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Astron. Astrophys. 328, 121-129 (1997)

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2. Conditions for fragmentation and molecularization

The analysis of linear perturbations of transverse motions on a three-dimensional shell expanding into a uniform medium has been performed by Elmegreen (1994) and Vishniac (1994). Taking into account the convergence of the perturbed flow, stretching of the perturbed region due to expansion, and its own gravity, the instantaneous maximum growth rate [FORMULA] of a transverse perturbation of a shell is given as

[EQUATION]

where R is the radius of the shell with mass column density [FORMULA]. v is its velocity of expansion relative to the ambient medium, c is the sound speed within the shell, and G is the constant of gravity. The instability occurs for

[EQUATION]

This is equivalent to

[EQUATION]

where [FORMULA] is a dimensionless parameter. This maximum growth rate corresponds to the transversal angular wavenumber

[EQUATION]

Another condition of the instability is that the wavelength of the fastest transversal perturbation [FORMULA] is within a fraction of the shell, or smaller than R, which may be written as

[EQUATION]

which is equivalent to

[EQUATION]

This may be rewritten as

[EQUATION]

where [FORMULA]. This is only a restriction to the [FORMULA] parameter if M is near unity. For [FORMULA], which is satisfied in almost all the cases examined, the minimum wavelength of a growing transverse perturbation is always within a fraction of the shell radius.

The fragmentation begins when both criteria are fulfilled for the first time, [FORMULA]. We follow the expansion even beyond this time and evaluate the fragmentation integral [FORMULA]:

[EQUATION]

At the time [FORMULA] when [FORMULA] the fragments are well developed, so that clouds form at a later time.

In the following sections, the models of an expanding shell are described, and the above instability criteria are tested. As soon as [FORMULA], the approximation of an expanding shell probably breaks down, and the fragments continue along individual galactic orbits.

If the particle column density N in a fragment surpasses the critical value (Franco & Cox, 1986)

[EQUATION]

(where Z is the metallicity and [FORMULA] is the solar value) the gas is shielded against the ionizing interstellar radiation, can recombine and later become molecular. If this condition is fulfilled during [FORMULA], we conclude that the expansion of the shell triggered the formation of a new molecular cloud.

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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998

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