The Black Hole–Bulge Mass Relation Including Dwarf Galaxies Hosting Active Galactic Nuclei

Before fitting the 2D light profiles of our target galaxies in GALFIT, we created point-spread functions (PSFs) to model the unresolved light coming from the central AGNs. We used StarFit (Hamilton 2014), which accounts for changes in the PSF due to telescope breathing and the changing position of sources on the detector between different observations. StarFit creates a TinyTim PSF model (Krist 1995) and matches the focus of the telescope by fitting the model to a source in the observation field (in our case, the central bright pixel of each galaxy). This process is performed on each individual frame in the dither pattern, and the resulting PSF models are drizzled together with the same parameters as the observations. We found that the PSFs generated in this manner worked well for our purposes, matching the increase in central light from the AGN accurately and allowing the other components to be fit freely. We used StarFit to create the PSFs for images taken in the F110W and F606W filters for each galaxy.

We approach the fitting process by initially modeling galaxies in the F110W images with a PSF to account for the light from the AGN and a single Sérsic component for the galaxy, in addition to a flat background sky. With this initial model, we were able to obtain reasonable estimates for the magnitudes of the PSF components and the sizes of the galaxies. During this initial modeling step, we also account for light from foreground stars and other objects in the images that are located close to the target galaxy. Dim objects are fit with a single Gaussian component to remove any light contamination to the galaxy of interest. In the case of RGG 9, there is a foreground star that is too bright to be robustly accounted for using a Gaussian component. In this case, we create a masked region following the procedure presented in the GALFIT manual (Peng et al. 2010), which adequately removes the excess light and allows for robust modeling of the galaxy.

We then applied a PSF, inner Sérsic, and outer Sérsic model to each galaxy, drawing on the information from the simpler model. We find that a three-component decomposition results in a significantly better fit than a two-component decomposition for five of the seven galaxies in our sample (e.g., ${\chi }_{\nu }^{2}$ reduced by ∼20% or more). The first exception is RGG 9, with a single Sérsic component and a PSF providing an adequate fit (${\chi }_{\nu }^{2}=3.283$). The second exception is RGG 127, which requires the inclusion of a bar (modeled with an additional Sérsic component with n ∼ 0.5). Inclusion of a bar allows for much cleaner residuals and an acceptable ${\chi }_{\nu }^{2}$ of 1.737.

In the case of RGG 48, the resolution of the F110W filter is not high enough for GALFIT to consistently settle on a three-component model. This is due to the presence of many asymmetric structures (e.g., spiral arms, stellar ring, bar) that are not sufficiently resolved in the F110W image, resulting in GALFIT models that are not robust. We therefore develop our GALFIT model for this system using the F606W image with superior resolution, which allows GALFIT to converge on a stable three-component model. The final GALFIT model parameters are shown in Table 2.

Table 2.  GALFIT Model Parameters (F110W)

ID	Component	mF110W	n	Re	q	PA	${\chi }_{\nu }^{2}$	Additional
				(kpc)		(deg E of N)		Components
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
	PSF	21.86 ± 0.02	⋯	⋯	⋯	⋯
RGG 1	Inner	19.52 ± 0.23	0.32 ± 0.09	0.71 ± 0.03	0.51	25.48	1.043	⋯
	Outer	17.41 ± 0.06	0.83 ± 0.13	1.61 ± 0.03	0.72	25.17
	PSF	19.80 ± 0.06	⋯	⋯	⋯	⋯
RGG 9	Inner	17.01 ± 0.04	2.30 ± 0.11	1.21 ± 0.26	0.86	0.26	3.283	⋯
	PSF	19.61 ± 0.03	⋯	⋯	⋯	⋯
RGG 11	Inner	18.40 ± 0.23	2.40 ± 0.18	0.13 ± 0.02	0.96	−90.73	1.484	⋯
	Outer	16.22 ± 0.18	1.69 ± 0.09	2.57 ± 0.30	0.78	−19.39
	PSF	18.45 ± 0.03	⋯	⋯	⋯	⋯
RGG 32	Inner	17.77 ± 0.25	1.62 ± 0.20	0.29 ± 0.02	0.90	−14.13	1.091	⋯
	Outer	16.07 ± 0.10	0.74 ± 0.03	2.03 ± 0.01	0.95	−72.75
	PSF	21.55 ± 0.01	⋯	⋯	⋯	⋯
RGG 48	Inner	19.75 ± 0.07	0.61 ± 0.08	0.29 ± 0.06	0.42	−33.85	1.363	⋯
	Outer	16.64 ± 0.06	0.29 ± 0.01	2.12 ± 0.30	0.48	−30.68
	PSF	18.94 ± 0.07	⋯	⋯	⋯	⋯
RGG 119	Inner	19.36 ± 0.21	2.55 ± 0.47	0.17 ± 0.01	0.46	6.18	1.798	⋯
	Outer	17.23 ± 0.05	0.91 ± 0.06	1.02 ± 0.01	0.78	−85.42
	PSF	19.94 ± 0.01	⋯	⋯	⋯	⋯
RGG 127	Inner	20.40 ± 0.01	0.95 ± 0.48	0.09 ± 0.02	0.53	−25.43	1.737	Bar (m${}_{\mathrm{bar}}^{{\rm{F}}110{\rm{W}}}$ = 17.75)
	Outer	18.13 ± 0.05	0.70 ± 0.23	1.25 ± 0.02	0.68	−32.77

Note. Column 1: identification number used in this paper. Column 2: basic components of the GALFIT model. Column 3: total AB magnitude of each model component reported by GALFIT for the F110W filter. Column 4: Sérsic index of each component. Column 5: half-light radius of each component. Column 6: axis ratio (b/a) for each Sérsic component. Column 7: position angle of each Sérsic component, measured in degrees east of north. Column 8: reduced χ² for the best-fit model in GALFIT. Column 9: additional components included in GALFIT model.

Download table as:  ASCIITypeset image

We also test the necessity of the inclusion of a central point source in our models. To test the robustness of including a PSF, we remove the PSF component from the final model for each galaxy and use these parameters as our initial estimates in GALFIT. This process results in GALFIT being unable to converge to a model in four of the seven systems. For the three systems in which models are achieved the reduced χ², ${\chi }_{\nu }^{2}$, worsens by an average of 10%. While this change is somewhat modest, we also observe signatures in the fit residuals (e.g., an overly bright central region surrounded by a dark region and bright ring) that indicate that the inner Sérsic component is converging to a profile that is much too steep near the center. Additionally, with the inclusion of a PSF component in our models, the fit residuals do not exceed ∼10% (residuals can be seen in Figure 1 and the Appendix), even in the central region of the images where surface brightness rises rapidly and has the largest values. As a final check for PSF necessity, we perform a visual inspection of the images for an unresolved nuclear source. In three of the seven galaxies in our sample (specifically RGG 1, RGG 48, and RGG 119), we find that an unresolved nuclear source is discernible by eye in the F110W and F606W images. Finally, we find that when the PSF is omitted from our models, the inner Sérsic index will often diverge or settle to nonphysical values (n > 10) to account for the rapid increase in surface brightness at the center of the galaxy. The combination of these factors indicates that a PSF component is justified in our models.

Zoom In Zoom Out Reset image size

Once our models are developed for each galaxy in our sample, we apply these results to the images taken in the F606W filter (F110W for RGG 48). To accomplish this, we fix the structural parameters reported by GALFIT in our original model (e.g., radii, Sérsic index, axis ratio), taking into account the differing pixel scales of the two filters. While the structural parameters are fixed in this step, we allow the magnitudes of each component to vary freely. This enables us to obtain magnitude measurements for each photometric component in both filters. In general, the models derived in the F110W images (F606W for RGG 48) agree well with the results from fitting the F606W images (F110W for RGG 48) and require little modification. Further discussion concerning the morphology of individual galaxies is provided in the Appendix.

To estimate the uncertainty of the magnitudes reported by GALFIT, we begin by fixing the background sky value in our model. The background sky value is determined by iteratively σ-clipping (with 10 iterations and σ = 3) the image used in our GALIT models to mask bright features and taking the median value of the resulting masked image. We then fit the image with the fixed background and take the uncertainty in the magnitude of each component to be the difference between the magnitudes from the best-fit and fixed background models. To estimate the error in the Sérsic index and effective radii, we replace the PSF in our best-fit model with an isolated bright star taken from the image used in the modeling. The error is then taken to be the difference between the resulting fit parameters and those from our best-fit model.

Overall, we find a median inner Sérsic index for our sample of 1.6 and a median outer Sérsic index of 0.79. The inner Sérsic indices span a large range of values (seen in the left panel of Figure 2). This indicates that the inner components range from a pseudo-bulge to a classical bulge morphology (we again refer the reader to the Appendix for more detail on the morphological classifications for individual galaxies). The outer Sérsic indices show much less variation (left panel of Figure 2) and are generally consistent with a Gaussian or exponential disk. For the entire sample, we find that six of the seven systems require two or more Sérsic components to produce an acceptable fit. When considering the systems with a detected outer Sérsic component, we find the ratio between the inner (bulge) component and total light (with AGN contribution excluded) to have a median value of 0.12 and a range from 0.05 to 0.17. These findings are in good agreement with work done by Jiang et al. (2011), who found the median bulge-to-total light ratio to be 0.16 when considering galaxies with a detected disk.

Zoom In Zoom Out Reset image size

To supplement our 2D GALFIT models, we perform elliptical isophotal fitting using the photutils package from the Astropy library (Astropy Collaboration et al. 2013, 2018). With this, we derive 1D radial surface brightness and intensity profiles for each of our galaxies and GALFIT models. An example can be seen in Figure 1 for the galaxy RGG 1. The models for the rest of our sample can be found in the Appendix.

It is of interest to briefly consider the structural parameters from GALFIT modeling in the context of other nonactive dwarf galaxy samples. Amorín et al. (2009) characterized the stellar host structure of 20 blue compact galaxies using similar 2D Sérsic models from GALFIT to model surface brightness profiles. They found that all galaxies but one have low Sérsic indexes (0.5 ≲ n ≲ 2), and the sample has a mean effective radius of ∼1.1 kpc. Janz et al. (2014) used GALFIT to study the stellar structure of 121 Virgo early-type dwarf galaxies using near-IR imaging. They performed single and multiple Sérsic component fits to find that surface brightness profiles tend to follow an overall exponential shape. This result holds for multiple Sérsic fits as well, with the inner component typically having an exponential profile as well (n < 1.2). Lian et al. (2015) studied the surface brightness profiles of 34 blue compact dwarf galaxies found in the Great Observatories Origins Deep Survey (GOODS) north and south fields from the Cosmic Assembly Near-IR Deep Extragalactic Legacy Survey (CANDELS; Grogin et al. 2011; Koekemoer et al. 2011). They performed one- and two-component Sérsic fits to find that approximately half of the galaxies in their sample are better fit with two components. Across the entire sample, they found the effective radius to be less than ∼4 kpc and that the inner and outer components are well fit by low Sérsic indices (n ≲ 1.5). We find similar structural parameters for our sample of active dwarf galaxies. First, a large fraction of the samples require two or more components to provide adequate fits to the surface brightness profile, and the inner component of the multicomponent fits often have a low Sérsic index (n ∼ 1). Additionally, the size of the galaxies across all of the samples is relatively consistent, with the effective radius of the outer Sérsic component being approximately 1 kpc ≲ r_e ≲ 4 kpc.

The HST magnitudes derived using the GALFIT models are comparable to broad V (F606W) and J (F110W) filters but are not equivalent to true Sloan or Johnson filters. The color-dependent mass-to-light ratio we employ from Zibetti et al. (2009) requires Sloan r − z colors and 2MASS J luminosities (see Equation (2)). To estimate the magnitudes in the 2MASS J, SDSS r, and SDSS z filters, we use the flux density measurements reported by GALFIT in the HST filters and fit a power law in log(f_λ) versus log(λ) space. We then obtain estimates of flux densities in the J, r, and z filters by evaluating the power-law fit at the appropriate pivot wavelengths (Gunn et al. 1998; Cohen et al. 2003). The flux densities are then converted to AB magnitudes to use in the Zibetti et al. (2009) relation (see Table 3 for magnitudes in all filters). This approach is motivated by stellar population models (e.g., Leitherer et al. 1999) that demonstrate that a power law is a good description of a stellar population spectrum redward of 4000 Å, which is our region of interest.

As a consistency check, we also investigate the method described in Läsker et al. (2016), where simple stellar population (SSP) models from the PARSEC code (Marigo et al. 2017) are used to generate magnitudes for an SSP in several different filter systems. The model results are then used to derive a relationship between the known (F110W and F606W) and desired (r, z, and J) magnitudes. When comparing the two methods, we find no significant change in the magnitudes or stellar masses produced, and we adopt the power-law fitting method in this work. When converting the derived r, z , and J apparent magnitudes to luminosities, the distance to each galaxy was estimated using the zdist parameter reported in the NSA that includes the peculiar velocity model of Willick et al. (1997; see Table 1).

With magnitudes determined with our GALFIT models and converted into the correct filters, we use the color-dependent mass-to-light ratios developed by Zibetti et al. (2009) found in Table B1 of their work. Specifically, we use J-band luminosity as a function of r − z color to determine mass-to-light ratios and compute the stellar masses of each structural component via the relation

$$\begin{eqnarray}&&\mathrm{log}(M/{L}_{J})=1.398(r-z)-1.271,\end{eqnarray} \tag{ 2 }$$

adopting a solar absolute magnitude of M_J = 4.54 in the AB system (Cohen et al. 2003). The resulting stellar masses can be found in Table 4. Errors in these stellar mass estimates are expected to be ∼0.3 dex and will be dominated by uncertainties in stellar evolution (Conroy et al. 2009).

Table 3.  Additional Apparent Magnitudes

ID	${m}_{\mathrm{psf}}^{{\rm{F}}606{\rm{W}}}$	${m}_{\mathrm{inner}}^{{\rm{F}}606{\rm{W}}}$	${m}_{\mathrm{outer}}^{{\rm{F}}606{\rm{W}}}$	${m}_{\mathrm{inner}}^{r}$	${m}_{\mathrm{outer}}^{r}$	${m}_{\mathrm{inner}}^{z}$	${m}_{\mathrm{outer}}^{z}$	${m}_{\mathrm{inner}}^{J}$	${m}_{\mathrm{outer}}^{J}$
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
RGG 1	23.11	20.66	18.39	20.58	18.32	19.95	17.78	19.40	17.30
RGG 9	20.49	17.80	⋯	17.75	⋯	17.31	⋯	16.92	⋯
RGG 11	19.93	19.01	17.07	18.96	17.01	18.63	16.54	1833	16.13
RGG 32	18.86	18.44	16.93	18.39	16.87	18.03	16.40	17.70	15.98
RGG 48	21.78	20.70	17.13	20.63	17.097	20.11	16.82	19.65	16.58
RGG 119	19.59	19.75	18.18	19.72	18.11	19.51	17.59	19.31	17.13
RGG 127	21.17	21.82	18.88	21.72	18.83	20.94	18.42	20.25	18.04

Note. All magnitudes are reported in the AB system. Column 1: identification number. Column 2: total magnitude of PSF component reported by GALFIT in F606W filter. Column 3: total magnitude of inner Sérsic component reported by GALFIT in F606W filter. Column 4: total magnitude of outer Sérsic component reported by GALFIT in F606W filter. Column 5: total magnitude of inner Sérsic component in SDSS r filter. Column 6: total magnitude of outer Sérsic component in SDSS r filter. Column 7: total magnitude of inner Sérsic component in SDSS z filter. Column 8: total magnitude of outer Sérsic component in SDSS z filter. Column 9: total magnitude of inner Sérsic component in 2MASS J filter. Column 10: total magnitude of outer Sérsic component in 2MASS J filter. Magnitudes reported for SDSS r, SDSS z, and 2MASS J filters are not reported by GALFIT but are determined using the power-law fitting procedure described in Section 4.1.

Download table as:  ASCIITypeset image

Table 4.  Masses and Luminosities

ID	$\mathrm{Log}({M}_{\mathrm{inner}}/{M}_{\odot })$	Log(Mouter/M⊙)	Log(Linner/L⊙)	Log(Louter/L⊙)	Minner/Mtotal	Linner/Ltotal
(1)	(2)	(3)	(4)	(5)	(6)	(7)
RGG 1	8.21	8.93	8.66	9.50	0.16	0.13
RGG 9	8.96	⋯	9.67	⋯	1	1
RGG 11	7.97	9.03	8.78	9.66	0.08	0.12
RGG 32	8.14	8.98	8.95	9.64	0.13	0.17
RGG 48	7.87	8.73	8.45	9.67	0.11	0.06
RGG 119	7.49	8.80	8.53	9.40	0.05	0.12
RGG 127	7.75	8.11	8.00	8.88	0.09	0.05

Note. Column 1: identification number. Column 2: stellar mass of the inner Sérsic (bulge) component. Column 3: stellar mass of the outer Sérsic component. Column 4: 2MASS J-band luminosity of the inner Sérsic component. Column 5: 2MASS J-band luminosity of the outer Sérsic component. Column 6: ratio of inner Sérsic component stellar mass to total stellar mass. Column 7: ratio of inner Sérsic component 2MASS J luminosity to total 2MASS J luminosity (not including light from AGNs).

Download table as:  ASCIITypeset image

We choose to use the color-dependent mass-to-light ratios from Zibetti et al. (2009), as they derive their relations using stellar population synthesis models with revised prescriptions for the TP-AGB evolutionary phase from Marigo & Girardi (2007) and Marigo et al. (2008). Additionally, Zibetti et al. (2009) developed their relations by taking into account the effect of dust and, more importantly, allowing for young stellar populations and including star formation bursts. They argued that on local scales, the effects of dust and variation in star formation history cannot be ignored, whereas in work such as that of Bell et al. (2003), who considered global properties, a smooth star formation for the entire galaxy can be employed. As we are investigating the structure and mass distribution in this study, the relations from Zibetti et al. (2009) are a natural choice for our purposes. Nevertheless, we also estimate the stellar masses using the relations from Bell et al. (2003) when placing them on the M_BH–M_bulge relation (see Section 6.1) to obtain a secondary estimate of the best-fit relation and exemplify the effect of the stellar mass estimation method on BH scaling relations.

When estimating stellar masses in this way, attention must be given to the different initial mass functions (IMFs) used to derive the mass-to-light ratios. Bell et al. (2003) used a "diet" Salpeter (1955) IMF, while the relations in Zibetti et al. (2009) are derived using a Chabrier IMF (Chabrier 2003). For our sample of dwarf galaxies, the Bell et al. (2003) relations predict stellar masses that have a median increase of 0.74 dex when compared to stellar masses found using the Zibetti et al. (2009) relation with the same colors. When the systems studied in Kormendy & Ho (2013; see Section 5) are considered in addition to our sample of dwarf galaxies, the median increase falls to 0.24 dex. This difference is consistent with findings by Reines & Volonteri (2015), where masses calculated with g − i colors using the Bell et al. (2003) relation are compared to the Zibetti et al. (2009) relation, as well as masses taken from the NSA that are computed using the kcorrect code (Blanton & Roweis 2007). We reiterate the importance of estimating stellar masses consistently across samples, as the choice of mass-to-light ratios can have a significant impact on stellar mass estimates.

In this work, we report stellar mass estimates for the inner and outer structural components of our target galaxies using the Zibetti et al. (2009) relations (see Table 4). The median stellar mass for the inner Sérsic components is 10^7.97 M_⊙, with a standard deviation of ${\sigma }_{{M}_{* },\mathrm{inner}}=0.43$ dex (see the right panel of Figure 2). Note that RGG 9, which has an inner component mass of ${10}^{9.0}{M}_{\odot }$, is modeled with a PSF and single Sérsic component, so the inner component mass will be equivalent to the total stellar mass. The outer Sérsic components have a median stellar mass of ${10}^{8.86}{M}_{\odot }$ and a standard deviation of ${\sigma }_{{M}_{* },\mathrm{inner}}=0.31$ dex, with the distribution of these masses seen in the right panel of Figure 2. With the stellar masses for each Sérsic component determined, we are able to estimate the bulge-to-total stellar mass ratio for the galaxies in our sample. For the systems that require two Sérsic components, we find the median bulge-to-total stellar mass ratio to be 0.11 with a range from 0.05 to 0.16, indicating that the stellar mass in these systems is dominated by the outer component.

When we compare the total mass of these systems to the estimates calculated by Reines & Volonteri (2015; given in column 5 of Table 1), we find that our estimates are smaller by a median value of ∼0.3 dex. This arises as the magnitudes from modeling with GALFIT are slightly dimmer than those in the NSA, which are used by Reines & Volonteri (2015) to estimate the total stellar masses for our sample. Given that the uncertainty in the method used to estimate both our stellar masses and the masses reported by Reines & Volonteri (2015) is ∼0.3 dex, and that the magnitudes we estimate for our sample are slightly dimmer than those used by Reines & Volonteri (2015), our results are consistent with their findings.

We obtain virial BH mass estimates for our sample (see Table 1) from Reines & Volonteri (2015, 2019), who derived BH masses from broad Hα emission detected in SDSS spectroscopy. Estimating BH masses from single-epoch spectroscopy is a commonly used method relying on the assumption that the broad-line region (BLR) kinematics are dominated by the gravity of the central BH. Under this assumption, the BH mass can be estimated by M_BH ∝ RΔV²/G. The average gas velocity is estimated from the width of a broad emission line, and the radius of the BLR is taken from the radius–luminosity relation based on reverberation-mapped AGNs (e.g., Vestergaard & Peterson 2006; Bentz et al. 2013). The constant of proportionality in estimating BH mass is dependent on the geometry and orientation of the BLR, which are in general unknown. While this parameter is known to vary from system to system (Bentz et al. 2009b; Barth et al. 2011), a single proportionality constant is adopted from calibrating reverberation-mapped BH masses to the M_BH–σ_* relation (Gebhardt et al. 2000; Greene & Ho 2005; Park et al. 2012; Ho & Kim 2014).

The BH masses estimated by Reines & Volonteri (2015) were found using Equation (1) in their work,

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}\left(\displaystyle \frac{{M}_{\mathrm{BH}}}{{M}_{\odot }}\right) & = & \mathrm{log}\epsilon +6.57+0.47\mathrm{log}\left(\displaystyle \frac{{L}_{{\rm{H}}\alpha }}{{10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}}\right)\\ & & +2.06\mathrm{log}\left(\displaystyle \frac{{\mathrm{FWHM}}_{{\rm{H}}\alpha }}{{10}^{3}\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right),\end{array}\end{eqnarray} \tag{ 3 }$$

which was derived with the methods of Greene & Ho (2005) and incorporates an updated radius–luminosity relationship from Bentz et al. (2013). Reines & Volonteri (2015) adopted  = 1.075, which corresponds to a mean virial factor $\left\langle f\right\rangle =4.3$ from Grier et al. (2013).⁴ Estimates of BH masses from single-epoch virial methods are very indirect and have uncertainties of ∼0.5 dex (Shen 2013).

First, we consider five additional dwarf galaxies hosting broad-line AGNs. The dwarf galaxy NGC 4395 (Filippenko & Sargent 1989; Filippenko & Ho 2003) hosts a Seyfert 1 nucleus, with its morphology well described by a disk and nuclear star cluster. As this galaxy does not have a well-defined bulge/pseudo-bulge component, we do not include it when fitting a linear regression to the data (see Section 6.1). We do, however, place this system on the plot of BH mass versus bulge mass using the total stellar mass from Reines & Volonteri (2015) as an upper limit and the reverberation-mapped BH mass estimate (${M}_{\star }^{\mathrm{NGC}\ 4395}={10}^{8.90}{M}_{\odot }$ and ${M}_{\mathrm{BH}}^{\mathrm{NGC}\ 4395}={10}^{5.45}{M}_{\odot }$). Though Woo et al. (2019) recently performed an updated reverberation mapping study of NGC 4395 to find a revised BH mass estimate of ${M}_{\mathrm{BH}}^{\mathrm{NGC}\ 4395}={10}^{3.96}{M}_{\odot }$, we choose to use the larger reverberation mapping mass, as it agrees with previous kinematic BH mass estimates. Additionally, we include the dwarf Seyfert 1 galaxy POX 52 (Barth et al. 2004; Thornton et al. 2008), which has no detected disk component and a Sérsic index of n = 4.0. For POX 52, we estimate the stellar mass using the photometry provided by Barth et al. (2004), specifically using the relation

$$\begin{eqnarray}&&\mathrm{log}\left(M/{L}_{I}\right)=1.475(B-V)-1.003\end{eqnarray} \tag{ 4 }$$

with a solar absolute I-band magnitude of 4.10 (Mann & von Braun 2015). We find a stellar mass of ${M}_{\star }^{\mathrm{POX}\ 52}={10}^{9.26}{M}_{\odot }$, which is in good agreement with the mass found by Thornton et al. (2008) of ${M}_{\star }^{\mathrm{POX}\ 52}={10}^{9.08}{M}_{\odot }$. We adopt the BH mass reported in Thornton et al. (2008) for POX 52 of ${M}_{\mathrm{BH}}^{\mathrm{POX}\ 52}\,={10}^{5.48}{M}_{\odot }$.

We also add the two remaining dwarf galaxies hosting broad-line AGNs from Reines et al. (2013), RGG 20 and RGG 123 (SDSS IDs J122342.81+581446.1 and J153425.59+040806.7, respectively). These systems were previously identified by Greene & Ho (2007), and the host galaxies were studied in detail by Jiang et al. (2011). They performed morphological decompositions using GALFIT on HST/WFPC2 images taken in the F814W (∼I-band) filter. To obtain bulge stellar mass estimates for these systems, we first take the bulge-to-total luminosity ratio calculated by Jiang et al. (2011; 0.93 for RGG 20 and 0.12 for RGG 123) and assume this to be constant across the SDSS r, i, and z filters. This allows us to take the Petrosian magnitudes from SDSS (the recommended magnitudes for estimating the total magnitude from an extended source such as a galaxy) and multiply the corresponding flux densities by the bulge-to-total ratio to obtain an estimate for the bulge component in these SDSS filters. With the scaled SDSS magnitudes, we calculate bulge stellar masses using the relation

$$\begin{eqnarray}&&\mathrm{log}\left(M/{L}_{i}\right)=1.797(r-z)-1.238,\end{eqnarray} \tag{ 5 }$$

with a solar absolute i-band magnitude of 4.53 mag (Gunn et al. 1998). Using this relation, we find the stellar bulge masses of RGG 20 and RGG 123 to be ${10}^{8.97}{M}_{\odot }$ and 10^8.11 M_⊙, respectively. We emphasize that these bulge mass estimates are not as robust as the other RGG dwarf galaxies with new HST observations, though their inclusion in the BH–bulge mass relation does not affect the outcome when calculating a best-fit linear regression (see Section 6.1). To differentiate these systems from the other dwarf galaxies when placing them on the scaling relation, they are plotted with triangles as opposed to stars (see Figure 4).

The final dwarf galaxy we include is RGG 118 (Reines et al. 2013; Baldassare et al. 2015, 2017b), a spiral system hosting one of the smallest nuclear BHs yet found, with a mass of ∼50,000 M_⊙. For RGG 118, we use the F160W luminosity and the F475W−F775W color (which are equivalent to 2MASS H-band luminosity and SDSS g − i color, respectively) provided in Baldassare et al. (2017b) to estimate the stellar mass with the following relation:

$$\begin{eqnarray}&&\mathrm{log}\left(M/{L}_{H}\right)=0.780(g-i)-1.222,\end{eqnarray} \tag{ 6 }$$

with a solar absolute F160W magnitude of 4.60 mag (Cohen et al. 2003). This gives a stellar mass for the bulge component of RGG 118 of ∼10^8.25 M_⊙. Our estimate is lower than the bulge stellar mass reported by Baldassare et al. (2017b) of ∼10^8.59 M_⊙ but in good agreement, as they use the color-dependent mass-to-light ratios from Bell et al. (2003), which are expected to predict a larger mass.

Kormendy & Ho (2013) compiled a sample of BH mass measurements that come from a variety of dynamical methods (stellar, CO molecular gas disk, ionized gas, and maser disk dynamics). We use 79 of these galaxies in our comparison sample for investigating the ${M}_{\mathrm{BH}}\mbox{--}{M}_{\mathrm{Bulge}}$ relation, which enables us to span the entire known mass range of nuclear BHs (M_BH ∼ 10⁵–10¹⁰ M_⊙).

To estimate the stellar mass of the bulge (or pseudo-bulge) component for galaxies with dynamical BH masses, we use the absolute K-band magnitudes and B − V colors provided by Kormendy & Ho (2013). We include all objects found in Kormendy & Ho (2013) except those with BH mass upper limits (two ellipticals and two spiral galaxies with pseudo-bulges) and galaxies without provided B − V colors, which are required for estimating stellar masses (two additional ellipticals and three spiral galaxies with pseudo-bulges). We use the relation (Zibetti et al. 2009)

$$\begin{eqnarray}&&\mathrm{log}\left(M/{L}_{K}\right)=1.176(B-V)-1.390\end{eqnarray} \tag{ 7 }$$

to estimate stellar masses for this sample, assuming a solar absolute K-band magnitude of 3.32 mag (Bell et al. 2003). As Kormendy & Ho (2013) reported only the integrated B − V colors for the entire galaxy, it should be noted that there will be some bias toward lower-mass estimates for disk galaxies in their sample arising from bluer colors of disks, which results in a smaller mass-to-light ratio.

As discussed in Reines & Volonteri (2015), Kormendy & Ho (2013) estimated M/L_K as a function of B − V color differently. The relation they derived is based on the mass-to-light ratio calibrations of Into & Portinari (2013). Using the method of Kormendy & Ho (2013) results in stellar masses that are systematically larger than the masses estimated using the Zibetti et al. (2009) relation by 0.33 dex (Reines & Volonteri 2015). We adopt stellar masses using the Zibetti et al. (2009) relations to maintain consistency between all samples used in this work.

In addition to the sample of dynamical BH mass measurements from Kormendy & Ho (2013), we also include the galaxies studied by Läsker et al. (2016). Läsker et al. (2016) performed detailed photometric structure decompositions using HST imaging for nine megamaser disk galaxies, yielding luminosities and colors for the identified bulge components. We use these nine late-type galaxies in our comparison sample, estimating the stellar mass using the following relation from Zibetti et al. (2009):

$$\begin{eqnarray}&&\mathrm{log}\left(M/{L}_{H}\right)=0.780(g-i)-1.222,\end{eqnarray} \tag{ 8 }$$

using the g − i colors and H-band luminosities provided in Tables 5 and 7, respectively, in Läsker et al. (2016).

It should be noted that since the work of Kormendy & Ho (2013), there have been a number of studies that have provided new or updated estimates of BH mass in a variety of systems (e.g., Seth et al. 2014; Saglia et al. 2016; Krajnović et al. 2018; Nguyen et al. 2019; Thater et al. 2019). It is a high priority to obtain comparable bulge–disk decompositions in multiple bands and place these systems on similar scaling relations.

The most accurate AGN BH masses are derived from reverberation mapping (e.g., Peterson et al. 2004; Bentz et al. 2009b; Barth et al. 2011). In this technique, the time lag between the continuum flux and broad-line emission variability gives the light travel time across the BLR of the AGN, and therefore the radius of the BLR. With the dynamics of the BLR being dominated by the gravity of the central supermassive BH, the radius and velocity of the BLR gas can be used to infer the mass of the central object.

In this work, we include 37 galaxies hosting AGNs with reverberation-mapped BH masses from Bentz & Manne-Nicholas (2018). Bentz & Manne-Nicholas (2018) performed detailed morphological decomposition using GALFIT (Peng et al. 2010) with the goal of determining the bulge properties of their sample. They reported stellar masses estimated using the M/L ratios from Bell & de Jong (2001) and Into & Portinari (2013). As the M/L ratios derived in Bell & de Jong (2001) and Into & Portinari (2013) use different IMFs than those derived by Zibetti et al. (2009), we cannot directly compare to the masses calculated in this work. Additionally, the colors used in the Bell & de Jong (2001) and Into & Portinari (2013) relations are different from those used in Zibetti et al. (2009), so we are unable to use the magnitudes reported by Bentz & Manne-Nicholas (2018) to recalculate stellar masses.

To address this, we use the Kormendy & Ho (2013) sample to calculate stellar masses using the M/L relation from Bell & de Jong (2001), given by

$$\begin{eqnarray}&&\mathrm{log}\left(M/{L}_{K}\right)=0.652(B-V)-0.692,\end{eqnarray} \tag{ 9 }$$

and then find the relation between these mass estimates and the stellar masses estimated using the Zibetti et al. (2009) M/L relations. We find that for the Kormendy & Ho (2013) sample, the Bell & de Jong (2001) relations predict stellar masses that have a median offset of 0.22 dex compared to the Zibetti et al. (2009) estimates with a scatter of σ = 0.09 dex. We then fit a linear regression to the different mass estimates, seen in Figure 3, and find

$$\begin{eqnarray}&&\mathrm{log}\left({M}_{\mathrm{Zibetti}}/{M}_{\odot }\right)=1.06\mathrm{log}\left({M}_{\mathrm{Bell}+\mathrm{deJong}}/{M}_{\odot }\right)-0.91.\end{eqnarray} \tag{ 10 }$$

Zoom In Zoom Out Reset image size

We use this result to transform the bulge masses reported by Bentz & Manne-Nicholas (2018) so that they are consistent with the stellar mass estimates found using the Zibetti et al. (2009) relations and include them as part of our comparison sample.

Figure 4 shows all 137 galaxies in our full sample on the M_BH–M_bulge relation. It is immediately clear that the dwarf galaxies in our study tend to fall on the extrapolation of the relation defined by more massive galaxies. Whether this is expected or not is an open question, with some recent works finding evidence of low-mass or late-type systems falling below the scaling relations derived using observations of high-mass elliptical and classical bulge systems (Kormendy & Ho 2013; Läsker et al. 2016). We discuss our results in the context of a variety of observational and theoretical studies in Section 7.

Zoom In Zoom Out Reset image size

We find that a linear relationship between log(M_BH) and log(M_bulge) is a good description of the data shown in Figure 4, with a Spearman correlation coefficient ρ = 0.81 and a probability p < 10⁻³¹ that no linear correlation is present. We therefore parameterize the scaling relation as

$$\begin{eqnarray}&&\mathrm{log}({M}_{\mathrm{BH}}/{M}_{\odot })=\alpha +\beta \mathrm{log}({M}_{\mathrm{bulge},\star }/{10}^{11}{M}_{\odot })\end{eqnarray} \tag{ 11 }$$

for direct comparison to other studies. We perform a linear regression using the Bayesian approach implemented in the LINMIXERR algorithm developed by Kelly (2007). This process allows for the inclusion of measurement errors in both variables, as well as accounting for a component of intrinsic random scatter. We report values for the slope, intercept, and scatter for this relationship that are the median values and 1σ widths of a large number of draws from the posterior distribution for each quantity. Using this method, we find best-fit parameters of

$$\begin{eqnarray}&&\alpha =8.80\pm 0.085;\beta =1.24\pm 0.081,\end{eqnarray} \tag{ 12 }$$

where the scatter about the relation has a standard deviation of σ = 0.68 dex.

To illustrate the impact of using different M/L ratios, we refit the relation using bulge stellar masses derived using the color-dependent mass-to-light ratios from Bell et al. (2003; using the same relations discussed in Sections 4.2 and 5 with coefficients found in Table A7 from Bell et al. 2003). We find a best fit of

$$\begin{eqnarray}&&\alpha =8.53\pm 0.076;\beta =1.41\pm 0.094,\end{eqnarray} \tag{ 13 }$$

where the scatter about the relation has a standard deviation of σ = 0.70 dex.

One of the earliest BH scaling relations to be discovered was the BH mass and bulge luminosity relation, M_BH–L_bulge (Kormendy & Richstone 1995; Magorrian et al. 1998). While this relationship is thought to arise as a consequence of the M_BH–σ and M_BH–M_bulge relations (Kormendy & Ho 2013), investigating this relation remains a relevant concern. Recent work has led to much tighter M_BH–L_bulge relations with reported scatter about the relation becoming similar to that found in the M_BH–σ relation (Marconi & Hunt 2003; Gültekin et al. 2009). Other work has also expanded the scope of this relation to include not only bulge-dominated classical and elliptical systems but also late-type galaxies (Wandel 2002; Bentz et al. 2009a; Bentz & Manne-Nicholas 2018). Given the variety of methods used to estimate bulge stellar mass and the accompanying uncertainties, M_BH–L_bulge provides useful information and additional constraints on BH–bulge scaling relations.

To investigate this relation in the context of our sample of dwarf galaxies, we estimate the V-band luminosity of the inner Sérsic component for our sample using the same photometric conversion technique described in Section 4.1. The V-band luminosities are used, as this allows us to use the absolute V-band magnitudes reported by Kormendy & Ho (2013) and the V-band luminosities reported in Bentz & Manne-Nicholas (2018) to act as a comprehensive comparison sample.

We find strong evidence of a linear correlation between the BH mass and the V-band luminosity, with a Spearman correlation coefficient ρ = 0.83 and a probability p < 10⁻³² that no correlation is present. With this consideration, we parameterize the scaling relation as

$$\begin{eqnarray}&&\mathrm{log}({M}_{\mathrm{BH}}/{M}_{\odot })=\alpha +\beta \mathrm{log}({L}_{V,\mathrm{bulge}}/{10}^{10}{L}_{\odot })\end{eqnarray} \tag{ 14 }$$

and find a best fit of

$$\begin{eqnarray}&&\alpha =7.93\pm 0.061;\beta =1.15\pm 0.075,\end{eqnarray} \tag{ 15 }$$

where the scatter about the relation has a standard deviation of σ = 0.67. This result, seen in Figure 5, is in good agreement with previous works that examine the relation between BH mass and optical bulge luminosity. Marconi & Hunt (2003) used a sample of 27 galaxies to investigate this scaling relation for the near-IR and optical spectral bands. When considering the optical B band, they found the slope of their relation to be β = 1.19 ± 0.12 with an intercept of α = 8.18 ± 0.08. McConnell & Ma (2013) studied 72 BHs and their host galaxies to study a variety of scaling relations. When considering the relation between BH mass and bulge V-band luminosity, they found a slope of β = 1.11 ± 0.13 and an intercept of α = 8.12 ± 0.10. Recently, Bentz & Manne-Nicholas (2018) studied this relation using the bulge V-band luminosities to from their sample and those from Kormendy & Ho (2013) to find a slope of β = 1.13 ± 0.08 and an intercept of α = 8.04 ± 0.06.

Zoom In Zoom Out Reset image size

Figure 1 shows RGG 1, an S0 galaxy that contains a disky outer component (n_outer ∼ 0.8) with a half-light radius of ∼1.6 kpc. The inner Sérsic component has a half-light radius of ∼0.7 kpc with a roughly Gaussian profile (n_inner ∼ 0.3). The disk component has an axis ratio of 0.7, and the inner component has an axis ratio of 0.5. The inner Sérsic component is subdominant to the disk at all radii, which may indicate a nuclear disk as opposed to a more classical bulge component. With these factors in mind, we classify RGG 1 as pseudo-bulge galaxy. The point source included to model the AGN has a low central surface brightness that is consistent with the X-ray observations from Baldassare et al. (2017a), who found the Eddington ratio to be L_Bol/L_Edd ∼ 0.001.

Figure 7 shows RGG 9, which appears to be a dwarf elliptical galaxy with a nuclear disk. The nuclear disk is most notable when examining the residuals seen in top right panel of Figure 7. Though the disk is clearly visible in the residuals in the F110W filter, this feature is more difficult to fit in the F606W filter, and including a component to account for the disk produces a mass for the nuclear disk that is overly large (M_ND ≈ 10⁹ M_⊙). When we choose to fit a single Sérsic component, we find that the magnitudes do not change significantly (${\rm{\Delta }}{m}_{\mathrm{ST}}^{\mathrm{elliptical}}\approx 0.05$) in either filter. Similarly, the Sérsic index of the elliptical component does not change drastically either, from n ≈ 4 to 2.5 when going from the nuclear disk/elliptical fit to a single Sérsic component. With this in mind, we use a single Sérsic component fit and find the elliptical portion to have a half-light radius of ∼1.21 kpc with an axis ratio of ∼0.85. As this is a dwarf elliptical galaxy, it is classified as a classical bulge.

Zoom In Zoom Out Reset image size

Figure 8 shows RGG 11, which appears to be an S0 galaxy with classical structure. The inner component is rounded and has a Sérsic index of n = 2.4, indicating a classical bulge. It has a half-light radius of 0.13 kpc. The outer component has a slightly higher than average Sérsic index with n = 1.69 but still closely resembles the classic exponential disk. The disk is one of the larger features in our sample of galaxies with a half-light radius of 2.57 kpc.

Zoom In Zoom Out Reset image size

Figure 9 shows RGG 32, an Sa galaxy with a dim ring/spiral structure in the disk, most readily seen in the residual panel of Figure 9. The bulge component has a Sérsic index of n ≈ 1.6 and a half-light radius of 0.29 kpc. While the Sérsic index of this component is less than 2, it has a round profile and dominates over the disk brightness at inner radii; we therefore classify it as a classical bulge. The disk component has a Sérsic index of n ≈ 0.75 with a half-light radius of 2.03 kpc.

Zoom In Zoom Out Reset image size

Figure 10 shows RGG 48, a disk-dominated spiral galaxy with a great deal of structure and star-forming regions. Most clearly seen in the residuals of Figure 10, there is a partially obscured ring around the small central bulge and AGN. This is accompanied by an asymmetric/obscured barred spiral, which is embedded in an asymmetric disk. While these features are readily picked out in the residual image, they are actually quite dim, and GALFIT has difficulty converging on a model that takes into account more than the inner bulge and the disk. We find that the inner component has a Sérsic index of n ≈ 0.6 and a half-light radius of 0.3 kpc. The low Sérsic index, flat profile, and presence of features such as the stellar ring allow this component to be confidently classified as a pseudo-bulge. The outer component has a Sérsic index of n ≈ 0.3 with a half-light radius of 2.12 kpc, fitting the classic description of an exponential disk. The disk is interesting, as it is quite asymmetric and slightly offset from the pseudo-bulge component.

Zoom In Zoom Out Reset image size

Figure 11 shows RGG 119, an S0 galaxy with evidence of a small stellar ring seen in the residuals. The inner component has a Sérsic index of n ≈ 2.5 with a half-light radius of 0.17 kpc. The relatively high Sérsic index, round profile, and dominance of the inner component at small radii clearly place this feature in the classical bulge category. The outer component has a Sérsic index of n ≈ 0.9 with a half-light radius of 1.02 kpc.

Zoom In Zoom Out Reset image size

Figure 12 shows RGG 127, which is more difficult to classify. Two obvious interpretations of its structure come to mind. First is that we are observing a disk galaxy edge-on, with the bright extended feature being the edge-on disk and the diffuse, rounder feature being an envelope/halo surrounding the disk. The second interpretation would be that the galaxy is being observed close to face-on and that the bright, thin feature is a bar, while the outer feature is a dimmer disk. In either interpretation, the galaxy requires three Sérsic components to be cleanly fit. The innermost component has a Sérsic index of n ≈ 1 with a half-light radius of 0.09 kpc. From the low Sérsic index of this component, the presence of a possible bar feature and the low central surface brightness indicate that this component should be classified as a pseudo-bulge. The bar structure has a Sérsic index of n ≈ 0.5 and a half-light radius of 0.88 kpc. The outer disk/envelope has a Sérsic index of 0.7 with a half-light radius of 1.25 kpc, indicating a roughly exponential disk.

Zoom In Zoom Out Reset image size