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5 Applications of the GA to more complicated
interaction geometries

 While the runs discussed in the previous section served to illustrate the basics of the algorithm, they were somewhat oversimplified compared to the cases encountered when actual interacting systems are studied. Thus, in this section, some more stringent tests of the algorithm will be presented.

\psfig {,height=6.0cm}

\psfig {,height=6.0cm}\end{figure} Figure 4: The observations corresponding to Run 6 (upper panel) and Run 8 (lower panel)
First, it is rare that observed interacting galaxies are exactly face-on, and that assumption will now be abandoned by allowing arbitrary disc plane inclinations and major axis position angles for both discs. Thus, four new parameters, hereafter denoted $i_{1}, i_{2}, {\rm PA}_{1}$, and ${\rm PA}_{2}$, need to be introduced. Now, for those interacting systems for which the method described here is best adapted, i.e. systems such that the outer parts of the galaxies are affected but the inner parts of the discs are left intact to a great extent, such that the scale lengths and scale heights of the discs can be measured, the parameters $i_{1}, i_{2}, {\rm PA}_{1}$, and ${\rm PA}_{2}$ can also normally be measured, albeit with some uncertainties. Hence, these angles will in general be assumed to be measurable and will be made part of the input data. A case in which the inclinations and position angles are instead made part of the set of variables will, however, also be discussed.
Table 3: Results from the run illustrated in Fig. 3. The upper row shows the orbital parameters of the best simulation in generation 100, and the lower row shows the orbital parameters corresponding to the observational data

Gen. & $a$\space & $e$\...
 ... 15.75 & 33.00 & 1 & 1 & 1.00 &
\\ \noalign{\smallskip}

Table 4: Results from Runs 5 to 8. In Runs 5 and 6, elliptical orbits were used, and in Runs 7 and 8, the orbits were hyperbolic. As shown in the last four columns of the table, at least one of the galaxies had a non-zero inclination in all cases. The information concerning Run 7 contains four rows, since it was rerun twice: once without making use of the velocity information, and once with the inclinations and position angles as variables. The observations corresponding to Runs 6 and 8 are shown in Fig. 4

Gen. & \multi...
 ...0 &
0.40 & 80.0 & $-$40.0 & 0.0 & 0.0\\ \noalign{\smallskip}

Second, the orbits used in Sect. 4 were all hyperbolic. In this section elliptical orbits will be considered as well. The elliptical orbits used will be fairly eccentric (e > 0.5) and will be chosen such that the discs are not severely damaged at pericentre passages.

Thus, the simulations to be described below constitute a greater challenge to the genetic algorithm. However, there is also an additional tool yet to be used, namely the radial velocity field. The use of the velocity field may be particularly useful for e.g. highly inclined systems for which the position data may yield ambiguous information.

The results from four runs with inclined discs are shown in Table 4. In these runs, both position and velocity data were used by the GA. As is evident from the table, the algorithm was able to find orbital parameters close to the correct ones in all cases. Note that the two first pairs shown in Table 4 correspond to elliptical orbits.

In order to test the importance of the velocity data, Run 7 was repeated using only position data, and the result from this run is shown in Table 4 as well. Clearly, the use of the velocity data improved the results, even though acceptable orbital parameters were found even in the case in which only position data were used.

For real systems, the inclinations and position angles are rarely known with high accuracy. In order to take this into account, the simulation program was modified to include the four angles $i_{1}, i_{2}, {\rm PA}_{1},$ and ${\rm PA}_{2}$ in the set of unknowns. Since the values of these angles can at least be estimated, they were allowed a range of variation of only $\pm 10$ degrees, in steps of 1 degree, thus increasing the size of the search space by a factor 214 = 194,481. Under these conditions, Run 7 was repeated once more and the results, including the inclinations and position angles found by the GA, are shown in the seventh row of Table 4. Due to the large increase in the size of the search space, the GA was allowed to run for 200 generations in this case, and, as can be seen from the table, acceptable orbital parameters were found in this case as well.

The observations corresponding to Runs 6 and 8 are shown in Fig. 4. Of particular interest is the fact that the algorithm is able to cope with the difficult case in which one of the discs is edge-on. Thus, the results reported in this section show that the GA method can successfully be applied to realistic interaction geometries.

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