Galactic Dynamics: Second Edition

“1, the vertical energy is approximately conserved for particles on nearly circular orbits (Binney & Tremaine 2008).”

“Close encounters can induce the BHB disruption over a typical evaporation time, defined as (Binney & Tremaine 2008; Stephan et al. 2016; Hoang et al. 2018) tev = √ 3σg 32 √ πGρg ln ΛaBHB mBHB m∗ , (22) where m∗ is the average stellar mass in the nucleus, ρg is the stellar density and ln Λ = 6.5 is…”

“We have implemented a subgrid DFmodel (Tremmel et al. 2015) based on the Chandrasekhar formula (Chandrasekhar 1943; Binney & Tremaine 2008) which estimates the density of stars and dark matter within the unresolved space and adds an additional acceleration, acting as drag on the MBH.”

“The scale length of the MW disc is 2–3 kpc and its full spatial extent is about 15–20 kpc (Binney & Tremaine 2008).”

“…we discuss below, also corresponds to an approximately phase-mixed distribution function, which is skewed because of the combined effect of the exponential density profile of the stellar disk and the decreasing velocity dispersion with Galactocentric radius in the disk (Binney & Tremaine 2008).”

“Nevertheless, we keep in mind that for more massive and extended systems (trh ∝ M1/2R3/2h , see e.g. Binney & Tremaine 2008) this may introduce biases if radial stellarpopulation gradients exist.”

“We assume that the stellar disc of the Milky Way is exponential with a scale length hR = 3.2 kpc (Binney & Tremaine 2008) and a maximum extension of Rm = 5 hR.”

“In order to use observables, the stellar component must be projected along the line of sight [90]:”

“To estimate the importance of two-body relaxation processes in the Fornax clusters, we calculate first the relaxation time as (Binney & Tremaine 2008, equation 1.38) trel ≈ N 8 ln(N γ ) tcr, (9) where N ≈ MGC/m and m is the mean stellar mass in a cluster.”

“By definition (e.g., Binney & Tremaine 2008), V2c = V 2 rot + V 2 AD,gas, with V2AD,gas the gas-asymmetric drift correction (e.g., Read et al. 2016; Iorio et al. 2017), a term to correct for pressure-supported motions.”

“In this convention, we specify coordinates in both the cartesian (x, y, z) and spherical (r, θ, φ) systems (see also Binney & Tremaine 2008).”

“We made a model assuming the local velocity of the Sun to be (U, V,W ) = (10.0, 5.2, 7.2) km s−1 (Dehnen & Binney 1998) and that the Galaxy is rotating with a constant circular speed outside of R1 = 2 kpc of 220 km s −1, decreasing linearly inside of that R1 (Binney & Tremaine 2008).”

“For spherical halos, the invariancy under velocity reversals is a direct corollary of the Jeans (1919) theorem (see also Binney & Tremaine 2008).”

“As stars age, repeated gravitational encounters increase the random component of their mean velocities (e.g. Binney & Tremaine 2008).”

“Binney & Tremaine (2008) give the following expression for the magnitude of the gravitational torque that the outer radius of the disk r exerts on the part of the disk interior to that radius in the WKB limit: ⎜ ⎟⎛⎝ ⎞ ⎠ p d p z d = S = S S S j j C r m Gr k m Gr tan , 34 G 2 2 2 2 3 2 2 ∣ ( )∣ ( ) (…”

“…kinematic ingredients required for dynamical modeling of a spherical system are (1) the rms radial velocity, vr,rms ≡ √ v2r , where vr = v · r/ |r|, and (2) the velocity anisotropy, commonly quantified through the parameter β (Binney & Tremaine 2008), which is defined as β = 1 − v2θ + v2φ 2v2r .”

“We derived them from the Milky Way model in table I-2 of Binney & Tremaine (2008).”

“For a spherically symmetric system in dynamical equilibrium the spherical Jeans equations reduce to a single differential equation (Binney & Tremaine 2008): dνσ2r dr + 2 r β (r)νσ2r +ν GM(r) r2 = 0 , (15) where ν(R) is the 3D stellar density distribution, M(r) is the mass distribution, σr is the…”

“(3) For a spherical system with an isotropic velocity distribution at each point, the Jeans equation reads d(nσ 2) dr = −nd dr , (4) where n is the number density of the constituents of interest and σ is their velocity dispersion (e.g. Binney & Tremaine 2008).”

“…simulations (Springel et al. 2005), and dark matter haloes falling into larger haloes experience tidal forces leading to the stripping of their constituent particles (Ghigna et al. 2000; Binney & Tremaine 2008), also see Natarajan et al. (2002), Gillis et al. (2013) for observational studies.”

“Taking ln Λ = 5 (see Binney & Tremaine 2008) and ξ = 1, we find (using Equation 4) tdf ≃ 0.3 Gyr.”

“In the direct N-body approach, each of the N stars of a cluster is modelled as a particle which experiences the gravitational acceleration from the N − 1 others (see Spitzer, 1987; Heggie and Hut, 2003; Binney and Tremaine, 2008, for more complete and detailed presentations).”

“This phase has a timescale given by (Binney & Tremaine 2008): tdyn = 19 Gyr ln Λ ( a 5 kpc )2 σ 200 km/s 108 M M2 , (1) where a is the semi-major axis of the binary, σ is the velocity dispersion of the stars, and M2 is the mass of the lighter SMBH.”

“When the anisotropy parameter β is constant but not necessarily equal to zero, the Jeans equation corresponds to a differential equation in ρσ 2r whose solution is σ 2r (r) = G r2βρ(r) ∫ Rvir r ρ(r ′)M(r ′)r ′2β−2dr ′ (23) assuming that limr→+∞ σ 2r = 0 (Binney & Tremaine 2008, equation 4.216).”

“This explains the 1/b3 scaling in the integrand in equation 8.53 of Binney & Tremaine (2008), which states that the Newtonian heating rate per unit dwarf mass is: ĖNewt = 14 3 √ 2π G2M2pnpr 2 h,dwarf√ 2σ︸ ︷︷ ︸ A ∫ ∞ rh,p db b3 = A 2r2h,p , (23) where A is a constant.”

“The reason for softening the interparticle force is mostly to avoid the need for shorter time steps during close approaches between particles (Binney & Tremaine 2008, hereafter BT08, p123).”

“Indeed, the effect of ǫ on the two-body relaxation process (see e.g. Theis 1998; Athanassoula et al. 2001) has to be taken into account when evaluating the relaxation time of the system, since its presence essentially increases the minimum impact parameter in the Coulomb logarithm expression (see Section 2.2).”

“(7.158) in Binney & Tremaine 2008).”

“In addition, the velocity distribution used in our expectation values includes an additional factor of v so that we calculate cross sections σint ≡ 〈vσ〉/vb, rather than 〈vσ〉 (Binney & Tremaine 2008).”

“B x x x xerf 2 exp 2( ) ( ) ( ) is the standard dynamical friction velocity expression (e.g., Binney & Tremaine 2008). ò is the eccentricity correction from Lacey & Cole (1993), defined as the ratio of the angular momentum of the particle to that of a particle on a circular orbit with the same…”

“We calculate ψ1 using cloud-in-cell interpolation (e.g. Binney & Tremaine 2008, Sect.”

“This introduces an additional geometric effect involving the velocity anisotropy (Binney & Tremaine 2008): σ2los(R) = 2 I∗(R) ∫ ∞ R ( 1 − βR 2 r2 ) ν∗σ 2 rr√ r2 −R2 dr, (7) where I∗(R) is the projected surface brightness profile of the tracer population.”

“In Section 4 we demonstrate how Rewinder can be used to measure properties of a non-trivial Galactic potential by performing several experiments with simulated observations of data from N-body simulations.”

“Binney & Tremaine (2008) have calculated thediffusion coefficients – the rate of change in the components ofv.”

“(4) the rotation curve is flat with both R and z in the solar neighbourhood (Binney & Tremaine 2008; Bovy & Tremaine 2012), i.e. the contribution of the circular velocity term is negligible.”

“…which are thought to be triaxial in shape (Kormendy & Bender 1996), stars on box orbits can cross the central region arbitrary close to the SMBH (Binney & Tremaine 2008) such that they become disrupted or swallowed as a whole for the case of a very massive SMBH. Merritt & Vasiliev (2010)…”

“Based on Chandrasekhar’s formula we calculate the dynamical friction timescales for star clusters in our galaxies using equation 8.12 of Binney & Tremaine (2008): tfric = 1.17 lnΛ r2i vc GM , (2) where lnΛ is the Coulomb logarithm, vc the typical velocity of the stars in the galaxy at radius ri,…”

“For example, in the epicyclic approximation (Binney & Tremaine 2008) where κ and a are the epicyclic frequency and amplitude, respectively, the radial motion can be written as r(t) = a cos(κt+ ψ) (8) where ψ is an arbitrary phase angle.”

“Although the Jacobi radius provides only a very approximate estimate of the tidal radius (Binney & Tremaine 2008), this suggests that stars in the outskirts of Pal 13 may not be bound to the cluster.”

“The crossing time can be approximated bytcross = (R3/GMtotal)1/2 (Binney & Tremaine 2008) where R is the distance between the two galaxies and M is the total mass (dark matter+ stellar) of the central galaxy, approximately 1013M⊙ in this case (Hatch et al. 2009).”

“…nested wave modes and their close correspondence with the phaseshift distribution is in direct contradiction with Lynden-Bell & Kalnajs (1972)’s and Binney & Tremaine (2008)’s conclusion of a constant angular momentum flux throughout the galactic disk: such a flux pattern does not deposit angular…”

“The anisotropy parameter can be defined by (Binney & Tremaine 2008) β ≡ 1− σ 2 t 2σ2r (9) whereσt andσt are the tangential and radial velocity dispersion, respectively.”

“The collision time in a system with isotropic velocity dispersion σ and number density n is given by (Binney & Tremaine 2008, equation 7.195) t−1c = 16 √ πnσr2 (1 + ) , (24) where ≡ Gm 2σ 2r (25) is the Safronov number.”

“For axisymmetric haloes, the invariancy under velocity reversals is a direct corollary of the Jeans (1919) theorem (see also Binney & Tremaine 2008).”

“Then, the tidal radius can be estimated by means of the equation (see Binney & Tremaine 2008): Rt = ( Msat 3 ·MDM,halo )1/3 ·D , (1) where Msat is the satellite mass (stellar mass + cold gas mass), MDM,halo is the dark matter mass of the parent halo, and D the satellite distance from the halo…”

“2.72b in Binney & Tremaine 2008).”

“The internal dynamics of how and why bars form has been addressed in many reviews (Toomre 1981; Sellwood & Wilkinson 1993; Binney & Tremaine 2008; Sellwood 2013).”

“We use the triaxiality parameter T ≡ a 2−b2 a2−c2 (Binney & Tremaine 2008) to describe the triaxial shape of a galaxy.”

“On the contrary, the equilibrium configuration for the collisionless dark matter component is provided by the Jeans equation (Binney & Tremaine 2008).”

“form. Specifying a velocity distribution would still be a challenge, buta simple proposal would be to employ the standard Eddington relation on the contracted halo (see the relevant exercises in ref. [70], for example). Such an approach would certainly not be self-consistent as both formalisms assume spherical symmetr…”

“A large number of studies have explored this process in the context of galaxy formation and early evolution (see e.g. Bertin 2000, Binney & Tremaine 2008 and references therein for reviews of the vast literature devoted to this topic).”

“For smaller mass ratios, the dynamical friction time scale exceeds the Hubble time (e.g., Binney & Tremaine 2008; Mo et al. 2010).”

“In the simplest approximation where both the host halo and its subhalo are considered as pointlike objects, and taking into account the centrifugal force, the tidal radius can be defined as [77, 79, 130, 131]”

“4 (see e.g. Binney & Tremaine 2008).”

“…(2015), we fix the velocity disper- sion of RC K-giant stars at the Sun position to be 30.3 km s−1, 23.6 km s−1and 16.6 km s−1in the radial, tangential and vertical directions, respectively (Binney & Tremaine 2008, and references therein), and a constant scale-height of 300 pc (Robin & Creze 1986).”

“The degeneracy between mass and anisotropy (see e.g. Binney & Tremaine 2008) then leads to significant uncertainties in the mass estimation.”

“The parameters ρb0, ab, αb, ρh0, ah, αh, βh, Σ0, Rd were adopted as given for Models I and II in Table 2.3 of Binney & Tremaine (2008), which fit the measured rotation curve of the Galaxy but bracket the range of allowable disk/halo mass ratio in the Solar circle (Model I describes a Galaxy…”

“We combine this mass with the 3D distance from M31 (367 kpc) and a mass for M31 of 2 × 1012M⊙ (intentionally in the high range of M31 masses) to estimate the Jacobi tidal radius of And XXVIII (Binney & Tremaine 2008).”

“Therefore the mass in halo stars that could be produced during BGC formation is between 3 and 8 × 108M , consistent with estimates of the old stellar halo mass (e.g., Binney & Tremaine 2008).”

“Their orbit decay with a characteristic time scale that can be estimated from the Chandrasekhar’s formula (Binney & Tremaine 2008) tfric = 1010 y ln Λ ( D 60 kpc )2( v 220 km/s )( 2 · 1010M Mh ) (15) where v is the velocity of the dwarf galaxy and D is its distance from the Milky-Way center.”