(1) E = E_{kin}+E_{pot} <0 .

(2) E_{kin} = kT (k=Boltzmann constant),

(3) E_{pot} = -GmM/R (G= gravitational constant).

(4) E_{kin} = - E_{pot}/2 ,

(5) E = E_{pot}/2 .

From Eqs.(1), (3) and (5), one can see that the cloud can therefore at best shrink to half its critical size (if it has initially a temperature T=0), i.e.

(6) R_{E} = - GmM/(2E) .

(7) mv^{2}/r = -GmM(r)/r^{2} = -Gm^{2}/r^{2} ^{.}4π^{.} _{0}∫^{r}dr' ^{.}r'^{2}n(r') ,

(8) n(r') ~ 1/r'^{2} .

(9) 1/n(r)^{.}dP(r)/dr = kT/n(r)^{.}dn(r)/dr = -GmM(r)/r^{2} = -Gm^{2}/r^{2}^{.}4π^{.} _{0}∫^{r}dr' ^{.}r'^{2}n(r') ,

Any further contraction, which is obviously required for star formation , is only possible if the cloud loses overall energy due to collisional excitation (this is further discussed under point c) below).

Note: since a density distribution 1/r

In view of the very fast cooling rate due the proposed excitation between highly excited atomic states (see below), these equilibrium considerations may however anyway be rather academic apart from on a local scale.

The only energy loss can therefore be provided by collisional excitation between highly excited atomic states (Rydberg states). Although the volume density of the latter is very small (about 10

For a gas cloud with a density of 100 cm^{-3}, a temperature of 10 ^{o}K and an ionization degree of 10^{-3}, the time scale for the energy loss through excitation of high atomic Rydberg states can be estimated to be of the order of 10^{4} years. As this is considerably less than the dynamical time scale for the gas cloud, the latter would practically collapse in free fall. However, as the time scale for elastic collisions is even shorter (about 100 years), energy equilibrium (including the 1/r^{2} density distribution) is re-established locally. Half of the energy gained during the collapse is therefore turned into thermal energy (Eq.(5)) and the acceleration consequently also halved. As the collisional timescale is inversely proportional to n(r)^{.}v(r), it decreases like r^{3.5} during contraction (the timescale for the excitation of Rydberg states behaves in a similar manner), whereas the dynamical timescale decreases only ~r^{1.5} (Kepler's law). One can therefore assume that the cloud contracts in sub-sonic free-fall all the way down to solar system dimensions.

In contrast, the assumption of energy loss through excitation of molecules would lead to a cooling time scale about 1 order of magnitude longer in the initial phase and even longer as the cloud contracts (see http://cfa-www.harvard.edu/swas/science1.html). Eventually, molecular cooling would completely cease once the temperature has reached a value which would prohibit the formation of molecules (this should certainly be the case if the energy reaches a few eV, i.e. if R has decreased by a factor 10^{-3}-10^{-4} (Eq.(3)).

Furthermore, because of the very rapid cooling associated with the excitation of highly excited atomic states, it is in fact the case that practically any gas cloud will become gravitationally unstable as even systems that have positive energy and do not satisfy Eq.(1) will lose their energy excess before they have time to expand and escape their own gravitational field.

Thomas Smid (M.Sc. Physics, Ph.D. Astronomy)

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