New Probability Distributions in Astrophysics: III. The Truncated Maxwell-Boltzmann Distribution

Abstract

The Maxwell-Boltzmann (MB) distribution for velocities in ideal gases is usually defined between zero and infinity. A double truncated MB distribution is here introduced and the probability density function, the distribution function, the average value, the rth moment about the origin, the root-mean-square speed and the variance are evaluated. Two applications are presented: 1) a numerical relationship between root-mean-square speed and temperature, and 2) a modification of the formula for the Jeans escape flux of molecules from an atmosphere.

Transport in Astrophysics: VI. Ultra-High Energy Cosmic Rays

“…The spherical nodes are marked by large green points. number of nuclei, the side of the box in Mpc, and the number of pixels, for example 1400, that are used to build the diagrams, see [31]. The parameters adopted in the simulation of the pseudo-Euclidean cosmology are given in Table 1.…”

New Probability Distributions in Astrophysics: IX. Truncation for Exponential, Half Gaussian and Sech-Square Distributions with Application to the Galactic Height

“…Another two distributions that have astrophysical interest are the sech-square distribution, corresponding to an isothermal self-gravitating disk, see equation (41) in [1], equation at page 441 in [2], equations (9.8) and (14.6) in [3], equation (2.31) in [4], and the normal or Gaussian distribution, see equation (5) in [5]. In the field of probability, the truncation of a distribution is a common topic of research and we report some of the approaches on the double applications to a numerical relation between the root-mean-square speed and temperature and to a modification of the formula for the Jeans escape flux of molecules from an atmosphere [14], the relativistic Maxwell-Boltzmann distribution, with applications to the synchrotron emission in the presence of a magnetic field and to relativistic electrons [15], the truncated Weibull distribution, with applications to the masses of the stars and to the luminosity functions for galaxies and quasars [16], the truncated two-parameter Sujatha distribution [17], the gamma-Pareto distribution, with application to cosmic rays [18], and the truncated Weibull-Pareto distribution, with applications to the initial mass function for stars, the luminosity function for galaxies of the Sloan Digital Sky Survey, the luminosity function for QSO, and the photometric maximum of galaxies of the 2MASS Redshift Survey. This paper analyses in Section 2 the exponential, the half-normal and the sech-square distributions defined in the interval [ ] 0, ∞ .…”