THE Ĝ SEARCH FOR EXTRATERRESTRIAL CIVILIZATIONS WITH LARGE ENERGY SUPPLIES. IV. THE SIGNATURES AND INFORMATION CONTENT OF TRANSITING MEGASTRUCTURES

Arnold (2005) considered the transit light curves of planet-sized artificial objects, such as those that might be used to intercept star light (i.e., "giant solar panels"). If such objects had non-disk (e.g., triangular) aspects, Arnold (2005) argued they could be distinguished from planets by their anomalous transit light curves. Arnold (2005) focused on the opportunities for such an object to serve as a beacon, and noted that the transmission power of such a beacon was provided by the star itself, potentially giving such beacons very low marginal transmission costs and very long lives compared to other proposed forms of beacons.

Arnold described three examples of such beacons. First, he noted that a megastructure with triangular aspect would have anomalous ingress and egress shapes compared to an exoplanet, and that Kepler would be able to distinguish the two cases for Jupiter-sized objects. Second, he considered a series of objects with identical periods whose transits resulted in clearly artificial patterns of spacings and transit depths. Finally, he considered a screen with louvres which could be rotated to modulate the fraction of stellar flux blocked, producing complex transit light curves between second and third contact that could transmit low bandwidth information, such as a sequence of prime numbers.

Korpela et al. (2015) considered the case of a fleet of structures in a halo around a planet, such as mirrors used to provide lighting to the night side. If these satellites, as an ensemble, had sufficient optical depth and orbital altitude then they would produce their own transit signature as the planet transited the disk of the star, much like a thick, gray atmosphere. Korpela et al. (2015) showed that Kepler would not but James Webb Space Telescope would, be able to detect the ingress, egress, and transit bottom anomalies from such a satellite swarm around terrestrial planet in the Habitable Zone (Kasting et al. 1993) of a $V\sim 11$ star.

A non-spherical megastructure would generate a non-standard light curve in reflection or emission, as well. For instance, a disk rotating synchronously to keep its surface normal to the incoming starlight (as, perhaps, a stellar energy collector) would present a circular aspect during transit, but a vanishing aspect at quadrature. Its phase curve, either in reflected or emitted light, would thus have zeros at quadrature, while a spherical object's flux would steadily increase through quadrature toward superior conjunction. If a transiting megastructure with non-circular aspect and high effective temperature or reflected flux were suspected via ingress and egress anomalies, then a high signal-to-noise ratio (S/N) detection of ingress and egress of its secondary eclipse could confirm its shape and break degeneracies with potentially poorly constrained limb-darkening parameters.

Real exoplanets and stars are appreciably accelerated only by the force of gravity, and so their transit times and durations must obey certain physical constraints parameterized by their impact parameter and the density of the star they orbit (Seager & Mallén-Ornelas 2003). Kipping (2014) described the technique of "asterodensity profiling" by which various properties of a star-planet system could be diagnosed via deviations of the stellar density derived via the Seager & Mallén-Ornelas (2003) relations from the true density of the star (which might be measured independently by other transiting objects or other methods, such as asteroseismology). Kipping (2014) identified six effects that would create such a discrepancy: orbital eccentricity, blends with other stars, starspots, dynamically generated transit timing variations (TTVs), transit duration variations (TDVs), and very massive planets.³  Zuluaga et al. (2015) added a seventh: the photo-ring effect.

Artificial structures, however, might be subject to non-gravitational forces, such as radiation pressure or active thrusts and torques for attitude control and station keeping. As such, their transit signatures might be distinguished by an "impossible" mismatch among the duration and period of the transits, and the stellar density. To Kipping (2014) and Zuluaga et al. (2015)'s list we therefore here add an eighth asterodensity profiling effect, presumably only applicable to megastructures: significant non-gravitational accelerations (the "photo-thrust effect").

In the limit of very small radial thrusts (as in, for instance, the case of radiation pressure on a solar collector), the photo-thrust effect results in an inferred stellar density too low by a fraction β equal to ratio of the thrust to the gravitational force from the star that would otherwise keep the structure on a purely Keplerian orbit ($\beta ={F}_{{\rm{thrust}}}/{F}_{{\rm{Kep}}},$ see Appendix A). In that appendix we also show that planet-sized megastructures with surface densities comparable to common thin metal foils would have photo-thrust effects detectable via asterodensity profiling.

The most extreme case of an anomalous orbit is a static shield, an object held stationary with respect to the star through the balance of thrust (via, for instance, radiation pressure) and gravitational accelerations (a "statite," McInnes & Simmons 1989; Forward 1993). Such a structure might exist only to collect energy, although the resulting imbalance of outgoing photon momentum would result in a small thrust on the statite-star system (resulting in a "class A stellar engine" (Badescu & Cathcart 2000) or "Shkadov thruster" (Shkadov 1987)), and a warming of the star itself. Although we currently know of no material with sufficiently low surface density and opacity that could create such a shield around a solar-type star (e.g., Kennard 2015), presumably the materials science problem of manufacturing such a substance is easier to solve than the engineering problem of constructing a planet-sized shield.

In this case there would be no transits to observe, but the shield might obscure a constant fraction of the stellar disk. Forgan (2013) noted that light curves of an exoplanetary transit of a star with such a shield would be anomalously short and asymmetric. In a reversal of the proposal of Arnold, here it is the apparent aspect of the star that is non-circular due to alien megastructures, not the transiting object. As a result, the transit shape anomalies are significantly larger in Forgan's model of an ordinary planet plus a static shield, than in Arnold's model of a non-circular megastructure.

A civilization that built one megastructure might be expected to build more (Wright et al. 2014b). Their host star might therefore be transited by many artificial structures of a variety of periods, sizes, and aspects—a "swarm." In the limit of a very large number of very small objects, the ensemble might appear as a transluscent screen, and not be easily detected. Large numbers of larger objects might contribute to a constant, low-level variability that could be mistaken for photospheric noise due to granulation (e.g., Bastien et al. 2013) or astereoseismic variations. Larger objects might generate light curves characterized by aperiodic events of almost arbitrary depth, duration, and complexity. Such a light curve might require highly contrived natural explanations (although given the number of stars surveyed to date by, for instance, Kepler, and the rarity of such signals, contrivances might be perfectly warranted).

The most extreme case of a transiting megastructure would be a structure or swarm so large and opaque that it completely occults the star. In this case there might be a very small amount of scattered light from other components of a swarm, but for the most part the star would go completely dark at optical wavelengths. In the limit that such a structure or swarm had complete coverage of the star, one has a "complete Dyson sphere" ($\alpha =1$ in the AGENT formalism of Wright et al. 2014a). Less complete swarms or structures (as in the case of Badescu and Shkadov's scenarios above) undergoing (perhaps non-Keplerian) orbital motion might lead to a star "winking out" as the structure moved between Earth and the star. In such scenarios, the occulting structure might be detectable at mid-infrared wavelengths if all of the intercepted stellar energy is ultimately disposed of as waste heat (that is, in the AGENT formalism, if $\epsilon \approx \alpha$ and α is of the order of 1).

An artificial structure might have very low mass—solid structures or swarms of structures could have very large collecting or radiating areas that block significant fractions of starlight, but have no appreciable gravitational influence on their star or planets orbiting it. Such megastructures would appear anomalous because of the very low densities astronomers would infer from mass measurements via, for instance, radial velocity or TTVs from other transiting objects in the system.

Most megastructure models invoke of geometric absorbers, and so predict nearly achromatic eclipses. In contrast, stars are luminous, and brown dwarfs and exoplanets have atmospheres (and, in some cases, dust trails) which can show spectral features in transmission (such as absorption lines and wavelength dependent scattering and absorption). If a transit signature were gray in both broadband and spectroscopic measurements, this would imply that the object has no detectable region of dust or gas in transmission, respectively. Note that even in the case of purely geometric absorbers there may be some wavelength dependence in transit depths from limb darkening (and, potentially, diffraction, e.g., Forgan 2013), but this effect is well understood and can be modeled to separate the chromatic effects of the source and transiting object.

In particular, the exoplanetary interpretation of observations of a large object with very low inferred density (as might be expected for megastructures), would have to invoke a very large atmospheric scale height (of the order of the radius of the object) and thick atmosphere. Any such real exoplanet would be an extremely favorable target for transmission spectroscopy, and so the lack of any spectral features or wavelength dependence in such an object's transit signature would be both easily established and extremely difficult to explain naturally.

Observing the object spectroscopically in secondary eclipse would potentially reveal its albedo and composition. Such measurements can be difficult, but, in combination with the other signatures above, might be sufficient to demonstrate that a megastructure had been detected.

Non-Keplerian motion in real exoplanets can be the result of planet–planet interactions. Such interactions can generate large TTVs, and even TDVs, especially if the perturbing planet and the transiting planet are in or near a mean motion resonance (MMR, Agol et al. 2005; Holman & Murray 2005) including the 1:1 (Trojan) resonance (Ford & Holman 2007). These interactions also lead to the photo-timing and photo-duration effects in asterodensity profiling (see Section 2.1.2).

These interactions can be diagnosed by the fact that they cannot produce arbitrary TTVs—their patterns are constrained by the families of orbital parameters consistent with long-term stability of the system.

Perturbers near an MMR generally generate a TTV signal that is dominated by a sinusoid, with a period that depends on the proximity of the orbits to an MMR, and an amplitude that depends on the orbital eccentricities and masses of the planets (e.g., Lithwick et al. 2012). Diagnosis is especially straightforward if both exoplanets transit and are near an MMR—in this case each exoplanet perturbs the other, and the two exoplanets exhibit, roughly, opposite TTV signals, with amplitudes that depend on their masses and eccentricities (Carter et al. 2012; Ford et al. 2012). More complex signals can be generated in systems with more than two strongly interacting planets (e.g., Jontof-Hutter et al. 2014), but the physical constraints that the system be dynamically stable prevent the natural generation of arbitrary TTVs.

The maximum observed TTVs and TDVs to date are those of the planets of KOI-142 (Nesvorný 2013; Barros et al. 2014), with approximately sinusoidal signals and semiamplitudes of ∼12h and ∼5 m, respectively (the higher frequency components of the TTVs have amplitudes ∼20 m). Amplitudes significantly in excess of this⁵ or patterns that deviate strongly from the patterns described above would require highly contrived and possibly unstable configurations of exoplanets.

Another, related effect is variations in transit times due to the displacement of the planet-star system by an outer planet or star, creating a varying light travel time for photons carrying the transit information to Earth (Montalto 2010). Such an effect is easily distinguished because its signal will be described well by the Keplerian motion of the outer planet, and any perturber massive enough to create noticeable TTVs would be generate a large, potentially observable radial velocity signal on the star.

Seager & Hui (2002) and Carter & Winn (2010) showed how exoplanetary oblateness (and, so rotation rates and tidal locking) could be probed via the ingress and egress shapes of a transit light curve. These effects are very small, and to date the only such detection is the marginal one that Zhu et al. (2014) describe. More extreme deviations from non-circularity, such as that due to a ring system (Arnold & Schneider 2004; Barnes & Fortney 2004; Dyudina et al. 2005; Ohta et al. 2009; Tusnski & Valio 2011; Braga-Ribas et al. 2014; Zuluaga et al. 2015) would be easier to detect, and easier to diagnose (for instance, ring systems should exhibit a high degree of symmetry about some axis, which may not be the orbital axis).

Stars, too, may be non-spherical. Rotation may make them oblate, and a massive nearby companion may make them prolate. The oblateness effect is usually diagnosed through estimates of the stellar rotation period via line widths (Cabrera et al. 2015) or the rotational modulation of the light curve via spots; the prolateness effect requires careful examination of the details of the light curve (Morris et al. 2013).

The dominant effect of a non-disk-like stellar aspect on transit light curves is to potentially generate an anomalous transit duration; the effects on ingress and egress shape are small. Gravity darkening, which makes the lower-gravity portions of the stellar disk dimmer than the other parts, can have a large effect on the transit curves of planets and stars with large spin–orbit misalignment, potentially producing transit light curves with large asymmetries and other in-transit features (first seen in the KOI-13 system, Barnes 2009; Barnes et al. 2011).

Another, less obvious effect of a non-spherical star is to induce precession in an eccentric orbit, leading to TDVs and transit depth variations (TδVs; such precession can also be caused by general relativistic effects, Miralda-Escudé 2002; Pál & Kocsis 2008).

Both gravity darkening and orbital precession can be in play at once, as in the TDVs of the KOI-13 system (Szabó et al. 2012). A more dramatic example seems to be the PTFO 8–8695 system (van Eyken et al. 2012; Barnes et al. 2013), which exhibits asymmetric transits of variable depth, variable duration, and variable in-transit shape. The diagnosis of PTFO 8–8695 was aided by its well-known age (∼2.65 Myr, aided by its association with the Orion star-forming region, Briceño et al. 2005) and its deep transits. Such dramatic effects would not be expected for older, more slowly rotating objects.

Starspots complicate transit light curves. When a transiting object occults a starspot, it blocks less light than it would in the absence of the spot, and the system appears to slightly brighten. Such an effect was seen by Pont et al. (2007) in Hubble Space Telescope observations of HD 189733 b. For poorly measured transits, such features can also introduce errors in transit time and duration measurements. Starspots are also responsible for the "photo-spot" effect in asterodensity profiling (see Section 2.1.2).

Fortunately, there are several diagnostics for starspots. One is that the shape of starspot anomalies is a well-known function of wavelength, allowing multi-band measurements to identify them. Another is that long-baseline precise light curves will reveal the spots' presence as they rotate into and out of view. If the spots are persistent or appear at common latitudes or longitudes, then repeated transits will reveal their nature. Indeed, such a technique has already allowed for spin–orbit misalignments to be measured for several systems (Deming et al. 2011; Nutzman et al. 2011; Sanchis-Ojeda & Winn 2011; Sanchis-Ojeda et al. 2011).

Even for a spot-free star, the ingress and egress shapes of a transit depend on the effects of limb darkening and the impact parameter of the transit. Misestimations of the host star's properties might lead to inappropriate estimates of limb-darkening parameters which, if held fixed in a fit to a transit light curve, might result in a poor fit to the data, misleading one into believing that the aspect of the transiting object is anomalous. Limb darkening is a wavelength-dependent effect, so misestimates of it might lead one to incorrectly measure how achromatic a transit depth or shape is.

Kipping et al. (2012a) describes many ways in which large moons orbiting exoplanets (or, in the extreme case, binary planets, Tusnski & Valio 2011; Ochiai et al. 2014; Lewis et al. 2015) can leave their signature in the transit timing and duration variations, as the exoplanet "wobbles" about its common center of mass with the moon. If the moon is physically large enough, it can produce its own transit events, creating ingress or egress anomalies, or mid-transit brightenings during mutual events. The orbital sampling effect can also alter the transit light curve and produce anomalous TTVs and TDVs (Heller 2014). Such effects can not generate arbitrary TTVs, TDVs, or light curves, and so can be diagnosed via consistency with the physical constraints of the exomoon model. To date, no such effect has been observed, so any effects of exomoons on light curves is likely to be very small.

Large occulters (i.e., those with at least one dimension similar to or larger than the size of the star) with extreme departures from circular aspects blur the distinctions among the some of the signatures described above, since there might be no clear delineation between ingress, transit, and egress. Their transit signatures will be highly complex, and, if their orbital periods are long, might take place over long time frames (months). Such systems would appear similar in many ways to swarms of artificial objects.

Two major categories of large occulters are ring systems and disks. A ring system or disk around a secondary object can cause complex or severe dimming, as in the cases of 1SWASP J140747.93–394542.6 (Mamajek et al. 2012; Kenworthy & Mamajek 2015), an apparent ringed proto-planet around a pre-main-sequence star, and EE Cep (Graczyk et al. 2003; Gałan et al. 2010), a Be star occulted every 5.6 y by an object with what appears to be a large, almost gray, disk. A disk that is warped, precessing, or that contains overdense regions can also produce occasional and potentially deep eclipses, as in the cases of UX Ori stars ("Uxors," Wenzel 1969; The 1994; Waters & Waelkens 1998; Dullemond et al. 2003) including AA Tau (Bouvier et al. 2013) and V409 Tau (Rodriguez et al. 2015). In the case of an inclined, warped, and/or precessing circumbinary disk, the stars' orbits might bring them behind the disk in a complex pattern, as in the case of KH 15 D (an eccentric binary star system occulted by a warped disk, Chiang & Murray-Clay 2004; Johnson & Winn 2004; Winn et al. 2004; Johnson et al. 2005).

Such systems can be very tricky to interpret; indeed each of the three listed above required an ad-hoc model (which in the 1SWASP J1407 case is not even entirely satisfactory). Adding to the weight of these explanations is that all of these systems are sufficiently young that circumstellar and circumplanetary disks are to be expected. If such a target were to be found to have a light curve so strange that the best natural explanations were highly implausible, and especially if the target star was clearly too old to host extensive circumstellar or circumplanetary disks, then the ETI hypothesis should be entertained and investigated more rigorously.

One contrived natural explanation for such a signature might be a debris field: a compact collection of small occulting objects. Such a field might be a short-lived collection of debris from a planetary collision, or might be collected at the Trojan points of a planet. Such scenarios might be diagnosed through long-term monitoring of the system, especially after one complete orbital period of the field, or, if the swarm contained sufficiently massive bodies, TTVs or radial velocity measurements (Ford & Gaudi 2006; Ford & Holman 2007).

We discuss a fourth category of large occulters, the extensive dust tails of evaporating planets such as that of KIC 12557548 (Rappaport et al. 2012), in Section 3.1.

Finally, disks can occult a portion of the stellar disk, creating an asymmetric transit shape analogous to Forgan's model of a static shield. Such a disk could be diagnosed via the youth of its host star, thermal emission from the disk appearing as an infrared excess, or direct imaging of the disk with interferometry or coronagraphy.

The orbital velocities of planets in eccentric orbits are a function of their orbital phase, so their transverse velocity during transit may be significantly different from that of a hypothetical exoplanet in a circular orbit with the same period. The "photo-eccentric effect" (PE, Dawson & Johnson 2012) is the resulting transit duration deviation from that expected from a circular orbit (or, equivalently, a component of asterodensity profiling). PE has been used to estimate the eccentricity distribution of exoplanets (Moorhead et al. 2011), validate and investigate multitransiting systems (Kipping et al. 2012b; Fabrycky et al. 2014; Morehead & Ford 2015), and determine the history of the eccentricities of exoplanets (Dawson et al. 2015).

Eccentricity can be diagnosed via secondary eclipse timing, stellar radial velocity variations, and dynamical models to TTVs (Nesvorný & Vokrouhlický 2014; Deck & Agol 2015).

Blends occur when one or more stars are either bound or coincidentally aligned on the sky with a star hosting a transiting object. Blends are a major source of false positives for exoplanetary transits, and so might also be for megastructures.

For blends involving planet-sized transiting objects (as opposed to blended eclipsing binary false positives), the primary effect of blending is to dilute the transit signal, which, if unrecognized, to first order leads to an underestimation of the size of the transiting object, but not its shape. There are second-order effects, however, including the "photo-blend" effect (which yields a erroneous stellar density estimate because of an inconsistency in the transiting object radius derived via the transit depth and the ingress/egress durations, Kipping 2014) and wavelength-dependent transit depths (if the stars have different effective temperatures, e.g., Torres et al. 2004).

Blends also confound diagnostics of other natural origins of signs of megastructures by providing a second source of photometric variability, starspots, and spectroscopic signatures such as line widths and chromospheric activity levels. Identification of many of the signatures mentioned here would be complicated by a blend scenario. For instance, a stable star with an equal mass background eclipsing binary might appear to show a planet-sized transiting object, but have zero Doppler acceleration (suggesting a sub-planetary mass).

Searches for transiting exoplanets have produced a comprehensive framework both for calculating the blend probability for a given source (e.g., Morton & Johnson 2011; Torres et al. 2011; Díaz et al. 2014; Santerne et al. 2015), and identifying individual blends, especially via high-resolution imagery, careful examination of spectra, multiband transit depth measurements, and single-band photocenter shifts (e.g., Torres et al. 2004; Léger et al. 2009; Lillo-Box et al. 2014; Désert et al. 2015; Everett et al. 2015; Gilliland et al. 2015).

Clouds in exoplanets can be nearly gray, opaque scatterers, and so a high cloud deck can serve to hide the spectroscopic and broadband signatures of the underlying gas in an exoplanetary atmosphere. A high surface gravity and an atmosphere composed primarily of molecules with large molecular weight will have a very small-scale height, and so spectra will be unable to probe the thin atmospheric annulus in transmission. In either case, an achromatic transmission spectrum would be observed (e.g., Bean et al. 2010, 2011; Knutson et al. 2014). A small-scale height would not preclude a secondary eclipse spectrum from revealing an object's composition, but clouds might introduce a complication.

Clouds, global circulation, and other weather can also produce time-variable or longitudinally dependent albedos, emissivities, and temperatures, giving planets asymmetric and potentially complex phase curves in reflected and emitted light (Knutson et al. 2012; Hu et al. 2015; Kataria et al. 2015; Koll & Abbot 2015).

In practice, objects with unexpectedly small densities have been found. Densities of $\lesssim 0.15$ g cm⁻³ have been found for multiple Kepler planets, including Kepler-7b (Latham et al. 2010), Kepler-79d (Jontof-Hutter et al. 2014), Kepler-51b (Masuda 2014), Kepler-87c (Ofir et al. 2014), and Kepler-12b (Fortney et al. 2011). These low densities may be related to the heavy insolation they receive from their host stars, and in any case the measured masses are all firmly in the planetary regime.

If a $R\sim {R}_{\unicode{0x2643}}$ megastructure were to be found in a short period orbit around a cool star amenable to precise Doppler work, masses as low as a few Earth masses could be ruled out definitively (the state of the art for planet detections is the possible detection of a 1 Earth-mass planet orbiting α Cen B, Dumusque et al. 2012). The implied density would then be $\lt {10}^{-3}$ g cm⁻³, and the implied escape velocity would be 8 km s⁻¹. These figures are inconsistent with a gas giant exoplanet: the Jeans escape temperature for hydrogen would be 3000 K, which would be similar to or less than the temperature of a short-period planet. An alternative natural interpretation could be that the object is a small, terrestrial object with an extended, opaque atmosphere.

In both cases, the natural interpretation would be amenable to testing via transmission and emission spectroscopy and broadband photometry.

In order to illustrate an "information free" DTS, we chose Kepler-4b, a $\sim \;4{R}_{\oplus }$ planet with a 3.21 day period orbiting a 1.2 ${M}_{\odot }$ star, and Kepler-5b, a $\sim \;1.4{R}_{\unicode{0x2643}}$ planet with a 3.5 day period orbiting a 1.4 ${M}_{\odot }$ star. Their transit depths are ${\mu }_{d}=728\pm 30\;{\rm{ppm}}$ and 6600 ± 60 ppm, respectively. Figure 3 (middle) shows the DTS for Kepler-4b, for quarters 1–17 (excepting 8, 12, and 16). From this we see that the Kepler-4b transits are very regular, and so we anticipate little information content in the time series. The gaps in the Kepler-4b DTS are due to missing long cadence data from quarters 8, 12, and 16, part of quarter 4, and regular instrument shutdown times. The mean transit depth of Kepler-4b is a bit lower than the mean transit depth of KIC 1255b, our benchmark "false positive" case, but since the star is brighter it has a comparable S/N to KIC 1255b which makes it an ideal comparison target.

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We generated the Kepler-4b and Kepler-5b DTS's from all available quarters Kepler long-cadence data for these targets. We downloaded the light-curves from the Mikulski Archive for Space Telescopes (MAST) and removed low-frequency variability using the Pyke function kepflatten (Still & Barclay 2012). We then ran the flattened light curves through the autoKep function of the Transit Analysis Package (Gazak et al. 2012) to identify the planetary transits, which we then folded and jointly fitted to a transit light curve model using exofast (Eastman et al. 2013) and the stellar parameters found on the Kepler Community Followup Observing Program (CFOP).¹¹ Having solved for the parameters of the system, we then re-fit each transit individually fixing all transit parameters (using very narrow priors) except transit depth in the exofast fitting. In a few cases, we identified anomalous fits (reduced ${\chi }^{2}\gt 5$), which we rejected.

In principle, hypothetical unseen planets in the Kepler-4b or Kepler-5b systems could affect the fitting of the transit DTS, since we have forced the transit centers to fit a strictly linear ephemeris. However, we are not motivated to perform more detailed investigations considering the large parameter space for undetected planets and the results of a Durbin–Watson test¹² that show no evidence for a significant positive or negative autocorrelation in either data set.

This method generated a DTS for Kepler-4b and Kepler-5b, with 282 and 351 transits, respectively, including depths, depth uncertainties, and transit center times. The DTS's and their PDF's of the Kepler-4b and Kepler-5b light curves are shown in Figures 3–4. Our DTSs are included in our supplemental electronic files associated with this paper.

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Figure 3 also shows the DTS of KIC 1255b for Q1-Q16 on the same vertical scale as the Kepler-4b and Kepler-5b DTS plots. Notable in this DTS are the two quiescent periods at the beginning and end of the DTS where the measured transit depth is nearly zero, and between those two quiescent areas where the depths are highly variable. This DTS was kindly provided by B. Croll (2015, private communication) who describes its construction in Croll et al. (2014). We rejected one highly negative depth in the KIC 1255b time series as unphysical.

Figure 5 shows the PDF for the KIC 1255b DTS. The PDF on the left was generated in the same manner as the Kepler-4b DTS, with a kernel width equal to the average measurement uncertainty of the transit depths (Croll et al. 2014). The smoothness of the PDF is due to the larger width of the convolving kernel, but the width of the PDF is notably much wider than for Kepler-4b and does stretch below ${\rm{depth}}=0$ owing to the quiescent periods. To illustrate the information content that could exist in a KIC 1255b-like system observed by Kepler, we simulated a system with the same measured depths, but at ∼10 times better precision (consistent with the S/N level of Kepler-5b, labeled as "KIC-1255b (high S/N)" in figures). The PDF generated from the convolution of the KIC 1255b DTS with a kernel with the width of the Kepler-5b uncertainties is shown on the right of the figure.

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A simple beacon that has been considered for decades is a sequence of repeating prime numbers (Sagan 1985), a simplified version of which constitutes Arnold's beacon: a series of co-orbital objects whose combined signature is a repeated series transits with depths following the pattern $\bar{[1,2,3,5]}.$ We test this signal at both a high S/N and a low S/N to simulate our current detection capabilities.

We generate the high-S/N DTS case for Beacon 1 (labeled as "B1-high S/N") by repeating $[1,2,3,5]/5\times \mathrm{max}(d)$ (where max(d) is the maximum depth of the Kepler-5b signal) an integer number of times to match as closely as possible the length of the Kepler-5b DTS, as would be observed by Arnold's suggestions, ignoring the gaps between the transits. We then add to this noise at the level of the Kepler-5b uncertainties by randomly sampling from the Gaussian distribution $N(\mu =0,\sigma ={\sigma }_{d,{\rm{K5}}}).$

The PDF for this case is shown on the right side of Figure 6. This was generated by convolving the B1-high S/N DTS with a Gaussian kernel with a width equal to ${\sigma }_{d,{\rm{K5}}}.$ We generate the low-S/N DTS case for Beacon 1 (labeled as "B1-low S/N") by repeating the same sequence an integer number of times to most closely match the length of the KIC 1255b DTS, this time adding to this noise at the level of the KIC 1255b uncertainties by randomly sampling from the Gaussian distribution $N(\mu =0,\sigma ={\sigma }_{d,{\rm{K1255}}}).$ The PDF for this case is shown on the left side of the figure. This was generated by convolving the B1-low S/N DTS with a Gaussian kernel with a width equal to ${\sigma }_{d,{\rm{K1255}}}.$

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Arnold's beacon also encoded the prime number sequence a second way: in the spacings of the transit events (see Figure 8 of Arnold 2005). We accommodate this with a second interpretation of the signal of Arnold's beacon, by constructing a DTS as a repeating sequence with events spaced by the narrowest gap between transits, and containing an appropriate number of null transits (depth = 0) between the more widely spaced transits: [1, 2, 0, 3, 0, 0, 5, 0, 0, 0, 0].

We again test this beacon at both a high and low S/N to simulate our current detection capabilities. The high-S/N and low-S/N synthetic DTS for Beacon 2 were generated in the same way as for Beacon 1, as described in Section 5.2.3.

The PDF for the high-S/N case is shown in the right side of Figure 7. This was generated by convolving the B2-high S/N DTS with a Gaussian kernel with a width equal to ${\sigma }_{d,{\rm{K5}}}.$ The PDF for the low-S/N case is shown on the left side of the figure. This was generated by convolving the B2-low S/N DTS with a Gaussian kernel with a width equal to ${\sigma }_{d,{\rm{K1255}}}.$

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Following the procedure described above, we calculated normalized information content M. Table 2 contains the values that went into calculating Equation (5) as well as the final statistic values for each case. Figure 10 (top) shows where the information content of each case falls on the statistic.

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Table 2.  Depth Time Series Relative Entropy Values

Case	max(d) (ppm)	${\sigma }_{d}$ (ppm)	Km	K0	${K}_{{\rm{max}}}$	M
Kepler-5b	6670	64	7.6833	7.8833 ± 0.0185	5.0355 ± 0.0113	0.0215 ± 0.0073
B1-high S/N	6670	64	6.4921	7.8837 ± 0.0188	5.0362 ± 0.0117	0.4887 ± 0.0039
B2-high S/N	6670	64	6.7204	7.8833 ± 0.0195	5.0374 ± 0.0126	0.4086 ± 0.0045
KIC 1255b-high S/N	10900a	32	4.8589	8.5507 ± 0.0079	4.5331 ± 0.0034	0.9189 ± 0.0008
Kepler-4b	790	40	8.2801	8.3412 ± 0.0207	7.0821 ± 0.0193	0.0483 ± 0.0157
KIC 1255b	10900a	561	4.7908	5.7207 ± 0.0083	4.3877 ± 0.0070	0.6976 ± 0.0043
B1-low S/N	10900	561	4.5466	5.7204 ± 0.0079	4.3876 ± 0.0071	0.8806 ± 0.0048
B2-low S/N	10900	561	4.7189	5.7211 ± 0.0080	4.3876 ± 0.0067	0.7516 ± 0.0041

Notes. Values listed for K₀, ${K}_{{\rm{max}}},$ and M are the mean and $1\sigma$ of the ensemble of values calculated.

^aThe median and 99th percentile depths for KIC 1255b are 3200 and 7950 ppm, giving S/N values of ∼100 and 250 in the high-S/N case, and ∼6 and 14 in the low-S/N case.

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From both of these we can see that the constant cases of Kepler-4b and Kepler-5b are well measured as having near-zero information content. Their non-zero values are likely due to small systematic errors in the photometry in excess of the (very low) shot noise, which broaden the PDF of the measured depths slightly in excess of that for an ideal Gaussian with a width given by the median of the formal measurement errors. The error bars do not encompass zero in part because we have not simulated measurement noise in our calculation of K_m (as described in Section B.2.4).

The high-S/N cases all have S/N $\approx 100,$ as the beacons would if they had been observed by Kepler and they had the depth, brightness, and transit frequency of Kepler-5b. We see that, as we anticipated, the beacons have intermediate information content, with M values near 0.5. The B2 case scores slightly lower because we have chosen to represent the gaps with many zeros, making the distribution of "depths" less uniform.

Also as we expected, KIC 1255b scores very high on the M statistic if we grant the measured depths false precision and assign them the very low measurement noise of Kepler-5b. What we are seeing here is that the measurement noise is information-rich in the sense that it spans many of the values within its range, unlike the beacons which take on only a few values.

The low-S/N cases give very different results. Because the S/N in this case is much lower (∼15), the beacons are no longer detected as having a discrete series of depths. Rather, they appear to span the range from $[0,\mathrm{max}({\rm{depth}})]$ rather uniformly, and so have very high M-values. Interestingly, at this S/N KIC 1255b actually scores lower than the beacons (or itself at high S/N) because we are now more sensitive to the non-uniformities in the depth PDF (the highest depths are underrepresented).

We conclude from this that the relative entropy statistic of the DTS data is a good way to distinguish constant stars, simple beacons, and "random" or information-rich signals if those signals are detected at high S/N. We also conclude that KIC 1255b cannot be yet be distinguished from a beacon (in its DTS) because it has not been measured at sufficient precision to exclude the possibility that the transits exhibit only a small number of discrete depths.

We present the results for the DFT calculation of the relative entropy statistic for the several cases explored in Table 3, and in Figure 10 (bottom). We present the solutions for the individual quarters in Table 4. As in our time series analysis, the constant cases show low information content, consistent with zero, and the beacons in the high-S/N case again have intermediate values of M, around 0.5.

Table 3.  Frequency Space Relative Entropy Values

Case	nd	Km	${K}_{0}$	${K}_{{\rm{max}}}$	M
Kepler-5b	413	5.3296	5.3312 ± 0.0001	4.6411 ± 0.0249	0.0023 ± 0.0002
B1-high S/N	412	4.9872	5.3312 ± 0.0001	4.6424 ± 0.0246	0.5001 ± 0.0179
B2-high S/N	407	4.8558	5.3166 ± 0.0001	4.6287 ± 0.0258	0.6708 ± 0.0253
Kepler-4b	456	5.3512	5.3843 ± 0.0029	4.4028 ± 0.0353	0.0337 ± 0.0032
KIC 1255b	2182	4.8283	6.6762 ± 0.0084	3.0761 ± 0.0340	0.5133 ± 0.0049
B1-low S/N	2180	5.8540	6.6750 ± 0.0087	3.0748 ± 0.0335	0.2280 ± 0.00294
B2-low S/N	2178	5.4401	6.6747 ± 0.0082	3.0742 ± 0.0334	0.3429 ± 0.0036

Note. Values listed for K₀, ${K}_{{\rm{max}}},$ and M are the mean and $1\sigma$ of the ensemble of values calculated. See Table 2 for ${\sigma }_{d}$ and max(d) values.

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Table 4.  M (Frequency Space) for Kepler Quarters

Quarter	Kepler-5b	Kepler-4b	KIC 1255b
Q0	−0.0247 ± 1.6156	0.6930 ± 9.1294	⋯
Q1	−0.0023 ± 0.0021	−0.0092 ± 0.0463	0.7296 ± 0.0796
Q2	0.0034 ± 0.0010	0.0378 ± 0.0158	$0.6870\pm 0351$
Q3	0.0029 ± 0.0009	0.0562 ± 0.0184	$0.6729\pm 0368$
Q4	0.0026 ± 0.0009	0.0368 ± 0.2640	$0.6219\pm 0302$
Q5	0.0026 ± 0.0008	0.0170 ± 0.0176	$0.7726\pm 0384$
Q6	0.0036 ± 0.0010	−0.0203 ± 0.0181	$0.6994\pm 0407$
Q7	0.0044 ± 0.0011	−0.0036 ± 0.0197	$0.6953\pm 0374$
Q8	0.0055 ± 0.0012	⋯	$0.6315\pm 0368$
Q9	0.0005 ± 0.0007	0.0467 ± 0.0183	$0.7278\pm 0347$
Q10	0.0036 ± 0.0009	−0.0219 ± 0.0190	$0.5613\pm 0264$
Q11	0.0022 ± 0.0008	0.0141 ± 0.0175	$0.6047\pm 0340$
Q12	0.0009 ± 0.0008	⋯	$0.6211\pm 0298$
Q13	−0.0003 ± 0.0008	−0.0077 ± 0.0188	$0.6847\pm 0309$
Q14	0.0042 ± 0.0010	0.0210 ± 0.0168	$0.5896\pm 0273$
Q15	0.0040 ± 0.0009	0.0414 ± 0.0191	$0.7009\pm 0355$
Q16	0.0001 ± 0.0008	⋯	$0.6281\pm 0365$
Q17	−0.0017 ± 0.0018	0.1151 ± 0.0583	⋯

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In contrast to the time-series analysis, our frequency analysis properly distinguishes KIC 1255b from the beacons, even at low S/N. This makes sense because our "low" S/N case actually contains five times as many data points as the high-S/N case, which enhances the power of the beacons in frequency space and makes their simple structure more prominent. The constant cases, the pseudo-random KIC 1255b case, and the simulated beacons thus land roughly where we expect them to.

The individual quarters group around their full time series average very well for the Kepler-4b and Kepler-5b cases, as we expected. However, for KIC 1255b the full series shows significantly less information constant than the average of the individual quarters—indeed the statistic values for the individual KIC 1255b quarters do not encompass the value for the full depth sequence. When we examine the DTS of KIC 1255b (Figure 3, bottom) we see that there are two quiescent periods at the start and end of the DTS, which adds power at low frequency in the power spectrum for the full sequence (see Figure 9) that is not apparent in the individual quarters.

Three quarters had fewer than 10 detected transits because they were shorter than usual: quarters 0 and 4 for Kepler-4b, and quarter 0 for Kepler-5b. Because of the extremely low n_d for these quarters, the statistic values were unreliable, and we do not include them in the plot.

Our instant application of the KL Divergence will be to Arnold beacons of Sections 5.2.3–5.2.4, and to the transit DTS of ordinary exoplanets (Kepler-4b and Kepler-5b) and one with variable transit depths (KIC 1255b). For these purposes we choose R = 1 and define our PDFs on the domain $[0,1]$ in order to encompass the physically meaningful range of possible transit depths. Applications to other signals may use different values for R.

We approximate our signal PDFs via KDE of a series of measurements. This procedure convolves the distribution of transit depths with a Gaussian kernel with width equal to the typical depth measurement uncertainty. This produces a continuous distribution, and removes the problems of false precision that come with binning small numbers of points more finely than measurement precision warrants.

The resulting PDF can extend outside the defined range $R=[0,1],$ both because the kernel widens the distribution and because real measurement noise can yield negative transit depths. Simply rejecting the parts of the PDF outside our range leads to problems with normalization and introduces artificial features in the PDF that can yield misleading measures of the information content. Since the signals we will consider here never have values near 1 (total obscuration) and our precision is high, we simply shift our depth measurements by a small constant to ensure that the resulting KDE falls entirely within our range R. A more robust approach would redefine our KL divergence to account for the wings of our PDF outside of our range due to measurement noise and our KDE procedure, but for high precision measurements the effects on the resulting statistic will be small, so we adopt this simpler procedure here for illustrative purposes.

There are many ways in which information might be encoded in a signal conveyed as a discrete sequence within a specified range, and our analysis of the time series only explores one way in which the signal might be simple or complex. For instance, while the KL divergence for a DTS as described above can measure how variable or discrete a sequence of transit depths may be, it does not distinguish between simple, repeating signals and stochastic signals that take on only discrete values.

For example, when using a joint distribution $q({\boldsymbol{x}})$ that factorizes as ${\displaystyle \prod }_{i=1}^{n}q({d}_{i}),$ as in the previous section, the sequence [1, 2, 3, 1, 2, 3, 1, 2, 3, ...] will apparently have the same relative information as the sequence [1, 3, 2, 2, 1, 2, 3, 3, 1, ...], even though the first is strictly repeating and the second is not. Therefore, we extend our metric into frequency space. This corresponds to an alternative choice of $q({\boldsymbol{x}})$ which is not separable in terms of d_i, but is separable in terms of the DFT coefficients ${\boldsymbol{c}},$ i.e., $q({\boldsymbol{x}})={\displaystyle \prod }_{i=1}^{n}q({c}_{i}).$

Because we are dealing with real-valued signals, we elect to fold our DFTs by using only the amplitudes of the positive frequencies to calculate the relative entropy. To account for the power contained within the negative frequencies, we double the power contained in the positive frequencies (except that for sequences with an even number of elements, we did not double the power at the Nyquist frequency, which has no negative counterpart). We then rescale the frequency axis to have domain [0, 1] (so, R = 1; note that in Figure 9 we show these power spectra in physical units of frequency, not from [0, 1]).

We would like to use this folded DFT as p(x) (in place of the PDF of the DTS) in our calculation of Equation (2), however there is a complication, because p(x) must be normalized to have unit area. The value for K in frequency space will thus be very sensitive to one's choice for the value of the DC (zero frequency) term. For instance, real constant signals measured with some amount of measurement uncertainty will have a power spectrum consistent with white noise, with typical amplitude determined by the precision of the measurement. The normalization procedure will rescale this white noise so that the entire DFT has unit area, making the final values at most frequencies dependent on the value of the DC term and the number of frequencies sampled.

Thus, when comparing values of frequency-space K from different signals (as we will do when we compute our normalized information content M) it is important that the DFTs of the signals have identical DC terms. We choose the mean transit depth of measured signal (squared, since we are using power spectra).

Our purpose in this section is to provide a quantitative, statistical description of "beacons" in a SETI context. In our development of the normalized information content, we have considered only bases in the time and frequency domains, since those are the ones in which most of the beacons proposed in the literature are simple and obviously artificial. Of course, an alien (or human) signal might encode information in some other basis than we have considered here, making neither the time nor frequency domains the appropriate ones for our information-content analysis. Applications of the KL divergence to many other domains can be developed straightforwardly, and at any rate the two we have developed here will suffice to illustrate their utility and dependence on some of the properties of a received signal.

We wish to normalize the KL divergence K onto a scale with range [0, 1], spanning the zero and maximal information cases. We thus define the KL divergence for a maximal distribution ${K}_{{\rm{max}}};$ the divergence for a constant signal K₀; and the divergence of a real, measured signal K_m. According to our formalism, ideal signals with no measurement noise will have ${K}_{{\rm{max,}}\;{\rm{ideal}}}=0$ and ${K}_{0,{\rm{ideal}}}=\infty .$

However random measurement noise and finite signal lengths will alter the information content of any real signal. This and the details of the construction of the underlying PDF (for instance, the KDE width) make the measured values of ${K}_{{\rm{max}}}$ and K₀ for maximal and empty signals non-zero and finite.

This allows us to rescale K_m from the values of K₀ to ${K}_{{\rm{max}}}$ that one would calculate from empty and maximal signals at the same S/N and same signal length as the measured signal. The normalized information content will thus be a function of the S/N of the signal.

The numerical values of K₀ and ${K}_{{\rm{max}}}$ in the presence of measurement noise depends on three quantities: the length of the measurement series, the precision of the measurements, and the range the values of those measurements can take. To compute them, we match these values to the equivalent values of the signal we used to compute K_m.

For instance, consider a real signal consisting of n_d discrete measurements d_i, having mean ${\mu }_{d},$ maximum $\mathrm{max}(d),$ measured with precision ${\sigma }_{d}.$ We compute K₀ from an artificial, constant signal as n_d random values drawn from a normal distribution $N({\mu }_{d},{\sigma }_{d});$ and we compute ${K}_{{\rm{max}}}$ from n_d random values drawn from a uniform distribution with Gaussian noise, $U(0,\mathrm{max}(d))+N(0,{\sigma }_{d}).$

For the computation of K in the time domain, we apply a Gaussian kernel with width ${\sigma }_{d}$ to the distributions of values from all three signals to construct a continuous distribution, $p(x),$ and numerically compute the KL divergence via Equation (2) (using 2¹⁶ points to ensure that the function is well sampled).

In the frequency domain, we fold the DFT of each series as described in Section B.1.2 and set the DC term in the in all three cases to ${\mu }_{d}^{2}$ (since we are using the power spectrum). We then normalize this function to unit area (i.e., the sum of the terms of the folded DFT will be the number of elements in it, which, since we have folded the DFT, is $({n}_{d}+1)/2$). This folded, DC-corrected, normalized DFT is our p(x) for Equation (2).

Since the KL divergence gives the relative entropy of a signal, the entropy of the measured signal compared to what we would measure from an empty signal is

$$\begin{eqnarray}&&{\rm{\Delta }}S={K}_{m}-{K}_{0}.\end{eqnarray} \tag{ 3 }$$

If ${\rm{\Delta }}S\approx 0$ (i.e., there is no difference between our divergences), then our signal is consistent with pure measurement noise, and we can detect no other source of information in our data. The maximum value of ${\rm{\Delta }}S$ is that from a maximal distribution, i.e.,

$$\begin{eqnarray}&&{\rm{\Delta }}{S}_{{\rm{max}}}={K}_{{\rm{max}}}-{K}_{0}.\end{eqnarray} \tag{ 4 }$$

We then normalize ${\rm{\Delta }}S$ to its maximal possible value, which allows us to construct our normalized statistic of information content, M, on a scale from [0, 1], as we desired:

$$\begin{eqnarray}&&M=\displaystyle \frac{{\rm{\Delta }}S}{{\rm{\Delta }}{S}_{{\rm{max}}}}=\displaystyle \frac{{K}_{m}-{K}_{0}}{{K}_{{\rm{max}}}-{K}_{0}}.\end{eqnarray} \tag{ 5 }$$

These are several sources of uncertainty in our information-content statistics.

The first is that measurement noise itself contributes some amount of entropy to the signal. We have adjusted for this to first order by comparing the K_m value we calculate with constant and uniform cases that include noise in the statistic M. The second is that the noise in the frequency-space case depends strongly on the length of the signal (the number of events observed). We account for this by measuring K₀ and ${K}_{{\rm{max}}}$ using the same number of points as K_m. But different realizations of the noise will lead to different values for ${K}_{m},{K}_{0},{K}_{{\rm{max}}},$ and therefore M. We account for these effects of noise on K₀ and ${K}_{{\rm{max}}}$ by recalculating these quantities for 1000 draws of the Gaussian noise. The ensemble of values for K₀, ${K}_{{\rm{max}}},$ and M that we calculated from these draws give us uncertainties on these statistics.

The effects of noise on K_m cannot be robustly calculated without knowledge of the underlying signal, which we cannot assume one has. This is related to the third source of uncertainty, which is that we may have only measured a small portion of the signal, and that other parts of the signal may have a different information content. For both reasons, we frame the problem as that of measuring the information content of the portion of the signal we have actually measured, understanding that if we repeated the measurement we would get a (perhaps slightly) different number, both because of measurement noise and because the underlying signal would be different.