Principal Component Analysis for Estimating Parameters of the L1287 Dense Core by Fitting Model Spectral Maps into Observed Ones

Abstract

An algorithm has been developed for finding the global minimum of a multidimensional error function by fitting model spectral maps into observed ones. Principal component analysis is applied to reduce the dimensionality of the model and the coupling degree between the parameters, and to determine the region of the minimum. The k-nearest neighbors method is used to calculate the optimal parameter values. The algorithm is used to estimate the physical parameters of the contracting dense star-forming core of L1287. Maps in the HCO⁺(1–0), H¹³CO⁺(1–0), HCN(1–0), and H¹³CN(1–0) lines, calculated within a 1D microturbulent model, are fitted into the observed ones. Estimates are obtained for the physical parameters of the core, including the radial profiles of density ($\propto {\kern 1pt} {{r}^{{ - 1.7}}}$), turbulent velocity ($\propto {\kern 1pt} {{r}^{{ - 0.4}}}$), and contraction velocity ($\propto {\kern 1pt} {{r}^{{ - 0.1}}}$). Confidence intervals are calculated for the parameter values. The power-law index of the contraction-velocity radial profile, considering the determination error, is lower in absolute terms than the expected one in the case of gas collapse onto the protostar in free fall. This result can serve as an argument in favor of a global contraction model for the L1287 core.

APPENDIX

1.1 MODEL DESCRIPTION

Excitation of rotational levels of the HCO⁺ and HCN molecules and their isotopes and the line profiles of the (1–0) transitions were calculated within a spherically symmetric microturbulent model. The model cloud is a set of concentric layers in which a certain physical parameter $P$ (density, kinetic temperature, turbulent and systematic velocity) was set constant, changing from one layer to another by the relationship $P = {{P}_{0}}{\text{/}}(1 + {{(r{\text{/}}{{R}_{0}})}^{{{{\alpha }_{p}}}}})$, where $r$ is the distance from the center and ${{R}_{0}}$ is the radius of the central layer. This functional dependence, which is a simplified form of the Plummer function, is used quite often as a model density profile (see, e.g., [22]) to avoid singularity at the center. In our model, this form of the dependence was used for all the parameters. The values of ${{P}_{0}}$ and ${{\alpha }_{p}}$ for each parameter were varied while fi-tting the model profiles into the observed ones. The  kinetic temperature profile was taken as T = $80\;{\text{K/}}(1 + {{(r{\text{/}}{{R}_{0}})}^{{0.3}}})$ and remained unchanged during calculations. It should be noted that kinetic temperature affects the intensities of the calculated HCO⁺(1–0) and HCN(1–0) lines to a lesser degree than density and concentration. Turbulent velocity was a parameter that gives an additional contribution—aside from the thermal one—to the local width of the lines. The relative molecular abundance was independent of radial distance. When calculating the excitation of HCN and H¹³CN, the hyperfine structure of the rotational spectrum and the overlapping of closely located hyperfine components [40, 71] was taken into account. The description of the model and calculation techniques for radiation transfer in the case of HCN is given in the Appendix to [40]. In our version of this model, the layer thickness increases by the power law with the distance from the center, and the radial profile of systematic velocity, which gives a Doppler shift to the local profile of the line, is taken into account. The calculations were conducted for 14 layers. The calculations used collisional probabilities of HCO⁺–H₂ [72] and HCN–H₂ taking into account hyperfine structure [73].

Excitation of rotational levels of a certain molecule was calculated by an iterative method, sequentially for one point in each layer, the radial distance of which is equal to the geometric mean of the inner and outer radii of the layer. To this end, a system of population balance equations was solved, while the populations in other layers were considered unchanged. After reaching the outer layer, the populations in each layer were compared with the values obtained in the previous iteration, and the process was repeated [40]. To increase the accuracy of calculating the radiation transfer in a moving medium, each layer was additionally divided into ten sublayers, with different systematic velocities. A test comparison of the calculated results for this model with the calculated results in [74] for a molecule with two energy levels showed that the calculated populations differ by no more than 0.4% in the case of line optical depth of $\lesssim {\kern 1pt} 60$.

The model code, written in Fortran, was controlled by means of a module written in Python. Model spectra were calculated for the different impact parameters. Using the astropy.convolve_fft procedure [75], the resulting maps were convoluted channel by channel with a two-dimensional Gaussian of width 40$''$ (the width of the main beam of the OSO-20m radio telescope at a frequency of $\sim {\kern 1pt} 90$ GHz). The model spectra were fitted into the observed ones using a PCA- and kNN-based algorithm (Section 2), written in Python.