymomentum tensor from the bulk velocity and the internal energy from the dispersion…”

“This epicyclic frequency is given by (see, e.g., Binney & Tremaine 2008 and references therein): κ2 = 2Ω [2Ω + r(dΩ/dr)] .”

“The inspiral time is proportional to M−1 (Binney & Tremaine 2008), implying that more massive clusters segregate inwards more quickly.”

“We may then calculate the gravitational torques τg on the test masses in the rings using equations for potentials of homogeneous spheroids given in tables 2.1 and 2.2 of Binney & Tremaine (2008).”

“The dynamical friction force is given by fDF = 4πG (2)μ(2)ρDM(r) ln Λ/v 2 where v is the velocity of the stellar mass object [40].”

“— For a spherical, self-gravitating, collisionless system in dynamical equilibrium, with radial density profile, ρ(r), the Boltzmann equation reduces to the well-known Jeans equation [138]: d(ρσ2 r ) dr + 2 β”

“In a spherical potential, the orbits are generally expected to be of the rosette type (Binney & Tremaine 2008), with most of the stars having only slightly negative total energis.”

“The radial dispersion σR is always significantly larger than σφ, which is significantly larger than σz, as also seen in local field stars (e.g., Binney & Tremaine 2008; Kuijken & Tremaine 1994; Anguiano et al. 2018).”

“The relative lack of highly bound orbits in haloes with ra- dial velocity anisotropy has immediate consequences for the responsiveness and hence vulnerability of the central regions t perturbations, such as an infalling massive satellite (seealso Binney & Tremaine 2008, p. 299).”

“The diffusion coefficients in the directions parallel and perpendicular to the test-particle velocity are (Binney & Tremaine 2008) D[∆v‖] =− 4πG2ρpmp log Λ σ2 G(X) (30) D[(∆v2‖)] = 4 √ 2πG2ρpmp log Λ σ G(X) X , (31) D[(∆v⊥) 2 ] = 4 √ 2πG2ρpmp log Λ σ [ erf(X)−G(X) X ] , (32) where X ≡ v/ √ 2σ…”

“The morphology of the spiral arms in the Galaxy is intrinsically connected to the mechanism responsible for their origin, which is still uncertain (Toomre 1977; Athanassoula 1984; Binney & Tremaine 2008; Shu 2016).”

“Depending on the masses of the accreted dark matter haloes they either merge quickly with the main galaxy or they spend a long time orbiting around it (Chandrasekhar 1943; Binney & Tremaine 2008; Boylan-Kolchin et al. 2008; Jiang et al. 2008).”

“In addition, the incompleteness of MW open cluster samples and the difficulty of measuring cluster concentration in crowded 5 c = log(rt /r0) for King model fits, Binney and Tremaine (2008), p. 307. fields means that we cannot rule out the existence of such clusters in the MW.”

“For barred galaxy models such as those used here, the energy of a particle in the rotating frame is a constant of the motion and is often referred to as the Jacobi constant,EJ or simply as the energy (Binney & Tremaine 2008).”

“…time-scales for a solar mass binary with separation of order 0.1 pc in each of these regimes, tdiff and tcat, are (see equations 8.65a and b of Binney & Tremaine 2008) tdiff ≈ vp 200 km s−1 100 M Mp 0.01 M pc−3 ρp 0.1 pc a Gyr (3) and tcat ≈ 3 ( 0.01 M pc−3 ρp ) ( 0.1 pc a )3/2 Gyr, (4) where…”

“The stellar component is collisionless so its velocity dispersion is anisotropic, while gas is collisional and has an isotropic velocity dispersion (see, e.g., Binney & Tremaine 2008).”

“Note also, that such a (approximate) symmetry of the tidal tails’ density profiles might simply be a direct consequence of the symmetry of the Hill’s surface (see e.g. Binney & Tremaine 2008).”

“…stars in the solar neighborhood with respect to the local standard of rest can be approximated as superposition of 3D Gaussian distributions (e.g., Binney & Tremaine 2008): p(v|τ) = exp [− 12 (v − v0) TΣ(τ)−1(v − v0)]√ 8π3|Σ(τ)| , (C9) whose mean and covariance matrix are determined by stellar…”

“…on the FWHM, as discussed in Section 4.4. gravity, which is r h V r rzd ( ) ( )s = for a compact “thick” or large “thin” disk (Genzel et al. 2008; Binney & Tremaine 2008; Davies et al. 2011);(ii) a mixing term, ms , arising from mixing the velocities along the line of sight for a…”

“Binney & Tremaine (2008) in their text Galactic Dynamics state unequivocally that "Our Galaxy is the nearest barred spiral".”

“Perturbations whose wavelength λ is less than λJ oscillate as sound waves with period T = 2πλ/cs (see e.g. Binney & Tremaine 2008).”

“From the Vlasov equation, we can derive a system of hydrodynamic equations called the Jeans equations [34].”

“…× fdes × fcol tH = (5) = 1.2× 10−4 ( N∗ 1.5× 1011 ) × ( fevolve 0.1 )( ftriple 0.1 )( fdes 0.012 )( fcol 0.09 ) yr−1, where N⋆ is the number of stars in the Galaxy (Binney & Tremaine 2008; taking the mean stellar mass to be 0.3 M⊙; Kroupa 2001), fevolve is the observed fraction of evolved stars…”

“In fact, for highly eccentric orbits, only the potential at pericentre Φ(rp) is relevant for the purpose of this work 2; at 30 <∼ rp <∼ 100 kpc this is reasonably well constrained for the MW (e.g. Binney & Tremaine 2008).”

“We model the dynamical properties of the BHB tracers using a distribution function (DF) method (see e.g. Binney & Tremaine 2008 and references therein).”

“…(2006), when a dark matter halo is marked as having merged, we utilize dynamical friction arguments to approximate the time taken for the orbit of the incoming galaxy to decay and the corresponding galaxy–galaxy merger to occur (Binney & Tremaine 2008): where it is standard to take α merge = 1 .”

“Thevφ is the asymmetric drift and is given by vφ 2(τ,R)= v2c (R) + σ 2 R × ( d ln ρ d lnR + d ln σ2R d lnR + 1− σ2φ σ2R + 1− σ2z σ2R ) (6) This follows from Equation 4.227 in Binney & Tremaine (2008) assumingvR vz = (v2R − v2z)(z/R).”

“(2.165) in Binney & Tremaine (2008), so the rotational density is ρrot(k) = − Σ?,0 16R?”

“The space density of stars in the Galactic Disc is ≈ 6× 10−1 pc−3 (Binney & Tremaine 2008, Pg. 3), implying that the fraction of the free-floating population which has arisen from dynamical instability soon after formation in alreadydead stars is on the order of 1%.”

“We also include dynamical friction on the orbit of M33 (Binney & Tremaine 2008), but the effect is again small because the galaxy never gets close enough to M31.”

“…infallig halos do not start from rest at the time of their last merger, and tidal forces will reduce the merger rate close to the host halo (Binney & Tremaine 2008)—it is instructive to calculate the expected radial distances for the last>1:5 merger based on the ratio of the time between…”

“For a spherically-symmetric system the Jeans equation (Binney & Tremaine 2008): GM(r) r = −σ2r [ d ln ρ d ln r + d ln σ2r d ln r + 2 ( 1− σ 2 ϑ σ2r ) ] , (2) relates M(r) the cumulative mass inside a radius r to the square of the velocity dispersion profiles in the radial (σ2r) and tangential (σ2ϑ)…”

“…dynamical friction time scale for a gravitating object on a Keplerian orbit in an isothermal halo is tdf ∝ (σ × r2i )/MCl, whereσ, r i and MCl are the local velocity dispersion (km/s), the initial distance from the centre (kpc) and the cluster mass (M⊙), respectively (e.g. Binney & Tremaine 2008).”

“The formation of binary systems by triple encounters (Binney & Tremaine 2008) transfers kinetic energy to some sink particles, causing them to be ejected from the highdensity region.”

“The standard treatment of the background gravity in the tidal limit (e.g. Binney & Tremaine 2008) consists in regrouping the last two terms in equation (1) and performing a linear expansion by considering that r′ ≪ RG, with RG the distance between the cluster and the galaxy’s barycentre.”

“The axisymmetric part of the Galactic potential is taken to be Model I of Binney & Tremaine (2008).”

“More information can come frommeasuring gas kinematics and using the asymmetric drift equation (Binney & Tremaine 2008) as an additional constraint.”

“What pro - erty of the satellites is most correlated with quenching before infall, and what will that tell us about the physical mechanism( ) for quenching before the satellites fall into the Galaxy?”

“The relative frequency of these three outcomes is a function of the ratio of the planetary orbital speeds to their surface escape speeds, the square of which is often referred to as the Safronov number (e.g. Safronov & Zvjagina 1969; Binney & Tremaine 2008) θS = 1 2 v2esc v2orb .”

“In this approximation, apocentre and pericentre are defined as the roots of the following equation of motion from Binney & Tremaine (2008), Chapter 3. u2 + 2[Φ(1/u)− E] L2 = 0 (14) Φ is the gravitational potential of the fixed massive object and E is the Hamiltonian for the system. u = 1/r, where r…”