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The impact of numerical derivatives on radial velocity extraction

This paper investigates the limitations of derivative-based radial velocity extraction methods when using low-SNR stellar models, finding that while numerical derivatives introduce meter-per-second level residuals at moderate SNRs, their impact diminishes significantly with higher signal-to-noise ratios and the inclusion of multiple spectral lines, eventually falling below the noise floor of state-of-the-art spectrographs.

Original authors: A. M. Silva, K. Al Moulla

Published 2026-06-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: A. M. Silva, K. Al Moulla

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to measure the speed of a car driving past you by listening to the change in the pitch of its engine (the Doppler effect). In astronomy, scientists do the exact same thing with stars. They look at the "notes" (spectral lines) in a star's light to see how fast the star is wobbling. If they can measure this wobble precisely enough, they can detect tiny planets, like Earth, orbiting that star.

To do this, astronomers need a perfect "reference note" (a template) to compare against the noisy, real-time sound of the star. This paper investigates a specific tool used to find that match and asks: Does the way we calculate the "slope" of that reference note introduce hidden errors?

Here is the breakdown of their findings using simple analogies:

1. The Two Ways to Measure the Slope

When astronomers align the star's light with their reference template, they need to know how steep the "hills and valleys" of the light spectrum are. Think of a spectral line as a smooth hill. To measure the star's speed, you need to know the slope of that hill.

The paper compares two methods to find this slope:

The "Analytical" Method (The Perfect Map): This is like having a mathematically perfect drawing of the hill where you know the exact slope at every single point. It's clean and error-free.
The "Numerical" Method (The Rough Sketch): This is what many modern, high-precision tools actually use. Instead of a perfect drawing, they take the real, noisy data and try to guess the slope by looking at the difference between two neighboring points. It's like trying to draw the slope of a hill by connecting two dots on a shaky, grainy photograph.

2. The Problem: Noise Makes the Sketch Shaky

The authors ran simulations using "perfect" hills (Gaussian profiles) and then added "static" (noise) to them, just like real starlight has noise.

The Finding: When the data is noisy (low signal-to-noise ratio), the "Rough Sketch" method gets confused. The static makes the calculated slope look jagged and wrong.
The Result: This jagged slope leads to a wrong calculation of the star's speed. At moderate noise levels (like a per-pixel signal-to-noise ratio of 100), this error can be as large as 1 meter per second.
Why it matters: An Earth-like planet orbiting a Sun-like star only causes a wobble of about 9 centimeters per second. An error of 1 meter per second is like trying to hear a whisper while someone is shouting next to you; it completely drowns out the tiny signal of an Earth twin.

3. The Solution: More Data Smooths the Roughness

The paper shows that this "shaky sketch" problem gets better in two ways:

Turn up the volume (Increase SNR): If you have a very clear, loud signal (high signal-to-noise ratio), the static becomes less important. The paper found that once the signal is very strong (over 1,000 times the noise), the error from the numerical method drops so low that it disappears beneath the current limits of our best telescopes.
Use more hills (More Spectral Lines): If you only look at one spectral line, the error is high. But if you look at hundreds or thousands of lines at once, the errors start to cancel each other out. It's like trying to guess the average height of a crowd by measuring one person (high error) versus measuring the whole crowd (low error).

4. Real-World Test: The Star K2-129

To prove this wasn't just a computer simulation, the authors looked at real data from a star called K2-129 using the ESPRESSO telescope.

They built a "reference template" by stacking observations of the star.
With few observations: The template was "noisy." The numerical method and the perfect method disagreed significantly.
With many observations: As they added more data to the template, the two methods started to agree much better.
The Catch: Even with 78 observations (a very good dataset), there was still a tiny difference of 12.5 centimeters per second between the two methods. While this is small, it is still larger than the 9 cm/s signal of an Earth twin.

The Bottom Line

The paper concludes that using numerical derivatives (the "rough sketch" method) to find a star's speed introduces a small but measurable error, especially when the data is noisy or when looking at a single line.

If you have a lot of data: The error is negligible.
If you have a little data: The error can be big enough to hide the discovery of Earth-like planets.

The authors suggest that to fix this, astronomers might need to "smooth out" the data or the calculated slopes before using them, ensuring that the "static" doesn't trick the computer into seeing a speed that isn't there. This is crucial for the next generation of planet hunting, where we are trying to find worlds that are almost identical to our own.

Technical Summary: The Impact of Numerical Derivatives on Radial Velocity Extraction

Problem Statement
Radial velocity (RV) measurements are critical for detecting exoplanets, particularly Earth-twins, where the target signal amplitude is approximately 9 cm s⁻¹. While template matching (TM) is a standard technique for RV extraction, many modern implementations—specifically line-by-line (LBL) methods—rely on the numerical computation of the derivative of a stellar template to determine the optimal velocity shift. This paper investigates a potential systematic error inherent to this approach: the contamination of the stellar template's derivative by noise. While the stellar template is often assumed to be noise-free in theoretical derivations, in practice, it is constructed from stacked observations. The authors posit that computing a numerical derivative (e.g., via finite differences) on a noisy template introduces biases and scatter into the RV extraction, potentially limiting the precision required for detecting low-mass planets.

Methodology
The study employs a comparative analysis between two RV extraction paradigms applied to the same data:

Classical Template Matching (TM): A $\chi^2$ minimization procedure aligning the observed spectrum with a stellar template, accounting for continuum variations via a first-degree polynomial.
Derivative-Based Method: A formulation based on a first-degree Taylor expansion (Connes 1985; Bouchy et al. 2001), where the RV shift is calculated using the ratio of the flux difference to the derivative of the template ( $\partial T/\partial \lambda$ ).

The authors evaluate these methods using two distinct approaches:

Simulated Data: They generate Gaussian profiles with injected Poisson noise to simulate spectral lines. They compare RVs derived using the analytical derivative of the Gaussian (ground truth) against those derived using a numerical derivative (finite differences). Simulations vary line depth, width, sampling (pixels per line width), and signal-to-noise ratio (SNR) from $10^2$ to $10^6$ . They also test "forests" of Gaussian profiles (up to 1000 lines) to assess the effect of line multiplicity.
Real Observations: The authors apply both methods to ESPRESSO observations of the M3 star K2-129. They construct a stellar template by sequentially adding observations (from 1 up to 78) to vary the template's SNR. They then extract RVs for the remaining observations using both the classical and derivative methods to analyze the residuals and uncertainty estimates as a function of template SNR.

Key Results

Simulation Findings (Single Line): On simulated Gaussian profiles, the use of a numerical derivative introduces RV residuals at the level of meters per second when the per-pixel SNR is 100. The magnitude of this impact is strongly correlated with SNR and line depth; shallower lines exhibit larger scatter. As the SNR increases, the discrepancy between numerical and analytical methods decreases, falling below the noise floor of state-of-the-art spectrographs for per-pixel SNR > 1000.
Simulation Findings (Multiple Lines): Including multiple spectral lines in the simulations leads to an overall decrease in the contamination across all SNR regimes, suggesting that the noise averages out when many lines are utilized.
Real Data Application: Analysis of K2-129 confirms the simulation trends. As the template SNR increases (by adding more observations), the RVs derived from the derivative method converge with those from the classical method. However, even with a template constructed from 78 observations (yielding a template SNR 7.5 times higher than individual observations), a residual difference of 12.5 cm s⁻¹ persists between the two methods.
Uncertainty Estimates: The study notes that the derivative method also affects the calculated RV uncertainties. In simulations, incorrect determination of the template gradient impacts uncertainty estimates at lower SNRs. In real data, the ratio of uncertainties between the two methods increases with template SNR, though the authors note that drawing definitive conclusions is complex because the two methods estimate uncertainties through different mathematical avenues.

Significance and Claims
The paper identifies the numerical computation of stellar derivatives in noisy regimes as a source of systematic error in RV extraction. The authors claim that for low-SNR regimes (typical of individual observations or stars with fewer available spectra), this numerical noise can introduce RV scatter on the order of meters per second or, in optimized multi-line scenarios, residuals around 12.5 cm s⁻¹.

The significance of this work lies in its demonstration that even with high-quality data, the method of derivative calculation can introduce biases comparable to the signature of an Earth-twin planet. The authors suggest that while increasing the number of lines and the template SNR mitigates the effect, it may not fully eliminate it for all observation campaigns. They propose that future mitigation strategies could involve smoothing the spectra or the numerical derivative to reduce the impact of high-frequency flux variations on velocity estimates. The paper concludes that this systematic must be considered when pushing RV precision toward the 10 cm s⁻¹ frontier.

1. The Two Ways to Measure the Slope

2. The Problem: Noise Makes the Sketch Shaky

3. The Solution: More Data Smooths the Roughness

4. Real-World Test: The Star K2-129

The Bottom Line

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