Galactic Dynamics: Second Edition

“The potential of the Galactic bar is adopted from J́ılková et al. (2012), where the bar is modeled by a Ferrers potential of an inhomogeneous triaxial ellipsoid (Binney & Tremaine 2008, where we choose n = 2).”

“…and (consistent with the fluctuation-dissipation theorem) suffers of a dissipative drag that produces an effective ‘cooling’ (see Chandrasekhar 1943; Kandrup 1980; Antonuccio-Delogu & Atrio-Barandela 1992; Del Popolo & Gambera 1997; Ma & Bertschinger 2004; for a review see Binney & Tremaine 2008).”

“For axisymmetric disturbances, the dispersion relation holds so long as |kR| 1 (Binney & Tremaine 2008), which is |2πR| λfsech2 and here fulfilled for R 63 pc and λfsech2 392.7.”

“…derived independently by Safronov (1960, 1980) and Toomre (1964) and subsequently by Lin & Shu (1966), Lin et al. (1969), Shu (1970), Polyachenko & Fridman (1972), Rohlfs (1977), Lovelace & Hohlfeld (1978), Fridman & Polyachenko (1984, Vol. 1, p. 242), Binney & Tremaine (2008), and others.”

“around the time of decoupling and freeze-out, the annihilating particles in the weak gravitational field of the galactic halo are moving at non-relativistic speeds, v ∼ 10−3c, and can be fully described by a Newtonian3 distribution function (DF) f such that f(x,v, t)d3xd3v is the probability that a particle χ has phas…”

“A stationary, axisymmetric disc has no vertex deviation, whereas a triaxial bar necessarily introduces a vertex deviation (Binney & Tremaine 2008).”

“…satellites is approximated by a spherically symmetric isothermal profile, such that 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…”

“We interpret this as the effect of the x1 orbits – the dominant bar-supporting orbits which rotate in the prograde sense (Binney & Tremaine 2008).”

“In a Galactocentric spherical coordinate system, anisotropy is defined (Binney 1980; Binney & Tremaine 2008) as ( ) ( ) ( )b s s s= - +q f1 2 , 1r2 2 2 where σ is the velocity dispersion within the velocity components of distance rgc from the Galactic center, polar angle, and azimuthal angle, ( )q…”

“Furthermore, the growth of the Galactic bar could trigger Inner Lindblad resonances, in which the stars are lifted into higher amplitude orbits (Quillen (2002); Binney & Tremaine (2008), page 539).”

“The mass-segregation time-scale can be expressed as [210]:”

“…satellite galaxy will merge with the central galaxy of the parent halo/subhalo after a time-scale motivated by dynamical friction arguments (Binney & Tremaine 2008): tmerge = 1.17 G Vvirr 2 sat msat ln(1 + Mvir/msat) , (5) where Vvir and Mvir are the virial velocity and mass of the parent…”

“The problem addressed here goes back to Binney & Spergel (1982, 1984), who Fourier transformed the time series x(ti) of individual coordinates and assigned to each line in the resulting spectrum appropriate integers nj so that ωt could be identified with ∑ j njΩjt.”

“Beyond rt, a star will feel a greater acceleration towards the galaxy centre than it feels towards the centre of the cluster, and will therefore escape (Binney & Tremaine 2008).”

“The distribution function for an anisotropic Osipkov–Merritt velocity profile (Osipkov 1979; Merritt 1985) was calculated by making use of equation 4.78a from Binney & Tremaine (2008).”

“Firstly, there is the vertical oscillation time of the galaxy; in the Milky Way at the solar radius this time is 84 Myr (Binney & Tremaine 2008).”

“Unlike dark matter particles, however, massive subhalos experience dynamical friction when moving through an ambient density of background particles [81, 82].”

“As discussed by Binney & Tremaine (2008) in the context of galactic warps (their section 6.6.1), neglecting the vertical velocity the evolution of such a distortion at a given radius R can be described as a propagation of two kinematic bending waves with pattern speeds ωp = Ω ± ν/2 (a different…”

“When the tidal perturbations are applied impulsively over the time scale of ∆T , S is a good measure of tidal strength (e.g., Section 8.2 of Binney & Tremaine 2008).”

“…a way that its action integrals Jr, Jφ and Jz will be constant, with the consequence that the density of dm particles in action space, f(J), is unaffected by the accumulation of baryons and the resulting evolution of the galaxy’s gravitational potential Φ(x) (e.g. Binney & Tremaine 2008, §4.6.1).”

“(6) Substituting the expressions in Equations 5 and 6 into the gravitational energy for a spherical body (Equation 2.32, Binney & Tremaine 2008) yields the relation 3 5 αvir = 3− β 5− 2β .”

“The dynamical friction timescale for galactic cannibalism is (Binney & Tremaine 2008) tfric = 2.7 Gyr ln Λ ri 30 kpc ( σM 200 km s−1 )2 ( 100 km s−1 σs )3 , (1) where Λ is the Coulomb logarithm, ri is the initial radius, σM is the typical velocity of these hosts through the halo, and σs is the…”

“In fact, (u, v) in Galactic Astronomy refer to the Cartesian velocities in the Local Standard of Rest (Binney & Tremaine 2008).”

“Integrals of motion refer to the angular momentum actions defined by Binney & Tremaine (2008) and total orbital energy of a star.”

“Gradually, the cumulative effect of these collisions causes particles to deviate from their original orbits in a process called two-body relaxation (see, e.g., chapter 1.2 of Binney & Tremaine 2008).”

“…is perhaps the least used among the kinematic spaces shown here, with a similar implementation first being used by Vasiliev (2019) (to our knowledge, although they recognize that the space is analogous to visualisations of three-dimensional action spaces presented by Binney & Tremaine 2008).”

“For the thin and thick disks, the number density is modeled as the following formula (Chen et al. 2001; Binney & Tremaine 2008): = ´ - - -n R Z n R R h Z h , exp exp , 10 R Z 0 ⎛ ⎝⎜ ⎞ ⎠⎟ ⎛ ⎝⎜ ⎞ ⎠⎟( ) ∣ ∣ ( ) where hZ and hR are the scale height and scale length of the disk, respectively.”

“As the mass segregation proceeds, the heaviest stars lose kinetic energy and reach the innermost region of the cluster, where they c© 2015 RAS form a dense, contracting nucleus (Spitzer 1969; Heggie & Hut 2003; Binney & Tremaine 2008).”

“The radius of influence is defined as the location from a black hole within which the potential from the black hole dominates (Binney & Tremaine 2008): rinfl = GMBH σ , (1) where MBH is the black hole mass and σ is the velocity dispersion at the effective radius, as measured in the MBH − σ…”

“In the third scheme, we sample uniformly, randomly from the integrated differential energy distribution (see Binney & Tremaine 2008) to obtain a particle’s energy, use the energy to determine the maximum radius it implies, allocate the particle’s spatial position within that radius using νE(r)…”

“…the simulations of Freitag, Rasio, & Baumgardt (2006), a runaway collision will occur if the cluster has an initial W0 (the dimensionless central potential of a King model) of 8 or higher, corresponding to a concentration parameter c = log(rt/rc) = 2 or higher (Binney & Tremaine 2008, figure 4.10).”

“The surface density of a Kuzmin disk is given by Binney & Tremaine (2008) to beΣK(R) = aM2π (R 2 + a2)−3/2.”

“The churning can be oscillatory in the case of an interaction with a lasting non-axisymmetric pattern such as long-lived spiral arms or long-lived bars (Sellwood & Binney 2002; Ceverino & Klypin 2007; Binney & Tremaine 2008).”

“…from stellar orbital anisotropy, we follow the same practice and adopt the definition of the 3D anisotropy parameter under spherical symmetry (Binney & Tremaine 2008): β = 1 − σ2φ + σ 2 θ 2σ2r , (2) where σr, σφ and σθ are the velocity dispersion within the stellar half mass radius in the…”

“The pitch angle evolves as (e.g., Binney & Tremaine 2008) j = ( ) At tan 1 2 , 3 where A is the first Oort constant and t=0 means that the arm extends radially outward across the disk (j=90°).”

“The pioneering study by Binney & Strimpel (1978) showed that even for spheroidal clusters highly inclined to the line-of-sight, deprojection assuming that the symmetry axis lies in the plane of the sky (effectively an assumption of spherical symmetry for large inclination angles) did not produce…”

“collapse is a fair approximation. amplification, with significant amplification of a mode, m, requiring the swing amplification parameter, Xm = Ω2R 2πGΣm , (11) to satisfy 1 < Xm < 3 (Binney & Tremaine 2008).”

“…data, this does not guarantee that this model is physical: it may not exist a well-defined DF (f > 0) in the potential that corresponds to the density ρ and that satisfies all the assumptions made in this section (steady state, axisymmetry, i–iv; see e.g. section 4.4.1 in Binney & Tremaine 2008).”

“First, a tidally stripped stellar population is expected to inhabit a broad region around the H i ridgeline of at least the width of the SMC (e.g., Read et al. 2006; Binney & Tremaine 2008).”

“Finally, we expect dissolution from two body relaxation to be very slow; the clusters have relaxation times of > 5 × 108 yr; dissolution due to dynamical evolution occurs over many relaxation times (Spitzer 1987; Binney & Tremaine 2008).”

“The re-mergers tend to have LOSVDs with positive h4, also an expected signature of violent relaxation, which generally results in a radially-biased distribution function (Binney & Tremaine 2008).”

“In the local approximation, the corresponding radial amplitude of the epicycle orbits is ∆R = ∆vR/κ (e.g., Binney & Tremaine 2008).”

“If the mass is stratified on concentric similar ellipsoids, the density profile drops out (Roberts 1962; Chandrasekhar & Lebovitz 1962) and the ratio of the ordered kinetic energy to the unordered one (or its square root V/σ) is a function only of the ellipticity ε (Binney 1978).”

“The corresponding gravitational potential Ψ? can be obtained using a standard formula for the potential of the spherically symmetric density distribution (see e.g. Binney & Tremaine 2008) Ψ?”

“This is a generalized form of the projected surface density of a Plummer profile; a value of β = 2 gives the familiar Plummer profile (Plummer 1911; Binney & Tremaine 2008), which has a well defined density-potential pair, and is commonly used to characterize star cluster profiles (e.g. Portegies…”

“An additional concern about a wide 3+1 system is its stability, as wide binaries tend to be disrupted by perturbers in the disks of their galaxies (Binney & Tremaine 2008).”

“In order to estimate the rate of inward transport of ring material [through AM conservation, or the angular Euler equation (equation 6.34 of Binney & Tremaine 2008, hereafter BT)], we wish to compute the relative AM change in the outer disc outside radius r due to torques exerted by the perturbed…”

“In fact, a necessary, but not sufficient, condition for the formation of a bar is that Q < 2–3 (e.g. Binney & Tremaine 2008); whether the bar forms or not depends on other structural parameters of the galactic disc and halo, as well as on numerical details such as the force and time resolutions…”

“Under this assumption of an exponential disk, C is about 2 (Binney & Tremaine 2008).”

“The limit on the projected separation of 1 pc corresponds to the separation where the Galactic tidal field typically starts to dominate the gravitational attraction of two stars (e.g. Binney & Tremaine 2008; El-Badry et al. 2021).”

“It is also possible, at least theoretically, to observationally constrainMDM and ρDM by using a method based on Jeans equation [130].”

“Therefore the mass of the MW/M31 galaxy pair can be estimated by the TA, as shown by Lynden-Bell (1981) and Binney & Tremaine (2008).”

“…the cylindrical coordinate axes, one can approximate δAD ≡ (V 2c − V 2 θ ) sin 2 i σ2∗ ≈ tan2 i γα2 ( 4R hσ + β2 − 1 ) (6) from the asymmetric drift (AD) equation (Binney & Tremaine 2008), where we use γ ≡ σ2∗ σ2z cos 2 i = 1 + tan2 i 2α2 (1 + β2) (7) as defined in Paper II and α and β are…”

“Binney & Tremaine (2008), for example, distinguish between bright-star (star-forming) arms, dust arms, mass (stellar density) arms, and (gravitational) potential arms, each of which may have a different distribution.”

“Jacobi’s integral is conserved (see Binney & Tremaine (2008)) to a relative accuracy of ≈ 10−2.”

“Taking the relation between velocity dispersion and rotation velocity σ = vrot/ √ 2 for the isothermal sphere (Binney & Tremaine 2008), the wind launch velocity can be written as vlaunch ∼ 2 vrot, similar to equation (10).”

“A fully analytic algorithm of this type is known only for the multi-dimensional harmonic oscillator and the isochrone potential (Henon 1959; Binney & Tremaine 2008) (which includes the Kepler potential as a limiting case).”

“…of the velocity dispersion tensor, also called velocity ellipsoid, of the local dynamically selected stellar halo, i.e. σ2i j ≡ 〈(vi − 〈vi〉) (v j − 〈v j〉)〉, (12) where 〈Q〉 ≡ (∫ du f Q ) / (∫ du f ) is the average of the quantity Q weighted by the phase-space DF (e.g. Binney & Tremaine 2008).”

“First, we transformed the proper motions from the measured equatorial reference to radial and tangential components according to the equatorial-polar coordinates relation in Binney & Tremaine (2008): where φ = arctan( y / x ), and x and y are the (local Cartesian) gnomonic projected coordinates.”

“…dynamical mass, Mdyn, assuming a thick (1:4) exponential disk for the KMOS3D rotation-dominated galaxies (disk criterion 1 and 2) from the rotational velocities using: Mddyn(r re) = v2dre α0G , (3) where the factor α0 is dependent on the mass distribution of the system (Binney & Tremaine 2008).”