The probability distribution function ( PDF ) of the mass surface density of molecular clouds provides essential information about the structure of molecular cloud gas and condensed structures out of which stars may form .
In general , the PDF shows two basic components : a broad distribution around the maximum with resemblance to a log-normal function , and a tail at high mass surface densities attributed to turbulence and self-gravity .
In a previous paper , the PDF of condensed structures has been analyzed and an analytical formula presented based on a truncated radial density profile , \rho ( r ) = \rho _ { c } / ( 1 + ( r / r _ { 0 } ) ^ { 2 } ) ^ { n / 2 } with central density \rho _ { c } and inner radius r _ { 0 } , widely used in astrophysics as a generalization of physical density profiles .
In this paper , the results are applied to analyze the PDF of self-gravitating , isothermal , pressurized , spherical ( Bonnor-Ebert spheres ) and cylindrical condensed structures with emphasis on the dependence of the PDF on the external pressure p _ { ext } and on the overpressure q ^ { -1 } = p _ { c } / p _ { ext } , where p _ { c } is the central pressure .
Apart from individual clouds , we also consider ensembles of spheres or cylinders , where effects caused by a variation of pressure ratio , a distribution of condensed cores within a turbulent gas , and ( in case of cylinders ) a distribution of inclination angles on the mean PDF are analyzed .
The probability distribution of pressure ratios q ^ { -1 } is assumed to be given by P ( q ^ { -1 } ) \propto q ^ { - k _ { 1 } } / ( 1 + ( q _ { 0 } / q ) ^ { \gamma } ) ^ { ( k _ { 1 } + k _ { 2 } ) / \gamma } , where k _ { 1 } , \gamma , k _ { 2 } , and q _ { 0 } are fixed parameters .
The PDF of individual spheres with overpressures below \sim 100 is well represented by the PDF of a sphere with an analytical density profile with n = 3 .
At higher pressure ratios , the PDF at mass surface densities \Sigma \ll \Sigma ( 0 ) , where \Sigma ( 0 ) is the central mass surface density , asymptotically approaches the PDF of a sphere with n = 2 .
Consequently , the power-law asymptote at mass surface densities above the peak steepens from P _ { sph } ( \Sigma ) \propto \Sigma ^ { -2 } to P _ { sph } ( \Sigma ) \propto \Sigma ^ { -3 } .
The corresponding asymptote of the PDF of cylinders for the large q ^ { -1 } is approximately given by P _ { cyl } ( \Sigma ) \propto \Sigma ^ { -4 / 3 } ( 1 - ( \Sigma / \Sigma ( 0 ) ) ^ { 2 / 3 } ) ^ { -1 / 2 } .
The distribution of overpressures q ^ { -1 } produces a power-law asymptote at high mass surface densities given by \left < P _ { sph } ( \Sigma ) \right > \propto \Sigma ^ { -2 k _ { 2 } -1 } ( spheres ) or \left < P _ { cyl } ( \Sigma ) \right > \propto \Sigma ^ { -2 k _ { 2 } } ( cylinders ) .