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Astron. Astrophys. 325, 961-971 (1997)

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1. Introduction

The star formation rate (SFR, usually designated as [FORMULA]) specifies how much mass of interstellar gas is converted into stars per unit time. Which physical properties of the interstellar medium it depends on, and what its functional form is - if there is a deterministic relation at all - is still rather unclear. As young objects are found in regions of enhanced gas density, it seems most likely that the SFR increases with the local gas density. From a comparison of the scale heights of young stars and H I gas in the Milky Way, Schmidt (1959, 1963) finds a quadratic dependence on the mass density: [FORMULA]. From UV photometry Donas et al. (1987) find a linear dependence of the total SFR of a galaxy and the mass of its atomic gas.

But it is not yet clear whether atomic hydrogen is a good star formation indicator at all: The lack of correlation between total SFR and total H I mass of galactic disks is noted by Kennicutt & Kent (1983) and Balkowski et al. (1986). The inclusion of the molecular component yields an exponent of the density of [FORMULA] (Guibert et al. 1978). Rana & Wilkinson (1986) find a better correlation of the SFR with surface density of molecular hydrogen alone than with either H I or total gas (H I + H2); the exponent is [FORMULA]. Studies of the relation of indicators for formation of massive stars (H [FORMULA] emission, H II regions, blue stars) with gas surface density give a variety of values for the exponent, ranging from 1 to 4 (e.g. Madore et al. 1974, Buat 1992).

There may well be other quantities the SFR depends on: From studies of the Milky Way galaxy and M 83, Talbot (1980) proposes that the SFR in the outer regions of a galaxy is proportional to the frequency with which the gas passes through spiral arms. Wyse & Silk (1989) generalize this approach by also allowing a dependence on the gas surface density. Dopita (1985), Dopita & Ryder (1994), and Ryder & Dopita (1994) suggest a dependence on the total mass surface density.

Moreover, there are both observational and theoretical indications for a minimum density or pressure for the SFR (cf. Kennicutt 1989, van der Hulst et al. 1993, Kennicutt et al. 1994, Wang & Silk 1994, Chamcham & Hendry 1996). Kennicutt (1989) finds a SFR which depends nearly linearly on total gas density (with an exponent of [FORMULA]) as long as the gas surface density exceeds some critical density which is of the order of the critical density for gravitational stability in the disk. A pressure threshold is predicted by Elmegreen & Parravano (1994), so that star formation should cease at nearly three radial scale lengths, whereas the H I disks usually extend beyond this.

The judgment of which law of star formation is the best one depends on how good an agreement can be achieved between observational data and theoretical predictions. Of course, one might get a perfect fit for any data, if only the model had a sufficiently large number of free parameters. Thus, extending a theory by including additional dependencies - which lead to more free parameters - would in general improve the fit, but makes the theory more complex. As the accuracy of the data is finite, one has to make a decision of what level of accuracy of the fit or what level of complexity of the theory is still reasonable. Usually, this decision is made more or less subjectively, based on one's feeling and experience. The metaphor of Occam's razor - of leaving out any unnecessary complexity in a model or theory - is well known as a philosophical concept, but can one devise a practical method that implements this concept when comparing several forms of the law of star formation with the observational data?

In this paper the Bayesian approach to statistics is shown to permit the construction of such a practical method. This method is described, tested with artificial data, and applied to the question of whether the presently available observational data permits a judgement on the form of the SFR. In particular we wish to decide whether the SFR is a power law of the local gas density alone or whether an additional dependence on galactocentric radius exists - or rather to what extend the data permit to make such a decision.

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© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998

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