# Untitled Document

1. 1.

As stated in Appendix B, when the optical and the HI estimate of the inclination and/or of the centre are largely discrepant we use the estimate coming from the HI data. In general the fit of the optical disc gives more robust estimate of the centre and of the inclination, but in some cases this can be biased by the presence of bar-like structure. Moreover the HI disc is often several time larger with respect to the stellar disc and it should be better and more self-consistent to derive the parameter of the disc directly from the HI data. The discrepancy between the disc parameters obtained with the optical and HI fit is a very interesting topic, but it deserves a very deep study that is beyond the scope of this paper. Concerning DDO 47, the inclination of the stellar disc is clearly to high compared with the iso-density contours of the HI map. Since the stellar disc could have a bar, we decided to estimate the galactic centre again from the fit of the HI contours, however the differences between the two centres are small ( 0.7 arcsec in RA and 2 arcsec in DEC) and they do not have a great influence on the final results. The reasons beyond the decision to target as not totally reliable the Vc for $$R<120$$ arcsec is that in this region the HI distribution shows clumpy regions of high density ad a large hole, the presence of these structures can bias the fit of the rotation curve since it tends to follow regions with very high density (as shown in the PVs of Fig. 6). Moreover, the presence of this early flattening could be due to the presence of strong non-circular motions. We changed the description of DDO 47 as follows:

The HI disc of DDO 47 (Fig. ??) is nearly face-on, while the stellar disc looks highly inclined at 64$${}^{\circ}$$ ??. An inclination of 64$${}^{\circ}$$ is not compatible with the HI contours of the total map, therefore we decided to estimate the $$i_{\text{ini}}$$ and the PA$${}_{\text{ini}}$$ by fitting ellipses on the contours of the total HI map (see Appendix ??). The discrepancy between the galactic centre estimated with the HI contours and the optical centre is fairly small (0.7 arcsec in RA and 2 arcsec in DEC), so we decided to use the HI centre given that the the stellar disc probably hosts a bar-like structure ??. The best-fit $$i$$ (see Tab. \ref{tab:sample_res}) is consistent with previous works (35$${}^{\circ}$$ in ??, 30$${}^{\circ}$$ in ?? and ??) and seems a reasonable upper limit for this galaxy (see ellipses in Box C in Fig. ??). The final rotation curve shows a flattening at $$R<120^{\prime\prime}$$: analysing the PVs and the channel maps we found that at these radii the emission is dominated by small regions with with high HI flux. The presence of such structures can bias the kinematic fit, since 3DB tends to weight more the kinematics of these regions with very high density. Moreover in the approaching half of the galaxy there is a large hole that can be clearly seen both on the total map and on the PV along the major axis, the presence of such hole could further bias the estimate of the gas rotation. For this reason, we retain the velocities for $$R<120^{\prime\prime}$$ not completely reliable (empty circles in Box A in Fig. ??).

2. 2.

We have not fully understood what are the quantitative information that the referee is asking. The sentence ’The i of the optical disc (47$${}^{\circ}$$) looks too high to be compatible with the HI contours’ is just saying that the inclination of the optical disc is not compatible with the one estimated with the flattening of the HI contours (similar to the case of DDO 47). We have changed the sentence to be hopefully more clear:

The $$i$$ estimated with the isophotal fitting of the optical disc (47$${}^{\circ}$$) is too high to be compatible with the flattening of the HI contours which favour an $$i<40^{\circ}$$.

3. 3.

As stated in Sec. 4.2, we fitted the PA and/or the inc with a non-constant function only if they show a clear regular radial trend. The presence of the radial trend is checked by eye (inspecting also the PVs and of the datacube channels) and it can be considered a subjective approach. However, we think that it is more robust than other automatic criteria (as for example evaluating the $$\chi^{2}$$ of the polynomial fit). We have added a short sentence ad the end of Sec. 4.2:

We chose to use the lowest polynomial order allowed by the data: in practice, we set the geometrical parameters to a constant value, unless there is a clear evidence of radial trends of $$i$$ and/or of PA as for example in DDO 133 and in DDO 154. The existence of the radial trend and the degree of the polynomial are set after a visual inspection of the radial profile of $$i$$ and/or PA and also of the HI map, of the velocity field and of the datacube.

4. 4.

We have extended the description of the PA fit in DDO 154:

In order to obtain a good description for the radial trend of the PA we used a fourth order polynomial. However, the best-fit polynomial shows a very steep gradient (about 35$${}^{\circ}$$) in a very small region (less than 1 kpc) that is not justified by the data. The points located in this region are compatible with a constant PA, so we decided to fix it to the mean values of the first three points (see Box B in Fig. \ref{fig:DDO154}).

5. 5.

Following the referee’s comment, We have changed the description of DDO 210:

DDO 210 (also know as Aquarius dIrr, Fig. \ref{fig:DDO210}) is the least massive galaxy in our sample and it is classified as a transitional dwarf galaxy ((citation not found: mc) and reference therein). The HI map is quite peculiar with isodensity contours that are not elliptical-shaped. As a consequence the estimate of the galactic centre using the HI data is very uncertain, hence we decided to set it to the optical value. The kinematics is dominated by the velocity dispersion, however a weak velocity gradient is visible. The velocity gradient looks misaligned with both the stellar and the HI disc, so we set the initial values of $$i$$ and PA (60$${}^{\circ}$$ and 65$${}^{\circ}$$, respectively) using a by-eye inspection of the velocity field. This procedure is quite arbitrary, but it is important to note that the final circular-velocity curve and the related errors are quite independent of our procedure since they are totally dominated by the asymmetric-drift correction (see Sec. \ref{sec:asy} and Sec. \ref{sec:fnotes}). The rotation in the first two rings (10 and 20 arcsec) is not well-constrained, so we decided to exclude them. Notice that along the minor axis there is an extended region with HI emission apparently not connected with the rotating disc. As in the case of DDO 53 this emission could trace an inflow/outflow.

6. 6.

The referee is right here. We have changed the description of DDO 50 as follows:

The gaseous disc of DDO 50 (Fig. \ref{fig:DDO50}) is quite peculiar: it is ’drilled’ by medium-large HI holes ranging from 100 pc to 1.7 kpc and it also shows clumpy regions with high density. Therefore, the HI surface density evaluated across the rings could be very discontinuous with very high peaks and/or regions without emission. As a consequence, the quoted errors on the intrinsic surface density profile (see Sec. \ref{sec:mapd}) are quite large and they are probably over-estimating the real statistical uncertainties, especially in the inner disc ($$R<3$$ kpc).

7. 7.

We agree that the description about the exclusion of the points is a bit vague. We prefer to not extend the Appedix B, but we have added a reference to a more detailed explanation at the end of Sec. 4:

In some cases the velocity dispersion found with 3DB is not well constrained, as a consequence there are galaxies where at certain radii the $$\sigma_{v}$$ is discrepant or peculiar (e.g. very small error) with respect to the global trend. In general, the presence of a single ‘rogue’ $$\sigma_{v}$$ (e.g. DDO 87, Fig. \ref{fig:DDO87} or DDO 133, Fig. \ref{fig:DDO133}) does not have a significant influence on the estimate of the $$\textrm{V}_{\textrm{A}}$$ and on the median of $$\sigma_{v}$$ (see Tab. \ref{tab:sample_res}). However, if the discrepant $$\sigma_{v}$$ is in a peculiar position (e.g. the last radius in DDO 210, Fig. \ref{fig:DDO210}) or the region with ‘rogues’ is extended more that one ring (e.g. DDO 216, empty circles in Fig. \ref{fig:DDO216}) the correction for the asymmetric drift and the estimate of the circular velocities are dramatically biased. In these cases we exclude the radii with discrepant $$\sigma_{v}$$ both from the calculation of the asymmetric-drift correction and from the calculation of the median.

8. 8.

We have added the sentence “treated with some caution” as very conservative analogy with the other nearly face-on galaxies. Actually, the estimate of the inclination for this galaxy seems more robust with respect to the other nearly face-on galaxies. Moreover, the position of this galaxy in the BTFR (Sec. 5) is compatible with the galaxy with well-estimated inclinations. For this reason we are enough confident on the reported rotation curve, so we have removed the warning.

9. 9.

Very interesting point. The best-fit rotation velocities found with 3DB (blue curve in Panel A of Fig. 16) do show point-to-point scatter. However, the final circular velocity curve of DDO 210 is totally shaped (both on values and errors) by the asymmetric-drift correction. So, the real point is why the VA does not show point-to-point scatter despite of the large errors. This is not only the case of DDO 210, the VA shows this feature in all the galaxies of our sample and this is directly related to the method we use to calculate the asymmetric-drift correction. We derive VA fitting functional forms to avoid abrupt scatter in the final corrected circular-rotation curve, as a consequence we are in a certain sense ’correlating’ data at different radii as the referee pointed out. In other terms, we built our method to force the asymmetric-drift correction to be as smooth as possible and probably this hides some intrinsic scatter. However, we are confident that the quoted errors truly trace the uncertainties introduced by the our method. In conclusion, we have added a new small sections after 4.3.3 with a general notes on the quoted errors for the asymmetric-drift correction

### Final notes

The method described in the above sections has been built to obtain the asymmetric-drift correction terms as smooth as possible. This ensures that our final rotation curves are not affected by non-physical and abrupt scatters due to the derivative term in Eq. \ref{eq:asy_asyfin}. The drawback of this approach is that intrinsic scatter on $$\textrm{V}_{\textrm{A}}$$ is hidden and the final errors $$\delta_{\textrm{A}}$$ could be several times larger with respect to the point-to-point scatter. However, we are confident that the quoted $$\delta_{\textrm{A}}$$ truly trace the degree of uncertainties introduced by the asymmetric-drift correction at different radii. In galaxies where the final circular velocity is totally dependent on the asymmetric-drift correction (e.g. DDO 210, see Sec. \ref{sec:gls} and Fig. \ref{fig:DDO210}) the scatter in $$\textrm{V}_{\textrm{rot}}$$ could be smoothed out, however in these cases also the final error are dominated by the errors on the asymmetric-drift and $$\delta_{\textrm{c}}$$ is still a good representation of the global uncertainties.

and we have modified a little bit the description of DDO 210:

This procedure is quite arbitrary, but it is important to note that the final circular-velocity curve and the related errors are quite independent of our procedure since they are totally dominated by the asymmetric-drift correction (see Sec. \ref{sec:asy} and Sec. \ref{sec:fnotes})

10. 10.

We have not fully understood what the referee means about using the same approach of Fig. 18 in Fig. 17, so we try to answer in the most general way hoping to catch the point of the referee. Altough DDO 216 (Fig. 17) and NGC 1569 (Fig. 18) are both galaxies with very peculiar kinematics, the ’kind of peculiarity’ is different. The inner region of NGC 1569 is totally a mess, without signs of rotation at all, so our intention was to separate the inner radii (open circles in Panel A) from the rest of the galaxy that shows some velocity gradient. In the case of DDO 216, we think there are no reasons to indicate that some points of the rotation curve are more reliable than others, instead the problem is that the gradient observed could be not entirely due to the rotational motion of the gas as extensively explained in Appendix C. We could have used only open circles in the rotation curve of DDO 216 to indicate the uncertain nature of the rotation curve, but we prefer to use these white points to indicate a warning of reliability in some peculiar region of the analyzed galaxies.

11. 11.

The referee is right and our sentence is not complete. We meant that $$\text{V}_{0}$$ is a measure of the maximum circular velocity in the considered radial range. We agree with the referee that it could be also a lower limit if the curve continues to rise beyond our last radius. However we prefer to avoid the use of the term lower limit, since it seems that the value to use in the BTFR could be truly anything above the reported $$\text{V}_{0}$$. So, we have just modified the sentence in:

For galaxies with a rising rotation curve V$${}_{\textrm{o}}$$ is an estimate of the maximum circular velocity within the considered radial range

12. 12.

The referee is right here, it is a typo, the right Eq. is:

$$\delta^{\text{min},\text{max}}_{\text{D}}=1-\left(\frac{\min(\text{D},\text{D}^{\text{min/max}}_{\text{NED}})}{\max(\text{D},\text{D}^{\text{min/max}}_{\text{NED}})}\right)^{2}\nonumber \\$$

The Eq. essentially comes from the relative difference between the mass obtained with our assumed distance and the one obtained with the minimum or maximum distance found on NED as explained in the text. We have added a short sentence and a new Eq. to explain the final form of former Eq. 15 (now Eq. 16):

Unfortunately, the works we used to take the stellar masses (citation not found: d47stellarmass) (citation not found: ZLT) and the distances (citation not found: LT) do not report the errors on their measures, so we assumed a conservative error of the 30$$\%$$ for $$\text{M}_{\textrm{bar}}$$. We also performed a deeper analysis for the distance uncertainties. The relative difference between the masses estimated assuming two different distances is

$$\label{eq:mass-distance} \label{eq:mass-distance}\delta_{\text{D}}=\frac{\text{M}(\text{D}_{1})-\text{M}(\text{D}_{2})}{\text{M}(\text{D}_{1})}=\frac{\text{D}^{2}_{1}-\text{D}^{2}_{2}}{\text{D}^{2}_{1}},\\$$

with $$\text{D}_{1}>\text{D}_{2}$$. For each galaxy in our sample we choose the best distance estimator11The scale of distance estimators is, from the best to the worst: Cepheids, RGB-Tip, CMD, Brightest-Stars, Tully-Fisher relation. available on NED (NASA/IPAC Extragalactic Database) and we considered the minimum ($$\text{D}^{\min}_{\textrm{NED}}$$) and the maximum ($$\text{D}^{\max}_{\textrm{NED}}$$) estimate of the distance, then using Eq. \ref{eq:mass-distance} we calculated

$$\delta^{\min/\max}_{\text{D}}=1-\left(\frac{\min(\text{D},\text{D}^{\min/\max}_{\textrm{NED}})}{\max(\text{D},\text{D}^{\min/\max}_{\textrm{NED}})}\right)^{2},\\$$

where D is the distance assumed in this work (see Tab. \ref{tab:sample_obs}). When $$\delta_{\text{D}}$$ is large the error on the total mass is dominated by the uncertainty on the distance. Three galaxies have $$\delta_{\text{D}}$$ larger than the $$60\%$$: DDO 47 ($$\delta^{\max}_{\text{D}}=62\%$$), DDO 101 ($$\delta^{\max}_{\text{D}}=85\%$$) and NGC 1569 ($$\delta^{\min}_{\text{D}}=68\%$$). For these galaxies we do not show the 1-$$\sigma$$ error on the baryonic mass, but a magenta bar indicating the interval of mass found assuming the distance D or the distance $$\text{D}^{\min/\max}_{\textrm{NED}}$$.

1. 1.

Done

2. 2.

Done

3. 3.

Done