ORIGIN AND GROWTH OF NUCLEAR STAR CLUSTERS AROUND MASSIVE BLACK HOLES

Many galaxies, over the whole Hubble sequence, show nucleated central regions often referred to as "nuclear star clusters" (NSCs). These systems are among the densest star clusters observed, with effective radii of a few parsecs and central luminosities up to ∼10⁷ L_☉ (Matthews & Gallagher 1997; Carollo et al. 1997, 1998; Böker et al. 2002, 2004; Balcells et al. 2007; Graham & Guzmán 2003; Côté et al. 2006; Turner et al. 2012). The total frequency of nucleation is as large as 80% for galaxies fainter than M_B = −19.5, while NSCs tend to disappear in galaxies brighter than this magnitude.

There is no consensus on how NSCs form. A dissipative origin bases on the hypothesis of radial gas inflow into the galactic center (GC) and requires efficient dissipation mechanisms to work (e.g., Loose et al. 1982). In this model, an NSC consists mostly of stars that formed locally (Schinnerer et al. 2006, 2008; Milosavljević 2004; Emsellem & van de Ven 2008; Shlosman & Begelman 1989; Bekki 2007).

Alternatively, NSCs could have a dissipationless origin in which massive stellar (globular-like) clusters migrate to the center due to dynamical friction and merge to form a dense nucleus growing up to the size of observed NSCs (Tremaine et al. 1975). Observations of NSCs in dE galaxies suggest that the majority of these systems could be the result of accumulating mass in the form of orbitally decayed globular clusters. Numerical studies have also shown that such a formation model is consistent with the measured sizes and luminosities of nuclei (Capuzzo-Dolcetta 1993; Bekki et al. 2004; Capuzzo-Dolcetta & Miocchi 2008; Hartmann et al. 2011). In addition, stellar population synthesis studies of NSC spectra suggest that most of the mass usually resides in stars as old as typical globular clusters (∼10 Gyr; Figer et al. 2004; Böker 2010). On the other hand, the observed correlation between colors and luminosities together with the complex star formation histories that often characterize the central region of galaxies may be difficult to explain on the basis of a dissipationless origin, unless there is some contribution from continuous or recurrent star formation in addition to the ancient globular cluster stars (Antonini et al. 2012). This suggests that dissipation and dissipationless processes are not exclusive, and NSCs might indeed originate from a combination of the two processes.

Due to their small sizes and their crowded stellar fields, galactic nuclei beyond the Local Group are typically unresolved, and the only quantities that can be determined are integrated properties such as half-mass radius and total luminosity. A radial density profile and velocity structure can be reliably determined only for the Milky Way (MW) NSC (Genzel et al. 2003; Schödel et al. 2007; Graham & Spitler 2009; Oh et al. 2009), which, due to its proximity (∼8 kpc), can be resolved into individual stars (Schödel et al. 2007, 2009). In addition to the MW, NGC 205 also has a spatially resolved NSC (Figure 2 of Merritt 2009). The MW NSC has an estimated mass of ∼10⁷ M_☉ (Launhardt et al. 2002; Schödel et al. 2008) and hosts a massive black hole (MBH) whose mass, ∼4.3 × 10⁶ M_☉, is uniquely well determined (Genzel et al. 2003; Ghez et al. 2008; Gillessen 2009). A number of other galaxies also contain both an NSC and an MBH (Seth et al. 2008; Graham & Driver 2007; Graham & Spitler 2009) that have comparable masses. In models of NSCs, the dynamical influence of an MBH should therefore be considered, at least in bulges brighter than about 10⁹ L_☉ which are believed to always contain an MBH (Ferrarese & Ford 2005).

More recently, we have presented large-scale N-body simulations of the inspiral and merger of massive clusters in the inner regions of the Galaxy (Antonini et al. 2012). We showed that current observational constraints are consistent with the hypothesis that a large fraction of the MW NSC mass is in old stars brought in by infalling globular clusters. In this paper, we expand on this previous work and use simple analytical considerations to investigate the possibility that globular clusters can migrate through dynamical friction in the center of galaxies and form compact stellar nuclei. In particular, we focus on how the presence of MBHs at the center of galaxies can impact the merger hypothesis for the formation of their NSC. In order to highlight the role of MBHs in NSC formation, we will systematically present our results for both cases of galaxy with and without central MBH.

This paper is organized as follows. We start in Section 2 by comparing some of the observed scaling relations for NSCs with the same relations predicted in both the merger and the gas model. In Sections 3 and 4 we develop a simple analytical model for studying the orbital decay of globular clusters in galaxies like the MW, and explore the effect that a central MBH in the galaxy has on the properties of the resulting NSC. In Section 5 we use an approach similar to that of Section 3 to follow the orbital evolution of globular clusters near massive MBHs in the central regions of bright elliptical galaxies. In Section 6 we discuss a variety of astrophysical implications of a dissipationless origin of NSCs and conclude in Section 7.

The study of NSCs has revealed a number of correlations between their masses and several global properties of their host galaxies, such as velocity dispersion and bulge mass. The existence of such correlations might indicate a direct link among large galactic spatial scales and the much smaller scale of the nuclear environment. While it was long believed that NSCs and MBHs followed similar scaling relations with their host galaxies (Ferrarese et al. 2006; Wehner & Harris 2006), it is now well established that NCs do not follow any of the scaling relations defined by MBHs (Balcells et al. 2007; Graham & Spitler 2009; Graham 2012a; Scott & Graham 2012; Leigh et al. 2012). Observations also reveal that, unlike globular clusters, NSC half-light radii are luminosity dependent, increasing with increasing total mass (Côté et al. 2006; Forbes et al. 2008). This relation might contain important information on the processes that shaped the central regions of galaxies and their NSCs.

We start here by comparing such observed correlations with predictions from both the dissipative and dissipationless formation models.

2.1. M_NSC − σ Relation

Of the global-to-nucleus relations, the most frequently referred to is the tight correlation between NSC mass, M_NSC, and the host galaxy's velocity dispersion, σ. Ferrarese et al. (2006) argued that such a correlation has a slope which is consistent with that of the well-known M_• − σ relation obeyed by the MBH mass, M_•. More recently, a series of papers reached a different conclusion, suggesting that the NSC scaling relations are instead substantially shallower than the corresponding MBH scaling relations (Graham & Spitler 2009; Graham 2012a; Scott & Graham 2012; Leigh et al. 2012). The version of the M_NSC − σ relation given in Graham (2012a) is

$$\begin{eqnarray} \log \left(\frac{M_{\rm NSC}}{M_{\odot }}\right) &=&(6.83\pm 0.07)\nonumber \\ &&+(1.57\pm 0.24)\log (\sigma /70\, {\rm km\, s^{-1}}). \end{eqnarray} \tag{ 1 }$$

We mention here that the M_NSC − σ relation might be a non-primary correlation, instead resulting from a projection of the fundamental plane, given the observed correlation between M_NSC and the total host galaxy mass (M_gx): M_NSC∝M^{1.18 ± 0.16}_gx (e.g., see Leigh et al. 2012).

2.1.1. Dissipationless Model

A predicted M_NSC − σ relation can be easily derived in the globular cluster merger model if we assume that the globular cluster distribution initially follows the stellar light and by using an isothermal sphere density model: ρ(r) = σ²/2πGr², where σ is the one-dimensional (1D) velocity dispersion and G is the gravitational constant. The cumulative mass within r is M(r) = rv²_c/G = 2rσ²/G, with v_c the velocity of a circular orbit. The dynamical friction coefficient for a stellar cluster of mass m_cl moving in an isothermal sphere model is (Chandrasekhar 1943)

$$\begin{equation} \boldsymbol{f}_{\rm df} = -4\pi G^2 m_{\rm cl} \rho (r) {{\boldsymbol{v}} \over v^3 } \ln \Lambda \left[{\rm erf}(X)-\frac{2X}{\sqrt{\pi }} e^{-X^2}\right], \end{equation} \tag{ 2 }$$

with ln Λ the Coulomb logarithm and $X=v/\sqrt{2}\sigma$. Noting that

$$\begin{equation} \frac{1}{r}{dr \over dt}=-|\boldsymbol{f}_{\rm df}| \left({{dL}\over {dr}}\right)^{-1}, \end{equation} \tag{ 3 }$$

where $L(=\sqrt{GM(r)r})$ is the orbital angular momentum, for a cluster moving on a circular orbit we obtain

$$\begin{equation} \frac{dr}{dt}=-Gm_{\rm cl} {\ln \Lambda \left[{\rm erf}(X)-\frac{2X}{\sqrt{\pi }} e^{-X^2}\right] }\Big/\sqrt{2}r\sigma. \end{equation} \tag{ 4 }$$

Integrating Equation (4) yields

$$\begin{equation} r_{{\rm in}}=\left(\sqrt{2}G m_{\rm cl} \ln \Lambda \left[{\rm erf}(X)-\frac{2X}{\sqrt{\pi }} e^{-X^2}\right] t/\sigma \right)^{1/2}. \end{equation} \tag{ 5 }$$

Clusters initially within r_in reach the center at a time ⩽t.

Assuming that the total mass accumulated in the center is equal to the total mass in globular clusters that are initially within r_in, then we obtain (e.g., Tremaine et al. 1975)

$$\begin{eqnarray} M_{\rm NSC}&=&2^{5/4} \left(\ln \Lambda \left[{\rm erf}(X)-\frac{2X}{\sqrt{\pi }} e^{-X^2}\right] \langle m_{\rm cl} \rangle t/G \right)^{1/2} \nonumber \\ &&\times f \frac{ \langle m_{\rm cl} \rangle }{m_\star } \sigma ^{3/2}, \end{eqnarray} \tag{ 6 }$$

where m_⋆ is the mass of the field stars, 〈m_cl〉 is the average globular cluster mass, and f is the initial number fraction of globular clusters (after galaxy formation) to the total number of stars in the galaxy. Since the term in square brackets in Equation (6) is a constant, the globular cluster merger model predicts M_NSC∝σ^3/2.

In the MW, the initial total number of globular clusters, N_cl(0), can be recovered based on their observed luminosity function and semi-analytical modeling of mass loss due to stellar evolution and due to tidal interaction with the Galactic environment. Kruijssen & Portegies Zwart (2009) derived a survival fraction of 0.004 for a minimum initial cluster mass M_min = 5000 M_☉. Assuming 100 present-day globular clusters, we have that the total initial number of clusters in the Galaxy is 10^4.4. Considering only those GCs that are associated with the bulge on a Hubble time, this gives N_cl(0) = 10^3.9, which, compared to the total stellar mass of the bulge, ∼10¹⁰ M_☉, yields f ≈ 10⁻⁶. We can rewrite Equation (6) as

$$\begin{eqnarray} M_{\rm NSC}&=& 3\times 10^{7} \, M_{\odot } \left(\frac{f}{10^{-6}} \right) \left(\frac{\ln \Lambda }{3} \right)\left(\frac{m_\star }{M_{\odot }}\right)\left(\frac{\langle m_{\rm cl} \rangle }{10^5 \, M_{\odot }}\right)^{3/2} \nonumber \\ && \times \left(\frac{t}{10^{10} \,{\rm yr} }\right)^{1/2} \left(\frac{\sigma }{50\, {\rm km\, s^{-1}}}\right)^{3/2}. \end{eqnarray} \tag{ 7 }$$

Despite its simplicity, our model reproduces the observed M_NSC − σ relation both in slope and normalization.

2.1.2. Dissipative Model

McLaughlin et al. (2006) proposed an NSC in situ star formation model regulated by momentum feedback. This model is an extension of the argument proposed by King (2003) to explain the M_• − σ relation, and invokes the formation of a massive nuclear cluster due to gas inflow and accumulation at the center of the galaxy during the early phases of galaxy evolution. Stellar winds and supernovae from a young nuclear cluster with a standard IMF produce an outflow with momentum flux given by

$$\begin{equation} \dot{\Pi }\approx \lambda L_{\rm Edd}/c=\frac{ \lambda 4\pi G M_{\rm NSC}}{k}, \end{equation} \tag{ 8 }$$

where L_Edd is the Eddington luminosity calculated by the stellar mass, c is the speed of light, λ ∼ 0.05 is related to the mass fraction of massive stars, and κ ≡ 0.398 cm² g⁻¹ is the electron scattering opacity. The outflow is initially momentum conserving and produces an outward force on the gas in the bulge, whose weight is W(r) = GM_gas(r)M(r)/r², with M_gas(r) the enclosed gas mass and M(r) the total enclosed mass of the galaxy. For an isothermal potential, one finds

$$\begin{equation} W=\frac{4f_g}{G}\sigma ^4, \end{equation} \tag{ 9 }$$

where f_g = 0.16 is the baryonic mass fraction (Spergel et al. 2003). Requiring that the momentum output produced by the nuclear cluster balances the weight of the gas leads to the relation

$$\begin{eqnarray} M_{\rm NSC}&=& \frac{f_g\kappa }{\lambda \pi G^2}\sigma ^4 = 3\times 10^{7} \, M_{\odot } \nonumber \\ &&\times \left(\frac{f_g}{0.16}\right)\left(\frac{\lambda }{0.05}\right)^{-1} \left(\frac{\sigma }{50\,{\rm km\, s^{-1}}}\right)^4. \end{eqnarray} \tag{ 10 }$$

A similar scaling relation can be derived in protogalaxies with non-isothermal dark matter halos (McQuillin & McLaughlin 2012).

This model is very attractive because it contains no free parameters. In addition, Equation (10) is in good agreement with the M_NSC − σ correlation reported by Ferrarese et al. (2006):

$$\begin{eqnarray} \log \left(\frac{M_{\rm NSC}}{M_{\odot }} \right)&=&(6.91\pm 0.11) \nonumber \\ && +(4.27\pm 0.61)\log \left(\frac{\sigma }{54\, {\rm km\, s^{-1}}}\right), \nonumber \\ \end{eqnarray} \tag{ 11 }$$

but, as previously mentioned, it is at odds with Graham (2012a), who finds NSC scaling relations considerably shallower than the corresponding MBH scaling relations. The difference between these two studies was due to the proper exclusion of nuclear disks in the sample of Graham (2012a) and the larger sample of NSCs used in this latter work. For these reasons we consider the results of Graham to be more robust and conclude that the McLaughlin et al. model provides a poor description of the observed M_NSC − σ correlation. This suggests that momentum feedback may be not relevant, which would be expected if the NSCs originated elsewhere and were subsequently deposited into their host galaxy centers.

2.2. R − M_NSC Relation

The size of galactic nuclei clearly correlates with their luminosity, in the sense that brighter NSCs have larger effective radii. The relation is approximately (Côté et al. 2006)

$$\begin{equation} R\propto \sqrt{\mathcal {L}_{\rm NSC}}, \end{equation} \tag{ 12 }$$

where R is the NSC effective radius (or half-mass radius) and L is its total luminosity. This relation is consistent with the extrapolation of the $R_{{\rm eff}}- \mathcal {L}$ relation for elliptical galaxies.

2.2.1. Dissipationless Model

In the merger model the radius of the nucleus increases with increasing total mass as globular clusters merge. The brighter, larger mass nuclei are therefore predicted to be spatially very extended. Antonini et al. (2012) used simple energy arguments to derive the size–mass relation for NSCs in galaxies with no central MBH. For the sake of completeness we repeat this simple calculation in what follows. After a merger the NSC energy, E_f = −GM²_f/2R_f, equals the energy of the nucleus before the merger, E_i, plus the energy brought in by the globular cluster:

$$\begin{equation} E_f=E_i+E_o+E_b, \end{equation} \tag{ 13 }$$

with E_b the internal binding energy of the cluster and E_o ≈ −Gm_clM_i/2R_i its orbital energy before the merger.

The equations above permit expressing the mass, energy, and radius of the nucleus recursively as

$$\begin{eqnarray} &&M_{j+1}=(j+1)M_{1}, \hspace{70pt} \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} &&jE_{j+1}=\left(j+1 \right)E_{j}+jE_1,\hspace{46pt} \end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray} &&\left(j+1\right)^2 R^{-1}_{j+1}=j\left(j+1\right)R^{-1}_j +R^{-1}_1,\quad j=1,2,3,\ldots \nonumber \\ \end{eqnarray} \tag{ 16 }$$

where the subscript 1 denotes the initial nucleus, and, by assumption, M₁ = m. Equations (14)–(16) imply R∝M^0.5 at the time the mass of the nucleus is still comparable to the mass of the infalling clusters. After many mergers the nucleus is much more massive than one globular cluster and the relation steepens to R∝M. After 25 mergers, Equations (14)–(16) imply R ≈ 5R₁. The half-mass radii of globular clusters are ∼3 pc (Jordán et al. 2005), irrespective of their luminosity, so for a nucleus assembled from 25 mergers, R ∼ 15 pc. Such a value is in reasonable agreement with the measured half-mass radii for the brightest nuclei. We conclude that a model that attributes the origin of NSCs to the mergers of globular clusters at the centers of galaxies is consistent with the sizes and luminosities of the nuclei.

In a number of galaxies NSCs are observed to coexist with MBHs. An example of such systems is the MW for which the mass of the NSC, M_NSC ∼ 10⁷ M_☉ (Launhardt et al. 2002; Schödel et al. 2008), is somewhat comparable to the mass of the central black hole (BH), M_• ∼ 4 × 10⁶ M_☉ (Ghez et al. 2008; Gillessen 2009). An MBH will disrupt clusters that pass within the radius

$$\begin{equation} r_\mathrm{disr} \approx 2 \left(\frac{\sigma _\mathrm{NSC}}{5\sigma _K}\right)^{2/3} \left(\frac{r_\mathrm{infl}}{r_K}\right)^{1/3} r_K, \end{equation} \tag{ 17 }$$

where σ_K is the 1D velocity dispersion of a globular cluster, r_K is its core radius, σ_NSC is the velocity dispersion in the NSC, and r_infl = GM_•/σ²_NSC is the influence radius of the MBH. In the presence of an MBH, the dependence of the half-mass radius R on M_NSC is substantially weaker than the observed correlation, R ∼ M^0.2_NSC (Antonini et al. 2012), due to the fact that the size of the NSC is determined by the fixed tidal field from the MBH. When a stellar cluster is disrupted, stars that were initially within the cluster core will redistribute locally at a distance r_disr from the MBH. The density profile of the NSC will therefore have a core of characteristic size ∼r_disr.

2.2.2. Dissipative Model

Although the gas model remains somewhat more qualitative there are some indications that the observed R − M_NSC relation might be difficult to reconcile with a purely dissipative scenario.

Bekki (2007) performed fully self-consistent chemodynamical simulations to investigate how NSCs can form through dissipative gas dynamics. He found that compact nuclei can be formed via dissipative, repeated merger of gaseous (or stellar) clumps that develop from nuclear gaseous spiral arms due to local gravitational instability. These computations showed that fainter NSCs are likely to have a more diffuse configuration due to more negative feedback from SNe II which in turn can prevent NSCs from forming in faint galaxies. The fact that NSC formation is more strongly suppressed by stronger feedback effects in less-luminous galaxies would explain why brighter dwarf galaxies (dE) are more likely to contain NSCs (van den Bergh 1986). However, simulations also show that more massive NSCs are less extended than their lower mass counterparts, which is the opposite of the observed trend (see Figure 15 of Bekki 2007).

Further investigation of the above inconsistency is needed in order to determine how the results depend on the resolution limit of the simulations and on the adopted initial conditions.

3.1. Phase-space Constraints

The above discussion gives some level of reliability to the hypothesis of a dissipationless, merging formation for NSCs. We now wish to test whether the merger of globular clusters can result in an NSC similar to that of the MW. As a first step one can derive the globular cluster parameters required to give a peak density, ρ_NSC, equal to the observed density of the NSC in the MW, and see whether such parameters are reasonable when compared to typical globular cluster properties.¹

The stellar cluster central phase-space density is given by

$$\begin{equation} \frac{\rho _k}{\sigma _K^3}=\frac{9}{4\pi G}\frac{1}{\sigma _Kr_K^2}, \end{equation} \tag{ 18 }$$

where ρ_K is the cluster core density. After inspiral, we require ρ_NSC/σ³_NSC ≲ ρ_k/σ³_K, or

$$\begin{equation} \rho _{\rm NSC}\lesssim \frac{9}{4\pi G}\frac{\sigma _{\rm NSC}^3}{\sigma _Kr_K^2}, \end{equation} \tag{ 19 }$$

so that

$$\begin{eqnarray} \rho _{\rm NSC}&\lesssim & 1.5\times 10^7 \, M_{\odot } \ {\rm pc^{-3}} \left(\frac{\sigma _{\rm NSC}}{100\, {\rm km\, s^{-1}}} \right)^3 \nonumber \\ \ms && \times \left(\frac{\sigma _{K}}{10\, {\rm km\, s^{-1}}} \right)^{-1}\left(\frac{r_K}{{\rm 1\,pc}}\right)^{-2}. \end{eqnarray} \tag{ 20 }$$

The observed NSC density within 0.5 pc at the GC is ρ_NSC ≈ 10⁶ M_☉ pc⁻³ (Merritt 2010). Comparing this with Equation (20), we obtain

$$\begin{equation} \left(\frac{\sigma _{K}}{10\, {\rm km\, s^{-1}}} \right)\left(\frac{r_K}{{\rm 1\,pc}}\right)^{2}\lesssim 15. \end{equation} \tag{ 21 }$$

This crude calculation shows that for an MW-like galaxy, the globular cluster parameters required to give the observed peak density of the NSC are quite reasonable.

3.2. NSC Formation via Cluster Migration

A simple analytical model is developed in what follows to evolve a population of stellar clusters subject to migration and dissolution in a galactic bulge and to calculate the influx of migrating clusters into the center of the galaxy. In this way we can address the possibility that a substantial fraction of the NSC mass in galaxies like the MW could have been assembled through cluster migration and mergers.

We assume that the central properties of a stellar cluster (i.e., σ_K and r_K) remain unchanged during inspiral and that r_t > r_k, where r_t is the cluster tidal (limiting) radius given by (King 1962)

$$\begin{equation} r_t= \alpha \frac{\sigma _K}{\sqrt{2}}\left(\frac{3}{r}\frac{ {d} \phi }{{d} r} -4\pi G \rho \right)^{-1/2}, \end{equation} \tag{ 22 }$$

with ϕ the galactic potential and α a "form factor" that depends on the density distribution within the cluster. For a King model with a large central concentration, α ≈ 1. Equation (22) includes the tidal force due to the radial gradient in the galaxy potential, the centrifugal force from the cluster's orbit and assume that the density of the cluster goes to zero roughly at the radius where the force acting to remove a star is balanced by the attracting force from the rest of the cluster.

The tidal radius of a King model orbiting in a galaxy with a power-law density profile, ρ(r) = ρ₀(r/r₀)^−γ, and containing an MBH at its center, is

$$\begin{equation} r_t\approx \frac{\sigma _K}{\sqrt{2}}\left[4\pi G \rho _0 \left(\frac{r}{r_0} \right)^{-\gamma } \frac{\gamma }{3-\gamma } +\frac{3GM_\bullet }{r^3}\right]^{-1/2}. \end{equation} \tag{ 23 }$$

The radius at which the tidal force from the MBH starts to dominate the tidal force from the stellar cusp (in galaxies containing both) is

$$\begin{equation} r=\left(\frac{3(3-\gamma)M_{\bullet }}{\gamma 4\pi \rho _0 r_0^{\gamma }}\right)^{\frac{1}{3-\gamma }}. \end{equation} \tag{ 24 }$$

A King model also satisfies the relation:

$$\begin{equation} Gm_t\approx \frac{\sigma _K^2r_t}{2}, \end{equation} \tag{ 25 }$$

where m_t is the truncated mass of the globular cluster whose radius is limited by the external tidal field.

Given the cluster central velocity dispersion, Equations (23) and (25) can be combined to evaluate, at any radius, the cluster mass permitted by the galaxy tidal field and the mass dispersed along the orbit. This corresponds to a density enhancement with respect to the galactic background of

$$\begin{equation} \Delta \rho (r)=\frac{ \sigma _K^2}{8\pi G r^2}\frac{dr_t}{dr}. \end{equation} \tag{ 26 }$$

The resulting radial dependence of the density profile of stars in the growing NSC is easily found to be

$$\begin{equation} \Delta \rho (r) \propto \sigma _K^3\times \left\lbrace \begin{array}{@{}ll}\sqrt{\gamma (3-\gamma)} r^{\frac{\gamma -6}{2}} & \mbox{cusp}, \\ \ms r^{-\frac{3}{2}}/\sqrt{M_{\bullet }} & \mbox{black hole}. \end{array} \right. \end{equation} \tag{ 27 }$$

Steeper density cusps in the distribution of background stars give lower values of the density slope for the resulting NSC. Larger values of M_• also give smaller densities near the center since the cluster models start being truncated at larger radii and their mass is dispersed over larger spatial scales.

3.2.1. Dynamical Friction

We obtain the cluster orbits by using the standard Chandrasekhar's dynamical friction formula (Chandrasekhar 1943) for a self-gravitating cusp (e.g., Merritt et al. 2004; Just et al. 2010):

$$\begin{eqnarray} {dr\over dt} &=& {-}2{(3-\gamma)^{3/2}\over 4-\gamma }\sqrt{G\over r_0} \nonumber \\ \ms && \times \left({r\over r_0}\right)^{\gamma /2-2}\frac{F(\gamma)\ln \Lambda }{\sqrt{4\pi \rho _0 r_0^3}}{m_{\rm cl}}. \end{eqnarray} \tag{ 28 }$$

The coefficient F is a function of γ, with F = (0.193, 0.302, 0.427) for γ = (1, 1.5, 2). We set an initial limiting radius of 40 pc. If r_t > 40 pc (i.e., the model is not truncated), integrating Equation (28) yields

$$\begin{eqnarray} r(t)&=& \Bigg[ r_{{\rm in}}^{\frac{6-\gamma }{2}} - \frac{(3-\gamma)^{3/2}(6-\gamma)}{(4-\gamma)} \nonumber \\ &&\times \sqrt{\frac{G}{4\pi \rho _0}}r_0^{-\gamma /2}F(\gamma) \ln \Lambda m_{\rm cl}\times t \Bigg ]^{\frac{2}{6-\gamma }}, \end{eqnarray} \tag{ 29 }$$

which gives a timescale to reach the center of the galaxy:

$$\begin{eqnarray} \tau _{\star } &=& {10^{10} \,{\rm yr}}\frac{(4-\gamma)}{(3-\gamma)^{3/2}(6-\gamma)F(\gamma)} \nonumber \\ && \times \ln \Lambda _{3} ^{-1}\sqrt{ \rho _{0,5}} r_{0,700}^3 m_{{\rm cl},6}^{-1} \left(\frac{r_{{\rm in}} }{r_0}\right)^{\frac{6-\gamma }{2}}, \end{eqnarray} \tag{ 30 }$$

with ρ_{0, 5} = ρ₀/5 M_☉ pc⁻³, r_{0, 700} = r₀/700 pc, ln Λ₃ = ln Λ/3 and m_{cl, 6} = m_cl/10⁶ M_☉. The coefficient depending on γ is approximately 1.

After a cluster starts being truncated by the galactic tidal field, its mass also becomes a function of radius. In this case, we computed the cluster orbit by setting m_cl = m_t in Equation (28), obtaining

$$\begin{eqnarray} &&r(t)=\left[ r_{{\rm in}}^{3-\gamma }-\frac{(3-\gamma)^3}{(4-\gamma)\sqrt{2\gamma }}\frac{ r_0^{-\gamma }}{4 \pi G \rho _0 }F(\gamma)\ln \Lambda \sigma _k^3 \times t \right]^{\frac{1}{3-\gamma }}. \nonumber \\ \end{eqnarray} \tag{ 31 }$$

This equation takes into account mass loss due to the interaction with the galactic tidal field, but ignores the possible presence of an MBH.² Equation (31) implies that the massive object comes to rest at the center of the stellar system in a time

$$\begin{eqnarray} \tau _{\star }&=& 3\times 10^{10}\,{\rm yr} \frac{(4-\gamma)\sqrt{\gamma }}{(3-\gamma)^3F(\gamma)} \nonumber \\ &&\times \ln \Lambda _{3} ^{-1} \rho _{0,5} r_{0,700}^3\sigma _{K,10}^{-3} \left(\frac{r_{{\rm in}}}{r_{0}}\right)^{3-\gamma }, \end{eqnarray} \tag{ 32 }$$

with σ_{K, 10} = σ_K/10 km s⁻¹ and the coefficient depending on γ equals to (2.8, 4.27, 9.4) for γ = (1, 1.5, 2).

These basic formalisms assume that mass loss from the clusters is entirely due to their interaction with the external tidal field and it omits mass loss (evaporation) due to internal dynamics. Since the only clusters that can contribute to the growth of an NSC are very massive systems with relaxation times longer than their dynamical friction timescale, this simplification is quite reasonable and does not alter our results in any important way. However, dynamical evaporation due to internal dynamics is also accounted for in the sense that we do not follow the evolution of clusters which are disrupted (evaporated) on timescales shorter than the relevant dynamical friction timescales.

3.2.2. Formation of the MW NSC

Luminosity profiles of galaxies are well approximated by power laws with 1 < γ < 2 at radii smaller than the effective radius of the stellar spheroid (R_eff; Terzić & Graham 2005). Since for MW-like galaxies the only clusters that can reach the GC in one Hubble time are those initially at r ≲ R_eff (Milosavljević 2004), and that within these radii the baryonic matter dominates the galactic potential, we represent the galaxy bulge by using a simple power-law density model: ρ(r) = ρ₀(r/r₀)^−γ, with ρ₀ = (3 − γ)M_sph/4πr³₀,³M_sph = 10¹⁰ M_☉, r₀ = 700 pc, and γ = 1 (e.g., McMillan & Dehnen 2007). The scale length, r₀, is related to the bulge effective radius via R_eff/r₀ = (1.8, 1.5, 1) for γ = (1, 1.5, 2). We assume that the clusters have initially the same distribution of stars in the galaxy (e.g., Agarwal & Milosavljević 2011) and we assign their masses using the cluster initial mass function (CIMF), dn/dM∝M⁻² (Bik et al. 2003; de Grijs et al. 2003), and limiting mass values of M_min = 10² M_☉, M_max = 10⁶ − 10⁷ M_☉.

The fraction of all star formation that occurs in gravitationally bound stellar clusters is usually referred to as "cluster formation efficiency." The cluster formation efficiency in the MW is ∼10% (Lada & Lada 2003). Accordingly, in our model we assume that the total mass in clusters is initially M_cl = 0.1 × M_sph.

We start by identifying a "dissolution time" for globular clusters due to dynamical evaporation as a function of their initial mass (Baumgardt & Makino 2003; Gieles & Baumgardt 2008; Lamers et al. 2010):

$$\begin{equation} t_{\rm dis}=t_0\left(\frac{m_{\rm cl}}{M_{\odot }} \right)^{\beta }, \end{equation} \tag{ 33 }$$

with β = 0.7, m_cl = m_t if the cluster is truncated, t₀ = 0.3 × |dΩ/dln r|⁻¹, and Ω the angular velocity at the galactocentric radius r. We then remove clusters with total lifetime, t_dis/β, shorter than the dynamical friction timescale, τ_⋆. This procedure reduces the initial total mass in globular clusters from 10⁹ M_☉ to ∼5 × 10⁷ M_☉ for M_max = 10⁶ M_☉, and to ∼2 × 10⁸ M_☉ for M_max = 10⁷ M_☉. By comparing Equation (32) with Equation (33) one can easily show that if γ ≲ 3/2 and t_dis/β > τ_⋆ initially, this latter condition remains satisfied during inspiral.

We set the core radius of the globular clusters to be r_K = 1 pc, roughly equal to the median value of the core radii listed in the Harris's compilation (Harris 1996) of Galactic globular clusters. If a cluster reached its tidal disruption radius its core mass is redistributed uniformly at the radius of disruption over a region of extent r_disp = 3 pc. But we note that our results do not strongly depend on the particular value for this parameter, which only affects the NSC density profile at very small radii (≲ 3 pc). Within these central regions we expect that other dynamical processes (e.g., two-body relaxation, mass segregation) will modify the NSC density profile with respect to the simple predictions of our Monte Carlo experiments (we discuss this point in more detail below). When a cluster enters the inner 1 pc, its mass was redistributed using a Plummer sphere model of total mass m_t (1 pc) and core radius r_K.

In addition, we make the reasonable assumption that the distribution of field stars is constant when computing the relevant dynamical friction timescales. Given that the total mass in clusters is only 10% of the bulge mass initially, the density profile of the galaxy at intermediate and large radii (≳ 30 pc; e.g., see Figure 4 of Agarwal & Milosavljević 2011) and consequently the dynamical friction timescales are not significantly affected by such simplification. However, to approximately account for the mass that is progressively accumulated into the nucleus, the orbital decay of clusters was computed in the order of increasing migration time; then, for any inspiral, we added the quantity 3GM_acc(< r)/r³ to the terms in square brackets in Equation (23), with M_acc(< r) the accumulated mass within r due to clusters with shorter orbital decay times.

Finally, we computed the NSC density at any radius by summing up the contribution, Δρ(r), of all clusters that within 10¹⁰ yr reached that radius. After this time the total mass left in stellar clusters is ∼10⁷ M_☉, almost independent on the initial value of M_max. The initial mass of the globular cluster system is therefore reduced from 10% to 0.1% of the total bulge mass after one Hubble time due to dynamical dissolution of the less massive systems, and also due to massive clusters that inspiral into center of the galaxy and dissolve locally to form the stellar nucleus. Figure 1 gives the results of such calculations.

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In agreement with Agarwal & Milosavljević (2011) we found that the mass of the forming NSC is mostly (≳ 90%) composed of clusters with initial masses ≳ 0.1 × M_max, suggesting that the low-mass clusters scarcely contribute to NSC formation in galaxies. Instead, our results very much depend on the value of M_max, since only very massive clusters can arrive in the central regions of the galaxy in a reasonable time without being destroyed by the galactic tidal forces in the process.

In galaxy models without an MBH, NSC formation is more efficient, and even for M_max = 10⁶ M_☉ an NSC forms in the central few parsecs of the galaxy. If an MBH is present at the center of the galaxy, star clusters that come closer than r_disr from the center are disrupted and, as a result, the NSC density is limited inside this radius (left panels in the Figure 1). For an MBH mass of 4 × 10⁶ M_☉ and M_max = 10⁶ M_☉, no clusters penetrate radii smaller than ∼7 pc. From the lower-left panel in Figure 1 we can see that, in this case, the presence of an MBH will strongly inhibit the formation of an NSC. Taking M_max = 10⁷ M_☉ (upper-left panel) results in a smaller radius of the NSC core and higher central densities, since the most massive clusters can reach smaller galactocentric radii (∼2 pc).

A model that starts with a high-mass truncation of M_max = 10⁷ M_☉ reproduces the observed mass density profile of stars in the Galactic NSC (red-dotted lines in Figure 1) outside 3 pc. We conclude that a scenario in which a large fraction of the mass of the MW NSC is due to infalling globular clusters is in good agreement with current observational constraints.

The results presented here should be interpreted with caution when comparing the detail of the NSC density profile to observations. Our modeling includes a series of simplifications that can have some impact on the inner structure of the resulting NSC. For example, consideration of the internal dynamics and mass spectrum in the globular clusters could enable mass segregation which can increase σ_K and then allow the cluster stars to reach much smaller radii. A realistic treatment of these effects will require careful N-body simulations and is beyond the scope of this paper. Nevertheless, we note that the results of our computations that include a central MBH are very similar to those obtained in Antonini et al. (2012) via N-body simulations. In both cases, the density profile that results after the final inspiral event is characterized by a core of roughly the tidal disruption radius of the most massive clusters, ∼3 pc, and an envelope with density that falls off as ρ ∼ r⁻². These properties are similar to those of the MW NSC, with the exception of the core size, which in the MW is somewhat smaller (∼0.5 pc). Merritt (2010) showed that such a core will shrink substantially via gravitational encounters in a time of 10 Gyr as the stellar distribution evolves toward a Bahcall–Wolf cusp (Bahcall & Wolf 1976). In our computations cluster inspiral occurs more or less continuously over the lifetime of the galaxy. The core resulting from the combined effects of cluster inspiral and relaxation would therefore be somewhat smaller than that we found above, and closer to the observed core size (we discuss this point in more details below in Section 6.1).

3.2.3. The Role of MBHs

Figure 2 shows the results of additional computations that explore the effect that varying M_• has on the structure of the NSC. In these plots we set M_max = 10⁷ M_☉ and we relate the mass of the galaxy to M_• through the relation log [M_•/M_☉] ≈ 8.4 + 1.9log [M_sph/7 × 10¹⁰ M_☉] (Graham 2012b). The galaxy velocity dispersion, σ, is derived from the M_• − σ relation (Ferrarese & Merritt 2000; Gebhardt et al. 2001), while the model scale length, r₀, is obtained from the effective radius: R_eff ≈ 1 kpc (M_sph/10¹⁰ M_☉)/(σ/100 km s⁻¹)², where here the normalization reproduces approximately the observed effective radii of elliptical galaxies (Forbes et al. 2008; Graham & Worley 2008). In the left panel of Figure 2 we show the increasing dominance of the central MBH over the NSC stars for more massive galaxies. Our results agree reasonably well with studies of galaxies at high-mass end with M_• > 10⁸ M_☉, and which exclude the presence of stellar nuclei in such systems (e.g., Häring & Rix 2004). The middle panel of Figure 2 displays how the total central mass in the nuclear component (i.e., MBH + NSC) divided by the stellar mass of the host spheroid varies with this latter quantity. Also in this case our results agree reasonably well with the observed correlation (dashed line in the plot).

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In order to reproduce the NSC masses obtained from observations, our model would require a high-mass truncation of the CIMF of M_max = 10⁷ M_☉ and that ≲ 10% of stars in galaxies originate in stellar clusters. Such a result is in agreement with the conclusion of Leigh et al. (2012) that the globular cluster infall model strongly underpredicts the observed NSC masses when assuming the present-day number of globular clusters in galaxies. This assumption might be incorrect, however, given that the number of globular clusters in a galaxy drops rapidly with time due to dynamical disruptions (e.g., Vesperini 1997, 1998; Fall & Zhang 2001; Gieles 2009). In addition, the cluster formation efficiency and the cluster disruption rate also vary substantially with cosmic time, peaking at early epochs, in gas-rich disk galaxies (Kruijssen et al. 2012). In fact, up to 30% of all stars in the universe could have been formed in bound stellar clusters (D. Kruijssen 2012).

The right panel of Figure 2 gives the density profile of an NSC that forms around an MBH of mass M_• = 10⁶ − 10⁸ M_☉ assuming an initial mass in clusters M_cl = 0.1 × M_sph. Smaller BH masses correspond to smaller tidal disruption radii for the infalling globular clusters. As a consequence, the NSC peak density becomes progressively larger and the NSC core radius smaller as M_• decreases. The mass delivered in the inner region of the galaxy is also a strong function of M_•. For M_• ≳ 10⁸ M_☉, very little mass (≲ 10⁶ M_☉) can be deposited in the inner ∼30 pc of the galaxy. In this case the NSC density profile remains below the stellar density of the background galaxy.

To summarize the results presented in this section, if we postulate that NSCs grow slowly through globular cluster migration and merging, then in order to form an NSC similar to that of the MW, our model will require M_max ∼ 10⁷ M_☉ and that ∼10% of the bulge mass originated in stellar clusters. We showed that this fraction reduces to roughly 0.1% of the total bulge mass after 10¹⁰ yr due to dynamical evaporation of the less massive clusters, and also because more massive clusters inspiral toward the GC and dissolve due to their interaction with the galactic tidal field.

The presence of a pre-existing MBH at the center of the galaxy can have a strong impact on the inner structure of the forming nucleus. If an MBH of mass M_• ∼ 10⁶ M_☉ sits at the center of the galaxy, massive globular clusters are disrupted at radii of the order of ∼3 pc and the NSC density profile will have a core of roughly this radius. MBHs less massive than about 10⁶ M_☉ do not have much influence the structure of the stellar nucleus during its formation since the tidal disruption radius of massive GCs is in this case of the order of, or less than, the core radius of the most massive clusters, and in either models, with or without MBH, the NSC density profile will have a central core of typical size r_k; in low-mass spheroids after a few Gyr due to two-body relaxation such a core will most likely relax to a Bahcall–Wolf cusp or, in the absence of MBH, undergo core collapse. The central peak density of the resulting NSC progressively declines as M_• increases. NSCs forming around MBHs with M_• ≳ 10⁸ M_☉ have such low central densities that they would be more difficult to observe as distinct galactic components; this appears to be in agreement with observations that reveal a lack of NSCs in stellar spheroids brighter than about 10^10.5 L_☉.

In the above discussion we have considered Dehnen's models with inner density profile slope dlog ρ/dlog r = −1. In Figure 3 we relax this assumption and explore the formation of NSCs in galaxy models with different values of γ. In these computations we adopt M_max = 10⁷ M_☉ and M_cl = 0.1 × M_sph. The structural parameters defining the galaxy model are M_sph = 10¹⁰ M_☉, R_eff = 1 kpc, and M_• = 0.

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On the basis of Equation (26) we would expect the final (after 10¹⁰ yr) densities in the inner ∼100 pc of the forming NSC to be lower in galaxy models with steeper density profiles, provided that the dynamical friction timescale of massive stellar clusters remains shorter than the Hubble time. The results illustrated in the upper panel of Figure 3 agree with this basic prediction, suggesting that, independently of γ, the sinking timescale of massive clusters in these models is short enough that they can always reach the center of the galaxy and form a compact nucleus.

The middle and bottom panels in Figure 3 compare the structural properties of our model NSCs to the properties of NSCs in early-type galaxies that were found to be nucleated in Côté et al. (2006). For each of the NSC models an approximation of its effective radius is obtained as the effective radius of the best-fitting Sérsic model to the NSC projected density profile within a galactocentric radius of 30 pc. Given R, the total NSC mass, M_NSC, is then obtained as the total mass within a radius twice the effective radius. A useful reference time is the relaxation time computed at R. Setting lnΛ = 12, and ignoring the possible presence of an MBH, the half-mass relaxation time is (Spitzer 1987)

$$\begin{equation} t_{\rm rh}=1.75 \times 10^5\frac{ [ r_h(\rm pc)]^{3/2} N^{1/2}}{(m/M_{\odot })^{1/2}}\,{\rm yr}, \end{equation} \tag{ 34 }$$

where N is the total number of stars and m is the mass of a single star (m = M_☉ in Figure 3). From Figure 3 it is evident that our NSC models have structural properties that are similar to those of real nuclei.

In the absence of a central MBH, the dynamical evolution of a nuclear cluster is a competition between core collapse, which causes densities to increase, and heat input from the surrounding galaxy, which causes the cluster to expand and densities to decrease (Kandrup 1990; Quinlan 1996; Merritt 2009). In a double-Plummer-law galaxy model, the maximum size of an NSC of mass M_NSC/M_sph = 10⁻³ in order to resist expansion is R/R_eff ≈ 0.02 (Quinlan 1996). This simple condition, when compared to the middle panel of Figure 3, implies that the core-collapse time of our models are always shorter than the time for the nucleus to absorb energy from the rest of the galaxy. However, when using more realistic Einasto profiles to describe the surrounding galaxy our NSC models appear to be close to the transition region between systems that are in the prompt core-collapse phase and systems for which the evolution is driven by heating from the galaxy. Our models will therefore either expand or contract depending on the "compactness" of the surrounding galaxy.

Stellar spheroids more luminous than ∼10^10.5 L_☉ often host high-mass BHs (M_• ≳ 10⁹ M_☉) at their center, while they show no evidence for nucleation. Rather, the density profile of such "giant" elliptical galaxies is observed to be flat inside the influence radius of the MBH (Kormendy 1987; Lauer et al. 2002). Two factors have been invoked to explain the observed lack of NSCs in such systems: (1) NSCs and MBHs grow in competition from the same gas reservoir (Escala 2007). If an MBH forms first this can prevent, through its feedback, an NSC from growing if the gas accretion rate is smaller than the Eddington rate (Nayakshin et al. 2009). (2) MBH binaries forming during the last galaxy–galaxy major merger destroyed their host NSCs by ejecting stars from the inner galactic regions (Bekki & Graham 2010).

Observations seem to point against both scenarios. Picture (1) seems to break down for NSC-dominated galaxies, given the extremely low accretion rates observed in bulgeless galaxies and that at least few of them also contain central MBHs. The model of Bekki & Graham (2010) is disfavored due to the fact that what sets NSC disappearance does not seem to be galaxy morphology. Rather, there is a limit to the MBH/NSC mass ratio that fixes a sharp transition from galaxies with NSC- and MBH-dominated galaxies (Neumayer & Walcher 2012).

Of course, it is possible that nuclei are absent in bright galaxies because some mechanism prevents them from forming in the first place, or because they did not have time to reform after they were destroyed by the scouring effect of binary BHs. In what follows, we show that NSCs might in fact be difficult to form in such galaxies, due to long dynamical friction timescales of star clusters and also the little mass that the clusters can deliver to the center due to the strong MBH tidal field. In order to do so, we repeat a similar analysis to that presented in Section 3.2.2 but for a galaxy like M87, a giant elliptical with no evidence for NSC. We stress that the problems discussed here and in Section 3.2.2 are substantially different. The influence radius of the MBH in a galaxy like M87 extends much further out (a few hundreds of parsecs from the center) than the tidal disruption radius of a massive globular cluster (∼10 pc), while for the MW we found r_disr ≲ r_infl. In the former case, the clusters will therefore move in a stellar background whose velocity distribution is directly influenced by the presence of an MBH, which will strongly affect the timescale for cluster inspiral as we now show.

The distribution of field-star velocities has the following form near an MBH:

$$\begin{equation} f(v_\star) =\frac{\Gamma (\gamma +1)}{\Gamma \left(\gamma -\frac{1}{2}\right)} \frac{1}{2^\gamma \pi ^{3/2}v_c^{2\gamma }} \big(2v_c^2 - v_\star ^2\big)^{\gamma -3/2}, \end{equation} \tag{ 35 }$$

where the normalizing constant corresponds to unit total number. This expression gives the local distribution of velocities at a radius where the circular velocity is v_c = (GM_•/r)^1/2, assuming that the density of field stars follows ${\rho }(r)=\tilde{\rho }(r/\tilde{r})^{-\gamma }$. The phase-space density is zero for v_⋆ ⩾ v_esc = 2^1/2v_c.

As γ → 1/2 the velocity distribution (Equation (35)) becomes progressively narrower, and for γ = 1/2 all stars have zero energy; in other words, the number of stars with v_⋆ < v_c goes to zero as γ approaches 1/2. In the case of a test particle moving in a circular orbit, the standard Chandrasekhar's formula will therefore predict zero frictional force. It turns out that in this situation the frictional force must be computed using a more general formulation that also includes the contribution from stars moving faster than the test particle:

$$\begin{eqnarray} \boldsymbol{f}_{\rm df} &\approx & \boldsymbol{f}^{(v_\star <v)}_{\rm df}+ \boldsymbol{f}^{(v_\star >v)}_{\rm df} \nonumber \\ &&= {-}4\pi G^2 m_{\rm cl} \rho (r) \frac{\boldsymbol{v}}{v^3} \times {\Big (} {\rm ln \Lambda }\int _{0}^v dv_\star 4\pi f(v_\star) v_\star ^2 \nonumber \\ &&+\int ^{\sqrt{-2 \phi (r)}}_v dv_\star 4\pi f(v_\star) v_\star ^2 \left[ {\rm ln} \left(\frac{v_\star +v}{v_\star -v} \right)-2\frac{v}{v_\star } \right] \Big), \nonumber \\ \end{eqnarray} \tag{ 36 }$$

where $\boldsymbol{v}$ is the velocity of the infalling cluster. The second term in parentheses represents the frictional force produced by stars moving faster than the massive particle. N-body experiments verify the accuracy of this formula (Antonini & Merritt 2012).

The fraction of stars that move slower than the local circular velocity can be computed as

$$\begin{eqnarray} I_{v_\star <v_c} &=& \int _{0}^{v_c} dv_\star 4\pi f(v_\star) v_\star ^2 \nonumber \\ &=& \frac{2}{\sqrt{\pi }}\frac{\Gamma (\gamma +1)}{\Gamma (\gamma -1/2)} \int _{1/2}^1dx\; x^{\gamma -3/2}\sqrt{1-x}, \end{eqnarray} \tag{ 37 }$$

This function varies smoothly from 0 when γ = 0.5 to 0.5 when γ = 2. For the fast moving stars an (ad hoc) approximation is

$$\begin{eqnarray*} I_{v_\star >v_c}&=& \int ^{\sqrt{-2 \phi (r)}}_{v_c} dv_\star 4\pi f(v_\star) v_\star ^2 \left[ {\rm ln} \left(\frac{v_\star +v_c}{v_\star -v_c} \right)-2\frac{v_c}{v_\star } \right] \nonumber \\ &\simeq &0.1721+0.5280\gamma -0.1812\gamma ^2 -529.2\exp (-14.46\gamma). \nonumber \end{eqnarray*}$$

For γ > 1, $I_{v_\star >v_c}\sim 0.53$, and when γ ∼ 2 the contribution from the slow stars is of order ∼lnΛ larger than the frictional force due to the fast moving stars.

Given these expressions, and setting $L=\sqrt{GM_{\bullet }r}$ in Equation (3) we have

$$\begin{eqnarray} &&{dr\over dt}={-}\frac{8\pi \sqrt{G} \tilde{\rho } \tilde{r} ^{\gamma }}{M_{\bullet }^{3/2}} \times [{\rm ln \Lambda } I_{v_\star <v_{\rm c}} +I_{v_\star >v_{\rm c}}]m_{\rm cl} r^{5/2-\gamma }. \nonumber \\ \end{eqnarray} \tag{ 38 }$$

This expression can be then used to compute orbits for both point-like and extended objects (e.g., a stellar cluster that experiences mass loss during inspiral) that move in a stellar cusp near an MBH. We identified our model with the center of a galaxy like M87. We adopted M_• = 3 × 10⁹ M_☉ (Macchetto et al. 1997; Gehbardt et al. 2011), a core velocity dispersion σ_h = 278 km s⁻¹ (Young et al. 1978; Lauer et al. 1992) and we used the relation σ²_h = 4πGρ_h(r_h/3)² with r_h = 600 pc to obtain the core density: ρ_h = 35 M_☉ pc⁻³. Taking γ = 0.5, the density at $\tilde{r}=r_{\rm infl}=300$ pc is $\tilde{\rho }=50\,M_\odot \,{\rm pc^{-3}}$.

For a test particle of mass m_cl, integrating Equation (38) yields

$$\begin{eqnarray} r(t)&=&\left [r_{{\rm in}}^{\gamma -3/2}-\frac{8\pi \sqrt{G} \tilde{\rho } \tilde{r} ^{\gamma }(\gamma -3/2) }{M_{\bullet }^{3/2}} \right. \nonumber \\ && \times [{\rm ln \Lambda } I_{v_\star <v_{\rm c}} +I_{v_\star >v_{\rm c}}] m_{\rm cl}\times t\Bigg]^{\frac{1}{\gamma -3/2}}, \end{eqnarray} \tag{ 39 }$$

for γ ≠ 3/2, and

$$\begin{eqnarray} &&r(t)=r_{{\rm in}}\exp \left(-\frac{8\pi \sqrt{G} \tilde{\rho } \tilde{r} ^{\gamma } }{M_{\bullet }^{3/2}} [{\rm ln \Lambda } I_{v_\star <v_{\rm c}} +I_{v_\star >v_{\rm c}}] m_{\rm cl}\times t \right) \nonumber \\ \end{eqnarray} \tag{ 40 }$$

for γ = 3/2.

Equation (39) corresponds to a characteristic decay time for angular momentum loss (for γ > 3/2):

$$\begin{eqnarray} \tau _{\bullet } &=& 4\times 10^8 \,{\rm yr}\frac{[{\rm ln \Lambda } I_{v_\star <v_{\rm c}} + I_{v_\star >v_{\rm c}}]^{-1}}{\gamma -3/2}\nonumber \\ && \times M_{\bullet,9.5}^{3/2}\tilde{\rho }_{50}^{-1} \tilde{r}_ {300}^{-3/2}m_{{\rm cl},6}^{-1}\left(\frac{r_{{\rm in}}}{\tilde{r}}\right)^{\gamma -3/2}, \end{eqnarray} \tag{ 41 }$$

where M_{•, 9.5} = M_•/3 × 10⁹ M_☉, $\tilde{\rho }_{50}=\tilde{\rho }/50 \, M_{\odot }\ {\rm pc^{-3}}$, and $\tilde{r}_ {300}=\tilde{r}/300\, {\rm pc}$. We note that, for γ < 3/2, τ_• becomes negative and, formally, Equation (38) gives an infinite decay time to the center. In this latter case the dynamical friction timescale can be re-defined as the time required to reach a radius which is some fraction, ζ, of its initial value: τ_• × (1 − ζ^{γ − 3/2}), with τ_• from Equation (41).

In the case of extended objects, we assume that the central properties of the globular cluster (i.e., σ_K and r_K) remain unchanged during inspiral. The cluster's limiting radius is given by Equation (23), which permits expressing the satellite mass as a function of radius. Setting m_cl = m_t in Equation (38) and assuming that the galactic potential is dominated by the MBH, we obtain

$$\begin{eqnarray} r(t)&=&\Bigg [r_{{\rm in}}^{\gamma -3}-\frac{2^{3/2}\pi }{\sqrt{3}}\frac{\tilde{\rho }\tilde{r}^\gamma (\gamma -3)}{GM_{\bullet }^2}\nonumber \\ && \times [{\rm \ln \Lambda } I_{v_\star <v_{\rm c}} +I_{v_\star >v_{\rm c}}]\sigma _K^3\times t\Bigg ]^{\frac{1}{\gamma -3}}, \end{eqnarray} \tag{ 42 }$$

and

$$\begin{eqnarray} \tau _{\bullet }&=&5\times 10^{9} \,{\rm yr} \frac{[{\rm ln \Lambda } I_{v_\star <v_{\rm c}} +I_{v_\star >v_{\rm c}}]^{-1}}{3-\gamma } (\zeta ^{\gamma -3}-1) \nonumber \\ && \times M_{\bullet,9.5}^{2} \tilde{\rho }_{50}^{-1}\tilde{r}_{300}^{-3} \sigma _{K,10}^{-3} \left(\frac{r_{{\rm in}}}{\tilde{r}}\right)^{\gamma -3}. \end{eqnarray} \tag{ 43 }$$

We used Equations (39) and (42) to trace the orbital evolution of massive stellar clusters in the core of our galaxy model. We take a limiting value for the mass of m_K = 4 × 10⁶ M_☉, corresponding to the non-truncated model, when the cluster is far from the center. The simulated orbits (upper panel of Figure 4) demonstrate that the dynamical friction timescale in the core of bright elliptical galaxies is extremely long. Also very massive clusters (σ_K ≳ 30 km s⁻¹) starting well within the galaxy core (r_in ∼ 300 pc) do not reach the center after one Hubble time.

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In the lower panel of Figure 4, we use Equations (23) and (26) to compute the density profile of an NSC resulting from the accumulation of 500 equally massive globular clusters in the center of our M87 galaxy model. At radii smaller than r_disr (marked by black arrows in the plot) the clusters are fully disrupted by the interaction with the MBH and our integrations are no longer valid. N-body simulations show that the density profile of stars will be very flat or even declining toward the MBH inside r_disr, as the stars accumulate near the radius of disruption (Kim & Morris 2003; Fujii et al. 2009, 2010).

In order to highlight the role of the central MBH in determining the structure of the growing nucleus, Figure (4) also shows the density profile of the nucleus when the mass of the MBH is set to zero. In this case, most of the cluster mass is delivered in the inner ∼100 pc and consequently the NSC is much more centrally concentrated. After 500 inspiral events, the NSC appears to be distinct from the galaxy background density profile. Apparently, NSCs assembled around MBHs have a spatially more diffuse configuration and lower densities than those forming in galaxies with no MBH. We conclude that the formation of NSCs in giant ellipticals might be inhibited by the presence of their central MBH for two basic reasons: (1) the long dynamical friction timescale of massive objects in the galaxy core and (2) clusters are disrupted by the strong tidal field of the MBH producing a merger remnant with density profile that rises only modestly above that of the background galaxy.

We note in passing that the latter argument may also apply to the case in which a giant elliptical galaxy accretes a smaller galaxy containing a dense central nucleus. The large cores observed in the central light profile of bright ellipticals are usually interpreted in terms of the scouring effect of MBH binaries forming during major merger events (Milosavljević & Merritt 2001). It is less clear, however, how such cores can be preserved up to the present epoch despite the large number of minor mergers that are predicted to occur by standard cosmological models. For instance, an M87-like galaxy is expected to have accreted few galaxies of the size of the MW in the last ∼5 Gyr (Fakhouri et al. 2010). A plausible explanation for the lack of central dense stellar concentrations in the brightest galaxies was provided by Merritt & Cruz (2001). These authors performed high-resolution N-body simulations of the accretion of high-density dwarf galaxies by low-density giant galaxies. They found that the cusp of the secondary galaxy is disrupted during the merger by the giant galaxy MBH tidal field, producing a remnant with a central density that is only slightly higher than that of the giant galaxy initially; removing the BH from the giant galaxy allowed the smaller galaxy to remain essentially intact and led to the formation of a high central density cusp, contrary to what is observed.

6.1. Long-term Evolution of NSCs around MBHs

In the previous sections we showed that NSCs resulting from the accumulation of globular clusters around MBHs have lower central densities and larger cores than those forming in galaxies without MBHs. Such a result is consistent with the results obtained using more sophisticated N-body simulations (Antonini et al. 2012).

Our analysis, however, did not account for the effects of internal dynamics occurring in the NSC during and after its formation. In a pre-existing NSC, the presence of an MBH would inhibit core collapse, causing instead the formation of a Bahcall & Wolf (1976) cusp on the two-body relaxation timescale, followed by a slow expansion as stars are tidally disrupted (Merritt 2009). Whether or not a Bahcall–Wolf cusp will form depends on the initial core size of the nucleus and on its relaxation time.

For nuclei similar to the MW NSC, relaxation times are too long to assume that they have reached such a collisionally relaxed state, but they are still sufficiently short that gravitational encounters would substantially affect their structure over the age of the galaxy. Two-body gravitational interactions will cause the central density of the nucleus to increase and the core to shrink as the stellar distribution evolves toward the Bahcall–Wolf form (e.g., Preto et al. 2004). The time evolution of the core radius can be approximately described by the expression (Antonini et al. 2012)

$$\begin{equation} r_{\rm core}(t)=1.57\, {\rm pc} \exp \left[ t/0.25t_{\rm relax} \right], \end{equation} \tag{ 44 }$$

where t_relax is the relaxation time computed at the radius of influence of the MBH. For the MW, t_relax ∼ 20 Gyr and M_• ≈ 4 × 10⁶ M_☉. A BH of this mass corresponds to a core of ∼3 pc (Figure 1). From Equation (44), we see that gravitational encounters occurring during and after the formation of the NSC will reduce the core size to ∼1 pc after 10¹⁰ yr. Such a value is more consistent with the size of the core observed in the distribution of late-type (old) stars in the inner parsec of the MW (Buchholz et al. 2009; Do et al. 2009; Bartko et al. 2010; Yusef-Zadeh et al. 2012; Nadeen et al. 2012). Whether NSCs in other "power-law" galaxies will turn out to have stellar cores similar to that of the MW remains to be seen.

6.2. The Distribution of Stellar Remnants Near MBHs

6.3. Dissipative NSC Formation

6.4. NSCs Morphology and Kinematics

6.5. The Lack of NSCs in Faint Galaxies

In this paper we considered a dissipationless formation model for NSCs where globular clusters orbital decay and merge at the center of a galaxy to form a compact nucleus. Our main results are summarized below.

• 1.  
  The observed scaling relation between NSC masses and the velocity dispersion of their host spheroids, the M_NSC − σ relation, is difficult to reconcile with a purely dissipative formation model for the nuclei. These models predict that M_NSC is a steeply rising function of σ. The observed M_NSC − σ relation is instead in agreement with the predictions of a dissipationless formation model. Dissipationless formation modes produce relations that are substantially shallower than the corresponding MBH scaling relations and are therefore more consistent with observations.
• 2.  
  The globular cluster merger model, in the absence of a central MBH, naturally reproduces the observed relation between the size of galactic nuclei and their total luminosity, $R\propto \sqrt{\mathcal {L}_{\rm NSC}}$. When an MBH is present, the dependence of the NSC radius on its mass is substantially weaker than the observed relation because the size of the NSC is mainly determined by the fixed tidal field of the MBH.
• 3.  
  We derived explicit expressions for the orbits of globular clusters owing to dynamical friction and subject to mass loss due to their tidal interaction with the galaxy (Equation (31)) or with a central MBH (Equation (42)). These expressions were used to (1) address the possibility that NSCs in galaxies similar to the MW could have been assembled via cluster migration and mergers; and (2) to follow the orbital evolution of globular clusters in the core of bright elliptical galaxies.
• 4.  
  NSCs that form through the mergers of globular clusters have a density profile characterized by a parsec-scale core and an envelope that falls off as ρ ∼ r⁻². These properties are similar to those of the MW NSC. An NSC with mass comparable to that of the MW NSC is obtained by assuming an initial total mass in stellar clusters which is consistent with the cluster formation efficiency inferred from observational studies of embedded clusters in Galactic molecular clouds (Figure 1).
• 5.  
  A pre-existing MBH at the center of the stellar spheroid has a strong impact on the structure of the growing stellar nucleus (Figure 2). Tidal stresses from an MBH disrupt the clusters when they pass within their tidal disruption radius, r_disr, limiting the density within that radius. Hence, the density profile of the resulting NSC will also have core size of ∼r_disr. Removing the MBH from the galaxy allows the stellar clusters to keep their inner structure almost unchanged leading to the formation of an NSC with higher peak densities. We find that separating the contribution of the nucleus from that of the galaxy could more difficult in stellar spheroids with more massive MBHs.
• 6.  
  For globular clusters orbiting in the core of a massive elliptical galaxy like M87, the timescale to reach the center is much longer than one Hubble time (Figure 4). The essential reason for this is that the phase-space density of a shallow density cusp of stars around an MBH falls to zero at low energies: inside the galaxy core there are no stars at any radius that move slower than the local circular velocity. The standard Chandrasekhar's formula predicts little or even no frictional force in this case. In addition, when an MBH is included, the tidal field near r_infl becomes much stronger, resulting in faster mass loss and fewer stars deposited to the center. Based on these facts, we conclude the presence of central MBHs might inhibit the formation of compact nuclei in the brightest galaxies, in agreement with what is observed.

I am grateful to D. Merritt for numerous discussions about the ideas presented in this paper and for his detailed comments to an earlier version of this manuscript. I thank R. Capuzzo-Dolcetta for encouraging me to study this problem, D. Kruijssen and A. Madigan for useful discussions, and M. Alvarez and A. Graham for their comments to an earlier version of this paper. This material is based upon work supported in part by the National Science Foundation Grant No. 1066293 and the hospitality of the Aspen Center for Physics.