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Mitigating stellar radial velocity jitter using orthogonal activity indices and a time-aware neural network

This paper introduces CANSTAR, a time-aware neural network that utilizes orthogonal activity indices derived from cross-correlation function distortions to effectively mitigate stellar radial velocity jitter, thereby improving the detection of Earth-like exoplanets and the precision of orbital parameter determination compared to traditional methods.

Original authors: Jordi Blanco-Pozo, Manuel Perger, Guillem Anglada-Escudé, Ignasi Ribas, David Baroch, Marina Lafarga, Juan Carlos Morales, Òscar Porqueras-León, Sophie Stucki, David Vallmanya Poch

Published 2026-05-07

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CC BY 4.0

Original authors: Jordi Blanco-Pozo, Manuel Perger, Guillem Anglada-Escudé, Ignasi Ribas, David Baroch, Marina Lafarga, Juan Carlos Morales, Òscar Porqueras-León, Sophie Stucki, David Vallmanya Poch

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 listen to a very quiet whisper (a planet orbiting a star) coming from a room where a loud, chaotic fan is spinning (the star's surface activity). The fan creates wind gusts that shake the microphone, making it hard to hear the whisper. This is the biggest problem in finding Earth-like planets: the stars themselves are "noisy," creating ripples in their light that look exactly like the wobble caused by a planet.

This paper introduces a new, high-tech way to silence that fan so we can finally hear the whisper. Here is how they did it, explained simply:

1. The Problem: The "Fan" is Shaping the Sound

Astronomers usually look at a star's light by taking a "fingerprint" of its spectrum (a cross-correlation function, or CCF). Normally, they assume this fingerprint is a smooth, perfect hill (a Gaussian shape). They measure how much the hill moves left or right to find the planet.

But stars aren't perfect. They have sunspots and magnetic storms that don't just move the hill; they squish, stretch, and twist it. It's like someone grabbing a smooth clay hill and pinching it into weird shapes. Traditional methods only look at the center of the hill, missing all the weird pinching happening on the edges.

2. The Solution: Unfolding the Clay

The authors developed a mathematical trick to "unfold" that squished clay. Instead of just looking at the whole hill, they broke the shape down into a set of building blocks (called an orthogonal basis).

Block 1: Measures how much the whole thing moved left or right (the planet signal).
Blocks 2, 3, 4...: Measure exactly how the shape was twisted, pinched, or stretched (the star's noise).

Think of it like a music mixer. Instead of just hearing the whole song, they separated the vocals (the planet) from the drums, bass, and guitar (the star's activity). They realized that for some stars (like the cool, red M-dwarf TZ Ari), the "clay" has two humps instead of one, so they used a special set of building blocks designed for that specific shape.

3. The Brain: A "Time-Aware" Detective

Once they had these "twist and pinch" measurements, they needed to figure out how much of the movement was actually caused by the star. They built a special computer brain called CANSTAR (Convolutional-Attention Network for STellar Activity Removal).

The Old Way (The "Snapshot" Brain): Previous methods looked at one day's data and tried to guess the noise based only on that single moment. It's like trying to predict the weather by looking at the sky for only one second.
The New Way (CANSTAR): This brain is "time-aware." It looks at the history of the star's shape changes. It understands that stars have rhythms, like a heartbeat. It uses a "self-attention" mechanism (like a detective connecting the dots between yesterday's storm and today's wind) to predict exactly how the star's noise will distort the signal right now.

They trained this brain using a super-accurate simulator called StarSim, which creates millions of fake stars with known noise patterns, teaching the brain to recognize the difference between a planet's wiggle and a star's tantrum.

4. The Results: Quieting the Fan

They tested this on two real stars: Epsilon Eridani and TZ Arietis.

For Epsilon Eridani: The new method reduced the "noise" (the jitter) by nearly half. It successfully filtered out the star's rotation signals, leaving behind a much cleaner signal where a potential planet could be found more easily.
For TZ Arietis: This star has a known planet, but the star's noise was so loud it was messing up the measurements of the planet's orbit. The new method cleaned up the data so well that the planet's orbit could be calculated with much higher precision than before. In fact, it did a better job than the current "gold standard" method (Gaussian Processes), which sometimes gets confused by the complex noise.

5. The Catch: The Simulator vs. Reality

The authors are honest about one limitation: The computer brain was trained on simulated stars. While the simulator is very good, real stars have tiny, messy imperfections (like atmospheric turbulence or instrument quirks) that the simulator doesn't perfectly copy yet. Because of this, the brain isn't perfect on real data, but it is still significantly better than what we had before.

The Bottom Line

This paper shows that by looking at the shape of a star's light in extreme detail and using a smart computer brain that understands time, we can filter out the star's own noise much better than before. This brings us one step closer to hearing the faint "whisper" of an Earth-like planet orbiting a Sun-like star.

Technical Summary: Mitigating Stellar Radial Velocity Jitter Using Orthogonal Activity Indices and a Time-Aware Neural Network

Problem Statement
Despite instrumental advancements in high-resolution spectrographs (e.g., HARPS, CARMENES, ESPRESSO) pushing precision toward the cm s⁻¹ regime, the detection of Earth-like exoplanets remains hindered by stellar magnetic activity. Surface inhomogeneities such as spots, faculae, and granulation induce radial velocity (RV) variations at the m s⁻¹ level, often mimicking or obscuring planetary signals. These effects distort spectral line shapes in ways that standard Gaussian fit approximations of the Cross-Correlation Function (CCF) fail to fully capture. Existing mitigation strategies, including Gaussian Process (GP) regression and standard activity indicators (e.g., BIS, FWHM), often struggle to disentangle complex, non-Gaussian line deformations from true Doppler shifts, particularly in active stars.

Methodology
The authors propose a two-stage framework to disentangle stellar activity from planetary signals:

Orthogonal Decomposition of CCFs:
Instead of relying solely on Gaussian fits, the authors decompose the CCF using a Gram-Schmidt orthogonal basis. This separates the observation into:
- A line-shift component ( $l(t)$ ), capturing the net velocity shift (Doppler + activity-induced apparent shift).
- Line-shape distortion components ( $g_n(t)$ ), derived from a distortion basis function.
  For Gaussian-like profiles (e.g., $\epsilon$ Eridani), a single Hermite basis is used. For non-Gaussian, double-humped profiles typical of cool M dwarfs (e.g., TZ Arietis), a multi-Hermite basis (a sum of two Gaussians) is employed. This decomposition yields time-series coefficients ( $c_1, l, g_2, g_3, \dots$ ) where the distortion coefficients ( $g_n$ ) are mathematically orthogonal to the line shift, serving as pure activity indicators.
Time-Aware Neural Network (CANSTAR):
The authors developed CANSTAR (Convolutional-Attention Network for STellar Activity Removal). Unlike standard fully connected networks (FCNs) that treat observations as independent snapshots, CANSTAR incorporates temporal context.
- Architecture: It utilizes 1D convolutional layers to extract local, high-frequency features from the distortion coefficient time series, followed by Transformer encoder layers with multi-head self-attention mechanisms to model long-term temporal dependencies.
- Training: The network is trained on synthetic data generated by StarSim, a physically motivated stellar activity simulator. The training set includes 500,000 simulations per target, with realistic photon noise injection and varying active region parameters (spots, faculae). The network learns to predict the activity-induced shift component ( $\kappa$ ) from the distortion coefficients.

Key Contributions

Orthogonal Basis for Activity Indicators: The paper introduces a rigorous mathematical framework to separate line shifts from shape distortions using orthogonal basis functions (Hermite and multi-Hermite), recovering classical indicators (CON, FWHM, BIS) as lower-order coefficients while extending sensitivity to higher-order distortions.
Temporal Modeling in Activity Correction: By integrating convolutional layers with Transformer attention mechanisms, CANSTAR explicitly models the time-correlated nature of stellar activity (evolution, migration, and reappearance of active regions), outperforming time-agnostic architectures.
Synthetic-to-Real Transfer: The study demonstrates a pipeline where a network trained exclusively on synthetic StarSim data (with realistic noise injection) successfully mitigates activity in real observational data from HARPS and CARMENES.

Results
The framework was validated on two active stars:

$\epsilon$ Eridani (K2 dwarf): CANSTAR reduced the RV Root Mean Square (RMS) to 52.5% of the uncorrected variability (from 4.69 m s⁻¹ to 2.46 m s⁻¹). The activity-induced peaks at the rotation period (11.2 d) and its harmonic were suppressed below the 10% False Alarm Probability (FAP) level. The detection limit for a planetary signal (at 10% precision) improved from 3.99 m s⁻¹ (uncorrected) to 2.51 m s⁻¹.
TZ Arietis (M5 dwarf): For this star, where a planetary signal (TZ Ari b) coexists with strong activity, CANSTAR was used in a joint fit with a Keplerian model. It reduced the RV RMS to 62.4% of the variability remaining after subtracting the Keplerian signal. Crucially, CANSTAR provided more precise orbital parameters for TZ Ari b compared to a Gaussian Process (GP) regression. While the GP model reduced scatter slightly more (due to flexibility/overfitting), it left significant residual periodicities (37.2 d and 41.4 d) likely caused by uncorrected activity aliases. CANSTAR effectively removed these signals, confirming the planetary origin of the long-period signal.

In simulated noiseless data, CANSTAR achieved near-perfect correction (Residual Relative Error ~3-4%), significantly outperforming FCNs. In noisy regimes, CANSTAR maintained superior performance, requiring fewer input coefficients to reach saturation compared to FCNs.

Significance and Claims
The authors claim that their results demonstrate that neural networks incorporating temporal context can outperform state-of-the-art methods (specifically GPs) in complex activity regimes. By leveraging high-order line-shape distortions and temporal evolution, CANSTAR offers a robust pathway to disentangling planetary signals from stellar activity.

The paper acknowledges a performance gap between synthetic and real data (RRE increased from ~13% in noisy simulations to ~52-62% in real data), attributing this to unmodeled instrumental systematics and the limitations of 1D atmospheric models in StarSim. However, the authors assert that future improvements—specifically integrating 3D magnetohydrodynamic (MHD) spectra and instrumental systematics into the training data—will bridge this gap. They conclude that this approach is currently capable of pushing detection limits toward the photon-noise floor, facilitating the discovery of lower-mass planets and exo-Earths around Sun-like stars.