4 Dynamical modelling

4.1 Modelling procedure

The modelling technique follows the lines described in Gerhard et al. ([1998]; G+98) and Saglia et al. ([2000]; S+2000), and consists of the following steps:

(i) Calculate the density and potential of the luminous matter from the smoothed SB-profile, assuming spherical symmetry and using a nonparametric regression algorithm based on generalized cross validation (GCV, Wahba & Wendelberger [1980]).

(ii) Specify a gravitational potential, consisting of the potential of the luminous matter and a nonsingular isothermal dark matter halo of the form

 

$$\Phi^{\rm DM} = \frac{1}{2} v_0^2 \ln \left(r^2 + r_0^2\right),$$ (1)

with core radius r₀ and circular speed v₀. By employing these halo potentials we implicitly maximize the luminous mass in the galaxy.

(iii) In a given potential, construct a set of basis functions for the phase-space distribution (DF), each reproducing the stellar density. The basis functions contain the isotropic model, a radially anisotropic model, and several series of tangentially anisotropic functions. The form of these basis functions is described in G+98.

(iv) Project each of these basis DFs into the space of kinematic observables.

(v) Determine the smoothing parameter $\lambda$ and the cumulative $\chi ^2$ distribution from Monte Carlo simulations of kinematic data similar to those for the galaxy under study.

(vi) Fit the composite sum of the basis DFs in the kinematic observables to the measured velocity dispersions $\sigma$ and Gauss-Hermite parameters h₄ of the galaxy, by a regularised least-square algorithm imposing non-negativity and smoothness of the composite DF. In this way we find the unique best DF for the given potential and the previously fixed $\lambda$.

(vii) By varying the parameters of the dark matter component and repeating steps (ii-vi) determine the range of potentials consistent with the data, using a $\chi ^2$-statistic.

In this procedure, it is advantageous to use a realistic model DF for setting the velocity scales with which the projected Gauss-Hermite moments of the basis DFs are computed. For this we have either taken the isotropic model or a radially anisotropic DF determined by a parametric fit. In most cases we have used "basis 2'', which consists of 59 functions including this radially anisotropic model. "Basis 1'' consists of the same 59 functions, but in this case the isotropic model was used for the velocity scales. Finally "basis 3'' does not contain the radially anisotropic function and thus has only 58 components. Which basis was used for which galaxy is shown in Tables 5 and 6. NGC 1399 was modelled using a similar but not identical basis as described in S+2000. For this galaxy we have also verified the robustness of the kinematic results by repeating the analysis with a very different set of basis functions, consisting of the isotropic and 39 radially anisotropic models; see Sect. 5.3.

In the non-parametric fit the degree of smoothness is directed by a penalty term whose weight $\lambda$ is determined by Monte Carlo simulations. For each of a series of values for $\lambda$, we analyse 100 Monte Carlo data sets. These datasets are generated from a parametric model which matches the galaxy data closely, and kinematic data points are generated as Gaussian random deviates with the observed error bars at the positions of the observed data points. The value of $\lambda$ which results in the smallest rms relative deviation between the known DF of the input model and the recovered DF, averaged over the 100 realizations, was chosen for the final fits (see G+98 and S+2000). Here the rms deviation between input and output DFs was determined on a grid out to three times the last kinematic data point. It turned out that $\lambda$ is only weakly dependent on the potential. The final values of $\lambda$ employed for our galaxies are given in Tables 5 and 6.

   
4.2 Estimation of confidence intervals

We estimate the confidence interval from the Monte Carlo simulations described above, for the final value of $\lambda$ (see also G+98). The 95% confidence level is taken to be at the value of $\chi^2_{\sigma+h_4}$ which is surpassed only by 5 of the 100 realizations. The results from this procedure do not depend much on whether the selfconsistent potential or one with a halo is used for the Monte Carlo simulations, and the average value from these two cases is used.

As in S+2000 we have encountered the problem of high values of $\chi ^2$ (> 1 per data point) for some of the galaxies, in particular those with the new, higher quality data. The reason is the same as described in that paper: the point to point variations in the real data are larger than the error bars derived from the simulations of the spectra. These point-to-point variations in the data appear to be caused by systematic effects in the data.

To obtain smaller values of the $\chi ^2$-statistic one can decrease the degree of smoothing on the DF. We have tested this for NGC 7626. In this case, the range of circular velocities at the last kinematic data point, $v_{\rm c}(R_{\rm max})$, determined from the fitting procedure with the best value of the smoothing parameter $\lambda$ as above, is $\sim 370-440{\rm\,km\,s^{-1}}$ at $95\%$ confidence (see below). Without any smoothing ($\lambda=0$), these models can be fitted with $\chi^2\simeq 0.7$ per point, and models with $v_{\rm c}(R_{\rm max})$ down to $350{\rm\,km\,s^{-1}}$ and up to more than $500{\rm\,km\,s^{-1}}$ are consistent with the data ( $\chi^2<1.27$). If $\lambda$ is chosen such that the best-fitting models have $\chi^2\simeq 1.0$, then a range in $v_{\rm c}(R_{\rm max})$ from $\simeq 360{\rm\,km\,s^{-1}}$ to $\sim 500{\rm\,km\,s^{-1}}$ have $\chi^2<1.27$, i.e., are consistent with the data at the $2\sigma$level. However, the price to be paid is a much less smooth DF. In the last case, the DF has several holes in radius (energy): in one example, it tends to zero at some small radius and at some large radius, and in addition dives by more than an order of magnitude relative to nearby regions at a number of other energies. For $\lambda=0$, the behaviour of the DF is even worse; in this case it is highly unsmooth, as expected, and has of order 5 holes in energy where it tends to zero. Such DFs are contrary to the smooth coarse-grained DFs of hot stellar systems that are predicted by violent relaxation. Moreover, we believe that decreasing $\lambda$ in order to improve the point-by-point fit of the dynamical models is also inappropriate because the point-to-point variations in the data often appear to be caused by systematic effects, such as line emission, dust absorption, residual template mismatches, and asymmetries between the two sides of the galaxy.

  Table 4. Calibration checks for the adopted SB-profiles. Column 1 is the NGC number; Col. 2 gives the total B-magnitude from the RC3 (de Vaucouleurs et al. [1991]); Col. 3 the aperture of the photoelectric measurements of Burstein et al. ([1987]), Col. 4 the corresponding aperture magnitude corrected for extinction and converted to the adopted distance (Table 3).Column 5 gives the integral of the adopted SB-profiles within the aperture of Burstein et al. ([1987]), and Col. 6 the difference between the integrated and the measured aperture magnitudes. Two galaxies, NGC 3193 and NGC 4486B, are not in the catalogue of Burstein. Column 7 gives the integrated magnitude of the SB-profiles out to 1 $R_{\rm e} - 0.75$ mag. For NGC 1399 and NGC 7507 the SB-profiles were corrected to the Burstein et al. photometric system

NGC	$B_{\rm T}^{\rm RC3}$	$R_{\rm A}$['']	$B(<R_{\rm A}$)	$B_{\rm int}$	$B_{\rm int}-B(<R_{\rm A}$)	$B_{\rm int}(R_{\rm e}) - 0.75$
315	-22.25	30.00	-21.42	-21.45	-0.03	-22.80
1399	-21.15	45.41	-20.40	-20.69	-0.29	-21.11
2434	-20.08	47.55	-20.32	-20.30	0.02	-20.62
3193	-21.06	--	--	--	-	-20.96
3379	-20.36	47.55	-19.54	-19.67	-0.13	-20.20
3640	-21.37	42.38	-20.85	-20.72	0.13	-21.32
4168	-21.00	78.91	-20.63	-20.73	-0.10	-21.17
4278	-20.68	47.55	-20.09	-20.09	0.00	-20.63
4374	-21.47	42.38	-20.38	-20.37	0.01	-21.32
4472	-22.19	78.91	-21.25	-21.38	-0.13	-22.35
4486	-21.97	47.55	-20.67	-20.70	-0.03	-22.07
4486B	-17.20	--	--	--	-	-17.41
4494	-19.44	78.91	-18.94	-18.98	-0.04	-19.39
4589	-21.65	64.14	-21.28	-21.23	0.05	-21.70
4636	-21.13	78.91	-20.44	-20.53	-0.09	-21.51
5846	-21.73	33.66	-20.52	-20.51	0.01	-21.88
6703	-19.92	42.38	-19.77	-19.79	-0.02	-20.35
7145	-20.23	47.55	-19.62	-19.59	0.03	-20.24
7192	-20.65	47.55	-20.15	-20.15	0.00	-20.73
7507	-20.79	33.66	-20.29	-19.91	0.38	-21.01
7626	-21.55	78.91	-21.26	-21.37	-0.11	-21.70

Table 5. Dynamical modelling parameters for the part of the sample galaxies with new kinematic data. NGC 1399 analyzed with the identical method but another basis set in S+2000 is included for completeness. Column 1 gives NGC number, Col. 2 the optimal value of $\lambda$, Col. 3 codes the employed basis, Col. 4 lists the resulting rms accuracy with which a smooth DF can be recovered from the data, Col. 5 gives the (probably systematic) error enlargement factor, Cols. 6 and 7 the alternative systematic error floors to achieve a $\chi^2=1$ per data point fit. Systematic errors in $\sigma$ are given in km s^-1. See Sect. 4 for details

NGC	$\lambda$	basis	rmsDF	err.	sys. $\sigma$	sys. h4
315	0.01	1	0.125	$43\%$	6	0.027
1399	0.02	-	0.126	$40\%$	6	0.01
2434	0.02	2	0.134	0	0	0
3379 B	0.02	2	0.116	0	0	0
3379 R	0.01	2	0.116	0	0	0
4374	0.03	2	0.118	$35\%$	3	0.011
5846	0.02	3	0.136	$8\%$	0	0
6703	0.006	2	0.176	$10\%$	1	0.01
7145	0.002	2	0.159	$80\%$	7	0.008
7192	0.006	3	0.156	0	0	0
7507	0.03	2	0.116	$13\%$	3	0.005
7626	0.03	2	0.123	$35\%$	5.5	0.01

Table 6. Same as Cols. 1-5 of Table 5, for the sample galaxies with kinematic data from BSG94. Larger values of the deviation of the DF indicate poorer quality of the kinematic data (both in terms of sampling and in terms of the sizes of the error bars). Especially NGC 4486B is less reliable

NGC	$\lambda$	basis	rmsDF	err.
3193	0.01	2	0.181	0
3640	0.01	2	0.129	30%
4168	0.01	2	0.168	0
4278	0.03	3	0.097	50%
4472	0.03	2	0.084	10%
4486	0.01	1	0.115	0
4486B	0.03	2	0.281	0
4494	0.02	2	0.107	60%
4589	0.002	2	0.170	0
4636	0.1	2	0.112	0

To derive a confidence interval despite the presence of systematic errors in the data, we have in most cases (see also S+2000) (i) increased the kinematic errors bars by a global factor until one obtains $\chi^2 \simeq 1$ per data point for the best model, (ii) used this revised data set for the Monte Carlo simulations as described above, giving a new value for the 95% value of $\chi ^2$, and (iii) finally rescaled this value upwards by the factor of the error enlargement in order to compare with the fits based on the original errors. These (where necessary) rescaled confidence values and the resulting confidence intervals are shown in Figs. 10 and 11. Tables 5 and 6 list the necessary error enlargements.

We have also investigated an alternative way of treating the problem, based on the observation that, in terms of the respective error bars, large point-to-point variations between neighbouring data points occur most often in those galaxies with new data and small errors, and within these in the central parts where the error bars are particularly small. This suggests that systematic effects influence all data points in nearly the same way and with a similar amplitude. To model this, we have for these galaxies, both in the model fits and in the Monte Carlo simulations, added in quadrature a constant error in $\sigma$ and h₄ to the statistical errorbars, until the best-fitting models reached $\chi^2=1$ per point. The resulting "systematic'' errors are rather small, ranging from 3 to 7 km s^-1 in $\sigma$ and 0.005 to 0.027 in h₄. Table 5 gives values for individual galaxies. For some objects, we have done the full analysis with both methods, and generally find rather little difference between the results. See Sect. 5.3 for a comparison plot. Below, we present our results using the first technique, except in the two galaxies NGC 315 and NGC 7626 where the second method gave somewhat better results.