Where does the simplified Stellar Contamination Model fail in Exoplanet Transmission Spectroscopy?

Abstract

Stellar photospheric heterogeneities (e.g., starspots, faculae) distort the stellar spectrum in transit and imprint wavelength-dependent biases on the planet-to-star radius ratio (Transit Light Source Effect, TLSE). The Rackham-TLSE (R-TLSE) prescription applies a disc-averaged correction based solely on filling factor and spectral contrast, but transmission spectroscopy also depends on limb darkening, active-region distribution, and transit geometry. We include these in a pixel-resolved framework, ECLIPSE-Xlambda, and run idealised noise-free model-model comparisons to R-TLSE. For LHS 1140 b, K2-18 b, and WASP-69 b, disc-averaged corrections differ from the pixel model by up to about 400 ppm in the optical for active hosts and non-equatorial transits, but stay below about 10 ppm in the near-infrared where limb darkening is weak. We then apply both approaches to the JWST/NIRISS SOSS spectrum of LHS 1140 b. With limb darkening set to zero, ECLIPSE-Xlambda recovers stellar-contamination parameters matching the reference R-TLSE solution, confirming consistency in the disc-averaged limit. With wavelength-dependent limb darkening, reproducing the short-wavelength slope via stellar contamination alone requires hot faculae (delta Tfac about 600 K; ffac about 0.35), equivalent to a circular facular region of radius about 0.6 Rstar (about 60% of the stellar radius) on the disc; such an extended unocculted region is physically unlikely even for an active M dwarf. Purely stellar contamination would therefore require extreme faculae, whereas a genuine atmospheric contribution complementing a more modest facular signal is more plausible. These results delineate the validity regime of R-TLSE and underscore the need for geometry-aware stellar-heterogeneity models including limb darkening in high-precision transmission spectroscopy.

Figures (9)

• Figure 1: Wavelength-dependent errors between the ECLIPSE-X$\lambda$ and the R-TLSE (\ref{['eq:err']}) for LHS 1140 b, computed using a fixed filling factor ($f_{\rm het} = 0.15$) and a temperature contrast ($\Delta T$) varying from $-300$ K (spots) to $+300$ K (faculae). The largest discrepancies arise in the optical range (500--800 nm), where steep limb-darkening gradients drive optical biases. The dashed lines mark reference levels at -50, -20, -10, 10, 20, and 50 ppm, serving as a visual aid to highlight variations in the data. By construction, positive values of the error indicate that R--TLSE predicts a shallower transit depth than ECLIPSE-X$\lambda$ (overcorrecting stellar contamination), while negative values correspond to a deeper R--TLSE transit (undercorrecting contamination).
• Figure 2: Discrepancies, defined by \ref{['eq:err']}, as a function of fixed temperature contrast ($T_\mathrm{star}\pm300$) and vary the filling factor ($f_\mathrm{het}$) from 0.01 to 0.15. Positive errors (R-TLSE overestimation) dominate for cool spots, while negative errors (R-TLSE underestimation) arise for hot faculae. Extreme cases (e.g., $f_{\rm het} = 0.15$, $\Delta T = \pm 300$ K) show maximal divergence. The dashed lines delineate boundaries at -50, -20, -10, 10, 20, and 50 ppm.
• Figure 3: Two-dimensional parameter space exploration showing the discrepancies (\ref{['eq:err']}) at a representative wavelength of 600 nm. Each panel corresponds to a different exoplanetary system: LHS 1140 b (left panel), K2-18 b (middle panel), and WASP-69 b (right panel). Color scales indicate the difference between our model and the R-TLSE correction equation, as a function of filling factor and temperature contrast.
• Figure 4: Effect of the active region's spatial position on the stellar disc for LHS 1140 b, illustrating the wavelength-dependent error between ECLIPSE-X$\lambda$ and R-TLSE. Each curve corresponds to a different temperature of the active region (spot or facula). The four panels examines positions gradually shifting from close to disc centre (15$^\circ$ latitude, 0$^\circ$ longitude) toward the limb (35$^\circ$ latitude, 35$^\circ$ longitude). A sign inversion emerges at intermediate positions beyond the optical wavelengths (right panel, first row). The lower panels depict more extreme cases with active regions placed even closer to the limb, showing that the sign-inversion boundary moves to shorter wavelengths, significantly altering and reducing errors in the optical regime, while further limb positioning exacerbates R-TLSE inaccuracies due to enhanced limb darkening and foreshortening. Positive errors mean that R--TLSE predicts a shallower transit than ECLIPSE-X$\lambda$ (overcorrecting contamination), whereas negative errors mean that R--TLSE predicts a deeper transit (undercorrecting contamination).
• Figure 5: Comparison between ECLIPSE-X$\lambda$ and the R--TLSE approximation for the JWST/NIRISS SOSS transmission spectrum of LHS 1140 b. Top: Combined $R\!\sim\!100$ NIRISS spectrum (grey points with error bars), together with the best-fitting stellar–contamination models. The magenta curve shows the self-consistent ECLIPSE-X$\lambda$ fit including wavelength-dependent limb darkening, the cyan curve shows the corresponding fit with limb darkening set to zero (LDCs = 0), and the black curve reproduces the TLS-only stellar–contamination model from the R--TLSE reference retrieval Cadieux.et.al.2024ApJ...970L...2C. The small-scale structure along the model curves follows atomic and molecular absorption bands in the underlying PHOENIX spectra: TiO/VO and metal lines dominate in the optical, while broad features in the near-infrared are mainly shaped by H$_2$O bands and CO absorption complexes longwards of $\sim$2.2 $\mu$m, i.e. precisely the wavelength ranges where exoplanet transmission spectra are usually interpreted in terms of H$_2$O, CH$_4$, and CO absorption. Bottom: One-dimensional marginal distributions for the stellar–contamination parameters in the ECLIPSE-X$\lambda$ fits, for the cases with limb darkening (top row, magenta) and with LDCs$=0$ (bottom row, cyan). From left to right, panels show the filling factor and temperature of spots ($f_{\mathrm{spot}}$, $T_{\mathrm{spot}}$) and faculae ($f_{\mathrm{fac}}$, $T_{\mathrm{fac}}$). Vertical lines indicate the median (solid) and 16th–84th percentile (dashed) intervals reported in Table \ref{['tab:stellar_contam']}.