Galactic Dynamics: Second Edition

“This distance can also be expressed via the tidal or Jacobi radius rJ in a two-body system, given by Binney & Tremaine (2008) as rJ ≈ ( m 3M )1/3 R0, (7) where m and M are the masses of the orbiting bodies and R0 their separation.”

“[70] J.”

“Binney & Tremaine (2008) approximate that in a galaxy cluster with the velocity dispersion of σ and galaxy number density of np of Mp mass galaxies, the disruption time for galacto-centric radius a, can be calculated as (see their Eq.”

“However, the coherent torques are mixed with NR noise (the classical NR as treated in Binney & Tremaine 2008), and as a result 1 + θ1 1 as well.”

“The disk mass estimate of Binney and Tremaine (2008) yields 4.”

“To investigate this, we estimate the tidal radius, rt, of our high tidal index dwarfs using equation 8.91 from Binney & Tremaine (2008): rt = ( m 3M )1/3 rp (6) where m is the dark halo mass of the primary dwarf (see Table A5), the pericentre distance, rp, is estimated as the current projected…”

“We use orbit actions to quantify the stellar orbital properties (Binney & Tremaine 2008).”

“We therefore assume an exponential disc profile and estimate a scale length from the position of the peak of the contribution of the stellar component of each galaxy (Binney & Tremaine 2008, section 2.6.1b).”

“(6) (HEE, Binney & Tremaine 2008).”

“Using an energy diffusion argument, Binney & Tremaine (2008) show that the time-scale for binary destruction both through instantaneous ionization through a single encounter and via multiple encounters is ∝ σ eclρ−1ecl a−1.”

“Galaxies of the latter type merge with the central galaxy of their host subhalo after a dynamical friction time-scale (Binney & Tremaine 2008).”

“This proof is analogous to the continuity equation from stellar dynamics, found in Binney & Tremaine (2008), page 348, equation 4.204 (see also Drory & Alvarez 2008, Franx in prep.)”

“In the standard (Chandrasekhar) model of two-body relaxation in stellar systems (e.g., Binney & Tremaine 2008), each star undergoes a random walk in Cartesian velocity space due to encounters with stars passing nearby.”

“By further assuming that the disc scale height is constant with radius and that the system is isotropic, that is, σR = σz, we can write the asymmetric drift velocity as (e.g. Binney & Tremaine 2008, Sect.”

“However, some uncertainty is introduced by the translation of the tracer’s velocity dispersion measurement into a mass profile, which usually implies solving the Jeans equations under some simplifying assumptions (Binney and Tremaine 2008).”

“Not accounting for the true 3-dimensional shape of the disk has only a very small effect on the resulting MWRC fits (Binney & Tremaine 2008).”

“For any individual set of parameters p ∈ P, the Jeans equation (Binney & Tremaine 2008), dν〈v2r 〉 dr + 2β r ν〈v2r 〉 + ν d dr = 0, (19) is used to calculate 〈v2r 〉(p) which may then be used in turn to calculate the LOS velocity dispersion 〈v2z 〉(p) via (Binney & Tremaine 2008) 〈v2z 〉(R) = 2 ∫ ∞ R (1…”

“Substructures that are accreted onto a host halo can survive for significant periods of time within the host halo (Chandrasekhar 1943; Klypin et al. 1999; Moore et al. 1999; Binney & Tremaine 2008; Boylan-Kolchin et al. 2008; Jiang et al. 2008).”

“…friction timescale, approximated by tmerge ≈ Co Mhalo/Msat ln(1 + Mhalo/Msat) tdyn (8) where tdyn = 0.1tHubble, tHubble = 1 H(z) , and Co ≈ 1 accounts for the ensemble averaged satellite orbital parameters (Conroy et al. 2007; Binney & Tremaine 2008; Boylan-Kolchin et al. 2008; Jiang et al. 2008).”

“In the simplest approximation that the galactic distribution function can be described by a Maxwellian, the df term can be calculated as (Binney & Tremaine 2008) dvM dt = −4πG2Mρ lnΛF vM v3M .”

“However the mean rotation velocity of a population of stars will fall below the circular velocity due to asymmetric drift, which is difficult to correct (see Chapter 4.8.2 of Binney & Tremaine 2008).”

“the distant star most likely to be a nonmember, is excluded. Modeling of Dark Matter Density Profile We model the density profile of the dark matter halo of Tucana II using a Jeans modeling procedure[94,95,96]. For completeness, we briefly outline the most important steps here but more details will be reported in Chan…”

“The dynamical friction time-scale is often estimated using the Chandrasekhar formula (Chandrasekhar 1943; Lacey & Cole 1993; Binney & Tremaine 2008): tdf = fdf orb ln Mhost Msat tdyn, (7) where Mhost and Msat are the masses for the host and satellite haloes, respectively, ln is the Coulomb…”

“In a Kepler potential, the radial orbital frequency of a particle can be expressed in terms of its actions as (Binney & Tremaine 2008, appendix E) r = (GM) 2 (Jr + L)3 (19) and therefore r (t) ∝ exp(2t/tscale).”

“Stars in galactic nuclei live in orbits of periodτorb whose energy and angular momentum change on timescalesτε and τJ, respectively. τε is the cluster’s local relaxation time (Binney & Tremaine 2008; Alexander 2005).”

“1 IN T RO D U C T I O N Toomre’s (1964) stability criterion, Q ≥ 1, is one of the pillars of disc galaxy dynamics (see e.g. Binney & Tremaine 2008).”

“(4.228) of Binney & Tremaine (2008) becomes, Va ≃ 1 2Vc [ σ2V + σ 2 W +Rσ 2 U ( 1 Rd + 2 Rσ − 2 R ) ] , (10) where the intrinsic velocity dispersions6 of the selected thin disc stars areσU = 18.98, σV = 11.05 andσW = 9.61 km s−1, respectively.”

“This bar mode induces the formation of a bisymmetric spiral structure apparently connected to the bar ends (e.g. Binney & Tremaine 2008, and references therein).”

“The latter has a rather uncertain tangential component V (see Binney & Tremaine 2008)3.”

“W ´∣˙ ∣ ∣ ∣ ( )r rE 1 2 1 2 , 7J rot 2 rot 2 expressed here in terms of the separation vector to the system center of mass in the corotating frame, rrot (Binney & Tremaine 2008).”

“An isothermal law with β = 2 provides a good description of a virialized system (e.g. Binney & Tremaine 2008).”

“The orbits of the fountain particles are integrated in a standard Galactic potential, Model I in Binney & Tremaine (2008), starting from initial conditions that can be varied to fit the data.”

“Stellar mass seed black holes form in shallow potential wells and their growth is hampered by the (negative) feedback due to photoionization heating and the energy and momentum injection due to supernovae (Haiman, Abel & Rees 2000; Haiman & Holder 2003; Greif et al. 2007; Johnson & Bromm 2007; Bromm, Ferrara & Heger 2…”

“For spherically symmetric systems, the anisotropy parameter β can be written as (Binney & Tremaine 2008) β = 1− V 2φ + V 2 θ 2V 2r = 1− σ 2 φ + σ 2 θ 2σ2r , (14) where V 2φ , V 2 θ and V 2 r are the second velocity moments, and σ2φ, σ 2 θ and σ 2 r are the velocity dispersions measured in the…”

“…of the Landau equation in the kinetic theory of stellar systems, while this equation does not appear in the works of Chandrasekhar (1942), Rosenbluth et al. (1957), King (1960), and Hénon (1961), nor in the standard textbooks of stellar dynamics by Spitzer (1987) and Binney and Tremaine (2008).”

“…by adopting a time-dependent, better motivated and easy to implement description for the epicyclic motion based on the epicyclic approximation (Binney & Tremaine 2008), where the oscillations of stars around their guiding radius are described by the resulting dispersion in their radial…”

“Each set of actions, (J1, ..., JN) (or frequencies), labels a torus, and regular orbits are sometimes referred to in terms of their orbital tori (see, e.g., Section 50 in Arnold 1978, Section 10-5 in Goldstein 1980, Section 3.5 in Binney & Tremaine 2008).”

“It is then migrated inwards on a path governed by dv dt = − [µ(1 + q) +M⋆( r)] r3 r− ωdfv (5) where M⋆( r) is the stellar mass interior to r, q ≤ 1 is the binary mass ratio and ωdf = 4π logΛ q µ ρ( v) v3 (6) characterizes the dynamical friction (Binney & Tremaine 2008).”

“Bars form, if, at fixed other parameters, the mass exceeds or the angular momentum falls below (due to loss or initial lack) a stability limit consistent with standard stability criteria for bar formation (see Binney & Tremaine 2008 Section 6.3).”

“This approach has the advantage that it results in more stable initial conditions than using Jeans equations (Binney & Tremaine 2008; Kazantzidis et al. 2004).”

“Pitch angles [73] evolve with time as the spirals wind up and eventually disappear, once the perturber has left the disc vicinity.”

“For a circular orbit and spherically symmetric host and subhalo, rt is given in Binney & Tremaine (2008) and rewritten here as rt = R0 ( m(rt) M(R0)(3− d lnMd lnR |R=R0) )1/3 , (8) where R0 is the radius of the satellite’s orbit, m(rt) is the mass enclosed by the satellite at a distance rt from its…”

“…clusters (e.g. Vikhlinin et al. 2006), we also experimented with a socalled “cored logarithmic” mass model, which has been widely used in studies of elliptical galaxies (Binney & Tremaine 2008), including some recent stellar dynamical studies (e.g. Thomas et al. 2007b; Gebhardt & Thomas 2009).”

“The radial excursions associated with epicyclic motions for thin disc stars at the Solar neighbourhood with σV ∼ 30 km s−1 are of the order 1 kpc (Binney & Tremaine 2008). variations are below 2σ, cannot allow us to discard either of the models.”

“Starting from the definition of the kinetic energy tensor (Binney & Tremaine 2008, Eq.”

“Thus, this kind of satellites are unlikely to reach the centre of the host system and to cause significant changes in its structure (VH08, Binney & Tremaine 2008; Kazantzidis et al. 2009).”

“As we will show, the NIFS data entirely determine our final BH mass, justifying our decision to not include a dark matterhalo. resulting density profile is then converted to a gravitational potential using an axisymmetric multipole expansion scheme (Binney & Tremaine 2008).”

“The expression for g e( , )g here is slightly different from the one found in Arca-Sedda & Capuzzo-Dolcetta (2014a) due to the fact that in that paper the authors mainly focused their attention on the study of the dynamical friction process in models with 1g ⩽ for both circular and radial orbits,…”

“The second is a power-law relation, σv(τ) = v10 ( τ + τ1 10 Gyr + τ1 )β (5) (cf. Binney & Tremaine 2008), where σv is the unweighted total velocity dispersion, and we used all six best-fit parameter sets in Aumer & Binney (2009, Table 2): 55.179 ≤ v10 ≤ 57.975 km s−1, 0.148 ≤ τ1 ≤ 0.261 Gyr and…”

“Once accreted, galaxies are subject to mass loss due to tidal forces and ram pressure stripping, while dynamical friction can drive them towards the centre of their host halo and therefore enhance the mass loss yet further (see, e.g., Binney & Tremaine 2008).”

“The quantity Nc,s declines steadily due to standard gravothermal evolution, in which the cluster core becomes denser and smaller in number with time (e.g., Binney & Tremaine 2008).”

“The relaxation time is for a single mass system conventionally defined as tr ≡ 0.34v3disp (Gm)2n ln Λ , (1) where m is the mass of a star, n is the number density, vdisp is the velocity dispersion, and ln Λ is the Coulomb logarithm (Spitzer & Hart 1971; Binney & Tremaine 2008).”

“(4.215) in Binney & Tremaine 2008) becomes: σ2v d ν(r) dr = − ν(r) GM(r) r2 (1) where r is the three-dimensional distance from the centre of the galaxy, ν(r) is the de-projected stellar density profile and M(r) is the enclosed mass.”

“Indeed, it is difficult to imagine that the ratio between σ and Vc for massive, stable systems evolves strongly with redshift and that it can be much different from √ 3 or √ 2 (see Binney & Tremaine 2008).”

“IAT orbits are unstable (e.g. Binney & Tremaine 2008, p. 263).”

“Furthermore, there is mounting evidence in the local Universe that the densest stellar clusters are significantly denser than the densest gas clouds (Longmore et al. 2014; Walker et al. 2015), suggesting that massive clusters are formed via hierarchical merging of smaller stellar clusters embedded in the parent gas cl…”

“…a globular cluster are WD-WD binaries (Shara & Hurley 2002); (5) the triple fraction relative to the binary fraction is the same as the field, i.e., 0.2 (Raghavan et al. 2010);5 and (6) the typical velocity dispersion of a globular cluster is 6 km s−1 (Harris 1996; Binney & Tremaine 2008, p. 31).”

“The Toomre criterion states that thin disks are gravitationally unstable when QT ≡ cΩ πGΣ 1 (3.2) where c is the random velocity (for a gas the sound speed), Ω is the orbital frequency6, and Σ is the surface density (see Shu, 1992 or Binney and Tremaine, 2008 for a derivation).”

“Additionally, using the orbit-averaged subhalo evolution model and artificial MNRAS 503, 4075–4091 (2021) D ow nloaded from https://academ ic.oup.com /m nras/article/503/3/4075/6166764 by C alifornia Institute of Technology user on 09 June 2021 disruption mechanism of Jiang & van den Bosch (2016), GB19 inferred that a…”