On the probability distribution function of the mass surface density of
  molecular clouds I
  The probability distribution function (PDF) of the mass surface density is an
essential characteristic of the structure of molecular clouds or the
interstellar medium in general. Observations of the PDF of molecular clouds
indicate a composition of a broad distribution around the maximum and a
decreasing tail at high mass surface densities. The first component is
attributed to the random distribution of gas which is modeled using a
log-normal function while the second component is attributed to condensed
structures modeled using a simple power-law. The aim of this paper is to
provide an analytical model of the PDF of condensed structures which can be
used by observers to extract information about the condensations. The condensed
structures are considered to be either spheres or cylinders with a truncated
radial density profile at cloud radius r_cl. The assumed profile is of the form
rho(r)=rho_c/(1+(r/r_0)^2)^{n/2} for arbitrary power n where rho_c and r_0 are
the central density and the inner radius, respectively. An implicit function is
obtained which either truncates (sphere) or has a pole (cylinder) at maximal
mass surface density. The PDF of spherical condensations and the asymptotic PDF
of cylinders in the limit of infinite overdensity rho_c/rho(r_cl) flattens for
steeper density profiles and has a power law asymptote at low and high mass
surface densities and a well defined maximum. The power index of the asymptote
Sigma^(-gamma) of the logarithmic PDF (Sigma x P(Sigma)) in the limit of high
mass surface densities is given by gamma = (n+1)/(n-1)-1 (spheres) or by
gamma=n/(n-1)-1 (cylinders in the limit of infinite overdensity).
