## 4. Mass modelsPrevious authors (Fillmore, Boroson & Dressler 1986; Kent 1988; Kormendy & Westpfahl 1989) noticed that in the bulge of early-type spirals falls below the predicted circular velocity. Such `slowly rising' gas RCs are explained by Bertola et al. (1995) with the argument that random (non-circular) motions are crucial for the dynamical support of the ionized gas: in some galaxies of their S0 sample they measured over an extended range of radii. We do observe the same phenomenon in some of our early-type spirals (see Fig. 5): this fact prevents us to adopt, for early-type disk galaxies, the inner portion of as the circular velocity on which to perform the mass decomposition. When the high values for the velocity dispersion of the gas are measured only in the very central parts (as in NGC 2179) we can not exclude that this is an effect of rotational broadening due to the seeing smearing of the steep velocity gradient. At larger radii where , the ionized gas can be considered a tracer of the actual circular velocity. But the limited extension, ( (see Table 1), of our makes the derivation of the halo parameters of early-type spirals more uncertain than for later types. In fact, on one hand we lack data at very large radii where only disk and halo affect the circular velocity, on the other hand at small radii (where we can not consider the gas in circular motion) not two (like for Sc-Sd galaxies), but three mass components will have locally similar behaviors (solid-body like). So, in absence of extended and complete RCs, an Sa mass solution would be degenerate. We therefore have to model the stellar kinematics to determine the galaxy's total gravitational potential. Then we need to check the derived mass decomposition by comparing the circular velocity, inferred from the model, with at large galactocentric radii. This is necessary to minimize uncertainties on the mass structure obtained from the stellar kinematics. In fact, the uncertain orbital structure of the spheroidal component, consistent with the observed kinematics leads to a degeneracy between velocity anisotropy and mass distribution, which can be solved only through the knowledge of the line-of-sight velocity distribution profiles (Gerhard 1993). Two galaxies of the observed sample, namely NGC 2179 and NGC 2775, are particularly suited to be studied with the three-component mass models, based on stellar photometry and kinematics, at our disposal. They were chosen for their nearly axisymmetric stellar pattern and to not contain kinematically decoupled (as found NGC 3593 and NGC 4698) or triaxial (as found in NGC 4845) stellar components. ## 4.1. The modeling techniqueWe apply the Jeans modeling technique introduced by Binney, Davies & Illingworth (1990), developed by van der Marel, Binney & Davies (1990) and van der Marel (1991), and extended to two-component galaxies by Cinzano & van der Marel (1994) and to galaxies with a DM halo by Cinzano (1995). (For a detailed description of the model and its assumptions, see the above references.) The galaxy is assumed to be axisymmetric. Its mass structure
results from the contributions of: The stellar distribution function To obtain the potentials of the bulge and the disk, we proceed
through several steps. We first solved the Jeans equations only for both the bulge and disk components in their total potential, to give in every point of the galaxy the velocity dispersions onto the meridional plane and the mean azimuthal squared velocities . To disentangle the respective contributions of the azimuthal velocity dispersion and the mean stellar motion to , for the bulge we made the same hypotheses of Binney et al. (1990) while for the disk we followed Cinzano & van der Marel (1994) respectively. Part of the second azimuthal velocity moment in the bulge is assigned to the streaming velocity as in Satoh (1980). The azimuthal velocity dispersion in the disk is assumed to be related to (which is assumed in turn to have an exponential fall-off with central value and scale-length ) according to the epicyclic theory (cfr. Binney & Tremaine 1987). In the framework of Cinzano & van der Marel (1994), we have to take into account the effects of seeing, of finite slit-width and pixel-size in data acquisition, and of Fourier filtering in data reduction (notably the wavenumber range, as discussed in Sect. 2.1.), in order to compare the sky-projected model predictions with the observed stellar kinematics. We interpret the discrepancy between the model's circular velocity
and the observed gas rotation in the outer regions as due to the
presence of a DM halo. In this case, the Jeans equations have to be
solved again, taking the halo into account, too. By introducing the DM
halo, the number of free parameters of the model increases to ten.
They are: The modeling technique described above derives the 3-D distribution of the luminous mass from the 3-D luminosity distribution inferred from the observed surface photometry. For this reason, in the central regions we take into account the seeing effects on the measured photometrical quantities (surface brightness, ellipticity and deviation profiles). We derive for NGC 2179 and NGC 2775 the seeing cutoffs and , defined by Peletier et al. (1990) as the radii beyond which the seeing-induced error on the profile is lower than, respectively, 0.05 in surface brightness and 0.02 in ellipticity. They have expressed and for a de Vaucouleurs profile as a function of the seeing FWHM, the effective radius and the ellipticity . For the bulge, we obtained and , and the corresponding seeing cutoffs and , following an iterative procedure. We started by performing a standard bulge-disk decomposition with a parametric fit (e.g. Kent 1985): we decomposed the observed surface-brightness profile on both the major and the minor axis as the sum of a de Vaucouleurs bulge of surface-brightness profile plus an exponential disk of surface-brightness profile We assumed the minor-axis profiles of each component to be the same
as the major-axis profiles, with values scaled by a factor
. A least-squares fit of the model
to the photometric data provided ,
and
of the bulge,
,
of the disk, and the galaxy inclination A least-squares fit to in the bulge-dominated region beyond was performed using the 2-D brightness distribution resulting from the 3-D luminosity density given for a spherical body by:
where and are respectively the central luminosity density and the core radius;
where and are the total luminosity and the half-light radius;
where and are the total luminosity and a scale radius. The best fit was achieved for NGC 2179 with a Hernquist
profile and for NGC 2775 with a modified Hubble profile. We used
them to extrapolate the profiles of
the two galaxies to . After
subtracting the disk contribution from the total surface brightness,
the 3-D luminosity density of the bulge was obtained starting Lucy's
iterations from a flattened Hernquist model for NGC 2179 and from
a flattened modified Hubble model for NGC 2775. The radial
profiles of these flattened models are derived respectively from
Eqs. 6 and 4 by replacing ## 4.2. The modeling resultsIn this section we present the mass models of NGC 2179 and NGC 2775. ## 4.2.1. NGC 2179In Fig. 6 we show the
The seeing cutoffs are and . The corresponding best-fit parameters obtained from the photometric decomposition are: , , for the de Vaucouleurs bulge; , , for the exponential disk. (Taking into account the photometric bulge-disk decomposition, the exponential disk yielding the best-fit model to the observed stellar kinematics has , kpc, and ; see dashed curve in Fig. 6). We then subtract the disk contribution from the total surface brightness. The residual surface brightness is the contribution of the spheroidal component. The difference between the surface brightness of the spheroid and that obtained projecting the 3-D luminosity distribution of each of the four Lucy iterations (including the initial flatted Hernquist model) is shown in Fig. 7 (right panel) along NGC 2179's major, minor and two intermediate axes. (The r.m.s. residual of the last Lucy iteration corresponds to 0.06509 ). The 3-D luminosity density of the final bulge model along the same four axes is also presented in Fig. 7 (left panel).
We fold and around their respective centers of symmetry. In order to determine the latter, we fit a suitable odd function to both RCs independently: this yields the position of the kinematical center of the curve, , and the heliocentric velocity of the galaxy, . We find for both gas and stars, and and . We then fold and around the kinematical center of the respective component. The best-fit model to the observed major-axis stellar kinematics is shown in Fig. 8. Its parameters are as follows. The bulge is an oblate isotropic rotator () with = 6.1 (M/L. The exponential disk has with scale-length kpc), and = 6.1 (M/L. The derived bulge and disk masses are M and M, adding up to a total (bulge + disk) luminous mass of M. The DM halo has and kpc, which correspond to an asymptotic rotation velocity ; its mass at the outermost observed radius is M. The ratio between the mass-to-light ratios of the stellar components in the models with and without the DM halo is 0.9.
The comparison between the observed rotation of the ionized gas and the true circular velocity, inferred from stellar kinematics, is given in the upper panel Fig. 9. It shows that a DM halo is unambiguously required to explain the rotation at large radii (). This result hinges on accuracy of the gas kinematics data beyond mostly derived from H line measurements. In NGC 2179, the gas rotation does provide the circular velocity at all radii. The contribution of the DM halo to NGC 2179 circular velocity as function of radius is plotted in lower panel of Fig. 9.
## 4.2.2. NGC 2775In Fig. 10 we show and along the major axis (Kent 1988). The seeing FWHM for the Kent (1988) data is . We derive the seeing cutoffs and . As no profile is available, we assume it is zero throughout.
The photometric model is improved by taking into account an outer dust lane surrounding the spiral pattern of the galaxy. (This appears as a thin dust ring at about from the center, and is visible on panels 78 and 87 of the CAG, see Sandage & Bedke 1994). As a fitting function for the disk component, we use an exponential profile weighted by an absorption ring. We assume the section of the dust ring to have a Gaussian radial profile, defined by central intensity maximum absorption , a center and a scale-length . This leads to a disk light contribution The best-fit parameters resulting from the improved photometric
decomposition are:
The surface brightness of the exponential disk in Fig. 10 is subtracted from the total surface brightness. The fit to the spheroidal component's deprojected surface brightness is obtained after four Lucy iterations from an initial flatted modified Hubble model (The r.m.s. residual is 0.03209 , see Fig. 11). The 3-D luminosity density of the final bulge model along the major, minor and two intermediate axes is also shown in Fig. 11.
We find and to have same center of symmetry at and . The velocity dispersion profiles are folded around the kinematical center. The best-fit model to the observed major-axis stellar kinematics is shown in Fig. 12. Its parameters are as follows. The bulge is an oblate isotropic rotator () with = 5.2 (M/L. The exponential disk has with scale-length kpc), and = 7.0 (M/L. The derived bulge and disk masses are M and M, so the total (bulge + disk) luminous mass is M. We kept the ratio between the bulge and disk fixed at the value 1.36 (see Kent 1988). The DM halo has central density and core radius kpc, which correspond to an asymptotic rotation velocity ; its mass at the outermost observed radius is M. (The latter should be considered an upper limit because the inner kinematics can be explained with no DM halo). Contrary to NGC 2179 in this case we kept the same mass-to-light ratios of the stellar components in the models with and without the DM halo.
For the presence on the equatorial plane of the dust ring, which reduces the light contribution of the (faster-rotating) disk stars of a further , causes the observed stellar kinematics to be more affected by the (slower-rotating) bulge stars. In this picture as shown in Fig. 12, our bulge model agrees with the observed drop in velocity and the rise in velocity dispersion. NGC 2775 is a dust-rich system, as can be inferred from its dust-to-HI mass ratio (Roberts et al. 1991), which is times larger than the mean S0/Sa values (Bregman, Hogg & Roberts 1992). Although the total luminous mass found by Kent (1988)
M
(with
Mpc Van der Marel et al. (1991) studied the effects of a deviation on the kinematics of NGC 4261. They found that changing the Fourier component from zero to produces variations of in velocity dispersion and in rotation velocity. Fig. 13 shows the stellar kinematics of NGC 2775 in the case of slightly disky () and slightly boxy () isophotes (solid and dotted line, respectively). The disky model rotates faster and has a lower than the boxy model in the inner (1.9 kpc). The differences in and in between the two models are : this means that, in the observed range of values, a difference of 0.04 in coefficients corresponds to a difference of in velocities and in velocity dispersions. However, these uncertainties are immaterial to our results on the mass structure of NGC 2775.
The comparison between and the true circular velocity inferred from the stellar kinematics is shown in the upper panel of Fig. 14. It shows that a DM halo is not strictly required to explain the rotation at large radii (). The contribution of the DM halo to NGC 2775 circular velocity as a function of radius is plotted in the lower panel of Fig. 14. Inside on the receding arm, and . This rules out the case that the gas kinematics is dominated by random motions, and leads us to speculate that we are looking at gas rotating on a non-equatorial plane. We suggest this is the signature of a past external acquisition (possibly from the companion galaxy NGC 2777) of gas still not completely settled onto the disk plane.
© European Southern Observatory (ESO) 1999 Online publication: February 23, 1999 |