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Sections 5.3–5.4 show how we generate the models for comparison to the data. The entire process beginning from Figure 5.2 and finally ending at Figure 5.12 constitutes one run of the program with one parameter set. Table 5.2 for example represents one parameter set. In one run we can extract all the required slits for a particular galaxy and compare the radial velocities in them to the data simultaneously. The compari- son at present is done by eye. There is no efficient statistical algorithm to compare the model and data. Several statistical tests have been tried in previous attempts with the results being unfruitful, because the statistical process generated too much wasted shaded regions. We will attempt no such test in this presentation. Start- ing input parameters to generate the models were taken from Crenshaw & Kraemer

Table 5.3: Parameters used to generate Figure 5.13.

Parameters zmax θinner θouter iaxis P Aaxis vmax rt

(pc) (deg) (deg) (deg) (deg) ( km s−1_{) (pc)}

Values 200 30 40 30 20 1000 200

(2000b) and Crenshaw et al. (2000) for NGC 1068 and NGC 4151 respectively. The starting input parameters by no means bias the final best-fit results, as the choice of input parameters are independent of final best-fit ones, as will be shown in the next chapter. For the actual comparison, we use the model radial velocity shaded plot as in Figure 5.12 and over-plot radial velocities identical to the ones shown in Fig- ures 4.1–4.5 and Figures 4.9–4.16. An example of a data-model comparison is shown in Figure 5.13. This figure was generated with parameters shown in Table 5.3 and showed an example of a ‘bad’ fit for slit 1 of NGC 1068. Most of the data points lie outside the shaded region. In such a case, we reran the modeling process and tweaked the input parameters to improve the fit. The process of model fitting is summarized as follows:

1) The best fit model parameters are obtained when model slits enclose the max- imum number of data points within a minimum shaded region and also match the trend (i.e. the increasing and decreasing velocity) in the data reasonably well.

2) Models should be consistent across all slits for a galaxy; i.e., input parameters cannot change across individual slits.

Figure 5.13: An example of a bad fit for slit 1 of NGC 1068. Most of the data points fall outside the shaded region. The colors represent the different flux components. since some points are most likely not in the NLR and hence not in the bicone geom- etry. For example the points in Figure 5.13 at∼ 0 km s−1 _{in radial velocity between}

∼ 2–700 _{are most likely emission from the host galactic disk.}

4) If a fit is found to be unsuitable, the parameters are adjusted intuitively, and the entire process of generation, slit extraction, velocity sampling, and plotting un- dergoes repeated runs until we determine an acceptable match to the data.

5) Errors for a fit are not quantitatively defined, because the model fitting itself is not quantitatively defined. Therefore the errors are subjective and are defined as a range of values over which each input parameter can vary without significantly af- fecting the fit. The inclination of the bicone axis is the most important parameter

because it determines the Seyfert type. Because of this, we will vary the inclination, and make adjustments to the rest of the parameters to offset any resulting changes. If for example the best fit model parameter set is [zmax, θinner, θouter, iaxis, P Aaxis, vmax, rt],

then if we can varyiaxisover the range [a, b], with the rest of the parameters changing

while maintaining a good fit, then the error invmax becomes ₋+(_(vv_maxb−v₋max_va)₎, where [va, vb] is the range in vmax that maintains a good fit. Similarly the error for iaxis would be +(ib−iaxis)

−(iaxis−ia), whereia andib are the extrema ofiaxis while maintaining a good fit. Errors

–

6

–

Kinematic Results

The process outlined in the previous chapter was applied to data for both NGC 4151 and NGC 1068. We did multiple program runs until we were satisfied with a best fit parameter set for each galaxy. The results for those fits are presented in this chapter. A brief outline of the model fitting process is shown to explain to the reader how the process works. We will vary some of the parameters and discuss reasons for changes, if any, to the input parameters until we converge upon the best-fit set. We will start with the center slit of NGC 4151, slit 1, and show how the choices of parameters in fitting slit 1 affect the outcome of the fit on a step by step basis. We start with an initial parameter set estimated from the observations, and vary the parameters appropriately with visual justification to show in the end that our best-fit set is not too different from that derived in the previous study of Crenshaw et al. (2000). After we have determined the best fit, we will vary the most important parameter, the inclination, and track the changes in the rest of the parameters to get a feel for the errors involved in the fitting (see _§5.5). Finally, we will show for each galaxy the bicone model that has been generated with our best-fit parameters, and how it would appear in the sky to an observer. The procedure for fitting the data of NGC 1068 is identical to that for NGC 4151, so we will not show the fitting and error processes for

NGC 1068; only its final models will be shown together with the errors. The radio comparison is also done in this chapter and the results for both galaxies are presented here, following their model fitting.