4 Results from simple applications of the method

 

4.1 Run 1

 In this section, the results of applying the method presented above will be given. Applications to systems with complicated interaction geometries will be deferred to Sect. 5. Thus, this section will serve as an introduction to the basic features of the method, and the discussion will be limited to simple face-on systems for which only position data are known.

First, the system shown in the left panel of Fig. 1 will be studied: In order to determine the corresponding orbital parameters, a run with population size $N_{\rm pop} = 500$ was carried out. The number of generations ($N_{\rm gen}$) was 100, the mutation rate was $p_{\rm mut} = 0.003$, and the number of grid cells was 49 (n_x = 7, n_y = 7). For Run 1 and all other runs described in this section, the values of m₁ and m₂ were constrained to lie between 0.3 and 3.0, $\Delta z$ between -50.0 and 50.0, and $\Delta v_{x}$ and $\Delta v_{y}$ between -0.999 and 0.999. All possible spin combinations were allowed. The total number of combinations of the unknown variables $m_{1}, m_{2}, \Delta z, \Delta v_{x}, \Delta v_{y}, s_{1}, s_{2}$ was then approximately $1.2 \ 10^{15}$.Mutated chromosomes were only accepted if their values of the unknown parameters were contained in the above intervals. For the observational data used in Run 1, the actual values of m₁, m₂, $\Delta z$, $\Delta v_{x}$, $\Delta v_{y}$, s₁, and s₂ were 1.0, 1.0, 3.0, -0.672, 0.839, 1, and 1, respectively. Table 1 presents the results of Run 1: The first 6 rows show the orbital parameters of the best simulation in generations 1, 5, 10, 20, 50, and 100, and the final row shows the actual orbital parameters of the "observed" system. As can be seen, the GA was able to find the orbital parameters with great accuracy. In fact, acceptable orbital parameters were obtained already after 20 generations. After 100 generations, the image of the data (left panel of Fig. 1) and the image obtained from the best simulation were almost identical. This is partly due to the fact that the same disc distributions were used both for the run and for generating the observation. However, the method does not require perfect data of the kind used in this run and it can, in fact, cope with quite high levels of noise, as discussed in Sect. 6.3 below.

  

Table 1: Results from Run 1. The first 6 rows show, for the best simulation of generations 1, 5, 10, 20, 50, and 100, the orbital parameters a, e, i, $\omega$, $\Omega$, and F₀, as well as the spins, s₁ and s₂, and the masses, m₁ and m₂, of the galaxies. The parameter F₀ is the hyperbolic anomaly at the final time step of each simulation, from which T, the time of the pericentre passage, can be computed. The final row shows the actual orbital parameters used for generating the left panel of Fig. 1

4.2 Additional runs

 The results of three additional runs are shown in Table 2. The upper row in each pair shows the orbital parameters of the best simulation in the final generation, and the lower row shows the actual orbital parameters of the (artificial) observation. The grids used for data comparison were again of size $7 \times 7$, and the population size was equal to 500 for all runs. 1000 particles per galaxy were used in Runs 2 and 3. In Run 4, 5000 particles per galaxy were used. The artificial observation used in Run 4 was generated with the same orbital parameters as those used for generating the observational data of Run 1, to facilitate comparison between the two runs. As is evident from the table, acceptable orbital parameters were found in all cases.

  

Table 2: Results from Runs 2 to 4. For each pair of rows, 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. Upper pair: Run 2, middle pair: Run 3, lower pair: Run 4

Not all determinations of orbits proceed as smoothly as, for example, Run 1. In order for the GA to be able to find the orbit, it is a requirement that the galaxies show some signs of interactions, i.e. distortions of some form. This was the case for Runs 1 to 4. However, in a case where the observation consisted of two more or less unperturbed discs, the GA was not able to find the orbit. The fact that signs of interactions are needed is rather obvious and does not, in practice, imply any restrictions. After all, the method is intended for interacting systems.

A more important problem is that, even for clearly interacting systems, the GA is not always able to find the correct orbit on the first attempt. Since the population size is far from infinite, there may simply never be sufficient variation in the genetic material to obtain the correct orbital parameters, at least if $p_{\rm mut}$ is small. $p_{\rm mut}$ can of course be increased, but the larger its value, the more the GA approaches a random search. Fortunately, it is always easy to distinguish between a run that fails and one that succeeds, namely through the fitness values. In a successful run, the fitness values continue to increase throughout the run, whereas in an unsuccessful run, the GA rather quickly gets stuck at low values.

For the excellent fit obtained for Run 1, the fitness of the best simulation was 0.430. The corresponding numbers for Runs 2 to 4 were 0.192, 0.339, and 0.475. In contrast, in a run that fails, fitness values above 0.1 are not reached, and usually the GA gets stuck at even lower values. Even in such situations improvements do occur, but at a very slow rate, and it is usually faster to restart the run, using different values of the random number generator seed and the mutation rate.

The results for Runs 1,2, and 4 were obtained on the first attempt, but Run 3 required two attempts. In the first attempt, with mutation rate $p_{\rm mut} = 0.003$, the GA got stuck at a suboptimal solution with fitness 0.0863. The mutation rate was then increased to 0.010 and the random number generator seed was changed, resulting in a maximum fitness of 0.339 in the second attempt.

4.3 Blocking out the inner regions of the galaxies

 As mentioned previously, tidal features such as bridges and tails are used by the GA when it attempts to determine the orbital parameters of an observed system. However, unlike the artificial data used here, real observational data is made complicated by the existence of features such as bars, rings, ovals, etc. in the inner regions of the galaxies. Furthermore, in an observation for which the tidal features are clearly seen, the inner regions may be saturated, i.e. overexposed. Thus, in order to avoid problems caused by the appearance of the inner regions, a modified version of the GA program was constructed such that any pixel in the grid could be blocked out and discarded. An example of a run with the modified program is illustrated in Fig. 3, where the two pixels with highest density were discarded.

  

Figure 3: The two regions of highest density were discarded, as indicated by the black squares. The original picture, before discarding the data in the two black squares, was identical to the left panel of Fig. 1

Thus, in this run, the deviation was computed using only the 47 (i.e. $n_{x}\times n_{y} - 2$) remaining pixels. Clearly, this problem is more difficult to solve for the GA (or anyone else!), since only partial information is accessible. The result of the run is shown in Table 3, the upper row as usual showing the orbital parameters of the best simulation and the lower row showing the observational data, which were the same as for Run 1. Even though the resulting fit is not as stunning as for Run 1, acceptable orbital parameters were obtained.