2D: Two disk (thin + thick) Fits

For our 2D Two Disk fits we made use of equation 12 as the Bessel function will return scalelength values which we can compare better with results from the literature. The results and boundary parameters and binning factor of the fits are listed in Table 11. A selection of the vertical profiles is shown in Appendix A in a set of four panels, containing the charactistic bending in the outer part of the vertical profile, resembling a thick disk, together with the two and one disk fit and the individual thin and thick disk component to the fit.
For each galaxy the number of datapoints and cuts are given. $\chi^{2}_{\nu,1}/\chi^{2}_{\nu,2}$ is the ratio of the reduced $\chi^{2}$ for the 1D Single Disk exponential fit for each cut of the dataset and the reduced $\chi^{2}$ for the 2D Two Disk fit over the whole dataset, and gives a measure of the quality of the thick disk component. Following Wadadekar et. al. wadadekar1999, we use this ratio and not the reduced $\chi^{2}_{\nu,x}$, because we want to compare the quality of the fitted thick disk components, as a six parameter fit will naturally provide a better fit. With each ratio a break ratio is given, which is the ratio between the total number of free parameters for each of the two fits. As the 1D Single Disk fit is not radially related the degrees of freedom accumulate depending on the amount of profiles and are thus different for each galaxy. For example, if there are 12 cuts the number of free parameters for the single disk fit will be 12 times 2 parameters; 24 in total. For the 2D Two Disk fit the parameters include one dataset containing all profiles and will thus be only 6. The break ratio is then 24/6 = 4. The higher the ratio is above the break point the better the two disk fit is compared to the one disk fit. At the break point both fits are equally good.
An import remark has to be made however. When using a $\mu_\textrm{\scriptsize cut}$ as a lower boundary for the fits, the elliptic shape of the galaxy causes the outer cuts to lose signal in the $z$-direction. This causes a systematic effect that the thick disk component becomes smaller at larger radius. This causes a systematic decrease of the quality of the contribution to $\chi^{2}_{\nu,1}/\chi^{2}_{\nu,2}$ towards the outer cuts, causing $\chi^{2}_{\nu,1}/\chi^{2}_{\nu,2}$ to be lowered as well. Since this effect is not truly systematic, and different for each galaxy, we cannot correct for this effect. Another issue is that a single disk fit would not fit itself to the dominating thin disk component but chose an intermediate slope to minimize the difference between the datapoints.
$R_{\textrm{\scriptsize min}}$ and $R_{\textrm{\scriptsize max}}$ are the boundaries of the inner disk, excluding the bulge region on the one side and the outer disk on the other side (see Section 3.4.1). The radial fit was taken within these boundaries. $z_{\textrm{\scriptsize min}}$ is where the vertical profile in the inner part showed flattening and we decided to remove those inner datapoints.
What one notices is that the values of $\mu_{n-k}$, $f_z$ and $f_h$ stay within certain limits. The $\mu_{n-k}$ values lie close together: 4.5 $\pm$ 0.7 mag arcsec$^{-2}$. $f_z$ varies considerably, giving an average value of 5.6 $\pm$ 1.9. Both values are higher than found in previous research. $f_h$ is set at 1.6 $\pm$ 0.8 which is nicely within literature boundaries. The result of NGC 4179 is not included in the averages, as it is a different galaxy type while we want to compare the typical results for late-type galaxies. Within these boundaries we can speak of an average thick disk with these three fundamental parameters, with which one will find most likely the correct thick disk component for late-type disk galaxies according to our sample and fitting method.
During the fits we noticed the Downhill Simplex method to show a certain coupling between $\mu_{n-k}$ and $f_h$ while trying to converge to the lowest $\chi^2$. In some cases it made $\mu_{n-k}$ smaller by decreasing $f_h$ and similarly it increased $\mu_{n-k}$ as it increased $f_h$, while the other parameters hardly changed. As it is not possible to determine in which cases this effect appears, it does allow us to explain certain low or high values for $\mu_{n-k}$ and $f_h$ in some cases in combination with their other parameter results.