Galactic Dynamics: Second Edition

“…profiles of massive elliptical galaxies have ⋆ E-mail: masamune.oguri@ipmu.jp been extensively studied using velocity dispersion measurements (see Binney & Tremaine 2008) and strong gravitational lensing (see Treu 2010), which indicates that the total mass profiles of elliptical galaxies are…”

“Fortunately, this was corrected in the second edition (Binney & Tremaine 2008, page 796, Equation D-39).”

“Over time, they densely fill a three-dimensional box-like region centered on the origin (de Zeeuw 1985; Statler 1987; Binney & Tremaine 2008).”

“Oh et al. 2008; Dobbs et al. 2010; Struck et al. 2011) as well as by the central stellar bars (Sellwood & Sparke 1988; Binney & Tremaine 2008, (section 6.4)).”

“Dynamical friction can slow down the motion of satellite galaxies inside haloes, and the effect is stronger for more massive satellites (Chandrasekhar 1942; Binney & Tremaine 2008).”

“…here are slightly longer-lived than our N-body galaxy, which could be because the bar is a powerful driver of spiral structure (e.g. Sparke & Sellwood 1987), and may help to maintain the feature for longer (e.g. Donner & Thomasson 1994 Binney & Tremaine 2008; Baba et al. 2009, Quillen et al. 2011).”

“…system, the global shape of the velocity dispersion tensor can be quantified by using the three anisotropy parameters (Cappellari et al. 2007; Binney & Tremaine 2008) β ≡ 1− Πzz ΠRR , (12) γ ≡ 1− Πϕϕ ΠRR , (13) and δ ≡ 1− 2Πzz ΠRR+ Πϕϕ = 2β − γ 2− γ , (14) where Πkk = ∫ ρσ2k d 3x…”

“…mixture coefficient, PH,norm(r|`, b) the normalised halo prior expressed in Equation (9), and PD,norm(r|`, b) the normalised thick disc prior defined by Binney & Tremaine (2008): PD(r|`, b) = r2ΣT 2zT exp ( −DGC(r, `, b) DT − |z| zT ) , (11) c© 0000 RAS, MNRAS 000, 000–000 with ΣT = 268.648M pc…”

“On top of that,previous studies have modelled the spiral arm potential with a cosine function, following the Tight-Winding Approximation (TWA, e.g. Binney & Tremaine 2008).”

“We note that at least three of the sources show prominent WolfRayet features indicating the presence of very massive stars 13 E(B − V )∗ = 0.44E(B − V )gas at a special phase only∼3–4 Myr after the onset of a massive burst (e.g. Brinchmann et al. 2008b).”

“The former is the radius at which the internal gravitational acceleration equals the tidal acceleration from the host galaxy (see e.g. Binney & Tremaine 2008).”

“In particular, we construct isothermal (T = 2 × 106 K) models of coronae in equilibrium in the axisymmetric gravitational potential of the Milky Way, as given by Binney & Tremaine (2008), with rotation law vϕ(R) = vs R R + Rs .”

“All our models use the constant velocity disc known as the Mestel disc (Binney & Tremaine 2008, hereafter BT08).”

“For an orbital radius r, tides exerted by the cluster mass M enclosed within r can yield substantial pruning of the satellite if mt/(3M(rt/r) 3)61 (Binney & Tremaine 2008).”

“(Binney & Tremaine 2008).”

“The evaporation time of a system of N = M/MPBH PBHs clustered in a region of size R and subject to the gravitational force is given by [29]”

“Such models4 are generally thought to be unstable (Binney & Tremaine 2008, section 5.5).”

“…other (opposite) case, and assume that these systems are rotating disks, where instability can be described by the Toomre parameter Q (Toomre 1964; Binney & Tremaine 2008) Qgas = σzκ πGΣgas (4) where σz is the velocity dispersion perpendicular to the disk, and Σgas is the average gas-mass surface…”

“…of the same gravitational potential Φ as the progenitor until the present day (t = 0), when it was observed: (x,v)(t = 0) =Hts(x p(−ts)+∆x,vp(−ts)+∆v), where H denotes the Hamiltonian flow from t = −ts to t = 0 (see Binney & Tremaine 2008); this is simply the orbit of the star between these times.”

“One, denoted M⋆, is derived fromKS luminosities usingMK⊙ = 3.28 (Binney & Tremaine 2008) and a typical stellar mass-to-light ratio of M⋆/LK = 0.82 in solar units (Bell et al. 2003).”

“The height z∗ = 300 pc is observed for the solar neighbourhood (Gilmore & Reid 1983; Binney & Tremaine 2008); we keep it unchanged for all our runs.”

“…of intense star formation are not infrequently observed in barred spirals and are ascribed to the presence of the inner Lindblad resonance (ILR; Binney & Tremaine 2008), with inflowing gas slowing down and accumulating between the outer and the inner ILR (having radius of ∼30′′ and ∼3′′ from…”

“Note that the derivation of F also requires the real part of the growth term to be positive so that an integral converges at negative infinite time (Binney & Tremaine 2008, Appendix 6A).”

“The constant a relates the epicyclic and circular frequency in different potentials such that a = √ 2 for a disc with a flat rotation curve (Binney & Tremaine 2008).”

“The Jacobi (tidal) radius of Tuc III is therefore rJ = 23000 (MTucIII/2.6×1011 M )1/3 pc (Binney & Tremaine 2008).”

“We consider frictional force on a star with mass m and velocity v in a matter field with a density profile ρ(r) (Binney & Tremaine 2008), and use the equation of angular momentum change (Fujii & Portegies Zwart 2014), to derive the dynamical friction time-scale tdf tdf = 0.186v3 G2m ln Λ′ ∫ ri 0 dr…”

“Linear theory predicts that the orientation of orbits shifts across the 2:1 OLR of the bar, from perpendicular to the bar major axis inside the OLR, to parallel with it outside the resonance (Binney and Tremaine 2008).”

“The corresponding gravitational potential at the disk midplane is Φdisk = − v20R0 √ R2 + R20 , (A2) (Binney & Tremaine 2008).”

“This model being well known and documented (see for example Binney & Tremaine 2008), we restrict its description to the main results used in this study.”

“2.6.2 in Binney & Tremaine 2008), Ψ b is the bar-perturbation and ζ ( t ) is the bar amplitude.”

“The phasespace distributions of these models can be determined numerically (Binney & Tremaine 2008).”

“…475 timestep, ts [Myr] 0.23 0.23 0.14 the total density, circular velocity, and potential are: ρtot ∝ r −α, (9) vc = v0 ( r r0 )1−α2 , (10) Φ(r) = v20 2− α ( r r0 )2−α2 , (11) where α 2, as is probably appropriate on ∼ 10– 100 kpc scales in M87 (cf. also section 2.2(f) of Binney & Tremaine 2008).”

“Following Revaz et al. (2009), we assume an isotropic velocity dispersion of dark matter particles, σ(r), for a spherical distribution (Hernquist 1993; Binney & Tremaine 2008), r r dr r r r 1 , 6 r 2 ( ) ( )( ) ( ) ( )òs r r= ¢ ¢ ¶F ¢ ¶ ¢ ¥ where Φ(r) is the gravitational potential.”

“…wavelength of λ<λcrit= σ 2/GΣdisk, where σ is the one-dimensional, inplane velocity dispersion of the disk and Σdisk its mass surface density (Toomre 1964; Binney & Tremaine 2008, chapter 6, 8 The Astrophysical Journal, 902:98 (29pp), 2020 October 20 Genzel et al. van der Kruit & Freeman 2011).”

“…including the dynamical friction term: d2~r dt2 = −GM(r) r2 ~r − 4πG 2 ln Λ ρMsat v2 f(x)~v (5) where f(x) ≡ [erf(x)− (2x/ √ π)e−x 2 ], x ≡ v/ √ 2σ, and the velocity dispersion σ is derived from the Jeans’ equation for the composite density profile, assuming isotropy (e.g. Binney & Tremaine 2008).”

“…the tidal radius (or Jacobi radius), which can be written as rtidal = ( msat f M( r) )1/3 r (1) where M is the mass of the host in which the progenitor orbits, and f ≡ 1− Ω−2d2Φ/dr2 = 3− d ln(M)/d ln(r) (2) and Ω = Vc(r)/r = (GM( r)/r 3)1/2 is the circular angular speed (Binney & Tremaine 2008).”

“…the effect of accretion is sub-dominant, so the energy losses are solely due to dynamical friction, ˙ E env = ˙ E DF , which takes the form [33, Appendix L]: where v is the orbital velocity, ξ ( v ) is the fraction of DM particles moving more slowly than v and log Λ is the Coulomb…”

“Orbits that move significant distances from the Galactic plane are much more nearly bounded by ellipses than straight vertical lines in the (R, z) plane (e.g. Binney & Tremaine 2008, fig.”

“…t)/τ(E, t) where n(E, t) is the number of stars of energy E per unit (specific) energy, while τ(E, t) is equal, in the spherical case, to τrlx ∝ σ3/ρ ∝ rγ−3/2 (Binney & Tremaine 2008) or, in the triaxial case, to the star’s orbital precession time τprec(E, t) ∝ M −1/2 ∗ (< r) ∝ r(3−γ)/2, for r…”

“When viewed this way, the fact that the eigenfrequencies for a given are independent of azimuthal number m is natural, as any of these elementary modes is equally likely to arise in a spherically symmetric accretion flow (see, e.g., Binney & Tremaine 2008 for a more rigorous argument).”

“The scaling between these two timescales depends on the size of the cluster (e.g., Binney & Tremaine 2008): Trelax ≈ 0.1N logN Tdyn .”

“(12) To calculate particle velocities, we use the Eddington formula for the distribution function (Eddington 1916; Binney & Tremaine 2008): f (E) = 1√ 8π2 [∫ E 0 d2ρ d 2 d √E − + 1√E ( dρ d ) =0 ] .”

“…of the Milky Way disk as a function of age and Galactocentric radius, using the approximation of an isothermal disk and the harmonic limit (e.g., Binney & Tremaine 2008): p(Z | R,R0, τ,pm) = 1 2hz(R,R0, τ) sech2 ( Z hz(R,R0, τ) ) , (15) where the scale-height is modelled as hz(R,R0, τ) = az √…”

“…of stars born at a lookback time of τ and at radius Rb, p(x,v|τ ,Rb), we use a distribution function of the following form (see Equation 4.147 from Binney & Tremaine 2008). f (ER,L,Ez)∝ F(L) σ2R exp ( − ER 2σ2R ) exp ( − Ez 2σ2z ) (13) Here, the potential Φ(R,z) is assumed to be linearly…”

“Given that the diagram is defined for edge-on orientations, while M33 has an inclination of i ≈ 49◦, we projected the (V/σ , ε) values for the NC to an edge-on view following Binney & Tremaine (2008).”

“…of sight, the mass profile relates to observable profiles, the projected stellar density, Σ(R) (Figure 2), and velocity dispersion, σ(R) (Figure 12), according to (Binney & Tremaine 2008) σ2(R)Σ(R) = 2 ∫ ∞ R ( 1 −βa R2 r2 ) ν〈v2r〉r√ r2 − R2 dr. (11) Equation 11 forms the basis for many methods of…”

“Orbits free of resonant trapping are well described by linear perturbation theory where any deviation from circular orbit is assumed to be small at the order of ∼ b/ 0 (see Binney & Tremaine 2008, pp.189–191 where the equation below is derived).”

“This is of the order of a few crossing times (Binney & Tremaine 2008).”

“(2.42) of Binney and Tremaine, 2008).”

“This does not affect the scalings presented here (see e.g. Courteau & Rix 1999; Binney & Tremaine 2008, and references therein).”

“…theorem, the components of a galaxy of mass M and radius r will have characteristic velocities of the order σV,cl = √ GM/Cr, where the factor C depends on the geometry and the actual mass distribution, as well as on whether the system is rotationor dispersion-dominated (Binney & Tremaine 2008).”

“We generate the disc initial conditions as discrete realizations of the phase-space Shu-Schwarzschild distribution function (Shu 1969; Bienayme & Sechaud 1997; Binney & Tremaine 2008), as explained in F14.”

“Two conditions must be simultaneously verified for the swing amplification to be effective: Q ∼ > 1 so that the disk is stable but still strongly responsive to density perturbations, and X ∼ 3 to prevent the density waves from being too tightly wound (see Binney & Tremaine 2008).”

“Note that in the approximation of a tightly wound bending wave Z(R,φ, t) = [Za(R)eim(φ−Ωpt)], where Za(R) = Z(R)ei ∫ kdR and Ωp the pattern speed (Binney & Tremaine 2008).”

“19R R2 2 Taking the “textbook” quantities for the radial epicyclic frequency k » - -37 km s kpc1 1 and s » -38 km sR 1 for old stars from Binney & Tremaine (2008), we find ( )á D ñ »JR 2 -55 kpc km s 1.”

“As explained by Binney & Tremaine (2008) and Binney & Merrifield (1998), one needs also to know which side of the galaxy is nearest to us.”

“In this region, a power-law is a better representation than a King profile for this cluster indicating a self-similar collapse (Figure 16; e.g. Heggie & Hut 2003; Binney & Tremaine 2008).”

“The ‘maximum-disk hypothesis’ (van Albada & Sancisi 1986; see also Binney & Tremaine 2008) bypasses this mass-decomposition (‘disk-halo’) degeneracy.”

“Concerning the height above the plane, we model it as an exponential with scale height h equal to the scale height of the thin disc (h = 0.167 kpc; Binney & Tremaine 2008).”

“For example, Binney & Tremaine (2008) estimate a disruption time of τd ≈ 3× 108 yr for clusters in the solar neighborhood, due to interactions with molecular clouds.”

“The second type has a spiral pattern that is not welldefined and is seen only in the optical bands, with little presence in redder bands commonly used to probe the stellar mass density (Elmegreen & Elmegreen 1984, Binney & Tremaine 2008, Zibetti et al. 2009).”

“A star’s actions and angles (hereafter AA) are combinations of the star’s positions and velocities, and they are particularly convenient phase-space coordinates for several reasons (see Binney & Tremaine 2008).”

“Furthermore, if we model the velocity dispersion using the equation for a thick or compact disk (Binney & Tremaine 2008); σ ∼ hzVc(r) r (9) where hz is the disk scale height, the expected model would give a constant, featureless dispersion map within the plateau radius (r 6 rp).”

“…radius, rt, for a satellite that moves in a circular orbit around a spherical host halo can be approximated as rt ≈ [ m 3M(r) ] 1 3 r (3) where m is the total mass of the satellite and M(r) is the total (baryonic+dark) matter mass within the orbit radius r (e.g. Binney & Tremaine 2008, p. 681).”