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Astron. Astrophys. 335, 329-340 (1998)

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7. Discussion

7.1. Comparison with the power spectrum

Since the gamma distribution function for the time-intervals in between the pulses is strongly peaked around the duration of the pulses, the time dependence of the driving can be approximated by the function

[EQUATION]

The power spectrum of this signal can be easily calculated analytically and is shown in Fig. 10. This figure reflects the remarkable form of the graph in Fig. 9 as was to be expected.

[FIGURE] Fig. 10. The power spectrum of the function f.

7.2. Comparison with the numerical simulations

As an additional check, we compare the mean value as calculated in Sect. 6.2 with the amplitude of the kinetic energy contributions which we obtained numerically after several simulations. We consider the loop to be driven by 5 identical pulses with in between them 40 different series of 4 time intervals each satisfying a [FORMULA] distribution. After each simulation we look for the final amplitude of [FORMULA] for three different eigenfrequencies, namely [FORMULA], [FORMULA] and [FORMULA].

[FORMULA]

The probabilistic calculations we did in the previous section predict [FORMULA] to oscillate with an average amplitude of approximately 0.286 (see Fig. 9).

In our numerical solutions we found for the final amplitude 40 different values which all lie between 0.22 and 0.37 with an average value around 0.27. The standard deviation

[EQUATION]

is 0.051. Thus our first check is very satisfying since the values we found all approximate the predicted average.

[FORMULA]

For this eigenfrequency of the coronal loop the calculations predict an average amplitude of 0.0616. We again did similar numerical simulations and we found 40 values lying between 0.045 and 0.072. The average of the values now equals approximately 0.060 and the standard deviation equals 0.0089. These results again seem to show that our probabilistic calculations are relevant.

[FORMULA]

The predicted value for the amplitude is 0.489 and this value is closely approximated by the different results of the numerical code as it should be. This time the standard deviation is larger, namely 0.20, but the average, in this case 0.52, again approximates the predicted value.

7.3. Influence of the gamma distribution

As mentioned before Figs. 9 and 10 are very similar because of the strongly peaked gamma distribution. Now we are interested to see how Fig. 9 changes if we consider a less peaked gamma distribution.

Fig. 11 shows five less peaked gamma distributions with mode 0.5 on the left hand side and on the right hand side the corresponding graphs for the average of the final amplitude of [FORMULA] after driving by five pulses characterized by a.

[FIGURE] Fig. 11. Five gamma distributions together with the corresponding graphs for the average of the final amplitude of [FORMULA] after driving the loop with five identical pulses with a frequency a.

As expected the fine structure in the variation of the mean value of the amplitude with respect to [FORMULA] disappears and the amplitude of the peaks decreases as the gamma distribution becomes wider and hence as the time intervals between the pulses vary more. The two outer dominant peaks of Fig. 9 first remain but they disappear as the gamma distribution widens. In addition we notice that the large contrasts in the figure seem to weaken first for large frequencies and only later on for the small frequencies. In the end, for gamma distributions with small parameter [FORMULA], the graph is reduced to one wide peak around [FORMULA], as shown on Fig. 11 in the case of [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: June 12, 1998

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