4 Models

In this Section, we present the methods used to model the photometry and kinematics of the central region of NGC 4621. The available data suggest that the very central region (100 pc) of NGC 4621 slightly departs from axisymmetry. The CRC seems off-centered in both the OASIS and STIS data. However, the off-centering is only $\sim$4 pc. At the resolution of the BSG94 data, this is obviously not resolved. Moreover, the position angle measured in the OHP photometry does not vary more than 2 degrees within 15$^\prime$. Axisymmetry is therefore a reasonable approximation for a first modelling. Axisymmetric, two-integral models then have the advantage to be semi-analytical. We first derived simple Jeans models to roughly constrain the input parameters (inclination, mass-to-light ratio). We then used the Hunter & Qian (1993) formalism to compute the two-integral distribution function f of NGC 4621, as a function of energy (E) and of the vertical component of angular momentum (L_z). The present modelling is only intended to provide a first view at the dynamics of NGC 4621, so we decided not to include a central dark component: this issue will be examined in detail in a forthcoming paper.

Figure 3: Inner isophotes of NGC 4621 (WFPC2 F555W, solid contours, step of 0.2 mag). The center (0,0) of the galaxy is defined as the center of the outer isophotes (outside 0 $.\!\!^{\prime\prime}$1). The open triangle and filled circle correspond to the center of the counter-rotating core (CRC) as measured in the OASIS and STIS data repectively (see Sect. 3.1). The vertical bars correspond to the OASIS and STIS spatial resolutions ($\sigma$). The dashed contours represent the V-I colour map from WFPC2: levels 1.36 and 1.37 mag. North is 73 degrees clockwise from the vertical axis.

Figure 4: V-I map (WFPC2, F555W and F814W filters). Left: convolved with a gaussian of $\sigma=0\hbox{$.\!\!^{\prime\prime}$ }1$. Right: convolved with the PSF of the OASIS data (0 $.\!\!^{\prime\prime}$51 FWHM). Contour separation is 0.005 mag. A y-axis elongated central structure can be observed at HST resolution (see also Fig. 3).

Figure 5: Top panels: multi Gaussian Expansion fit (thick contours) of NGC 4621 superimposed on the V band isophotes (thin contours). Top left: OHP V-band photometry. Top right : HST/WFPC2 F555W band image (isophote step of 0.4 mag/arcsec2). Notice the nuclear disc in the HST data. Bottom left panel: NGC 4621 light profiles along r2=x2/a2+y2/b2, where a and b are the semi major and minor axes of fitted ellipse respectively. Crosses correspond to the WFPC2 image, circles to the deconvolved WFPC1 data from Byun et al. (1996). The dashed and dotted curves correspond to the convolved and deconvolved MGE models respectively. Bottom right panel: ellipticity profile, with Michard's (solid bold line) and WFPC2 (solid hairline) data, along with the MGE model (dotted line).

4.1 MGE

 

 

Table 1: MGE photometric model of NGC 4621 in the V band. The flattened inner disc components are emphasized in bold.

	${I} (L_{\odot}~{\rm pc}^{-2})$	${\sigma}(\hbox{$^{\prime\prime}$ })$	q
1	4.502 $\times$ 105	0.040	0.860
2	7.713 $\times$ 104	0.112	0.610
3	4.792 $\times$ 104	0.201	0.941
4	1.161 $\times$ 104	0.438	0.344
5	1.475 $\times$ 104	0.516	0.919
6	4.638 $\times$ 103	1.036	0.325
7	6.317 $\times$ 103	1.280	0.872
8	3.357 $\times$ 103	2.486	0.275
9	2.700 $\times$ 103	3.211	0.658
10	1.326 $\times$ 103	5.698	0.817
11	6.313 $\times$ 102	6.926	0.377
12	6.417 $\times$ 102	12.468	0.639
13	3.295 $\times$ 102	25.674	0.627
14	8.208 $\times$ 101	57.091	0.633
15	1.517 $\times$ 101	128.782	1.000

The first step in modelling NGC 4621 was to build a luminosity model which properly reproduced the observed photometry. We used the Multi Gaussian Expansion (MGE) formalism proposed by Monnet et al. (1992) and Emsellem et al. (1994), which expresses the surface brightness as a sum of two-dimensional Gaussians. Assuming the spatial luminosity is also a sum of (three-dimensional) Gaussians, given the choice of viewing angles, and using an MGE model for the PSF, we could then deconvolve and deproject the MGE model uniquely and analytically.

The procedure was performed stepwise. We began fitting the wide field V-band image. We then subtracted the outer gaussian components from the high resolution WFPC2 image, and fitted the residual image (central 15 $^{\prime\prime}$). In this step, we had to exclude the innermost 0 $.\!\!^{\prime\prime}$4, to avoid convergence problems probably due to the slightly asymmetric central feature (Fig. 3). We finally fitted the very central arcsecond. Gathering these three parts, we thus obtained a 15 Gaussian components model, with the same center and PA, the parameters of which are given in Table 1. The goodness of the fit is illustrated in Fig. 5.

4.2 Two-integral models

We then made use of the Hunter & Qian (1993) formalism to build the two-integral distribution function of the galaxy using the best fit value for the mass-to-light ratio of $\Upsilon_V=6.6\ M_\odot/L_\odot$ found from simple Jeans models. We used a default value of $i = 90\hbox{$^\circ$ }$, which produced a marginally better fit.

The DF is divided into two parts, which are respectively even and odd in L_z. The even part ( $f_{\mbox{\tiny even}}$) is uniquely determined by the input MGE mass model. This involves the calculation of a path integral in the complex plane, as described by Hunter & Qian (1993). The odd part of the distribution function ( $f_{\mbox{\tiny odd}}$) is then parametrized, and adjusted to fit the kinematical data. We chose the parametrization proposed by van der Marel et al. (1994), and modified it to account for the CRC. The original parametrization corresponds to Eq. (1), and we additionally allowed $\eta$ to be function of $E_{\rm p}=E/E_{\mbox{\tiny max}}$(Fig. 6). The analytical form of this $\eta$ function is the same as the one in Eq. (2), with an additional variable change. We can adjust the energy $E_{\mbox{\tiny crit}}$ above which the stars begin to counter-rotate, as well as the smoothness of the transition ($a=\infty$: abrupt transition, a=1: smooth transition, see Fig. 6).

  

$$% f_{\mbox{\tiny odd}}(E,L_z) (2\eta-1)\ h_{\alpha}(L_z/L_{z,\mbox{\tiny max}})\cdot f_{\mbox{\tiny even}}(E,L_z)$$ = (1)

$$h_{\alpha}(x) \left\{ \begin{array}{ll} \tanh(\alpha x/2)/\tanh(\alpha/2) & (\a... ...\\ (2/\alpha)\mbox{arctanh}[x\tanh(\alpha/2)] & (\alpha<0). \end{array}\right.$$ = (2)

The best fit model reproduces the BSG94 velocity profiles reasonably well (Fig. 7) with values of $\alpha$ ranging from 8 outside 10 $^{\prime\prime}$ to -2 in the central part. The higher resolution OASIS (Figs. 8 and 1), and STIS (Fig. 2) velocity measurements, revealing the counter-rotating core, are also well fitted by this two-integral model. The best fitting model uses a core with a diameter of 1 $.\!\!^{\prime\prime}$1 ( $E_{\tiny\mbox{crit}}=0.62$), and an abrupt transition (a=100 i.e. almost all stars having $E_{\rm p}>E_{\tiny\mbox{crit}}$ are counter-rotating, see bold line in Fig. 6). A rough estimate of the CRC mass can be made by selecting stars counter-rotating in the central part. This is performed by integrating the DF weighted by a function which is 0 for $E < E_{\tiny\mbox{crit}}$ and L_z > 0 and 1 otherwise. The total mass of NGC 4621, which is given by the mass-to-light ratio and the deprojected MGE-model is $1.78 \times 10^{11}\ M_{\odot}$. The total mass of the CRC is $2.13 \times 10^{8}\ M_{\odot}$, yielding a mass fraction of 0.12%.

Figure 6: Parametrization of $\eta$. $E_{\rm p}$ is the normalized energy $E_{\rm p}=E/E_{\mbox{\tiny max}}$. $E_{\mbox{\tiny crit}}$ is the energy at which the counter-rotation starts, e.g. when $\eta <0.5$, $f_{\mbox{\tiny odd}}<0$ (see Eq. (1)). The bold line corresponds to the best-fitting model.

The dispersion profiles are well reproduced by the model outside the central few arcseconds. The central values of the dispersion predicted by the models are however systematically too low compared to the BSG94 observations. This is confirmed by the OASIS and STIS dispersion values: the model predicts a central dispersion of $\sim$220 km s^-1, to be compared with the central observed STIS dispersion of $320\pm27$ km s^-1. We were finally unable to fit the higher order moments, even at large radii. The h₃ values predicted by the model are thus systematically too high, by a factor of almost two. This discrepancy could not be solved even by changing the parameters of the odd part of the DF.

These two discrepancies do indicate that we need a more general dynamical model for NGC 4621. First, we should remove the constraint imposed by the two-integral model by allowing a third integral of motion. There may then still be the need for an additional central dark mass to explain the observed dispersion values. Such a model will be examined in a forthcoming paper (Wernli et al., in preparation).

Figure 7: Best fit two-integral DF model. Left: major axis. Right: Minor axis. Inclination is 90 degrees, no black hole and $\Upsilon _V=6.06$. Points are the BSG94 data, and the curve is the model. The fit is reasonable but indicates the need for a black hole (low central $\sigma$) and maybe a third integral (overestimated h3).

Figure 8: Best fit two-integral DF model for the comparison with OASIS. Only the velocity map is shown: central dispersion values are too low due to the absense of a central dark mass. Grey levels range from -50 to +50 km s-1.