Galactic Dynamics: Second Edition

“For more details on how to compute these quantities and integrals see Chandrasekhar (1961); Binney & Tremaine (2008); Cui et al. (2017).”

“We are interested in the distribution of the tidal debris across the Galaxy since it gives us a direct insight into the actual orbit of the progenitor (Binney & Tremaine 2008).”

“S ¶ S¢ - ¶ F¢fv v D 2 1 , 20t r r r ( ) k ¶ ¢ + W ¢ =fv v 2 0, 21t r 2 ( ) and the usual pF¢ = S¢G k2 ∣ ∣ for the perturbed potential of a razor-thin disk (Binney & Tremaine 2008), where the delta function for the vertical density stratification is implemented as r = S -k e 2 .”

“…systems both in and out of equilibrium, and (iii) actions are adiabatic invariants, that is, they are constant under slow changes of the potential (see Binney & Tremaine 2008). f (J) models have been somewhat underused in galactic dynamics, mainly because actions generally cannot be expressed…”

“…the inner gravitational potential around the galaxy, we can expect a galaxy elongated along the major axis of the triaxial halo, because the shape of a collisionless stellar system is mainly determinated by the shape of the underlying triaxial potential (Binney & Tremaine 2008, chapters 1 and 2).”

“The main criterion used in this compilation is purely dynamical: the pair separation must be less than three times the average value of the tidal radius for clusters in the Milky Way disk (10 pc, Binney & Tremaine 2008).”

“Our objective is to observe clusters born within a tail environment that typically disperses before log age(yr) ∼ 8.5–9 (Binney & Tremaine 2008), this is a suitable result.”

“Given a choice of the DM density profile in the halo ρχ(r), and a mass model for the Milky Way, Eddington’s inversion formula [38] is used in Ref.”

“/(Rstr · Lorb)—where the stream is spread over an area7 deter- 7 Rstr · Lorb ends up being comparable to the full ellipse area, ≈ aMbm. mined by the stream radius Rstr and orbital circumference Lorb— we can calculate the Toomre parameter (Toomre 1964; Binney & Tremaine 2008), Q ≡ cs,str Ω/(πG Σ).”

“A Cartesian frame with the x coordinate is aligned in the radial direction (Binney & Tremaine 2008) measured from the center of the progenitor.”

“Thus as the satellite gets disrupted, its stars remain somewhat constrained in azimuth, which results in the observed m = 2 deviations, in rather close analogy to what happens as a result of the radial orbit instability (Binney & Tremaine 2008).”

“For an axisymmetric thin disk, the Poisson equation then reduces to (Binney & Tremaine 2008): d2Φg dz2 = 4πGρg.”

“If the mass associated to the disc and the bulge were to be included, the velocity dispersions would be significantly increased, since we may relate the velocity dispersion σ2 to the circular velocity V 2c and V 2c ∝ M( r) = Mdisc +Mhalo, and Mdisc ∼ Mhalo near the Sun (Binney & Tremaine 2008).”

“s inside a DM halo. The authors of [58] have shown the tidal radius to be a good proxy of the virial radius, this property holding even in the presence of baryons [105]. The tidal radius is dened as [181] rtid = m(rtid) [2 dlnMMW=dlnR] MMW(R) (1=3) R; (D.1) where MMW(R) is the mass of the MW enclosed within the galact…”

“7.1 from Binney & Tremaine (2008) and ∼22 Gyr, considering only the most massive galaxies and using eqn.”

“This is technically not correct and tends to overestimate j⋆ (El-Badry et al. 2018; Fall & Romanowsky 2018; Posti et al. 2018a) because it neglects asymmetric drift corrections, which are significant where σ⋆ >∼ vc (see, e.g., Binney & Tremaine 2008).”

“In a classic paper, Schwarzschild [1] described a method for selfconsistently constructing such a halo with a desired density profile: compute a library of particle orbits in the corresponding gravitational potential, and choose an appropriate weighting of the orbits to reproduce the input density profile, thereby ded…”

“The angle–action variables also provide a basis for the development of a perturbative solution to the equations of motion (see Binney & Tremaine 2008 for a much fuller discussion of the merits of angle–action variables).”

“The Coulomb factor comes from the approximate expression Λ ∼ (Rclump/Rgal,tot)(Mgal,tot/Mc) in Binney & Tremaine (2008), where Rclump and Mc are the clump radius and mass, and Rgal,tot and Mgal,tot are the total galaxy radius and mass.”

“While the NFW model is theoretically motivated, we also experiemented with the so-called “cored logarithmic” mass model (Binney & Tremaine 2008).”

“The dominant linear global modes, known as cavity modes, in bar unstable discs are standing waves between the centre and corotation that must have a high enough pattern speed to avoid any inner Lindblad resonances (Toomre 1981; Binney & Tremaine 2008, pp. 508-518).”

“On the other hand, the typical random velocity of clouds is ≈7 km s−1 (Binney & Tremaine 2008).”

“The stellar specific angular momentum was measured by P18 via radial integration, imposing a convergence criterion on the cumulative j⋆( R) profile and including asymmetric drift corrections (see, e.g., Binney & Tremaine 2008, chap.”

“…two initial conditions with an initially radially anisotropic velocity distribution with anisotropy following an OsipkovMerritt profile (see e.g. Binney & Tremaine 2008), β = 1− (σ2θ +σ 2 φ )/(2σ 2 r ) = 1/(1+ r 2 a/r 2) where σθ , σφ , and σr are the three components of the velocity dispersion…”

“Assuming, as in Cresci et al. (2009), that the stars and the ionized gas are distributed in thin rotating disks, the scale height of the disks (hz) can be derived as (Binney & Tremaine 2008) hz = σ2 × R V(R)2 .”

“The unique isotropic distribution function is given by the Eddington inversion formula (Binney & Tremaine 2008): f (E) = 1√ 8 π2 ∫ 0 E d2ρ d 2 d √ − E .”

“These trends are counter-intuitive because the dynamical friction force is an increasing function of Msub/Mhost, the mass ratio between the subhalo and its host (Binney & Tremaine 2008).”

“However, this seems to be a minor point, because even for stars that move in axisymmetric flattened potentials,|L| is approximately conserved and similar to the third integral I3, for which no analytic expression exists (Binney and Tremaine 2008, § 3.2.2).”

“…vertical velocity dispersion to maintain; for a stellar mass surface density in the disk of 75 M⊙ pc −2 (as for the Milky Way at the solar circle, Binney & Tremaine 2008; this is a reasonable assumption given the similar total luminosity and disk scale length of the two galaxies), σz = 104 km…”

“…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…”

“̄?. (C1) Defining the angular velocity as Ω(𝑟) = 𝑣c (𝑟)/𝑟 , where 𝑣c (𝑟) is the circular velocity at the galactocentric radius 𝑟 (e.g. Binney & Tremaine 2008), we can relate it to the change in the potential as Ω2 = 𝑣2c (𝑟) 𝑟2 = 𝐺𝑀 ( 𝑟) 𝑟3 = 4𝜋𝐺 3 ?”

“This is exactly what would be expected if angular momentum were carried by waves, since the angular momentum density of a wave undergoes a change in sign at the corotation radius (Binney & Tremaine 2008).”

“In the axisymmetric case, the virial theorem in tensor form (Binney 2005) gives: v20 σ 20 = 2(1 − δ)Wxx Wzz − 2, (8) where Wij are the components of the potential-energy tensor (Binney & Tremaine 2008).”

“In Newtonian theory, the relationship between the velocity and density fields is fully described by three nonlinear equations: the continuity, Euler, and Poisson equations [51].”

“Angular 2 This is the ‘donkey effect’, see Lynden-Bell & Kalnajs (1972) or Box 3.3 in Binney & Tremaine (2008).”

“MNRAS 000, 1–15 (2013)”

“The vertical velocity dispersion can be increased by massive objects in the disc such as giant molecular clouds rather than spiral arms (Carlberg 1987; Jenkins & Binney 1990; Binney & Tremaine 2008).”

“The background potential is logarithmic (Binney & Tremaine 2008) with a constant circular velocity of 200 km s−1 at large radii (r > 2 kpc).”

“The gravitational potential due to the perturbed density is given by [74] with G = 6 .”

“…profiles of discs will depend on their gravitational potential and more specifically on the radial acceleration at the midplane, a varying scale height will produce a different circular speed than razor-thin discs and even than thick discs with constant thickness (e.g. Binney & Tremaine 2008).”

“In the case of spirals we should use the generalized equations for uaR and u a φ from Lin & Shu (1964), including reduction factors depending on the velocity dispersion of the background stellar population (Binney & Tremaine 2008).”

“Finally, deviations from axisymmetry due to, for example, spiral structure are encoded in the velocity ellipsoid components (Binney & Tremaine 2008).”

“For instance, for the used relation it is assumed that the clusters are on circular orbits or that both the galaxy and the cluster are spherical (Binney & Tremaine 2008).”

“The tidal forces of the SMBH disrupt a binary with a separation (A color version of this figure is available in the online journal.) larger than the Hill radius (Binney & Tremaine 2008) during pericenter passage.”

“Except where stated otherwise, orbits and reconstructions are calculated using the Galactic potential of Model II in Binney & Tremaine (2008), which is a slightly modified version of a halodominated potential described by Dehnen & Binney (1998a).”

“For b 6= 0, these models give stellar distributions with constant velocity anisotropy (for a discussion see, e.g., Binney & Tremaine 2008).”

“(1) reduces to the usual dispersion relation for an infinitesimally thin gas disc (for a derivation, see e.g. Binney & Tremaine 2008).”

“We define the tidal radius (Rt) according to Binney & Tremaine (2008) as the distance between the Lagrange point, L3, and the center of the satellite.”

“…system, the vertical part of the Jeans equation in cylindrical coordinates is ∂ ∂z (νσ2zz) + 1 R ν ∂ ∂R (Rνσ2Rz) = −ν ∂Φ ∂z , (4) where σ2ij = vivj−v̄iv̄j is the velocity dispersion tensor (Binney & Tremaine 2008), Φ the Galactic potential, ν the vertical stellar density profile of the tracers.”

“The fundamental plane reflects the correspondence between the potential and kinetic energy of the system vis-a-vis the virial theorem (Binney & Tremaine 2008).”

“…eccentricity, but the radial motion becomes more nearly harmonic as the orbit eccentricity decreases, and the angular frequencies tend to the familiar definitions introduced long ago by Lindblad (e.g. Binney & Tremaine 2008): Ωφ → Ωc and ΩR → κ, (3) where RΩ2c = dΦ/dR and κ 2 = 4Ω2c +RdΩ 2 c/dR.”

“The Galactic potential will influence the destruction of low-mass star-forming regions much more than high-mass regions (Binney & Tremaine 2008), which take longer to dissolve into the Galactic field (and some remain as long-lived open clusters).”

“If we assume that the LMC luminous component is tidally truncated by the MW potential we can use the simple theoretical tidal radius formula (Binney & Tremaine 2008), given by: Rt = dLMC ( MLMC 2MMW(d dLMC) ) 1 3 (4) we find the following relation for the LMC mass (MLMC) and the MW mass (MMW)…”

“Although the energy 𝐸 and 𝑧 component of angular momentum 𝐿𝑧 of a particle can vary, a linear combination of the two known as the Jacobi integral is conserved (Binney & Tremaine 2008).”

“…within each galaxy disk, the non-axisymmetric potential induced during the merger results in the appearance of radial orbits which do not conserve angular momentum and funnel the gas toward the centre of the galaxy (Binney & Tremaine 2008), where it compresses, shocks and gives rise to new stars.”

“(40) A more detailed description on the growth rate and the dispersion relation can be found in Binney & Tremaine (2008).”

“velocity dispersion which one can derive from adiabatic scaling arguments [118] given by σ∝ m1/3 . (34) This scaling relation can be easily derived in the case of an isothermal distribution function [137], since then ρ(r) = σ2 2πGr2 , (35) from which equation (34) immediately follows. This scaling relation has been fo…”

“Traditionally the second order Jeans equation (Binney & Tremaine 2008) is used to generate the velocity dispersions for a set of input parameters including the potential which is then fitted to the dSph data with a likelihood analysis by radially binning the line of sight velocities for the…”

“While for the Sellwood & Binney (2002) mechanism to work short-lived transient spirals are required, in barred galaxies, such as the MW, spirals are more likely coupled with the bar, and are thus longer-lived (Binney & Tremaine 2008; Quillen et al. 2011).”

“φD(R, z) = v20 2 ln ( r20 +R 2 + z2 q2φ ) (9) (Binney & Tremaine 2008).”

“An ‘isotropic’ system has all velocity dispersions equal, while an ‘anisotropic’ system is a system where the velocity dispersions along the principle axes mutually differ (Binney & Tremaine 2008).”

“…smooth potential and steady state), and the further assumption of axisymmetry, the Jeans equations may be written in cylindrical coordinates as ρv2R−ρv2φ R + ∂ (ρv2R) ∂R + ∂ (ρvRvz) ∂ z = −ρ ∂Φ ∂R , (13) ρvRvz R + ∂ (ρv2z ) ∂ z + ∂ (ρvRvz) ∂R = −ρ ∂Φ ∂ z (14) (Jeans 1922; Binney & Tremaine 2008).”

“(16) Taking the ratio between the FA and the FG , and define the critical wave number kcrit (Binney & Tremaine 2008) as, kcrit = Ω2 2πGΣ = Q 2 Ω cs = Q 2h , (17) MNRAS 000, 1–21 (2019) we get, FA FG = 2 (Q2 4 |k | kcrit − 1 ) = 2 ( |k | kcross − 1 ) , (18) where kcross = 4kcrit/Q2 = 2/Qh.”

“They offer a unique possibility to constrain simultaneously the total (dark and luminous) mass profile and the orbital structure of galaxies which is commonly quantified by the anisotropy of velocity dispersion tensor (Binney & Tremaine 2008).”

“More recently, Posti et al. (2018) used the cross-match between Gaia DR1 and RAVE to select, through Action Angle distributions (Binney & Tremaine 2008), the local stellar halo.”

“Considering the velocity dependence of the thermally averaged cross-section 〈σv〉, we estimate the particle velocity distribution function of the Milky Way DM halo and the spike halo around PBHs, by using Eddington’s formula [84, 85].”

“…around the guiding radius (Schönrich & Binney 2009), while it cannot overcome the migrating effect of churning, that is the change of Rg due to interactions with a lasting non-axisymmetric pattern such as longlived spiral arms or long-lived bars (Sellwood & Binney 2002; Binney & Tremaine 2008).”

“We assigned the initial semimajor axis, a , of the binary sampling from a distribution p ( a ) ∝ a 9 / 2 (Binney & Tremaine 2008; Tagawa et al. 2020b) for a ∈ [ a min , a max ], where a min = 1 R (cid:12) and a max is equal to the Hill radius, computed as where m 1 and m 2 are the primary and…”

“The origin of the LSR is placed at 8kpc from the Galactic centre (Binney & Tremaine, 2008).”