Transit Light Source Effect in Exoplanet Studies

• Transit-light-source effect is the bias in transit observations caused by differing spectral contributions from active stellar regions like cool spots and hot faculae.
• This effect alters transit depth measurements, leading to systematic errors in inferred planetary radii, densities, and atmospheric signals, sometimes reaching biases of several percent.
• Researchers mitigate these biases by combining multi-wavelength photometry, high-resolution spectroscopy, and detailed stellar surface models to correct for heterogeneity.

The transit-light-source effect is the bias introduced into transit photometry and transmission spectroscopy when the spectrum of the portion of the stellar disk occulted by a planet differs from the disk-integrated out-of-transit stellar spectrum. In that circumstance, the stellar light source sampled by the planetary transit chord is not identical to the apparent source spectrum inferred from the unresolved star, and stellar photospheric heterogeneity—most notably cool spots and hot faculae—can imprint wavelength-dependent structure on the measured transit depth. In the literature summarized here, the effect is treated as a contamination of the observed transit signal that can mimic, amplify, or mask atmospheric features and can also bias inferred planetary radii, densities, transit durations, and transit timings (Rackham et al., 2017, Rackham et al., 2018).

1. Definition and phenomenology

In transmission spectroscopy, the measured quantity is the in-minus-out transit depth,

$$D_{\lambda,\rm obs}=\frac{F_{\rm oot}(\lambda)-F_{\rm in\ transit}(\lambda)}{F_{\rm oot}(\lambda)}.$$

The standard interpretation assumes that the occulted stellar chord has the same spectrum as the disk-integrated stellar emission. The transit-light-source effect arises when this assumption fails because the stellar photosphere is heterogeneous. If unocculted spots or faculae alter the out-of-transit baseline, the apparent transit depth becomes a convolution of planetary opacity and stellar contamination rather than a purely planetary signal (Rackham et al., 2017).

Two observational regimes are distinguished in the source material. Unocculted heterogeneities modify the out-of-transit reference spectrum and therefore bias the measured transit depth as a function of wavelength. Occulted heterogeneities produce localized anomalies during transit, such as the “bump” generated when a planet crosses a dark spot. Oshagh et al. showed that such spot anomalies can lead to the estimate of a planet radius that is 4% smaller than the real value, that the effects on the transit duration can also be of the order of 4%, and that transit timing variations with signal amplitudes of 200 seconds can be produced when the spot is completely dark and as large as the largest Sun spot (Oshagh et al., 2013).

A central misconception addressed in this literature is that low rotational variability implies negligible contamination. The M-dwarf analysis explicitly finds that large ranges of spot and faculae covering fractions are consistent with observations and that corrections assuming a linear relation between variability amplitude and covering fractions generally underestimate the stellar contamination (Rackham et al., 2017). A related misconception is that a non-detection of contamination in transit light curves excludes stellar activity. In the TRAPPIST-1 Spitzer analysis, none of the planets show statistically significant evidence for self-contamination, but small-scale magnetic activity could still be lurking in the transit photometry undetected (Morris et al., 2018).

2. Mathematical description

A standard formalism writes the disk-integrated stellar spectrum as a flux-weighted sum of immaculate photosphere, cool spots, and hot faculae:

$$F_{\rm obs}(\lambda)=(1-f_{\rm spot}-f_{\rm fac})\,F_{\rm phot}(\lambda)+f_{\rm spot}\,F_{\rm spot}(\lambda)+f_{\rm fac}\,F_{\rm fac}(\lambda).$$

If the transit chord is assumed to sample only the immaculate photosphere, the true atmospheric transmission depth $D_\lambda$ is related to the observed depth by

$D_{\lambda,\rm obs}=D_\lambda\,\epsilon_\lambda,$

with contamination factor

$$\epsilon_\lambda= \Bigl[1-f_{\rm spot}\bigl(1-\tfrac{F_{\rm spot}}{F_{\rm phot}}\bigr)-f_{\rm fac}\bigl(1-\tfrac{F_{\rm fac}}{F_{\rm phot}}\bigr)\Bigr]^{-1}.$$

The contamination-induced change in transit depth is then

$$\Delta D(\lambda)=D_{\lambda,\rm obs}-D_\lambda=D_\lambda(\epsilon_\lambda-1).$$

This formulation is the basis for the M-dwarf and FGK-dwarf contamination spectra reported by Rackham and collaborators (Rackham et al., 2017, Rackham et al., 2018).

Waalkes et al. express the AU Mic case in closely related form:

$$D_{\rm obs}(\lambda)=\left(\frac{R_p}{R_*}\right)^2+\Delta D_{\rm atm}(\lambda)+\Delta D_{\rm spot}(\lambda),$$

and, for unocculted cool spots with no spot crossing so that $f_{\rm spot,tra}=0$,

$$\Delta D_{\rm spot}(\lambda)= \left(\frac{R_p}{R_*}\right)^2 \left[ \frac{1}{(1-f_{\rm spot})+f_{\rm spot}\,I_{\rm spot}(\lambda)/I_{\rm phot}(\lambda)}-1 \right].$$

In the small-contrast limit,

$$\Delta D_{\rm spot}(\lambda)\simeq \left(\frac{R_p}{R_*}\right)^2 f_{\rm spot} \frac{I_{\rm spot}(\lambda)-I_{\rm phot}(\lambda)}{I_{\rm total}(\lambda)},$$

where

$$F_{\rm obs}(\lambda)=(1-f_{\rm spot}-f_{\rm fac})\,F_{\rm phot}(\lambda)+f_{\rm spot}\,F_{\rm spot}(\lambda)+f_{\rm fac}\,F_{\rm fac}(\lambda).$$0

For AU Mic, the step-by-step calculation adopts $$F_{\rm obs}(\lambda)=(1-f_{\rm spot}-f_{\rm fac})\,F_{\rm phot}(\lambda)+f_{\rm spot}\,F_{\rm spot}(\lambda)+f_{\rm fac}\,F_{\rm fac}(\lambda).$$1 and $$F_{\rm obs}(\lambda)=(1-f_{\rm spot}-f_{\rm fac})\,F_{\rm phot}(\lambda)+f_{\rm spot}\,F_{\rm spot}(\lambda)+f_{\rm fac}\,F_{\rm fac}(\lambda).$$2 from PHOENIX model spectra, computes

$$F_{\rm obs}(\lambda)=(1-f_{\rm spot}-f_{\rm fac})\,F_{\rm phot}(\lambda)+f_{\rm spot}\,F_{\rm spot}(\lambda)+f_{\rm fac}\,F_{\rm fac}(\lambda).$$3

assumes a spot-free chord, and evaluates the resulting contamination from optical through near-IR wavelengths (Waalkes et al., 2023).

The same literature makes explicit that contamination propagates into bulk-parameter inferences. Because $$F_{\rm obs}(\lambda)=(1-f_{\rm spot}-f_{\rm fac})\,F_{\rm phot}(\lambda)+f_{\rm spot}\,F_{\rm spot}(\lambda)+f_{\rm fac}\,F_{\rm fac}(\lambda).$$4, a multiplicative contamination factor produces

$$F_{\rm obs}(\lambda)=(1-f_{\rm spot}-f_{\rm fac})\,F_{\rm phot}(\lambda)+f_{\rm spot}\,F_{\rm spot}(\lambda)+f_{\rm fac}\,F_{\rm fac}(\lambda).$$5

and, since $$F_{\rm obs}(\lambda)=(1-f_{\rm spot}-f_{\rm fac})\,F_{\rm phot}(\lambda)+f_{\rm spot}\,F_{\rm spot}(\lambda)+f_{\rm fac}\,F_{\rm fac}(\lambda).$$6,

$$F_{\rm obs}(\lambda)=(1-f_{\rm spot}-f_{\rm fac})\,F_{\rm phot}(\lambda)+f_{\rm spot}\,F_{\rm spot}(\lambda)+f_{\rm fac}\,F_{\rm fac}(\lambda).$$7

These relations are used to quantify radius and density shifts for small planets around active stars (Rackham et al., 2017).

3. Connection to stellar heterogeneity and variability

The stellar quantities that enter transit-light-source calculations are the spectra and covering fractions of the distinct photospheric components. In the formalism for M dwarfs and FGK dwarfs, the quiet photosphere spectrum is denoted $$F_{\rm obs}(\lambda)=(1-f_{\rm spot}-f_{\rm fac})\,F_{\rm phot}(\lambda)+f_{\rm spot}\,F_{\rm spot}(\lambda)+f_{\rm fac}\,F_{\rm fac}(\lambda).$$8, the cool spots spectrum $$F_{\rm obs}(\lambda)=(1-f_{\rm spot}-f_{\rm fac})\,F_{\rm phot}(\lambda)+f_{\rm spot}\,F_{\rm spot}(\lambda)+f_{\rm fac}\,F_{\rm fac}(\lambda).$$9, and the hot faculae spectrum $D_\lambda$0, parameterized by their temperatures $D_\lambda$1, $D_\lambda$2, and $D_\lambda$3 (Rackham et al., 2018).

A recurrent result is that rotational variability traces only the non-axisymmetric component of surface heterogeneity. Forward models of randomly placed active regions give

$D_\lambda$4

rather than a linear scaling. For M dwarfs, typical fits over M0–M9 yield $D_\lambda$5, and the analysis states that a naive linear assumption systematically underestimates $D_\lambda$6 because it ignores cancellation among multiple spots (Rackham et al., 2017). For FGK dwarfs, the Kepler-band relation is summarized as $D_\lambda$7 with fitted $D_\lambda$8 $D_\lambda$9 for F5V–K9V dwarfs (Rackham et al., 2018).

The inferred covering fractions depend strongly on spectral type and activity level. For median Kepler variability amplitudes, spots-only models for FGK dwarfs yield mean $D_{\lambda,\rm obs}=D_\lambda\,\epsilon_\lambda,$0 for F dwarfs, increasing smoothly to $D_{\lambda,\rm obs}=D_\lambda\,\epsilon_\lambda,$1 for late-K dwarfs. Spots-plus-faculae models with a 10:1 facula-to-spot area ratio give $D_{\lambda,\rm obs}=D_\lambda\,\epsilon_\lambda,$2 and $D_{\lambda,\rm obs}=D_\lambda\,\epsilon_\lambda,$3 (Rackham et al., 2018). For M dwarfs, the TRAPPIST-1 case study finds that $D_{\lambda,\rm obs}=D_\lambda\,\epsilon_\lambda,$4 I+z-band variability is consistent with $D_{\lambda,\rm obs}=D_\lambda\,\epsilon_\lambda,$5 and $D_{\lambda,\rm obs}=D_\lambda\,\epsilon_\lambda,$6, versus the naive linear-scaling estimate $D_{\lambda,\rm obs}=D_\lambda\,\epsilon_\lambda,$7 (Rackham et al., 2017).

This body of work therefore separates photometric variability amplitude from absolute coverage fraction. A plausible implication is that a low-amplitude light curve does not uniquely determine a small contamination spectrum, because many spot and facula configurations can produce comparable rotational modulation while yielding materially different disk-integrated stellar spectra.

4. Observational signatures and diagnostic frameworks

The most direct signatures of occulted heterogeneities are localized distortions in transit light curves. In Oshagh et al., the anomalous bump during a spot crossing is written as the difference between blocking a normal photospheric patch and blocking the dimmer spot. Their simulations show that, for a dark spot with $D_{\lambda,\rm obs}=D_\lambda\,\epsilon_\lambda,$8 and a small planet $D_{\lambda,\rm obs}=D_\lambda\,\epsilon_\lambda,$9, the inferred planet radius can be underestimated by about 4% when the overlap occurs near disk center. They also report numerically up to

$$\epsilon_\lambda= \Bigl[1-f_{\rm spot}\bigl(1-\tfrac{F_{\rm spot}}{F_{\rm phot}}\bigr)-f_{\rm fac}\bigl(1-\tfrac{F_{\rm fac}}{F_{\rm phot}}\bigr)\Bigr]^{-1}.$$0

and a linear fit for spot-induced timing shifts,

$$\epsilon_\lambda= \Bigl[1-f_{\rm spot}\bigl(1-\tfrac{F_{\rm spot}}{F_{\rm phot}}\bigr)-f_{\rm fac}\bigl(1-\tfrac{F_{\rm fac}}{F_{\rm phot}}\bigr)\Bigr]^{-1}.$$1

with $$\epsilon_\lambda= \Bigl[1-f_{\rm spot}\bigl(1-\tfrac{F_{\rm spot}}{F_{\rm phot}}\bigr)-f_{\rm fac}\bigl(1-\tfrac{F_{\rm fac}}{F_{\rm phot}}\bigr)\Bigr]^{-1}.$$2 for $$\epsilon_\lambda= \Bigl[1-f_{\rm spot}\bigl(1-\tfrac{F_{\rm spot}}{F_{\rm phot}}\bigr)-f_{\rm fac}\bigl(1-\tfrac{F_{\rm fac}}{F_{\rm phot}}\bigr)\Bigr]^{-1}.$$3, $$\epsilon_\lambda= \Bigl[1-f_{\rm spot}\bigl(1-\tfrac{F_{\rm spot}}{F_{\rm phot}}\bigr)-f_{\rm fac}\bigl(1-\tfrac{F_{\rm fac}}{F_{\rm phot}}\bigr)\Bigr]^{-1}.$$4 for $$\epsilon_\lambda= \Bigl[1-f_{\rm spot}\bigl(1-\tfrac{F_{\rm spot}}{F_{\rm phot}}\bigr)-f_{\rm fac}\bigl(1-\tfrac{F_{\rm fac}}{F_{\rm phot}}\bigr)\Bigr]^{-1}.$$5, and $$\epsilon_\lambda= \Bigl[1-f_{\rm spot}\bigl(1-\tfrac{F_{\rm spot}}{F_{\rm phot}}\bigr)-f_{\rm fac}\bigl(1-\tfrac{F_{\rm fac}}{F_{\rm phot}}\bigr)\Bigr]^{-1}.$$6 for $$\epsilon_\lambda= \Bigl[1-f_{\rm spot}\bigl(1-\tfrac{F_{\rm spot}}{F_{\rm phot}}\bigr)-f_{\rm fac}\bigl(1-\tfrac{F_{\rm fac}}{F_{\rm phot}}\bigr)\Bigr]^{-1}.$$7 (Oshagh et al., 2013).

For unocculted spots, the same source gives the baseline-dilution relation

$$\epsilon_\lambda= \Bigl[1-f_{\rm spot}\bigl(1-\tfrac{F_{\rm spot}}{F_{\rm phot}}\bigr)-f_{\rm fac}\bigl(1-\tfrac{F_{\rm fac}}{F_{\rm phot}}\bigr)\Bigr]^{-1}.$$8

which implies a positive bias in $$\epsilon_\lambda= \Bigl[1-f_{\rm spot}\bigl(1-\tfrac{F_{\rm spot}}{F_{\rm phot}}\bigr)-f_{\rm fac}\bigl(1-\tfrac{F_{\rm fac}}{F_{\rm phot}}\bigr)\Bigr]^{-1}.$$9 and a spurious color-dependent slope in transmission spectra if the spot contrast $$\Delta D(\lambda)=D_{\lambda,\rm obs}-D_\lambda=D_\lambda(\epsilon_\lambda-1).$$0 varies with wavelength (Oshagh et al., 2013). This is the regime most often associated with false spectral features in atmospheric studies.

A complementary diagnostic is the self-contamination method applied by Morris et al. to TRAPPIST-1. In that framework, $$\Delta D(\lambda)=D_{\lambda,\rm obs}-D_\lambda=D_\lambda(\epsilon_\lambda-1).$$1 is the true planet-to-star radius ratio inferred from ingress/egress, and $$\Delta D(\lambda)=D_{\lambda,\rm obs}-D_\lambda=D_\lambda(\epsilon_\lambda-1).$$2 is the apparent radius ratio derived from the observed transit depth. The method tests whether the mean surface brightness inside the transit chord differs from the mean brightness of the rest of the visible stellar disk. Bright regions inside the chord make the planet look smaller; dark regions make it look larger (Morris et al., 2018).

Applied to 2018 Spitzer photometry, the self-contamination analysis found that all seven TRAPPIST-1 planets yielded posterior distributions with $$\Delta D(\lambda)=D_{\lambda,\rm obs}-D_\lambda=D_\lambda(\epsilon_\lambda-1).$$3 within uncertainties, and none show a statistically significant $$\Delta D(\lambda)=D_{\lambda,\rm obs}-D_\lambda=D_\lambda(\epsilon_\lambda-1).$$4. The most constraining cases were planet b at $$\Delta D(\lambda)=D_{\lambda,\rm obs}-D_\lambda=D_\lambda(\epsilon_\lambda-1).$$5m, with $$\Delta D(\lambda)=D_{\lambda,\rm obs}-D_\lambda=D_\lambda(\epsilon_\lambda-1).$$6, and planet g at $$\Delta D(\lambda)=D_{\lambda,\rm obs}-D_\lambda=D_\lambda(\epsilon_\lambda-1).$$7m, with $$\Delta D(\lambda)=D_{\lambda,\rm obs}-D_\lambda=D_\lambda(\epsilon_\lambda-1).$$8; in every case $$\Delta D(\lambda)=D_{\lambda,\rm obs}-D_\lambda=D_\lambda(\epsilon_\lambda-1).$$9 and $$D_{\rm obs}(\lambda)=\left(\frac{R_p}{R_*}\right)^2+\Delta D_{\rm atm}(\lambda)+\Delta D_{\rm spot}(\lambda),$$0 agree at the $$D_{\rm obs}(\lambda)=\left(\frac{R_p}{R_*}\right)^2+\Delta D_{\rm atm}(\lambda)+\Delta D_{\rm spot}(\lambda),$$1 level. However, STSP spot-occultation simulations with $$D_{\rm obs}(\lambda)=\left(\frac{R_p}{R_*}\right)^2+\Delta D_{\rm atm}(\lambda)+\Delta D_{\rm spot}(\lambda),$$2 K spot contrast showed that spots with $$D_{\rm obs}(\lambda)=\left(\frac{R_p}{R_*}\right)^2+\Delta D_{\rm atm}(\lambda)+\Delta D_{\rm spot}(\lambda),$$3 would produce only $$D_{\rm obs}(\lambda)=\left(\frac{R_p}{R_*}\right)^2+\Delta D_{\rm atm}(\lambda)+\Delta D_{\rm spot}(\lambda),$$4 features in the b-planet light curve and thus could hide undetected (Morris et al., 2018).

5. Host-star regimes and representative systems

For M dwarfs, the predicted contamination spectra can be large compared with atmospheric signals. Using PHOENIX/DRIFT-PHOENIX spectra and Monte Carlo spot models anchored to a 1% I-band variability, Rackham et al. found that in a giant-spot scenario the mean contamination is modest, whereas in a solar-like spot scenario spot-only models yield $$D_{\rm obs}(\lambda)=\left(\frac{R_p}{R_*}\right)^2+\Delta D_{\rm atm}(\lambda)+\Delta D_{\rm spot}(\lambda),$$5 with range 2.6–16%, and spots-plus-faculae yield $$D_{\rm obs}(\lambda)=\left(\frac{R_p}{R_*}\right)^2+\Delta D_{\rm atm}(\lambda)+\Delta D_{\rm spot}(\lambda),$$6 with range 2.9–16%. At near-visual and near-IR wavelengths of common molecular bands, these stellar-induced changes can be up to 10 times larger than a 5-scale-height planetary feature in an Earth-like atmosphere (Rackham et al., 2017).

The TRAPPIST-1 system is treated as a specific M-dwarf case. The associated stellar contamination signals alter transit depths of the TRAPPIST-1 planets at wavelengths of interest for planetary atmospheric species by roughly 1–15 times the strength of planetary features. Integrating over Spitzer/IRAC $$D_{\rm obs}(\lambda)=\left(\frac{R_p}{R_*}\right)^2+\Delta D_{\rm atm}(\lambda)+\Delta D_{\rm spot}(\lambda),$$7m gives $$D_{\rm obs}(\lambda)=\left(\frac{R_p}{R_*}\right)^2+\Delta D_{\rm atm}(\lambda)+\Delta D_{\rm spot}(\lambda),$$8 and hence $$D_{\rm obs}(\lambda)=\left(\frac{R_p}{R_*}\right)^2+\Delta D_{\rm atm}(\lambda)+\Delta D_{\rm spot}(\lambda),$$9 (Rackham et al., 2017). The later Spitzer self-contamination study did not find convincing evidence for contamination in the transit chords, but it explicitly noted that active regions at higher latitudes would not be crossed and that very small spots could still be present undetected (Morris et al., 2018).

For broadly Sun-like FGK dwarfs, the effect is weaker but not negligible. Relative to M dwarfs, the typical variabilities of FGK dwarfs imply lower spot covering fractions, though they generally increase with later spectral types, from $f_{\rm spot,tra}=0$0 for F dwarfs to 2–4% for late-K dwarfs. The largest signatures occur in the UV and visual: spots-only models produce positive visual slopes, and spots-plus-faculae models produce strongly decreasing blueward slopes with $f_{\rm spot,tra}=0$1 in the UV/blue. For typically active FGK dwarfs, band-averaged offsets at molecular bands for CH$f_{\rm spot,tra}=0$2, CO, CO$f_{\rm spot,tra}=0$3, H$f_{\rm spot,tra}=0$4O, N$f_{\rm spot,tra}=0$5O, O$f_{\rm spot,tra}=0$6, and O$f_{\rm spot,tra}=0$7 are not detectable, but unocculted faculae in K dwarfs can appreciably alter transit depths around the Na D doublet, and stellar TiO/VO features are potentially detectable for typically active late-K dwarfs (Rackham et al., 2018).

AU Mic provides a detailed quantified example of the effect in a young, active system. AU Mic is described as an active 24 Myr pre-main sequence M dwarf at $f_{\rm spot,tra}=0$8 pc with a rotation period of 4.86 days, hosting the transiting planets AU Mic b and c on 8.5 and 18.9 day periods. From multi-color photometry and high-resolution spectra, Waalkes et al. parameterized the integrated stellar spectrum by three spectral components and their coverage fractions. They found at least two robust components: a $f_{\rm spot,tra}=0$9 K ambient photosphere and cool spots at $$\Delta D_{\rm spot}(\lambda)= \left(\frac{R_p}{R_*}\right)^2 \left[ \frac{1}{(1-f_{\rm spot})+f_{\rm spot}\,I_{\rm spot}(\lambda)/I_{\rm phot}(\lambda)}-1 \right].$$0 K covering $$\Delta D_{\rm spot}(\lambda)= \left(\frac{R_p}{R_*}\right)^2 \left[ \frac{1}{(1-f_{\rm spot})+f_{\rm spot}\,I_{\rm spot}(\lambda)/I_{\rm phot}(\lambda)}-1 \right].$$1 of the surface, increasing and decreasing by 5% from the average throughout a rotation. They also detected a third flux component with filling factor less than 0.5% and a largely uncertain temperature, attributed to flare flux not entirely omitted in the time-averaged spectra. The 2-temperature and 3-temperature models agreed strongly, with the 2-T model giving $$\Delta D_{\rm spot}(\lambda)= \left(\frac{R_p}{R_*}\right)^2 \left[ \frac{1}{(1-f_{\rm spot})+f_{\rm spot}\,I_{\rm spot}(\lambda)/I_{\rm phot}(\lambda)}-1 \right].$$2 K and $$\Delta D_{\rm spot}(\lambda)= \left(\frac{R_p}{R_*}\right)^2 \left[ \frac{1}{(1-f_{\rm spot})+f_{\rm spot}\,I_{\rm spot}(\lambda)/I_{\rm phot}(\lambda)}-1 \right].$$3, and the 3-T model giving $$\Delta D_{\rm spot}(\lambda)= \left(\frac{R_p}{R_*}\right)^2 \left[ \frac{1}{(1-f_{\rm spot})+f_{\rm spot}\,I_{\rm spot}(\lambda)/I_{\rm phot}(\lambda)}-1 \right].$$4 K and $$\Delta D_{\rm spot}(\lambda)= \left(\frac{R_p}{R_*}\right)^2 \left[ \frac{1}{(1-f_{\rm spot})+f_{\rm spot}\,I_{\rm spot}(\lambda)/I_{\rm phot}(\lambda)}-1 \right].$$5 (Waalkes et al., 2023).

Using these parameters and assuming a spot-free transit chord, the AU Mic contamination spectrum was evaluated from optical $$\Delta D_{\rm spot}(\lambda)= \left(\frac{R_p}{R_*}\right)^2 \left[ \frac{1}{(1-f_{\rm spot})+f_{\rm spot}\,I_{\rm spot}(\lambda)/I_{\rm phot}(\lambda)}-1 \right].$$6 through near-IR $$\Delta D_{\rm spot}(\lambda)= \left(\frac{R_p}{R_*}\right)^2 \left[ \frac{1}{(1-f_{\rm spot})+f_{\rm spot}\,I_{\rm spot}(\lambda)/I_{\rm phot}(\lambda)}-1 \right].$$7. For AU Mic b, $$\Delta D_{\rm spot}(\lambda)= \left(\frac{R_p}{R_*}\right)^2 \left[ \frac{1}{(1-f_{\rm spot})+f_{\rm spot}\,I_{\rm spot}(\lambda)/I_{\rm phot}(\lambda)}-1 \right].$$8, so $$\Delta D_{\rm spot}(\lambda)= \left(\frac{R_p}{R_*}\right)^2 \left[ \frac{1}{(1-f_{\rm spot})+f_{\rm spot}\,I_{\rm spot}(\lambda)/I_{\rm phot}(\lambda)}-1 \right].$$9 ppm. The resulting $$\Delta D_{\rm spot}(\lambda)\simeq \left(\frac{R_p}{R_*}\right)^2 f_{\rm spot} \frac{I_{\rm spot}(\lambda)-I_{\rm phot}(\lambda)}{I_{\rm total}(\lambda)},$$0 is approximately 1200 ppm at $$\Delta D_{\rm spot}(\lambda)\simeq \left(\frac{R_p}{R_*}\right)^2 f_{\rm spot} \frac{I_{\rm spot}(\lambda)-I_{\rm phot}(\lambda)}{I_{\rm total}(\lambda)},$$1m, approximately 750 ppm in the HST/WFC3 band $$\Delta D_{\rm spot}(\lambda)\simeq \left(\frac{R_p}{R_*}\right)^2 f_{\rm spot} \frac{I_{\rm spot}(\lambda)-I_{\rm phot}(\lambda)}{I_{\rm total}(\lambda)},$$2, and approximately 300 ppm by $$\Delta D_{\rm spot}(\lambda)\simeq \left(\frac{R_p}{R_*}\right)^2 f_{\rm spot} \frac{I_{\rm spot}(\lambda)-I_{\rm phot}(\lambda)}{I_{\rm total}(\lambda)},$$3m. The reported implication is that AU Mic b’s nominal transit depth of about 1875 ppm suffers an unocculted-spot bias equal to 25–75% of the signal, while AU Mic c’s depth of about 980 ppm suffers a bias of 40–75%; without spot correction planetary radii will be overestimated by about 0.6–1.7 $$\Delta D_{\rm spot}(\lambda)\simeq \left(\frac{R_p}{R_*}\right)^2 f_{\rm spot} \frac{I_{\rm spot}(\lambda)-I_{\rm phot}(\lambda)}{I_{\rm total}(\lambda)},$$4 (Waalkes et al., 2023).

6. Consequences for inference and mitigation

The principal consequence of the transit-light-source effect is that stellar contamination can rival or exceed the planetary atmospheric signal of interest. For rocky planets around M dwarfs, the contamination can be more than $$\Delta D_{\rm spot}(\lambda)\simeq \left(\frac{R_p}{R_*}\right)^2 f_{\rm spot} \frac{I_{\rm spot}(\lambda)-I_{\rm phot}(\lambda)}{I_{\rm total}(\lambda)},$$5 larger than transit depth changes expected for atmospheric features, and in the AU Mic analysis planetary atmospheric features of approximately 50 ppm per H$$\Delta D_{\rm spot}(\lambda)\simeq \left(\frac{R_p}{R_*}\right)^2 f_{\rm spot} \frac{I_{\rm spot}(\lambda)-I_{\rm phot}(\lambda)}{I_{\rm total}(\lambda)},$$6 scale height are stated to be much smaller than the spot-induced variations of hundreds of ppm (Rackham et al., 2017, Waalkes et al., 2023). In such regimes, apparent spectral slopes, molecular bands, and line features cannot be assigned to planetary atmospheres without simultaneous constraints on stellar heterogeneity.

The effect also propagates into planet bulk properties. Rackham et al. state that stellar spectral contamination can lead to systematic errors in radius and therefore the derived density of small planets, and in the TRAPPIST-1 case the densities may be underestimated by a few percent, up to approximately 10%, biasing inferences of volatile content (Rackham et al., 2017). Oshagh et al. add that spot anomalies can bias transit duration and timing if each individual transit is fitted without an explicit activity model, and they conclude that including more free parameters in the fitting procedure to measure the transit time of each individual transit will not produce accurate results in the case of active stars (Oshagh et al., 2013).

The mitigation strategies reported across these studies are consistent. They include contemporaneous multi-band or multi-color photometric monitoring to constrain the time-variable spot covering fraction, high-resolution spectroscopy or high-resolved line profiles to constrain temperature contrasts and spectral components, direct modeling of spot-crossing events to infer the transit-chord contrast, and preferential use of longer wavelengths where the contamination spectrum decreases. Waalkes et al. specifically recommend contemporaneous multi-color photometry plus high-resolution spectroscopy to pin down $$\Delta D_{\rm spot}(\lambda)\simeq \left(\frac{R_p}{R_*}\right)^2 f_{\rm spot} \frac{I_{\rm spot}(\lambda)-I_{\rm phot}(\lambda)}{I_{\rm total}(\lambda)},$$7 and $$\Delta D_{\rm spot}(\lambda)\simeq \left(\frac{R_p}{R_*}\right)^2 f_{\rm spot} \frac{I_{\rm spot}(\lambda)-I_{\rm phot}(\lambda)}{I_{\rm total}(\lambda)},$$8, observing at longer wavelengths where $$\Delta D_{\rm spot}(\lambda)\simeq \left(\frac{R_p}{R_*}\right)^2 f_{\rm spot} \frac{I_{\rm spot}(\lambda)-I_{\rm phot}(\lambda)}{I_{\rm total}(\lambda)},$$9 drops, and directly fitting spot-crossing events if they occur (Waalkes et al., 2023). The FGK-dwarf analysis likewise recommends modeling the contamination factor $$F_{\rm obs}(\lambda)=(1-f_{\rm spot}-f_{\rm fac})\,F_{\rm phot}(\lambda)+f_{\rm spot}\,F_{\rm spot}(\lambda)+f_{\rm fac}\,F_{\rm fac}(\lambda).$$00 in forward models or retrievals and using correlated trends such as the combination of visual slope and Na line offset as indicators of stellar origin (Rackham et al., 2018).

Taken together, these results establish the transit-light-source effect as a stellar-photosphere problem embedded within exoplanet inference. Its severity depends on host-star activity, spectral type, active-region covering fractions, and whether the transit chord is representative of the disk-integrated stellar spectrum. The strongest consequences reported here occur for active M dwarfs, but the effect remains relevant for active G and K dwarfs in the UV and visual, and it remains diagnostically important even when self-contamination is not formally detected in the transit photometry (Morris et al., 2018, Rackham et al., 2018).