UPDATED MASS SCALING RELATIONS FOR NUCLEAR STAR CLUSTERS AND A COMPARISON TO SUPERMASSIVE BLACK HOLES

GS09 provide stellar masses for their nuclei, whereas F06 and BGP07 tabulate only magnitudes. For the F06 objects we derived nuclear stellar object masses following F06. We multiplied the total CMO g-band magnitude by a mass-to-light ratio, M/L, determined from the single stellar population models of Bruzual & Charlot (2003), using the nuclear cluster colors given in Côté et al. (2006) and a stellar population age t = 5 Gyr. For the BGP07 objects we derived masses following BGP07, by multiplying the total CMO K-band magnitude by M/L_K = 0.8 (Bell & de Jong 2001) based on typical colors of the bulge population. The uncertainties on the nuclear object masses for the F06, BGP07, and GS09 data are given by the respective authors as 45%, 33%, and a factor of two, respectively. In passing, we note that if NCs are related to ultra compact dwarf galaxies (e.g., Kroupa et al. 2010) they may have a high stellar M/L due to either a bottom-heavy (Mieske et al. 2008) or top-heavy initial mass function (Dabringhausen et al. 2009).

BGP07 distinguish between extended nuclear components (11 objects) and unresolved nuclear components (also 11 objects—5 galaxies contain both a resolved and unresolved nuclear component), finding that the extended components are well fit with an exponential profile and are thus likely to be nuclear disks (or possibly nuclear bars), whereas the unresolved components are probably nuclear star clusters. They revealed that the disks and clusters follow quite different relations, in terms of, for example, how the nuclear disk luminosity scales with host galaxy σ. For this reason it is important to distinguish between nuclear disks and clusters when examining the scaling relations of nuclear objects with their host galaxy.

F06 identified three of their objects as containing small-scale stellar disks. Based on their published HST surface photometry we identify one further object, NGC 4550 (VCC 1619) as likely containing a nuclear stellar disk. These four nuclear disks are also the most extended nuclear components in the F06 sample, having half-light radii ranging from 26 to 63 pc (for comparison, the mean half-light radii for the F06 nuclear objects is 4 pc). This provides final samples of 76 and 15 nuclear clusters and nuclear disks, respectively, from the galaxy samples of F06, BGP07, and GS09. Five galaxies contain both a nuclear star cluster and a nuclear disk, and thus there are 86 unique galaxies with a stellar CMO.

We were not able to obtain every galaxy and spheroid property for every object from the literature. The number of objects for which we were able to obtain a given property is indicated in the final row of Table 1. Velocity dispersions were obtained from F06, Falcón-Barroso et al. (2002, for the BGP07 galaxies), and Graham et al. (2011, for the GS09 galaxies), giving 51/76 nuclear star cluster galaxies and 15/15 nuclear disk galaxies having measured σ. The velocity dispersions were measured in inhomogeneous apertures and we do not attempt to correct the measurements to a common aperture here. The F06 σ values were measured from long-slit observations within a 1 R_e aperture. The values from Falcón-Barroso et al. (2002) were obtained within a central 1.1 arcsec² aperture and corrected to a "standard" aperture defined to be equivalent to a circular aperture with radius 1.7 arcsec at the distance of Coma (as established by Jorgensen et al. 1995). The σ values presented in Graham et al. (2011) were originally drawn from the HyperLeda database (Paturel et al. 2003)² and represent a disparate set of measurements corrected to the same "standard" aperture as in Falcón-Barroso et al. (2002). We adopt an uncertainty of 10% for all σ measurements.

We considered correcting the standard aperture measurements to R_e measurements based on Equation (1) of Cappellari et al. (2006), from which we derived a mean correction to the σ measurements reported here of 2.6% (ranging from +4% to −6% for individual objects). However, the error on the derived correction for individual objects is significant, ∼10%, and much larger than the typical correction of ∼2.5% for a given measurement. The correction is not correlated with host galaxy σ, and hence will not introduce a systematic error in our uncorrected σ values. Given this uncertainty, and that we were unable to derive an aperture correction for all our objects due to missing R_e measurements, we opted not to apply any aperture correction.

We obtained total apparent B-band magnitudes for all galaxies following F06. For the BGP07 and GS09 galaxies we obtained m_B from de Vaucouleurs et al. (1991, RC3). For the F06 galaxies we obtained m_B from Binggeli et al. (1985), reduced to the RC3 system using the relation given in the HyperLeda database. We note that this approach fails to fully correct for dust in disk galaxies (see Graham & Worley 2008). We therefore also obtained total K-band magnitudes, m_K, for 80/86 galaxies from the Two Micron All Sky Survey (2MASS) Extended Source Catalogue (Jarrett et al. 2000). To convert to absolute magnitudes we used the distances from Mei et al. (2007) for the F06 sample, and from BGP07 and GS09 for the corresponding galaxies. We adopt an uncertainty of 0.25 mag for all absolute magnitudes.

We derive dynamical masses using the simple but popular virial estimator M_dyn = ασ²_eR_e/G, where R_e is the effective half-light radius and σ_e the luminosity-weighted velocity dispersion measured within a 1 R_e aperture. Following F06, we used a value of α = 5 for the F06 galaxy sample. The virial factor α can take on a range of values (Bertin et al. 2002) depending on the radial mass distribution. By comparing virial estimator derived masses to the results of more sophisticated dynamical models, Cappellari et al. (2006) found that, in practical situations when working with real data, α = 5 provides a virtually unbiased estimate of a galaxy's dynamical mass within 1 R_e. They found this to be true for galaxies with a broad range of Sérsic indices, n = 2–10 and bulge-to-total ratios, B/T ∼ 0.2–1.0. They caution (p. 1140), however, that their result "strictly applies to virial measurements derived ... using 'classic' determination of R_e and L via R^1/4 growth curves, and with σ_e measured in a large aperture." Based on their findings we conclude that the virial estimator is a reasonable approximation of the dynamical mass for galaxies of types Sa and earlier—we do not determine M_{Gal, dyn} for galaxies of morphological type Sb or later, as these are heavily disk-dominated systems for which the virial estimator has not been calibrated.

For the BGP07 and GS09 objects we use R_e values from the RC3 (which are determined from R^1/4 curve-of-growth fits to the surface brightness profile). For the F06 objects we use R_e values from Ferrarese et al. (2006b) which are derived from Sérsic R^1/n fits to the observed surface brightness profile. For these 51 objects F06 report a range in n from 0.8 to 4.6 (with 78% of galaxies with n in the range 1–2.5). We compared Sérsic-based R_e,s for the subset of F06 galaxies for which RC3 R^1/4-based R_{e, deV} were available (22 objects) and found a one-to-one correlation, with the F06 R_e,s being systematically 30% ± 7% larger than the R_{e, deV} values. For these comparison galaxies n ranged from 1.1 to 4.6 (with 72 % of galaxies with n in the range 1–2.5), representative of the full 51 objects. This suggests that, after we apply this correction to the F06 R_e,s (R_{e, deV} = 0.77R_{e, F06}), the use of Sérsic fit based R_e,s will not significantly bias M_{Gal, dyn} for the F06 objects. While we find good agreement for this small sample of galaxies for the specific methods used to determine R_e by the respective authors, we caution that in general Sérsic and R^1/4 based R_e typically show significant differences (Trujillo et al. 2001). After excluding disk-dominated galaxies of Sb type or later we were able to derive M_{Gal, dyn} for 48/86 galaxies; all objects for which a σ and R_e measurement was available. Typical errors on M_{Gal, dyn} are ∼50%.

We additionally determine stellar masses for the full galaxy, M_{Gal, *}, and for the spheroidal component, M_{Sph, *}. To determine M_{Gal, *} we multiplied the total galaxy luminosity of each object by the appropriate mass-to-light ratio. For all objects we used M_K from 2MASS (excluding galaxies with no 2MASS M_K total magnitude) and, following BGP07, assumed a standard mass-to-light ratio, M/L_K = 0.8 (see also Bell & de Jong 2001). M_{Gal, *} was determined for 80/86 galaxies. All magnitudes and colors were corrected for Galactic extinction following Schlegel et al. (1998). The data were not corrected for internal extinction, though we note that for most galaxies M_K and M/L_K are minimally affected by dust extinction.

M_{Sph, *} was determined by multiplying the total spheroid magnitude of each object by the appropriate mass-to-light ratio as described above. We use the spheroid masses provided by GS09 for their galaxies. We adopt the same spheroid masses as BGP07, obtained by multiplying their K-band spheroid magnitudes by M/L_K = 0.8. For the F06 galaxies we use the galaxy stellar masses described above, excluding 16 galaxies classified as S0 or dS0 as they are likely to contain a large scale disk and no bulge-disk decomposition is available. This resulted in M_{Sph, *} for 66/86 galaxies—all galaxies for which a spheroid mass or spheroid magnitude and an optical color were available. Typical errors on M_{Gal, *} are ∼25% and on M_{Sph, *} ∼ 40% due to the increased uncertainty in separating the spheroid component of the galaxy's light.

To compare to our nuclear star cluster sample, we take the SMBH sample of Graham et al. (2011). This sample consists of 64 galaxies with directly measured SMBH masses. The host galaxy velocity dispersions for this sample are presented in Graham (2012b), the host galaxy B- and K-band luminosities in Graham & Scott (2012) and the distance to each object in Graham et al. (2011).

We also determine derived quantities, M_{Gal, dyn}, M_{Gal, *}, and M_{Sph, *} for the SMBH host galaxies following the approach for the nuclear star cluster host galaxies described above. Briefly, we derive M_{Gal, dyn} from the Virial estimator, using the velocity dispersions from Graham (2012b) and R_e from the RC3. This allowed us to derive M_dyn for 40/64 galaxies. We derive galaxy stellar masses, M_{Gal, *} for the SMBH galaxies as for the nuclear star cluster galaxies, using the galaxy K-band magnitude and an assumed M/L = 0.8. We derive M_{Gal, *} for 59/64 galaxies—all objects with an available K-band magnitude. To derive spheroid stellar masses we make use of the spheroid magnitudes, m_Sph, presented in Marconi & Hunt (2003) (with the exception of NGC 2778 and NGC 4564; see Graham 2007) and Häring & Rix (2004).³ These were then multiplied by an appropriate M/L determined using the relations presented in Bell et al. (2003), with optical colors obtained from the HyperLeda database. We determined M_{Sph, *} for 39/64 SMBH galaxies—all objects with an available m_Sph and optical color.

We have derived scaling relations connecting the nuclear star cluster mass to various properties of their host galaxies: B- and K-band luminosity, M_B and M_K, velocity dispersion σ, galaxy dynamical mass M_{Gal, dyn}, and galaxy and spheroid stellar masses, M_{Gal, *} and M_{Sph, *}. These linear scaling relations are presented in Table 2. At present, given the relatively small samples and the significant errors on the derived nuclear star cluster masses, there is no compelling reason to fit more complicated broken or nonlinear scaling relations to our data, though future studies with improved data may reveal additional complexity. Our Figure 1 builds on Figure 2 from F06 by presenting linear fits of M_NC against: host galaxy B-band magnitude M_B, velocity dispersion σ, and virial mass M_{Gal, dyn}. In Figure 2, we present fits of M_NC against M_K, M_{Gal, *,} and M_{Sph, *}.

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We find a slope of 2.11 ± 0.31 for the M_NC–σ relation. This is significantly shallower than that reported by F06 (4.27 ± 0.61), though in better agreement with recently reported slopes of 1.57 ± 0.24 and 2.73 ± 0.29 (Graham 2012a; Leigh et al. 2012). The principal difference between the F06 study and the recent findings of a shallower slope of ∼2 is the inclusion of nuclear star clusters in more massive galaxies with σ > 200 km  s⁻¹ (the F06 sample was limited to nuclear star clusters in host galaxies with σ < 150 km  s⁻¹). The exclusion of NDs from our fits (open blue symbols in all figures), which are typically an order of magnitude more massive than NCs, also contributes to our flatter slope, and accounts for the difference between our slope and that of Leigh et al. (2012).

We find a good correlation between M_NC and host galaxy luminosity in both the B and K band, with M_NC∝L^{∼0.6 ± 0.1}_K. We find a strong correlation of M_NC with M_{Gal, *} and a somewhat weaker correlation with M_{Sph, *}—this is consistent with the findings of Erwin & Gadotti (2012), though we find a smaller difference between the strength of the correlations than they report (0.72 and 0.65, compared to their 0.76 and 0.38).

We find a shallow slope of ∼0.5 for the M_NC–M_{Gal, dyn} relation, which is significantly flatter than the slope 1.32 ± 0.25 reported by F06. We again attribute this difference to the inclusion of many more massive galaxies in our sample (though we caution that the Virial-estimator based dynamical masses used here and in F06 have significant errors). Bearing in mind that the relations were not constructed to minimize the scatter in the M_CMO direction, the M_NC–M_{Gal, dyn} relation has the lowest rms scatter (in the vertical M_CMO direction) of any of the NC scaling relations (though it is not significantly tighter than either the M_NC–M_K or M_NC–M_{Gal, *} relations).

In this subsection, we derive a set of six scaling relations involving black hole masses. Except for the M_BH–σ relation, which involves galaxies of all types and is essentially a copy from Graham et al. (2011), due to available data these relations predominantly involve massive galaxies and spheroids. As such we have not included the developments which reveal a bent nature to the other five relations at lower masses (e.g., Graham 2012a; Graham & Scott 2012). However, these "bends" are such that the lower mass systems define steeper relations than shown here, which only emphasises the differences with the NC scaling relations discussed in the following section.

We emphasise here that most of the relations we derive for SMBH are for comparison only and do not represent the state-of-the-art in SMBH scaling relations. For this reason we derive only simple linear fits to the available SMBH sample for each variable. We do not distinguish between barred and unbarred galaxies (see Graham 2008, for a discussion of the offset nature of barred galaxies in the SMBH–σ relation), nor do we fit broken relations that better describe the scaling of SMBH mass with host luminosity (Graham & Scott 2012) or mass (Graham 2012a). Whether the M_BH–M_{Gal, *} and M_BH–M_{Sph, *} relations are also bent is beyond the scope of this work but will be addressed in a future paper in this series.

The linear relations we derive are presented in Table 2 and are shown as the thick black lines in Figures 1 and 2. While we emphasise again that these linear SMBH scaling relations are for comparison only, we briefly discuss their consistency with similar scaling relations presented in the literature. We note that our M_BH–σ relation has a slope ∼6, whereas a slope ∼4–5 had typically been reported in the literature (Merritt & Ferrarese 2001; Tremaine et al. 2002). However, our SMBH sample now includes a significant number of barred galaxies which, as Graham (2008) first observed, are offset from the unbarred relation. As Graham et al. (2011) noted, including barred galaxies in one's sample increases the slope of the M_BH–σ relation—Graham et al. (2011) and Graham (2012b) report a slope of 5.95 ± 0.44 and 5.76 ± 1.54, respectively, for their full samples of both barred and unbarred galaxies, consistent with our value. While the value we report is biased by the inclusion of the barred galaxies, their inclusion is appropriate as we do not distinguish between barred and unbarred galaxies in the corresponding M_NC–σ relation.

The M_BH–M_{Gal, dyn} and M_BH–M_{Sph, *} relations we present, with slopes ∼1, are typical of those reported in the literature (e.g., Marconi & Hunt 2003; Häring & Rix 2004). The correlation with spheroid stellar mass is significantly tighter than with total stellar mass, consistent with other studies (e.g., Kormendy 2001; McLure & Dunlop 2002). Our relations are dominated by SMBHs with M_BH ≳ 10⁸ M_☉, and Graham (2012b) has shown that above this rough threshold the M_BH/M_{Gal, dyn} ratio is fairly constant, while at lower masses the M_BH/M_{Gal, dyn} ratio is not constant but increasingly smaller, a result which can also be seen in our Figure 1, where the data points fall below the extrapolation of the solid black line at lower masses, causing the steepening of our M_BH–M_{Gal, dyn} relation. We note that the correlation between M_BH and host galaxy luminosity for the full sample is poor, with Spearmann r coefficients ∼  − 0.4. However, if we include only purely spheroidal systems (E and dE) this correlation improves markedly (Spearmann r = −0.81 and −0.70 in the B and K bands, respectively), which is again unsurprising given that M_BH is known to correlate with the properties of the spheroid.

In contrast to F06 and WH06 we find that NCs and SMBHs do not follow common scaling relations. In all six diagrams that we have considered, the NCs and SMBHs appear to follow different relations (see Table 2). Our most significant finding is that the M_NC–σ relation (with a slope ∼2) is significantly flatter than the M_BH–σ relation (with a slope ∼6 for all galaxies, or ∼5 for barless galaxies: Graham et al. 2011). This is in agreement with Graham (2012b), but in contrast to F06 who find an M_NC–σ relation parallel to the M_BH–σ relation. The difference is due to: (1) the exclusion of NDs from our NC sample and (2) the inclusion of NCs that have masses higher than the SMBH/NC threshold of 10⁷ M_☉ suggested by WH06. NDs are significantly more massive than NCs in comparable host galaxies and follow significantly different scaling relations (Balcells et al. 2007). Scorza & van den Bosch (1998) showed that NDs follow galaxy-scale stellar disk scaling relations, extending those relations to much lower mass.

In the middle panel of Figure 1, at a CMO mass of ∼10⁷ M_☉, NCs are, on average, found in galaxies of significantly lower σ and M_{Gal, dyn} than SMBHs of the same mass. Graham (2012a) has revealed that the M_BH–M_{Gal, dyn} relation steepens from a slope of ∼1 at the high-mass end (M_BH ≳ 2 × 10⁸ M_☉) to a slope of ∼2 at lower masses (our slope of 1.37–1.55 is intermediate to these values because we fit a single linear relation to the high- and low-mass ends of what is a bent relation). The steepening in slope at the low-mass end is in the opposite sense to that observed for the NCs, which have a flatter slope of 0.55 ± 0.15.

When considering the scaling of M_CMO with the stellar mass content of its host we find that M_NC appears to be driven by the total stellar mass, whereas M_BH is more closely associated with only the spheroidal component. Combining this finding with the result that M_NC follows much flatter relations with σ and M_{Gal, dyn} than M_BH does, suggests that the physical processes that lead to the build-up of a nuclear stellar cluster may be significantly different to those that drive the formation of SMBHs. A complementary view is provided by Figure 3, where we plot the NC mass fraction as a function of host galaxy dynamical mass and spheroid stellar mass, i.e., M_NC/M_{Gal, dyn} versus M_{Gal, dyn} and M_NC/M_{Sph, *} versus M_{Sph, *}. The ratio M_NC/M_{Gal, dyn} shows a clear trend with M_{Gal, dyn}, in the sense that the NC mass fraction decreases smoothly with M_{Gal, dyn}. We note that M_NC/M_{Sph, *} is also not constant, spanning ∼2 orders of magnitude from 0.02% to 2% (see also Figure 7 of Turner et al. 2012).

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The GS09 sample of galaxies contain both an NC and an SMBH. It is likely that the SMBH and NC in these galaxies interacted in some way during their formation, and hence additional physical processes may have influenced their scaling with their host galaxy. It is unclear what exactly the result of any interaction may be: it has been suggested that (1) the presence of a single SMBH may evaporate the NC (Ebisuzaki et al. 2001; O'Leary et al. 2006), (2) a binary SMBH may heat and erode the NC (Bekki & Graham 2010), and that (3) some SMBHs are, in part, built up by the collision of NCs (Kochanek et al. 1987; Merritt & Poon 2004). Additionally, given the existence of some dual-CMO galaxies, it is likely that some of the supposed NC- or SMBH-only galaxies in our sample contain an undetected SMBH or NC, respectively. Because of this it is unclear whether it is correct to include or exclude the GS09 galaxies from our main NC sample. More importantly, while including the GS09 galaxies does affect the NC scaling relations our conclusions do not depend on whether or not we include them.