A Hermite–Gaussian based exoplanet radial velocity estimation method

Abstract

As the first successful technique used to detect exoplanets orbiting distant stars, the radial velocity method aims to detect a periodic Doppler shift in a stellar spectrum due to the star’s motion along the line sight. We introduce a new, mathematically rigorous approach to detect such a signal that accounts for the smooth functional relationship of neighboring wavelengths in the spectrum, minimizes the role of wavelength interpolation, accounts for heteroskedastic noise and easily allows for accurate calculation of the estimated radial velocity standard error. Using Hermite–Gaussian functions, we show that the problem of detecting a Doppler shift in the spectrum can be reduced to linear regression in many settings. A simulation study demonstrates that the proposed method is able to accurately estimate an individual spectrum’s radial velocity with precision below $0.3{\text{m s}}^{-1}$, corresponding to a Doppler shift much smaller than the size of a spectral pixel. Furthermore, the new method outperforms the traditional cross-correlation function approach for estimating the radial velocity by reducing the root mean squared error up to $15\text{cm}{\text{s}}^{-1}$. The proposed method is also demonstrated on a new set of observations from the EXtreme PREcision Spectrometer (EXPRES) for the host star 51 Pegasi, and successfully recovers estimates of the planetary companion’s parameters that agree well with previous studies. The method is implemented in the R package rvmethod, and supplemental Python code is also available.