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Appendix A: Quantification of the bias introduced in v_max

Using the technique described in Sect. 5.2 to measure the maximal rotation speed, it seems that none of the galaxies have zero rotation (look at the v_max presented in Table 4). However, rotation curves like PGC1154903 or VCC1087 (Fig. 6) appear to be statistically consistent with no rotation. That we always find positive maximum velocities is a consequence of the method used to measure it. To quantify the bias introduced by this technique, we ran some simulations. We took a zero-rotation object with errors typical of those galaxies that statistically are non-rotators. We took a typical rotation curve with 11 bins, at 0, 2′′, 4′′, 7′′, 12′′, and 16′′, with symmetrical errors of 2, 5, 7, 10, 12, and 15 km s^-1 respectively. From these errors we generated 100 simulated rotation curves assigning to each radius a random number with a Gaussian distribution, of which the width is the error associated to that radius. After folding these simulated rotation curves, shown in Fig. A.1, we calculated v_max exactly following the same technique as the one we used for the target galaxies. We then obtained a mean value for the 100 values of v_max and its scatter, 9  ±  6 km s^-1, shown as a thick black line and a grey shaded area in Fig. A.1. As a result, we consider that those galaxies with v_max < 9 km s^-1 are not rotating, based on our data. For the other galaxies the rotation is significant (three times the standard deviation, except for VCC1122 and VCC1261, see Table 4, 3rd column), so that the systematic bias described here is much less relevant. For VCC1122 (17.3  ±  7.7 km s^-1) and VCC1261 (13.9  ±  5.2 km s^-1), the rotation is marginal, as also shown by their low v_max/σ.

Fig. A.1 Simulations performed to quantify the bias that can be introduced in the measurement of vmax. Plotted are the 100 folded rotation curves computed with random numbers distributed as a Gaussian with width the typical errors of the target galaxies for that radius. Red squares and black dots are the left and right arms of the unfolded rotation curves, respectively. The thick black line is the mean vmax for all the simulations. The grey shaded area is the scatter for this mean.

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Appendix B: Absolute I band magnitudes, optical and half-light radii, and ellipticity

The I-band images for our sample of 21 dwarf galaxies were drawn from the Sloan Digital Sky Survey (SDSS, York et al. 2000) data release 6 (DR6 Adelman-McCarthy et al. 2008) and converted to the Johnson-Cousins system following Appendix C. The photometric parameters were calculated using the IRAF task ellipse.

To remove the stars from the images, we used the IRAF task fixpix. This task allows us to remove the stars interpolating the surrounding galaxy area. To improve the final outcome, we averaged the results of interpolating along the horizontal and vertical directions for each star. For those galaxies that were on the edge of the FITS images or had a very bright nearby star, a more careful procedure was implemented. Taking advantage of the fact that the galaxies are ellipticals, so that they have smooth and axisymmetric surface brightness profiles, the bmodel task of IRAF can be used to replace the affected areas of the galaxy by the azimuthal average of the unaffected ones. The output from bmodel was only used to replace a small fraction of pixels, so this procedure was not affected by the possible presence of more subtle features, such as bars or spiral arms, which could not be reproduced by the model provided by the bmodel task. Once extremely bright nearby stars had been removed, the same procedure as above was followed with ellipse and fixpix.

The procedure followed to run ellipse depends on the parameters we want to measure. First of all we ran ellipse by fixing only the centre of the galaxy and assuming a step between isophotes of 1 pixel, while the rest of the parameters were left free. We also made the masks for the stars to be removed as described above. With the aim of measuring the absolute magnitude, R_opt and R_eff we ran ellipse, again fixing the centre of the galaxy, the ellipticity, and the position angle (PA) to avoid overlap between consecutive isophotes. The adopted ϵ and PA in this case are the typical values in the outer parts of the galaxy (beyond 1.5–2R_eff, region where these two parameters stabilise). To measure ϵ and C₄ we again ran ellipse after removing the stars, leaving only the centre of the galaxy fixed.

Asymptotic magnitudes and the radii were derived as in Gil de Paz et al. (2007). We first computed the accumulated flux and the gradient in the accumulated flux (i.e., the slope of the growth curve) at each radius, considering the major-axis value provided by ellipse as radius. After choosing an appropriate radial range, we performed a linear fit to the accumulated flux as a function of the slope of the growth curve. The asymptotic magnitude of the galaxy was the Y-intercept or, equivalenly, the extrapolation of the growth curve to infinity. Once the asymptotic magnitude was known, the optical and effective radii of each galaxy were obtained as the major axis of an elliptical isophote containing 83% and 50% of the total flux respectively. Different sources of error have been considered for the asymptotic magnitudes (see Appendix C). The resulting uncertainty is  ~0.02 mag.

The ellipticities (ϵ) were measured as the mean value between 3′′ and the R_eff, the galaxy region covered by our spectroscopic observations.

Appendix C: Errors in magnitudes

The zero points (ZP) and the errors in the i-band magnitudes were computed as described in SDSS documentation. The ZP were obtained from F₀, which is the flux that a source produces in counts per second in the image. It is calculated as a function of three parameters (aa, kk, and airmass) and defined as (C.1)where the exposure time (t_exp) is the same for all the SDSS images (53.91 s). The uncertainties in the i-band magnitudes are affected by different sources of error: firstly, the errors in the flux, that can be calculated following the equation (C.2)where F is the total flux in counts, the sky and Δsky are the background sky and its error (in counts), the gain and the dark   variance are given in the header, and N_pix is the number of pixels in the largest aperture where the flux is measured. This error was typically 10^-3 mag. Other error sources are the error introduced in the fit to the growth curve (between 10^-3 mag and 6 × 10^-3 mag), and the error due to photometric zero point differences between the different scans of SDSS, which might lead to an error of 0.01 mag. SDSS i-band magnitudes are not exactly in the AB system, so an error of 0.01 mag might also be introduced (see SDSS documentation about the photometric flux calibration). And finally, we transformed our data from the SDSS i-band to the Johnson-Cousins I-band assuming m_I = m_i − 0.52 ± 0.01 mag Fukugita et al. (1995) given that their r − i colour ranges from 0.23 mag to 0.57 mag.

After adding all these sources of error quadratically, the final estimated error is 0.02 mag for the apparent I-band magnitudes.

Appendix D: The C₄ boxyness/diskyness parameter

The boxyness/diskyness parameter is defined as the fourth moment in the Fourier series as follows (D.1)where I(Φ) is the intensity measured in each isophote. The first two moments in this series completely describe an ellipse. Higher order moments (k ≥ 3) define deviations in the isophotes from ellipses. The third-order moments (S₃ and C₃) represent isophotes with three-fold deviations from ellipses (e.g. egg-shaped or heart-shaped), while the fourth-order moments (S₄ and C₄) represent four-fold deviations. Rhomboidal or diamond shaped isophotes have nonzero S₄. For galaxies that are not distorted by interactions, C₄ is the most meaningful moment, indicating the disky/boxy shapes of the isophotes (see Fig. 1 from Peletier et al. 1990 for an example of these different shapes).

	Fig. D.1 Examples of C4 vs. radius for three galaxies: VCC397, VCC308 and VCC1695. The dashed purple lines indicate the region between 3′′ and the 3Reff. The dotted purple line shows the Reff.
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C₄ is measured in i-band SDSS images using ellipse that performs equation D.1 along the radius of the galaxy fixing only the centre of the galaxy and leaving the rest of ellipse parameters free, as described in Sect. 6.1. Figure D.1 shows C₄ as a function of radius for 3 dEs. In the upper panel one can see that if an average value is used between the dashed lines, C₄ will be compatible with zero and, as a consequence, the prominent disky structure will be smeared out by the adjacent regions. In contrast, in the middle panel a galaxy with no clear disky structures is shown. In this latter case the errors are larger. The bump in C₄ only covers  ~3′′, while the rest of the galaxy is boxy. In this situation a mean value of the C₄ and its scatter is more representative. In the lower panel VCC1695 is an example of a typical boxy shaped galaxy. Due to the usual large changes of C₄ with radii, taking an averaged value is therefore not the best way to detect disks in these galaxies, especially if they only cover a limited range in radius. We have thus adopted the following procedure.

If at least one prominent bump is detected, which has a width larger than  ~6′′ inside three effective radii (above this radius the scatter of C₄ and its error become too large to be reliable), we consider the galaxy to be disky and assign the maximum C₄ between 3′′ and 3R_eff to the global C₄. The error in this measurement has been estimated by dividing the photometric error of the maximum of C₄ by the square root of the number of points that describe the disky structure in order to quantify the reliability of the bump considered. If the bump is described by a large number of points, it is highly likely that the bump is truly there so the error will be small, but if the number of points is small but the photometry is of high quality then the error will again be small. In any other case, the error will be large and the result must be used cautiously.

Otherwise, if the values oscillate around C₄  ~  0 (lower panel of Fig. D.1), are always negative, or if there is a bump with a small radial coverage (see middle panel Fig. D.1), we assign a mean value and its scatter between 3′′ and 3R_eff to the global C₄. In this case, the RMS quantifies the quality of the photometry simultaneously and the possible presence of small bumps (as it is the case of VCC308, middle panel of Fig. D.1).

The results obtained for this parameter are listed in Table 6 and plotted vs. the anisotropic parameter, (v_max/σ) ^∗ , in Fig. 10. The agreement between the C₄ classification in disky/boxy galaxies and the morphological classification from Lisker et al. (2006b) is evident but apart from three red dots. These correspond to VCC917, a rotationally supported galaxy with strong disky isophotes but no structure found by Lisker et al. (2006a); VCC1122, a dE that is not rotationally supported but that shows appreciable rotation in Fig. 6 and a significant disky structure in the inner R_eff not detected by Lisker et al. (2006a); and VCC1912, inside half the effective radius a moderate rotation is found in this system with a clear disky feature that peaks at 1.5R_eff. For this galaxy, however, no underlying structure was found by Lisker et al. (2006a). More importantly, VCC308 and VCC856, two rotationally supported galaxies with boxy C₄, present prominent spiral arms in Lisker et al. (2006a). In Ferrarese et al. (2006), based on ACS-HST images, VCC856 also shows spiral arms but their analysis of the isophotes’ shapes shows that they are boxy too. Looking at Table 6 we see that both galaxies are nearly face-on, which means that the isophotes are boxy since face-on disks are round and not disky. As a consequence, we emphasise that boxy isophotes could miss disk features (mainly if the galaxies are face-on), but not the other way round (see VCC917).

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Fig. 7 Comparison of our kinematic profiles with other works. Each diagram shows the rotation curve (upper panel) and the velocity dispersion profile (lower panel). The bottom X-axis is measured in arcseconds and the upper X-axis is measured as a fraction of the effective radius (Reff) of each galaxy in i band (see Sect. 6). For Van Zee et al. (2004) we only present their velocity profiles for MgbI, more similar in wavelength to our data.

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© ESO, 2011