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8 Decomposition of the rotation curve

The observed rotation curve can be decomposed into the contributions of the individual galactic mass constituents (van Albada et al. 1985; Kent 1986, 1987; Begeman et al. 1991). As usual, for the luminous components a constant M/L ratio is adopted. Then for each luminous component a mass model can be constructed proportional to the light distribution which is given by the photometry. The gravitational influence of the gas is included and a dark halo is needed to explain the large rotation in the outer parts of galaxies. The decomposition of the observed rotation curve can then be achieved by a least squares fitting procedure where the individual rotational contributions of disc, bulge, gas, and dark halo are determined. Such a fit is, however, far from unique (van Albada et al. 1986); in general a range of M/L ratios for the disc (and bulge) gives an equally good fit to the data. Additional information or assumptions are needed. One such assumption is the maximum disc hypothesis which states that the contribution of the disc to the rotation should be scaled up as high as possible and so assigning the maximum possible M/L ratio to the disc. On the other hand, observations of the stellar velocity dispersions of galactics discs suggest that the maximum contribution of the disc to the rotation is, on average, 63% at the position where the disc has its maximum velocity (Bottema 1993). As discussed in the introduction, there is other evidence that for normal and specifically for the low surface brightness galaxies the maximum disc hypothesis cannot hold. Below we will investigate the possibilities for NGC 3992.

Photometry in the B, R, I, and $K^{\prime }$ band is given by Tully et al. (1996), of which the I and $K^{\prime }$ data are reproduced in Fig. 12. Since the $K^{\prime }$ data do not extent very far out, the I band data are used for the mass modelling. Note that in the inner regions the I and $K^{\prime }$ photometry are nearly identical and hence there are no large population and dust gradients in the inner regions and the I band gives a good representation of the actual mass distribution. The total luminosity of NGC 3992 in the I band is identical to within the errors with the total luminosity derived by Héraudau & Simien (1996) which gives confidence that the absolute calibration of the surface brightness is correct. The photometry has been cut off at a radius of 250$\arcsec$ but the calculated rotation curve does not depend on the exact position of the cutoff as long as it is at or beyond this 250$\arcsec$.


  \begin{figure}
\par\resizebox{\hsize}{!}{\includegraphics{H3038F12.ps}}\end{figure} Figure 12: Observed photometric profiles in the I and $K^{\prime }$ band of NGC 3992 by Tully et al. (1996). The I profile is used for the rotation curve decomposition. A bulge/disc decomposition has been made by assuming a disc profile (dashed line) which has been subtracted from the observed profile to get the bulge light (dotted line).

For the gaseous component a surface distribution equal to that of the H  I is taken (see Fig. 6), multiplied with a factor 1.4 to account for helium. The gas disc is adopted to be infinitely thin. For the dark halo a pseudo isothermal sphere generally provides a satisfactory model (Carignan & Freeman 1985) for which the density distribution ${\rho}(R)$ is given by

\begin{displaymath}{\rho}_{\rm h} = {\rho}_{\rm h}^0 \left[ 1 + \frac{R^2}{R^2_{\rm core}}
\right]^{-1},
\end{displaymath} (2)

with a rotation law

\begin{displaymath}v_{\rm h} = v_{\rm h}^{\max} \sqrt{
1 - \frac{R_{\rm core}}{R} \arctan \left( \frac{R}{R_{\rm core}}
\right) },
\end{displaymath} (3)

where $R_{\rm core}$ is the core radius related to the maximum rotation of the halo $v_{\rm h}^{\rm max}$ by

\begin{displaymath}v_{\rm h}^{\rm max} = \sqrt{ 4\pi G {\rho}_{\rm h}^0
R^2_{\rm core} }.
\end{displaymath} (4)

In this section two situations will be considered. At first one where the luminous mass is all in a disc like distribution. The disc is given a locally isothermal sech-squared vertical distribution (Van der Kruit & Searle 1981) with a z0 scale height parameter of 700 pc. Secondly a decomposition of the photometric profile is made in a bulge and a disc. This is done by extrapolating the disc inwards from the relative exponential section between radii of 50 to 130$\arcsec$. This disc is subtracted from the total light to obtain the bulge surface brightness for radii less than 47$\arcsec$ as has been illustrated in Fig. 12. Note that the bar extends out to 72$\farcs$5 and hence in the bar region the bulge slowly starts to build up coming closer to the centre as suggested by the optical image. For the disc again the same sech-squared density distribution is assumed while the bulge is taken to be spherical. Purpose of the bulge/disc decomposition is to investigate whether and how much fit parameters will change compared to a pure disc fit. To that aim the decomposition need not be very precise. There is a difference between the bulge and disc concerning the conversion to face-on brightnesses. The photometric profile of the disc along the major axis is converted to face-on magnitudes by multiplying with a cosine(inclination) factor. Since the bulge is assumed to be spherical no conversion to face-on is needed. At this stage, no corrections for internal and galactic absorption have been made and consequently all derived M/L ratios are those as observed. The rotation of the stellar and gas disc is calculated by the prescription of Casertano (1983) while the rotation of the bulge is given by the equations of Kent (1986).

The decomposition of the observed rotation curve is performed by fitting the sum of the rotation curves of the components to the observed data in a least squares sense. In that way the best fit is designated as the situation of minimum ${\chi}^2$ value. However, a least squares fit procedure assumes that the fitting function is known a priori and the data points scatter in a Gaussian way around that function. For rotation curves that is not valid. Firstly a rotational functionality for the halo is adopted which need not be correct. Secondly the procedure to determine the rotation is an approximation in the sense that azimuthal symmetry is assumed with no in or outflow. For example spiral arms can produce small irregularities, which may lead to small systematic deviations from the actual rotation law. Because of these matters the resulting minimum ${\chi}^2$ value is only a limited indicator of the quality of the fit. In general one has to make an inspection by eye to judge the quality of the fit, taking the errors of the individual data points into account.

First the decomposition results for a disc only situation will be considered. Three parameters are free and have been fitted simultaneously: the core radius and maximum rotation of the dark halo, and the M/L ratio of the disc. The result is illustrated in Fig. 13a and has a disc with an I-band mass-to-light ratio of $1.79 \pm 0.19$. Further numerical values are given now and for the following fits in Table 4.

 

 
Table 4: Decomposition of the rotation curve of NGC 3992.

Situation
panel in red disc (M/L)$_{\rm d}$ bulge (M/L)$_{\rm b}$ $R_{\rm core}$ $v_{\rm h}^{\max}$
  Fig. 13 ${\chi}^2$ mass   mass      
      (10 $^9~ M_{\odot}$) ( $M_{\odot}/L_{\odot}^I$) (10 $^9~ M_{\odot}$) ( $M_{\odot}/L_{\odot}^I$) (kpc) (km s-1)

D only, best fit
a 1.22 73.7 1.79 $\pm$ 0.19 - - 1.16 $\pm$ 0.35 230 $\pm$ 98
D only, max disc b 1.94 194.1 4.71 $\pm$ 0.11 - - 44.9 $\pm$ 17 482 $\pm$ 188
D + B, equal M/L c 1.22 64.9 2.03 $\pm$ 0.21 18.7 2.03 1.79 $\pm$ 0.35 230 $\pm$ 64
D + B, $v^{\rm max}_{\rm b}$ = 240 - 1.25 71.3 2.23 $\pm$ 0.26 36.9 4.0 3.7 $\pm$ 0.6 233 $\pm$ 57
D + B, max d 1.08 134.6 4.2 $\pm$ 0.3 47.1 5.1 $\pm$ 0.5 23.2 $\pm$ 5.7 327 $\pm$ 91


Surprisingly the fit finds a least squares minimum for a non maximum disc situation. This is surprising because in most cases when rotation curves are decomposed a least squares fit tends to the maximum disc solution while other solutions are nearly equally as good (van Albada & Sancisi 1986). The maximum rotation of the disc is 162 km s-1 which amounts to 60% of the observed maximum rotation. This is close to the 63% found from the study of stellar velocity dispersions by Bottema (1993). If the disc rotational contribution is forced below 50% the agreement between data and model rotation rapidly becomes worse. For such a situation features in the photometry cannot be reconciled adequately with the accompanying features in the observed rotation curve.


  \begin{figure}
\par\resizebox{\hsize}{!}{\includegraphics{H3038F13.ps}}\end{figure} Figure 13: a) to d) Various rotation curve decompositions. The dots are the observed rotational data. The fit to these is indicated by the full drawn line. Individual contributions of the bulge (dotted line), disc (long dashed line), gas (short dashed line), and dark halo (dash - dot line) are also given. Details are given in the text and in Table 4. a) All the mass is assumed to be in a disc like distribution. The best fit is for a disc contributing $\sim $60% at most to the total rotation. b) As a), but now for a secondary minimum in the least squares fitting procedure. This is a maximum disc fit. c) For a separate bulge and disc mass distribution, where the M/L ratios of both are constrained to be equal. d) As c), but the M/L ratios of bulge and disc are both unconstrained.

The photometric radial profile clearly indicates that NGC 3992 is highly non exponential, contrary to claims by Elmegreen & Elmegreen (1985). This galaxy is of an extreme Freeman type II (Freeman 1970) and the maximum of the rotation does not occur at the usual 2.2 scalelengths, if a scalelength can be defined at all. Instead the relative exponential part between 50 and 130$\arcsec$ followed by the more steep decline in brightness beyond, results in a specific shape of the rotation curve of the luminous matter as shown by the long dashed line in Fig. 13. This curve has a maximum around 13 kpc and a long linearly decreasing section between 17 and 24 kpc. The observed drop in the rotation curve is then not a consequence of a truncation feature of the disc as suggested by G84 but simply a consequence of the non-exponentiality of the disc. Also the photometry does not suggest a sudden drop in brightness at a radius of $\sim $19 kpc = 211$\arcsec$ needed to explain such a drop at the observed radii between 19 and 24 kpc. Some experimenting has been done by taking cut offs in the photometry at 250$\arcsec$ = 22.5 kpc and at larger radii. But this is already so far out that an associated drop in the rotation curve cannot be noticed any more (Casertano 1983).

There is a secondary minimum in the least squares fit, which is illustrated in Fig. 13b. It is in principle a maximum disc fit with a dark halo having a core radius comparable to the maximum radius to where the rotation curve is determined. Compared to the previous fit, the outer data points seem to be better represented. In the stellar disc regions however, the fit is worse and at certain positions not compatible with the data. The reduced minimum ${\chi}^2$ value for this fit is 1.94, for the previous fit it was 1.22. The mass-to-light ratio in the I band, uncorrected for absorption amounts to $4.71 \pm 0.11$ for this maximum disc fit.

When a bulge is added to the system there is one more free parameter, namely the M/L ratio of the bulge. This parameter has been constrained in two ways; a situation where the M/L ratio is taken equal to that of the disc (see Fig. 13c) and a situation where the maximum rotation of the bulge is fixed at the flat level of the rotation curve. The fit is performed and dark halo parameters and disc M/L ratios follow and are given in Table 4. The reduced ${\chi}^2$ value for both bulge M/L ratios are equal. What one can achieve by adding a bulge is some mass transfer from the dark halo to the bulge resulting in a larger core radius compared to the disc only case. Presently core radii are 1.8 and 3.7 kpc compared to the 1.16 kpc when only a disc is present. When the M/L ratio of the bulge is not constrained the fitting procedure generates a result given in Fig. 13d which is analogous to the maximum disc fit in Fig. 13b. This fit is actually better then the one where the M/L ratio of the bulge is constrained in the sense that the reduced ${\chi}^2$ value is slightly lower. However the M/L ratios are substantial; 5.1 and 4.2 for the bulge and disc respectively. So adding a bulge has the side effect that one is again close to the situation for other rotation curve decompositions: nearly equally good fits can be made for a whole range of disc contributions to the total rotation. Other indicators are then needed to determine the M/L ratio of the disc, like observations of disc stellar velocity dispersions or population synthesis arguments.

One can conclude that the features in the rotation curve are generated by the strong non-exponentiality of the photometry. Still, even with the determined detailed rotation curve of NGC 3992 it is not possible to get a tight constraint on the disc contribution to the total rotation or on the M/L ratios of disc and bulge. There is only a slight preference for the disc to be sub maximal.


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