We describe an accurate, one-dimensional, spherically symmetric Lagrangian hydrodynamics/gravity code, designed to study the effects of radiative cooling and photoionization on the formation of protogalaxies. The code can treat an arbitrary number of fluid shells (representing baryons) and collisionless shells (representing cold dark matter). As a test of the code, we reproduce analytic solutions for the pulsation behavior of a poly-trope and for the self-similar collapse of a spherically symmetric, cosmological perturbation. In this paper we concentrate on the effects of radiative cooling, examining the ability of collapsing perturbations to cool within the age of the universe. In contrast to some studies based on order-of-magnitude estimates, we find that cooling arguments alone cannot explain the sharp upper cutoff observed in the galaxy luminosity function.
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