V.6 No 1

23

Some improvements to the definition of entropy of macrosystem

To calculate, let us take some spherical volume of gas – e.g., molecular hydrogen, satisfying the Shklovsky condition (8). Premise the mass of this gas M_cloud = 10 M = 1,99110³⁴ g  and its initial temperature T = 4^o K. Then the radius of this sphere is

Premising that before compression the gas was evenly distributed in the volume, we can determine the initial gas pressure; we will need this value in calculation below. Noting the gas in interstellar clouds is very rarefied, we may think the gas ideal and determine its pressure with the laws of gas dynamics

where R = 8,31696 j/ dgmole is the gas constant and m_m = 21,0079 g/ mole is the molar mass of molecular hydrogen. Substituting the known values to (18), yield p₀ = 2,1510^-14 N/ m² = 2,8610^-12  mm of mercury. The yielded value of initial pressure corroborates the validity of initial premise to think the gas ideal.

  Now suppose that the gas in the volume is not affected externally, it is quasi-statically compressed to the centre exceptionally under affection of its own gravity field. As the gas is isolated from external affection (as stipulated) and during the compression no one parameter of the gas continuum is constant, we can premise the adiabatic pattern of process. With it, at least at the stage of primary compression, the condition

will be true, where is the Poisson constant. For a two-atomic gas like molecular hydrogen = 1,4.

It will be desirable for further calculation to transform (19) to the relationship between the pressure and density of substance. Select some infinitesimal volume v of compressed gas. Noting (19), the equality [19, p. 21] will be true for it:

where p₀ is the initial pressure in the selected volume v₀ . With it, as the mass of gas in the volume did not change, the equality

is true, where ₀ is the initial gas density in the volume v₀ and m is the mass if this gas volume. Joining (20) and (21), yield

which is the sought relation.

In finding this relation, we premised that all selected small volumes of the cloud are transformed during the substance redistribution. And we have to account, such transformations have to lead to the related shifts of these volumes along the radius. To account this factor, take some small selected volume with the height r_0i located at the distance r_0i from the centre to which the cloud’s substance is shrunk, as in Fig. 4, and consider its shift to the radius  r_1i with the transformation taking place at this time, to the height r_1i , noting the central symmetry of the problem.

  Fig. 4. The calculation scheme of the selected volume in the cloud substance redistribution

As the problem is centrally symmetrical and the selected volume is a part of some thin shell with the height r_0i located at the distance r_0i from centre, we may broaden the analysis for this shell. We can write for it: if its height was initially r_0i and after the substance redistribution r_1i , and the shell has shifted to r_1i , then in accordance with the adiabatic pattern of process

where p_0i and p_1i are the pressures in the volume before and after the cloud substance redistribution. Naturally, all elementary volumes of which the shell consists will transform after the same law; so we may further use (23) to study the elementary volume.

  Then notice, r_1i in (23) accounts that simultaneously with the selected volume transformation, there occurs also the transformation of all volumes along the cloud radius, as in absence of this accompanying transformation, the selected volume did not shift along the radius but only changed its height. But if after the redistribution the height of volumes located lower than that ith became r_1j , we can write

where

and is taken positive in the compression of selected volume.