Stellar kinematics and dynamical model for NGC 4258.

The benchmark black hole in NGC 4258: dynamical models from high-resolution two-dimensional stellar kinematics

Abstract

NGC 4258 is the galaxy with the most accurate (maser-based) determination for the mass of the supermassive black hole (SMBH) in its nucleus. In this work we present a two-dimensional mapping of the stellar kinematics in the inner 3.0  3.0 arcsec = 100  100 pc of NGC 4258 using adaptative-optics observations obtained with the Near-Infrared Integral Field Spectrograph of the GEMINI North telescope at a  0.11 arcsec (4 pc) angular resolution. The observations resolve the radius of influence of the SMBH, revealing an abrupt increase in the stellar velocity dispersion within  10 pc from the nucleus, consistent with the presence of a SMBH there. Assuming that the galaxy nucleus is in a steady state and that the velocity dispersion ellipsoid is aligned with a cylindrical coordinate system, we constructed a Jeans anisotropic dynamical model to fit the observed kinematics distribution. Our dynamical model assumes that the galaxy has axial symmetry and is constructed using the multi-gaussian expansion method to parametrize the observed surface brightness distribution. The Jeans dynamical model has three free parameters: the mass of the central SMBH (), the mass-luminosity ratio () of the galaxy and the anisotropy of the velocity distribution. We test two types of models: one with constant velocity anisotropy, and another with variable anisotropy. The model that best reproduces the observed kinematics was obtained considering that the galaxy has radially varying anisotropy, being the best-fitting parameters with 3 significance and . This value for the mass of the SMBH is just 25 per cent larger than that of the maser determination and 50 per cent larger that a previous stellar dynamical determination obtained via Schwarzschild models for long-slit data that provides a SMBH mass 15 per cent lower than the maser value.

keywords:
galaxies: kinematics and dynamics – galaxies: supermassive black holes – galaxies (individual): NGC 4258
12

1 Introduction

NGC 4258 (M 106) is a spiral galaxy with Hubble type SABbc, at a distance of 7.280.3 Mpc (Herrnstein et al., 1999), which harbours one of the closest active galactic nucleus (AGN), classified as Seyfert 1.9. This galaxy is well known from previous studies for harbouring the supermassive black hole (SMBH) with the best constrained mass after that of the Milky Way, with a value of , obtained from resolved kinematics of a rotating H0 maser disc within 0.13 pc from the nucleus (Miyoshi et al., 1995; Greenhill et al., 1995; Herrnstein et al., 1999). It is also well known for its anomalous arms, which resemble spiral arms but are more diffuse than regular spiral arms (Wilson, Yang & Cecil, 2001). The anomalous arms span 5 arcmin in optical line emission and 10 arcmin in radio continuum emission (Cecil et al., 2000).

A nuclear non-stellar continuum and broad optical emission lines have been observed in polarized light (Wilkes et al., 1995), supporting the presence of an obscured AGN, from which a radio jet is observed propagating perpendicularly to the maser disc (Cecil, Wilson & Tully, 1992). The jet is oriented at a position angle (hereafter PA) PA=, consistent with the projected spin axis orientation of the maser disc, which has a major axis PA=86 (Cecil et al., 2000). Farther from the nucleus, interactions between the radio jet and the interstellar gas are probably the origin of the anomalous arms (Wilson, Yang & Cecil, 2001, and references therein).

In the infrared (hereafter IR), the nuclear continuum has been found to be well reproduced by a power-law , which seems to extend through the optical to the ultraviolet (Yuan et al., 2002). In X-rays, the nuclear spectrum presents several components: two power-laws, a thermal component and the Fe K emission-line (Yang et al., 2007). The nuclear source is not resolved in the IR (1 to spectral region) in observations with angular resolution and presents total luminosity of (Chary et al., 2000).

The host galaxy disc has the photometric major axis at PA=, as obtained from H i observations (van Albada, 1980), who have also obtained an inclination for the disc of . A previous H study by van der Kruit (1974) gave values of PA=146 and . More recently Sawada-Satoh et al. (2007) have mapped the large scale molecular CO(2-1) velocity field and concluded it is dominated by rotation in a disc with major axis PA= and .

Other studies include the one of Pastorini et al. (2007), who measured the gas kinematics close to the nucleus using long-slit spectra obtained with the Space Telescope Imaging Spectrograph (HST-STIS) and that of Herrnstein et al. (2005), who constrained even further the SMBH mass using new maser observations. There is also a more recent study by Siopis et al. (2009) which uses the long-slit spectra from HST-STIS to obtain the stellar velocity distribution using the Ca ii triplet absorption lines from to along the major axis and to along the minor axis of the galaxy, obtaining a direct determination of the mass of the SMBH.

Adopting the SMBH mass obtained from the kinematics of the maser disc, and using a velocity dispersion for the galaxy bulge of 3, one can calculate the radius of influence of the SMBH 15 pc. For a scale of 35 pc per arcsecond, this radius corresponds to 0.42 arcsec, thus resolvable with ground-based adaptative optics observations, that typically reach 0.1 arcsec in the near-infrared. We thus decided to observe NGC 4258 with the Gemini Near-Infrared Integral Field Spectrograph (NIFS) in order to map the stellar kinematics and investigate the effect of the presence of the SMBH in the inner dynamics of the galaxy. Although we also use stellar kinematics to probe the presence of the SMBH as done by Siopis et al. (2009), our K-band data are less affected by reddening, allowing the mapping of stellar kinematics much closer to the nucleus than previous observations. In addition, our NIFS data provide a two-dimensional coverage instead of the limited spatial coverage of long-slit observations.

Although the use of dynamical models based on the Schwarzschild orbit superposition method (Schwarzschild, 1979) is currently the most popular method to determine the masses of SMBHs in the centre of galaxies, the Jeans dynamical models have proved capable of reproducing the values of the masses of the SMBH in good agreement with those determined by the Schwarzschild models (Cappellari et al., 2010; Krajnović et al., 2009). The Jeans models are also useful when one wants to have more physical insight to the problem. In addition, a Schwarzschild orbit superposition model of NGC 4258 already exists (Siopis et al., 2009), providing a value for the mass of the SMBH of , which is comparable to the better constrained value () derived from maser observations Herrnstein et al. (2005).

We have thus decided to study the dynamics of the nuclear region of NGC 4258 using our NIFS data and Jeans models in order to determine the dynamical quantities that govern the system. Our goal is also to investigate how well these models reproduce the observed stellar kinematics and how large are the differences between the values of the mass of the SMBH of NGC 4258 determined by different methods. The use of the Jeans anisotropic dynamical model has allowed us to understand how variations in the model parameters – such as the velocity anisotropy and the mass-to-light ratio – affect the stellar kinematics and the determination of the mass of the SMBH in NGC 4258.

Figure 1: Top Left: the NICMOS PSF modelled with the tiny tim software. Top Right: the central of the NICMOS image of the galaxy presenting the the features of the NICMOS PSF. Intensity units are normalized counts. The bottom panels show cuts along the two diagonals (dashed line and cyan continuum line) across the upper panels, in logarithmic scale.
Figure 2: Top: the 2MASS Ks-band image. Bottom: the K-band image from NICMOS. The rectangle in the bottom panel shows the field of view of our NIFS observations presented in Fig 3.

This paper is organized as follows. In section 2 we present our IFU observations and the photometric data used to construct the dynamical model. In Section 3 we discuss the derivation of the two-dimensional line of sight velocity distribution (LOSVD). Section 4 is dedicated to the description of the dynamical model and the discussion of the results and in section 5 we present our concluding remarks.

2 Observations and data reduction

The Integral Field spectroscopic data were obtained in the near-IR K-band with the instrument NIFS (McGregor et al., 2003) operating with the ALTAIR adaptive optics module at the 8-m Gemini North telescope in April 2007 under the science program GN-2007A-Q-25. The K-band was selected for this study because it contains the absorption CO bands at which can be used to derive the stellar kinematics. In order to obtain the surface brightness distribution to model the galaxy mass density distribution (and the gravitational potential) we used two sets of archive images: the first of them obtained with the Near-Infrared Camera and Multi Object Spectrometer (NICMOS) NIC2 camera of the HST. The second is a large scale image from the Two Micron All Sky Survey (2MASS). In the next sections we describe these sets of data and the reduction procedures.

2.1 The photometric data

Figure 3: Left: Continuum map (reconstructed image) from our NIFS spectroscopy. Right: Spectra at the positions A, N and B marked at the left panel with the Br emission-line and CO absorption bands identified. Flux units are erg s cm and the extraction apertures are 0.05 arcsec x 0.05 arcsec.

Nicmos

Near-infrared K-band images were obtained from the Multimission Archive at STScI (MAST) corresponding to data sets N46801050 for object observations and N46801060 for sky observations, from the proposal ID 7230 (Scoville, 1997). These observations were obtained with the NIC2 camera of NICMOS using the F222M filter. The NIC2 camera has pixels, with scales of in and , providing a field of view of . The F222M filter covers the wavelength range , with central wavelength at . The galaxy observation have a total exposure time of seconds performed in multiaccum mode with a four point spiral dither; the sky observations were obtained with a three point spiral dither.

The data reduction was done with standard iraf4 and pyraf tasks following the same steps as in Chary et. al. (2000). We used the multidrizzle task to combine the dithered images following the standard sequence for NICMOS dithered observations described in the multidrizzle handbook. Inspecting the data quality files we noted that due the emission of the AGN, some of the pixels at the centre of the galaxy are incorrectly identified as cosmic rays in the four dithered images. We thus corrected this in the data quality files and created a static mask to correct for a spurious “bar” feature in the column 128. Then a separated drizzled image was created for each dither position of the detector. To determine the need for extra offsets we have done cross correlation measurements between the images with the crossdriz task and the extra shifts have them been computed with the shiftfind task. Then we ran again the multidrizzle with the extra shifts and created new separately drizzled images as well as a well-aligned median image. Finally, we ran multidrizzle again to derive the cosmic ray masks and combine the images in a final drizzled image, clean of cosmic rays and detector artifacts.

The compact nuclear emission due to the AGN its evident in the reduced image. This unresolved emission appears in the image as a point source introducing the spurious features of the NICMOS point spread function (PSF) in the image. In the top left panel of Fig. 1 we show the NICMOS PSF, and in the right panel the central of the image of NGC 4258. In the bottom panels of this figure we display cuts along the two diagonals of each of the the images in the top panels (black dashed and cyan continuous lines). The arrows indicate the strong signature of the PSF in the image of the galaxy. As our interest is to determine the luminosity distribution and gravitational potential generated by the stars, we need to subtract the contribution from the AGN. In order to do this, we generated a large image of the NICMOS PSF and centred it at the same position of the galaxy centre in the detector. We then normalized the PSF image so that its peak flux coincided with that of the galaxy. After several tests in which we multiplied the normalized PSF by different fractions and subtracted it from the galaxy image, we concluded that the smoothest residual (without an “inverted peak” due to over-subtraction) was obtained with a contribution of the PSF of 70 per cent to the peak of the brightness distribution. The resulting image after reduction and AGN subtraction is shown in the bottom panel of Figure 2. The rectangle in the centre of the image represents the field of view of the NIFS observations. The total amounth of light subracted corresponds to  1.9 10. In Apendix  C we perform some tests to evaluate the effects of a possible over- or under-subtraction of the AGN on the determination of the mass of the SMBH.

2mass

In order to characterize the surface brightness distribution of the galaxy beyond the inner we used a large scale image from the 2MASS telescope in the K-short-band (Ks-band) from Jarrett et al. (2003). The 2MASS image has a scale of and the Ks-filter covers the wavelength range from 1.93 to with central wavelength of . The central of this image is shown in the top panel of Fig. 2. The black rectangle in the centre of the panel represents the field of view of the NICMOS image.

Figure 4: pPXF fits to the galaxy spectra. Examples of the resulting fit of the pPXf to the galaxy spectra on the positions A (top panel), N (middle panel) and B (bottom panel) indicated in Fig. 3; the cyan lines are the galaxy spectra, the black lines are the resulting fits of pPXF to the galaxy spectra and the red lines are the residuals.

2.2 The NIFS data

NIFS has a square field of view of , divided into 29 slices with an angular sampling of in and in . The observing procedures followed the standard Object-Sky-Object-Sky-Object-Sky-Object dither sequence, with off-source sky positions since the target is extended, and individual exposure times of centred at with a total of 10 individual exposures at the galaxy. The IFU was oriented with the slices along the position angle PA°. We have used the grating and the filter , which resulted in an arc lamp line full width at half maximum (FWHM) of (thus R5300).

The data reduction was accomplished using tasks contained in the nifs package which is part of gemini iraf package as well as generic iraf tasks. The reduction procedure included trimming of the images, flat-fielding, sky subtraction, wavelength and s-distortion calibrations. We have also removed the telluric bands and flux calibrated the frames by interpolating a black body function to the spectrum of the telluric standard star. Small shifts between exposures due to guiding problems were corrected by mosaicing the individual data cubes into a final one. The final data cube contains 4300 spectra, each corresponding to an angular coverage of , which translates into at the galaxy. The NIFS angular resolution with ALTAIR in the K-band is as measured from the FWHM of the spatial profile of the telluric standard stars.

In the left panel of Fig. 3 we present a reconstructed image obtained from the data cube collapsing the spectra in the same wavelength range of the NICMOS F222M filter, i.e. from to . The right panel of the figure presents three characteristic spectra extracted from the data cube: The nuclear spectrum (position N in the continuum map), a spectrum from a location at NW of the nucleus (position A) and another from SE of the nucleus (position B). The only emission-line present in the nuclear spectrum is the broad Br at . The CO absorption bands used to obtain the stellar kinematics are identified in the spectrum from position B.

3 Line of sight velocity distribution

In order to obtain the line-of-sight velocity distribution (LOSVD) we have used the penalized Pixel-Fitting (pPXF) method5 of Cappellari & Emsellem (2004) to fit the stellar absorptions bands present in the -band spectra. The algorithm finds the best fit to a galaxy spectrum by convolving template stellar spectra with the corresponding LOSVD. This procedure gives as output the radial velocity , velocity dispersion and higher-order Gauss-Hermite moments h and h. The pPXF method allows the use of several template stellar spectra and to vary the weights of the contribution of the different templates to obtain the best fit, minimizing the template mismatch problem. We use the templates from the spectroscopic library of late spectral type stars6 observed with the Gemini Near Infrared Spectrograph (Winge, Riffel & Storchi-Bergmann, 2009), which have an almost identical spectral resolution to that of our data. We have used 60 templates in the pPXF fits, and for all spaxels of the Monte-Carlo simulations. We have also restricted the fit to the interval shown in Fig. 4, as for smaller wavelengths there are no features to constrain the stellar kinematics and there may be some contamination from emission lines.

In Fig. 4 we present the resulting pPXF fits to the galaxy spectra from the positions A (top panel), N (middle panel) and B (bottom panel) indicated in Fig. 3. The cyan lines are the galaxy spectra, the black lines are the pPXF fits to the galaxy spectra and the red lines are the residuals. The high signal-to-noise ratio of the IFU data, in the range of to with a average signal-to-noise ratio of , allows us to obtain fits of very high quality to the observed galaxy spectra without the need for spatial binning. The resulting kinematic maps of the velocity distribution are presented in Fig. 5. The top left panel shows the velocity field after subtraction of the galaxy systemic velocity of 453 km s, determined as the stellar velocity at the location of the peak of the continuum. The centroid velocity field presents a maximum of  km s and a rotation pattern with the line of nodes oriented along the position angle ° and with the SE side approaching and the NW side receding. The top right panel shows the map of the stellar velocity dispersion, varying in the range  km s, with a maximum at the location of the peak of the continuum. The average velocity dispersion over the whole field of view is  km s. In appendix B we make a comparison of the resulting kinematics with that presented in the paper of Siopis et al. (2009), obtained with the HST/STIS instrument. As pointed out in the Introduction, the radius of influence of the SMBH in NGC4258 is R15 pc, that corresponds to 0.42 arcsec at the galaxy distance. Fig. 5 shows that inside this radius there is indeed a sharp rise in the values as expected for the region where the SMBH dominates the gravitational potential, what is probably better seen in Fig. 6.The lower panels show the Gauss-Hermite and moments with values ranging from to .

Figure 5: Two-dimensional stellar kinematic maps of NGC 4258. Top left: centroid velocity field. Top right: velocity dispersion. Bottom: Gauss-Hermite moments (left) and (right). Orientation as in Fig. 3 where the negative y-axis is at 145 degrees relative to the North direction.

The errors in the centroid velocities and velocity dispersions were calculated from 850 Monte Carlo simulations and are illustrated in Fig. 6. The left panel of this figure presents the values of the measured centroid velocity () obtained from a cut along the galaxy major axis (black points) and the 1 error bar from the Monte Carlo simulation (shaded band). The average standard deviation of the velocities is  km s, with a maximum of  km s in the central pixel. Ignoring the region around the galaxy minor axis, where the velocities are approximately zero, the average velocity error is lower than . The right panel shows de values of the measured velocity dispersions () (black points) and the 1 error bar (shaded band) along the galaxy major axis. The average standard deviation in the velocity dispersion is  km s, with a maximum of in the central pixel, an average error corresponding to . The errors in the central region of the data cube are higher than at the edges because of the presence of absorption features stronger in the nucleus than outside, that we attribute to the effect of dust and dilution by a red continuum from the AGN dusty torus.

Figure 6: Measured values and Monte Carlo error estimation for the velocities and velocity dispersions along the galaxy major axis. Left: Measured values of the velocities with the 1 error bar obtained from 850 Monte Carlo simulations. Right: Measured values of the velocity dispersions with the 1 error bar.

4 Dynamical Modelling

4.1 Surface brightness distribution

In order to model the galaxy surface brightness distribution we use the mgefitsectors package7 (Cappellari, 2002) which performs a Multi-Gaussian Expansion (MGE) parametrization for the observed surface brightness of the galaxy (Emsellem, Monnet & Bacon, 1994). In what follows, we present the most relevant steps in applying the MGE model to the galaxy NGC 4258 (see Cappellari (2008) for a complete description of the method).

The surface brightness distribution in the plane of the sky can be described by the sum of a set of two-dimensional concentric Gaussians as

(1)

where is the number of the Gaussians, each with total luminosity , axial ratio between and dispersion along the major axis (). In this reference system, the axis is aligned with the galaxy photometric major axis, derived from the photometry of the 2MASS image, which is oriented at PA=°, and points to the line of sight. Before comparing (1) with the observed surface brightness distribution it is convolved with a MGE model for the NICMOS PSF. The NICMOS PSF was obtained with the tiny tim software (Krist et al., 2010). The MGE parameters of the modelled NICMOS PSF are presented in Tab. 4 of Appendix A. In Fig. 14 we present a comparison of the MGE model and the tiny tim PSF. To examine how important are the effects of the small differences between the MGE model and the tiny tim PSF in the convolution procedure, we present in Fig. 15 a comparison between the surface brightness distribution convolved with the tiny tim PSF and the one convolved with the MGE model for the PSF. Fortunately the two convolved surface brightness distributions are very similar. This comparison show that the two are practically indistinguishable.

Note: The first column lists the surface brightnesses of each Gaussian in units L; the second column list the Gaussian dispersions along the major axis in arcseconds; the third column lists the axial ratios of the Gaussians.

Table 1: Gaussian parameters of the MGE model.

In order to reproduce the surface brightness distribution of NGC 4258 we fit together the 2MASS and NICMOS K-band images using the mgefitsectors package. NGC 4258 has an angular size of . The NICMOS image has an angular size of and is used to fit the surface brightness distribution in the central region of the galaxy. Due to the large 2MASS PSF we do not use the the 2MASS image in the fit of the nuclear region (). From the radius to the edges of the galaxy we fit the two images (NICMOS and 2MASS) together. As NGC 4258 is a spiral galaxy with a small bulge and a large disc component we implemented a two-component MGE parametrization by separating the Gaussians in two sets with different constraints for the axial ratios (). The first with to model the disc component, and the second set with more flexible constraints, , to model the other galaxy components.

After several attempts we obtained a satisfying fit of the galaxy surface brightness distribution using a set of 12 Gaussians centred in the galaxy nucleus and with the major axis aligned with the galaxy photometric major axis. In Tab. 1 we present the parameters of the Gaussians of our best fit: the first column lists the surface brightnesses () in units of , the second column lists the Gaussian dispersions in and the third column lists the axial ratios . In Fig. 7 we present linear cuts across the galaxy centre: along the galaxy photometric major axis in the left panel and along the photometric minor axis in the right panel. The cyan continuous lines correspond to the MGE surface brightness distribution, the black open circles correspond to the observed brightness distribution in the NICMOS image and the red points correspond to the MGE after convolution with the NICMOS PSF (PSF). In both panels the modelled convolved surface brightness distribution provides a good reproduction of the observed one for the inner 6 arcseconds of the galaxy that is approximately two times the size of the region where we have kinematic measurements. Due the presence of the spiral arms and of isophotal twist the surface brightness distribution in the outer region of the galaxy can not be well reproduced by the MGE model.

Figure 7: MGE model for the central of NGC 4258 compared to the NICMOS image. Left: cuts along galaxy major axis of the modelled surface brightness distribution (blue line), observed surface brightness distribution (black open circles) and the modelled convolved surface brightness distribution (red points). Right: same as in the left for the galaxy minor axis.

The good agreement of the red points with the open circles shows the good quality of the MGE model in describing the galaxy surface brightness distribution.

Mass density and gravitational potential

In order to obtain the deprojected three-dimensional luminosity density we adopt the approximation that the galaxy is axisymmetric. In this case the luminosity density can be obtained from the parameters that describe the projected surface luminosity density. Assuming that the galaxy is oblate-axisymmetric, the luminosity density can be obtained from

(2)

where, , and is the galaxy inclination (° being edge-on). It is important to make clear that this approach can not solve the intrinsic degeneracy of the deprojection.

The galaxy mass density can be described by a set of Gaussians as

(3)

In the self-consistent case, i. e., if only stars contribute, we can obtain the galaxy mass density by the multiplication of the luminous density by a dynamical mass-to-light ratio. In this case the Gaussians in (3) are the same as in (2) with . The gravitational potential generated by this density can be obtained through the Poison equation, .

The contribution of the SMBH to the gravitational potential is modelled by the approximation that the corresponding distribution of matter is given by an extremely narrow spherical Gaussian. The dispersion of the Gaussian is constrained by the resolution of the kinematic data, such that , where is the smallest distance from the black hole that one needs to model.

4.2 The Jeans anisotropic dynamical model

Figure 8: Determination of the best-fitting parameters and the minimization for the model C. Left: the plane for b = 1.1 ( = 0.09). Central: the plane for  = 4.1. Right: the plane for . The values shown at the contours are for the quantity where =25.48. The blue lines indicate the best-fitting parameters in each plane. The two inner contours shown correspond to the values of  = 2.3 and  = 11.8 that are the boundaries of the 1 and 3 confidence regions for the two parameters together (Press et all. (2007)).

A complete description of the derivation of the Jeans equations (Jeans, 1922), can be found in the book of Binney & Tremaine (2008). Here we only provide an overview of some fundamental assumptions of the model. The model assumes that the galaxy is a large system of stars with positions () and velocities () described by a distribution function (DF). If this system is in a steady state under the influence of a smooth gravitational potential, the DF must satisfy the steady-state collisionless Boltzmann equation. Under the assumption that the galaxy has axial symmetry we obtain the axisymmetric Jeans equations in cylindrical coordinates (Binney & Tremaine (2008), equations 4-29). The solutions of these equations provide the velocity moments of the DF and can be compared with the observed kinematics of the galaxy.

Semi-isotropic Jeans dynamical models (Jeans, 1922) of galaxies have been applied for different purposes (Nagai & Miyamoto, 1976; Satoh, 1980; Binney et al., 1990; van der Marel, Binney & Davies, 1990; Emsellem, Monnet & Bacon, 1994; van Albada, Bertin & Stiavelli, 1995; Riciputi et al., 2005). Two of the most interesting applications are to estimate the dynamical mass-to-light ratio of galaxies (van der Marel, 1991; Statler, Dejonghe & Smecker-Hane, 1999; Cappellari et al., 2006; Cortés, Kenney & Hardy, 2008) and to obtain the masses of the SMBH in the nuclei of galaxies (Magorrian et al., 1998; van der Marel et al., 1998; Cretton & van den Bosch, 1999; Joseph et al., 2001).

Cappellari (2008) has presented an anisotropic generalization of the axisymmetric semi-isotropic Jeans formalism. Making the assumption that the velocity ellipsoid is aligned with a cylindrical coordinate system () and that the anisotropy is constant and quantified by , the Jeans equations reduce to (eqs. (8) and (9) of Cappellari (2008)):

(4)
(5)

One advantage of this approach is that, if a MGE model for the galaxy surface brightness is available, the solutions of the Jeans equations, (i. e., the first and second moments in velocity of the distribution function (DF) of the system, and the projections of these moments in the plane of the sky) are given as a function of the Gaussian parameters of the luminosity density (2) and mass density (3). Cappellari (2008) applied this method to determine the mass-to-light ratio and inclination of galaxies classified as fast-rotators in the SAURON survey (de Zeeuw et al., 2002). The effectiveness of this Jeans Anisotropic Multi-Gaussian expansion dynamical model (JAM)8 is shown in Cappellari et al. (2009) where they model the observed kinematics of Centaurus A with the same parameters () obtained from the Schwarzschild orbit superposition method. Another test of the JAM models is presented by Lablanche et al. (2012) where the authors demonstrate the ability of the models in reproducing the anisotropy profile and mass-to-light ratios of realistic N-body collisionless simulations of barred and unbarred galaxies. In Medling et al. (2011) the JAM models are used to determine the mass of the SMBH in he southern component of the pair of interacting galaxies in NGC 6240. In Neumayer & Walcher (2012) they established upper limits for the black hole masses of 9 late type galaxies using the JAM models. Raimundo et al. (2013) uses the JAM models to obtain an upper limit of the black hole mass in MCG-6-30-15. Emsellem (2013) used the JAM models together with N-body simulations to review the hypothesis of the presence of an overmassive black hole in NGC 1277. Another frequent application of the JAM models is the determination of intermediate-mass black holes in globular clusters (Seth et al., 2010; Lützgendorf et al., 2011, 2012, 2013; Feldmeier et al., 2013). The largest applications of the JAM method to date are the studies of the stellar initial mass function (Cappellari et al., 2012) and dynamical scaling relations (Cappellari et al., 2013) of the 260 early-type galaxies of the ATLAS survey.

The anisotropic Jeans method used in this work is significantly different from Schwarzschild method. The former describes the orbital distribution via a few anisotropy parameters, while the latter makes virtually no assumptions on the orbital distribution. A drawback of the Jeans approach is that it can potentially bias the BH mass determination. However this issue is largely overcome by allowing for a variation in the ansiotropy, which can be constrained by integral-field kinematics, as we do here. Previous tests agree with the result of this paper, showing a general consistency between the Schwarzschild and JAM approach Cappellari et al. (2010); Seth et al. (2014). An advantage of the Jeans approach is its good predictive power: It is unable to fit the noise of the systematics in the data and can often be used to detect problems with the data or flag bad data (e.g. Cappellari et al. (2013)). The Jeans solutions are also relatively easy to qualitatively understand. The Schwarzschild approach, given its generality in the adopted orbital distribution, does not suffer from potential bias in the BH mass. However the method can easily fit noise and bad data without raising any concern. The method can easily create unphysical orbital distributions outside the region constrained by the kinematics (e.g. Fig. 2 of Cappellari & McDermid (2005)). For this reason it provides better constraints to the SMBH masses when integral-field data are available out to larger radii (Verolme et al., 2002). Overall, the two methods are complementary and sufficiently different to motivate the re-determination of BH mass presented in this paper, using Jean rather than the previously published Schwarzschild approach.

The velocity second moment

For a galaxy with a surface brightness distribution parametrized by a MGE model as in sec. 4.1, the projection in the plane of the sky of the second moment, of the DF is given in terms of the parameters of the Gaussians (eq. (28) of Cappellari (2008)). This quantity is a function of three free parameters: the galaxy mass-to-light ratio (), the anisotropy () and the mass of the SMBH (). The comparison of the modelled with the measured , where and are shown in Fig. 5, provide the values of the best-fitting parameters for the galaxy.

We start by assuming that the galaxy has a constant anisotropy in the velocity dispersions. Thus the space of parameters has three dimensions: . We then perform a minimization in this space searching for the values of and that best fit the observed kinematics. We weight the minimization by assigning errors to the kinematic measurements that are inversely proportional to the galaxy surface brightness at each location. This is done in an attempt to give comparable contributions to the for all radii sampled by the NIFS kinematics. Without this weighting, the would be artificially dominated by the numerous pixels at large radii, which contain virtually no information on the mass of the central supermassive black hole. In our trials without weighting the kinematics in the minimization we have obtained a significantly larger value for the mass of the SMBH (), but the modelled values of the velocity second moment clearly did not reproduce the values of the observed for the central region of the galaxy.

We considered in a first trial that the galaxy has an inclination of °  (Model A). This inclination is obtained considering that the outer disc of the galaxy () is thin (van Albada, 1980; Siopis et al., 2009). The best-fitting model for this inclination with  = 27.21 is obtained with the parameters ,  = 4.3 and  = -0.11). This model does not reproduce satisfactorily the measured kinematics along the galaxy minor axis.

After several attempts with different values for the galaxy inclination we obtained a satisfactory model for °  (Model C) . The minimum obtained was  = 25.48 for ,  = 4.1 and  = 0.09). In Fig. 8 we show the minimization in the space of parameters. In Tab. 2 we list the best-fitting parameters for three models with different inclinations: the models A and C described before and a third model with an inclination of °  for which the best-fitting parameters are ,  = 4.3 and  = -0.05), that provide a minimum of 26.84. Systematic variations in the best-fitting parameters are observed as the galaxy inclination decreases: the values of the mass of the SMBH and the mass-to-light ratio decrease but the value of the anisotropy parameters increase.

Model [] b ()
(1) (2) (3) (4) (5) (6)
A 72° 5.2 4.3 0.90 (-0.11) 27.21
B 68° 5.0 4.3 0.95 (-0.05) 26.84
C 64° 4.7 4.1 1.1 (0.09) 25.48

Note. Column (1): model designation. Column (2): galaxy inclination. Column (3): best-fitting mass of the SMBH. Column (4): best-fitting K-band mass-to-light ratio. Column (5): best-fitting anisotropy. Column (6): lowest value of the obtained.

Table 2: Summary of the models with constant anisotropy.

In Fig. 9 we present the minimization for for models A (black dotted line), B (open circles) and C (green solid line). The values of the parameters and are the best-fitting values of each model. The vertical line indicates the value of the maser determination, . The green shaded region corresponds of the values that are within 3 confidence intervals for the model with °. The best-fitting mass of the SMBH with 3 of confidence is .

Figure 9: Variation of with the SMBH mass for the models with constant anisotropy. The green solid line represents the minimization of as a function of for the best-fitting model with ° (Model C). The values of in the shaded region correspond to the 3 uncertainty in this parameter, thus the best-fitting mass of the SMBH is . The open circles represent the minimization for the model with ° (Model B). The black dashed line represents the minimization for the model with °  (Model A). The vertical line indicates the value of the maser determination, .

Effects of the PSF in the best-fitting parameters

One important issue in the dynamical models of galaxies is that before comparing the modelled velocities with the measured ones we need to convolve the modelled velocities with the PSF of the kinematic observations. For the Jeans anisotropic dynamical models the convolution is implemented as in Appendix A of Cappellari (2008). We used a model for the NIFS PSF that is obtained from a MGE fit to the observations of three stars observed in the same night of the galaxy observations, as described in sec. A.2. As the time needed for the galaxy observations are longer (600 s) than that of the stars observations (15 s) there is the possibility that we are underestimating the NIFS PSF.

In order to verify the influence of a possible broader PSF (larger FWHM) in the modelled kinematics and best-fitting value for we ran again the model C with , and and performed the minimization for . We used two broader PSF models obtained from the model described in sec. A.2: in the first one we increase the dispersions (listed in second column of Tab. 5) of the Gaussians of the MGE model of the PSF by a factor of 1.5 and in the second we duplicated the values of the dispersions of the Gaussians.

Figure 10: Effects of variations in the PSF width in the best-fitting mass of the SMBH. The green solid line represents the variation of with the mass of the SMBH for the model C. The dashed black line shows the variation of using a model for the NIFS PSF with a dispersion that is larger than that presented in App. A.2, which results in a for . The open circles shows the variation of using a model for the NIFS PSF with a dispersion that is larger than the original and results in a for . The shaded region shows the range of values of that are inside the 3  uncertainty of the best-fitting value of of the reference model (Model C). The vertical line indicates the value of the maser determination, .

In Fig. 10 we present the modifications in the minimization and in the best-fitting value of introduced by the increase of the NIFS PSF by a factor of 1.5 (represented by the black dotted line) and by a factor of 2.0 (represented by the open circles). The continuous line is for model C which provides a for . For the model with the PSF 1.5 times broader we obtained for which is within the 1 uncertainties of the best-fitting model. For the model with the PSF 2 times broader for . For both cases the values for the mass of the SMBH that best reproduce the measured kinematics are inside the 3 confidence interval of the best-fitting model and the modelled velocities still reproduce the observed ones.

Models with variable anisotropy

Figure 11: Comparison between the observed and the best-fitting (Model D). Top left: the observed after symmetrization; the contours correspond to the isovelocity curves of and . Top central: the best-fitting model presenting the superposition of the isovelocity curves of the modelled (red solid lines) and observed (black dotted lines) velocities. Top right: the residual in . Bottom left: linear cuts along the galaxy major axis (), where the black points show the observed symmetrized , the red solid lines are the result of our best-fitting model, the shaded bands are the 1 errors in the velocity measurements. Bottom central and right panels show the same quantities as the left panel along the galaxy minor axis and along the diagonal direction, respectively.

As the solutions of the Jeans equations are presented for each Gaussian individually and the total second moment is the quadratic sum of the contributions of each Gaussian, it is possible to assign different values of anisotropy for each Gaussian of the MGE model that describes the galaxy luminosity density. As exemplified by Krajnović et al. (2009) for the galaxies NGC 2549 and NGC 524 and by Gebhardt et al. (2003), spatial variations in the velocity anisotropy are frequent. In order to test how better the measured kinematics can be reproduced by models with variable anisotropy, we considered two scenarios. In the first we assumed that there is a radial variation in the anisotropy. We considered two different values for the anisotropy parameters , one for the four inner Gaussians that have and another value for the remaining outer Gaussians. Then we performed the minimization for two different inclinations °  (model D) and ° (model E). In a second scenario we assigned one anisotropy value for the Gaussians with axial ratio representative of the disc component and another anisotropy value for the Gaussians with axial ratio representative of the spheroidal component. This minimization was also performed for two values of the galaxy inclination: ° (model F) and ° (model G). For all the models with variable anisotropy we considered a constant mass-to-light ratio of . The best-fitting parameters for the models with variable anisotropy are presented in Tab. 3.

Model [] b () b ()
(1) (2) (3) (4) (5) (6)
D 64° 4.8 1.10 (0.09) 1.05 (0.05) 25.24
E 72° 5.3 1.00 (0.00) 0.90 (-0.11) 28.65
b () b ()
F 64° 4.8 1.25 (0.20) 1.05 (0.05) 25.34
G 72° 5.3 1.30 (0.23) 0.85 (-0.17) 27.90

Note. Column (1): model designation. Column (2): galaxy inclination. Column (3): best-fitting mass of the SMBH. Column (4): for models (D) and (E) these are these are the values of the anisotropy parameter of the inner 4 gaussians, for models (F) and (G) these are the values of the anisotropy parameter of the Gaussians with axial ratio lower than 0.47. Column (5): for models (D) and (E) these are the values of the anisotropy parameter of the outer 8 Gaussians, for models (F) and (G) these are the values of the anisotropy parameter of the Gaussians with axial ratio higher than 0.5. Column (6): lowest value of the obtained.

Table 3: Summary of the models with variable anisotropy

The best-fitting model with was obtained for the inclination °  and with a radially variable anisotropy (Model D). The dynamical parameters of this model are , , and . The average absolute error in the second moment over all pixels of the kinematic field is of . The results of the model for the second moment and the comparison with the observed are shown in Fig. 11.

Figure 12: Variation of the with the SMBH mass for the models with variable anisotropy. The green solid line represents the minimization of as a function of the for the best-fitting model with ° (Model D). The values of in the shaded region corresponds to the 3 confidence interval for this parameter, thus the best-fitting mass of the SMBH is . The black dashed line is the minimization for the model with ° (Model E).

The top left panel presents the observed after bi-symmetrization. The minimum value for is and occurs at the most distant locations from the galaxy centre along the photometric minor axis. The maximum value is at the galaxy centre. The top central panel shows the distribution for the best-fitting model (model D), showing very good agreement with the measured . The top right panel shows the residuals . The differences between the measured and modelled velocities are small, with the highest positive residual being (which represents a error of in relation to the measured velocity in this position), and highest negative residual of representing an error of . The average error over the whole field is . The good agreement between the modelled and measured velocities is also clear in the superposition of the isovelocity curves of the model on those of the measured velocity field presented in the top central panel.

In the bottom panels we present linear cuts across the galaxy centre in three different directions: along the galaxy major axis in the left panel, the minor axis in the central panel and along a diagonal cut in the right panel. The black points show the measured values for with 1 error bars represented by the shaded region. The red solid lines show our best-fitting model for the projected second moment of the velocity (). The red dotted lines show the residuals between the observations and the best-fitting model. Along the galaxy major axis the modelled velocities reproduced very well the observed ones. Along the galaxy minor axis the largest differences occur in the most distant regions from the galaxy centre.

In Fig. 12 we present the minimization for for the models D (green solid line) and E (black dashed line). The vertical line indicates the value of the maser determination, . The green shaded region corresponds to the values that are inside the 3 confidence interval for the model with ° (Model D). The best-fitting mass of the SMBH with 1 of confidence is .

Comparison with previous results

We can now compare the value of the SMBH mass that we have obtained with the previous ones from the literature. When compared with the maser determination (Greenhill et al., 1995; Herrnstein et al., 1999) of our value of is 25 per cent larger.

For comparison, the previous stellar dynamical determination obtained via Schwarzschild models from long-slit data by Siopis et al. (2009) () is 15 per cent lower than the maser value. Making a direct comparison between the two dynamical determinations, there is a difference of in the best-fitting values for the mass of the SMBH. The main factors that contribute to this difference are:

  • As we have shown in Fig. 17 of appendix B the measured velocity dispersions of our NIFS data are considerably higher than those tabulated for the STIS data, being the differences larger than the error bars for most of the points analysed. The differences between the values of the V are consistent with the differences introduced in the modelled velocity second moment by the difference of the mass of the SMBH from the models.

  • Another possible cause for these difference in the mass of the SMBH can be an over-subtraction of the contribution from the AGN to the surface brightness distribution in the NICMOS image. The uncertainties introduced in the model by this factor are discussed in the appendix C.

  • A third factor that could cause the difference in the SMBH mass derived using the two methods are the intrinsic differences between the dynamical models used (Schwarszchild vs. Jeans models). The adopted JAM method makes more restrictive assumptions than Schwarszchild method. Alought the two methods have been shown to generally agree quite well in real galaxies, some differences are not unexpected.

The velocity first moment

In order to model the velocity first moment it is necessary to make extra assumptions in the Jeans equations to specify how the second moment is composed in terms of ordered and random motions. We use the approximation that the tangential component of the velocity first moment of each Gaussian is a function of the difference between the tangential and radial components of the velocity second moment of each Gaussian, . (See Sec. 3.2.1 of Cappellari (2008) for a more complete explanation).

Adopting the above assumption, we modelled the velocity first moment using the best-fitting parameters of model E. A comparison between the observed rotation velocity field and the modelled first moment for this model is presented in Fig. 13.

Figure 13: Comparison between the measured centroid velocity V and the first moment for model D. Top left: measured velocity ; the contours are for the isovelocity curves of . Top central: modelled with the best-fitting values of , , and obtained from the fit of the second moment and . Top right: residuals in . Bottom right: errors in . Bottom left: linear cuts across the galaxy centre along the galaxy major axis. Bottom central: linear cuts across the galaxy centre in a diagonal direction.

The top left panel shows the measured velocity field for . The top central panel shows the modelled velocity first moment with and the best-fitting values of model D. The top right panel shows the residuals. The bottom right panel shows the residuals in . The bottom left and central panels compare the measured and modelled velocity fields along the galaxy major axis and along a diagonal cut. It can be observed that the maximum residuals reach about which correspond to , as they are observed in the regions along the minor axis where the velocities are lowest. The residuals are higher than within the inner . In the outer regions the differences between the model and measurements are lower than . But the bottom panels highlight that the modelled velocity first moment does not reproduce the observed rotation mostly in the region between and (3.5 pc and 30 pc).

The discrepancy between the modelled velocity first moment field and the measured centroid velocity field, does not affect our conclusion about the mass of the SMBH, that is mostly based on the fit of the second moment. The first moment derived from the model depends on an extra assumption: that the tangential anisotropy is the same everywhere, what may not be true. A more likely explanation for the discrepancy is however the inaccuracy in the deprojection of the luminous stellar density from the observed surface brightness. The deprojection is known to be mathematically degenerate (Rybicki, 1987; Gerhard & Binney, 1996), with the effect becoming quite significant at low inclination (Romanowsky & Kochanek, 1997). Our galaxy is not seen close to edge-on and could easily hide a weak nuclear disc which would be invisible in the projected photometry, could be showing up in the velocity field, leading to the observed discrepancy between the model and the data. This statement is based on numerical experiments where we tried to force the model to have a flat disk by making the Gaussians of the MGE to have the same axial ratio. Only in this way one can lower the model inclination and make the inner disk intrinsically very thin within the NIFS FoV. With this forced MGE one can better reproduce the fast increase in rotation and the difference in V between the major and minor axis. Unfortunately the galaxy overall is not well reproduced by an MGE with constant projected ellipticity. For this reason our test only shows that the existence of a very thin disk within the NIFS FoV would improve the fit to the kinematics.

5 Summary and conclusions

We have presented a two-dimensional mapping of the stellar kinematics of the inner () of the galaxy NGC 4258 using NIFS K-band data with a signal-to-noise ratio higher than 50 over most of the observed field, spectral resolution of 5300 and spatial resolution of  4 , allowing to resolve the radius of influence of the SMBH ( 15 pc). The centroid velocity field presents a rotational pattern with a maximum velocity of , with the SE side approaching and the NW side receding. The stellar velocity dispersion presents an abrupt increase within the inner (10 pc), reaching a value of at the nucleus, consistent with the presence of a SMBH there.

In order to model the stellar kinematics we built a Multi-Gaussian Expansion Jeans Anisotropic Dynamical Model. In this model the velocity second moment is a function of three free parameters: the galaxy mass-to-light ratio , the anisotropy in the velocity and the mass of the SMBH . We performed a minimization in this space of parameters in order to search for the values that best reproduce the measured . In order to ensure that all regions of the galaxy, independently of its radial position, have approximately the same relevance in the minimization process we have weighted the minimization by assigning errors to the kinematic measurements that are inversely proportional to the galaxy surface brightness at each location. Without this weighting the best-fitting model does not reproduce the measured in the nuclear region of the galaxy and provides a a wrong (too high) value for the mass of the SMBH.

We have tried models with only one value for the velocity anisotropy and models with two values. The best-fitting model was obtained adopting a galaxy inclination °  and considering that the galaxy has a radially variable anisotropy in the velocity second moment, being the values of the anisotropy parameter for the inner 4 Gaussians and for the remaining 8 outer Gaussians .

Considering the 3 confidence intervals, we have obtained a mass-to-light ratio of and the best-fitting SMBH mass of . These 3 uncertainties of our model are comparable to the typical values for the uncertainties of stellar dynamical models for other galaxies, that usually are lower than for the mass of the SMBH McConnell et al. (2011). This provides further confirmation of the robustness of the stellar dynamical determinations. It is worth noting that an accurate BH masses was obtained even when using a simple model and only fitting the kinematics within the innermost few arcseconds, without the need for a large-scale model of the galaxy dynamics.

Acknowledgements

Based on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Science and Technology Facilities Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciência e Tecnologia (Brazil) and SECYT (Argentina). This work has been partially supported by the Brazilian institution CNPq. MC acknowledges support from a Royal Society University Research Fellowship. We thank the referee, Remco van den Bosch for the valuable suggestions that helped to improve the paper.

Appendix A MGE Parameters for the NICMOS and NIFS PSF

In this appendix we present the MGE models for the NICMOS and NIFS PSF’s and the agreement of the MGE model of the NICMOS PSF with the on of the tiny tim model.

a.1 The NICMOS PSF

We modelled the NICMOS PSF by a set of four approximately circular Gaussians using the MGE method. The resulting Gaussian parameters are presented in Tab. 4.

Table 4: Gaussian parameters of the MGE model for the NICMOS PSF.

In Fig. 14 we present the comparison between the luminosity profiles of the tiny tim PSF showed by the black open circles and our MGE modelled PSF showed by the blue solid line.

Figure 14: NICMOS PSF fit. Left: linear cut along the detector horizontal direction showing the radial profile of the NICMOS PSF of the tiny tim model (black open circles) and of the resulting MGE model for the PSF (blue solid line). Right: Same as the left but for the detector vertical direction.

In order to verify the effects of the small differences between the MGE model and the real PSF in the convolution procedure we present in Fig. 15 a comparison of the convolution of the surface brightness distribution of NGC 4252 presented in sec. 4.1 with the tiny tim PSF (black open circles) and with our MGE model for the NICMOS PSF (red solid line) for the central of the galaxy. The left panel shows a linear cut across the galaxy centre and along the photometric major axis and the right panel along the galaxy minor axis. In both directions the differences are very small in most of the regions.

Figure 15: Comparison of the convolution of the MGE model for the surface brightness distribution with the tiny tim PSF and with the MGE model for the PSF. Left: cuts along galaxy major axis. Right: cuts along galaxy minor axis.

a.2 The NIFS PSF

The MGE model of the NIFS PSF is performed using the average image of three reconstructed images from the data cubes of three stars observed with NIFS in the same night of the observations of the galaxy. The Gaussian parameters of the MGE model for the NIFS PSF are presented in Tab. 5.

Table 5: Gaussian parameters of the MGE model for the NIFS PSF.

In Fig. 16 we present a comparison between the resulting MGE model and the NIFS PSF. In the left panel we show a linear cut along the detector horizontal axis of the NIFS PSF (black open circles) and of the MGE model (blue solid line). In the right panel we show a linear cut along the detector vertical axis.

Figure 16: NIFS PSF fit. Left: linear cut along the detector horizontal direction showing the radial profile of the NIFS PSF obtained from observations of three stars (black open circles) and of the resulting MGE model for the PSF (blue solid line). Right: Same as the left but for the detector vertical direction.

Appendix B Comparison between the kinematics measurements of NIFS and STIS

Figure 17: Comparison between STIS and NIFS kinematics along the galaxy major axis. Top: The blue diamonds are the measures of the centroid velocities from our NIFS data with the 1 error bar represented by the shaded region, the black circles with error bars are the measurements from STIS observations. Middle: Velocity dispersions from the NIFS and STIS observations. Bottom: The blue diamonds and black circles are the NIFS and STIS measured V respectively. The dashed line is our best-fitting model for the velocity second moment, the continuous line is the modelled velocity second moment using the mass of the SMBH from the maser determination and the dashed-dotted line is the model using the mass of the SMBH from Siopis et al. (2009).

In this section we present a comparison between the measured velocities and velocity dispersions of our NIFS data and the previous published long-slit data from the HST/STIS instrument used by Siopis et al. (2009) to determine the mass of the SMBH in the nucleus of NGC 4258 through Schwarzschild dynamical orbit superposition. In Tab. 6 we present kinematic measurements with 1 errors from the STIS data (Siopis et al., 2009) for the positions along the galaxy major axis coincident with our NIFS measurements. In column one we list the positions of the extractions; in column two, the values of the centroid velocities obtained from the STIS data; in column three, the values of the centroid velocities from the NIFS data; in columns four and five we present the values of the velocity dispersion measurements from the STIS and NIFS data respectively.


Note: Column 1 are the positions of the extractions along the galaxy major axis; Column 2 are the centroid velocities from STIS observations as tabulate in Table 3 of Siopis et al. (2009); Column 3 are the centroid velocities from the NIFS observations; Column 4 are the velocity dispersions from the STIS observations; Column 5 are the velocity dispersions from the NIFS observations.

Table 6: Comparison between NIFS and STIS kinematic measurements along the galaxy major axis.

The differences between the measured velocities from both instruments are shown in Fig. 17. The top panel shows the centroid velocities: the blue diamonds are the NIFS data with the 1 errors being represent by the shaded grey band, the open circles with the error bars are the STIS data. In the region inside 0.2 arc seconds the STIS data shows a steeper velocity curve than the NIFS data, while for the outer region the error bars overlap. We attribute this difference to the somewhat better angular resolution of the STIS when compared to that of NIFS.

The middle panel shows the velocity dispersions: the NIFS data have values  20  km  s higher than those from the STIS data. We attribute this difference to the smaller FWHM of the STIS PSF and to the different methods and templates used for the determinations: while we have used the stellar templates library of Winge, Riffel & Storchi-Bergmann (2009), Siopis et al. (2009) used only one stellar template (HR 7615).

In the bottom panel we show the resulting V compared to different models. The black dashed line is the velocity second moment along the galaxy major axis from our best-fitting model with the parameters , , and ; the black continuous line is the second moment from a model with the same anisotropy and mass-to-light ratio of the best-fitting model but using the value of ; and the red dashed-dotted line is a model using the value of . While our best-fitting model gives a somewhat larger than that of the maser determination, the Siopis et al. (2009) gives a somewhat smaller, but the NIFS measurements gives V values closer to the one from the model with  = .

Appendix C Subtraction of the point source

Figure 18: Effects of the point source subtraction on the modelled second moment - Test 1. Left panel: The continuous line is a cut along the galaxy major axis of our best-fitting model, the dotted line shows the modelled second moment using the the same dynamical parameters of our best-fitting model and subtracting 10 per cent ( 0.27  10 L) more luminosity in the AGN contribution. The black fil circles represent the resulting model subtracting 10 per cent less luminosity in the AGN subtraction. In the two cases the luminosity is subtract from the innermost gaussian of the MGE model. The shaded band are the 1 confidence intervals for the V. Right panel: Same as the left panel but for the galaxy minor axis.
Figure 19: Effects of the point source subtraction on the modelled second moment - Test 2. Left: the continuous line is a cut along the galaxy major axis showing the values of the velocity second moment for the parameters of our best-fitting model and subtracting 10 per cent less luminosity ( 0.27  10 L ) in the AGN contribution, The dotted line corresponds to the model using the value of the mass of the SMBH of the maser determination (3.82  10 M) and the black bullets is the model using the value of the mass of the SMBH from the previous stellar dynamical determination of Siopis et al. (2009) (3.3  10 M).The shaded band are the 1 confidence intervals for the V.

In section 2.1, we have determined that the luminosity of the AGN contributes with approximately 70 per cent to the total intensity of the central pixel of the NICMOS image. Thus the total luminosity subtracted from the galaxy image in order to eliminate the contribution of the AGN is of the order of  1.9 10. Using the mass-to-light ratio of = 4.1 as obtained from our best-fitting model this corresponds to a mass of  7.8 10. In order of investigate how a possible over- or under-subtraction of the AGN contribution affects the determination of the mass of the SMBH we performed two simplified tests.

In the first test we assume that only the innermost gaussian component of the MGE model is affected by the subtraction of the point source. This is a plausible assumption as all the other Gaussians components of the model extend beyond the first Airy ring of the PSF and after the convolution of the surface brightness distribution with the PSF the Gaussian components become even flatter. We then added and subtracted  0.27 10 (or  1.12 10 ) from the innermost gaussian. In practice this is equivalent to instead of subtracting 70 per cent of the intensity of the central pixel to subtract 60 per cent and 80 per cent respectively. The effects on the modelled velocity second moment for the three different subtractions of the AGN contribution are shown in Fig. 18, that presents cuts along the galaxy major and minor axis showing the variation of the modelled velocity second moments. The continuous line is our original best-fitting model (Model D), the dashed line is a model with the same dynamical parameters of the best-fitting model but with a additional mass of  1.12 10 to the central gaussian, and the shaded band is the 1 confidence region for the kinematic measurements. The black filled circles are the model where we subtracted the same mass from the innermost gaussian. The resulting variation in the velocity second moment of the central pixel of the galaxy for this variation in the mass is of the order of 12 .

In the second test we assumed that the innermost gaussian has a luminosity of  0.27 10 higher, (this corresponds to subtract only 60 percent from the intensity of central pixel, this test is motivated by the fact that this amounth of light corresponds approximately to a stellar mass that is equal to the difference in the values for the mass of the SMBH obtained from our best-fitting model and the value from the maser determination) and we use the parameters of anisotropy and mass-to-light ratio of our best-fitting models to investigate how well the three different determinations for the mass of the SMBH (Maser, STIS, ours) reproduce the observed kinematics. The resulting modelled velocity second moments along the galaxy major and minor axiz are presented respectively in the left and right panels of Fig. 19. The continuous line corresponds to our best-fitting model, the dashed line is the model using the value for the mass of the SMBH from the maser determination and the black filled circles are the model with the value for the mass of the SMBH determined by Siopis et al. (2009). The shaded band is the 1 confidence region for the kinematic measurements. Under these assumptions, the best-fit is obtained for the model with . The model using a mass of for the SMBH shows a poor but a similar fit to the data to that of our model only that our model are close to the upper envelope of the 1 band while the of Siopis et al. (2009) are close to the lower envelope.

But we would to point out that the uncertainty in the determination of the AGN is snaller than 10 per cent. Actualy even an over-subtraction of  5  per cent already leaves a signature on the NICMOS PSF in the image. Our choice of  5  per cent is because it corresponds im mass to the difference between the value of the SMBH mass of our best-fitting model and that of the maser determination.

Footnotes

  1. pagerange: The benchmark black hole in NGC 4258: dynamical models from high-resolution two-dimensional stellar kinematicsLABEL:lastpage
  2. pubyear: 2015
  3. This is the average value of the velocity dispersion in the whole NIFS field of view and not the luminosity weighted velocity dispersion inside a single aperture that contains the whole galaxy bulge (see Sec. 3).
  4. IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
  5. Available from http://purl.org/cappellari/software
  6. Available from: http://www.gemini.edu/sciops/instruments/nearir-resources/?q=node/ 10167
  7. Available from http://purl.org/cappellari/software
  8. Available from http://purl.org/cappellari/software

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