Using the Modified Lognormal Power-law Distribution to Model the Mass Function of NGC 1711
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A determination of the mass function (MF) of stellar clusters can be quite dependent on the range of measured masses, the fitting technique, and the analytic function that is being fit to the data. Here, we use Hubble Space Telescope/WFPC2 data of NGC 1711, a stellar cluster in the Large Magellanic Cloud, as a test case to explore a range of possible determinations of the MF from a single data set. We employ the analytic modified lognormal power-law (MLP) distribution, a hybrid function that has a peaked lognormal-like body and a power-law tail at intermediate and high masses. A fit with the MLP has the advantage that the resulting best-fit function can be either a hybrid function, a pure lognormal, or a pure power law, in different limits of the function. The completeness limit for the observations means that the data contains masses above ∼0.90 M o˙. In this case, the MLP fits yield essentially a pure power-law MF. We demonstrate that the nonlinear regression/least-squares approach is not justified since the underlying assumptions are not satisfied. By using maximum-likelihood estimation, which is independent of binning, we find a best-fit functional form, where α = 1.72 0.05 or 1.75 0.05 for two different theoretical isochrone models, respectively. Furthermore, we explore the possibility of systematic errors in the determination of the power-law index due to the depth of the observations. When we combine the observational data with artificially generated data from the lognormal Chabrier initial MF for masses below 0.90M o˙, the best-fit MLP is a hybrid function but with a steeper asymptotic slope i.e., α = 2.04 0.07. This illustrates the systematic uncertainties in commonly used MF parameters that can depend on the range of data that is fitted.