Spot-Crossing Events

The so-called "spot-crossing events" occur when a transiting planet occults a starspot, which is cooler and therefore darker than the surrounding photosphere (see below). This produces a temporary increase in the observed flux. If uncorrected, spot-crossing events can bias transit parameters and the inferred planetary properties. However, they can also be exploited to measure stellar obliquity, the angle between the star’s rotation axis and the planet’s orbital axis, which in turn provides valuable insight into the dynamical history of the planetary system.

The basic idea of inferring stellar obliquity is straightforward. In a well-aligned (low-obliquity) system, the planet’s transit chord overlaps with the trajectory of a starspot, allowing the planet to occult the same spot across multiple transits. The recurrence of spot-crossing anomalies in the light curve thus indicates low stellar obliquity. A prime example is Qatar-2b, a 1.34-day hot Jupiter orbiting a K dwarf. Short-cadence K2 observations revealed dozens of spot-crossing anomalies, and by modeling five consecutive recurring events, we ( Dai et al 2017) showed that the system has a stellar obliquity less than 10 deg. ( Esposito et al (2017)) later confirmed the low obliquity (λ = 0 ± 10 deg) with Rossiter-McLaughlin measurement from HARPS-N.

Left: The recurrence of spot-crossing events indicate a low stellar obliquity. Right: An example is Qatar-2b, a 1.34-day hot Jupiter on a well-aligned orbit around a K dwarf ( Dai et al 2017).

Conversely, when spot-crossing events fail to recur as expected, one can place a lower limit on the stellar obliquity. Provided that starspots persist long enough, the absence of repeat events implies a minimum obliquity such that the transit chord misses the spot. This appears to be the case for WASP-107b, a super-Neptune (M = 0.12 Jupiter mass, a/Rs =18.2) around a K dwarf. The short-cadence K2 light curve of WASP-107 showed three isolated clear spot-crossing events. These spot-crossing events did not recur in neighboring transits as would be expected for a low-obliquity geometry. Using Monte Carlo simulations, we ( Dai et al 2017) demonstrated that WASP-107 has a high obliquity in the range of 40-140˚. The high obliquity has been confirmed by direct Rossiter-McLaughlin measurement ( Rubenzahl, Dai et al (2021)). Interestingly, WASP-107 may belong to the recently identified class of "hoptunes" ( Dong et al 2018) i.e. singly-transiting, Neptune-sized planets. Many hoptunes (WASP-107b, HAT-P-11b, Kepler-63b) appear to be misaligned with their host stars, a phenomenon often seen for their bigger brothers: hot Jupiters. The striking similarities between these two groups may point to a shared dynamical origin ( Yu & Dai (2024)).

Left: Non-recurrent spot-crossing events suggest high stellar obliquity. Right: This seems to be the case for WASP-107b, a polar-orbit, Neptune-mass planet (M = 0.12 Jupiter mass, a/Rs =18.2) around a K dwarf ( Dai et al 2017).

Transit Chord Correlation

In Dai et al 2018, we moved away from parameteric modeling of spot-crossing events to a more general and more efficient statistical approach which we termed Transit Chord Correlation.

The basic idea is still the same. For a transiting planet with stellar obliquity near zero, its trajectory across the stellar disk traces a line of nearly constant latitude (top panel). If active regions (spots or faculae) are present along this latitude, they imprint systematic anomalies in the transit light curves, since these regions differ in brightness from the average photosphere. Darker/brighter features respectively produce increases/decreases in the observed flux (Middle panel). Provided the active regions persist for several orbital periods, a low-obliquity planet will repeatedly occult the same latitude and thus the same active regions. In the bottom panel, by transforming the light curve from time to stellar longitude, a coherent pattern emerges, serving as a proxy for low stellar obliquity. In contrast, for high-obliquity orbits (right panels), active regions rotate out of the transit chord quickly, and residual flux variations arise from different regions each time. As a result, no correlation appears across neighboring transits.

We applied this method to several dozen CoRoT, Kepler, and K2 transiting planets with the highest signal-to-noise light curves. We identified 10 planetary systems that most likely have well-aligned orbits, as indicated by strong correlations. Due to their deep and frequent transits, all of these low-obliquity detections correspond to hot Jupiters. Notably, Kepler-45 is only the second M dwarf with an obliquity measurement. The traditional Rossiter McLaughlin method has not had much success with M dwarf because of their higher stellar variability, slower stellar rotation, and fainter optical magnitudes. We also applied the method to about 30 eclipsing binaries, and we found 8 systems that likely have well-aligned orbits.

Once a low obliquity is established, i.e. the planet probes a particular stellar latitude repeatedly. One can then reverse the argument. By analyzing the photometric signatures of active regions, we can investigate their typical lifetimes, spatial distributions, and migration patterns on the host star. The figure below illustrates the surface magnetic activity of Kepler-17 as revealed through its transiting planet.

Residual flux of Kepler-17b as a function of stellar longitude over consecutive transits c.f. the bottom panel of the previous figure. Visually, we can already see a repeating pattern in the residual flux indicating a well-aligned system geometry. Right —The same data shown as a heat map, with positive/negative residuals color-coded red/blue. Clustering of residuals traces the host star’s active regions, revealing their typical lifetimes, sizes, spatial distributions, and intensities. For Kepler-17, active regions usually span tens of degrees in longitude and persist for 100–200 days before dissipating or leaving the latitudes probed by the planet. Their intensities evolve gradually rather than appearing instantaneously at full strength. The regions remain relatively stationary in longitude, with a slight tendency toward prograde migration (longitude increasing over time). Given the impact parameter (b ≈ 0.27), these active regions are located near ~16° stellar latitude.

We summarize and compare the properties of the active regions of a few planet hosts with the highest SNR. The Sun is also shown for comparison. Listed in this table are several key properties of stellar activity such as rotation period, size and lifetime of active regions, the active latitude. In many respects, the active regions of these planet hosts are quite similar to our Sun. Such measurements provide valuable constraints for stellar dynamo theories.