Improving CMD Areal Density Analysis: Algorithms and Strategies

Essential ideas, successes, and difficulties of Areal Density Analysis (ADA) for color-magnitude diagrams (CMD's) of resolved stellar populations are examined, with explanation of various algorithms and strategies for optimal performance. A CMD-generation program computes theoretical datasets with simulated observational error and a solution program inverts the problem by the method of Differential Corrections (DC) so as to compute parameter values from observed magnitudes and colors, with standard error estimates and correlation coefficients. ADA promises not only impersonal results, but also significant saving of labor, especially where a given dataset is analyzed with several evolution models. Observational errors and multiple star systems, along with various single star characteristics and phenomena, are modeled directly via the Functional Statistics Algorithm (FSA). Unlike Monte Carlo, FSA is not dependent on a random number generator. Discussions include difficulties and overall requirements, such as need for fast evolutionary computation and realization of goals within machine memory limits. Degradation of results due to influence of pixelization on derivatives, Initial Mass Function (IMF) quantization, IMF steepness, low Areal Densities ($\mathcal{A}$), and large variation in $\mathcal{A}$ are reduced or eliminated through a variety of schemes that are explained sufficiently for general application. The Levenberg-Marquardt and MMS algorithms for improvement of solution convergence are contained within the DC program. An example of convergence, which typically is very good, is shown in tabular form. A number of theoretical and practical solution issues are discussed, as are prospects for further development.