Improving the full spectrum fitting method: accurate convolution with Gauss-Hermite functions

TL;DR

This paper introduces an improved convolution method for the pPXF spectral fitting tool, enabling accurate velocity measurements even when the velocity dispersion is below the sampling resolution, by analytically computing Fourier transforms of Gauss-Hermite kernels.

Contribution

The paper presents a novel, more accurate convolution technique using analytic Fourier transforms of Gauss-Hermite functions, enhancing pPXF's ability to measure low velocity dispersions.

Findings

Accurate velocity extraction for <<, even when </2.

Implementation of the method in pPXF improves kinematic measurements.

Potential to improve Gaussian convolution algorithms in other software.

Abstract

I start by providing an updated summary of the penalized pixel-fitting (pPXF) method, which is used to extract the stellar and gas kinematics, as well as the stellar population of galaxies, via full spectrum fitting. I then focus on the problem of extracting the kinematic when the velocity dispersion $\sigma$ is smaller than the velocity sampling $\Delta V$, which is generally, by design, close to the instrumental dispersion $\sigma_{\rm inst}$. The standard approach consists of convolving templates with a discretized kernel, while fitting for its parameters. This is obviously very inaccurate when $\sigma<\Delta V/2$, due to undersampling. Oversampling can prevent this, but it has drawbacks. Here I present a more accurate and efficient alternative. It avoids the evaluation of the under-sampled kernel, and instead directly computes its well-sampled analytic Fourier transform, for use with the convolution theorem. A simple analytic transform exists when the kernel is described by the popular Gauss-Hermite parametrization (which includes the Gaussian as special case) for the line-of-sight velocity distribution. I describe how this idea was implemented in a significant upgrade to the publicly available pPXF software. The key advantage of the new approach is that it provides accurate velocities regardless of $\sigma$. This is important e.g. for spectroscopic surveys targeting galaxies with $\sigma\ll\sigma_{\rm inst}$, for galaxy redshift determinations, or for measuring line-of-sight velocities of individual stars. The proposed method could also be used to fix Gaussian convolution algorithms used in today's popular software packages.