Here we perform some simple tests to assess whether the observational errors produce any bias in the derived models parameters. To this end we convolve the simulations of Sect. 3 with typical RAVE errors. In particular, we use the median errors of the radial bins that we use in our analysis. That is we take a relative error in distance of 25%, error in proper motion (both in α and δ) of 1.9 mas/yr, and error in radial velocity of 0.9 km s-1.
More in detail, we first orient the disc of the simulation so that the desired band with different position angles for the bar of 20,40,60 or 80 deg has the same orientation with respect to the solar position (6 deg) as our RAVE band. We select particles in a band with the same angular width as in the observations and we then convolve the positions and velocities with the mentioned errors (assuming they are Gaussian). Afterwards we bin the simulation in the same way as the data. We select 4 bins located in the same (relative) position with respect to the simulated Sun as in the data to have particles with a similar distance distribution (and, therefore, similar errors in distance and in transverse velocity).
An example of the final velocity distribution for the 4 bins for the band of 40 deg is shown in Fig. B.1 (middle). The new bins are smaller in extent and therefore have significantly less particles compared to Fig. 5. In order to do a proper comparison, we include here also the original simulation (i.e. without error convolution) in the same radial bins (left column). In the upper left corner we indicate the median direction of the transverse movement with an arrow. The blurring of the substructures occurs along this direction as this is the one influenced by errors in distance and proper motion, which are significantly larger than errors in line of sight velocity. The velocity distribution is distorted along this direction as explained in A12 and in Fig. 9 of McMillan (2013). For some of the bins (outside the range presented here), the blurring is large enough that the Hercules gap is no longer detected. The final determinations of vφ,OLR for all bands (black symbols in Fig. B.2) fall close to the expected theoretical lines in most of the bins. Because of this no bias is observed in our final bar’s parameter determination.
We now perform another test where we consider also particles originally located farther away in the disc but that, due to distance errors, end up in the selected band after error convolution. For this we first select particles with a maximum distance of 2.4 kpc from the Sun’s position, we convolve with the RAVE errors, and finally we take the subset of particles in the bands. The limit of 2.4 kpc corresponds to the maximum distance at which a red clump star with absolute magnitude of MJ = −0.87 would be observed by the RAVE survey assuming that the upper magnitude limit of the survey is J ~ 11. With this limit we avoid including particles in our band that were originally very far from the Sun and therefore, that would have never been observed because of the magnitude limit of the survey.
The example for the band of 40 deg is shown in the right column of Fig. B.1. The final determinations of vφ,OLR for all bands are the colour symbols in Fig. B.2. We observe in this case a slight tendency to obtain smaller estimates of vφ,OLR for the bins that are far from the simulated Sun (the ones at larger radius) and that seems to increase with distance. However, as the RAVE bins are still quite close to the Sun with a maximum distance of 1.67 kpc, this bias is not very significant.
Scatter plot of the velocities in bins in radius as indicated in the top right part of the panels for the band at 40 deg with no error convolution (left), with RAVE error convolution (middle) and with error convultion and allowing contamination from stars at different distances (right, see text). The numbers and the errors in the top left part of the panels are the median and dispersion of the direction of the transverse movement, taken as a counterclockwise angle with respect to the vR axis. The arrows indicate this median direction.
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Results of the fits for the toy model when we add RAVE errors and contamination from stars at other distances.
The recovered parameters for all bands after RAVE error convolution are shown in Table B.1. We see how in most of the cases the correct model parameters both in angle and in pattern speed are recovered within the error bars. However, the recovered correlation between the angle and the pattern speed does present a bias. In particular, the recovered pattern speeds obtained with the linear relation using the correct (true) value of the bar’s orientation are slightly larger than expected. In all cases the difference between the recovered value and the correct one is between 0.6 and 1.1 km s-1 kpc-1, increasing with bar’s orientation. This is in all cases equal or less than 2σ. We remark that we do not see this bias in the expectation values of the likelihood.
Several measurements for the bands at different bar angles as in Fig. 6 when we add RAVE errors (black symbols) and when we also allow for contamination from stars at other distances (colour symbols).
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© ESO, 2014