Size distribution of supernova remnants and the interstellar medium: the
  case of M33
  The size distribution of supernova remnants (SNRs) can help to clarify the
various aspects of their evolution and interaction with the interstellar medium
(ISM). Since the observed samples of SNRs are a collection of objects with very
different ages and origin that evolve in different conditions of the ISM,
statistical Monte Carlo methods can be used to model their statistical
distributions. Based on very general assumptions on the evolution, we have
modeled samples of SNRs at various initial and environmental conditions, which
were then compared with observed collections of SNRs. In the evolution of SNRs
the pressure of the ISM is taken into account, which determines their maximum
sizes and lifetimes. When comparing the modeled and observed distributions, it
is very important to have homogeneous observational data free from selection
effects. We found that a recently published collection of SNRs in M33 (Long et
al. 2010, ApJS,187,495) satisfies this requirement if we select the X-ray SNRs
with hardness ratios in a limited range of values. An excellent agreement
between distributions of this subset of SNRs and the subset of modeled SNRs was
reached for a volume filling-factor of the warm phase of the ISM (partly
ionized gas with $n_{\rm H}\sim 0.2-0.5~ \rm {cm}^{-3}; T \sim 8000-10000~K $)
in M33 of $\sim\ 90%$. The statistical distributions constructed in this way,
which reproduce practically all the statistical properties of observed SNRs,
allowed us to obtain one of the important parameters of M33: the birthrate is
one SNR every $ {140} - {150}$ yr, and the total number of SNRs with a shock
Mach number $M_{s} \geq 2$ is larger than $\sim 1000$.
