Stellar Activity Grid for Exoplanets (SAGE)

• SAGE is a comprehensive framework that quantifies and corrects stellar activity impacts on exoplanet measurements using pixelated surface models and empirical noise regression.
• It simulates transmission spectra, radial velocity series, and spot evolution to predict chromatic biases and transit depth errors with high precision.
• The platform further enhances exoplanet studies by forecasting stellar activity cycles and differential rotation, thereby optimizing observation planning.

The Stellar Activity Grid for Exoplanets (SAGE) is a comprehensive suite of methodologies, empirical grids, and simulation tools for quantifying and correcting the impact of stellar magnetic activity on the detection and characterization of exoplanets, with special emphasis on transmission spectroscopy, radial velocity (RV) time series, photometric variability, and the modeling of chromospheric indices. SAGE unifies pixelated surface modeling, empirical noise regressions, stellar activity cycle forecasting, rotation-activity relations, and differential rotation, enabling self-consistent, time- and wavelength-dependent correction strategies for exoplanet transit and RV measurements across a broad range of stellar types and activity regimes.

1. Pixelated Surface Models and Transmission Spectroscopy Correction

SAGE approaches the forward modeling of stellar activity-induced contamination in transmission spectra by discretizing the visible stellar disk into an $N \times N$ Cartesian grid of equal-area pixels, each assigned a specific intensity spectrum $F_i(\lambda)$ from stellar atmosphere libraries (PHOENIX, ATLAS9, Coelho). For each pixel, rotational Doppler shift $v_i$ and limb-darkening weight $I(\mu_i,\lambda)/I(1,\lambda)$ are computed; active regions (spots, faculae) are inserted as circular patches specified by latitude, longitude, angular radius (or absolute filling factor $f_{\rm abs} = {\rm area}_{\rm patch}/4\pi R_*^2$), and spectral contrast $$\rho(\lambda) = F_{\rm active}(\lambda)/F_{\rm clear}(\lambda)$$.

The aggregate disk-integrated spectrum at time $t$ is

$$F_{\rm spotted}(\lambda, t) = \sum_{i=1}^{N^2} F_i(\lambda) \frac{I(\mu_i,\lambda)}{I(1,\lambda)} \Delta A_i\, \delta[\lambda - \lambda(1+v_i/c)]$$

with the stellar contamination factor

$$\epsilon_\lambda(t) = \frac{F_{\rm clear}(\lambda)}{F_{\rm spotted}(\lambda, t)}$$

and the change in apparent transit depth,

$$\Delta F(\lambda) = \epsilon_\lambda - 1 \simeq f_{\rm proj}(t) \frac{F_{\rm clear}(\lambda) - F_{\rm active}(\lambda)}{F_{\rm clear}(\lambda)}$$

in the small covering-fraction approximation. Here, $F_i(\lambda)$0 is the instantaneous projected areal coverage of active regions, explicitly dependent on spot/facula location, size, and stellar inclination.

Limb-darkening is incorporated via a quadratic law,

$F_i(\lambda)$1

with wavelength-dependent coefficients derived from model atmospheres. Rotational line broadening is introduced either through per-pixel Doppler shifts or convolution with analytic kernels, reproducing the full disk-integrated profile at the requisite spectral resolution ($F_i(\lambda)$2–$F_i(\lambda)$3).

The model can simultaneously fit time-series photometry (e.g., TESS) to retrieve spot maps using MCMC, connecting contemporaneous variability to transmission spectra and yielding epoch-specific correction curves for multi-transit datasets. Application to WASP-69 demonstrates that two mid-latitude spots with a combined $F_i(\lambda)$41% filling factor explain observed light curve modulations and predict contamination spectra that can vary by tens of ppm across the optical (Chakraborty et al., 2023).

2. Empirical Stellar Noise Grids and Chromospheric Activity

A core constituent of SAGE is the empirical mapping of intrinsic photometric noise to observable stellar properties. Using a large LAMOST–Kepler training set, an XGBoost regression infers the 6-hr root-robust stellar noise $F_i(\lambda)$5 as a function of S-index ($F_i(\lambda)$6), $F_i(\lambda)$7, $F_i(\lambda)$8, [Fe/H], and Kepler $F_i(\lambda)$9 magnitude—attaining a precision of $v_i$020 ppm over $v_i$1 FGK stars.

The mapping,

$v_i$2

allows direct interpolation or model prediction of the photometric floor for any target. Residuals are unbiased versus stellar labels, and the resultant 4D grid is organized for rapid look-up in SAGE pipelines, supplying $v_i$3 for SNR calculations in yield forecasting and transit detectability. The parameter ranges (3800 K < $v_i$4 < 6500 K, $v_i$5) ensure robust applicability to FGKM dwarfs (Zhang et al., 2024).

3. Activity Cycles, Rotation, and Chromospheric Diagnostics

SAGE systematically incorporates stellar activity cycles using archival Ca II H&K data (S-index, $v_i$6) and RV time series, as exemplified by the California Legacy Survey (Isaacson et al., 2024) and HWO ARC (Fetherolf et al., 21 May 2026). For the 710-star CLS sample, activity cycles are detected in 49% of stars with adequate temporal coverage; robust cycles ($v_i$7 yr, amplitudes $v_i$8 dex) are prevalent in stars with $v_i$9 K and $I(\mu_i,\lambda)/I(1,\lambda)$0. Empirical relations link $I(\mu_i,\lambda)/I(1,\lambda)$1 and amplitude to $I(\mu_i,\lambda)/I(1,\lambda)$2 within this regime, e.g.,

$I(\mu_i,\lambda)/I(1,\lambda)$3

and $I(\mu_i,\lambda)/I(1,\lambda)$4, with $I(\mu_i,\lambda)/I(1,\lambda)$5 the S-index amplitude.

The HWO/ARC catalog details $I(\mu_i,\lambda)/I(1,\lambda)$6-index and $I(\mu_i,\lambda)/I(1,\lambda)$7, $I(\mu_i,\lambda)/I(1,\lambda)$8, $I(\mu_i,\lambda)/I(1,\lambda)$9 (from photometry and spectroscopy), and, where available, cycle periods for $f_{\rm abs} = {\rm area}_{\rm patch}/4\pi R_*^2$08,000 direct imaging targets. Rotation–activity–age and cycle–rotation–Rossby number parameterizations are explicitly tabulated for inclusion in SAGE, supporting the prediction of activity regime and cycle phase for mission planning (Fetherolf et al., 21 May 2026).

4. Differential Rotation and Latitudinal Spot Evolution

SAGE accounts for differential rotation of starspots, critical for simulating spot-crossing signatures in exoplanet transit light curves and their modulation over time. The latitude-dependent angular velocity is mapped as

$f_{\rm abs} = {\rm area}_{\rm patch}/4\pi R_*^2$1

with solar α-sunspot coefficients $f_{\rm abs} = {\rm area}_{\rm patch}/4\pi R_*^2$2, $f_{\rm abs} = {\rm area}_{\rm patch}/4\pi R_*^2$3, $f_{\rm abs} = {\rm area}_{\rm patch}/4\pi R_*^2$4. For other stars, SAGE generalizes via scaling parameters: $f_{\rm abs} = {\rm area}_{\rm patch}/4\pi R_*^2$5 where $f_{\rm abs} = {\rm area}_{\rm patch}/4\pi R_*^2$6 is the rigid rotation rate and $f_{\rm abs} = {\rm area}_{\rm patch}/4\pi R_*^2$7 a dimensionless shear factor. The evolution of spot longitudes is updated at each timestep by the local angular velocity. Varying $f_{\rm abs} = {\rm area}_{\rm patch}/4\pi R_*^2$8 from $f_{\rm abs} = {\rm area}_{\rm patch}/4\pi R_*^2$9 (rigid rotator) to $$\rho(\lambda) = F_{\rm active}(\lambda)/F_{\rm clear}(\lambda)$$0 (solar shear) to $$\rho(\lambda) = F_{\rm active}(\lambda)/F_{\rm clear}(\lambda)$$1 (enhanced shear) enables SAGE to model the beat patterns and amplitude modulations observed in spotted-star light curves and to evaluate their impact on transit and RV analysis (Lößnitz et al., 11 Aug 2025).

5. Forward Modeling of Activity-Induced Confounders in RV, Photometry, and Transit Depths

SAGE uses physical and empirical frameworks for generating synthetic time series of RV, photometric, astrometric, and chromospheric signals across grids of activity scenarios. In the StarSim-based methodology (Herrero et al., 2015), the observed flux is synthesized as a sum over small surface elements, each characterized by specific intensity, limb-darkening, and filling-fraction by spot or facula, with time-variable positions determined by the rotation law (including differential rotation). The corresponding RV perturbations are extracted from the disk-integrated cross-correlation function. Noise—instrumental and astrophysical (granulation, oscillations)—is added at realistic amplitudes to emulate observed time-series properties.

Grid-based approaches (Meunier et al., 2019) span stellar types (F6–K4) and chromospheric activity indices, building higher-level grids of S-index, $$\rho(\lambda) = F_{\rm active}(\lambda)/F_{\rm clear}(\lambda)$$2, spot contrast, rotation period, inclination, and cycle parameters for thousands of realizations per parameter set. These grids support injection–recovery tests for exoplanet detectability, performance benchmarking of activity-correction algorithms, and the translation of photometric–RV correlations into predicted transmission spectra contamination.

6. Activity Forecasting and Observation Optimization

SAGE includes a framework for forecasting stellar activity cycles using long-baseline Ca II H&K activity time series and photometry (Sairam et al., 2022). Lomb–Scargle periodograms and MCMC-fitted sinusoids yield per-star cycle periods, amplitudes, phase (e.g., timing of minima/maxima), and uncertainty propagation for the prediction of optimal observing windows. Typical minimum-timing uncertainties are $$\rho(\lambda) = F_{\rm active}(\lambda)/F_{\rm clear}(\lambda)$$30.5 years, enabling observation planning at epochs of minimized stellar activity. This forecasting demonstrably enhances RV and transmission spectroscopy sensitivity, reducing RV noise floors and transmission contamination by up to an order of magnitude in high-precision regimes. SAGE thus proposes an integrated public forecast service, with ingest of new time-series data, continuous cycle re-computation, and user-accessible per-target forecasts for mission and proposal support.

7. Quantification of Stellar Contamination and Associated Biases

The impact of surface inhomogeneities on transmission spectroscopy is quantified both analytically and through direct surface-flux simulations (Schrijver, 2020). The characteristic amplitude of activity-induced bias in the observed planet-to-star radius ratio $$\rho(\lambda) = F_{\rm active}(\lambda)/F_{\rm clear}(\lambda)$$4 can be hundreds to thousands of kilometers, depending on mean magnetic flux density ($$\rho(\lambda) = F_{\rm active}(\lambda)/F_{\rm clear}(\lambda)$$5), spot/facula filling factor, and wavelength. Chromatic signatures arise because facular contrast increases toward the limb and in the blue, and spot contrast varies with both temperature difference and wavelength; convective blueshift inhibition further modulates the RV and photometric signals in a subtype- and inclination-dependent manner. SAGE provides tabulations and interpolative relations for these biases as a function of activity level and wavelength, offering users direct grid-based corrections to observed light curves and RVs.

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SAGE constitutes a unifying data-driven and physically motivated infrastructure for mitigating the diverse impacts of stellar activity on exoplanet atmospheric and dynamical characterization. Its multi-level structure—incorporating pixelated surface modeling, empirical noise grids, time-dependent activity forecasting, spot-differential rotation, and chromospheric diagnostics—enables both observation planning and analysis correction at the precision level demanded by contemporary and future exoplanet surveys (Chakraborty et al., 2023, Zhang et al., 2024, Isaacson et al., 2024, Lößnitz et al., 11 Aug 2025, Sairam et al., 2022, Fetherolf et al., 21 May 2026, Herrero et al., 2015, Meunier et al., 2019, Schrijver, 2020).