5 Results

For each of the sample galaxies we calculated dynamical models using the method described above, for a series of potentials (between 20 and 150 per galaxy). A given potential can be parametrized by the value of the circular velocity at the radius of the last kinematic data point, $v_{\rm c}(R_{\rm max})$. $\chi ^2$ diagrams (plots of resulting model $\chi ^2$against $v_{\rm c}(R_{\rm max})$) together with confidence intervals as described in Sect. 4.2 are shown in Figs. 10 (for the sample with new kinematic data) and 11 (for the sample with spectroscopic data from BSG94). This grouping is maintained in subsequent plots because of the different radial range covered by the two sets of kinematic data. From these figures we select three models for each galaxy: two from the boundaries of the confidence interval and one of the best-fitting models from the middle of the confidence range. These models are depicted with the same linestyles in all the plots to follow: dashed lines for the two boundary models, and a solid line for the respective "best'' model. Note that this "best'' model does not necessarily have the smallest value of $\chi ^2$ since it is chosen in the middle of the range of valid models. In figures showing model fits to the kinematic data, or in circular velocity curve plots, the self-consistent (SC) models are additionally shown as dotted lines.

Figures 12 to 17 present the model fits to the kinematic data for all galaxies analysed. For each galaxy three panels are shown: the upper panel shows the velocity dispersion $\sigma$, the middle panel shows the h₄ parameter, and the lower panel shows the intrinsic anisotropy parameter $\beta$ defined as

$$\beta = 1 - \frac{\sigma_{\rm t}^2}{\sigma_{\rm r}^2}$$ (2)

where $\sigma_{\rm t}$ and $\sigma_{\rm r}$ are the one-dimensional tangential and radial velocity dispersion components.

The derived anisotropy profiles are surprisingly uniform. For most galaxies, the best-fit models are moderately anisotropic at $R_{\rm e}/2$, with typical , in a few cases . Towards the center most turn towards isotropy; a clear exception is NGC 2434. In a number of galaxies the transition to a near-isotropic center occurs in the inner 3'' and is uncertain because of the low spatial resolution of the data; in others, however, it starts at $R\simeq 10''$ and is therefore real (NGC 4374, 3379, 4168, 4472, 4486). We have also checked, using the data for NGC 1399, that the shape of the inner anisotropy profile is not dependent on the set of basis functions used in the modelling; see Fig. 12 and Sect. 5.3. In the outer parts, the $\beta$-profiles also generally turn back towards isotropy or even tangential anisotropy, but due to the finite range of the kinematic data, the detailed values near the outer edge depend on the halo mass distribution and the errors on $\beta$ there are large.

The circular velocity curves (CVCs) resulting from these model potentials are shown in Figs. 18 and 19. They are plotted as a function of $R/R_{\rm e}$, and extend to the radius of the outermost kinematic data point. All CVCs are consistent with being flat outside $R/R_{\rm e}\sim 0.3$ at the $\sim 10\%$ level. Extended flat rotation curves, however, can only be demonstrated for about eight of the galaxies with new kinematic data. In some of these, e.g., NGC 1399, 3379, the CVC rises to a maximum at $R/R_{\rm e}\sim 0.3 \sim 10''$, before falling by about $10\%$ to reach the flat part at $R/R_{\rm e}\sim 1$. In others, e.g., NGC 2434, 7626, no such maximum is seen before the flat part of the CVC is reached. In some, e.g., NGC 315, the situation is unclear because the data do not extend far enough. The same is true for many of the galaxies from the BSG94 sample, where also in some cases the poorer quality of the data (e.g., in NGC 4486B) causes additional uncertainty.

Figures 20 and 21 show the corresponding cumulative mass-to-light ratios in the B-band for all sample galaxies. For each case, the permitted range is indicated by plotting the two boundary models and the "best'' model from the central region of the confidence interval. Numerical values for the central and total mass-to-light ratios within the range of the kinematic data are given in Table 7.

 

 

Table 7: Circular velocities at the outermost kinematic data point, central B-band M/L ratios, and ratios of the B-band M/L at the outermost kinematic data point over the central value. All data are given for the "best-fitting'' model as well as for two models from both sides of the confidence interval (low and high). Column 1 lists the NGC number, Cols. 2-4 the circular velocity at the last kinematic data point, Col. 5 gives the M/L of the SC model for those galaxies for which this is within the confidence interval, Cols. 6-8 the central M/L values, and Cols. 9-11 the M/L ratios

	low	best	high		low	best	high	low	best	high
NGC	$v_{\rm c}(R_{\rm max})$	$v_{\rm c}(R_{\rm max})$	$v_{\rm c}(R_{\rm max})$	$M/L_{\rm SC}$	$M/L_{\rm B}^{\rm c}$	$M/L_{\rm B}^{\rm c}$	$M/L_{\rm B}^{\rm c}$	$M/L_{\rm B}$ ratio	$M/L_{\rm B}$ ratio	$M/L_{\rm B}$ ratio
315	521.6	568.7	639.7		11.0	10.5	9.6	1.468	1.830	2.533
1399	373.8	424.1	466.1		10.9	10.6	10.1	1.060	1.392	1.769
2434	288.2	331.0	371.6		6.2	6.0	6.4	1.417	1.918	2.253
3193	283.9	303.0	334.7		5.0	4.5	3.6	1.000	1.396	2.129
3379	241.5	259.3	286.7		4.7	4.5	4.2	1.000	1.219	1.602
3640	270.8	279.2	288.2	3.7	3.6	3.7	3.7	1.004	1.030	1.097
4168	264.7	286.8	306.2	6.5	6.1	5.8	5.4	1.058	1.295	1.602
4278	398.2	415.9	424.4	8.8	8.6	7.6	6.4	1.005	1.244	1.544
4374	380.8	409.7	441.8	9.5	8.7	8.8	8.3	1.023	1.168	1.457
4472	429.9	463.7	502.0	7.8	6.7	7.0	6.5	1.022	1.134	1.450
4486	491.9	506.5	566.9	11.8	10.9	9.4	8.1	1.044	1.280	1.853
4486B	233.9	248.7	268.5		7.2	7.2	6.2	1.000	1.128	1.543
4494	238.2	260.8	287.5	7.4	7.3	6.5	5.9	1.008	1.369	1.829
4589	311.7	333.2	355.7	6.0	5.4	5.4	5.4	1.088	1.258	1.457
4636	325.6	341.3	351.8	9.6	8.7	8.9	7.8	1.027	1.109	1.336
5846	328.5	338.3	346.0		10.1	9.7	9.5	1.053	1.164	1.244
6703	190.5	221.8	248.4		5.6	5.2	5.0	1.000	1.452	1.885
7145	193.0	210.2	254.5		3.5	3.9	3.7	1.922	2.026	3.162
7192	256.6	269.9	292.0		5.1	4.6	4.7	1.332	1.643	1.876
7507	350.6	399.0	471.8		6.2	5.9	5.9	1.417	1.937	2.700
7626	372.9	401.3	437.5		9.0	8.1	7.8	1.406	1.814	2.223

 

 

Table 8: Comparison with previous mass determinations. Columns 2 and 3 show the newly determined mass-to-light ratios at the innermost (Col. 2) and outermost (Col. 3) kinematic data points and at the half-luminosity radius (Col. 4) determined by Saglia et al. ([1992]), from the respective "best'' model, for part of the sample galaxies. Previous M/L values converted to the B-band and to the same galaxy distances are listed from van der Marel ([1991]; Col. 5), Lauer ([1985]; Col. 6), Saglia et al. ([1992], luminous component, Col. 7, luminous plus dark component at the half-luminosity radius determined by the fit, Col. 8)

NGC	$M/L_{\rm cent}$	$M/L_{\rm max}$	M/L1/2	$M/L_{\rm B}$	$M/L_{\rm B}$	$M_{\rm L}/L$	M/L1/2
				vdM	Lauer	SBS	SBS
1399	10.6	14.7	15.5	8.8	-	10.2	16.4
3379	4.5	5.5	5.1	4.7	8.3	6.0	10.2
4278	7.6	9.5	12.5	-	-	8.8	11.1
4374	8.8	10.3	11.1	9.7	9.4	7.0	17.5
4472	7.0	8.0	11.9	8.0	10.9	6.3	9.6
4486	9.4	12.0	26.1	10.2	-	7.5	11.3
4494	6.5	8.9	-	11.6	-	-	-
4636	8.9	9.8	17.6	9.1	13.3	10.5	14.4
5846	9.7	11.3	-	10.7	12.7	-	-
6703	5.2	7.6	-	-	6.4	-	-
7145	3.9	8.0	-	6.4	-	-	-
7507	5.9	11.3	10.1	5.5	-	5.21	11.31
7626	8.1	14.7	12.7	8.3	10.3	7.7	10.5
(1) From Bertin et al. ([1994]) using the same modelling.

Figure 12: Fits to the kinematic data of NGC 3379 using B-band photometry (top left), NGC 3379 again but using R-band photometry (top right), NGC 315 (bottom left), and NGC 1399 (bottom right). In each panel the uppermost of the three plots shows the velocity dispersion, the middle plot the h4 parameter, and the lower plot the intrinisic velocity anisotropy parameter $\beta$. Note that the velocity dispersion scale changes between different galaxies. For each galaxy we show a "best'' halo model from the middle of the confidence range (full line), two models from the boundary of the confidence interval (dashed lines), and the self-consistent model with constant M/L (dotted line). When only one dashed line is shown, the lower boundary of the confidence interval is represented by the constant M/L model. For NGC 1399 we show an additional model (long dash - dotted line), constructed from a very different set of basis functions in the same "best'' halo potential; this shows that the type of basis used does not significantly influence the derived kinematics (see also Sect. 5.3)

Figure 13: As in Fig. 12 for NGC 2434 (top left), NGC 4374 (top right), NGC 5846 (bottom right) and NGC 6703 (bottom right)

Figure 14: As in Fig. 12 for NGC 7145 (top left), NGC 7192 (top right), NGC 7507 (bottom left) and NGC 7626 (bottom right)

Figure 15: As in Fig. 12 for NGC 3193 (top left), NGC 3640 (top right), NGC 4168 (bottom left) and NGC 4278 (bottom right)

Figure 16: As in Fig. 12 for NGC 4472 (top left), NGC 4486 (top right), NGC 4486B (bottom left) and NGC 4494 (bottom right)

Figure 17: As in Fig. 12 for NGC 4589 (top left), and NGC 4636 (top right)

Figure 18: Inferred circular velocity curves, plotted out to the radius of the last kinematic data point. The "best'' model from the middle of the confidence range is shown as the solid line. The two dashed lines show models at the boundaries of the $95\%$ confidence interval and mark the implied uncertainty within our range of halo models. The self-consistent (SC) model with only luminous mass is shown as the dotted line. In cases where the SC model is the one at the low $v_{\rm c}$ end of the confidence interval, the evidence for extra dark mass is less than $2\sigma$ and only three lines are shown

Figure 19: Circular velocity curves for the sample with kinematic data from BSG94, as in Fig. 18

Figure 20: Inferred mass-to-light ratios in the B-band as function of radius. For NGC 3379 we also show the corresponding R-band plot. The "best'' model from the middle of the confidence range is shown as the solid line. The two dashed lines show models at the boundaries of the $95\%$ confidence interval and mark the implied uncertainty within our range of halo models. When the confidence interval includes the self-consistent model with constant M/L, this is shown as the dotted line. In some cases it is identical with the lower-mass confidence boundary and then replaces the lower dashed line. In other cases it may have slightly more mass than the lower boundary halo model because of a different velocity scaling resulting from the fit to the kinematic data. In such cases the evidence for extra dark mass is less than $2\sigma$

Figure 21: Same as Fig. 20 for the sample of galaxies with kinematics from BSG94

For all galaxies, the respective "best'' model shows an outward increase of M/L. However, the increase ranges from very small (NGC 3379) to about a factor of 2 (NGC 2434). Because of the finite radial range of the data, the modelling will allow some artificially massive halo models (see G+98 for an explanation). Three of the galaxies in Fig. 20 (NGC 3379, NGC 4374, NGC 6703) are consistent with a flat M/L over the range of even the extended kinematic data (the self-consistent constant-M/L model lies inside the $95\%$-confidence line), although in each case the M/L of the "best'' model increases outwards. Both in NGC 1399, NGC 5846 the constant-M/L model lies outside the $95\%$-confidence line. In NGC 5846 we consider this only marginal evidence, given the small radial range covered by the data and the similarity between the models. In NGC 1399 the weak evidence for dark matter is confirmed by X-ray and other data; see below. The rising M/L shown by the models in NGC 7145 is influenced strongly by this galaxy's peculiar inner kinematics. The clearest candidates for radially increasing mass-to-light ratios in this sample are therefore NGC 315, NGC 2434, NGC 7507, and NGC 7626, where in each case the self-consistent model is a poor fit to the data.

In Table 8 we compare our derived mass-to-light ratios with those determined by van der Marel ([1991]), Lauer ([1985]) and Saglia et al. ([1992]). The values given by the first author were transformed to the B-band, assuming a mean extinction-corrected Johnson colour of B-R=1.80. All values were scaled to the distances used in the present paper. The three $M/L_{\rm B}$ values given for each galaxy from our analysis are the central value of the respective "best'' model, corresponding to the "maximal stellar mass'' $M/L_{\rm B}$of the stellar population, the "best'' cumulative $M/L_{\rm B}$ inside the last kinematic data point, and the "best'' cumulative $M/L_{\rm B}$ inside the half-luminosity radii determined by Saglia et al. ([1992]). The latter two values thus refer to the total mass (luminous and dark) out to these radii.

Comparing our central values with those of van der Marel ([1991]) shows that the new mass-to-light ratios are similar in the mean with individual differences generally less than $20\%$ and particularly large discrepancies of $\sim 50\%$ for NGC 7145 and by a factor of two for NGC 4494. The latter galaxy was off in all of van der Marel's correlation plots. van der Marel used axisymmetric two-integral models but did not take into account line-profile shape information. Because these models have the property of flattening the mass distribution by equatorial near-circular orbits (Dehnen & Gerhard [1993]), they generally predicted too much motion on the major axis and thus van der Marel's M/L values were taken to be weighted averages of the minor and major axis fits.

The mass-to-light ratios given by Lauer ([1985]) are estimates based on the standard King formula for isotropic cores, and are always higher than our new determinations. Differences less than $20\%$ are observed between our central M/L values and those for the luminous component of Saglia et al. ([1992]). The M/L ratios computed at the half-luminosity radii used by Saglia et al. ([1992]) also agree reasonably well. However, for NGC 3379 and NGC 4374 Saglia et al. ([1992]) fit a rather massive dark halo not detected here, while for NGC 4486 we find a rather large extrapolated M/L also seen in the X-ray data.

For a few of our sample galaxies other data are available besides the absorption line kinematics, which considerately strengthen the evidence for dark halos in some of these galaxies. Several galaxies have velocity measurements of "discrete tracers'' (planetary nebulae, PNe; globular clusters, GC; dwarf galaxies bound to the respective galaxy, D) at intermediate to large radii; see Table 9 for a list. In these cases we performed a maximum likelihood analysis along the lines of S+2000 to test the consistency of our mass models with these data and select the most probable ones. Results are shown in Figs. 24, 25, and 27 and are discussed in the individual galaxy subsections below. Mass-profiles derived from X-ray data are still rare. S+2000 compared their models for NGC 1399 with mass-profiles derived from ASCA measurements by Ikebe et al. ([1996]). For three further galaxies of our sample, such profiles are also available: Besides the two brightest ellipticals of the Virgo-cluster, NGC 4486 and NGC 4472, for which ROSAT data were analysed by Schindler et al. ([1999]), NGC 4636 has an X-ray mass profile from ASCA-data (Matsushita et al. [1998]). For these three galaxies we compare the mass profiles of our absorption-line models to the X-ray mass profiles: NGC 4486 is shown in Fig. 28, NGC 4472 in Fig. 26 and NGC 4636 in Fig. 29. See the notes on individual galaxies below.

   
5.1 Estimate of uncertainties due to rotation and non-sphericity

A number of galaxies in our sample show slight rotation, and some may be flattened and seen along special viewing directions such as to appear round on the sky. Since our modelling assumes spherical symmetry and no rotation, we need to assess the uncertainties in the results associated with these assumptions.

The most rapidly rotating galaxy in this sample is NGC 3379, which reaches a maximum $(v/\sigma)_{\rm max} \simeq 0.3$. In the modelling reported above, we have simply ignored v and h₃. As an alternative, we have fitted an even velocity distribution to the spectral line shapes, forcing v=h₃=0, and have then tested the effect this has on the modelling results. In this case, the velocity dispersions near maximum rotation are larger by about $10\%$, while the h₄ values become slightly more negative. From simple arguments we would have expected an increase in $\sigma$ by $\simeq (1+0.3^2)^{1/2}$, i.e., about $5\%$. In these kinematics derived from symmetrized line profiles there are significant systematic deviations between both sides of the galaxy.

Modelling these data as before, with the same regularization parameter, gives the following results. The anisotropy profile remains similar. The mass scale increases slightly; in the model without halo, the mass-to-light ratio thus increases to M/L=5.44 (previously 4.7), while the central value in the best-fit halo potential is now M/L=5.27 (previously 4.5). In the three-integral models of Gebhardt et al. (2000) the stellar mass-to-light ratio is in the range 4.94-5.51when corrected to B-band (using B-V=0.94 from the profiles) and to our distance (13.2 instead of 10.4 Mpc). For NGC 3379, the uncertainty in M/L due to the neglect of rotation is therefore $\sim 15\%$. If we scale from these results, in the next most rapidly rotating galaxies in the sample, with $(v/\sigma)_{\rm max} \simeq 0.15$, we will underestimate the rms velocity $\sigma$ by $\simeq 3\%$ and the mass-to-light ratio by $5\%$. For $(v/\sigma)_{\rm max} \simeq 0.1$the corresponding errors will be negligible.

 

  Table 9: Galaxies with kinematics for discrete tracer populations. The number of PNe (planetary nebulae), GC (globular clusters), and D dwarf galaxy velocities is listed for each galaxy. Both for NGC 4472 and NGC 4486 there are 5 confirmed dwarf galaxy companions and 3 further candidates. In these cases all 8 velocities were used for calculating the velocity dispersion. The 20 dwarfs around NGC 5846 (Zabludoff & Mulchaey [1998]) lie too far from the galaxy to constrain our models

NGC	Sort
1399	37 PN, 74 GC
3379	29 PN
4472	57 GC, 8 (5) D
4486	224 GC, 8 (5) D
5846	20 D

We now turn to the effects of a possible flattening along the line-of-sight. From the results of Tremblay & Merritt ([1996]), who modelled the apparent shape distribution of a large sample of ellipticals, we can estimate the typical intrinsic flattening of our sample galaxies. These are bright elliptical galaxies: the large majority falls into the highest luminosity bin of Tremblay & Merritt, even after correcting for the difference in assumed distance. Taking their result for a distribution of triaxial shapes with constant triaxiality parameter (1-b/a)/(1-c/a)=0.3, and approximating their derived distribution of c/a in the bottom panel of their Fig. 4 by a Gaussian, we find that the mean intrinsic short-to-long axis ratio c/a of all luminous galaxies that project to E1 (0.9) or rounder is 0.79. (The corresponding value for the faint ellipticals with $M_{\rm B}=-18.5$, assuming oblate symmetry and using Fig. 3 of Tremblay & Merritt, would be 0.73.) The mean flattening along the line of sight is then somewhat smaller, because not all galaxies that project to E1 or rounder have their short axes along the line-of-sight.

If an apparently round galaxy is flattened along the line-of-sight, this will have the following effects on our results. (i) The line-of-sight velocity dispersion may be less than the dispersion in the plane of the sky if the system is flattened by anisotropy, and it will also not measure any kinetic energy corresponding to rotation ${\bar v}$ in the plane of the sky. This leads to an underestimate of the mass-to-light ratio. We can estimate the magnitude of this effect from the tensor virial theorem, in the form $({\bar v}^2 + 2\sigma_x^2)/\sigma_z^2 \simeq 2(a/c)^{0.9}$ (Gerhard [1994], Eq. 5.10). For a mean c/a=0.8 the total kinetic energy thus increases by a factor 1.15, for constant M/L. At the same time the flattened potential well is deeper for fixed mass and M/L; using relations from Binney & Tremaine ([1987], Sect. 2.5) the total potential energy increases by a factor 1.07. Since the former is linearly proportional to M/L and the latter scales like (M/L)², reestablishing virial equilibrium implies an increase in M/L by about 8%. (For a galaxy with intrinsic c/a=0.6, the corresponding value would be 21%.)

(ii) A more indirect effect is through the shape of the absorption line profiles. This depends critically on the stellar distribution function and hence the mechanism through which the galaxy is flattened (see Dehnen & Gerhard [1993]). If the galaxy is flattened by equatorial near-circular orbits, as is the case in two-integral models f(E,L_z), then these orbits seen face-on near $v\simeq 0$ generate a more peaked line profile with larger h₄, which could mimic some radial anisotropy when interpreted through a spherical model. This effect was investigated by Dehnen & Gerhard ([1994]) and by Magorrian (2000). From Fig. 12 of Dehnen & Gerhard ([1994]) for a de Vaucouleurs-like luminosity model inside $\sim R_{\rm e}$, the increase in h₄ from c/a=1 to c/a=0.8 is $\simeq 0.01$, the density profile being still relatively shallow there (d $\log\rho/$ d $\log r \mathrel{\hbox to 0pt{\lower 3pt\hbox{$\mathchar''218$ }\hss} \raise 2.0pt\hbox{$\mathchar''13E$ }}-2$). There is a simultaneous decrease in the velocity dispersion by about 3-4%, which will cause another upward correction of M/L. The implied $\Delta h_4$ is significantly smaller than the h₄ values responsible for the radial anisotropy in our models. In addition, because most of our galaxies are luminous ellipticals, they are not well-described by two-integral models (e.g., van der Marel [1991]), and for other flattening mechanisms the effect on the line profile shapes will be smaller.

More significant could be the effects of hidden, sufficiently luminous and cold face-on disks, which might lead to a larger increase in h₄. Using Fig. 1 of BSG94, we estimate that such a disk of disk-to-bulge ratio $\epsilon$ would generate $h_4\approx 0.5 \epsilon$, as a compromise between the increased flux at zero velocities due to the disk and the lack of effect on the large velocity tail of the profile also measured by h₄. The histogram of disk-to-bulge ratios for the known kinematically decoupled disks or torus-like structures (Bender & Saglia [1999]) shows that for boxy elliptical galaxies and for disky ellipticals , with some cases (NGC 4660) with $\epsilon$ up to 0.3. Generally these components are fairly small, extending to typical radii $r_{\rm b}/R_{\rm e}\approx 0.1-0.3$ (Mehlert et al. [1998]). They can be detected photometrically only if seen nearly edge-on, but might be present in virtually all ellipticals. The three galaxies of our sample appearing in Mehlert et al., NGC 4472, NGC 4494, NGC 7626, have $r_{\rm b}=6, 7, 8$ arcsec, respectively. Therefore we conclude that the anisotropy profiles should not suffer from the possible presence of disks at radii but might overestimate the real anisotropy in the inner regions of the galaxies by $\approx 0.2$.

Finally, we note that the analysis of the two flattened, non-rotating elliptical galaxies NGC 1600 (E3.5, near edge-on) and NGC 2300 (E2, edge-on or inclined) with three-integral models by Matthias & Gerhard ([1999]) and Kaeppeli ([1999]) resulted in similar anisotropy profiles to those we have obtained in this paper. As a further test, we have for these two galaxies taken the major axis photometric profile and major axis kinematics and applied our spherical modelling to these data, ignoring the flattening. The resulting models slightly underestimate the radial anisotropy as compared to the edge-on three-integral models, and the V-band M/L_Vcome out 5.5 and 6.5, respectively, compared to 6.0 and 6.25 in the axisymmetric analysis, a $5-10\%$ difference.

5.2 NGC 315

This galaxy has a shallow luminosity profile and considerable structure in the kinematic profiles. In the analysis the errors of the two outermost h₄ data points were set to unity to make sure that they do not influence the fit. These points are uncertain due to low signal-to-noise ratio. The dip in the velocity dispersion profile together with the clustering of the data points and the substantial scatter between the inner data points as compared to the kinematic errors (Table 5) make this dataset difficult to model. We obtained more regular results with the second method of treating the presumably systematic errors (see Sect. 4); the models shown in Figs. 12, 18 and 20 are obtained with this method. The $95\%$ confidence line is then at $\chi^2=1.2$.

Our simple halo models appear to be not very well suited for this galaxy. Together with the shallow distribution of the luminous mass these halo potentials favour rising velocity dispersion profiles in the outer parts when the inner data are well-fitted. There is little evidence in the data for this. Therefore while the upper boundary of the confidence interval appears reliable from our model results, the lower boundary should probably include models with a flat velocity curve with $v_{\rm c}\simeq 520{\rm\,km\,s^{-1}}$. Nonetheless, the self-consistent model falls short of fitting the data from about 40''.

   
5.3 NGC 1399

This galaxy is modelled in detail by S+2000. As described there the line-of-sight velocities of 37 PNe from Arnaboldi et al. ([1994]) and 74 GCs from Kissler-Patig et al. ([1998]) favour the high-mass halo models in Fig. 18. These are also consistent with X-ray mass profiles from Ikebe et al. ([1996]). The abrupt increase of the dispersion profile seen in the PNe and GCs just beyond the end of the absorption line data needs to be confirmed with larger PNe samples.

Since the absorption line data only weakly imply dark matter in this galaxy, we have used NGC 1399 to show that the details of how to determine the $95\%$ confidence line do not significantly influence the result; see Sect. 4 and Fig. 22.

Finally we have used this galaxy to check that within the uncertainties the derived kinematic properties are independent of the choice of basis functions employed in the modelling. The long dashed - dotted lines in the panel for NGC 1399 in Fig. 12 show the fit to the kinematic data using a distribution function composed from a basis of 39 radially anisotropic plus the isotropic basis functions in the same best-fit halo potential as for the full line with the basis used in S+2000, which contained 55 tangentially anisotropic, one radially anisotropic, and the isotropic models. Notice the close similarity of these two models in both the velocity dispersion and h₄ profiles, although the new model has a significantly higher $\chi ^2$. Both models also agree in the shape of the derived anisotropy profile, showing a similar radially anisotropy main body between 5''-50'' and turning towards a more isotropic structure both in the centre and in the outer parts. The most significant difference is the slightly lower maximum anisotropy resulting with the radial basis. This agreement in the inherently less accurate intrinsic quantity $\beta$ is gratifying.

Figure: 22 $\chi ^2$ results for NGC 1399, testing the two methods of accounting for systematic errors in the kinematic data. Crosses show $\chi ^2$ results from fitting a set of models near the best-fitting region to the original errors shown in S+2000. The dotted line shows the corresponding rescaled $95\%$ confidence line according to Sect. 4. Filled triangles are the $\chi ^2$ values obtained when fitting the models to the data with errors globally increased by $40\%$. The open stars result from fitting the models to the data after adding systematic "floors'' to the error bars according to Table 5. The full line shows the $95\%$ confidence line corresponding to these values of $\chi ^2$, again from Monte Carlo simulations. The range of allowed models is almost independent of which method is used, and the self-consistent model (circled) is ruled out in all three cases

5.4 NGC 2434

NGC 2434 was modelled extensively by Rix et al. ([1997]) with a modified Schwarzschild method. We have used the same photometric and kinematic data from Carollo & Danziger ([1994]), but have ignored the kinematic data inside R < 1 arcsec for the fit. The steep increase in $\sigma$and the relatively high values of h₄ may indicate a central dark mass.

The results for this galaxy allow a direct comparison of both methods. In both analyses the SC models are ruled out by a large margin in $\chi ^2$. Also, the circular velocity curves of the respective best models are very similar (see Figs. 13 and 8 of Rix et al. [1997]). Our value of the central $M/L_{\rm B}$ = 6.00 $\pm$ 0.3 whereas Rix et al. obtained $M/L_{\rm B}$ = $\pm$ 3.9 $\pm$ 0.35, probably because their different halo models contributed more dark mass near the center. Our "best'' value for the circular velocity at the outermost kinematic data point ( $331 {\rm\,km\,s^{-1}}$ at 62'') agrees well with their best-fitting models. Another difference concerns the intrinsic velocity anisotropy. Rix et al. found an almost constant anisotropy of $\beta \simeq 0.5$ which is slightly more than our maximum value at $\simeq$ 0.5 $R_{\rm e}$. Outwards of 10 arcsec our anisotropy falls, reaching a value of zero at 40 arcsec (roughly 2 $R_{\rm e}$) from whereon outwards it is consistent with isotropic. Comparing both figures suggests that Rix et al. may slightly overestimate the outer anisotropy while our model may slightly underestimate it; so both results may not be inconsistent, emphasizing the larger uncertainties in any derived intrinsic quantities.

   
5.5 NGC 3379

For this galaxy we have additional data available and have used these to do two additional tests. The first relates to the possible influence of a color gradient (0.3 mag measured between 2 arcsec and 120 arcsec), the existence of which implies that the luminous mass distributions based on the B-band surface brightness profile will be slightly different from that based on the R-band profile. To see how much this changes the dynamical models and the inferred circular velocity curves we have done the analysis separately for both bands. Therefore Figs. 10, 12, 18 and 20 each contain two panels for NGC 3379, referring to the B-Band and R-band models, respectively. From these plots it can be seen that the implied differences in the fits to the kinematic data, the intrinsic velocity anisotropy, and the circular velocity curves are small.

Secondly, independent kinematic data along four slit directions are available from Statler & Smecker-Hane ([1999]). The corresponding velocity dispersion and h₄ profiles differ slightly from each other, possibly indicating a weakly triaxial shape of the galaxy. To see how this might influence our model results, we have done the analysis independently for all of these position angles. Figure 23 shows the four fits to their data in comparison with the fit derived earlier for the data shown in Sect. 2. The data taken from Statler & Smecker-Hane ([1999]) have larger error bars and can be fitted with smaller $\chi ^2$ than our data, which result in $\chi^2 = 3.205$ due to large point-to-point variations. Nonetheless all models for the five data sets show good agreement in the fits to $\sigma$ and h₄, and even in the more sensitive intrinsic anisotropy parameter $\beta$ the differences are only $\sim \pm0.1$. The resulting cumulative mass and M/L curves agree to better than five percent.

Figure 23: Comparison of the best-fit models for NGC 3379 obtained from our data (solid line) and that from Statler & Smecker-Hane ([1999]) along several slits (dotted, dashed, dot-dashed and long-dashed lines for major axis, minor axis, PA = 25, and PA = 115, respectively). Top panel: Velocity dispersion versus radius. Middle panel: h4 profile. Bottom panel: the inferred velocity anisotropy $\beta$

In addition, radial velocity measurements for 29 PNe are available for NGC 3379 (Ciardullo et al. [1993]). The maximum likelihood analysis (using the B-band models) shows that the SC model is consistent with the data as already noted by Ciardullo et al. ([1993]). However, since the PNe sample is very small the likelihood analysis does not constrain the range of the valid models strongly. Thus models with circular velocities in the range of 240 km s^-1 to 350 km s^-1 are within the $95\%$ confidence interval; see Fig. 24.

Figure 24: The 29 PN of NGC 3379 binned (triangles) and plotted with the absorption line data (filled squares). The same three models as in Fig. 12 are plotted

5.6 NGC 6703

NGC 6703 was analysed by G+98 as a prototype for the analysis method used here. They made the simplifying assumption of using a Jaffe-model for the photometry, with $R_{\rm J} = 46.5$ arcsec and $L_{\rm B} = 4.16 \ 10^{10} h_{50}^{-2} L_{\odot, B}$, and employed a set of 20 basis functions in the analysis. They found that potentials with $v_{\rm c} = 250 \pm 40$ km s^-1 at the outermost kinematic data point fitted the data well in a $\chi ^2$-sense and the SC-model was ruled out by a large margin. We redid the analysis using the same kinematic data, but inverting the SB-profile as described in Sect. 4 and using the extended basis set of 59 DFs as for the other galaxies of the present sample. The new results are shown in Figs. 10, 12, 18 and 20 and supersede those of G+98.

Comparison with the older results shows the following. The agreement of the deprojected density profile with the Jaffe profile is very good except for the very inner parts and a few small bumps round $R_{\rm e}$ also seen in the SB profile. Nevertheless the new models employing the extended basis tend to favor smaller $v_{\rm c}$ by 20-30 km s^-1, and the SC model, although near the lower boundary of the confidence intervall, fits the data well in a $\chi ^2$ sense, even though it still lies below many of the h₄ data points at intermediate radii. To test the origin of this difference we have also computed models with the new basis and smoothing parameter $\lambda$ but for the old Jaffe density profile. This showed that the greater part of the change is due to the larger basis used here. Apparently the smaller basis still implied residual smoothing through the basis functions, which is not negligible given the significant amount of substructure seen in the NGC 6703 kinematic profiles (in particular, the sharp dip in h₄near 5'', and the structure near 30'' in $\sigma$), and thus influences the $\chi ^2$-analysis. The model profiles on which the tests of G+98 were carried out, and many of the other galaxies analysed here do not have such sharp substructures. The shape of the circular velocity curve, the anisotropy profile, and the stellar mass-to-light ratio are not strongly affected (note that the value given in Table 7 is for a different distance than the values given by G+98).

5.7 NGC 7145

The drop in the velocity dispersion inside of R = 10 arcsec possibly indicates the presence of central disk seen face-on. Although it is possible to produce a satisfying fit to the kinematics, the resulting rise in M/L could be due not to the dark matter halo but to the transition from the disk-dominated to a random motion-dominated region.

5.8 NGC 7626

For this galaxy we again used both methods described in Sect. 4 for deriving $\chi ^2$ values when the scatter in the data exceeds the observational errors. The results from the second method were marginally better than those obtained with globally increasing the error bars, and they are therefore shown in the figures.

5.9 NGC 3193

The BSG data for this galaxy have two very high velocity dispersion points with large errors on one side of the galaxy at $R\mathrel{\hbox to 0pt{\lower 3pt\hbox{$\mathchar''218$ }\hss} \raise 2.0pt\hbox{$\mathchar''13E$ }}20''$. For the modelling we have assumed that there is some problem with these points, and have used only the data points on the other side with significantly lower dispersions and smaller errors.

5.10 NGC 3640

The outermost pairs of points for $\sigma$ and h₄ in the original data deviate from each other by much more than their error bars. Thus the errors of these points were replaced by one half the separation (see Fig. 15). Even afterwards the fits yield values of $\chi ^2$larger than unity ( $\chi^2 = 1.7$ for the best fitting model). Thus the confidence interval was determined by increasing the errors by $30\%$.

5.11 NGC 4278

For this galaxy, Bertola et al. ([1993]) estimated the mass-to-light ratio from the kinematics of an extended gas ring. Converted to the distance used in the present paper, they found values for the $M/L_{\rm B}$of 2.9 at 0.5 $R_{\rm e}$ and 33 at 10 $R_{\rm e}$. The corresponding values at these radii of our "best'' model are 9.6 and 19.4, respectively, i.e., our profile is less steep than theirs.

5.12 NGC 4472

We used the major axis data for the fit. BSG show data points out to 43''. From the spectra, two further velocity dispersion points on both sides of the nucleus could be obtained, extending the profile to $R\simeq 55''$. However, it was not possible to derive corresponding h₄ values. The data points within R < 3 arcsec were not fitted. The increase of $\sigma$ and h₄ towards the center may be the kinematic signature of a central dark mass.

Sharples et al. ([1998]) provide a list of 57 GC velocities for NGC 4472. The maximum likelihood analysis result is that the GC-data allow a range of $v_{\rm c} = 400 - 660$ km s^-1 and thus do not constrain the potentials as strongly as the absorption-line data do, again because of the small number of velocities. To compare the data "by eye'' with our models we calculated the GC velocity dispersion for 5 bins containing 10 and 17 GCs. The same was done for 8 satellite galaxies of NGC 4472 taken from Binggeli ([1993]) of which 5 are believed to be "real'' and 3 "possible'' satellites. Figure 25 shows the combined velocity dispersion data together with "best'' and boundary models for the absorption line data. The dispersion for the satellite galaxies should be considered as an upper limit. A model with a somewhat less massive halo than included in the "best'' model, but well within the allowed range, would fit the combined data well.

In Fig. 26 we plot the X-ray mass-profile from Schindler et al. ([1999]) over our models. The "best'' model from the absorption line kinematics provides a very good match to the X-ray mass profile at large radii. Here, the significant result is the agreement in the normalisation. The functional form assumed for our halo models implies an outer mass profile linearly increasing with the radial distance. In the case of NGC 4472 (but see NGC 1399, S+2000, and NGC 4636 below) this is similar to the radial profile derived from the X-ray emission under the assumption of isothermality.

Figure 25: GC (triangles) and dwarf galaxy (open square) velocity dispersions in NGC 4472 plotted over the stellar kinematic velocity dispersions (solid squares) and corresponding dynamical models. Solid line: "best'' model; dashed lines: models at the boundaries of the confidence interval; dotted line: SC model. Note that the dwarf galaxies mark an upper limit since they contain three candidates with high line-of-sight velocities which are possibly not satellites of NGC 4472

Figure 26: Mass profiles for NGC 4472 derived from ROSAT X-ray measurements for an isothermal gas (long dashed-dotted curve) compared to the models plotted in Figs. 25 and  16. The solid line is again the "best'' model and provides a very good match to the X-ray mass profile at large radii. The dotted line is the constant mass-to-light model. The vertical line to the right of "stars'' denotes the outer boundary of the stellar absorption line data, the one to the right of "GCs'' the corresponding limit of the globular cluster data, and the line next to the "D'' marks the mean position of the dwarf galaxies

5.13 NGC 4486

For NGC 4486 there is a large sample of GC velocities compiled by Cohen & Ryzhov ([1997]) and Cohen ([1999]), containing 224 clusters. The maximum likelihood analysis for this sample shows that a range of potentials corresponding to a circular velocity between 490 km s^-1 and 600 km s^-1 at R = 42 arcsec, the outermost stellar-kinematic data point, is consistent with the GC velocity data. As noted by Kissler-Patig & Gebhardt ([1998]) the outer GCs of NGC 4486 show considerable rotation ( $\bar{v} = 377.76$ versus $\sigma = 438.20~{\rm\,km\,s^{-1}}$ for the outermost bin with R > 400 arcsec). At smaller radii, $v/\sigma \leq 0.2$, however. If we exclude the GCs outside 400 arcsec, the range of valid potentials is narrowed to 490 - 570 km s^-1.

As for NGC 4472 we binned the data and calculated the velocity dispersion (bin-size 20, 24 for the outermost bin) for the GCs and for 8 satellite galaxies from Binggeli ([1993]). Also in this case 5 of the galaxies are bound to NGC 4486 and 3 are uncertain; thus the dispersion point for the galaxies is again an upper limit. In Fig. 27 we show the combined data together with the models derived from the stellar kinematic data. The data points fit perfectly into the confidence interval. Note that the SC model fails to match the GC data whereas it is within the confidence interval set by the absorption-line data.

In Fig. 28 we plot the X-ray mass-profile from Schindler et al. (1999) over our models. Their profile was calculated under the assumption of an isothermal gas distribution. The "best'' model from the absorption line kinematics provides a very good match to the X-ray mass profile at large radii. Like in NGC 4472 the significant part of this good match is the agreement in the normalisation.

Figure 27: GC (triangles) and dwarf galaxy (open square) velocity dispersions in NGC 4486 plotted over the stellar kinematics and the corresponding dynamical models. Solid line: "best'' model; dashed lines: models at the boundaries of the confidence interval; dotted line: SC model. Note that the dwarf galaxies mark an upper limit since they contain three candidates with high line-of-sight velocities which may not be bound to NGC 4486

Figure 28: Mass profiles for NGC 4486 derived from ROSAT X-ray measurements for an isothermal gas (long dashed-dotted curve) compared to the models plotted in Figs. 27 and  16. The solid line is again the "best'' model and provides a very good match to the X-ray mass profile at large radii. The dotted line is the constant mass-to-light model. The vertical lines have the same meaning as in Fig. 26

5.14 NGC 4486B

For this galaxy we combined the major and minor axis data of BSG94. The errors for the pairs of velocity dispersion data points at 2 and 3.5 arcsec were set to one-half their respective separations. The reconstructed model DFs have large rms deviations indicating that the models constructed for this galaxy are the least reliable in the sample.

5.15 NGC 4494

Using the value of the smoothing parameter determined by MC simulations results in $\chi^2=2.49$. The main contribution to this large $\chi ^2$ stems from the large point-to-point variations in $\sigma$ between 5 and 10 arcsec, compared to the quoted errors. Increasing the errors by $60 \%$ allows the estimation of a confidence interval.

5.16 NGC 4636

The kinematic data have rather large errors. Both of the h₄ data points at 32 arcsec are probably spurious. Thus their errors were set to unity to ensure they do not influence the fit.

For NGC 4636 mass-profiles derived from ASCA X-ray data are available from Matsushita et al. ([1998]). Even in this case where the stellar kinematical models are not as extended and not as reliable as for the previous two galaxies with X-ray data, the agreement is still reasonably good.

Figure 29: Comparison of the ASCA X-ray mass-profile with the stellar kinematic mass models for NGC 4636. The vertical line marks the outer boundary of the stellar kinematics. The two long dashed-dotted lines denote the range of acceptable X-ray mass-profiles. Note the "bending'' of the mass profile which was also observed for NGC 1399