Starspot-Crossing Events in Exoplanet Transits

• Starspot-Crossing Events (SCEs) are instances when transiting planets pass over cooler, magnetically active regions on the stellar surface, causing localized upward anomalies in light curves.
• Methodologies employ geometric and mathematical models with MCMC fitting to extract spot sizes, contrasts, and positions, thereby enhancing characterization of exoplanet host stars.
• SCE analysis refines measurements of spin–orbit alignment and stellar rotation while addressing limitations like spot evolution, transit depth uncertainties, and contamination in transmission spectroscopy.

A starspot-crossing event (SCE) occurs when a transiting planet passes over a cooler, magnetically active region on the stellar photosphere—a starspot—producing a localized upward anomaly (“bump”) in the observed transit light curve. These features provide powerful constraints on stellar surface inhomogeneities, spin–orbit architectures, differential rotation, and the physical properties of exoplanetary host stars. SCEs are most prevalent and most diagnostically valuable in cool stars (T_eff ≲ 6100 K) where the Rossiter–McLaughlin effect is less effective. This article surveys the geometric origin, mathematical modeling, and empirical exploitation of SCEs in transit and microlensing photometry, detailing methodologies, limitations, and catalogued system results.

1. Geometric and Physical Origin of Starspot-Crossing Events

SCEs arise when the sky-projected transit chord of a planet intersects a region on the stellar disk with reduced brightness due to magnetically suppressed convection. As the planet moves across a spot, it temporarily occults an area emitting less flux than the surrounding photosphere, producing a net increase in observed flux relative to a spot-free transit—manifesting as an “upward blip” in the light curve (Southworth et al., 2016). The geometry is parameterized in a right-handed coordinate system, with planet center (x_p(t), y_p(t)) determined by $a/R_\star$ (scaled semi-major axis), impact parameter $b$, and orbital phase $\phi$, and spot by angular radius $r_\mathrm{spot}$, stellar latitude $\theta$, and longitude $\phi$. The instantaneous overlap area $A_\mathrm{overlap}(t)$ is computed using standard formulae for circle–circle intersections.

2. Modeling SCEs and Spot Parameter Extraction

The transit light curve in the absence of spots is modeled as $F_0(t) = 1 - \delta(t)$, with $\delta(t)$ from analytic transit models including quadratic limb-darkening ($$I_\mathrm{phot}(\mu) = I_0 [1 - u_1 (1 - \mu) - u_2 (1 - \mu)^2]$$). When the planet occults a spot of local specific intensity $I_\mathrm{spot}$, the perturbation is $$\Delta F_\mathrm{spot}(t) = [I_\mathrm{phot}(\mu) - I_\mathrm{spot}(\mu)] A_\mathrm{overlap}(t) / F_\mathrm{tot}$$, or equivalently $(1-c) A_\mathrm{overlap} / A_\star$ for contrast $c = I_\mathrm{spot} / I_\mathrm{phot}$ (Southworth et al., 2016). Fitting the shape of the anomaly recovers:

• Spot radius $r_\mathrm{spot}$ from duration,
• Spot contrast $c$ from amplitude,
• Spot projected coordinates $(x_\mathrm{spot}, y_\mathrm{spot})$ from anomaly timing relative to mid-transit.

These map to physical spot latitude and longitude via $$\mu = \sqrt{1 - x_\mathrm{spot}^2 - y_\mathrm{spot}^2}$$, $\theta = \arcsin(y_\mathrm{spot}/\mu)$, $\phi = \arctan2(x_\mathrm{spot}/\mu)$ (Southworth et al., 2016). MCMC fitting frameworks (GEMC, emcee, or PlanetPack) are employed to sample posterior distributions and extract uncertainties (Baluev et al., 2021).

3. Spin-Orbit Architecture: Rotation and Obliquity from Multiple SCEs

Tracking recurring SCEs across successive transits enables precise measurement of stellar rotation ($P_\mathrm{rot}$) and spin–orbit misalignment ($\lambda$, projected obliquity). The longitudinal drift of a spot between transits separated by $\Delta t$ is $\Delta\phi = 2\pi \Delta t / P_\mathrm{rot}$, yielding $P_\mathrm{rot} = 2\pi \Delta t / \Delta\phi$ (Southworth et al., 2016). For multiple spot crossings at distinct longitudes, simultaneous fitting constrains the inclination $i_\star$ of the stellar spin axis and yields the true 3D obliquity ($\psi$) via $$\cos\psi = \cos i_o \cos i_\star + \sin i_o \sin i_\star \cos\lambda$$, with $i_o \simeq 90^\circ$ in transiting systems. Empirically, SCE analysis has delivered $\lambda$ for 12 systems and $\psi$ for one (WASP-52), with typical uncertainties of $3$–$15^\circ$ on $\lambda$ and $20$–$50^\circ$ on $\psi$ (Southworth et al., 2016, Mancini et al., 2016).

4. Population Statistics, Detection Limits, and Event Frequency

Rigorous statistical pipelines (e.g., PlanetPack (Baluev et al., 2021)) have detected SCEs in $\simeq$7% of transits analyzed across 26 monitored exoplanet systems. Events are modeled as Gaussian anomalies, with amplitude $K$ and width $\sigma$ shown empirically to correlate (slope $a = 0.29 \pm 0.10$ in log–log regression), consistent with spot size scaling (Baluev et al., 2021). The spot/facula crossing ratio $\sim 2:1$, not explained solely by noise, reflects physical asymmetry in magnetically active regions. Simulations with PRISM (Tregloan-Reed et al., 2019) indicate that for TESS 2 min cadence light curves, spot anomalies can be detected down to radii of $1900$–$15900$ km in cool dwarfs, with smallest anomalies ($\Delta F_\mathrm{spot,min} = 0.00015 \pm 0.00001$) at $k = R_p/R_\star = 0.082 \pm 0.004$.

Host type	$r_\mathrm{spot,min}$ [km]	$\Delta F_\mathrm{spot,min}$	$k_\mathrm{min}$
M4V	$4900 \pm 1700$	$0.00015 \pm 0.00001$	$0.082 \pm 0.004$
M1V	$13800 \pm 6000$	$0.00015 \pm 0.00001$	$0.082 \pm 0.004$
K5V	$15900 \pm 6800$	$0.00015 \pm 0.00001$	$0.082 \pm 0.004$

Event frequency is insufficient to explain transit timing jitter ($\sim$1–3 min) in active stars, as spot crossings are detected in only a small fraction of possible transits (Baluev et al., 2021).

5. Applications to Stellar Spot Evolution, Differential Rotation, and Mapping

Continuous photometric monitoring (e.g., from Kepler) of SCEs enables joint modeling of in- and out-of-transit features, breaking degeneracies between spot size, latitude, and contrast (Davenport et al., 2014). For planetary hosts, SCEs probe small-scale features at fixed latitude, facilitating precise latitude and size measurements, and constraining spot decay rates and differential rotation (e.g., Kepler 17: $P_\mathrm{rot}=11.90\pm0.02$ d, spot lifetimes $\tau_\mathrm{diff}\sim30\pm10$ d, differential rotation $\Delta\omega=0.055\pm0.015$ rad d$^{-1}$). For non-transiting stars observed by Kepler, spot tracking yields extremely slow differential rotation rates ($\Delta\omega=0.0042\pm0.0008$ rad d$^{-1}$ in GJ 1243) (Davenport et al., 2014). In Qatar-2, SCEs spanning 58 d allow empirical lower limits on spot lifetime and precise measurement of stellar rotation ($P_\mathrm{rot}=18.03\pm0.24$ d), supporting the recurrence and longevity of active regions (Močnik et al., 2016).

6. SCEs Beyond Transits: Microlensing Anomalies

In gravitational microlensing, a rotating source star’s surface inhomogeneities (spots) can be crossed by the lensing caustic, producing localized light-curve anomalies. These are modeled as modifications to the limb-darkened intensity profile, with spot contrast $c$ and angular radius $\rho_\mathrm{spot}$ (Giordano et al., 2015). The spot-induced magnification excess for a single lens is $$\Delta\mu_i \simeq (c_i–1)[A_\mathrm{center}–A_\mathrm{base}](\rho_i/\rho)^2$$, with duration $\Delta t_i \simeq 2\rho_i t_E/v_\mathrm{eff}$. Event rates for OGLE-IV–like monitoring are low ($N_s \simeq 4.7$ yr$^{-1}$), limited by caustic geometry and fraction of magnetically active giants. Detectability requires mmag precision and high cadence ($\lesssim$minutes) during caustic crossings (Giordano et al., 2015).

7. Methodological Limitations and Practical Considerations

SCE exploitation is limited by several factors:

• Spot longevity: must persist coherently through $\Delta t$ between observed crossings.
• Rigid rotation assumption can be violated by differential rotation; sometimes a latitude-dependent term $$\Omega(\theta)=\Omega_\mathrm{eq}-\Delta\Omega\sin^2\theta$$ is included.
• Spot uniqueness: ambiguous identification arises if multiple spots exist at similar latitudes.
• Large spot size can blur anomaly timing, introducing uncertainties ($\gtrsim$0.05 in normalized transit units).
• Degeneracies among spot latitude, radius, and contrast complicate fitting; longitude is most tightly constrained, with 80% of high-SNR SCEs recovered within $1^\circ$ (Murray et al., 4 Nov 2025).
• For low-contrast ($<5\%$) or small ($<2\%$ covering fraction) spots, fitting may over-correct the transit depth bias compared to simple masking.
• SCEs can inflate recovered transit depth uncertainties, especially at high precisions typical of JWST-like data (Murray et al., 4 Nov 2025).

8. Impact on Systematics and Transmission Spectroscopy

SCEs have direct implications for transmission spectra: occulted spots imprint wavelength-dependent “bumps” which, if uncorrected, mimic planetary atmospheric features or contaminate inferred opacities (Bruno et al., 2021, Mancini et al., 2016). Spectrophotometric analysis allows retrieval of spot temperatures to $100$–$300$ K accuracy using JWST NIRSpec/Prism and NIRCam, provided sufficient contrast and brightness ($K \leq 12.5$ mag for K/M hosts). For transmission modeling, the amplitude of the SCE ($\Delta D(\lambda)$) is proportional to $(1-C(\lambda))A_{\rm spot}/\pi R_*^2$, with $C(\lambda)$ the spot–photosphere contrast ratio (Bruno et al., 2021). Empirical results show that neither modeled nor unocculted SCEs introduce detectable slopes or features in the WASP-52 b transmission spectrum beyond uncertainties (Mancini et al., 2016). SCE fitting is essential when unocculted-spot contamination exceeds $1.3\%$, or SCE SNR $\geq 4$, but masking is preferred for small/low-contrast cases (Murray et al., 4 Nov 2025).

9. Catalogued Systems and Spin-Orbit Alignment Results

SCE analyses have provided spin–orbit architecture constraints for twelve systems (including HAT-P-11, HATS-2, Kepler-17, WASP-4, WASP-6, WASP-19, WASP-41, WASP-52, WASP-85, and Qatar-2), with measured $\lambda$ values predominantly consistent with well-aligned orbits, and one system (WASP-52) with a true obliquity measurement ($\psi = 20^\circ \pm 50^\circ$) (Southworth et al., 2016, Mancini et al., 2016). For TOI-3884 b, multi-epoch SCE variation modeling reveals a highly misaligned orbit ($\psi_\star = 76.1^{+2.2}_{-2.6}\,\!^\circ$) about a nearly pole-on ($i_\star = 24.4^{+1.8}_{-1.4}\,\!^\circ$) M dwarf with a persistent polar spot ($r_\mathrm{spot}=33.1^{+2.0}_{-1.9}\,\!^\circ$ for $>7$ years) (Tamburo et al., 13 Jun 2025).

In summary, SCEs are a powerful diagnostic of both star–planet alignment and stellar surface inhomogeneities, especially on cool, magnetically active host stars. The available catalog demonstrates low obliquity and long-lived spot recurrence in most cool-star systems, supporting tidal-realignment theories, and providing critical input for exoplanetary system architecture models and transmission spectroscopy contamination correction.