A Linearized Approach to Radial-Velocity Extraction. II: Shot-Noise-Limited Precision via Spectral Factorization

TL;DR

We address the challenge of extracting precise radial velocities when spectral variability obscures Doppler shifts. We generalize the STFT-based RV extraction by factorizing a spectroscopic time series into principal spectra and time-dependent kernels, recovering both jointly via per-$\zeta$ SVD in Fourier space, with Doppler shifts encoded as phase differences across epochs. Validation on synthetic data, SOAP simulations, and EXPRES observations of HD 34411 and $\tau$ Ceti shows the method reaches shot-noise–limited precision around $30$ cm s$^{-1}$ and can detect signals with semi-amplitudes down to $\sim 50$ cm s$^{-1}$ with $\lesssim 10$ cm s$^{-1}$ uncertainty. The data-driven approach separates genuine Doppler shifts from spectral variability, offering a practical route toward Earth-like planet characterization around solar-type stars, while highlighting sampling requirements, activity leakage, and cross-talk that motivate future refinements.

Abstract

We generalize the short-time Fourier transform (STFT) formalism for radial velocity extraction to cases where the underlying spectral components are unknown. The method factorizes a spectroscopic time series into principal spectra and time-dependent kernels, enabling simultaneous recovery of both. In Fourier space, each inverse-wavelength slice is decomposed by singular value decomposition, and radial velocity shifts are inferred from phase differences between epochs. In the high-SNR regime, this provides a linearized and statistically tractable estimate of differential velocities. The method is validated on synthetic and SOAP simulations and applied to EXPRES observations of HD 34411 and $τ$ Ceti, recovering coherent signals and reaching the instrumental precision limit of ~30 cm/s. Apart from p-mode modulation, the residuals show no significant long-term correlations and allow the detection of signals with semi-amplitudes down to ~50 cm/s with $\lesssim10$ cm/s uncertainty. The framework thus enables extreme-precision radial velocity measurements in the presence of spectral variability, representing a step toward detecting and characterizing Earth-like planets around solar-type stars.

Figures (11)

• Figure 1: Top panel---a selected segment from one of the simulated spectra used in Section \ref{['sec: phase ratios']}, centered around the MgB lines ($T_{\rm eff}=5{,}800 \, {\rm K}$; $\log g = 4.5$; ${\rm[Fe/H]}=0$; $v \sin i = 2\,\,{\rm km\, s}^{-1}$; and $\mathcal{R}=10^5$). Bottom left panel---the zeroth-order estimated of the principal kernels obtained from $\mathbf{U}_\zeta$, for the first seven spectra. The kernels are normalized such that their peak value is one and sorted one on top of the other. The arrows represent the Doppler shift used for each spectrum. Bottom right panel---same as the bottom-left panel, but using the reweighted matrices $\mathbf{U}_\zeta \Sigma_\zeta$.
• Figure 2: Top panel---the normalized information in the first three principal spectra (corresponding to $\Sigma_{00}$, $\Sigma_{11}$ and $\Sigma_{22}$) versus their corresponding inverse-wavelength, $\zeta$, for the simulated white-noise spectroscopic time-series. All values are normalized to $\Sigma_{00}$ at $\zeta=0$. Bottom panel---relative phases between the first and fourth simulated spectra, shown as a solid black line. The red stripes correspond to the expected relative phase due to the velocity difference between the two observations, i.e., $2\pi\,\zeta\,\times\,(v_4-v_1)$. Notably, the relative phase follows the expected trend for inverse wavelength below $\sim 0.2$ s km-1, where the information content falls below the noise level.
• Figure 3: Top panel---Velocities extracted for the white noise simulation. The simulated spectroscopic dataset is based on a synthetic spectrum of a Sun-like star, and the noise is assumed to be white and Gaussian, with an SNR of 30. The injected radial velocity signal is a solid red curve in the background. Bottom panel---The residuals between the injected and inferred velocities. The scatter of the residuals is 58 cm s-1, consistent with the estimated error.
• Figure 4: The normalized zeroth- and first-order principal values versus inverse wavelength, for the SOAP simulation presented in Section \ref{['sec: SOAP']}. The normalization constant given by an estimate of $\Sigma_\infty$, estimated as the median singular value for $\zeta > 0.3$ s km-1. The white dots show the ratio between the zeroth and first order terms, i.e., $\Sigma_{00}/\Sigma_{11}$.
• Figure 5: Top panels---The rectified velocities, $v_{\rm c}$, extracted for the SOAP simulation described in Section \ref{['sec: SOAP']}. The simulated spectroscopic dataset is based on a synthetic spectrum of a Sun-like star, and the noise is assumed to be white and Gaussian. The injected radial velocity signal is a solid red curve. Middle panels---the extracted velocities from the zeroth-order kernel, $U^{(0)}$. These velocities show a combination of the injected Doppler shift and parasitic signal induced by stellar activity. Bottom panels---the extracted velocities from the first-order kernel, $U^{(1)}$, dominated by stellar activity. The solid red curve represents the fitted 5-harmonic model fitted to the extracted velocities (see text).