3 Position-velocity diagrams

   
3.1 Galaxy continuum subtraction

We subtracted the stellar continua from the spectra to study the two-dimensional shape of the emission lines. Without a large library of stellar (or even galaxy) absorption-line spectra obtained with the same observing setup of the spectra of the sample galaxies we were unable to apply the technique of template subtraction for an optimal correction for the starlight contamination (e.g. Ho et al. 1997).

The galaxy continua have been removed from MMT spectra by applying the technique outlined by Bender et al. (1994). The following procedure was applied to the each row of the galaxy spectrum. First we fit a sixth-to-tenth-order polynomial to the observed spectrum and calculated the rms variation, $\sigma$, of the spectrum around the polynomial. Then, the fit was repeated including only those pixels with values falling within $-1\sigma$ to $0\sigma$ of the first fit in order to avoid both emission and strong absorption lines. The new polynomial fit was adopted as the galaxy continuum and subtracted from the observed spectrum.

We were prevented from adopting the same method for the INT and ESO spectra because of their shorter wavelength range. Our major concern, with the stellar continuum subtraction was avoiding the creation of spurious features. For this reason we adopted a very simple but robust approach. Specifically, we made the reasonable assumption that the underlying observed stellar profile is the same at all wavelengths in the small observed range. An average profile was determined in regions free from emission-line flux and this same profile, properly scaled and subtracted from all the columns of the spectrum. The stellar continuum under the emission features was approximated by linear interpolation.

For our purposes, the above techniques give a satisfactory approximation of the galaxy continuum in the spectral range centered on the relevant emission lines we measure, specifically [O  III] $~\lambda5007$, [N  II] $~\lambda6583$, and H$\alpha$ for the MMT, INT and ESO spectra, respectively. In Fig. 2 we show the continuum-subtracted spectra of the sample galaxies as well as the isodensity contour plots (i.e. the PV diagram) of the emission lines we measure.

 

 

Table 4: Measured parameters of the sample galaxies.

Object	$(\Delta V / \Delta r)_{\rm in}$	$(\Delta V / \Delta r)_{\rm out}$	$\Gamma$	$\sigma_0$	$\sigma_{\rm out}$	$\sigma_0/\sigma_{\rm out}$	$F_{0}/F_{\rm out}$	Type
[name]	[ $\rm km~s^{-1}$ arcsec-1]	[ $\rm km~s^{-1}$ arcsec-1]		[ $\rm km~s^{-1}$]	[ $\rm km~s^{-1}$]
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
NGC 470	$33\pm 4$	$23\pm 7$	1.4+0.9-0.5	$101\pm15$	$103\pm41$	1.0+0.9-0.4	$4.0\pm0.1$	III
NGC 772	$32\pm 3$	$33\pm19$	1.0+1.5-0.4	$131\pm19$	$138\pm51$	0.9+0.8-0.4	$3.7\pm0.1$	III
NGC 949	$6\pm 3$	$6\pm 4$	1.0+3.5-0.7	$57\pm13$	$42\pm23$	1.4+2.3-0.7	$2.7\pm0.1$	III
NGC 980	$108\pm12$	$52\pm11$	2.1+0.8-0.6	$259\pm34$	$129\pm95$	2.0+6.6-1.0	$6.9\pm0.1$	I
NGC 1160	$17\pm14$	$33\pm24$	0.5+2.9-0.5	$66\pm21$	$32\pm11$	2.0+2.1-1.0	$1.2\pm0.1$	II
NGC 2179	$86\pm 4$	$41\pm 6$	2.1+0.5-0.4	$170\pm29$	$42\pm11$	4.0+2.4-1.4	$4.2\pm0.1$	I
NGC 2541	$7\pm 1$	$4\pm 2$	1.8+2.3-0.8	$48\pm18$	$55\pm13$	0.9+0.7-0.4	$5.2\pm0.2$	III
NGC 2683	$36\pm 6$	$12\pm 8$	3.0+7.5-1.5	$109\pm13$	$163\pm31$	0.7+0.3-0.2	$3.2\pm0.1$	*
NGC 2768	$20\pm 2$	$7\pm 4$	2.9+4.5-1.2	$174\pm12$	$141\pm48$	1.2+0.8-0.4	$10.8\pm0.1$	III
NGC 2815	$49\pm 1$	$49\pm 6$	1.0+0.2-0.1	$149\pm16$	$94\pm 9$	1.6+0.4-0.3	$15.0\pm0.5$	III
NGC 2841	$15\pm 7$	$17\pm 2$	0.9+0.6-0.5	$135\pm18$	$144\pm19$	0.9+0.3-0.2	$2.5\pm0.1$	III
NGC 3031	$21\pm 4$	$7\pm 3$	3.0+3.3-1.3	$237\pm51$	$57\pm41$	4.2+13.8-2.3	$65.6\pm0.4$	III
NGC 3281	$34\pm 5$	$18\pm 9$	1.9+2.4-0.8	$119\pm20$	$92\pm12$	1.3+0.4-0.3	$19.9\pm0.5$	III
NGC 3368	$35\pm12$	$32\pm 8$	1.1+0.9-0.5	$113\pm19$	$69\pm12$	1.6+0.7-0.5	$6.1\pm0.2$	III
NGC 3521	$21\pm10$	$18\pm11$	1.2+3.3-0.8	$180\pm39$	$146\pm66$	1.2+1.5-0.6	$8.9\pm0.1$	III
NGC 3705	$22\pm 1$	$14\pm10$	1.6+4.2-0.7	$110\pm14$	$55\pm42$	2.0+7.5-1.0	$13.3\pm0.4$	III
NGC 3898	$13\pm 1$	$26\pm 3$	0.5+0.1-0.1	$131\pm21$	$110\pm27$	1.2+0.6-0.4	$4.6\pm0.1$	III
NGC 4419	$7\pm 1$	$14\pm 2$	0.5+0.2-0.1	$117\pm11$	$83\pm21$	1.4+0.7-0.4	$13.6\pm0.1$	III
NGC 4698	$8\pm 5$	$3\pm 1$	2.7+3.8-1.9	$86\pm 8$	$103\pm 7$	0.8+0.1-0.1	$6.0\pm0.1$	III
NGC 5064	$74\pm11$	$56\pm11$	1.3+0.6-0.4	$52\pm22$	$43\pm11$	1.2+1.1-0.7	$0.9\pm0.1$	II
NGC 7320	$6\pm 1$	$5\pm 3$	1.2+2.3-0.6	$12\pm10$	$10\pm7$	1.2+6.1-1.1	$0.3\pm0.1$	II
NGC 7331	$16\pm 4$	$15\pm11$	1.1+3.9-0.6	$102\pm11$	$130\pm65$	0.8+1.0-0.3	$3.7\pm0.1$	III
NGC 7782	$170\pm25$	$56\pm11$	3.0+1.3-0.9	$151\pm19$	$96\pm29$	1.6+1.0-0.5	$5.3\pm0.1$	I

Notes - Column 2: inner velocity gradient at $r\simeq\pm1''$. Column 3: outer velocity gradient at $r\simeq\pm4''$. Column 4: inner-to-outer velocity gradient ratio. Column 5: central velocity dispersion. Column 6: outer velocity dispersion at $r\simeq\pm4''$. Column 7: central-to-outer velocity dispersion ratio. Column 8: central-to-outer integrated-flux ratio. F₀ and $F_{\rm out}$ have been measured at r=0'' and $r\simeq\pm4''$, respectively. Column 9: type of PV diagram according to our classification; * = figure-of-eight PV diagram (see appendix).

Figure 2: Optical images, spectra and PV diagrams of the sample galaxies. We show from left to right: a) an optical image of the galaxy taken from the Digitized Sky Survey with the slit position and angular scale superimposed. The orientation of the image is north up and east to the left. b) The galaxy spectrum after continuum removal with wavelength, radial distance from the nucleus, and orientation marked. Color cuts are chosen to show the fainter structures and the radial extension of the emission lines. In the INT and ESO spectra the nuclear continuum is the residual after subtraction of about $90\%$ of the continuum. c) The galaxy PV diagram derived from [O  III] $~\lambda5007$, [N  II] $~\lambda6583$, and the H$\alpha$ emission line for the spectra obtained at the MMT, INT, and 3.6-m ESO telescopes, respectively. In the PV diagram the intensities of the plotted contours correspond to $5\%, 15\%, 25\%,$ ..., $95\%$ of the maximum emission-line intensity. The plotted region in the PV diagram corresponds to the rectangular box marked in the galaxy spectrum. PV diagrams are shown with the same scale for the observed radii and velocities, but we also indicate the distance from the center in kpc to aid comparison.

Figure 2: continued.

Figure 2: continued.

Figure 2: continued.

Figure 2: continued.

Figure 2: continued.

Figure 2: continued.

Figure 3: Inner vs. outer velocity gradients (left panel) and central vs. outer velocity dispersions of the sample galaxies (right panel). Observed velocity gradients collected in Table 4 have been corrected for galaxy inclination and distance given in Table 1.

   
3.2 Measuring the position-velocity diagrams

The PV diagrams plotted in Fig. 2 show the variety of two-dimensional shapes of the optical emission lines in the inner regions of disk galaxies. This finding is in agreement with the earlier results of Rubin et al. (1997) and Sofue et al. (1998). We suggest a classification of these PV diagrams based on the comparison of the velocity gradients, velocity dispersions and fluxes measured at different distances from the center. Our goal is to identify galaxies hosting a CNKD.

We measured the line-of-sight velocity, V, of the ionized gas at $r\simeq\pm1''$ and $r\simeq\pm4''$ by fitting a Gaussian to the relevant emission line. The central wavelength of the Gaussian fit was converted to velocity in the optical convention and then the standard heliocentric correction was applied to obtain V. The radii, $r\simeq\pm1''$, used for measuring the inner velocity gradient $(\Delta V / \Delta r)_{\rm in}$, are dictated by the spatial resolution limit imposed by seeing on our data ( $0\farcs8 \la {\it FWHM} \la 1\farcs5$). Choosing the smallest possible radii according to the seeing limit assures us that if a central mass concentration is present, the observed inner velocity gradient is maximized. The outer velocity gradient, $(\Delta V / \Delta r)_{\rm out}$, measured at $r\simeq\pm4''$, serves as a reference. In fact, for each sample galaxy the radius of influence, $\theta_\bullet$, of the possible central mass concentration predicted using the $M_\bullet-\sigma$ relation (Merritt & Ferrarese 2001a, see Table 1) is $\theta_{\bullet} \ll 4''$. Therefore, $(\Delta V / \Delta r)_{\rm out}$ is essentially determined by the contribution of galaxy stellar component to the potential.

We checked that velocity gradients did not significatively change if the $\Delta V$ are computed from the difference of line-of-sight velocity distribution maxima instead of the centers of the fitting Gaussian fit. Moreover, to test the robustness of our measurements and to estimate the associated uncertainties, we compute $(\Delta V / \Delta r)_{\rm in}$ at $r_{\rm in} \simeq \pm0\farcs7$ and at $\pm1$$\farcs3$ and we compute $(\Delta V / \Delta r)_{\rm out}$ at $r_{\rm out} \simeq \pm3''$ and at $\pm$5'', respectively. The errors on $(\Delta V / \Delta r)_{\rm in}$ have been estimated from the maximum difference between the values measured at $\pm0\farcs7$ and $\pm1\farcs3$ with respect to those measured at $\pm1''$. Similarly, the errors on $(\Delta V / \Delta r)_{\rm out}$ have been estimated from the maximum difference between the values obtained at $\pm3''$ and $\pm5''$ with respect to those measured at $\pm4''$. The measured values of $(\Delta V / \Delta r)_{\rm in}$ and $(\Delta V / \Delta r)_{\rm out}$ are given in Table 4.

The inner-to-outer velocity gradient ratio $\Gamma$ is independent of galaxy inclination and has been adopted to indicate which galaxies are characterized by rapidly-rotating gas in the inner regions. However, to allow a direct comparison of their absolute values, we plotted the inner velocity gradients as a function of the corresponding outer velocity gradients in Fig. 3, after correcting for galaxy inclination and distance given in Table 1. NGC 980, NGC 2179, NGC 2683, NGC 3031 and NGC 7782 are clearcut cases of galaxies characterized by $\Gamma\pm\Delta\Gamma>2$.

To characterize the velocity-dispersion and surface-brightness radial profiles of each gaseous disk, we measured the velocity dispersion and integrated flux of the ionized gas in the galaxy center and at $r\simeq\pm4''$, using a Gaussian fit to the line adopted for the velocity measurements. The FWHM of Gaussian fit was corrected for instrumental FWHM and converted into the velocity dispersion, $\sigma$. The formal error of the fit has been adopted as the error on the central value of velocity dispersion, while the errors on the outer values have been estimated using the maximum difference between the measurements obtained at $\pm3''$ and $\pm5''$ with respect to those at $\pm4''$. The integrated flux was assumed to be the area of the Gaussian fit and the associated error was estimated from Poisson statistics. We considered only the central-to-outer integrated-flux ratio since spectra were not flux calibrated. This process results in line fluxes of different objects observed with different setups that are not directly comparable. The measured values of the velocity dispersion and the central-to-outer integrated-flux ratio are given in Table 4. The central velocity dispersions are shown as a function of the outer velocity dispersions in Fig. 3. NGC 980, NGC 2179, and NGC 3031 exhibit the sharpest rises in observed velocity dispersion towards their centers.

   
3.3 A classification of the position-velocity diagrams

We propose an operational classification of the PV diagrams of Fig. 2 based on the properties we measured in Sect. 3.2 and based on the analogy between the shapes of the PV diagrams we observed and the shapes predicted for the spectrum of a ionized thin gaseous disk, under the assumption that the gas moves in circular orbits, in the plane of the galaxy. Although the model assumptions may not be accurate in practice, the classification is useful to identify the effects of rapid gas rotation. We do not pretend to draw any general conclusions on the phenomenology of PVdiagrams from such a classification; it is just a tool adopted to select the sample galaxies which possibly host CNKDs.

Under the model assumptions, two parameters are crucial in determining the observed shape of the PV diagrams; they are the value of the central mass concentration and the steepness of the intrinsic surface-brightness distribution of the gaseous disk. To investigate the change in the PV diagrams resulting from these two effects, we have used the IDL modeling software developed in Bertola et al. (1998). We refer the reader to that paper for further details on the model. A slit width and a seeing FWHM of 1$\arcsec$ have been adopted, with a spectrograph velocity scale of 10 km s^-1 pixel^-1 and a spatial scale of $0\farcs3$ pixel^-1. The underlying galaxy potential is assumed to result in a rigid rotation of 0.4 km s^-1 pc^-1 in the plane of the disk, which is "observed'' at 60$^\circ$ inclination ( $i=90^\circ$ corresponding to edge-on). A distance of 17 Mpc was adopted for the modeling, corresponding to the distance of the Virgo cluster.

The predicted effect on the PV diagram from an increase of the central mass concentration is presented in the upper panels of Fig. 4. In this case, we assume an exponential the surface-brightness profile superposed on a constant term: $I(R)=I_0+I_1 \exp(-R/R_I)$, with I₀=1 and I₁=5 (in arbitrary units) and $R_I=1\arcsec$, where the central mass is given by $M_{\bullet}=0,10^8,10^9$ $M_{\odot }$ in panels (a), (b) and (c) respectively.

The PV change that results from a variation in the brightness of a central unresolved source is shown in the bottom panels of Fig. 4. In these panels the adopted surface-brightness profile is assumed to be an essentially unresolved Gaussian superposed on a constant term: $I(R)=I_0+I_1 \exp[-R^2/(2\sigma_I^2)]$, with I₀=1, $\sigma_I=0\farcs3$ and I₁=0,20,100 (in arbitrary units) in panel (a), (b) and (c) respectively.

By comparing the models of Fig. 4 with the observed PV diagrams, and inspecting the measured values of $\Gamma$, $\sigma_0/\sigma_{\rm out}$ and $F_{0}/F_{\rm out}$ we identify three different types of PV diagrams (see Fig. 5).

Type I. This type of PV diagrams suggests the presence of two distinct kinematical gaseous components. This results from the sharp increase of $\Delta V / \Delta r$ towards small radii, which indicates the presence of a rapidly rotating gas in the innermost region of the galaxy. The inner-to-outer velocity gradient ratio is $\Gamma>2$ and the intensity distribution along the line shows two symmetric peaks with respect to the center.

The PV diagram of the Sa galaxy NGC 2179 (Fig. 5) can be considered the prototype of this class. As we showed in Bertola et al. (1998), the peculiar shape and intensity distribution of this PVdiagram can be modeled as a unique gaseous component that is rotating in the combined potential of a central mass concentration embedded in an extended stellar disk. Therefore the galaxies that exhibit this kind of PV diagram (NGC 980, NGC 2179, and NGC 7782) are good candidates to host a CNKD rotating around a central mass concentration. They are ideal targets for HST spectroscopic follow-up to constrain the mass of the possible SMBH. A good estimate of this mass requires that the innermost kinematical points be within the radius of influence (e.g. Merritt & Ferrarese 2001b). The three galaxies meet this criterion, since the expected angular extension of the radii of influence of their SMBHs are comparable to the pixel size of the Space Telescope Imaging Spectrograph $(\theta_{\bullet} \approx 0\farcs05)$. An increase in the velocity dispersion ( $\sigma_0 \ga 150$  $\rm km~s^{-1}$), associated with a large increase in the velocity gradient (as in NGC 980 and NGC 2179) is expected in the presence of a nuclear mass concentration. It could result from differential motion within the aperture or from intrinsic turbulence in the gaseous disk. On the other hand, an increase in velocity dispersion that is not associated with an increase in the velocity gradient may indicate that the gas is not in a disk. However, a central mass concentration may still be the cause of this increase in the velocity dispersion.

Type II. This class of PV diagram is characterized by a single velocity component which is in rigid-body rotation, as indicated by $\Gamma\approx1$. $\sigma_0/\sigma_{\rm out}\approx1$ and $F_{0}/F_{\rm out}\approx1$ are characteristic of these PV diagrams too.

We consider the PV diagram of the Sa galaxy NGC 5064 to be the prototypical example of this kind of PV diagram (Fig. 5). In Bertola et al. (1998), we pointed out that in this galaxy either the unresolved Keplerian part of the gaseous disk does not result in a detectable contribution or the central mass concentration is lower than $5\times10^7$ $M_{\odot }$. Therefore we suggest that galaxies exhibiting this type of PV diagram (see Table 4) may harbor low-mass SMBHs, although high spatial resolution spectroscopy and dynamical modelling of the stellar kinematics are required to distinguish this possibility from the effects of a peculiar gas distribution.

Type III. PV diagrams of this type are characterized by an apparently broad nuclear emission-line component superimposed on a normal velocity curve. This results from the sharp increase of the line flux toward the center, as indicated by $F_{0}/F_{\rm out}>1$.

The best example of this type of PV diagram is that of the S0 NGC 2768 (Fig. 5). Most of the sample galaxies exhibit a PVdiagram belonging to this class because of a selection effect. They have been observed because of their strong emission lines.

Figure 4: Upper panels: the shapes of PV diagrams as a function the central mass concentration. The three panels represent the emission lines of CNKDs rotating around central pointlike sources of a)  $M_\bullet = 0$ $M_{\odot }$; b)  $M_\bullet = 10^8$ $M_{\odot }$; and c)  $M_\bullet = 10^9$ $M_{\odot }$. The PV diagram in panel c) is representative of Type I. Lower panels: the shape of PV diagrams as a function of the intrinsic surface brightness of the gas. The three panels represent the emission lines of gaseous disks in rigid-body rotation with a projected velocity gradients ( $\Delta V / \Delta r)_{\rm in}=(\Delta V / \Delta r)_{\rm out}=27$  $\rm km~s^{-1}$ arcsec-1, observed velocity dispersions $\sigma _{0}=\sigma _{\rm out}=100$  $\rm km~s^{-1}$ , and nuclear fluxes a)  $F_0=F_{\rm out}$; b)  $F_0=20\times F_{\rm out}$; and c)  $F_0=100\times F_{\rm out}$. The PV diagrams in panels a) and c) are representative of Type II and III, respectively.

Figure 5: Contour plots of the prototypical examples of the three types of PV diagrams. Left panel: NGC 2179 (Type I). Middle panel: NGC 5064 (Type II). Right panel: NGC 2768 (Type III).

The classification and the peculiarities of the PV diagrams of all the sample galaxies are discussed in the appendix.