Molecular Line Profile Fitting with Analytic Radiative Transfer Models

We present a study of analytic models of starless cores whose line profiles have ``infall asymmetry,'' or blue-skewed shapes indicative of contracting motions. We compare the ability of two types of analytical radiative transfer models to reproduce the line profiles and infall speeds of centrally condensed starless cores whose infall speeds are spatially constant and range between 0 and 0.2 km s<sup>-1</sup>. The model line profiles of HCO<sup>+</sup> (J=1-->0) and HCO<sup>+</sup> (J=3-->2) are produced by a self-consistent Monte Carlo radiative transfer code. The analytic models assume that the excitation temperature in the front of the cloud is either constant (``two-layer'' model) or increases inward as a linear function of optical depth (``hill'' model). Each analytic model is matched to the line profile by rapid least-squares fitting. The blue-asymmetric line profiles with two peaks, or with a blueshifted peak and a red shifted shoulder, can be well fitted by one or both of the analytic models. Two-peak profiles are best matched by the ``HILL5'' model (a five parameter version of the hill model), with an rms error of 0.01 km s^-1, while the ``TWOLAYER6'' model underestimates the infall speed by a factor of ~2. For red-shoulder profiles, the HILL5 and TWOLAYER6 fits reproduce infall speeds equally well, with an rms error of 0.04 km s^-1. The fits are most accurate when the line has a brightness temperature greater than 3 K. Our most accurate models tend to not only reproduce the line profile shape but also match the excitation conditions along the line of sight. A better match to the line profile shape does not necessarily imply a better match to the infall speed. We provide guidance on how to minimize the risk of obtaining a poor infall speed fit. A peak signal-to-noise ratio of at least 30 in the molecular line observations is required for performing these analytic radiative transfer fits to the line profiles. Moderate amounts of depletion and beam smoothing do not adversely affect the accuracy of the infall speeds obtained from these models.