Identifying the rotation rate and the presence of dynamic weather on extrasolar Earth-like planets from photometric observations
E. Palle, Eric B. Ford, S. Seager, P. Montanes-Rodriguez, M. Vazquez
aa r X i v : . [ a s t r o - ph ] F e b N Identifying the rotation rate and the presence of dynamic weatheron extrasolar Earth-like planets from photometric observations.
E. Pall´e , Eric B. Ford , S. Seager , P. Monta˜n´es-Rodr´ıguez , M. Vazquez Instituto de Astrofisica de Canarias, La Laguna, E38200, Spain. [email protected], [email protected], [email protected]
Department of Astronomy, University of Florida, 211 Bryant Space Science Center, POBox 112055 Gainesville, FL, 32611-2055, USA [email protected]
EAPS, Massachusetts Institute of Technology, Cambridge, MA02139-4307, USA [email protected]
ABSTRACT
With the recent discoveries of hundreds of extrasolar planets, the search forplanets like Earth and life in the universe, is quickly gaining momentum. In thefuture, large space observatories could directly detect the light scattered fromrocky planets, but they would not be able to spatially resolve a planet’s surface.Using reflectance models and real cloud data from satellite observations, here weshow that, despite Earth’s dynamic weather patterns, the light scattered by theEarth to a hypothetical distant observer as a function of time contains sufficientinformation to accurately measure Earth’s rotation period. This is because oceancurrents and continents result in relatively stable averaged global cloud patterns.The accuracy of these measurements will vary with the viewing geometry andother observational constraints. If the rotation period can be measured withaccuracy, data spanning several months could be coherently combined to ob-tain spectroscopic information about individual regions of the planetary surface.Moreover, deviations from a periodic signal can be used to infer the presenceof relatively short-live structures in its atmosphere (i.e., clouds). This couldprovide a useful technique for recognizing exoplanets that have active weathersystems, changing on a timescale comparable to their rotation. Such variabilityis likely to be related to the atmospheric temperature and pressure being near aphase transition and could support the possibility of liquid water on the planet’ssurface. 2 –
Subject headings: exoplanets, Earth, albedo, earthshine, rotation, astrobiology
1. Introduction
Over the past two decades, more than 240 planets have been discovered orbiting starsother than the Sun. To date all planets discovered around main sequence stars are signif-icantly more massive than the rocky planets of the solar system. Radial velocity surveys,however, are starting to detect rocky planet candidates below 10 Earth masses (Rivera et al.,2006; Udry et al, 2007) and, for the coming decades, ambitious space missions are being pro-posed that would be able to detect nearby planets with physical properties similar to Earth(see, e.g., Lindensmith , 2003; Kaltenegger, 2005; Fridlund, 2004; Cash, 2005; Schneider etal, 2006).Among other important physical properties, the identification of the rotation rate ofan exoplanet with relatively high accuracy will be important for several reasons (Laskarand Correia, 2004). First, measuring the rotation rate can help to understand the forma-tion mechanisms and dynamical evolution of extrasolar planetary systems (Agnor et al.,1999; Chambers, 2001; Goldreich et al., 2004). For example, are planetary rotation peri-ods smoothly varying as a function of the planet mass and semi-major axis, as would beexpected if the planet’s angular momentum is dominated by the gradual accretion of smallplanetesimals? Or are planet’s rotation periods essentially uncorrelated with their mass andorbital properties, as would be the case if the planet’s angular momentum is dominated bythe late accretion of a few large impactors? The rotation periods of a sample of planetscould be directly compared to numerical simulations of planetary formation that track thespin evolution of planets, to probe the late stages of planetary accretion (Schlichting andSari, 2007).A precise determination of the rotation rate can also help improve our analysis of futuredirect detections of exoplanets, including photometric, spectroscopic, and potentially polari-metric observations (Gaidos and Williams, 2004; Tinetti et al, 2006; Monta˜n´es-Rodr´ıguez etal., 2005; Stam et al, 2006; Williams and Gaidos, 2007). For practical viewing geometries,most of the light scattered by an Earth-like planet comes from a small portion of the planet,and contains information about weather patterns, surface features, i.e. lands and oceans.While even the most ambitious space telescopes will not be able to spatially resolve thesurface of an extrasolar planet, the temporal variability contains information about regionalsurface and/or atmospheric features, possibly including localized biomarkers (Ford et al.,2003; Seager et al, 2005; Monta˜n´es-Rodr´ıguez et al., 2006). Determining the planet rotationperiod is necessary in order to know the rotational phase for a time series of observations. 3 –The precision with which the rotation period can be measured determines the time span ofobservations that can be coherently averaged.We will see in this paper how the deviations from a periodic photometric signal canhelp to identify active weather on an exoplanet. This could prove a useful technique forrecognizing exoplanets that have weather systems with inhomogeneous cloud patterns.Finally, the observations of our solar system bodies suggest that the presence of a plane-tary magnetic field, generated by dynamo processes, is mainly a function of two parameters:its composition (mass) and the rotation speed (Vallee, 1998; Russell, 2006). If the planetmass is known, a fast rotation speed of the planet could suggest the presence of a significantmagnetic field. One must note however that there will be a large list of caveats to thispossibility, given our current understanding of dynamos and planetary evolution (Bushbyand Mason, 2004; Grießmeier, 2007).In this paper we have determined the changes in photometric albedo that we would seeif Earth was observed as an extrasolar planet. First, we perform an accurate and realisticsimulation of the flux changes in reflected light from the planet’s surface and atmosphere.Second, we perform a periodicity analysis to determine under what conditions the rotationrate can be determined. Third, we explore how the accuracy and precision of the measuredrotation rate depend on four variables: the temporal resolution of observations (i.e., exposuretime), the total duration of observations, the signal-to-noise ratio, and the viewing geometry.We also discuss the role of clouds in altering the reflected light flux from Earth, and howto detect them in an exoplanet’s atmosphere. Finally, we discuss the implications for thedesign of future space missions to characterize extrasolar planets via direct detection.
2. Methods2.1. Planet Light Scattering Model
The albedo of each surface element, a , depends on the surface type, cloud and snow/icecover and solar zenith angle. Further, there is an anisotropic factor, L , that gives theangular distribution of the reflected radiation and depends upon the reflected zenith angleand azimuth. The anisotropy function, also known as bidirectional reflectance function(BDRF), generally depends on surface type, cloud cover, zenith angle and relative azimuthof the Sun. L is defined so that it is unity for a Lambert surface (Pall´e et al, 2003). Inmodeling the reflectance properties, a and L , of the Earth, we used scene models based onthe Earth Radiation Budget Experiment (ERBE) observations (Suttles et al., 1988), definedas the mean over the broad shortwave interval from 200 to 4000 nm . The parameters a and 4 – L are tabulated for twelve model scenes.The model of the Earth uses daily satellite observations of total cloud amount at eachsurface location from the International Satellite Cloud Climatology Project (ISCCP) asinput (Rossow et al., 1996). Four cloudiness levels (0-5%,5-50%, 50-95% and 95-100%),are considered for each of the 12 different ERBE scenes. For the snow/ice cover, we usedsimulations from the Canadian Center for Climate Modeling and Analysis (CCCM II). Themodel has already been validated by observations of Earthshine (Pall´e et al, 2003).Our model allows us to simulate the Earth’s reflectance observed from any viewinggeometry. For example, looking at the exoplanet (our modeled Earth) always from thenorth pole or along the ecliptic. In the context of observing extrasolar planets, this is similarto fixing the orbital inclination of the orbit with respect to the observation point. Thus,the Earth’s reflectance in the direction of β , where β is defined as the angle between theSun-Earth and Earth-Observer vectors, can be expressed as p e f e ( β ) = 1 πR e Z ( ˆ R · ˆ S, ˆ R · ˆ M ) ≥ d R ( ˆ R · ˆ S ) a ( ˆ R · ˆ M ) L, (1)where ˆ R is the unit vector pointing from the center of the Earth to a patch of Earth’s surface,ˆ S is the unit vector pointing from the Earth to the star, and ˆ M is the unit vector pointingfrom the Earth toward the observer. The integral is over all of the Earth’s surface elementsfor which the sun is above the horizon (i.e., ˆ R · ˆ S ) and the surface element is visible fromthe observer’s perspective (i.e., ˆ R · ˆ M ≥ R e is the radius of the Earth, p e is thegeometrical albedo of the Earth, and f e ( β ) is the Earth’s phase function (defined such that f e (0) = 1).The total reflected flux in a given direction, β , can be calculate using F e ( β ) = SπR e p e f e ( β ) , (2)where S is the solar flux at the top of the Earth’s atmosphere (1370 W/m ). There is asystematic variation of p e f e ( β ) throughout the Earth’s orbital period (sidereal year), andfluctuations of p e f e ( β ) about its systematic behavior are caused by varying terrestrial con-ditions, including weather and seasons (Pall´e et al, 2004).Comparing F e to the flux of sunlight for the same observer, yields contrast ratios oforder 10 − . This presents the main challenge in directly detecting an Earth-like planet.In comparison, the amplitude of the diurnal cycle of the Earth observed in our broadband(200 − nm ) simulations (Figure 1) is of the order of 0 . × − , but varies greatlydepending on wavelength. 5 –At present, broadband coronagraphic experiments are able to reach contrasts of 10 − only (Mawet et al., 2006). However, advances in the development of coronagraphs anddeformable mirrors are expected to enable such observations in the future. For example,Trauger and Traub (2007) have shown how contrast ratios of the order of 1 × − can beachieved with coronagraphs in the laboratory, using a laser beam at monochromatic visiblewavelength. In this paper, however, a wide bandwidth is considered, in order to have enoughphotons in each observation. The use of a wide bandwidth in coronagraphy will require avery good achromatization of the coronagraph to achieve a high light rejection, workingtowards a viable visible-wavelength direct imaging technique. In order to simulate the observations of the Earth as if it were a distant planet, wemust specify the viewing geometry of the simulated observations. An observer that looksat the Sun-Earth system from along the ecliptic will be looking at the Earth from a nearlyequatorial perspective. During a year the Earth will appear to go through phases from afully lit Earth to a fully dark Earth. For this case, the Earth would pass inside of the Sun’sglare twice per year. On the contrary an observer looking at the Sun-Earth system from adirection perpendicular to the plane of the Earth’s orbit, would see only the northern (orsouthern) hemisphere of the Earth. At any given time, approximately half of the Earthwould be illuminated and visible to the distant observer.In order to determine the sensitivity of our results to viewing angles, we have chosenfive different viewing geometries of the Earth which we will refer to as: the equatorialview, the north/south polar view, and the (primarily) northern/southern hemisphere view.Technically, we are choosing the inclination ( i ) of the line of sight with respect to to theecliptic plane: 0 ◦ , ± ◦ , and ± ◦ . In order to visualize the viewing geometries, Figure 2shows the Earth for a single date and time, as seen from each of the five viewing perspectivesthat we consider. The date corresponds to a day in November, when the Earth would presenta phase angle of approximately 90 ◦ (as seen from each of our viewpoints). Note that thefigure is misleading in the sense that clouds, which will play a major role in the photometricalbedo, are not represented.The quantity p e f e ( β ) is affected by three factors. First, as the Earth’s revolves aroundthe Sun, p e f e ( β ) will change due to a changing β . At the same time, due to the Earth’srotation, the portion of the Earth’s surface visible to the observer will also change, leadingto changes in the albedo diurnal cycle. Finally, the large-scale cloud patterns will changefrom day to day, adding short-term variability to the observations. In Figure 1 the yearly 6 –evolution of the flux ratio between the Earth and the Sun, taking into account these variouseffects, are represented.We generate photometric time series of the light scattered by the planet toward anobserver that include the effects of both the planet’s rotation and the planet’s orbital motion(as well as changing cloud and snow/ice cover). While our simulated data is centered on aphase angle of β = 90 ◦ , the phase angle deviates from 90 ◦ due to the orbital motion (e.g.,up to ≃ ◦ for an eight week time series with the equatorial viewing geometry). Several considerations need to be taken into account before we can realistically analyzeour simulations in terms of the Earth as an exoplanet. A space telescope intending to thesearch for exoplanets will have a long list of target stars to observe during the planned missionlife time (of order a few years). If a small number of remarkable Earth-like planet candidatesare identified, then multiple months of observations time could be devoted to characterizingindividual targets. On the other hand, if many Earth-like planet candidates are found,then the amount of observing time available for follow-up observations of most targets couldbe much more limited. Therefore, we have considered simulated observational data setsspanning 2, 4, and 8 weeks. Similarly, we have simulated observations made with severaltemporal resolutions (exposures times), ranging from 0.1 to 10 hours. Finally, we have addedPoisson noise to the data, to simulate signal-to-noise (S/N) ratios, for each exposure time,ranging from 3 to 1000. While such a large S/N is unrealistic for a first generation TPF-C mission, these calculations are relevant for determining if very high-precision rotationmeasurements are possible or if the stochastic nature of clouds results in a limit on theprecision of rotation periods that is independent of the S/N.The orbital position of the planet will also limit our observing capabilities. Ideally onewould like to observe the planet at full phase when its brightness, as compared to that ofthe parent star, is larger. However, observations at these phase angles are nearly impossibledue to the small angular distance between the planet and the star. In this work, we focuson observations made near a phase angle of 90 ◦ , when the planet-star separation is near itsmaximum. The best case scenario for measuring the rotation period of a planet occurs foran orbital plane nearly perpendicular to the line of sight, so that the planet remains at aphase angle of nearly 90 o (maximum angular separation) for it’s entire orbit orbit. 7 –
3. Results
We simulate several time series of the Earth’s scattered light towards a hypotheticalobserver. For each time series, we perform an autocorrelation analysis. For example, inFigure 3, the black curve shows the autocorrelation as a function of the time lag basedon a simulated data series for an Earth without any cloud cover. We assume the i = 90 ◦ viewing geometry described in § § ∼
20 every ∼ §
5, we will further discuss the capabilities of such a TPF-C mission, as well as missionsof alternative sizes. Based on the TPF Target List Database (v1; http://sco.stsci.edu; seealso Turnbull and Tarter, 2003), we find that 29 such main sequence K-A stars that haveaccurate parallax, B-V color, no companion stars within 10 arcsec , and show no indicationsof variability. Eliminating A stars reduces the number of such targets to 15. Note that thisis more than the 14 and 5 target stars included in the TPF Target List Database ‘extended”(including A stars) and “core” (excluding A stars) lists that apply several additional cutsbased on a notion of habitability (e.g., eliminating young stars that may be too young forsignificant biological alteration of the atmosphere).
By definition, the maximum autocorrelation equals unity at zero lag. The next greatestautocorrelation occurs at 24 hours, very near the true rotation period of the Earth. In thiscase, we find that the amplitude of the autocorrelation is very similar at integer multiplesof the Earth’s rotation period, since the only changes are due to the slow variations of theviewing geometry and phase angle ( β ) resulting from the orbital motion of the Earth.For a cloudless Earth, we find that there is a second series of local maxima in the 8 –autocorrelation function near twelve hours. This is not due to a fundamental property ofthe autocorrelation analysis (e.g., the blue curve for the cloudy Earth has no significantamplitude at 12 hours), but rather is due to the distribution of continents and oceans on theEarth. For this data set, the difference in the amplitude between the local maxima at 12 hoursand 24 hours would indicate that the peak at 24 hours corresponds to the rotation period.However, the possibility of the continental distribution leading to a significant autocorrelationat alternative lags could complicate efforts to identify the rotation period.We now consider a cloudy Earth using Earth’s actual cloud cover randomly selectedfor eight weeks in 1985. The blue curve in Figure 3 shows the results of an autocorrelationanalysis similar to the one for the cloud-free Earth, assuming the same viewing geometryand observational parameters as above. Aside from the maximum at zero lag, the maximumautocorrelation occurs at 24 hours, very near the true rotation period of the Earth. Theadditional local maxima of the autocorrelation that occur at integer multiples of 24 hoursare due to the viewing geometry repeating after multiple rotations of the Earth. In thiscase, the the autocorrelation decreases at larger multiples of the rotation period, since thevariations in the cloud patterns are typically greater on these longer time scales. Here, we explore how the precision of the measured rotation period depends on vari-ous observational parameters, such as the signal-to-noise ratio, the temporal resolution ofobservations, the total duration of the observational campaign, and the viewing geometry.In Figure 4 (top), we show the mean absolute value of the difference between the actualand the derived rotation period of the Earth based on 21 data sets, each for a different year(Global cloud coverage measurements from ISCCP satellite observations are only availableover the period 1984-2005, i.e. 21 years). Here we assume 8 weeks of observations with atemporal resolution of 6 minutes. Each curve corresponds to a different viewing geometry.For the equatorial and primarily northern/southern hemisphere views, we conclude that aS/N ratio of 10-20 is necessary to determine the rotational period with an error of about 1hour (5% of the 24-hour rotation period). With a S/N ratio of about 30, we find a precisionin the rotation determination of approximately 10 minutes (0.7%). On the other hand, thedetermination of the rotational period from a polar perspectives has a larger error. Evenwith increasing S/N, the rotational period that one obtains from a polar perspectives doesnot always converge to 24 hours but to a shorter periodicity (see § Here, we explore the outcome of performing a periodicity analysis to our simulatedphotometric time series, using a Fourier-based technique, the classical periodogram. InFigure 5 we have plotted the periodogram of the time series resulting from simulations ofthe real (cloudy) Earth, as viewed from the five different viewpoints and in two differentyears. For each case, the periodogram is calculated using time series spanning 2, 4 and 8weeks of observations.According to the periodogram analyses, the 24-hour periodicity is not always the strongest,and it is missing altogether for some of the series (depending on the specific cloud data).The observations from the nearly equatorial perspective seem to be the most affected bythe Earth’s particular continental distribution, as there are strong peaks at 12-hours. If thedistribution of continents on our planet were different (as it has been in the past), then thederived periodicities would also be different. The viewing geometry also plays a role. Forexample, our southern pole viewing geometry results in the Earth appearing to have a singlecontinent in the center surrounded by ocean.We compare the accuracy and precision of two types of periodicity analysis: the au-tocorrelation function and Fourier analysis. In Table 1, we show the frequency with whicheach type of analysis results in a determination of the rotation period near the true rotation 10 –period (24 ± ∆ hours), half the rotation period (12 ± ∆ hours), or other alternative values. Ineach case ± ∆ is taken as the exposure time (or sampling resolution). In other words, a 95%value in the 24-hour periodicity for the autocorrelation method, means that for 20 of the 21available years (cloud configurations) the primary periodicity retrieved by the autocorrela-tion method is 24 ± ∆. In Table 2, we present the the same quantities as in Table 1, butthis time for an Earth completely free of clouds, so that the detected periodicities are due tosurface albedo variations only. The Fourier analysis often results in the largest peak near 12hours (see Fig. 5), particularly for viewing geometries with orbital inclinations of 45 ◦ , 90 ◦ ,and 135 ◦ . Our autocorrelation analysis never makes this error. Thus, we conclude that theautocorrelation function provides a more robust and more accurate tool for characterizingthe rotation period of a planet using photometric time series data.
4. The Effect of Clouds
Clouds are common on solar system planets, and even satellites with dense atmospheres.Clouds are also inferred from observations of free-floating substellar mass objects (Ackermanand Marley, 2001). Hence, cloudiness appears to be a very common phenomenon.Clouds on Earth are continuously forming and disappearing, covering an average ofabout 60% of the Earth’s surface. This feature is unique in the solar system to Earth: Somesolar system planets are completely covered by clouds, while others have very few. Only theEarth has large-scale cloud patterns that partially cover the planet and change on timescalescomparable to the rotational period. This is because the temperature and pressure on theEarth’s surface allow for water to change phase with relative ease from solid to liquid to gas.In principle, weather patterns and/or the orbital motion of the Earth could pose a fun-damental limitation that prevents an accurate determination of the Earth’s rotation periodfrom the scattered light. Since the scattered light is dominated by clouds, it might be im-possible to determine the rotation period if the weather patterns were completely random.Alternatively, even if the atmospheric patterns were stable over many rotation periods, ob-servational determinations of the rotation period might not correspond to the rotation periodof the planet’s surface, if the atmosphere were rotating at a very different rate (e.g., Venus).In fact, we find that scattered light observations of the Earth could accurately identifythe rotation period of the Earth’s surface. This is because large-scale time averaged cloudpatterns are tied to the surface features of Earth, such as continents and ocean currents.This relatively fixed nature of clouds (illustrated in Figure 6) is the key point that wouldallow Earth’s rotation period to be determined from afar. 11 –Figure 6 shows the averaged distribution of clouds over the Earth’s surface for the year2004. The figure also shows the variability in the cloud cover during a period of two weeksand over the whole year 2004. The lifetime of large-scale cloud systems on Earth is typicallyof about 1-2 weeks (roughly 10 times the rotational period). In the latitude band around60 o south, there is a large stability produced by the vast, uninterrupted oceanic areas. In Figure 7, we show the folded light curve of the Earth in terms of the albedo anomaly,both with and without clouds. Albedo anomaly is defined as the standard deviation (rms)from the mean value over the entire 8-week dataset (e.g., an anomaly value of 0.7 meansthat the albedo in 30% lower than the mean). Here we assume an exposure time of 1 hourand S/N=30. The real Earth presents a much more muted light curve due to the smoothingeffect of clouds, but the overall albedo is higher. Note that the Y scale in the figure areanomalies and not the absolute albedo values.In the top panels of Figure 7 data from 8 weeks of continuous observations are foldedinto a single light curve. In the middle and lower panels, this 8 week period is subdividedand plotted in 3 and 6 periods of 18.6 and 9.3 days, respectively. For a cloudless Earth (leftpanels), the error in the albedo anomaly at a given phase decreases as shorter durations aretaken, because changes in phase and illuminated area decrease.On the contrary, for the real cloudy Earth (right panels), as the data is subdividedin smaller integration periods, the size of the error bar in the albedo anomalies does notdecrease, because of the random influence of clouds at short time scales. In the lower rightpanel of Figure 7, the light curves of consecutive 9-day integration periods vary arbitrarilyin shape from one to the next.Thus, the variability in the averaged light curve is primarily the result of short-termvariability in the cloud cover, a fact that can be exploited in future exoplanet observations.Once the rotational period has been determined, one can measure the average light curve ofan exoplanet, and the excess scatter for different consecutive periods. If the excess scatteringdoes not decrease at short time periods, and the changes are not smooth in time, such ananalysis can indicate the presence of clouds in its atmosphere. However, distinguishing thechanges in the exoplanets light curve from the observational noise will require very stringentS/N ratios. Fortunately, there might be a better way to probe for cloudiness in an exoplanet’satmosphere that we discuss in the following section. 12 – For extrasolar planet observations, a long time series could be subdivided into severalsubsets. Each can be analyzed for significant periodicities as in Figure 8. The data spanningfor 8 weeks is subdivided in several equal-length subperiods (e.g., six periods each of about9 days) and analyzed independently, so that the changes in β and illumination area are min-imized. In this case, several peaks appear in the Fourier periodograms and autocorrelationfunctions near 12 and 24-hours. The autocorrelation analysis show much greater correlationnear 24 hours. For our Earth model with clouds, the best-fit rotation period shifts slightlyto shorter periods. The shifts in the best-fit periodicity from the true periodicity are com-pletely absent when considering an Earth model free of clouds for the same dates and times,even when including added noise. Therefore, we conclude that they are produced by variablecloud cover.The shifts are introduced by the large-scale wind and cloud patterns (Houghton, 2002).Since clouds are displaced toward the west (in the same direction of the Earth’s rotation)by the equatorial trade winds (and to a minor extent by the polar easterlies) the apparentrotational period should be shorter than the rotation period of the surface. On the otherhand, when clouds are moved toward the east (in the opposite direction of the Earth’srotation) by the westerly winds at mid-latitudes, the apparent rotational period should belonger than the rotation period of the surface.In principle, both longer and shorter periodicities could be present in the periodograms,depending on the particular weather patterns. In our models however, we often find shorterapparent rotation rates, but not longer. The explanation probably lies in the differentmechanisms of cloud formation on Earth. In the tropical regions most of the clouds developthrough deep convection. This deep convective clouds have a very active cycle and a shortlifetime, in other words, these cloud systems do not travel far. At mid-latitudes however,deep convection does not occur, and large weather and cloud systems remain stable (andmoving) for weeks (Xie, 2004).Thus, both observing changes (anomalies) in the apparent rotational period and theamount of scatter about the phase-averaged light curve, one can recognize variable cloudcover and distinguish it from the presence of strong surface inhomogeneities, and the presenceof a cloud layer. Thus, photometric observations could be used to infer the presence of a‘variable’ surface (i.e. clouds), even in the absence of spectroscopic data. This would stronglysuggest the presence of liquid water on the planet’s surface and/or in the planet’s atmosphere,especially if the mean temperature of the planet were also determined. This could be an earlystep in selecting the most desirable targets for more intensive follow-up and/or observationswith future more advance missions with more powerful spectroscopic capabilities. 13 –
5. Implications for Future Missions
Finally, we consider the implication for future space missions. We have shown that theintegrated scattered light from the Earth contains enough information to determine Earth’srotation period. However, realistic space missions will likely be photon-starved. Here, weaddress whether precise measurements of the rotation period might be practical with next-generation observatories. First, we will ask for what mission specifications and target starswould it be possible to measure the rotation period of an Earth-close to ∼
2% precision. Thischoice is based on our simulated analysis of the Earth’s light curve that show the rotationperiod can be determined to an average of ∼
2% from data spanning 56 days with a signal-to-noise ratios of ∼
20 or greater and an integration times no longer than ∼ × ∼ ∼
11 such stars included in the possible TPF-C targetlist of Brown (2005) around which an Earth clone’s rotation period could be measured to ∼ × ∼ ∼
35 stars in the sampletarget list of Brown (2005) for which an Earth clone’s rotation period could be measuredto ∼ ∼ ∼
35 or ∼
90 stars, for the two mission scenarios. We caution thatthese last two figures are very approximate, since the expressions of Brown (2005) breakdown for large fractional bandpasses.It would be somewhat easier to achieve the needed signal-to-noise ratios for a planet thatrotates more slowly than the Earth. If we were to ignore the effects of the planet revolvingaround the host star, then our results could be scaled to apply to an Earth-like planet with 14 –a rotation period of P rot . For such a planet, the threshold for achieving a rotation periodprecision of ∼
2% would require achieving signal-to-noise ratio of 20 with integration timesof no more than ∼ P rot / ∼
2% or better.The above estimates assume that the cloud patterns on the Earth would not be affectedby the alternative rotation period. Further, the above estimates also assume that the du-ration of the time series scales with the rotation period of the planet. A single continuoustime series would be impossible for a planetary system viewed nearly edge-on, since theplanet would periodically pass inside the glare of the star (or inner working angle of thecoronagraph). Further research is needed to determine how well the rotation period couldbe measured by combining multiple shorter photometric time series, and which are the mostsuitable spectral ranges. For a planetary system with an orbital plane nearly in the planeof the sky, it would be possible to obtain photometric time series spanning 56 × P rot , evenfor slowly rotating planets. Depending on what other planets have been found, it might ormight not be practical to devote so much mission time to a single planetary system. Wealso caution that for planets with extremely slow rotation periods that approach the orbitalperiod (e.g., Venus), our assumed scaling may break down due to seasonal effects and thelarge changes in the viewing geometry.
6. Conclusions
Exoplanets are expected to deviate widely in their physical characteristics and not allexoplanets will have photometric periodicities. Some planets, such as Venus, are 100% cloudcovered and show no significant photometric variability with time. A variable photometricdata set with no autocorrelation signal may be indicative of slow rotation or chaotic weather.On Earth, the presence of continents and ocean currents results in relatively stableglobal cloud patterns, despite large variability on short time and length scales. Here wehave shown that, despite Earth’s dynamic weather patterns, the light scattered by the Earthto a hypothetical distant observer as a function of time contains sufficient information tomeasure Earth’s rotation period to within a minute, on the most favorable cases. Theaccuracy in the rotational period determination is a function of the viewing geometry, S/Nratio, temporal sampling and the duration of our simulated time series. The rotation periodcould be directly compared to numerical simulations of planetary formation, to probe the 15 –late stages of planetary accretion.According to our calculations, the duration of the observations is comparable to theintegration times needed for spectroscopic observations to search for multiple atmosphericbiomarkers (Traub et al, 2006). Thus, we recommend that a photometric time series spanningweeks to months be carried out simultaneously with planet spectral characterization, via“spectrophotometry”. Photon counting CCDs have no read noise and are being adoptedin mission concept studies for TPF-C and related missions (Woodgate et al, 2006). Suchphoton counting CCDs tag photon arrival at different wavelengths, and allows later binningin different ways. Observations of an exoplanet spanning several weeks could be binnedover the entire observational period to retrieve a low-resolution spectra and characterize itsatmospheric composition. Additionally, the data could also be binned in shorter time periodsover all wavelengths in order to retrieve the rotation rate and explore the presence of activeweather.We have shown in this paper that, if the rotation period of an Earth-like planet can bedetermined accurately, one can then fold the photometric light curves at the rotation periodto study regional properties of the planet’s surface and/or atmosphere. Most significantly wecould learn if dynamic weather is present on an Earth-like exoplanet, from deviations froma fixed phase curve. In contrast, a cloud-free planet with continents and oceans would notshow such light curve deviations. With phased light curves we could study local surface oratmospheric properties with follow-up photometry, spectroscopy, and polarimetry, to detectsurface and atmospheric inhomogeneities and to improve the sensitivity to localized biomark-ers. Finally, we have also provided guidance for the necessary specifications for future spacemissions.Research by E. Pall´e was supported by a ‘Ramon y Cajal’ fellowship. Support for E.B.Ford was provided by NASA through Hubble Fellowship grant HST-HF-01195.01A awardedby the Space Telescope Science Institute, which is operated by the Association of Universitiesfor Research in Astronomy, Inc., for NASA, under contract NAS 5-26555.
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19 –Table 1: The existence of 21 years of global cloud observations allow us to simulate thephotometric time series of Earth for each of these years, with the exact same geometricalconfigurations, with only clouds changing. For each year we have calculated the main pe-riodicities resulting from the photometric simulations analysis. In the table, we show thepercentage of years in which the main periodicity is found to be (i) 24 ± ∆ hours, (ii) 12 ± ∆hours or (iii) other periods. For example a 95% value in a given periodicity, means that for20 of the 21 available years (cloud configurations) that was the primary periodicity. Fourierand autocorrelation analysis results are both shown. In all cases ± ∆ is taken as the exposuretime.Area 24 12 other 24 12 otherFourier AutoS/N=20, Exp=0.5h, Followup=2wN. Pole 38 4 58 100 0 0Lat +45 9 76 15 100 0 0Equator 0 71 29 100 0 0Lat -45 4 57 39 100 0 0S. Pole 9 23 68 100 0 0S/N=20, Exp=0.5h, Followup=8wN. Pole 66 4 30 100 0 0Lat +45 4 71 25 100 0 0Equator 0 80 20 100 0 0Lat -45 0 76 24 100 0 0S. Pole 23 42 35 100 0 0S/N=5, Exp=0.5h, Followup=2wN. Pole 4 0 96 33 0 67Lat +45 0 9 91 52 0 48Equator 0 33 67 76 0 24Lat -45 0 28 72 80 0 20S. Pole 0 0 100 23 0 77S/N=20, Exp=1.7h, Followup=2wN. Pole 85 4 11 80 0 20Lat +45 66 28 6 85 0 15Equator 38 47 15 85 0 15Lat -45 57 33 10 90 0 10S. Pole 71 9 20 80 0 20 20 –Table 2: Same as Table 1, but the calculations are done for a cloud-free Earth.Area 24 12 other 24 12 otherFourier AutoS/N=20, Exp=0.5h, Followup=2wN. Pole 95 4 1 100 0 0Lat +45 19 80 1 100 0 0Equator 4 95 1 100 0 0Lat -45 4 95 1 100 0 0S. Pole 0 100 0 100 0 0S/N=20, Exp=0.5h, Followup=8wN. Pole 95 4 1 0 0 100Lat +45 4 95 1 4 0 96Equator 0 100 0 100 0 0Lat -45 4 95 1 100 0 0S. Pole 0 100 0 95 0 5S/N=5, Exp=0.5h, Followup=2wN. Pole 52 0 48 100 0 0Lat +45 28 71 1 100 0 0Equator 4 95 1 100 0 0Lat -45 4 95 1 100 0 0S. Pole 0 95 5 100 0 0S/N=20, Exp=1.7h, Followup=2wN. Pole 100 0 0 100 0 0Lat +45 100 0 0 100 0 0Equator 4 95 1 100 0 0Lat -45 9 90 1 100 0 0S. Pole 76 23 1 100 0 0 21 –Fig. 1.— The yearly evolution of relative flux of the Earth with respect to the Sun fromfive different viewing geometries. The equatorial veiw is marked in red, the primarilynorther/southern hemisphere views are in green and pink (respectively) and the north andsouth polar views are in dark and light blue (respectively). 22 –Fig. 2.— The Earth from space. The several images shown the viewing geometry of Earthfor the exact same day and time (2003/11/19 at 10:00 UT) but from our five differentviewpoints: from 90 ◦ above the ecliptic (north polar view) (1), from 90 ◦ below the ecliptic(south polar view) (2), from +45 ◦ north of the ecliptic (primarily northern hemisphere inview) (3), from − ◦ below the ecliptic (primarily southern hemisphere in view) (4), andfrom within the ecliptic (5). Note how the scenery from the different viewpoints, could wellhave been taken from different planets. 23 –Fig. 3.— Autocorrelation function of the scattered light by the real Earth (blue) and acloud-free Earth (black). An 8-week time series with S/N ratio of 40 and 0.1 hour observingcadence has been chosen, using cloud data from 1985. 24 –Fig. 4.— Top panel: The plot represents the error that one would get in estimating theEarth’s rotation rate from the globally integrated photometric light curve. Each point is theerror of the averaged rotational period found for 21 years with different (real) cloud patternsfor the same geometries. The five different colors indicate five different viewing angles (i.eequator means the observer is looking at the Sun-Earth system from the ecliptic plane, theNorth pole indicates the observer is looking at the Sun-Earth system from 90 ◦ above theecliptic). All calculations are given for a 90 ◦ phase angle in the orbit (i.e. one would see aquarter of the Earth’s surface illuminated). In the plot, the top broken line represents anaccuracy in determining the rotational period of 10 minutes , and the lower one of 1 minute .Lower panel: Same as in the top panel, but this time the S/N is fixed and the exposuretime is allowed to vary. As in the top panel, an object follow up of two months (8 weeks) isconsidered. 25 –Fig. 5.— Periodogram analysis of the Earth’s p e f e ( β ) times series as seen from five differentviewpoints, at phase angle 90 o . From top to bottom, the five viewpoints are: the northpolar view (a), primarily northern hemisphere view (b), the equatorial view (c), primarilysouthern hemisphere view (d), and the south polar view (e). The right column represents theperiodograms for the year 2000, while the left column represents the periodograms for year2004. The geometry is exactly the same for the two years, only the clouds have changed. Inall panels, the periodogram is shown for data lasting for a period of 2 (black line), 4 (blueline) and 8 weeks (red line) around phase 90 o . In all panels a thin vertical line indicates the“real” 24-hour periodicity. 26 –Fig. 6.— Large-scale cloud variability during the year 2004. In panel (a) the 2004 yearlymean cloud amount, expressed in percentage coverage, is shown. In panels (b) and (c) cloudcoverage variability (ranging also from 0 to 100%), is illustrated over a period of 2 weeks and1 year, respectively. Over the course of 2 weeks, the presence of clouds at a given locationis highly correlated. Note how the cloud variability is larger at weekly time scales in thetropical and mid-latitude regions rather than at high latitudes. Over the course of a wholeyear the variability is closer to 100% over the whole planet (i.e., at each point of the Earththere is at least a completely clear and a completely overcast day per year). One exceptionto that occurs at the latitude band near − o , an area with heavy cloud cover, where thevariability is smaller, i.e., the stability of clouds is larger. 27 –Fig. 7.— Light curves of the Earth observed from the ecliptic plane at phase 90 o (half phase).Left column are the light curves of a cloud-free Earth and right columns are the light curvesfor the real Earth. The Y-scale in the right and left panels is different because of the moremuted variability in the albedo introduced in the real Earth by clouds. Fifty six days (twomonths) of continuous observations are divided from top to bottom in 1, 3, and 6 sub-series,and folded over the 24-hour rotational period of the Earth for analysis. Note the contrastbetween the uniformity of the light curves of an ideal (cloudless) Earth and the real Earthlight curves. Also note how the change in the shape of the light curve is smooth (ordered intime) from one series to the next in the case of a cloudless Earth, but it is random for thereal Earth. The size of the error bars are the standard deviation of the mean. 28 –Fig. 8.— Left: Periodogram analysis of the Earth’s p e f e ( β ) equatorial time series. Right:Autocorrelation function of the same time series. For this figure we select 8 weeks of data(56 days) and calculate the periodograms and autocorrelation functions (top panels). S/Nratio are set here to 50 for clarity purposes. Then we subdivide these data in three (middlepanels) and six (bottom panels) equally-long time series and we again calculate the separateperiodograms and autocorrelations. In the figures, different colors indicate different datasubperiods. Note the appreciable decrease in the retrieved rotation rate for some of the timeseries in the bottom panels, detectable with both autocorrelation and Fourier analysis. 29 –Fig. 9.— Here we show the threshold host star magnitude and planet rotation period forwhich a signal-to-noise of ∼
20 or greater can be obtained for each integration of ∼ P rot / × P rot would typically result in measuring the rotation period to ∼
2% for an Earth-like planet.Higher precision measurements of the rotation period would be obtained for V and P rotrot