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on the Scenario Up: The
Scenario Machine: Operational Previous: The
Scenario Machine: Operational
The evolution of a binary system can be described as the movement of
a point in the parameter space 
, 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
.
The most typical questions to be answered with the help of our Scenario Machine are, among others:
of the space
?
?
), and what kind of system can result from it? Clearly, the first two questions lead to the computation of some integrals
over the volume
. 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
of the parameter space, we must calculate the ratio of the time 
the binary remains inside
, averaged over all tracks, to the mean duration of all tracks 
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 tracks stay inside
; 
required to calculate the average duration of the tracks
;
required to evaluate the accuracy of calculations (using variance D).
Then the fraction of binaries lying inside
is expressed by
where 
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 
by the ratio of the total number of stars in the modeled galaxy 
to the actual number of modeled tracks N:
To obtain the mean value of some parameter x
for the binaries from the volume
of the parameter space
for N modeled tracks, one needs to calculate
quantities 
and 
. Then the mean value 
is:
where 
is a confidence level, and the variance 
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]).