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Notes on Monte-Carlo Simulations


The evolution of a binary system can be described as the movement of a point in the parameter space tex2html_wrap_inline10168 , constituting both physical quantities (such as masses of binary components, their radii, luminosities, orbital period, etc., independently of direct observational measurability of these values), and related ``logical'' statements (such as whether the system is binary or not, whether the NS is a radio- or X-ray pulsar or not, whether the radiation from the NS is observable or fully absorbed by a stellar wind, and so on). Each binary traces a path (which we will call a ``track'') in this space, in a way strictly determined by the evolutionary scenario. Each such track emerges with a certain probability, which can be expressed through the initial distribution functions of binary systems, in other terms, through the probability for a binary to be formed with given initial parameters and age. Thus, one can introduce a probability density p in the parameter space tex2html_wrap_inline10168 .

The most typical questions to be answered with the help of our Scenario Machine are, among others:

  1. what is the number or fraction of objects (binaries) lying inside a certain volume tex2html_wrap_inline10174 of the space tex2html_wrap_inline10168 ?
  2. what is the mean value of a specific parameter x for the binaries from tex2html_wrap_inline10174 ?
  3. how do the results depend on the scenario parameters and initial distributions?
  4. what type of binary system was a progenitor of the binary with given parameters (i.e. lying within a small volume of the space tex2html_wrap_inline10168 ), and what kind of system can result from it?

Clearly, the first two questions lead to the computation of some integrals over the volume tex2html_wrap_inline10174 . One possible method which is convenient enough for this task is the Monte-Carlo method.  To determine what fraction of binaries of a modeled galaxy will be found in volume tex2html_wrap_inline10174 of the parameter space, we must calculate the ratio of the time tex2html_wrap_inline10188 the binary remains inside tex2html_wrap_inline10174 , averaged over all tracks, to the mean duration of all tracks tex2html_wrap_inline10192


For this task to be fulfilled accurately, a large enough number N of evolutionary tracks within a wide range of the initial parameters distributed in accordance with given functions of probability distribution needs to be calculated, and the following quantities must be specified:

the total time tex2html_wrap_inline10196 the tracks stay inside tex2html_wrap_inline10174 ;
the total duration of all tracks tex2html_wrap_inline10200 required to calculate the average duration of the tracks tex2html_wrap_inline10192 ;
the sum of squares of durations tex2html_wrap_inline10188 required to evaluate the accuracy of calculations (using variance D).

Then the fraction of binaries lying inside tex2html_wrap_inline10174 is expressed by


where tex2html_wrap_inline10210 is a coefficient defined by a confidence level, and the dispersion D is defined as


To obtain the absolute number of specified binary types, one needs to multiply the quantity tex2html_wrap_inline10214 by the ratio of the total number of stars in the modeled galaxy tex2html_wrap_inline10216 to the actual number of modeled tracks N:


To obtain the mean value of some parameter x for the binaries from the volume tex2html_wrap_inline10174 of the parameter space tex2html_wrap_inline10168 for N modeled tracks, one needs to calculate quantities tex2html_wrap_inline10228 and tex2html_wrap_inline10230 . Then the mean value tex2html_wrap_inline10232 is:


where tex2html_wrap_inline10234 is a confidence level, and the variance tex2html_wrap_inline10236 is


Note that all expressions written above can be obtained within the framework of a standard Monte-Carlo method  (see, e.g., Sobol' 1973[183]).

next up previous contents index
Next: Dependence on the Scenario Up: The Scenario Machine: Operational Previous: The Scenario Machine: Operational

Mike E. Prokhorov
Sat Feb 22 18:38:13 MSK 1997