Stacked Periodograms and Compact Systems

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Exoplanet systems detected by Kepler with multiplicity four or greater. These systems exhibit surprising uniformity in mass, radius and orbital separation.. Image by S. Cabot, adapted from Weiss et al. (2018) and Millholland et al. (2017).

An interesting and unexpected finding from Kepler is that multi-planet systems (comprising sub-Neptunes down to Earth-sized planets) exhibit uniformity in mass, radius, and orbital separation (Weiss et al. 2018; Millholland et al. 2017). The origin of these fascinating architectures remains unknown, but efforts toward understanding them benefit from an extensive catalog of multi-planet systems, and baseline characterization of the population as a whole; and radial velocities are the only method for probing the vast number of non-transiting compact systems. 

We developed a novel framework for stacking periodograms of archival, residual RV timeseries which can reveal the aggregate signal of an exoplanet population, even when individual planets have insufficient K to detect individually (Cabot & Laughlin 2022). The California Legacy Survey (CLS, Rosenthal et al. 2021) revealed yearly systematics in their residual RVs by summing their periodograms, which motivated us to develop a stacking methodology to isolate Keplerian signals. Furthermore, the process of stacking periodograms to enhance signals has precedent from Caminha-Maciel (2020), who summed the Fourier power of benthic oxygen isotope ratios (Lisiecki & Raymo 2005) to enhance peaks at Milankovitch frequencies. This power arises from secular variations in Earth's obliquity and eccentricity, which manifest as climate variations and temporal patterns in Oxygen fractionation.

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The relative fractionation of Oxygen 18 to Oxygen 16 in benthic core samples from around the globe. Long-term variations reflect secular changes to Earth's orbit and orientation called Milankovitch cycles. Figure adapted from Lisiecki & Raymo (2005)

Our study investigated stacked periodograms analytically, and with simulations of RV timeseries to demonstrate its efficacy. Finally, it was applied to the extensive Lick-Carnegie Exoplanet Survey. We used a cumulative distribution function (CDF) approach, which shows the fraction of accumulated power as a function of frequency. The CDF contains structure arising from an underlying planet population occupying a narrow period range, which is revealed by comparing it to a baseline CDF derived from a periodogram with constant power at all frequencies. 

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Our framework for stacking periodograms, illustrated with synthetic RV residuals (top left). An enhancement is visible in their stack; however, individual peaks below the threshold of detection. The distribution of Fourier power is well described by a Gamma distribution (top right), which allows nominal analytic estimates of the enhancement. Our CDF approach is highlighted (bottom left). The difference between the simulated CDF and the baseline reveals structure arising from the injected Keplerians (bottom right). Noise realizations are shown for comparison. Figure from Cabot & Laughlin (2022).

The stacked periodogram's structure was influenced heavily by interference between quasi-periodic stellar activity signals, systematics, and the semi-structured window function. We developed a process to mitigate these components. First, periodograms were binned together according to the host star's stellar activity index (log R'hk), and smoothed with a narrow kernel. Second, the power at each frequency was fit with an exponential, as a function of activity index. Third, the fits were evaluated at the lowest activity index in the sample. This interpolation leverages information from all periodograms to provide a single stacked periodogram in the absence of stellar activity. While RV activity should be correlated with the activity index, the planet burden should not.

Applying the CDF method to this resulting periodogram reveals an enhancement in the 3-7 day period range. It is consistent with 1-3 planets per system, each of 6-7 Earth masses. While the detection significance is marginal (1.6 sigma), the enhancement is consistent with the Kepler multi systems, and this finding suggests that such compact systems could be widespread in our galaxy. 

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Exponential fits to binned periodograms from the LCES residual RVs. The color scale corresponds to Fourier power. Note the strong power at multi-year timescales resulting from magnetic activity cycles, as well as moderate power in the 10-100 day regime arising from stellar rotation. Aliasing at one sidereal day is apparent. Figure from Cabot & Laughlin (2022)

While our stacked periodogram method probes the entire close-in population, there are still degeneracies between mass, period, and the number of planets per system. System-by-system investigations are necessary to carefully map the space of orbital parameters. I am a member of Debra Fischer's research group which manages the Extreme Precision Spectrometer (EXPRES), a latest-generation radial velocity instrument installed at the 4.3m Lowell-Discovery Telescope (LDT). As EXPRES continues its multi-year campaign for low-mass planets, I have worked on special set of high-activity stars which serve as a testbed for new activity mitigation methods.

I contributed a stability analysis to the first EXPRES science paper on HD 3651b (Brewer et al. 2020). I also led the development of our in-house Gaussian Process (GP) framework (Cabot et al. 2020) which we applied to HD 101501. I investigated the effect of observing cadence on our capacity to detect low-mass, short-period exoplanets. Our findings indicated that a high-cadence observing schedule is most conducive for GP modeling of quasi-periodic activity signals over timescales of tens of days. We also introduced the concept of studying contemporaneous surface maps of stars toward constraining the activity contribution.

The next flagship EXPRES paper presented a new stellar activity radial velocity model (Roettenbacher et al. 2020), in which I contributed a new framework for translating stellar surface maps into radial velocities over a full rotation. Surface maps can be derived from lightcurve inversion or interferometry. In this case, we observed Epsilon Eridani contemporaneously with TESS Sector 31. Our new activity model successfully reduced the RMS scatter by 2 m/s, and we are currently refining the methodology and identifying new targets for its application.

A new radial velocity activity model (Roettenbacher et al. 2022). Lightcurve inversion reveals the actual spot distribution on the stellar surface (left). Our framework converts the surface map into a 2D projection and pixelated grid, and simulates an integrated spectrum, from which we extract an RV curve. Our model closely matches the EXPRES data (right)