Galactic Dynamics: Second Edition

“on ˙ tot includes three terms (added in quadrature). It includes (i) the local isotropic velocity dispersion ˙ d driven by the disk self-gravity is ˙ d(r)=h z = V(r)=rfor a dynamically hot thick disk (45,53,54), (ii) a mixing term, ˙ m, arising from mixing the velocities along the line-of-sight for a geometrically thi…”

“The known baryonic mass in the Milky Way (stars + cold gas + dust) is between 6 − 7 × 1010M⊙ (Binney & Tremaine 2008).”

“Current estimates for the overall structural parameters of the Milky Way are compiled in Table 1.2 of Binney & Tremaine (2008); more specifically estimates for the mass of the Disk are ≃5× 1010 M⊙ (Flynn et al. 2006; McMillan 2011), though the most recent data sets have not yet been brought to bear…”

“In spherical potentials, the rosette pattern of an orbit can be uniquely described by the size and eccentricity of its “petals” – or equivalently the apocentric and pericentric distances and the azimuthal precession rate (see e.g. Binney & Tremaine 2008).”

“It is customary to describe its shape by means of the three global anisotropy parameters (Cappellari et al. 2007; Binney & Tremaine 2008) β ≡ 1− Πzz ΠRR , γ ≡ 1− Πϕϕ ΠRR and δ ≡ 1− 2Πzz ΠRR+ Πϕϕ , (10) where we denote the total unordered kinetic energy along thecoordinate directionk as Πkk = ∫ ρ∗σ…”

“Arguments based on the Jeans equation in virialized systems result in the relation Vc = Kσ, where typical values for K are √ 2− √ 3 (Binney & Tremaine 2008).”

“Typically√ 2σ (rc)/vc ≈ 1.2, which increases the time-scale tdf by a factor ∼2 over the standard definition (with B(1) and σ = vc/ √ 2, Binney & Tremaine 2008).”

“For example, Binney & Tremaine (2008) shows that C = 2.25 is an average value of known galactic mass distribution models.”

“In the classical version of dynamical formation in a dense stellar environment the black holes, being heavier than other cluster objects, sink toward the centre of the cluster through mass segregation (Spitzer, 1969; Binney and Tremaine, 2008).”

“According to Binney & Tremaine (2008), Rg can be found by equalising the potential energy (7) with the equation U = 4πG Rout∫ 0 M(r) r ρ(r)r2dr, (9) where M(r) is the mass within a sphere r and Rout is the outer limit of the system.”

“The rotation velocity of IC 5063 is about 240 km s−1, which means that the escape velocity in the inner regions is likely to be above 550 km s−1 (Binney & Tremaine 2008).”

“…and entropy is transferred to stars in the outer regions of the galaxy; Lynden-Bell 1967) and stellar evaporation from galaxies (whereby stars are ejected altogether, carrying with them energy, angular momentum and entropy, and allowing what remains behind to contract; e.g. Binney & Tremaine 2008).”

“The timescale for a small BH (∼ 103 M ) to settle in the galaxy centre is long, of order of a few hundred Myr to Gyr (Binney & Tremaine 2008), and, during the orbital decay, the rapidly moving BH cannot efficiently accrete gas from its surroundings.”

“Equation (2) corresponds to the total disk mass in the spherical approximation (for an infinitely thin Freeman disk, the Mdyn values would scale down by a factor of 0.8; Binney & Tremaine 2008).”

“Seventy percent of galaxies in the nearby Universe are characterized by a disk with prominent spiral arms, but our understanding of the origin of these patterns is incomplete, even after decades of theoretical study (Toomre 1977; Athanassoula 1984; Binney & Tremaine 2008; Sellwood 2011).”

“The known baryonic mass in the Milky Way (stars + cold gas) is approximately 5 × 1010 M (Binney & Tremaine 2008).”

“Many subsequent papers provided overwhelming evidence in support of this picture, most notably Bensby et al. (2003) and Bensby et al. (2014) with follow-up studies of more elements at higher sensitivity. Bensby et al. (2011) first suggested a shorter scalelength for the α-rich disk compared to the dominant α-poor disk…”

“If we include these forces, the tidal radius (also called the Jacobi or Roche radius, Binney & Tremaine (2008), Eqn.”

“The precession rate is not solely dependent on the mass decay rate in the host, it is also a weak function of the orbital energy and agular momentum (see equation 3.18 in Binney & Tremaine 2008).”

“To compute the force generated by such a spheroidal system we use Equations 2.129 a and b of Binney & Tremaine (2008).”

“A rough estimate of the fraction of stars lost by evaporation each trh can be calculated from the Maxwellian velocity distribution (Spitzer 1987; Binney & Tremaine 2008).”

“The isochrone model is defined by the potential (Binney & Tremaine 2008) (r) = − GNM b + √b2 + r2 .”

“…azimuthal wavelength,|kR| ≫ m, with R the cylindrical radius) we have D(ω, k,m) = (ω − mΩ)2 − c2sk2 + 2πGΣ|k| − κ2 = 0, (1) whereΩ(R) is the angular speed, andκ(R) is the epicyclic frequency,cs(R) is the sound speed andΣ(R) is the surface density of the umperturbed disc (Binney & Tremaine 2008).”

“…and the kinematics is supported by an assumption that rotation influences the shape of the object y flattening it and, for isotropic models, the rotation speed rsponsible for flattening of the shape is related to the shape of the stellar distribution as∼ √ǫ (Binney & Tremaine 2008, Section 4.8.2).”

“Such a halo is fully consistent with all of the constraints we have identified, but it does not contain more than a few percent of the missing baryons from the Galaxy.”

“We estimate these from the disc stars only by correcting for asymmetric drift using the following equation (Binney and Tremaine, 2008): Vc = V 2 φ + σ 2 φ − σ2r ( 1+ d(ln ν) d(ln r) ) − r dσ2r dr , (3) whereVc is the circular speed andVφ is the measured rotational velocity of stars for each bin.”

“Dynamical two-body relaxation processes, such as mass segregation and evaporation, are mass dependent and will alter the initial stellar mass function over a relaxation timescale, which we calculate according to Equation 1.38 in Binney & Tremaine (2008): trelax ∼ (N/log10N) ∗ (R/σ).”

“…correction: v2c = v 2 rot,gas = v 2 φ,gas + σ 2 D (9) where v2rot,gas is the ‘asymmetric drift corrected rotation curve’; σ2D = − R Σgas d dR ( Σgasσ 2 gas ) ; (10) and Σgas and σgas are the gas surface density and gas ‘effective velocity dispersion’, respectively (e.g. Binney & Tremaine 2008).”

“For a full analytic discussion see, for example, the comparison between the circular velocities of oblate spheroids as a function of q and mass profile in section 2.5.2 of Binney & Tremaine (2008).”

“From the tensor virial theorem (Binney & Tremaine 2008) we know that for 2I-models υ2Φ > υ 2 R in order to produce the flattening.”

“…of halo populations (assuming isotropically distributed), the radial velocity dispersionσr, number density ν and velocity anisotropy parameterβ ≡ 1 − σ2t /σ2r , are linked to the circular velocityVc through the Jeans equation (see, e.g. Binney & Tremaine 2008, pp. 349) for spherical systems.”

“We account for this effect by adopting the Chandrasekhar formula (Binney & Tremaine 2008): Ffric,G = − 4πG2M2mc ln(Λ)ρdm(r) v2 [erf(X)−2X√ π exp(−X2)]v v , (6) where Mmc is the mass of either the LMC or the SMC (MC), v is the velocity of the MC, and X = v/( √ 2σ), where σ is the one-dimensional…”

“…scattering of particles that pull back on the satellite, causing the satellite to lose angular momentum and energy in Corresponding author: Nicolás Garavito-Camargo jngaravitoc@email.arizona.edu ∗ CITA National Fellow a process referred to as dynamical friction (Binney & Tremaine 2008).”

“The epicycle frequency κ is computed numerically: κ(R)2 = R dΩ2/ dR+4Ω2, where R represents the distance to the galactic center, in the plane of the disc (Binney & Tremaine 2008).”

“…consistent with density profiles described by single power laws with logarithmic slopes d logρ/d logr > −3 — indeed, despite its unphysically infinite mass, the “isothermal sphere”, characterized by ρ(r)∝ r−2, has long been used to model kinematics of spheroidal galaxies (Binney & Tremaine 2008).”

“…and also these parameter values for the two stellar components are not the most sophisticated or up-to-date choices (see e.g. Dehnen & Binney 1998; Binney & Tremaine 2008; Bovy & Rix 2013; Irrgang et al. 2013; Licquia & Newman 2014), but we chose them here for comparability to past studies.”

“For a thin gas-dominated disk in a background potential (of dark matter and an old stellar component) Q is related to the local gas velocity dispersion σ 0 (assuming isotropy), circular velocity vc, epicyclic frequency κ (κ2 = 2(vc/R)2+ (vc/R) dvc/dR), and gas surface density Σgas at radius R via the relation (Wang &…”

“Dynamical Friction We include the effect of dynamical friction on cluster orbits by gradually reducing the radiusr as suggested by Binney & Tremaine (2008), equation (8.24): dr2 dt = − r2 tdf(r,M) , (7) tdf = 0.45Gyr ( r kpc )2( Vc(r) kms−1 )( M(t) 105 M⊙ )−1 fǫ, (8) where we assume a Coulomb…”

“Even absent coherent galactic winds, thermal and turbulent pressure support (from e.g., supernovae, cosmic rays, or the photoionizing background) in the warm (T ∼ 104 K) ISM keeps gas velocities at a significant fraction MNRAS 000, 1–24 (2017) of the local circular velocity in very low-mass halos (see Figure 3), so th…”

“Approximately 90% of those stars lie in the disk, 9% in the bulge, and 1% in the halo (Binney & Tremaine 2008).”

“This provides additional damping beyond dissipation by dynamical friction against the gas and stars, gdfo [M●(t)/Mc]O0 (39), which is balanced by the two-body fluctuations.”

“…(ii) above, spirals do not need to be short-lived transients for efficient migration; relatively long-lived waves are expected to exist in barred discs (Binney and Tremaine, 2008; Quillen et al., 2011; Minchev et al., 2012a), but are also found in non-barred simulations (e.g. D’Onghia et al. 2012).”

“(see Binney & Tremaine 2008, equation (8.46)).”

“…and negligible rotational support, the second-order Jeans equation is obtained by integrating moments of the phase-space distribution function (Binney & Tremaine 2008): 1 ν d dr (νv̄2r) + 2 βani(r)v̄2r r = −GM(r) r2 , (1) where ν(r), v̄2r(r), and βani(r) ≡ 1− v̄2θ/v̄2r are the stellar number…”

“Assuming that the stellar density distribution of each satellite can be approximated by a spherically symmetric isothermal profile, we can estimate the tidal radius by means of the equation Rt = ( Msat 3MDM,halo )1/3 D (5) (Binney & 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…”

“Sparke & Sellwood 1987), and thus could be longer lived (e.g. Donner & Thomasson 1994 Binney & Tremaine 2008; Baba et al. 2009, Quillen et al. 2011).”

“The tidal effects of LMC and SMC on 2 The disk model withb = 0 is also otherwise known as Toomre’s model or Kuzmin disk (Toomre 1963; Binney & Tremaine 2008) 0 20 40 60 80 100 120 140 160 180 r (kpc) 0 50 100 150 200 250 300 350 v c ir c ( km s− 1 ) X08 W99S09 G10 W10 S11 D12b Figure 11.”

“The Ja- cobi integral,Ej = E − Ωpjz is conserved in the rotating frame of a single, steady perturbation (Binney & Tremaine 2008).”

“…that we call dark matter and dark energy: the first residing in bound objects as nonluminous matter (Persic et al. 1996; Boriello and Salucci 2001; Binney & Tremaine 2008), the latter in the form of a zero-point energy that pervades the whole Universe (Peebles & Ratra 2003; Padmanabhan 2003).”

“Dynamical friction, the force exerted by the gravitational wake caused by a massive object moving in an extended medium (Chandrasekhar 1943; Binney & Tremaine 2008) causes the orbits of SMBHs to decay towards the center of massive galaxies (Governato et al. 1994; Kazantzidis et al. 2005).”

“Lacking any information about the galaxies’ density profiles, we adopt C = 2, appropriate to a singular isothermal sphere (Binney & Tremaine 2008), but the value of C remains a source of systematic uncertainty.”

“The evaporation time can be estimated as (Binney & Tremaine 2008): τevap= (m1 +m2)σ 16 √ πGainm2ν ln Λ = 3× 108 ( ln Λ 2 )−1 × ( m1 +m2 20M⊙ )( MBH 4× 106M⊙ )−1 ( σ 100km s−1 ) × ( ain 10AU )−1 ( mν 5× 105M⊙ pc−3 )−1 yr.”

“Observed polluted WDs can provide the inspiration for extensions to our work, with numerical simulations tailored to particular observational samples or campaigns.”

“1, which is similar to what is found in idealised simulations of isolated discs in which transient, recurring spiral arms continually form and disrupt (e.g., Grand et al. 2012a; Wada et al. 2011; Baba et al. 2013; Grand et al. 2013; Sellwood & Carlberg 2014; Dobbs & Baba 2014).”

“This fact is illustrated in Figures 3 and 4, where the orbital phases of the masers are shown for a logarithmic potential Φ = V 2c ln r (e.g., Equation (3.14) in Binney & Tremaine 2008) assuming R0 = 8.2 kpc and Vc = 244 km s−1.”

“The galaxy anisotropy has classically been measured through the parameter β that gives the relative ratio of the tangential to the radial velocity dispersion for a given dynamical tracer (Binney & Tremaine 2008).”

“4.215 of Binney & Tremaine 2008).”

“Tz is the effective lifetime of the binary: this is the sum of the dynamical friction (Binney & Tremaine 2008) (tdf ) and stellar hardening (Sesana & Khan 2015) (tsh) timescales (details in Methods).”

“To calculate the relaxation time, we use the standard equation (Binney & Tremaine 2008) t N N t 6 ln 2 , 8relax cross( ) ( )= where N is the number of stars and tcross is the crossing time.”

“Considering the one-component approximation for a gas disk, we can write (Binney & Tremaine 2008; Escala & Larson 2008; Dekel et al. 2009b) s k p s p = S = S Q G G av r r .”

“To answer this question we make use of the axisymmetric version of the Jeans equations (Binney & Tremaine 2008) v̄2R(R, z) = v̄ 2 z(R, z)= 1 ρ∗(R, z) ∫ ∞ z dz′ρ∗(R, z ′) ∂Φ ∂z′ (B1) v̄2φ(R, z)= v̄ 2 R + R ρ∗ ∂(ρ8v̄2R) ∂R +R ∂Φ ∂R , (B2) obtained assuming a distribution of stars of the form f(E,Lz).”

“To match this value, we corrected the interstellar matter mass density and attributed to it 0.05 M pc−3, in agreement with Binney & Tremaine (2008) and still within the error bars given by observational constraints.”

“Among the internal processes that dissolve clusters are evaporation via two-body scatterings and collectively amplified fluctuations, or ejection from three-body interactions with binary stars (e.g., Binney & Tremaine 2008).”

“The Jeans equation inside the clump then reads (e.g. Binney & Tremaine 2008, eq.”

“(8.102) of Binney & Tremaine (2008), which relate the orbital parameters and the phasespace coordinates to the energy and angular momentum in the epicycle approximation.”

“This approximation is accurate to 7% for r 4rc (Binney & Tremaine 2008).”

“For both the cored and cusped initial density profiles, and for each MC model, we generate the corresponding ergodic distribution function by performing an Eddington inversion (Eddington 1916; Binney & Tremaine 2008) and checking that the distribution function is nowhere negative.”