The benchmark black hole in NGC 4258: dynamical models from highresolution twodimensional stellar kinematics
Abstract
NGC 4258 is the galaxy with the most accurate (maserbased) determination for the mass of the supermassive black hole (SMBH) in its nucleus. In this work we present a twodimensional mapping of the stellar kinematics in the inner 3.0 3.0 arcsec = 100 100 pc of NGC 4258 using adaptativeoptics observations obtained with the NearInfrared 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 multigaussian expansion method to parametrize the observed surface brightness distribution. The Jeans dynamical model has three free parameters: the mass of the central SMBH (), the massluminosity 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 bestfitting 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 longslit 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 42581 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 nonstellar 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 powerlaw , which seems to extend through the optical to the ultraviolet (Yuan et al., 2002). In Xrays, the nuclear spectrum presents several components: two powerlaws, a thermal component and the Fe K emissionline (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 SawadaSatoh et al. (2007) have mapped the large scale molecular CO(21) 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 longslit spectra obtained with the Space Telescope Imaging Spectrograph (HSTSTIS) 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 longslit spectra from HSTSTIS 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
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 masstolight ratio – affect the stellar kinematics and the determination of the mass of the SMBH in NGC 4258.
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 twodimensional 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 nearIR Kband with the instrument NIFS (McGregor et al., 2003) operating with the ALTAIR adaptive optics module at the 8m Gemini North telescope in April 2007 under the science program GN2007AQ25. The Kband 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 NearInfrared 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
Nicmos
Nearinfrared Kband 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 iraf
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 oversubtraction) 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 undersubtraction 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 Kshortband (Ksband) from Jarrett et al. (2003). The 2MASS image has a scale of and the Ksfilter 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.
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 ObjectSkyObjectSkyObjectSkyObject dither sequence, with offsource 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, flatfielding, sky subtraction, wavelength and sdistortion 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 Kband 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 emissionline 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 lineofsight velocity distribution (LOSVD)
we have used the penalized PixelFitting (pPXF) method
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 signaltonoise ratio of the IFU data, in the range of to with a average signaltonoise 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 GaussHermite and moments with values ranging from to .
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.
4 Dynamical Modelling
4.1 Surface brightness distribution
In order to model the galaxy surface brightness distribution we use
the mgefitsectors package
The surface brightness distribution in the plane of the sky can be described by the sum of a set of twodimensional 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.
In order to reproduce the surface brightness distribution of NGC 4258 we fit together the 2MASS and NICMOS Kband 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 twocomponent 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.
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 threedimensional 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 oblateaxisymmetric, the luminosity density can be obtained from
(2) 
where, , and is the galaxy inclination (° being edgeon). 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 selfconsistent case, i. e., if only stars contribute, we can obtain the galaxy mass density by the multiplication of the luminous density by a dynamical masstolight 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
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 steadystate 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 429). The solutions of these equations provide the velocity moments of the DF and can be compared with the observed kinematics of the galaxy.
Semiisotropic 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 masstolight ratio of galaxies (van der Marel, 1991; Statler, Dejonghe & SmeckerHane, 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 semiisotropic 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 masstolight ratio and inclination of galaxies
classified as fastrotators in the SAURON survey (de Zeeuw et al., 2002). The
effectiveness of this Jeans Anisotropic MultiGaussian expansion dynamical model
(JAM)
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 integralfield 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 integralfield data are available out to larger radii (Verolme et al., 2002). Overall, the two methods are complementary and sufficiently different to motivate the redetermination 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 masstolight 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 bestfitting 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 bestfitting 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 bestfitting 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 bestfitting parameters are , = 4.3 and = 0.05), that provide a minimum of 26.84. Systematic variations in the bestfitting parameters are observed as the galaxy inclination decreases: the values of the mass of the SMBH and the masstolight 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): bestfitting mass of the SMBH. Column (4): bestfitting Kband masstolight ratio. Column (5): bestfitting anisotropy. Column (6): lowest value of the obtained.
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 bestfitting 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 bestfitting mass of the SMBH with 3 of confidence is .
Effects of the PSF in the bestfitting 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 bestfitting 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.
In Fig. 10 we present the modifications in the minimization and in the bestfitting 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 bestfitting 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 bestfitting model and the modelled velocities still reproduce the observed ones.
Models with variable anisotropy
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 masstolight ratio of . The bestfitting 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): bestfitting 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.
The bestfitting 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.
The top left panel presents the observed after bisymmetrization. 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 bestfitting 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 bestfitting model for the projected second moment of the velocity (). The red dotted lines show the residuals between the observations and the bestfitting 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 bestfitting 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 longslit 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 bestfitting 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 oversubtraction 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 bestfitting 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.
The top left panel shows the measured velocity field for . The top central panel shows the modelled velocity first moment with and the bestfitting 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 edgeon 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 twodimensional mapping of the stellar kinematics of the inner () of the galaxy NGC 4258 using NIFS Kband data with a signaltonoise 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 MultiGaussian Expansion Jeans Anisotropic Dynamical Model. In this model the velocity second moment is a function of three free parameters: the galaxy masstolight 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 bestfitting 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 bestfitting 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 masstolight ratio of and the bestfitting 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 largescale 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.
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.
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.
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.
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.
Appendix B Comparison between the kinematics measurements of NIFS and STIS
In this section we present a comparison between the measured velocities and velocity dispersions of our NIFS data and the previous published longslit 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.
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 & StorchiBergmann (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 bestfitting model with the parameters , , and ; the black continuous line is the second moment from a model with the same anisotropy and masstolight ratio of the bestfitting model but using the value of ; and the red dasheddotted line is a model using the value of . While our bestfitting 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
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 masstolight ratio of = 4.1 as obtained from our bestfitting model this corresponds to a mass of 7.8 10. In order of investigate how a possible over or undersubtraction 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 bestfitting model (Model D), the dashed line is a model with the same dynamical parameters of the bestfitting 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 bestfitting model and the value from the maser determination) and we use the parameters of anisotropy and masstolight ratio of our bestfitting 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 bestfitting 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 bestfit 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 oversubtraction 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 bestfitting model and that of the maser determination.
Footnotes
 pagerange: The benchmark black hole in NGC 4258: dynamical models from highresolution twodimensional stellar kinematics–LABEL:lastpage
 pubyear: 2015
 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).
 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.
 Available from http://purl.org/cappellari/software
 Available from: http://www.gemini.edu/sciops/instruments/nearirresources/?q=node/ 10167
 Available from http://purl.org/cappellari/software
 Available from http://purl.org/cappellari/software
References
 Binney J., Tremaine S., 2008, Galactic Dynamics: Second Edition, Princeton University Press, Princeton, NJ USA
 Binney J., Davies R. L., Illingworth G. D., 1990, ApJ, 361, 78
 Cappellari M., Emsellem E., 2004, PASP, 116, 138
 Cappellari, M., 2002, MNRAS, 333, 400
 Cappellari M., McDermid, R. M. 2005, Class. Quantum Gravity, 22, 347
 Cappellari M., Bacon R., Bureau M., et al. 2006, MNRAS, 366, 1126
 Cappellari M. 2008, MNRAS, 390, 71
 Cappellari M., Neumayer N., Reunanen J., et al., 2009, MNRAS, 394, 660
 Cappellari M. et al., 2010, in American Institute of Physics Conference Series, 1240, 211
 Cappellari, M. et al., 2012, Nature, 484, 485
 Cappellari, M., McDermid, R. M., Alatalo, K., et al. 2013, MNRAS, 432, 1862
 Cappellari M., et al. 2013, MNRAS, 432, 1709
 Cecil G., Wilson A. S., Tully R. B., 1992, ApJ, 390, 365
 Cecil G., Greenhill L. J., DePree C. G., Nagar N., Wilson A. S., Dopita M. A., PérezFournon I., Argon A. L., Moran J. M., 2000, ApJ, 536, 675
 Chary R., Becklin E. E., Evans A. S., Neugebauer G., Scoville N. Z., Matthews K., Ressler M. E. 2000, ApJ, 531, 756
 Chandrasekhar S., 1969, The Silliman Foundation Lectures, New Haven: Yale University Press
 Cortés J. R., Kenney J. D. P., Hardy E., 2008, ApJ, 683, 78
 Cretton N., van den Bosch, F. C., 1999, ApJ, 514, 704
 de Zeeuw P. T. et al., 2002, MNRAS, 329, 513
 Emsellem E., 2013, MNRAS, 433, 1862
 Emsellem E., Monnet G., Bacon R., 1994, A&A, 285, 723
 Feldmeier A. et al., 2013, A&A, 554, A63
 Gerhard O. E., Binney, J. J., 1996, MNRAS, 279, 993
 Gebhardt K., et al., 2003, ApJ, 583, 92
 Greenhill L. J., Jiang D. R., Moran J. M., Reid M. J., Lo K. Y., Claussen M. J., 1995, ApJ, 440, 419
 Herrnstein J. R., Moran J. M., Greenhill L. J., Trotter A. S., 2005, ApJ, 629, 719
 Herrnstein J. R., 1999, Nature, 400, 539
 Jarrett T. H., Chester T., Cutri R., Schneider S. E., Huchra J. P., 2003, AJ, 125, 525
 Jeans, J. H., 1922, MNRAS, 82, 122
 Joseph, C. L. et al., 2001, ApJ, 550, 668
 Krajnović D., McDermid R. M., Cappellari M., Davies R. L., 2009, MNRAS, 399, 1839
 Krist J., Hook R., Stoehr F., 2010, Astrophysics Source Code Library, 10057
 Lablanche, P. Y. et al. 2012, MNRAS, 424, 1495
 Lucy L. B. 1974, AJ, 79, 745
 Lützgendorf, N., KisslerPatig, M., Noyola, E., Jalali B., de Zeeuw P. T., Gebhardt K., Baumgardt H., 2011, A&A, 533, A36
 Lützgendorf N., KisslerPatig M., Gebhardt K., Baumgardt H., Noyola, E., Jalali B., de Zeeuw P. T., Neumayer, N., 2012, A&A, 542, A129
 Lützgendorf N. et al. 2012, A&A, 543, A82
 Lützgendorf N. et al. 2013, A&A, 552, A49
 Lyubenova M. et al., 2013, MNRAS, 431, 3364
 Magorrian J. et al., 1998, AJ, 115, 2285
 McConnell N. J., Ma C.P., Gebhardt K., Lauer Tod R., Graham James R. 2011, Nature, 480, 215
 McGregor P. J.et al., 2003, Proc. SPIE, 4841, p. 1581
 Medling A. M., Ammons S. M., Max C. E., Davies R. I., Engel H., Canalizo G., 2011, ApJ, 743, 32
 Miyoshi M., Moran J., Herrnstein J., Greenhill L., Nakai N., Diamond P., Inoue M., 1995, Nature, 373, 127
 Nagai R., Miyamoto, M., 1976, PASJ, 28, 1
 Neumayer N., Walcher, C. J., 2012, Advances in Astronomy, 2012
 Pastorini, G. et al., 2007, å, 469, 405
 Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P., 2007, Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press
 Raimundo S. I., Davies R. I., Gandhi P., Fabian A. C., Canning R. E. A., Ivanov V. D., 2013, MNRAS, 431, 2294
 Richardson, W. H., 1972, JOSA (19171983), 62, 55
 Riciputi A., Lanzoni B., Bonoli S., Ciotti L., 2005, A&A, 443, 133
 Riffel, R. A., StorchiBergmann, T., Winge, C., McGregor, P., Brck, T., Schmitt, H., 2008, MNRAS, 385, 1129
 Romanowsky A. J., Kochanek C. S., 1997, MNRAS, 287, 35
 Rybicki G. B., 1987, in IAU Symposium, Vol. 127, Structure and Dynamics of Elliptical Galaxies, de Zeeuw P. T., ed., p. 397
 Satoh C. 1980, PASJ, 32, 41
 SawadaSatoh S., Ho P. T. P., Muller S., Matsushita S., Lim J., 2007, ApJ, 658, 851
 Schwarzschild M., 1979, ApJ, 232, 236
 Scoville N., 1997, HST Proposal, 7230
 Seth A. C. et al., 2014, NATURE, 513, 398
 Seth A. et al., 2010, AIP Conference Series, 1240, 227
 Siopis C. et al., 2009, ApJ, 693, 946
 Statler T. S., Dejonghe H., SmeckerHane T., 1999, AJ, 117, 126
 Tremaine, S., Richstone, D. O., Byun, Y.I., et al. 1994, AJ, 107, 634
 van Albada G. D. 1980, å, 90, 123
 van Albada T. S., Bertin G., Stiavelli M., 1995, MNRAS, 276, 1255
 van der Kruit P. C., 1974, ApJ, 192, 1
 van der Marel R. P., Binney J., Davies R. L., 1990, MNRAS, 245, 582
 van der Marel R. P., 1991, MNRAS, 253, 710
 van der Marel R. P., Cretton N., de Zeeuw P. T., Rix H.W., 1998, ApJ, 493, 613
 Verolme E. K., Cappellari M., Copin Y., et al. 2002, MNRAS, 335, 517
 Wilkes B. J., Schmidt G. D., Smith P. S., Mathur S., McLeod K. K., 1995, ApJ, 455L, 13
 Wilson A. S., Yang Y., Cecil G., 2001, ApJ, 560, 689
 Winge C., Riffel R. A., StorchiBergmann, T., 2009, ApJS, 185, 186
 Yang Y., Li B., Wilson A. S., Reynolds C. S., 2007, ApJ, 660, 1106
 Yuan F., Markoff S., Falcke H., Biermann, P. L., 2002, å, 391, 139