Physics of Star Formation Revisited

Gravitational Stability and the Molecular Cooling Myth

a) Introduction - Energy Equilibrium Considerations

There can be little doubt about the theory that stars are formed by means of gravitational contraction of gas clouds in space. In fact, it is practically a logical consequence of the gravitational force once certain conditions are met. The first condition is that the atoms in the gas cloud must be gravitationally bound to each other, i.e. the total energy

(1)       E = Ekin+Epot <0    .

In the absence of turbulent or rotational movements, the average kinetic energy per atom is given by the thermal energy at temperature T

(2)       Ekin = kT       (k=Boltzmann constant),

whereas the potential energy of the atom of mass m within the cloud with mass M and radius R is given by

(3)       Epot = -GmM/R      (G= gravitational constant).

If one considers the mass M as given, the only variables that determine Ekin and Epot are the temperature T and the radius R of the cloud. These can in principle take on any value that satisfies Eq.(1). However, one can show (virial theorem) that for a system of particles bound by an inverse square law force (like gravitation) the kinetic and potential energy are on the average (or in the state of equilibrium) related by

(4)       Ekin = - Epot/2   ,

i.e. the total energy

(5)       E = Epot/2   .

However, for a closed system, the total energy has to be constant (Eq.(1)), and it follows consequently from Eqs.(3) and (5) that the radius R has to be constant as well (at least on average).
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)       RE = - GmM/(2E)   .

So it is clear from this that the usually given 'Jeans criterion' for gravitational collapse (which relates the mass, radius and temperature of the gas cloud (and is essentially given by Eqs.(2)-(4))) is not a sufficent condition for star formation, because the collapse would soon come to a halt as the temperature and density increases. The second condition for star formation is therefore that the cloud must permanently lose overall energy (i.e. E must become negatively larger for RE to become smaller). This is discussed in more detail in section c) below (which by the way also suggests that the Jeans criterion is actually not a necessary condition for star formation either, as practically any cloud should lose energy at a rate fast enough to lead to a collapse).

b) Radial Density Distribution in Equilibrium

In most theoretical treatments, the stability problem for interstellar gas clouds is only considered from the energy point of view as addressed above. The density distribution could indeed have any form if one assumes a corresponding spatial variation of the kinetic energy (temperature). However, for realistic gases, collisions between the atoms will tend to level out any temperature differences. A gas cloud in a stable equilibrium will therefore necessarily be isothermal (in the absence of any localized heat sources or sinks). The radial density distribution for a spherically symmetric cloud can be found by equating the centrifugal force with the gravitational force acting on the atom, i.e.

(7)       mv2/r   =   -GmM(r)/r2 = -Gm2/r2 .. 0rdr' .r'2n(r')   ,

where n(r') is the radial density distribution. Since the condition of an isothermal gas implies that v is independent of r, the left hand side of the equation is proportional to 1/r. This means that the integral on the right hand side must be proportional to r, which requires

(8)       n(r') ~ 1/r'2   .

Although this result has been derived from a purely dynamical consideration, it satisfies also the equation for hydrostatic equilibrium (which should be used for collisional gases)

(9)      1/n(r).dP(r)/dr = kT/n(r).dn(r)/dr  =  -GmM(r)/r2 = -Gm2/r2.. 0rdr' .r'2n(r')   ,

In equilibrium, the density of any isothermal spherical gas cloud will consequently decrease like 1/r2. Any other density distribution can not be stable. If the density decreases slower than 1/r2, the cloud will contract until the equilibrium distribution is reached, if it decreases faster than 1/r2, it will expand accordingly.
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/r2 produces a gravitational force ~1/r within the cloud, the gravitational potential varies only logarithmically (i.e. ~ln(r)) . Even if during the contraction the non-uniform potential gradient leads to a differential heating of the cloud (which one can assume because of the relatively long time scale for heat transfer), the resulting temperature gradient will therefore be small enough to justify the assumption that the cloud will stay isothermal throughout the collapse. This has to be seen in contrast to the usually assumed homologous collapse of a cloud with uniform density which would have a gravitational force ~ r (i.e. a potential ~r2) and could therefore hardly maintain its structure (which of course could not be stable in the first place as the pressure force due to the temperature gradient would be directed inwards).
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.

c) Energy Loss Processes

As shown above, the contraction of a gravitationally unstable gas cloud would soon stop and could therefore not lead to star formation unless the gas cloud continuously loses energy. For a cloud isolated in space this can only be caused by atoms losing their kinetic energy through inelastic collisions with other atoms, with the excitation energy then being radiated away. The problem is that temperature of interstellar gas clouds is much too low (of the order of 10 oK (=10-3 eV)) to excite atoms from the ground state. Present theoretical models usually assume therefore a presence of molecules, but even for these the excitation probability (for the corresponding vibrational and rotational states) is very small, whereas in the latter stages, when R is small, the energy is so high (see Eqs.(2)-(4) ), that molecules can not exist any more.
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 10-10 cm-3 initially), the excitation cross section is very high (up to 10-2 cm2 ) and yields therefore a relatively high excitation probability (the latter increases even drastically as the cloud becomes smaller because the increasing plasma density increases the level population of the excited atoms and also the excitation cross section due to the increased level broadening by plasma field fluctuations) (see the abstracts regarding the Scattering of Radio Waves by High Atomic Rydberg States and Airglow Excitation on the Research Home Page for more related details).

For a gas cloud with a density of 100 cm-3, a temperature of 10 oK 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 104 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/r2 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 r3.5 during contraction (the timescale for the excitation of Rydberg states behaves in a similar manner), whereas the dynamical timescale decreases only ~r1.5 (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 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.

In view of these aspects, it seems rather unlikely that molecules have any siginificant influence on the star formation process at all. Molecular clouds could in fact be the result rather than the cause of star formation, because the suggested cooling by excitation of highly excited Rydberg states is likely to be much less effective for molecules (in particular heavier and/or more complex ones) than for atoms, i.e. only for the latter does the cooling and hence the gravitational collapse occur, whereas the molecules are largely left behind.

Of course, once the density has become so high that no individual atoms can exist anymore (which is at a density of the order of n=1023cm-3) the collapse will stop, as no collisional excitation can take place anymore. The only energy loss will then be from the less dense atmosphere, which is too small to lead to a noticeabel contraction, but which explains the low temperature of the solar photosphere in comparison to the temperature to be expected from the gravitational energy of the sun (see the page regarding Coronal Heating).

d) Rotating Gas Clouds

As indicated in section a), the rotation of a gas cloud tends to counteract any contraction because of the additional kinetic (centrifugal) energy involved. However, as a certain amount of the gas will be ionized, the rotation would inevitably produce a magnetic field (dynamo effect), which, under certain circumstances, could act as a 'brake' for the rotation of the inner region of the cloud, the outer region taking up the corresponding energy and angular momentum (I have discussed this mechanism elsewhere in connection with the problem of galactic rotation and the related Dark Matter assumption).

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