The Interpolation Constraint in the RV Analysis of M-Dwarfs Using Empirical Templates

TL;DR

This work shows that template-based radial velocity measurements for M-dwarfs in the near-infrared suffer a fundamental interpolation-noise floor arising from how empirical templates are constructed at finite detector sampling. Using a SPIRou-like numerical simulation with PHOENIX spectra, the authors build templates from upsampled, BERV-aligned observations and compare template-based RVs to those from the intrinsic spectrum, revealing that interpolation distortions bias RVs, especially for cooler M-dwarfs with dense line spectra. The study quantifies the RV floor (roughly $0.5$–$0.8$ m/s for SPIRou) and shows that higher sampling beyond Nyquist can dramatically reduce this floor (to ~0.07 m/s in idealized high-sampling cases), highlighting the need for better spectral modeling and detector sampling to reach sub-$1$ m/s precision. The authors also provide a practical expression for the interpolation noise $\sigma_{\text{interp}}$ as a function of the spectrum quality factor $Q$ and $S/N$, offering a path to incorporate this floor into RV uncertainty budgets and guiding instrument design and data-analysis strategies.

Abstract

Precise radial velocity (pRV) measurements of M-dwarfs in the near-infrared (NIR) rely on empirical templates due to the lack of accurate stellar spectral models in this regime. Templates are assumed to approximate the true spectrum when constructed from many observations or in the high signal-to-noise limit. We develop a numerical simulation that generates SPIRou-like pRV observations from PHOENIX spectra, constructs empirical templates, and estimates radial velocities. This simulation solely considers photon noise and evaluates when empirical templates remain reliable for pRV analysis. Our results reveal a previously unrecognized noise source in templates, establishing a fundamental floor for template-based pRV measurements. We find that templates inherently include distortions in stellar line shapes due to imperfect interpolation at the detector's sampling resolution. The magnitude of this interpolation error depends on sampling resolution and RV content. Consequently, while stars with a higher RV content, such as cooler M-dwarfs are expected to yield lower RV uncertainties, their dense spectral features can amplify interpolation errors, potentially biasing RV estimates. For a typical M4V star, SPIRou's spectral and sampling resolution imposes an RV uncertainty floor of 0.5-0.8 m/s, independent of the star's magnitude or the telescope's aperture. These findings reveal a limitation of template-based pRV methods, underscoring the need for improved spectral modeling and better-than-Nyquist detector sampling to reach the next level of RV precision.

Figures (6)

• Figure 1: This flowchart describes the steps taken in the numerical simulation to create synthetic RV observations and the template that will then be used for RV estimation.
• Figure 2: Top: An intrinsic stellar spectrum which is Doppler-shifted and then convolved. Middle: The convolved spectrum is down-sampled to SPIRou’s resolution, but no photon noise is added, thus representing a noiseless observation. Here, each black point represents a SPIRou pixel. This synthetic observation is then upsampled using spline interpolation as a precursor to constructing the template. Bottom: Discrepancies between the upsampled and convolved spectrum, indicating that the template will contain altered stellar features.
• Figure 3: Top: Each color represents a separate noiseless synthetic SPIRou observation along with its corresponding upsampled (spline interpolated) spectrum. For clarity, only three observations are shown. Middle: The upsampled spectra are BERV-registered to reduce the line shift to a fraction of their width. The template shown as the dashed black line is the median of 10 BERV-registered spectra. Bottom: Discrepancies between the template and instrumentally broadened intrinsic spectrum, demonstrating that the template contains altered stellar features.
• Figure 4: The standard deviation of the Z-scores between the observation and the template is shown for different cases. A standard deviation greater than one indicates interpolation errors. Top: The significance of interpolation error depends on both sampling resolution and photon noise, with each line representing a different sampling resolution. Bottom: The standard deviation of the Z-scores for different M-dwarf types at SPIRou’s native resolution shows that interpolation errors increase for cooler M-dwarfs with higher RV content.
• Figure 5: The root mean squared error of the estimated RVs for $100$ evaluation observations for templates created with a varying number of observations, $N_{\mathrm{temp}}$, and with the intrinsic stellar spectrum. In the first panel, we plot the RMSE for a hypothetical instrument with $4$ pixels per FWHM observing an M9 star. We see that the RMSE of the template follows Poisson statistics. In the second panel, we observe the same M9 star but at the sampling resolution of SPIRou. Here we see an RMSE plateau at high S/N due to interpolation error. In the last panel, we compare this to observations of an M5 star with SPIRou.