The Evolution of Galaxies. I. Formulation and Mathematical Behavior of the One-Zone Model

Abstract

A procedure is discussed for calculating the evolution of a closed system of gas and stars. Numerical and analytical solutions are given for a simple set of prescriptions for the stellar birth rate and the evolutionary end state of stars. The various features of the numerical results are easily understood through the analytic solutions. More significantly, the analytic solutions clearly display how the results will change when the astrophysicai input is varied. A fundamental result, which previous investigators have not stressed, is that the metal content Z of the gas in a galaxy need not be a monotonically increasing function of time even if the system is homogeneous in space. We show that under conditions which lead to only a small fraction of mass in the form of gas, there generally may exist an early period with Z much larger than the present value. We propose that the very old super-metal-rich stars under investigation by Spinrad and others could have been formed during such a period. I. INTRODUCTION A theoretical description of the evolution of a galaxy requires the specification of initial conditions for the various constituents, prescriptions for star formation and the ejecta from stars, and a scheme for following the properties of the system in time and space. Several attempts toward solution of this problem have been made (). All of these investigations have considered the case of an initial mixture of gas (or gas plus stars) which remains homogeneous in space throughout the calculation. This simplification, which we shall call the one-zone model, allows formulation of the problem with time as a single independent variable. Some of these investigations have attempted a physical description of the stellar birth rate through a parametrization in terms of physical variables (such as the gas density). Others have assumed simple analytic prescriptions which possess the qualitative features of the investigator's physical picture of the process. The accuracy of the nucleosynthesis aspects of the early papers was greatly hampered by wide gaps in the theory of stellar evolution. In this paper we will carefully reconsider the formulation of the one-zone model, and examine its mathematical behavior by numerical and by analytical means. In particular we will demonstrate the existence of an effect which has not been emphasized by previous investigators. In some cases we find that the metal abundance in the interstellar gas does not increase monotonically with time. This paper will deal with the mathematical aspects of this effect for relatively simple astrophysicai assumptions; the intent is to provide simple analytical solutions with which to understand the mathematical behavior of the one-zone model. These analytical solutions will aid in the interpretation of the numerical results for the more realistic astrophysicai assumptions that we will employ in future work. * Alfred P. Sloan Foundation Fellow. t Present address: University of Texas, Austin. 409