MODELING INDICATIONS OF TECHNOLOGY IN PLANETARY TRANSIT LIGHT CURVES—DARK-SIDE ILLUMINATION

The most common type of star in our Galaxy is the M dwarf. These stars have very long lives. Because of their low luminosity, the HZ around M dwarfs is narrow and close to the star. Due to tidal considerations, such planets will probably always present one face to the star, except in unusual cases. (Tarter et al. 2007). Although young ($\lt 100$ Myr) dM stars have active chromospheres and high UV output that could be damaging to biological organisms, such activity decreases rapidly with age (Engle et al. 2009). There is also a wide variation in activity at all ages (Shkolnik & Barman 2014), and its precise impact on exoplanet atmospheres, water content, magnetospheres, and habitability is a subject of active research (e.g., Grenfell et al. 2012; Zuluaga et al. 2013; Mulders et al. 2014; Tian 2014). Since the specific conditions required for the evolution of an ecosystem are not well-defined, and given that M and K dwarfs are 80% of stars in the solar neighborhood (Henry et al. 2006), it seems plausible that most nearby habitable planets are in synchronous rotation in orbits around low-mass stars. A significant fraction of M dwarfs do host small planets within their HZs, with the detailed estimates depending on the choice of HZ (Dressing & Charbonneau 2013; Gaidos 2013; Kopparapu 2013).

Life around an M dwarf could pose unique challenges for life and for the evolution of intelligence. However, once a civilization has either evolved in planeta or migrated from elsewhere, it might wish to utilize portions of the planet that do not receive illumination from the star. After careful consideration of the climatic effects, the first deliberate planetary scale engineering project might be the illumination of the planet's dark side. The most straightforward method is via orbiting thin mirrors, either placing a single large mirror at the L2 Lagrange point or, more likely, launching a fleet of smaller mirrors into lower orbits. To provide significant illumination, these mirrors must have a total reflective surface area approximately equal to the disk area of the planet. This could render them detectable in a planetary transit.

If the primary motivation is dark-side illumination, we anticipate such structures would be found primarily around synchronously rotating planets, most likely orbiting M stars. As we discuss in Section 3.1, habitable worlds in certain orbits around earlier-type stars (e.g., G stars) might also be in synchronous rotation at the current time. In addition, an alien civilization might wish to utilize a previously uninhabited planet in their star system, or simply ensure illumination of the night side of a planet in non-synchronous rotation. Other possible uses of fleets of comparable area include photoelectric power generation. Thus, although in what follows we focus our attention on worlds with a perpetual dark side, such engineering projects might occur in other situations, and around many types of stars.

Dyson (1960) suggested that an extraterrestrial civilization might build a giant structure to capture and utilize most or all of the visible emissions of a star, radiating only waste heat. A Dyson sphere would be detectable as a stellar-luminosity $\sim 300\;{\rm{K}}$ featureless blackbody source if the structure completely encases the star, or as a star with a significant infrared (IR) excess in the case of a partial sphere. If civilizations capable of such feats of engineering were common, we could likely detect them in IR surveys. Searches of the IR sky sensitive to both complete (Carrigan 2009) and partial (Jugaku & Nishimura 2004; Conroy & Werthimer 2003, unpublished;³ Globus et al. 2003, unpublished⁴ ). Dyson spheres have been conducted using the IRAS data set. While these have detected a few interesting candidates, none have been shown to indicate intelligent origin. Two high-sensitivity searches for Dyson spheres are ongoing, one searching for IR signatures of Dyson spheres in the WISE survey (Wright et al. 2014b, 2014a; Griffith et al. 2015), and another examining Kepler light curves for unusual transit eclipses that could indicate a partial Dyson sphere (Marcy et al. 2013).

A civilization advanced enough to trap most of the available output of their host star ($\sim 4\;\times {10}^{33}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$) meets the definition of Kardashev Type II on Kardashev's (1964) scale of technological advancement of a civilization. A Type I civilization uses energy on the scale of current terrestrial civilization ($\sim 4\;\times {10}^{19}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$). Given the possibility of extinction, it seems likely that Kardashev Type II civilizations are much rarer than those capable of sending small spacecraft into orbit and that a continuum of possible engineering stages should exist. We might consider a civilization with significant ability to modify its planetary environment, but not capable of building AU scale structures as Type I.5.

Carrigan (2012) recently summarized several other potential archaeological signatures for civilizations at various levels on the Kardashev scale, concluding that, apart from electromagnetic communications, these are generally undetectable with current human technologies. Arnold (2005) suggests that an extraterrestrial civilization could deliberately create planetary scale structures with unique shapes or periodicities that could be detected when the structure transits the star, but could be distinguished from a planetary transit with current technology. Our proposed engineering project is of a similar scale, but with a useful function besides communication and with a form that follows from its function.

There are multiple methods by which an alien civilization could illuminate the dark side of a synchronously rotating planet. Conceptually, the simplest is placing a single large mirror at the L2 Lagrange point (Figure 1(a)). To provide illumination equivalent to the bright-side stellar illumination, it would need to be a large disk or annulus somewhat larger than the planetary diameter. Since the mirror is locked in the anti-stellar direction it would provide non-uniform directional illumination on the ground, with regions near the terminator receiving illumination at higher incidence angles than regions near the anti-stellar point. One notable difficulty is maintaining positional and directional stability at the unstable L2 point. A structure and control system that could allow pointing and positional control of a 6000–25000 km diameter mirror would be a significant engineering challenge.

Zoom In Zoom Out Reset image size

In the case of a single large (${R}_{m}\gt {R}_{p}$) mirror at the L2 Lagrange point, the optical transit light curve will last longer and be deeper than if only the planet blocked the starlight. If the mirror has been engineered so all intercepted light is redirected to the planet, and none toward our line of sight, the planetary eclipse phase will be the same as if the mirror were not present. Thus the difference in transit and eclipse time span would be the main clue to the presence of such a structure, but a similar timing effect could occur for a planet in a highly elliptical orbit with periastron at the time of eclipse and apastron at the time of transit.

A less technically challenging method would be to deploy a fleet of independently steerable smaller mirrors. Figure 1(b) schematically illustrates this possibility in cross-section for mirrors in a circular orbit. Elongated mirror orbits, as depicted in Figure 1(c), might be preferable, since they spend a greater fraction of their orbital period in a position to usefully illuminate the planet's dark side.

For a fleet of many individual reflectors, the translucent fleet of mirrors would block some starlight, deepening and lengthening the transit. Before modeling in detail the effects of a fleet of space-based mirrors on transit light curves, we first discuss technical and engineering considerations involved in its implementation.

Current materials technologies allow for the construction of metalized plastic films with surface mass densities of 0.7–1.5 $\mathrm{mg}\;{\mathrm{cm}}^{-2}$; neglecting supporting structures, the mass of a 1 km² mirror would be 7000–15000 kg (Bryant et al. 2014). Future materials development (carbon nanotube fabrics: Lima et al. 2011, space-based fabrication of metal films: Wright 1992) could result in materials with surface mass densities from $0.01\mathrm{mg}\;{\mathrm{cm}}^{-2}$ to as little as $0.002\mathrm{mg}\;{\mathrm{cm}}^{-2}$. This provides the potential to allow in situ creation of 1 km² reflective surfaces with masses of 20–100 kg. For these, structure and propulsion/control systems will likely be the dominant mass. Each reflector could be pointed based upon orbital position, and controlled independently and autonomously.

We will not attempt a detailed discussion of the dynamics of such an installation in this paper. However, we can utilize the planet's L2 Lagrangian point to estimate the largest stable orbits in the effective planetary gravitational well. For a planet in orbit at the inner edge of the HZ of an M8 star (as estimated below in Section 3.1), the Lagrange point is beyond 10 planetary radii, and is even more distant for earlier stellar types. Therefore, for mirror flotillas with orbits of modest extent (≲5 planetary radii), the mirror orbits should be relatively stable and controllable.

Providing a reflective area equivalent to Earth's disk area $(1.3\;\times {10}^{8}\;{\mathrm{km}}^{2})$ requires deployment of more than 130 million mirrors of 1 km² each, or 1.3 million mirrors of 100 km². At 1000 kg km⁻² even a 100 km² reflector lies within current human launch capabilities. However, even with optimistic ($2200 kg⁻¹) estimates,⁵ launch costs for such a fleet exceed $\$2.8\;\times {10}^{16}{\mathrm{USD}}_{2015}$.

There will undoubtedly be significant environmental impacts due to pollution from manufacturing, launch exhaust products, and altering the energy balance of the planet. However, we leave the details of engineering a climatic balance, combating pollution, and the possible sacrifice of a millennium of economic output to the inhabitants of these worlds.

It is worth highlighting certain considerations affecting implementation of dark-side illumination. For example, heating the dark side would lower the day–night temperature differential and slow day-to-night heat transfer, warming the day side. The mirrors would also obscure some starlight for day-side inhabitants. Optimally the mirrors could be turned edge-on to our transit line of sight quickly as they exit the zone of usefulness, in order to minimize obscuration, unnecessary exposure to solar radiation, and forces that would alter the orbit. However, some obscuration may be desirable to reduce the stellar flux to the day side and keep planetary temperatures from rising excessively. Obscuration of the day side is not overly large even if the mirrors are not directed edge on, lying between $\sim {R}_{p}^{3}/{R}_{m}^{3}$ and $\sim {R}_{p}^{2}/{R}_{m}^{2}$ depending on the orbital arrangement, where ${R}_{m}$ is the orbital radius of the mirror fleet. Satellites in a 3 ${R}_{p}$ orbit would decrease the flux by approximately 4%–11%, while a 10 ${R}_{p}$ orbit reduces flux by 0.1%–1%.

As a result of these and other considerations, the detailed distribution and orientation of mirrors within the orbiting fleet are likely quite complicated. The orbital distribution and variable incidence angle of the mirrors would affect the actual mirror surface area required, as well as the observed transmittance of the mirror fleet. However, for most of this paper we restrict ourselves to the simplest case and consider the observable impact of a constant-transmittance fleet of mirrors surrounding an extrasolar planet. Near the end of Section 4, we briefly consider alternative transmittance models, while deferring more detailed modeling of mirror fleet transmittance to another time.

Table 1 summarizes the properties of stars used in our transit models. Columns 1–5 contain the spectral type and basic stellar properties from Zombeck (1990). Values for surface gravity log($g$) were estimated in two ways. For Method A, we retrieved information on Kepler stars hosting exoplanets from the Exoplanet Data Explorer (Wright et al. 2011) at http://exoplanets.org/table, fit the data with $\mathrm{log}(g)={c}_{1}+{c}_{2}{T}_{\mathrm{eff}}$, then utilized this function to estimate $\mathrm{log}(g)$. For Method B, we scaled the surface gravity from the solar value, assuming g $\propto M/{R}^{2}$. In several test runs, the differences in the resulting light curves were at most 1% of the transit depth, and generally less than 0.2%. For all of our standard runs, we used the values from Method A, as shown in column 6.

Table 1.  Stellar Parameters

Sp Type	T ${}_{\mathrm{eff}}(K)$	M/M ${}_{\mathrm{Sun}}$	R/R ${}_{\mathrm{Sun}}$	L/L ${}_{\mathrm{Sun}}$	log($g$)
M8	2660	0.10	0.13	0.0008	4.96
M5	3120	0.21	0.32	0.0079	4.87
M0	3920	0.47	0.63	0.063	4.72
K5	4410	0.69	0.74	0.16	4.62
K0	5240	0.78	0.85	0.40	4.46
G5	5610	0.93	0.93	0.79	4.39
G2	5780	1.00	1.00	1.00	4.35
G0	5920	1.10	1.05	1.26	4.33

Download table as:  ASCIITypeset image

For our standard runs, we assumed heavy-element abundances half that of the Sun (log([Fe/H]${}_{\mathrm{ratio}}$) = −0.3), and calculated quadratic limb-darkening parameters ${\gamma }_{1}\mathrm{and}{\gamma }_{2}$ using the interface of Eastman et al. (2013; http://astroutils.astronomy.ohio-state.edu/exofast/limbdark.shtml), which interpolates over the values of Claret & Bloemen (2011). The relatively low element abundance value was chosen because we anticipate advanced civilizations around older stars, which tend to be metal-poor.

We assumed circular planetary orbits in all our models. Edge-on orbits were standard, but we also explored other orbit inclinations. To determine appropriate orbit sizes, we estimated each star's HZ using the optimistic values of Kopparapu et al. (2013), accessed via http://depts.washington.edu/naivpl/content/hz-calculator. We note that substantial disagreement exists on the precise range of orbits that lie in a star's HZ. For a detailed discussion, see Petigura et al. (2013), who compare the conservative (rather than optimistic) choices made by Kopparapu et al. (2013) with those of other authors (Kasting et al. 1993; Pierrehumbert & Gaidos 2011; Zsom et al. 2013). However, in this paper, we are concerned mainly with illustrating the potential impacts of a fleet of orbiting mirrors on transit light curves, using one plausible choice for the HZ.

Table 2 summarizes the properties of planets used in our transit models. For each stellar spectral type (column 1), the inner edge (${\mathrm{HZ}}_{\mathrm{in}}$) and center (${\mathrm{HZ}}_{\mathrm{mid}}$) of the chosen HZ are indicated in columns 2 and 5. Corresponding planetary orbit periods for circular orbits (${P}_{\mathrm{orb},\mathrm{in}}$ and ${P}_{\mathrm{orb},\mathrm{mid}}$), determined from stellar mass and orbit size, are in columns 3 and 6. Because we anticipate that mirror fleets might be implemented around planets in synchronous rotation, columns 4 and 7 contain estimates of the tidal synchronization timescale (${\tau }_{\mathrm{syn}}$) for Earth-like ($M={M}_{E},R={R}_{E}$) planets in these orbits. Planets that would not synchronize within $\sim 10$ Gyr are marked with an asterisk (*).

Table 2.  Planet Parameters

Sp Type	${\mathrm{HZ}}_{\mathrm{in}}$ (AU)	${P}_{\mathrm{orb},\mathrm{in}}$ (d)	${\tau }_{\mathrm{syn},\mathrm{in}}$ (Gyr)	${\mathrm{HZ}}_{\mathrm{mid}}$ (AU)	${P}_{\mathrm{orb},\mathrm{mid}}$ (d)	${\tau }_{\mathrm{syn},\mathrm{mid}}$ (Gyr)
M8	0.023	3.27	3 $\times {10}^{-7}$	0.043	9.24	1 $\times {10}^{-5}$
M5	0.073	14.8	7 $\times {10}^{-5}$	0.13	37.4	2 $\times {10}^{-3}$
M0	0.20	47.7	6 $\times {10}^{-3}$	0.36	115	0.2
K5	0.32	79.6	5 $\times {10}^{-2}$	0.55	179	1
K0	0.48	138	0.4	0.83	313	11*
G5	0.67	208	2	1.13	455	50*
G2	0.75	237	4	1.26	517	90*
G0	0.84	268	6	1.40	577	100*

Note.

^*Earth-like planets in these orbits do not synchronize their rotation within 10 Gyr.

Download table as:  ASCIITypeset image

To estimate ${\tau }_{\mathrm{syn}}$, we used Equation (3) of Forget & Leconte (2014), based on the equilibrium theory of tides (Darwin 1880; Hut 1981; Leconte et al. 2010). We followed their lead in estimating tidal dissipation values according to Lunar Laser Ranging experiments of Neron de Surgy & Laskar (1997), which yield

$$\begin{eqnarray}&&{\tau }_{\mathrm{syn}}(\mathrm{Gyr})=21.71\displaystyle \frac{({M}_{p}/{M}_{E})a{(\mathrm{AU})}^{6}}{G{({M}_{s}/{M}_{\mathrm{Sun}})}^{2}{({R}_{p}/{R}_{E})}^{3}}.\end{eqnarray} \tag{ 3 }$$

Terrestrial planets at the inner edge of the chosen HZ (${\mathrm{HZ}}_{\mathrm{in}}$) synchronize their rotation within 6 Gyr for all G, K, and M main sequence stars. However, ${\tau }_{\mathrm{syn}}$ is longer for terrestrial planets elsewhere in the HZ. For example, for planets at its center (${\mathrm{HZ}}_{\mathrm{mid}}$), synchronous rotation is achieved within ∼1 Gyr for main sequence stars K5 and later, but not within 10 Gyr for stars K0 and earlier. Tidal synchronization timescales are uncertain at best. Estimates made by Kasting et al. (1993) suggest that for planets orbiting at ${\mathrm{HZ}}_{\mathrm{mid}}$, only those around M stars would be in synchronous rotation at the current epoch, while those orbiting at ${\mathrm{HZ}}_{\mathrm{in}}$ around most K stars could also be in this state. Although detailed estimates vary, planets in an M star's HZ are extremely likely to be in synchronous rotation after a several gigayear biological evolution timescale. However, note that a planet's climate and habitability may also be altered by effects such as tidal heating or orbital migration (see Barnes et al. 2008, 2013). Because they are most likely to be in synchronous rotation, we focus on planets around M stars for the majority of our models, but do explore other options, since Earth-like planets orbiting a G star near ${\mathrm{HZ}}_{\mathrm{in}}$ might be in synchronous rotation after sufficient time, and as noted earlier, civilizations on Earth analogues might choose to utilize a fleet of mirrors for reasons other than dark-side illumination of a synchronously rotating planet.

In each orbit, we simulated transits for planetary radii of ${R}_{P}/{R}_{\mathrm{Earth}}$ = [0.5, 1, 2], corresponding to $M/{M}_{E}$ = [0.125, 1, 8] if all have the same average density as Earth, or $M/{M}_{E}$ = [0.53, 1, 5.1] using the relationships of Weiss & Marcy (2014). Effects on the transit due to brightness of the planet's dark side were ignored. For each planet, we simulated mirror fleets extending to three different distances from the planet center: ${R}_{m}/{R}_{P}$ = [2, 3, 10]. Mirror fleets that are substantially smaller (${R}_{m}/{R}_{P}=\sqrt{2}$) will be nearly opaque according to Equation (2), while those that are substantially larger may have less stable orbits and may suffer from inefficiencies in redirecting starlight.

For all runs, in addition to determining the light curve for the planet surrounded by its mirror fleet ($P+M$), we also determined the light curve for the planet alone ($P$) and for a larger planet (LP) sans mirrors that produces the same eclipse depth as $P+M$.

Since the mirror fleet mainly affects transit entry and exit, we questioned whether differences in limb-darkening could exhibit the same signature. One source of uncertainty in limb-darkening arises from abundance variations. In runs for identical planets (${R}_{P}=2{R}_{\mathrm{Earth}},{R}_{m}=3{R}_{P}$), varying the stellar abundance from half-solar to tenth-solar or to full-solar impacted the light curve depth by 0.5%–4%, depending on stellar type. This is of similar magnitude as the effects of a mirror fleet. We therefore created light curves for isolated planets transiting M5 stars with tenth-solar and full-solar abundances, with sizes generating transit depths equal to that for the reference simulation. The solid line in Figure 4 shows the difference of the mirror plus planet and large planet light curves from Figure 3, while the dotted and dashed lines show the difference of the light curves for the isolated planets for various stellar abundances. Different stellar abundances cannot be mistaken for the effects of a transiting planet surrounded by a fleet of mirrors because limb-darkening variations have a fundamentally different signature.

Zoom In Zoom Out Reset image size

Although not displayed, we simulated transits of planets orbiting the full range of stellar types displayed in Table 1. Transit light curve details differ somewhat for stars of earlier spectral types. In particular, (1) differences in limb-darkening affect the detailed shape of the light curve near entry and exit. (2) For larger stars, the same planet plus mirror fleet blocks a smaller fraction of light, resulting in a shallower transit, and the mirror fleet's effects last a smaller fraction of the transit time span. Compared with an isolated planet transit of the same depth, the impact of the mirrors is of the order of ${10}^{-4}{I}_{\mathrm{star}}$ for an M0 star, roughly a factor of 2 smaller for a G2 star, and several times larger for an M5 star. (3) Longer orbital periods, corresponding to the larger HZ orbits around earlier-type stars, increase the time span of transits for these planets. Despite these differences, simulating transits of other stars does not significantly change the bipolar signature of the mirror fleet in transit residuals. In addition, the maximum residuals remain approximately the same relative to transit depth for all stellar types (e.g., ∼4%–6% for ${R}_{P}=2{R}_{\mathrm{Earth}},{R}_{m}=3{R}_{P}$).

The top panel of Figure 5 displays light curves for transits that are identical to the reference simulation apart from the mirror fleet's extent. Larger fleets affect the light curve for a larger fraction of the transit duration, with more dramatic departures from the isolated planet situation. The dashed curve (${R}_{m}=10{R}_{P}$) has a slightly different transit depth because ${R}_{P}/{R}_{\mathrm{star}}=0.057{\text{}}{{soR}}_{m}/{R}_{\mathrm{star}}\;=$ 0.57; the mirror fleet extends across a significant fraction of the stellar surface, which has different intensities at different locations due to limb-darkening. Because all mirror fleets have the same effective area, the larger fleet has a lower absorptance. So at z = 0 the smaller, more opaque systems block light primarily from the star's brighter central regions, while the largest, most transparent one transmits more of that light, instead blocking less intense outer portions of the star. The bottom panel illustrates that, all else being equal, mirror fleets with large spatial extents would be more apparent in transit light curve abnormalities than those with smaller spatial extent, both temporally and in the magnitude of the effect on transit residuals.

Zoom In Zoom Out Reset image size

For mirror fleets such as those in the reference model, we explored the effects of varying planet size. Figure 6 displays the difference between the $P+M$ and LP light curves, relative to transit depth ${I}_{\mathrm{star}}-{I}_{\mathrm{min}}$, for three sizes of terrestrial planets, each orbiting at ${\mathrm{HZ}}_{\mathrm{mid}}$ around an M5 star. Smaller planets result in transits of substantially shallower depth; however, in all cases, the mirror fleet has a significant impact (a few percent of the transit depth) on the entrance and exit light curves. The duration of the largest residuals is shorter for smaller planets.

Zoom In Zoom Out Reset image size

All previous results are for edge-on orbits, so the minimum projected distance between star and planet centers is ${z}_{\mathrm{min}}=0$. The top panel of Figure 7 compares this situation (solid line) with orbits that are not exactly edge-on (${z}_{\mathrm{min}\;}\ne$ 0). For these, a planet with ${R}_{P}=2{R}_{E}$ and ${R}_{m}=10{R}_{P}$ is located at ${\mathrm{HZ}}_{\mathrm{mid}}$ around an M5 star. The dashed line planet transits at ${z}_{\mathrm{min}}=0.75$, so part of the mirror fleet never occludes the star (${R}_{m}/{R}_{\mathrm{star}}\;=$ 0.57). In the dotted line case, the planet center crosses the edge of the star (${z}_{\mathrm{min}}=1$), and for the dash–dotted light curve (${z}_{\mathrm{min}}\;=$ 1.25) the planet itself never crosses in front of the star, but the mirrors still create a transit. For each impact parameter ${z}_{\mathrm{min}}$, the bottom panel of Figure 7 shows the difference of the $P+M$ and LP light curves. This figure reveals that as ${z}_{\mathrm{min}}$ increases, the mirror effects are evident during a greater fraction of the transit timescale. As long as more than half of the planet crosses in front of the star, i.e., ${z}_{\mathrm{min}\;}\lt$ 1, the transit residuals display a bipolar signature of the mirror fleet's presence. For ${z}_{\mathrm{min}\;}\gt$ 1, the residuals are no longer bipolar, but the mirror fleet produces transits that last much longer than would be appropriate for the transit depth in the absence of mirrors, and the light curve minimum is not at transit center.

Zoom In Zoom Out Reset image size

Having explored the first-order effects of a fleet of mirrors, we considered alternative transmittance models based on actual distributions of steerable mirrors, which required our brute-force approach. The simplest such situation is with mirrors evenly distributed (density ${\rho }_{\mathrm{mirr}}$) within a spherical shell (${R}_{i}\leqslant r\leqslant {R}_{o}$) around the planet. Although not necessarily realistic, this scenario lets us explore the importance of our constant-absorptance assumption. We calculated the absorptance at each projected radial distance based on the projected density of the fleet:

$$\begin{eqnarray}{\rho }_{\mathrm{proj}} & = & 2{\rho }_{\mathrm{mirr}}(\sqrt{{R}_{o}^{2}-{r}^{2}}-\sqrt{{R}_{i}^{2}-{r}^{2}})\quad (r\lt {R}_{i})\\ {\rho }_{\mathrm{proj}} & = & 2{\rho }_{\mathrm{mirr}}(\sqrt{{R}_{o}^{2}-{r}^{2}})\quad ({R}_{i}\leqslant r\leqslant {R}_{o}).\end{eqnarray} \tag{ 4 }$$

The absorptance was again normalized so the total light intercepted equals that normally incident on the planet, while simultaneously ensuring absorptance $\alpha \leqslant 1$. We considered two extremes: a thick uniform shell of mirrors extending outward from just above the planet's surface, and mirrors all at nearly the same altitude. In the former case, the mirror fleet's collective absorptance is largest near the planet's surface (near ${R}_{i}$), gradually decreasing to zero at the fleet's outer extent. For the thin-shell case, the absorptance is small near the planet's surface, increasing with distance, then sharply peaked near the fleet's altitude. Figure 8 illustrates the extreme differences in absorptance profiles for the thick-shell (top panel) and thin-shell (bottom panel) cases.

Zoom In Zoom Out Reset image size

These simulations illustrate the relative insensitivity of transit light curves to varying mirror-fleet absorptance profiles. Figure 9 compares transit light curves for these two cases with previously discussed results. The constant-absorptance and large planet light curves are the same as those shown in Figure 3. The dotted line results from a thin shell of mirrors (${R}_{i}=2.99{R}_{p},{R}_{m}=3{R}_{p}$), and the dashed line from a thick shell of mirrors extending from ${R}_{i}=1.01{R}_{p}$ to ${R}_{m}=3{R}_{p}$. The inner radii in both extremes were chosen based on the resolution of our brute-force simulations (planet radius = 100 pixels). From this figure, it is apparent that light curves for various absorptance distributions are more similar to each other than they are to that for an isolated planet. Our simple constant-absorptance model therefore captures the dominant effects of this method of dark-side illumination.

Zoom In Zoom Out Reset image size

Until now we have neglected the reality that a mirror fleet's effects will be partially suppressed by transit fitting routines through variations in orbit inclination or stellar limb-darkening. To assess this, we re-sampled our $P+M$ model onto Kepler's short cadence (SC) timescale for three transits. We used the Transit Analysis Package (TAP, Gazak et al. 2012) to fit an isolated planet model to the simulated light curve, restricting limb-darkening parameters to a range appropriate for stars with effective temperatures within 750 K of a G2 star. Because the fleet is translucent ($\alpha \sim 0.01\;\mathrm{for}\;{R}_{m}=10{R}_{P}$), predicted residuals ($\sim {10}^{-5}{I}_{\mathrm{star}}$) are substantially smaller than those for the Jupiter-scale opaque structures of Arnold (2005), and thus too small to detect in Kepler data.

To confirm this, we simulated "best-case scenario" Kepler data for our $P+M$ model and compared the residuals for the previously determined isolated planet model with the difference between the $P+M$ model and the isolated-planet TAP model determined above (hereafter TAP IPM). Given the relatively long orbital period (∼237 days) for a planet orbiting a G2 star at HZ_in (a = 0.75 AU), at most six transits of data are available. Kepler 10 is a relatively bright (Kp = 10.96) G2 star in the Kepler sample, with instrumental and photon noise estimated by the Kepler pipeline as ∼200 parts-per-million (ppm) for SC data. We simulated six transits of SC data with Gaussian white noise at this noise level, although we note that Batalha et al. (2011) report more long-cadence noise (∼62 ppm) than the Kepler pipeline estimate (∼36 ppm).

The top panel of Figure 10 displays the phase-folded simulated data, where we retain the folded timescale for convenience in later analysis and t = 0 represents the center of each transit. For clarity, only half of the transit is displayed. The superimposed solid line is the TAP IPM described above, i.e., the one that best fits the noiseless $P+M$ model used to simulated the data. The residuals between our noisy data and the TAP IPM, smoothed for visual clarity, are shown in the middle panel. They are overlaid (solid line) by the smoothed difference between our $P+M$ model and the TAP IPM. Clearly, noise in these data overwhelms any potential mirror fleet signal. In the bottom panel, the convolution of the (unsmoothed) simulated and predicted residuals confirms that they are uncorrelated.

Zoom In Zoom Out Reset image size

M stars are substantially dimmer at visible wavelengths, resulting in noisier data. A typical Kepler catalog star has a magnitude of ∼14, based on information retrieved from the Exoplanet Data Explorer (Wright et al. 2011), but the median M star is fainter than ${Kp}=15$. Kepler 138, hosting multiple terrestrial planets (Kipping et al. 2014; Rowe et al. 2014), is an unusually bright Kepler M0-dwarf at Kp = 12.93, but still two magnitudes fainter than the G2 star simulated below. Thus the expected noise level would be ∼2.5 times greater. Because of the shorter orbital period for a planet at ${\mathrm{HZ}}_{\mathrm{in}}$ around an M0 star, up to five times more transits of Kepler data could be available, mitigating the additional noise. However, mirror fleets around these stars would still be undetectable in Kepler data.

The James Webb Space Telescope (JWST) will have a collecting area of 25 ${{\rm{m}}}^{2}$, 35 times larger than Kepler (0.708 m${}^{2}$). Using area, quantum efficiency, and filter transmission parameters provided in Van Cleve & Caldwell (2009) and http://ircamera.as.arizona.edu/nircam/features.html, along with the Planck blackbody function for a G2 star, we computed the expected count rate of the NIRCAM F150W2 filter relative to the Kepler full bandpass. The JWST F150W2 count rate will be ∼45 times that for Kepler, reducing noise levels to $\sim 200/6.7\;=$ 30 ppm.

M stars emit a larger fraction of their light at IR wavelengths, so we used the effective temperatures and radii of Table 1 to calculate the relative expected JWST photon rates for G2, M0, and M5 stars at equal distances. In the F150W2 bandpass, we expect 6.7 and 53 times more photons from a G2 star than for M0 and M5 stars, respectively. Data for M stars at similar distances will be noisier, requiring more transits to achieve detections, so only the closest and brightest M stars will be feasible targets for JWST.

As before, we simulated data with SC sampling for an Earth-like planet orbiting at a G2 star's ${\mathrm{HZ}}_{\mathrm{in}}$ using our $P+M$ model, but at JWST's expected noise level of 30 ppm. We compared the residuals between the simulated data and the TAP IPM with the difference between the $P+M$ model (used to simulate the data) and the TAP IPM. We did this comparison for 100 separate realizations of noise for each of 2, 3, 4, 5, and 10 transits of JWST data. Although we would ideally base our comparisons on isolated-planet TAP models fit to each simulated noise realization, this simplified first-order analysis is sufficient to illustrate the key points regarding mirror fleet detectability.

The top panel of Figure 11 shows the phase-folded data for a four-transit noise realization, along with the TAP IPM. For visual clarity, only half of the transit is displayed; however, all analyses were performed using the full-transit simulated data. The smoothed residuals in the middle panel reveal the signature of the mirror fleet; residuals of the simulated data and TAP IPM (dots) are compared with predicted residuals (solid line; difference between simulated noiseless $P+M$ model and TAP IPM). Both are smoothed by half of the timescale ${t}_{1/2}$ (defined below) for visual clarity. The convolution in the bottom panel demonstrates the significant correlation between the (unsmoothed) simulated residuals and the predicted model difference. Vertical dotted lines delineate points satisfying $| {\rm{\Delta }}t| \leqslant {t}_{1/2}$, where ${t}_{1/2}$ is the timescale for the isolated planet model light curve to decrease from maximum to half of the transit depth. The peak is 5.49$\sigma$ above the convolution function's mean, with both mean and rms ($\sigma$) determined by data outside the region delineated by $| {t}_{1/2}|$, i.e., where we do not expect a significant signal. We display this instance of noise because its convolution peak significance (5.49$\sigma$) equals the median significance for the 100 separate four-transit noise realizations.

Zoom In Zoom Out Reset image size

In the 165 points with $| {\rm{\Delta }}t| \leqslant {t}_{1/2}$, the probability of the convolution having a peak of at least this significance occurring by chance is $4.4\times {10}^{-4}$%. Of the 100 instances of four transits with white Gaussian noise of 30 ppm, the convolution peak was at least 4σ 94 times. In our five-transit simulations, 99/100 have convolution peaks of 4σ or higher, with a median of 6.10$\sigma$, and a probability of random occurrence no greater than ∼2% in all 100 realizations. Thus for a star like Kepler-10, we anticipate successful detection of such a mirror fleet with data for only a moderate number of transits.

Could the effects of dark-side illumination be detectable with fewer transits? For the 100 separate three-transit realizations, the median significance of the convolution peak was 4.78$\sigma$, with 83/100 occurring at 4$\sigma$ or above. For the two-transit realizations, the median peak significance was 3.90σ, with only 44/100 reaching the 4σ threshold. We conclude that at this 30-ppm noise level, spurious low-significance detections become increasingly likely with fewer than four transits of data.

An initial impression of the four-transit residuals as shown might suggest the isolated planet fit is adequate, therefore, mirror fleet signatures could go unnoticed without a correlation analysis similar to that carried out above. However, for 10-transit simulations, the convolution analysis results in a peak above 4$\sigma$ 100% of the time, with a probability of the weakest peak occurring by chance of <0.06%. Figure 12 displays the results of a 10-transit simulation with a peak of median significance ($7.74\sigma$). In this situation, the simulated smoothed residuals do indicate that the isolated planet model is insufficient, and the high significance of this signature is made apparent by the strong peak in the convolution shown in the bottom panel. The probability of this peak occurring at random is only 2.7 $\times {10}^{-10}$%.

Zoom In Zoom Out Reset image size

Residuals could be further suppressed if the isolated-planet TAP models do not restrict the range of limb-darkening parameters to values appropriate for similar stars. However, in the majority of cases the stellar temperature and spectral type are known sufficiently well to support the use of some restrictions.

The Transiting Exoplanet Survey Satellite (TESS), observing at 600–1000 nm, may identify transiting exoplanets for which targeted JWST observations would be sufficient for our purpose (Ricker 2014; Ricker et al. 2014). TESS will look at stars 30 to 100 times brighter than Kepler $(5\lesssim m\lesssim 12)$, across the full sky. Primarily sensitive to orbital periods of up to 60 days, it can detect longer orbital periods in regions of the sky coinciding with JWST's continuous viewing zone (CVZ). It will study primarily F, G, and K stars, but also all of the M dwarfs within 200 light years. It is anticipated that TESS will identify ∼300 transiting super-Earths and tens of Earth-sized planets, with $\sim 1/3$ of them in or near the CVZ. TESS might well discover Earth-sized exoplanets in the HZs of K or M stars bright enough for detailed follow-up observations with JWST or future ground-based 30 m class telescopes. Since high-quality transit data on such planets are of great interest for studying properties such as their atmospheres, data adequate for the task of identifying or ruling out fleets of orbiting mirrors will become available.

Since orbiting mirrors illuminate the planet's dark side, we explored the importance of planetary reflection of this light. We assume the planetary contribution to photons received is constant, and that the amount of starlight reaching the day side equals that reaching the "dark" side via mirrors. The planet to stellar flux ratio is therefore on the order of

$$\begin{eqnarray*}&&\displaystyle \frac{{F}_{P}}{{F}_{\mathrm{star}}}={(\displaystyle \frac{{R}_{P}}{{R}_{\mathrm{star}}})}^{2}(\displaystyle \frac{A}{2\pi })(1-\displaystyle \frac{\alpha }{2}).\end{eqnarray*}$$

The first factor represents the fraction of total starlight intercepted by the cross-sectional area of the planet. $A$ is the planet's albedo, i.e., the fraction of starlight that is scattered, with division by $2\pi$ because it scatters into $2\pi$ sr. The final factor accounts for dimming of scattered planetary light by the mirror fleet; absorptance is reduced because only half the mirrors lie on our side of the planet. Greater effects occur for planets that are large relative to the stellar size. For small M8 stars, a planet with ${R}_{P}=2{R}_{\mathrm{Earth}}$ has radius ${R}_{P}=0.14{R}_{{\rm{M}}8}$. For a mirror fleet extending to ${R}_{m}=3{R}_{P}$, the absorptance (given our assumptions) is α = 0.125. Assuming a planetary albedo comparable to Earth's (A = 0.3), this contribution alters the observed transit depth by at most 0.089%, significantly less than the 4%–6% due to mirror fleets (see end of Section 4.1). Thus scattered light from the now-illuminated dark side produces only a small effect on a transit light curve.

Since JWST and other future telescopes will observe primarily in the IR, we must consider thermal radiation from the now-warmed dark side. The planet to stellar flux ratio at any particular wavelength is

$$\begin{eqnarray*}&&\displaystyle \frac{{F}_{P}(\lambda )}{{F}_{\mathrm{star}}(\lambda )}=\displaystyle \frac{\epsilon {B}_{\lambda }({T}_{P})}{{B}_{\lambda }({T}_{\mathrm{star}})}{(\displaystyle \frac{{R}_{P}}{{R}_{\mathrm{star}}})}^{2}\end{eqnarray*}$$

where ${B}_{\lambda }(T)$ is the Planck thermal blackbody radiation, and $\epsilon$ is the planet's emissivity. An upper limit is achieved by assuming $\epsilon =1$ and choosing the coolest, smallest star (M8). For an M8 star, ${R}_{P}/{R}_{\mathrm{star}}=0.14$ and ${T}_{\mathrm{star}}=2660\;{\rm{K}}$; we assume a habitable planet with ${T}_{P}=$ 300 K. In the JWST bandpass (0.6–28.5 μm), the flux ratio reaches a maximum of 0.09% at a wavelength of 28.5 μm. At the wavelength of strongest planetary emission (∼10 μm), the flux ratio is only 0.01%. The change in transit depth is significantly smaller than the 4%–6% effects of orbiting mirrors. Even if ${T}_{P}=1000\;{\rm{K}}$, planetary thermal radiation only alters the transit depth by 0.7%.

In general, any potential detection of intelligence must be confirmed using an entirely different method. We must consider whether natural phenomena could mimic signs of intelligence. In particular, a face-on ring system is extremely similar to the translucent projected annulus of our putative fleet of mirrors. Several groups have modeled transit detectability of ring systems (Barnes & Fortney 2004; Ohta et al. 2009; Tusnski & Valio 2011), generally focusing on gas giant rings. No known terrestrial planet has a ring system, and we question the stability of such a system around a terrestrial planet in tidally induced synchronous rotation, particularly since Ohta et al. (2009) found that rings around all close-in planets are likely to be short-lived and nearly edge-on, rendering them more difficult to detect. More importantly, ring particles are likely warmer than the mirrors, emitting significantly in the IR thus reducing the IR transit depth. Comparison of IR and optical light curves will therefore help discriminate between natural and engineered systems, given sufficient sensitivity. Ring systems may also be distinguished from mirror fleets by the effects of forward scattering of light by ring particles, causing flux to brighten just before and after planetary transit (Barnes & Fortney 2004).

Planets in an M star's HZ might experience atmospheric erosion, due to strong stellar activity and weak planetary magnetic moments (e.g., Lammer et al. 2007 and discussions in Scalo et al. 2007 and Tarter et al. 2007). The excess absorption around the planet could produce a similar transit light-curve signature; however, mirrors are equally opaque at all visible wavelengths, while atmospheric transmission exhibits wavelength-dependent features. Extensive optical/UV transit spectroscopy observations of atmospheric escape from hot jupiters HD 209458b reveal that escaping material contains elements such as H, C, O, and Na (see for example, Charbonneau et al. 2002; Linsky et al. 2010; Ehrenreich 2011), effects not expected for engineered structures. Lyα transit observations of HD 189733b show temporal variability in the evaporating atmosphere (Lecavelier des Etangs et al. 2012), whereas a stable mirror fleet will produce no such fluctuations. As a related example, Rappaport et al. (2012) explain transit depth variability for KIC 12557548 b with a trailing cloud of material escaping a disintegrating super-Mercury; this phase-folded transit light curve exhibits substantial asymmetries. Here transit depth should depend on wavelength, although multi-wavelength studies have not yet detected this (Croll et al. 2014). In detailed modeling by Brogi et al. (2012), a pre-transit flux excess is due to forward scattering by dust, another feature not anticipated with a mirror fleet designed for optimal efficiency.

Kipping (2014) finds that bipolar residual signatures (somewhat different in functional form from those discussed in Section 4) can result from uncorrected low-amplitude transit timing variations or from uncorrected variations in transit duration. In principle, through trial and error it should be possible to offset variations in either transit timing or transit duration and sharpen the transit entry and exit for the folded light curve, making residuals consistent with those for an isolated planet. However, if the bipolar residuals are due to a mirror fleet, they will not be improved by any small adjustments to the timing or duration of each eclipse. Each subsesquent transit would make the difference more pronounced. Substantial technological improvements should eventually yield sensitivity sufficient for detailed analysis of individual transit light curves, in which case the mirror fleet would be clearly distinguishable from these effects.