Modeling the Observed Relationship

Before we can make use of this new relationship to constrain models, we must first convert from observable quantities (surface brightnesses) to the sorts of parameters (stellar surface density, star formation rate) tracked by the models. To get from surface luminosity density to surface mass density of stars, we make use of the finding by Buchhorn (1992), that the mass-to-light ratio in the I-band (for Sc galaxies at least) has a value of . Similarly, by combining the calculations of Wilson (1983) for the production rate of UV photons as a function of stellar lifetime and mass with the standard Caseá B recombination in HIIá regions, it can be shown that a surface luminosity density (extinction corrected) of 1á á pc in H equates to a formation rate for massive (M>10á M) stars of 3á Má pcá Gyr; using the Initial Mass Function (IMF) of Kennicutt (1983) to extrapolate over all masses (0.1á M - 100á M) gives a total star formation rate of 23á Má pcá Gyr. Although there have been suggestions that the stellar IMF may vary with age, metallicity, etc., no systematic trends have been identified (Gilmore 1989), and we therefore assume a constant IMF in order to minimise the number of free parameters in the model.

Our model traces the evolution of a series of radial ``zones'' having asymptotic total mass surface densities of 8, 16, 32, , 2048á Má pc. No mass exchange between these zones is allowed. Gas is assumed to fall into these zones at an exponentially decreasing rate. Following the suggestion of Dopita (1985, 1990) that the star formation rate may be a function of both the total and the gas surface densities, we adopt a ``compound'' Schmidt-type law for star formation:

For n=0, this defaults to the ``classical'' Schmidt Law. The efficiency parameter is chosen to give solar neighbourhood conditions (, ) after 13á Gyr of evolution.

The resulting evolutionary tracks for each zone, as well as isochrones, are shown superimposed on our calibrated observational correlation in Figureá 4 for the case of (n=0, m=2). By running a series of models with varying combinations of n and m, we were able to constrain their sum to 1.5<(n+m)<2.5, but cannot rule out particular values of the indices on the basis of the star formation rate - stellar surface brightness relationship alone.

á 
Figure 4: á Evolution of our model galactic disk, overlaid on the calibrated observational relationship between star formation rate and stellar surface density (crosses), for the ``classical'' Schmidt Law with a second-order dependence on gas surface density. Evolutionary tracks are shown for each mass zone, as well as isochrones, expressed in terms of the gas depletion timescale (Dopita & Ryder 1994).

á© Copyright Astronomical Society of Australia 1997