Gaussian-Process Detrending: Separating Planet Signals From Stellar Activity

What GP detrending is and why activity is the enemy

Radial-velocity (RV) planet detection measures the tiny wobble a planet induces on its star: Earth pulls the Sun at just 0.09 m/s, a hot super-Earth like CoRoT-7b at a few m/s. Modern spectrographs (HARPS, ESPRESSO) reach instrumental precision below 0.5 m/s, so the detection limit is no longer the instrument — it is the star itself.

Magnetic activity produces three RV effects: dark starspots break the rotational Doppler balance across the disk; bright faculae do the opposite; and the suppression of convective blueshift in magnetized regions shifts lines by up to several m/s. Crucially, this jitter is not white noise — it is correlated on the stellar rotation timescale (days to weeks) and persists over the spot-evolution timescale (weeks to months).

• A Gaussian process treats the whole activity signal as one random function with a specified covariance, rather than fitting individual spots.
• Because the covariance encodes rotation, the GP absorbs quasi-periodic wandering but leaves a clean Keplerian planet signal intact.

The mechanism: a prior over functions and the quasi-periodic kernel

A Gaussian process is defined by a mean function (often zero after removing the systemic velocity) and a covariance kernel k(t_i, t_j) that says how correlated two observations are given their time separation. The data likelihood is a single multivariate Gaussian whose covariance matrix K is built entirely from the kernel.

The workhorse is the quasi-periodic kernel, the product of a squared-exponential and an exp-sine-squared term:

k(Δt) = A² · exp[ −Δt²/(2ℓ²) − Γ·sin²(π·Δt / P_rot) ]

• A — amplitude of the activity signal.
• P_rot — stellar rotation period; the periodic term repeats the correlation every rotation.
• ℓ — the evolution (or decay) timescale; the Gaussian envelope damps correlation as spots grow and dissolve, so activity is periodic only for a few cycles.
• Γ — harmonic complexity, controlling how many wiggles appear within one rotation (spot-pattern complexity).

Fitting proceeds by maximizing the GP marginal likelihood (or MCMC over hyperparameters), inverting K, and using the conditional Gaussian to predict — and subtract — the activity component.

Key quantities and a worked example

The magic of GP detrending is that the four hyperparameters are physically interpretable. Take a Sun-like spotted star:

• P_rot ≈ 25 days — matched to the photometric rotation signal.
• ℓ ≈ 20–50 days — a few rotation cycles; short ℓ means spots evolve fast, long ℓ means a stable pattern.
• A ≈ 2–5 m/s for a moderately active G/K dwarf; young or M dwarfs can hit tens of m/s.
• Γ ≈ 0.5–3 — higher values for multi-spot, structured light curves.

A concrete case: CoRoT-7, a K-dwarf with P_rot ≈ 23 days. Its RV activity amplitude (~4–10 m/s) exceeded the ~4 m/s planet b signal. Haywood et al. (2014) trained a GP on the CoRoT light curve, transferred its covariance to the HARPS RVs, and recovered M_b = 4.73 ± 0.95 M⊕ and M_c = 13.56 ± 1.08 M⊕ — a rocky density of ~6.6 g/cm³, resolving years of conflicting mass estimates.

How it's observed and implemented in practice

GP detrending is a data-analysis layer sitting on top of standard RV pipelines, but it works best when fed ancillary activity indicators that share the star's frequency structure but contain no planet signal.

• Photometry: simultaneous light curves (CoRoT, Kepler, TESS) constrain P_rot and ℓ directly. The FF' method (Aigrain, Pont & Zucker 2012) predicts RV jitter from the flux F and its time derivative F′ with only two free parameters.
• Spectroscopic indicators: the bisector inverse slope, FWHM of the cross-correlation function, and chromospheric log R′_HK or Hα trace activity without a Doppler-reflex component.
• Multidimensional GPs (Rajpaul et al. 2015) tie the RVs and each indicator to one latent activity function G(t) and its derivative G′(t), sharing hyperparameters across all series. This dramatically breaks the degeneracy between activity and a planet at the rotation period.

Software: george, celerite/celerite2 (fast O(N) evaluation), exoplanet, and juliet make the O(N³) matrix inversion tractable for thousands of points.

How it compares to other detrending regimes

GP detrending is one of several stellar-activity mitigation strategies, each with trade-offs:

• Sinusoid / harmonic fits: model activity as sines at P_rot and its harmonics. Cheap, but assume a strictly periodic, non-evolving signal — they fail when spots decay, and can over-fit and swallow real planets.
• High-pass filtering / moving averages: remove slow trends but also erode short-period planets and treat noise as white.
• FF′ and SOAP-style spot models: physically motivated but require assumptions about spot number, contrast, and geometry.
• Gaussian processes: non-parametric and flexible, capturing quasi-periodic evolution the others cannot — at the cost of interpretability risk and heavy computation.

The central danger, shared with sinusoid fits, is over-flexibility: a GP given loose priors can absorb a genuine planetary signal, especially near the rotation period or its harmonics. This is why constrained, physically-informed priors on P_rot and ℓ (from photometry) and multidimensional sharing are essential for trustworthy detections.

Significance, famous cases, and open questions

GP detrending has become the field standard for activity-dominated systems and underpins many of the smallest RV-mass measurements to date. Landmark applications include CoRoT-7 (Haywood et al. 2014), K2-18b, the Kepler-10 system, and numerous TESS follow-ups where activity rivaled the planet.

Yet real debates remain:

• Over-fitting vs. detection: how do you prove the GP hasn't eaten the planet? Injection-recovery tests and model-comparison metrics (Bayesian evidence, cross-validation) are active research.
• Non-stationarity: the QP kernel assumes fixed hyperparameters, but activity amplitude changes over a star's magnetic cycle (the Sun's 11-year cycle changes RV jitter markedly).
• Faculae vs. spots: the RV signature differs, and simple kernels don't distinguish them; line-by-line and 2D line-profile GP models (Rajpaul et al. 2020, 2024) are pushing toward physically richer covariances.

The ultimate prize — an Earth twin at ~0.09 m/s — will demand activity modeling an order of magnitude better than today's, making GP detrending one of the most consequential open problems in exoplanet science.

Common covariance kernels used in stellar-activity Gaussian-process detrending and what they assume

Kernel	Functional form (plain text)	Free hyperparameters	Best suited to
Squared-exponential (RBF)	k = A² · exp(−Δt²/(2ℓ²))	Amplitude A, length scale ℓ	Smooth, non-periodic trends; instrumental drift
Exp-sine-squared (periodic)	k = A² · exp(−Γ·sin²(πΔt/P))	A, harmonic complexity Γ, period P	Strictly repeating signals; no spot evolution
Quasi-periodic (QP)	k = A² · exp(−Δt²/(2ℓ²) − Γ·sin²(πΔt/P_rot))	A, ℓ, Γ, P_rot	Evolving starspots — the field standard
Matérn-3/2	k = A²(1+√3·Δt/ℓ)·exp(−√3·Δt/ℓ)	A, ℓ	Rougher, less-differentiable granulation noise
QP + cosine (QPC)	QP × extra cosine harmonic term	QP set + 2nd-harmonic amplitude	Double-dip spot light curves