Galactic Dynamics: Second Edition

“The Milky Way, like other spiral galaxies, has a large fraction of its normal (baryonic) matter arranged in the shape of a flattened disk, with the density falling off exponentially over a characteristic distance of 3 kpc in the radial direction but in a much shorter characteristic distance of about 300 parsecs in the…”

“A good overview of Jeans models can be found e.g. in Binney & Tremaine (2008).”

“implementation of the well-established Jeans analysis [18] to extract DM profiles from the kinematics of baryonic tracers.”

“It’s also worth noticing that the number density of NSCs with SMBHs in them can be up to 40% higher due to ultracompact dwarf galaxies some of which are believed to be stripped NSCs of low-mass galaxies (Voggel et al. 2018).”

“The evolution of vertical energy (Ez ∼ σ2z) due to gravitational orbit scattering is dEz/dt ∼ Σpert/Ez, where Σpert is the surface density of the orbit perturbers (Binney & Tremaine 2008).”

“There have been many attempts using hydrodynamical models to match the observed gas kinematics in the MW (Englmaier & Gerhard 1999; Fux 1999; Baba et al. 2010; Sormani et al. 2015a; Pettitt et al. 2020) using various potentials and spiral/bar components.”

“This last constraint is chosen in such a way that the nominal 2σ interval coincides with the range of possible values 2-3 kpc quoted by Binney & Tremaine (2008).”

“In the ‘tight winding’ limit k X̃ ′/X̃, the perturbation is highly oscillatory, and the WKB approximation may be used to yield the well-known (Binney & Tremaine 2008) dispersion relation for tightly wound waves in a Newtonian gaseous disk: 8 Ω2 −m2(Ω− ΩP )2 + c2sk2 = 0 .”

“However, gravitational scattering is not a local phenomenon: it is an integrated effect resulting from a small number of short-range interactions, and a large number of distant ones (Binney & Tremaine 2008; Ludlow et al. 2019b).”

“The two are related through Iαβ = Tr(M)δαβ − Mαβ (e.g. Binney & Tremaine 2008).”

“…bulge-to-disc (B/D) flux ratio is ∼14 per cent, which is in agreement with expectations for Sbc-type galaxies (B/D = 0.138+0.164−0.083) like the MW (Binney & Tremaine 2008) and Sc-type galaxies (B/D = 0.087+0.2760.041 ) (median values ±68/2 per cent of the distribution on either side of the…”

“The MOND distribution function is obtained numerically with an Eddington inversion (e.g. Binney & Tremaine 2008, §4.3.1) fM (E) = 1√ 8π2 d dE ∫ ∞ E dρ∗ dφ dφ√ φ − E , (3) where the upper integration limit reflects the far-field logarithmic behaviour of the MOND potential.”

“In a spherical DM halo, the orbits of DM particles are planar rosettes (Binney & Tremaine 2008) conserving energy as well as all three components of angular momentum; consequently the central hole (or “empty loss cone”) that results from DM annihilation cannot be repopulated once it is depleted.”

“Given these assumptions, the most rigorous way to initialize galaxy velocities is to directly construct the distribution function from the density profile, using Eddington’s formula (Binney & Tremaine 2008; Kazantzidis et al. 2004).”

“…GMsv2typ ) ) , (2) where rgalaxy is the effective “size” of the galaxy, and the minimum impact parameter is the larger of the size of the satellite and the impact parameter for which a star interacting at a “typical velocity”, vtyp, is deflected by 90 degrees (Binney & Tremaine 2008).”

“Orphans If a substructure disappears before reaching the central region of its host halo, YSAM calculates its mass (Battin 1987) and orbit (Binney & Tremaine 2008) analytically until approaching the very central regions.”

“R z R z h, , , exp , 14z1 1 where we introduced the disk’s scale height hz (see Binney & Tremaine 2008, Section 2.6).”

“Sellwood & Wilkinson (1993) and Binney & Tremaine (2008) have summarized much of the old theoretical work that had led to considerable insight into the structure of bars.”

“Following Binney & Tremaine (2008) and Shu (1992), we note that for an axisymmetric system with gravitational potential field Φ(R, z), we can define the circular frequency Ω2(R) = 1 R ( ∂Φ ∂R ) (R,0) , (A1) and epicyclic frequency κ2(R) = 2Ω R ( 2RΩ +R2 dΩ dR ) .”

“This is a common functional form used to characterize the Galactic distribution of stars (see Section 1.1.2 of Binney & Tremaine 2008), and has already been applied in previous OC studies (Bonatto et al. 2006; Piskunov et al. 2006).”

“Because of the presence of significant random motions, we cannot simply use the mean streaming speed of the stars, but must correct these motions for asymmetric drift via the equation V 2c = V 2 + σ 2φ − σ 2r ( 1 + d ln ν d ln r + d ln σ 2 r d ln r ) (2) (Binney & Tremaine 2008).”

“These values are compatible with the assumption that the total stellar mass in the disc and bulge of the MW (5 × 10 10 M (cid:2) , Binney & Tremaine 2008) has assembled continuously during the last 13.7 Gyr with an SFR of 3.6 M (cid:2) yr − 1 .”

“The angle θz describes the oscillation phase perpendicular to the orbital plane, whilst θR and θφ − φ are dependent and describe the circulation about the guiding centre (Binney & Tremaine 2008, equation 3.265).”

“…& 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…”

“The Q-criterion is a measure of the competition between stabilizing pressure, stabilizing rotation, and de-stabilizing self-gravity (see, e.g., Binney & Tremaine 2008).”

“…of a given thin triaxial mass shell with axis ratios q and s (the homeoid) is constant within the shell and has a trixial shape outside of isopotential surfaces defined by the ellipsoid (e.g., Binney & Tremaine 2008): r2 = x 2 τ + 1 + y2 τ + q2 + z2 τ + s2 , where τ is the label of the surface.”

“In the past we used the simplified formula of the asymmetric drift proposed by Binney & Tremaine (2008), which is valid in the Galactic plane.”

“The rotation velocity for the potential in Equation C1 is Vrot(r,z) = (GM)1/2r{ r2 + [ r0 + ( z2 + z20 )1/2]2}3/4 (C2) where V2rot = r ∂φ(r,z) ∂r (e.g. Binney & Tremaine 2008).”

“For the velocity distribution, we assume that stars with the same true age τ have a Gaussian velocity distribution ( ( ) ( ))t tS~ v v ,0 relative to the Sun (e.g., Binney & Tremaine 2008).”

“It is difficult to identify which mechanism dominates spiral structure in a specific galaxy (e.g., Binney & Tremaine 2008).”

“Logarithmic spirals have pitch angles, φ, given by (Binney & Tremaine 2008): tanφ = ΔR dθ , (8) where the distance, dθ, is the spatial distance of the density enhancement in the azimuthal direction defined as dθ = RΔθ.”

“he DS mass at this point is O(103)M after a lifetime of ˘300;000 years, and the amount of DM consumed has only amounted to ˘1M . In a spherical DM halo, the orbits of DM particles are planar rosettes [68], conserving energy as well as all three components of angular momentum; consequently the central hole (or \empty l…”

“…the 1.6 × 1012 M Milky Way mass model from Carlin & Sand (2018) and assuming that (1) the tidal radius is equal to the Jacobi radius as defined by Binney & Tremaine (2008) and (2) no mass is present beyond the half-light radius, the tidal radius of Gru II at its current location is 296 (σ/2.0…”

“We could also simply use σ = vc/ √ 3 (e.g., see problem 4.35 in Binney & Tremaine 2008) to estimate σ, which only induces a slight difference in estimating the BHMF. Relation (11) may not hold for σ 80km s−1, which corresponds to M•,0 .”

“This mass segregation is a well-studied effect on long timescales (Binney & Tremaine 2008).”

“Strong Jeans theorem [30] implies that the n, as the required least number of independent isolating integrals for identifying the integrability, corresponds to the case of all regular with incommensurable frequencies.”

“We adopt the dynamical friction prescription introduced by Binney & Tremaine (2008): d~v dt dynf = − GMsat(t) r2 lnΛ ( Vc v )2 { erf ( v Vc ) − √ π 2 ( v Vc ) exp [ − ( v Vc )2 ]} ~ev, (2) where Msat is the mass of a subhalo, which is initially defined as the mass at a previous time step after…”

“For potentials which are mildly flattened, we will show that orbital planes effectively precess and nutate (see also Binney & Tremaine 2008) and derive the average precession and nutation rate in two limits.”

“The frequencyωz(RG) of vertical oscillations about the GP is equal to ωz(RG)2 = ( ∂2Φ ∂z2 ) RG ,0 (1) HereΦ is the Galactic potential (Binney & Tremaine 2008) that obeys the Poisson equation: 4πGρ = 1 R ∂ ∂R ( ∂Φ ∂R ) + ∂2Φ ∂z2 , (2) whereρ is local mass density.”

“…mass. Object 30545 is compact and round and we estimate its dynamical mass from its restframe optical size and velocity dispersion σv using the isotropic virial estimator from Förster Schreiber et al. (2009) and Förster Schreiber et al. (2006), who derived it from Binney & Tremaine (2008).”

“As a further validation of our results, we considered an improved estimate of the velocity distribution based on the Eddington formalism [64].”

“On the other hand, kinematic departures from regular rotation can be due to internal instabilities such as spiral arms or bars (Binney & Tremaine 2008; Sellwood 2013).”

“For this result, among the various models, we will focus only on the bar-less Allen & Santillan (1991) model and the Ferrers n = 2 model (Binney & Tremaine 2008), which has the maximum bar strength.”

“This approach is often used in studies of planetesimal and galactic dynamics (Binney & Tremaine 2008) and forms a basis of the so-called Hill approximation (Hénon & Petit 1986; Hasegawa & Nakazawa 1990).”

“…rate of b-EMRIs, we first recall that in a relaxed stellar system it takes approximately a timescale of Trlx for a star to explore all possible orbital angular momentum around a SMBH, and the fraction of the orbits that have been explored scales linearly with time (Binney & Tremaine 2008).”

“In that case the local velocity distribution function can be uniquely determined by using the Eddington equation [26], see also [27, 28] for similar approaches.”

“(The dimensionless Toomre parameter Q = κσ/(3.36GΣ) for stars, with surface density Σ, epicyclic frequency κ, and velocity dispersion σ, implies local gravitational instability at values less than about 1.0 (see Binney & Tremaine 2008).)”

“…the catastrophic regime, td depends only on rh and S ;H2 but in order to determine whether this case is applicable, rather than the opposite limit, the diffusive regime, one must know the values of all the other quantities listed above (see Binney & Tremaine 2008 for a clear and thorough analysis).”

“First, (i) the BHB can shrink and hence become harder, meaning that the kinetic energy of the BHB is higher than the average in the system (see e.g. Binney & Tremaine 2008); also (ii) the BHB 10 2 10 3 10 4 10 5 t/PBHB 10 3 10 2 a (p c) A00 B00 C00 D00 10 2 10 3 10 4 10 5 t/PBHB 10 3 10 2 a…”

“(z = 0) = 4.5× 1010 M (e.g., Binney & Tremaine 2008; Bovy & Rix 2013).”

“Toomre’s stability parameter Q (Toomre 1964; Binney & Tremaine 2008) at a reference radius (we adopt 2.2Rd), Q0, is controlled by the central velocity dispersion of the disk (σR0).”

“A potential concern in multi-mass collisionless simulations is the evolution towards mass segregation of 𝑁-body particles in equipartitioned systems (Binney & Tremaine 2008) due to spurious collisional relaxation.”

“Chandrasekhar’s dynamical friction formula for a satellite of mass MS is often written as (Binney & Tremaine 2008): dvS dt = −4πG2MSρ log(Λ)f(v∗) vS vS3 , (1) where log(Λ) is the Coulomb logarithm equal to log (bmax/bmin), vS is the satellite velocity, vS = |vS| and ρ is the local background…”

“The two tensors are related throughI i j = Tr(M)δi j − Mi j , and the quadrupole tensor is often referred to as the inertia tensorin the literature (Binney & Tremaine 2008; Zhang et al. 2009; Bett et al. 2007, 2010).”

“The flattening of the logarithmic dark matter potential may be very roughly estimated as 2/3 + qdm/3 (Binney & Tremaine 2008, p. 77), where qdm is the axial ratio of the dark matter halo.”

“Mass interior to some distancem2 = R2+z2/Q2 can be estimated in the following steps (See Binney & Tremaine 2008).”

“…any stellar population are often most conveniently described by its velocity dispersion tensor where the subscript indices denote one of the orthogonal coordinate directions, and the angled brackets represent averaging over the phase space distribution function (see e.g., Binney & Tremaine 2008).”

“Within a timescale ∆t, the closest expected encounter distance, or impact parameter b, is (see e.g. Binney & Tremaine 2008) b(R) ≈ [ 4π ( ρstars(R) 〈Mstars〉 ) U(R)∆t ]−1/2 , (26) where the quantity in the large parenthesis is the stellar number density.”

“Another observationally useful characteristic radius is the gravitational “radius of in- fluence” (Ferrarese & Ford 2005; Binney & Tremaine 2008) defined as the radius at which the Kepler velocity around a central mass M equals the transverse velocity dispersion, σ‖, of surrounding stars, Rinfl =…”

“…disc is much larger than both the internal relaxation time-scale and the time of the simulation (∼4 Gyr): theat trel tsim, where trel is given by (Binney & Tremaine 2008): trel = nreltcross = n 8 log bmax vtyp , (19) where vtyp = √ GM/R is the typical velocity at the solar position, R = 8.5…”

“On the other hand, if such clusters 5 Binary formation rate due to three body encounters scales with σ−9∗ , which is greatly affected by a massive black hole (Binney & Tremaine 2008). remain isolated, their relics may in principle be clearly identified in the local universe.”

“2020a, 2020b; Belokurov et al. 2020), and demonstrated that the metal-weak thick disk (MWTD) is an independent structure from the canonical thick disk (see also Carollo et al. 2019). Many recent works have used full space motions, based on RVs and astrometry (parallaxes and proper motions) provided by Gaiaʼs Data Rele…”

“…symmetry, and negligible rotational support, the second-order Jeans equation can be obtained from the collisionless Boltzmann equation and reads (Binney & Tremaine 2008): 1 ν d dr (νv̄2r) + 2 βani(r)v̄2r r = −GM(r) r2 , (2) with ν(r) the stellar number density, v̄2r(r) the radial velocity…”

“The legitimacy of assuming that dfs depend only on J is assured by Jeans’ theorem (Binney & Tremaine 2008).”

“We use the values of U(b/rh) given in Binney & Tremaine (2008), approximat- ing the density profiles of the systems as spherical Hernquist (1990) profiles.”

“We use a simple model for the disk potential, a Mestel disk (Mestel 1963; Binney & Tremaine 2008), which has a flat rotation curve and is uniquely characterized by the circular velocity; we assume a circular velocity at the Sun of Vc = 235 km s −1 and calculate Galactocentric distances using R0 =…”