RECIPE TO OBTAIN LOSVDS FROM K -BAND SINFONI SPECTRA

equivalent width (this is usually the case as MPL automatically chooses the best template combination), almost independent ofα. Values forv andσ can be found in the literature for most galaxies, which are usually a good first guess to determine the initial velocity profile. If the kinematics is extracted from unbinned data, the S/N per pixel is low (between a few tens in the centre and a few in the outer parts, with S/N<30 for most of the field of view), thus to determine a first good esti-mate of v and σ without too much scatter a high α ∼50 should be chosen (see Fig. 3.5).

Once having the S/N information the spectra can be binned into higher S/N bins, either using Voronoi binning or the radial and angular binning scheme used for the dynamical models described in Chapter 4. The size of the spatial bins should be a compromise between a S/N which is as high as possible and good enough spatial resolution to resolve the sphere of influence. This usually means that the S/N of the central spaxels is taken as a reference such that the central bins comprise only one spaxel and the size of the other bins increases with radius. If the central S/N is very low, the size of the central bins might be increased, but should not exceed the size of the PSF or (as a strict upper limit) the size of the sphere of influence. From Fig. 3.5it becomes clear that, in order to get relatively noise-free LOSVDs with reasonably small error bars, the S/N should be at least

∼40. The luminosity-weighted spatial binning of the spectra was done using the IRAF¹ task^imombine.

For the binned spectra the S/N then should also be exactly determined for each bin. This is not so much important for the determination of the best smoothing parameter, for which an approximate value would be sufficient. The exact S/N is needed for the determination of the LOSVD error bars, which is done by cre-ating 100 spectra from the templates convolved with the measured LOSVDs and appropriate amounts of noise added to them. Adding too much or too few noise would falsify the error bars. The average velocity and velocity dispersion deter-mined from the unbinned data can be used to define the velocity range and bin width. α=10 is a reasonable upper limit for the high-S/N, binned spectra for this first iteration, based on the simulations in the previous section. In case there are residual sky lines, bad pixels, or cosmics in the region between or blue or redwards of the CO bandheads the estimated S/N is too low. In these cases it is advisable to restrict the wavelength range or to use only one bandhead.

1IRAF (http://iraf.n oa o. ed u/) is distributed by the National Optical Astronomy Obser-vatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science foundation.

CHAPTER 3. STELLAR KINEMATICS

S/N

20 40 60 80 100 120 140 160

smoothing

1 10

Figure 3.12: “Best” smoothing parameter αas a function of S/N forσ =250 km s⁻¹ (red), 80 km s⁻¹(green) and 50 km s⁻¹(blue).αderived from theχ²difference between measured and correctσare plotted as thin solid lines, single or double exponential fits to these data points are plotted as thick solid lines.αderived in the same way fromh₄and

the according fits are plotted as dashed thin and thick lines.

Having a good estimate for the S/N and σ, it is now the time to analyse the simulations to determine the most appropriate value ofαfor each bin of each data set. The measuredσ andh₄ are the only parameters that depend onα, as long as the galaxy h₃ and h₄ do not differ too much from 0. Thus the best value for α can be chosen as where σ_out−σ_in =0 or h_4,out−h_4,in =0. The problem is, that these two values can differ substantially. Fig. 3.12shows for high and lowσ the

“best” smoothing parameters as a function of S/N determined fromσ_outand from h_4,out. This “best”α has been determined as the minimumχ²difference between the measured and the correct σ or h₄ forα <40. The scatter is quite large, but the overall trend with S/N can be described by a single or double exponential. α_σ is always larger thanα_h4. Both values decrease with decreasing σ_in and increasing S/N of the galaxy. However, the bias in σ_out is small and within the errors of the real value when the chosen α is smaller than α_σ for most S/N values of our SINFONI observations. If, on the contrary, the chosen α is larger than α_h4, the bias in h₄ could be large and not within the errors of the real h₄, as h₄ at some point starts to decrease very strongly. Therefore the final α should be closer to α than toα . In addition to the Gauss-Hermite parameters it is also important

3.5. RECIPE TO OBTAIN LOSVDS FROMK-BAND SINFONI SPECTRA

=0.2 α

−300 0 300

=5 α

=0.6 α

−300 0 300

=6 α

=1 α

−300 0 300

=7 α

=2 α

−300 0 300

=8 α

=3 α

−300 0 300

=9 α

=4 α

−300 0 300

=10 α

v (km/s)

Figure 3.13: LOSVDs (σ=80 km s⁻¹) forα=0.2−10 at a S/N of 70. The noise in the LOSVDs decreases with increasingα.

to look at the LOSVDs. The purpose of the smoothing is to reduce the noise as much as possible, thus blindly choosing the bestα indicated by h₄ does not help if it is still so small that the LOSVDs are noisy. Fig. 3.13 shows some example LOSVDs (σ =80 km s⁻¹) for differentα at S/N=70. For α 61 they are very noisy, but for largerαthe noise is insignificant, soα_h4 seems to be a good choice.

Note that real data may have different noise patterns (e.g. due to sky residuals, bad pixels etc.), thus it might be necessary to chooseα > α_h4. Forα_h46α®10 the measuredh₄is usually in agreement with the correct h₄within the errors, as long asσ is not too small.

For velocity dispersions smaller than 80 km s⁻¹the situation is much more dif-ficult, because h₄ is biased for all α to smaller values. The reason is the velocity bin width, which is smaller than the instrumental resolution, when using a non-parametric fit. As changing parameters such as the velocity bin width does not help to reduce the bias, one possible solution is to (1) derive theh₄bias for the ve-locity dispersion range of the galaxy from simulations, (2) subtract the bias from h₄ in the simulations and determine an appropriate α from the bias-corrected h₄ values, (3) determine the LOSVDs of the galaxy spectra with thisα, (4) do a Gauss-Hermite fit to the LOSVDs, (5) subtract the previously determined bias from the measured h₄ values and (6) calculate corrected LOSVDs from the corrected Gauss-Hermite parameters. If the last three steps are omitted and the uncorrected

CHAPTER 3. STELLAR KINEMATICS