10 Analysis of simulated photon flux

In the case of X-rays from AGN there might be a physical source of deterministic variations of the photon flux. Pacholczyk & Stoeger Pacholczyk94 propose "building blocks'' in the X-ray photon flux from active galactic nuclei. In the present study each building block is assumed to be formed by a Poisson process. A Poisson process may be simulated [Råde & Westergren1995] using a series of random numbers according to the formula

$$t_i = t_{i-1} - \frac{\ln X_{ni}}{C}$$ (12)

where t_i is time of event i, X_ni is a random number between 0 and 1 and C is the Poisson parameter, called also the process intensity. The event series is then converted into an event rate plot by counting events in consecutive time bins. In the present case it is assumed that each block is a fragment of a Poisson process with the process intensity within each block, C decreasing with time according to:

 

C = C_o t_j^-k (13)

where $C_{\rm {o}}$ is the starting process intensity, t_j is the time of the j-th photon measured from the beginning of the building block and k is the exponent determining the rate of decay of the block. The simulations were performed assuming random values for the number of photons in each block, NP, the number of building blocks during the simulation period, NB, and a pair of values for $C_{\rm {o}}$ and k. The results of simulations have shown that there will be a significant difference in the character of time scale spectrum depending on if building blocks are superimposed in a semi-regular manner or randomly. Figure 18 (left diagram) shows the time scale spectrum for randomly superimposed blocks (NB = 35, NP = 214, $C_{\rm o}=12.6$ and k=2.27), while in the right diagram the building blocks were superimposed in a semi-regular manner (NB = 8, NP = 238, $C_{\rm {o}} =5.0$, k=0.615 and the average period between blocks is 200 $\pm$ 40 s). Note the linear scale of x-axis on both diagrams.

Figure 18: Time scale spectrum of simulated superposition of building blocks: random (left) and semi-regular (average period 200 $\pm$ 40 s) (right diagram). Linear scale of x-axis

It may be seen that the character of superposition of Poisson-like building blocks may be studied using the present method. Also number of building blocks per time unit may be determined.