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Astron. Astrophys. 342, 671-686 (1999)

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4. Mass models

Previous authors (Fillmore, Boroson & Dressler 1986; Kent 1988; Kormendy & Westpfahl 1989) noticed that in the bulge of early-type spirals [FORMULA] 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 [FORMULA] [FORMULA] 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 [FORMULA] 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 [FORMULA] [FORMULA], the ionized gas can be considered a tracer of the actual circular velocity. But the limited extension, ([FORMULA] (see Table 1), of our [FORMULA] 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 [FORMULA] 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 technique

We 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: (i) a spheroidal component; (ii) an infinitesimally thin exponential disk; and (iii) a spherical pseudo-isothermal dark halo with density distribution [FORMULA]. The mass contribution of the ionized gas is assumed to be negligible at all radii. The spheroidal and disk components are supposed to have constant [FORMULA] ratios. The total potential is the sum of the (numerically derived) potential of the spheroid plus the (analytical) potentials of the disk and the halo.

The stellar distribution function f is assumed to depend only on two integral of motion [i.e., [FORMULA]]. In these hypotheses the Jeans equations for hydrostatic equilibrium form a closed set that, once solved in the total potential, yields the dynamical quantities to be compared with the observed kinematics, once projected onto the sky plane.

To obtain the potentials of the bulge and the disk, we proceed through several steps. (a) First, the bulge surface brightness is derived from the total one by subtracting the disk. Then, it is deprojected by means of Lucy's algorithm to yield the 3-D luminosity density which, via the [FORMULA] ratio, gives the 3-D mass density of the bulge. Finally, solving Poisson's equation through multipole expansion, we derive the bulge potential (Binney et al. 1990). (b) The exponential disk parameters (scale length [FORMULA], central surface brightness [FORMULA], and inclination i) are chosen according to the best-fit photometric decomposition. If [FORMULA], [FORMULA] and [FORMULA] are the disk parameters resulting from the photometric decomposition, the best-fit model to the observed stellar kinematics was obtained considering exponential disks with [FORMULA], [FORMULA] [FORMULA] [FORMULA], and [FORMULA]. These parameters determine the surface brightness of the disk. Through the disk [FORMULA] ratio we obtain the surface mass density of the disk, and then its potential (Binney & Tremaine 1987).

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 [FORMULA] and the mean azimuthal squared velocities [FORMULA].

To disentangle the respective contributions of the azimuthal velocity dispersion [FORMULA] and the mean stellar motion [FORMULA] to [FORMULA], 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 [FORMULA] in the bulge is assigned to the streaming velocity [FORMULA] as in Satoh (1980). The azimuthal velocity dispersion [FORMULA] in the disk is assumed to be related to [FORMULA] (which is assumed in turn to have an exponential fall-off with central value [FORMULA] and scale-length [FORMULA]) 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: k, the local rotation anisotropy parameter of the bulge; the [FORMULA] ratios of the bulge and the disk; the disk central surface brightness, scale length and inclination; the central value and scale length of the disk's second radial velocity moment; and the halo's central mass density and core radius. To reduce the number of free parameters, in the following we consider only three-component models having same best-fit parameters as no-halo models except for the bulge and disk [FORMULA] ratios. We choose the fit parameters in order to simultaneously reproduce the stellar kinematics at all radii as well as [FORMULA] at large radii.

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 [FORMULA] deviation profiles).

We derive for NGC 2179 and NGC 2775 the seeing cutoffs [FORMULA] and [FORMULA], defined by Peletier et al. (1990) as the radii beyond which the seeing-induced error on the profile is lower than, respectively, 0.05 [FORMULA] in surface brightness and 0.02 in ellipticity. They have expressed [FORMULA] and [FORMULA] for a de Vaucouleurs profile as a function of the seeing FWHM, the effective radius [FORMULA] and the ellipticity [FORMULA]. For the bulge, we obtained [FORMULA] and [FORMULA], and the corresponding seeing cutoffs [FORMULA] and [FORMULA], 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

[EQUATION]

plus an exponential disk of surface-brightness profile

[EQUATION]

We assumed the minor-axis profiles of each component to be the same as the major-axis profiles, with values scaled by a factor [FORMULA]. A least-squares fit of the model to the photometric data provided [FORMULA], [FORMULA] and [FORMULA] of the bulge, [FORMULA], [FORMULA] of the disk, and the galaxy inclination i. The values of [FORMULA] and [FORMULA] were used as a starting input to derive [FORMULA] and [FORMULA]. Following van der Marel (1991), the ellipticity [FORMULA] and the [FORMULA] Fourier coefficients were kept constant within [FORMULA] to their value at [FORMULA], and the surface-brightness profile was truncated at its value at [FORMULA]. A new parametric bulge-disk decomposition was then performed on the truncated photometric data. The resulting new values of the effective radius and ellipticity of the bulge were in turn used to obtain a further estimate of [FORMULA] and [FORMULA]. The surface photometry was again modified according to these new values, and then another parametric fitting was done. The process was repeated up to convergence.

A least-squares fit to [FORMULA] in the bulge-dominated region beyond [FORMULA] was performed using the 2-D brightness distribution resulting from the 3-D luminosity density given for a spherical body by:

(i) a modified Hubble law (Rood et al. 1972):

[EQUATION]

where [FORMULA] and [FORMULA] are respectively the central luminosity density and the core radius;

(ii) a Jaffe law (Jaffe 1983):

[EQUATION]

where [FORMULA] and [FORMULA] are the total luminosity and the half-light radius;

(iii) a Hernquist law (Hernquist 1990):

[EQUATION]

where [FORMULA] and [FORMULA] 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 [FORMULA] profiles of the two galaxies to [FORMULA]. 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 r with [FORMULA] where q is the flattening and [FORMULA] cylindrical coordinates.

4.2. The modeling results

In this section we present the mass models of NGC 2179 and NGC 2775.

4.2.1. NGC 2179

In Fig. 6 we show the R-band surface brightness ([FORMULA]), ellipticity ([FORMULA]) and [FORMULA] Fourier coefficient of the isophote deviations from elliptical, as a function of radius along the major axis.

[FIGURE] Fig. 6. The NGC 2179 [FORMULA]band surface brightness, ellipticity and [FORMULA] coefficient profiles as a function of radius along the major axis. The arrows in the surface brightness and in the ellipticity panels indicate the position of the seeing cutoff radii [FORMULA] and [FORMULA]. The dashed curve in the upper panel is the surface brightness exponential profile of the disk component of the best fit model to the stellar kinematics

The seeing cutoffs are [FORMULA] and [FORMULA]. The corresponding best-fit parameters obtained from the photometric decomposition are: [FORMULA] [FORMULA][FORMULA], [FORMULA], [FORMULA] for the de Vaucouleurs bulge; [FORMULA] [FORMULA][FORMULA], [FORMULA], [FORMULA] 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 [FORMULA] [FORMULA][FORMULA], [FORMULA] kpc, and [FORMULA]; 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 [FORMULA]). The 3-D luminosity density of the final bulge model along the same four axes is also presented in Fig. 7 (left panel).

[FIGURE] Fig. 7. The deprojection of the surface brightness of the spheroidal component of NGC 2179. The right panel shows the difference [FORMULA] after each Lucy iteration (dashed lines ) starting from an initial Hernquist fit to the actual NGC 2179 bulge brightness. The residuals are shown for four axes (major through minor axis: top through bottom). For each set of curves: the solid line corresponds to the projected adopted model for 3-D luminosity density; and the dotted line corresponds to a perfect deprojection. At each iteration if the model is brighter than the galaxy, the corresponding dashed or continuous curve is below the dotted line . In the left panel the final 3-D luminosity density profile of the spheroidal components of NGC 2179 is given in units of [FORMULA] for the same four axes (minor through major axis: innermost through outermost curve)

We fold [FORMULA] and [FORMULA] 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, [FORMULA], and the heliocentric velocity of the galaxy, [FORMULA]. We find [FORMULA] [FORMULA] for both gas and stars, and [FORMULA] and [FORMULA]. We then fold [FORMULA] and [FORMULA] 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 ([FORMULA]) with [FORMULA] = 6.1 (M/L[FORMULA]. The exponential disk has [FORMULA] [FORMULA] with scale-length [FORMULA] kpc), and [FORMULA] = 6.1 (M/L[FORMULA]. The derived bulge and disk masses are [FORMULA] M[FORMULA] and [FORMULA] M[FORMULA], adding up to a total (bulge + disk) luminous mass of [FORMULA] M[FORMULA]. The DM halo has [FORMULA] [FORMULA] and [FORMULA] kpc, which correspond to an asymptotic rotation velocity [FORMULA] [FORMULA]; its mass at the outermost observed radius is [FORMULA] M[FORMULA]. The ratio between the mass-to-light ratios of the stellar components in the models with and without the DM halo is 0.9.

[FIGURE] Fig. 8. Predicted and observed stellar kinematics of NGC 2179. The filled triangles and the open squares represent data derived for the approaching SE side and for the receding NW side respectively. The continuous and the dashed lines are the model stellar velocities (upper panel ) and velocity dispersions (lower panel ) obtained with and without the dark halo component respectively

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 ([FORMULA]). This result hinges on accuracy of the gas kinematics data beyond [FORMULA] mostly derived from H[FORMULA] 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.

[FIGURE] Fig. 9a and b. In the upper panel the predicted circular velocity and the observed gas rotation velocity of NGC 2179 are shown. The filled triangles and the open squares represent data derived for the approaching side and for the receding side respectively. The continuous and the dashed lines are the total circular velocities derived from the stellar kinematics with and without the dark halo component respectively. In the lower panel the maximum contribution of the luminous matter (dashed line ) to the total circular velocity (continuous line ) is plotted if a dark matter halo (dotted line ) is considered

4.2.2. NGC 2775

In Fig. 10 we show [FORMULA] and [FORMULA] along the major axis (Kent 1988). The seeing FWHM for the Kent (1988) data is [FORMULA]. We derive the seeing cutoffs [FORMULA] and [FORMULA]. As no [FORMULA] profile is available, we assume it is zero throughout.

[FIGURE] Fig. 10. The NGC 2775 [FORMULA]band surface brightness and the ellipticity profiles as a function of radius along the major axis (Kent 1988). The dashed curve in the top panel is the surface brightness exponential profile of the disk component of the best fit model to the stellar kinematics

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 [FORMULA] 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 [FORMULA], a center [FORMULA] and a scale-length [FORMULA]. This leads to a disk light contribution

[EQUATION]

The best-fit parameters resulting from the improved photometric decomposition are: [FORMULA] r-[FORMULA], [FORMULA], [FORMULA] for the de Vaucouleurs bulge; [FORMULA] r-[FORMULA], [FORMULA], [FORMULA] for the exponential disk; and [FORMULA], [FORMULA] and [FORMULA] for the dust ring. The exponential disk yielding to the best-fit model to the observed stellar kinematics has [FORMULA] r-[FORMULA], [FORMULA] kpc, and [FORMULA] (see Fig. 10).

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 [FORMULA], 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.

[FIGURE] Fig. 11. Same as Fig. 7, but for NGC 2775 and starting from a flattened modified Hubble model for the spheroidal component

We find [FORMULA] and [FORMULA] to have same center of symmetry at [FORMULA] and [FORMULA] [FORMULA]. 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 ([FORMULA]) with [FORMULA] = 5.2 (M/L[FORMULA]. The exponential disk has [FORMULA] [FORMULA] with scale-length [FORMULA] kpc), and [FORMULA] = 7.0 (M/L[FORMULA]. The derived bulge and disk masses are [FORMULA] M[FORMULA] and [FORMULA] M[FORMULA], so the total (bulge + disk) luminous mass is [FORMULA] M[FORMULA]. We kept the ratio between the bulge and disk [FORMULA] fixed at the value 1.36 (see Kent 1988). The DM halo has central density [FORMULA] [FORMULA] and core radius [FORMULA] kpc, which correspond to an asymptotic rotation velocity [FORMULA] [FORMULA]; its mass at the outermost observed radius is [FORMULA] M[FORMULA]. (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.

[FIGURE] Fig. 12. Same as Fig. 8, but for NGC 2775. The filled triangles and the open squares represent data derived for the receding SE side and for the approaching NW side respectively. The dotted line is the model kinematics for the spheroidal component

For [FORMULA] the presence on the equatorial plane of the dust ring, which reduces the light contribution of the (faster-rotating) disk stars of a further [FORMULA], 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 [FORMULA] times larger than the mean S0/Sa values (Bregman, Hogg & Roberts 1992).

Although the total luminous mass found by Kent (1988) [FORMULA] M[FORMULA] (with [FORMULA] [FORMULA] Mpc-1) is in good agreement with ours, his mass decomposition differs from ours. He assumed only the bulge to have an analytical [FORMULA] (a de Vaucouleurs law with [FORMULA] [FORMULA], [FORMULA], [FORMULA]), while the disk [FORMULA] was taken to be the bulge-subtracted major-axis profile. Kent's approach is opposite to ours. We assume the disk to have an analytical [FORMULA], and the bulge [FORMULA] to be the residual surface brightness after subtracting the disk (assuming no a priori analytical expression or fixed axis ratio). Scaled to our assumed distance, the luminosity of Kent's bulge is [FORMULA] of ours, while our disk luminosity is [FORMULA] of Kent's.

Van der Marel et al. (1991) studied the effects of a [FORMULA] deviation on the kinematics of NGC 4261. They found that changing the [FORMULA] Fourier component from zero to [FORMULA] produces variations of [FORMULA] in velocity dispersion and [FORMULA] in rotation velocity. Fig. 13 shows the stellar kinematics of NGC 2775 in the case of slightly disky ([FORMULA]) and slightly boxy ([FORMULA]) isophotes (solid and dotted line, respectively). The disky model rotates faster and has a lower [FORMULA] than the boxy model in the inner [FORMULA] (1.9 kpc). The differences in [FORMULA] and in [FORMULA] between the two models are [FORMULA] [FORMULA]: this means that, in the observed range of values, a difference of 0.04 in [FORMULA] coefficients corresponds to a difference of [FORMULA] in velocities and [FORMULA] in velocity dispersions. However, these uncertainties are immaterial to our results on the mass structure of NGC 2775.

[FIGURE] Fig. 13. Predicted stellar kinematics for NGC 2775 with fixed [FORMULA] (solid line ) and fixed [FORMULA] (dotted line ) at all radii

The comparison between [FORMULA] 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 ([FORMULA]). 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 [FORMULA] on the receding arm, [FORMULA] and [FORMULA] [FORMULA]. 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.

[FIGURE] Fig. 14a and b. Same as Fig. 9, but for NGC 2775. The filled triangles and the open squares represent data derived for the receding SE side and for the approaching NW side respectively

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Online publication: February 23, 1999
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