5 Rotation curves (RCs)

In investigating the dynamics of the BCGs we prefer presenting a rotation curve (rather than just a position velocity cut) in order to indicate the real amplitude of the deprojected velocity field and to take into account data from a large fraction of the whole velocity field (and not only along a cut). However, we will see that the RCs are not always good tracers of the mass distribution in the galaxies. The RC for each galaxy has been drawn by taking into account all velocity points within $\pm S$ degrees (in the sky plane) from the kinematical major axis (i.e. the PA). This parameter was chosen to be as big as possible, still allowing a regular RC. The half sector, S, used for each galaxy is indicated in the caption of the figure showing its RC and in Table 3. The RCs are given in Figs. 1 to 14 at the bottom of the pages. Table 6 gives the RCs in tabular form (this table is only published electronically). The "cloud'' of small points seen in each RC, are all the velocity points within $\pm 30 \hbox{$^\circ$}$ of the major axis in the sky plane. Although these points represent only a portion of the data (usually more data points are included in constructing the RC since normally $S \gt 30 \hbox{$^\circ$}$), they give a fair idea of the amount, dispersion and quality of the data and the difference between the velocity on the major axis and the mean velocity. If the discrepancy is large, it means that circular motion is not likely. The error bars represent the $\pm 1 \sigma$ dispersion in the velocity at each radius in the RC, and is a combination of the intrinsic dispersion and observational and reduction errors. In general the intrinsic dispersion is larger than the observational errors. Thus the dispersion in the RC is in general caused by real irregularities in the velocity fields. The rotation velocity scale has not been adjusted by the cosmological correction (1+z).

The RCs assume axisymmetric objects with circular rotation and are sensitive to the choice of inclination (Incl), position angle (PA) and dynamical centre. The PA is defined as the angle (measured from north towards east) of the receding side of the kinematical major axis, and in general lies along velocity gradient, orthogonal to the isovelocity contours. The PA was determined from the orientation of the velocity field with a typical accuracy of 5 to 10 degrees. The inclination was determined as to minimise the residuals and the dispersion in the RC. The centre coordinates and systemic velocity were chosen to give a RC with good agreement between the receding and approaching sides and to minimise the dispersion. The centre coordinates could be determined with a typical accuracy of half a pixel and in general the displacement between the dynamical centre and the broad band photometric centre (determined from the 1st statistical moment) is within one pixel. The inclination is in general the most uncertain parameter. Since many velocity fields look perturbed, a wide range in Incl may give comparably good fits overall. A typical accuracy for Incl is $10\hbox{$^\circ$}$. Secondary components have in general less well determined Incl. The uncertainties in Incl will enter in all mass estimates based on the RCs.

  

Table 3: Parameters for the rotation curves (RCs). For those galaxies where we have more than one dynamical component, or have drawn more than one RC, "Component" indicates which component the values refer to (see Sect. 6) and "RC" indicates the name of the corresponding RC. The inclination (Incl) and position angle (PA) are the ones used in constructing (and iteratively determined from) the RCs. S is the half sector (measured from the major axis) of the velocity field included in calculating the RCs. PA$_{\rm phot}$ and Incl$_{\rm phot}$ is the photometrically determined position angle and inclination respectively; the uncertainties are of the same order as for the kinematically determined properties. The absolute magnitudes and R_25,B are based on the quoted radial velocities in Table 2 and a Hubble parameter H₀ = 75 km s^-1/Mpc. $v_{\rm hel}$ is the determined systemic velocity in the heliocentric restframe which has a typical accuracy of 1 km s^-1. However $v_{\rm hel}$ may be systematically offset with  $\pm$  one FSR (117 km s^-1). The R_25,B values are corrected for inclination (using the photometric value) and given in arcseconds and kpc. A colon after the value indicates that it is uncertain

The RC gives V(R), the rotational velocity as a function of radius, and in essence the rotational velocity is the velocity in the sky plane divided by sin(Incl). Thus the lower Incl, the more sensitive will the derived velocities will be to uncertainties in Incl. The possible occurrence of warps or oval distortions and irregularities limit the validity of the RC, since its derivation is based on the assumption of a circular motions.

A rough estimate of the dynamical mass can be made using the simple model by Lequeux (1983) in which the mass within a radius R is:

$$\begin{eqnarray} M(R) = f \times R \times V^2(R)\times G^{-1}\end{eqnarray}$$ (1)

where V(R) is the rotational velocity at R, G the gravitational constant and f is a constant which has a value between 0.5 and 1.0. This formula is valid for any galaxy (in equilibrium) supported by rotation. For a disc with flat RC f=0.6 while for a spherical distribution (e.g. a galaxy dominated by a dark halo) f=1.0. For a disc with Keplerian decreasing RC outside R, f=0.5. Thus, according to this model, for any rotating galaxy f should lie in the range: f=0.5 to 1.0, independent of the presence of a massive halo. Mass estimates based on this equation are given in Table 4, where we assumed f=0.8. The assumption f=0.8 is not based on any physical reason, but was chosen simply to lie in the middle of the allowed interval. In addition to the intrinsic uncertainty in this model, all the uncertainties above affects the accurateness of this estimate. In Paper II we will discuss these, and more refined, mass estimates of the observed galaxies. Of course, the mass estimate provided by Eq. (1), will only be a good approximation of the true dynamical mass if the galaxy is supported by rotation. If on the other hand the galaxy is mainly supported by random motions, the presented mass will be a severe underestimate.

The RC is based on the assumption of circular rotation. If this is not true, the RC will not give a good description of the dynamics and mass of a galaxy. Nevertheless, even when the velocity field is perturbed and the derived RC looks weird (as for ESO 338-IG04, Fig. 5), the RC gives some insights to the dynamics of a galaxy. Even the failure of constructing a symmetric and tight RC is interesting, since it indicates that the system is complex or perturbed. In Table 3 we give some parameters for the RCs. In some cases where we have extracted more than one component, we provide information for both. We also provide some photometric information, like Incl$_{\rm phot}$ and PA$_{\rm phot}$, the photometric inclination and position angle, respectively; and R_25,B, the radius at which the B surface brightness drops to 25 magnitudes per square arcsecond, corrected for inclination (cf. e.g. Bergvall et al. 1999) and Galactic reddening. This information is not used in itself in the present investigation but provide complementary information. The R_25,B is given in the RCs to give an idea of the size of the galaxy and the extent of the RC. When we did not have B-band data, we scaled V- or R-band data assuming crudely B-V=0.5 and V-R=0.35. The position angle and inclination derived from kinematics and photometry do not always agree which could be due to dynamical disturbances and instabilities in the systems or internal absorption. Anyway, the only thing we used PA$_{\rm phot}$ and Incl$_{\rm phot}$ for in this investigation was to calculate R_25,B. Detailed surface photometry of some galaxies in this sample will be presented separately (Bergvall & Östlin 1999).

  

Table 4: Order of magnitude mass estimates. The first column gives the target name. In those cases where the target seems to be composed of more than one dynamical component, Col. 2 indicates which component is considered: "Total" means that it is the total non-decomposed velocity field, "Main" means that it is the main component of the galaxy, "2:nd" that it is the secondary (normally weaker), and "Comp" means that it is a companion galaxy. Column 3 gives the rotational velocity at the radius R, at which the mass is calculated. Column 5 contains the mass estimate, ${\cal M}$, where we have assumed f = 0.8 in Eq. (1). See Sect. 5 for a discussion on the accuracy of these mass estimates. Sometimes there are more than one mass estimate for each galaxy (component); this is when the mass have been calculated at several radii, e.g. at the radius of maximum rotational velocity (indicated by $V_{\max}$) and at the last point in the RC (indicated by $R_{\max}$), if these do not coincide. In the other cases, the maximum rotational velocity ($V_{\max}$) occurs at the last point ($R_{\max}$) in the RC and these are also the values of V(R) and R given in the Table below. The note "Both" indicates that the mass has been evaluated at the last point in the RC where there is data both from the receding and approaching sides

  

Table 5: Estimated H$\alpha$ fluxes ($f_{\rm H\alpha}$), luminosities ($L_{\rm H\alpha}$) and ionised gas masses(${\cal M_{\rm ion}}$). The H$\alpha$ luminosity was calculated using the quoted radial velocities in Table 2. The mass was calculated assuming an electron density of $n_{\rm e} = 10$ cm^-3, an electron temperature $T_{\rm e} = 10\,000$ K, and a mean molecular weight of $\mu = 1.23$. The H$\alpha$ recombination coefficient was taken from Osterbrock, Case B (1989)