Using polarimetry to retrieve the cloud coverage of Earth-like exoplanets
AAstronomy & Astrophysics manuscript no. paper c (cid:13)
ESO 2018May 23, 2018
Using polarimetry to retrieve the cloud coverageof Earth-like exoplanets
L. Rossi (cid:63) and D. M. Stam
Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The NetherlandsReceived February 09, 2017; accepted July 12, 2017
ABSTRACT
Context.
Clouds have already been detected in exoplanetary atmospheres. They play crucial roles in a planet’s atmosphere and climateand can also create ambiguities in the determination of atmospheric parameters such as trace gas mixing ratios. Knowledge of cloudproperties is required when assessing the habitability of a planet.
Aims.
We aim to show that various types of cloud cover such as polar cusps, subsolar clouds, and patchy clouds on Earth-likeexoplanets can be distinguished from each other using the polarization and flux of light that is reflected by the planet.
Methods.
We have computed the flux and polarization of reflected starlight for di ff erent types of (liquid water) cloud covers on Earth-like model planets using the adding–doubling method, that fully includes multiple scattering and polarization. Variations in cloud-topaltitudes and planet–wide cloud cover percentages were taken into account. Results.
We find that the di ff erent types of cloud cover (polar cusps, subsolar clouds, and patchy clouds) can be distinguished fromeach other and that the percentage of cloud cover can be estimated within 10 %. Conclusions.
Using our proposed observational strategy, one should be able to determine basic orbital parameters of a planet such asorbital inclination and estimate cloud coverage with reduced ambiguities from the planet’s polarization signals along its orbit.
Key words. polarimetry – exoplanets – atmospheres – clouds
1. Introduction
After two decades with huge successes in exoplanet detection,the next step in exoplanetary science is the characterization ofplanets around other stars and the comparison of their propertieswith those of the planets in the solar system. Such a comparisonwill undoubtedly lead to new understandings and insights in thephysical processes that form and shape planets, their surfaces,atmospheres, climates, and that determine habitability. Despitethe fast developments in dedicated telescopes and instruments,such as the Gemini Planet Imager (GPI) (Macintosh et al. 2014)and SPHERE on ESO’s VLT (Beuzit et al. 2006), directly ob-serving exoplanets is still an immensely di ffi cult task even forthe exoplanets closest to us and will remain so for several years,because of the low flux of an exoplanet compared to the highflux of the parent star that is very close by in angular distance asseen from the Earth.A significant contribution to the planetary signal will comefrom clouds in the planetary atmosphere. Clouds can influence aplanetary atmosphere and surface in several ways (see e.g., Mar-ley et al. 2013). Firstly, clouds scatter and absorb incident lightof the parent star and thermal radiation of the planet itself, andwith that they play a crucial role in the radiative balance of theplanet. Clouds thus influence the climate, the surface tempera-ture, and in particular the presence of liquid surface water. Thelatter is generally assumed to be essential for the habitability ofa planet (Kitzmann et al. 2010; Yang et al. 2013). Neglecting thepresence of clouds in atmospheric modeling can lead to under-estimating the surface temperature of a planet (Kitzmann et al. (cid:63) e-mail: [email protected] . µ m for planets around type F stars.At visible wavelengths, the presence, altitude and horizontal dis-tribution of clouds can change the observable depth of absorptionbands in spectra of reflected starlight, by scattering light back tospace before it reaches the absorber, and / or by increasing the av-erage optical path through the atmosphere (Fauchez et al. 2017).Clouds can also hide biosignatures from the surface, in particularthe so–called red-edge, the steep increase in the albedo of veg-etation between the visible and the near infrared (Tinetti et al.2006; Montañés-Rodríguez et al. 2006; Seager et al. 2005). Andfinally, in transit observations, Line & Parmentier (2016) showedthat when analyzing the stellar spectrum that is filtered throughthe upper layers of a planetary atmosphere during a planetarytransit, clouds along the limb will not only influence the retrievedamount of absorbing gas, by blocking stellar light, but their in-fluence on the measured spectrum can also mimic the signals ofa high mean molecular mass of the atmosphere. Article number, page 1 of 14 a r X i v : . [ a s t r o - ph . E P ] A ug & A proofs: manuscript no. paper
In this article, we investigate the influence of cloud proper-ties on the degree and direction of polarization of starlight thatis reflected by a planet, focusing on the influence of cloud-toppressure, the cloud coverage fraction, and the spatial distribu-tion of the clouds across the planet. Polarimetry promises to bea very powerful method in the detection and especially in thecharacterization of exoplanets. Polarimetry can be used to detectexoplanets because integrated across their disk, the light of so-lar type stars can be considered to be unpolarized (Kemp et al.1987) while the starlight that has been reflected by a planet willgenerally be polarized. Polarimetry thus enhances the contrastbetween a star and its planet.The degree and direction of polarization of a planet dependnot only on the illumination and viewing directions, and thus onthe planet’s phase angle (the angle between the observer and starmeasured from the center of the planet), but also on the com-position and structure of the planetary atmosphere and surface(if present) (Seager et al. 2000; Stam et al. 2004), and measure-ments of the degree and direction of polarization can be used toretrieve the atmospheric and surface properties. A famous exam-ple of this use of polarimetry is the derivation of the composi-tion and size of the particles constituting Venus’s upper cloudsfrom Earth–based observations of the planet’s disk–integratedpolarization at a few wavelengths and across a wide phase an-gle range (Hansen & Hovenier 1974). From Earth, the solar sys-tem’s outer planets can only be observed at a narrow phase an-gle range around 0 ◦ , where the degree of polarization is usuallyvery small, because mainly backscattered light is observed (seee.g., McLean et al. 2017, and references therein). Exoplanets willusually be observable at a large phase angle range (except if theplanetary orbit is seen face-on, thus with an inclination close to0 ◦ : then the phase angle will always be around 90 ◦ ). An exo-planet’s polarization signal will thus usually vary as the planetorbits its star.The structure of this article is as follows. In Sect. 2, we in-troduce the algorithms we use for the radiative transfer compu-tations and the integration of flux and polarization signals acrossthe visible and illuminated part of a planetary disk. In Sect. 3,we describe the model atmospheres and the cloud properties forour model planets. In Sect. 4, we present our numerical resultsand compare the influence of di ff erent types of cloud covers onthe flux and polarization signals of reflected light from a planetacross all phase angles, including a discussion on ambiguitiesthat can arise when retrieving cloud properties from measuredsignals. Finally, in Sect. 5, we discuss an observational strategythat could be used to derive cloud coverage with reduced ambi-guities, and in Sect. 6, we present our conclusions.
2. Numerical algorithms
We describe the starlight that is incident on a planet and thestarlight that is reflected by the planet by Stokes vectors, as fol-lows (see, e.g., Hansen & Travis 1974; Hovenier et al. 2004) F = FQUV , (1)where F is the total flux, Q and U are the linearly polarizedfluxes and V is the circularly polarized flux. These fluxes areusually expressed in W m − or, for example, in W m − nm − when used spectrally resolved. We have assumed the starlight that is incident on a planet tobe unpolarized (see Kemp et al. 1987). This incident light willbe described by F = F , with π F the stellar flux measuredperpendicular to the direction of propagation, and the unit col-umn vector. Starlight that is reflected by an orbiting planet willusually be polarized. Stokes parameters Q and U of this lightare defined with respect to a reference plane, for which we usethe planetary scattering plane, that is, the plane that contains thestar, the planet and the observer. We have ignored the circularlypolarized flux V as its values are very small compared to Q and U , and ignoring V causes no significant errors in the computedvalues of F , Q , and U (Stam & Hovenier 2005).The degree of linear polarization is defined as P (cid:96) = (cid:112) Q + U F . (2)For a planet that is symmetric with respect to the planetary scat-tering plane, the disk–integrated flux U will be zero, and the de-gree of linear polarization can then be defined as P s = − QF . (3)This definition includes the direction of polarization: P s is posi-tive if the polarization is perpendicular to the planetary scatteringplane and negative if it is parallel to the plane.The degree of polarization that we present in this paper per-tains to a planet that is observed spatially resolved from its par-ent star, without any background starlight. In actual observa-tions, even a spatially resolved planet will be surrounded bysome background starlight, depending on for example, the dis-tance between the planet and its star, the brightness of the star,the distance between the observer and the exoplanetary system,and the telescope and the instrument capabilities, such as coron-agraphs and / or adaptive optics. The amount of background lightwill also depend on the wavelength. In the presence of back-ground starlight, the observable degree of polarization in the de-tector pixel that contains the planet would be given by P (cid:96) ∗ = (cid:112) Q + U F + F ∗ , (4)with F ∗ the background stellar flux (that is assumed to be un-polarized) in the pixel. Seager et al. (2000) show simulations of P (cid:96) ∗ of spatially unresolved hot Jupiters where F ∗ is the full stel-lar flux. Because of the huge di ff erence in F and F ∗ , P (cid:96) ∗ is thenof the order of 10 − , depending on the size of the planet, its at-mospheric composition and the phase angle. Detections of suchspatially unresolved exoplanets in polarimetry have not yet beenconfirmed, but first attempts seem promising (Bott et al. 2016,and references therein). Instruments like EPICS on the E-ELT(Kasper et al. 2010; Keller et al. 2010) will combine corona-graphs and extreme adaptive optics to limit F ∗ and increase P (cid:96) ∗ .In the following, we assume F ∗ =
0. The polarization values inour results should thus be regarded as upper limits.
All the computations in this study were conducted under the as-sumption that the model planet is in an ‘edge-on’-orbit as seen bythe observer (the inclination angle i of the orbit is thus 90 ◦ ). Theplanet itself is assumed to be spherical and with no obliquity, sothe sub-observer point was always located on the equator of theplanet. As the planet’s orbital plane was assumed to be alignedwith the horizontal axis in the observer’s reference frame, thespin axis of the planet is vertical as seen by the observer. Article number, page 2 of 14. Rossi and D. M. Stam: Using polarimetry to retrieve the cloud coverage of Earth-like exoplanets
While our model planets are spatially resolved from their star,they themselves are spatially unresolved, meaning that eachplanet is observed as a single speck of light. We computedthe disk–integrated Stokes parameters and polarization P (cid:96) of amodel planet with the following four steps. Step 1.
We projected the planetary disk as seen by the ob-server on the plane of the sky and divide the disk–circumscribingsquare into an n pix × n pix grid of square, equally sized pixelswith the planet’s equator aligned with the (horizontal) x − axis.We projected the center of each pixel onto the spherical planet(discarding pixels with their centers outside the planetary disk)to identify the location on the planet for which we will com-pute the locally reflected Stokes vector. Increasing n pix (i.e., thespatial resolution on the planet) increases the accuracy of ourcomputations, but it also increases the computation time.The number of pixels required to reach a given accuracy in-creases with increasing phase angle α (the angle between thestar and the observer as measured from the center of the planet).Indeed, when the planet is close to ‘full’, the value for n pix re-quired for an accurate result is much smaller than when the sig-nal comes from a narrow crescent of the planet. As a compromisebetween su ffi cient resolution and acceptable computation time,we have used an adaptable value for n pix , given by the followingequation: n pix ( α ) = n pix (0 ◦ ) (cid:104) + sin ( α/ (cid:105) , (5)with n pix rounded up to the nearest integer. Unless stated other-wise, n pix (0 ◦ ) =
40, and thus n pix (180 ◦ ) =
80 (see Appendix Afor a discussion on optimizing n pix ). With this pixel approach,we could straightforwardly model horizontally inhomoge-neous planets by choosing a di ff erent atmosphere and / or surfacemodel for di ff erent pixels. These models are described in Sect. 3. Step 2.
For each projected pixel on the planet, we determinedthe following angles: θ , the angle between the local zenithdirection and the direction toward the star, θ , the angle betweenthe local zenith direction and the direction toward the observer,and the azimuthal di ff erence angle φ − φ , the angle betweenthe plane containing the local zenith direction and the directiontoward the star and the plane containing the local zenith direc-tion and the direction toward the observer (see de Haan et al.1987). The angles depend on the latitude and longitude of theprojected location on the planet (the sub–observer longitude andlatitude equal 0 ◦ ), and θ and φ − φ also depend on phase angle α . We note that we used the azimuthal di ff erence angle ratherthan φ and φ separately because the planetary atmosphereand surface of each pixel are rotationally symmetric withrespect to the local zenith direction. Pixels with θ > ◦ areassumed to be black, as the parent star is below the local horizon. Step 3.
For each projected pixel on the planet and the localatmosphere-surface model, we calculated the Stokes vector ofthe reflected starlight according to (see, e.g., Hansen & Travis1974) F ( θ, θ , φ − φ ) = cos θ R ( θ, θ , φ − φ ) F , (6)with R the first column of the 3 × R with an adding–doubling algorithm that fully includes polarization for all orders of scattering (based on de Haan et al. 1987). Ratherthan embarking on a separate radiative transfer computationfor every pixel, we first computed and store the coe ffi cients R m ( θ , θ ) of the expansion of R ( θ , θ, φ − φ ) into a Fourierseries (0 ≤ m < M , with M the total number of coe ffi cients)for the di ff erent atmosphere–surface models (typically two) onthe model planet. Our adding–doubling algorithm computesthese coe ffi cients at values of cos θ and cos θ that coincide withGaussian abscissae, the total number of which is user–defined.For increased accuracy in the disk–integration, we also com-puted coe ffi cients at cos θ = θ = ◦ ) and cos θ = θ = ◦ ).Given a pixel with local values of θ , θ , and φ − φ , we wereable to e ffi ciently compute its R by summing up the Fouriercoe ffi cients stored for the appropriate atmosphere–surfacemodel, interpolating when necessary. Step 4.
A locally reflected Stokes vector as computed using ouradding-doubling algorithm is defined with respect to the localmeridian plane, which contains both the local zenith directionand the direction toward the observer. We had to redefine eachlocally defined vector to the common reference plane, that is,the planetary scattering plane, with a rotation matrix (see Hov-enier & van der Mee 1983) and the local rotation angle mea-sured between the local meridian plane and the planetary scat-tering plane (for details, see Appendix A). Then we computedthe disk–integrated Stokes vector by summing up the local, re-defined Stokes vectors. The actual area on the three dimensionalplanet that is covered by a projected square pixel varies withthe latitude and longitude, but because all square pixels have thesame size, their respective Stokes vectors as calculated by Eq. 6contribute equally to the disk–integrated planetary signal.We finally normalized each disk–integrated Stokes vectorsuch that at α = ◦ , flux F equals the planet’s geometric albedo.The degree of polarization that we computed from the disk–integrated Stokes vector was independent of this normalizationbecause it is a relative measure (see Eqs. 2 and 3).
3. Atmosphere and surface models
Locally, the atmospheres of our model planets are composedof stacks of horizontally homogeneous layers, filled with gasand, optionally, cloud particles, above a flat, Lambertian (i.e.,isotropic and unpolarized) reflecting surface with albedo a surf .We assumed an Earth–like gas mixture in each layer, with a de-polarization factor δ = .
03 and a molecular mass of 29 g / mol.We do not consider absorption by the gas. Table 1 lists the at-mospheric parameters. Our cloud models are described in moredetail below. We used liquid water clouds. The refractive index of the cloudparticles is n r = . + − i (Hale & Querry 1973). The particlesize distribution is a two–parameter gamma distribution (Hansen& Hovenier 1974) with r e ff = . µ m and ν e ff = .
1, basedon Earth cloud values from Han et al. (1994). All clouds havean optical thickness of 6.0 (Warren et al. 2007), independent ofthe wavelength. We have not investigated the e ff ect of varyingthe particle size distribution and / or the optical thickness of theclouds as this has been studied by Karalidi et al. (2011, 2012).The clouds span a vertical extent of 100 mb and we set theiraltitude to represent low- to mid-altitude clouds correspondingto cumulus, stratus and stratocumulus (see Rossow & Schi ff er1999; Hahn et al. 2001). Table 1 includes the cloud parameters. Article number, page 3 of 14 & A proofs: manuscript no. paper
Parameter Symbol ValueSurface albedo a surf p surf δ / mol] m g / s ] g ff ective radius [ µ m] r e ff ff ective variance ν e ff τ c p c Table 1.
Parameters of our standard model atmosphere and surface.
We investigated three di ff erent types of cloud coverages:sub-solar clouds, polar–cusps, and patchy clouds. We modeledthese clouds by assigning specific pixels to be cloudy. All otherpixels on the planet are cloud–free. Sub-solar clouds are relevant for tidally–locked exoplanets(Yang et al. 2013). To model these clouds, the pixel grid wasfilled such that only the region on a planet with the local solarzenith angle θ smaller than a given angle σ c is cloudy (seeFig. 1a). Polar–cusps are clouds that form where the daily averagedincident stellar flux is below a certain threshold. In this model,the cloudy pixels are located above a threshold latitude L t on theplanet (see Fig. 1b). Patchy clouds can be anywhere on the planet. They are describedby F c , the fraction of all pixels on the whole disk that are cloudy,and the actual spatial distribution of cloudy pixels across theplanet (see Fig. 1c). We generated patchy clouds by drawing 50values from a 2D Gaussian distribution centered on a locationrandomly chosen within the n pix × n pix grid. The covariance ma-trix is given by Σ = n pix (cid:34) x scale y scale (cid:35) , (7)where x scale and y scale are used to fine–tune the shape of thecloud patches along the north–south and east–west axes. Weused x scale = . y scale = .
01 as nominal values in orderto generate clouds with a streaky, zonal–oriented pattern similarto that observed on Earth. Cloud patches are generated acrossthe planetary disk until the desired F c is reached. We defined F c at α = ◦ , because the planetary–wide cloud coverage is morerelevant in terms of climatology than the coverage seen by theobserver. The actual cloud fraction observed at a given angle α larger than 0 ◦ can thus di ff er from the specified value of F c .
4. Polarization signatures of different cloud covers
In this section, we compare the disk–integrated polarization ofstarlight reflected by our model planets for the di ff erent types ofcloud cover defined above. Figure 2 shows the degree of linear polarization P (cid:96) at 500 nm fordi ff erent angular sizes σ c of the sub-solar cloud as a function of α (recall that the actual range of phase angles that an exoplanet x-pixels y - p i x e l s Subsolar cloud, σ = ◦ , α = ◦ x-pixels y - p i x e l s Polar cusps, latitude = 50 ◦ , α = ◦ x-pixels y - p i x e l s Patchy clouds, 42% cover, α = ◦ Fig. 1.
Examples of our three types of cloud cover on a 80 ×
80 pixelgrid: (a) Sub-solar clouds for σ c = ◦ and α = ◦ ; (b) Polar cusps for L t = ◦ and α = ◦ ; (c) Patchy clouds for F c = .
42 and α = ◦ . can be observed at, depends on the orbital inclination angle i ; thisis the largest range occurring along the orbit with i = ◦ ). Therelation between the values of σ c that are used and the e ff ectivecloud coverage F e ff are given in Table 2.As expected, di ff erent values of σ c , and thus di ff erent cloudfractions, yield di ff erent curves with common features. First, as σ c increases, the (primary) rainbow feature near α = ◦ that isdue to light scattered by spherical water cloud droplets (Karalidiet al. 2012; Bailey 2007) becomes more distinct (its maximumvalue decreases slightly). The angular location of the rainbowis determined by the micro–physical properties, mostly the re-fractive index, of the clouds particles (see e.g., Hansen & Travis1974). The small bump in P (cid:96) at phase angles below 10 ◦ is dueto the glory that arises from light that is backscattered by thespherical cloud particles (see Hansen & Travis 1974).Patchy clouds Sub-solar clouds Polar cusps F c σ c ( ◦ ) F e ff L t ( ◦ ) F e ff Table 2.
Cloud covers of equivalent cloud fraction with the parame-ters used to generate them. F c is the fraction of the planet covered bypatchy clouds, σ c is the angular width of the sub-solar clouds, L t is thethreshold latitude of polar cusps and F e ff is the e ff ective coverage i.e.,the actual coverage for the considered distribution of sub-solar cloudsand polar cusps.Article number, page 4 of 14. Rossi and D. M. Stam: Using polarimetry to retrieve the cloud coverage of Earth-like exoplanets Furthermore, while the di ff erent model planets show di ff er-ent values of P (cid:96) at small α (except at α = ◦ , where P (cid:96) = P (cid:96) are very simi-lar at large phase angles. In fact, P (cid:96) of these planets is dominatedby that of the cloud as long as it is on the illuminated and visi-ble part of the planet. As the tidally locked planets move alongtheir orbit, the clouds disappear from the observer’s view at aphase angle that depends on σ c and on a possible o ff set of thecloud with respect to the sub-solar point (not shown in Fig. 2)(as seems to be the case for Kepler 7b, García Muñoz & Isaak2015), leaving only the P (cid:96) due to the gas. The maximum polar-ization due to the Rayleigh scattering gas is very high (around α = ◦ ) because at this wavelength, there is little multiple scat-tering and the surface is black. We note that the oscillations of P (cid:96) just before the clouds disappears completely from view aredue to the pixellation of the cloud, which becomes more appar-ent when the visible part of the cloud narrows while the cloudis disappearing across the limb of the planet. These oscillationsdecrease when the number of pixels is increased. Phase angle [ ◦ ] P ‘ [ % ] σ c = ◦ σ c = ◦ σ c = ◦ σ c = ◦ Fig. 2.
Degree of linear polarization P (cid:96) at λ =
500 nm as a functionof phase angle α for a sub-solar cloud with p c =
800 mb, for di ff erentvalues of σ c . For σ c = ◦ , the planet is cloud–free. The angular featurearound α = ◦ is the (primary) rainbow. Figure 3 is similar to Fig. 2, except for polar cusps for dif-ferent values of the threshold latitude L t . (the relation between L t and the e ff ective clouds coverage F e ff is given in Table 2).Polar cusps clouds exhibit a more continuous behavior of thepolarization than sub–solar clouds, because they remain in viewas our model planets rotate. Like with the sub-solar clouds, therainbow feature is clearly visible near α = ◦ . The peak of P (cid:96) around α = ◦ , is again due to Rayleigh scattering. The smaller L t , thus the larger F e ff , the stronger the rainbow and the lowerthe peak of polarization near 90 ◦ , because the smaller the contri-bution of (highly polarized) Rayleigh scattered light. Patchy clouds are interesting because each pixel on the planethas its specific illumination and viewing geometries (eventhough phase angle α is the same for all pixels) and thereforecontributes its own polarization signal to the disk–integrated sig-nal. The precise locations of the cloudy pixels on the disk thusinfluence P (cid:96) of the planet, and can give rise to di ff erent P (cid:96) valuesfor the same cloud coverage fraction F c . Because of this, for agiven value of F c and for each phase angle considered, we gener- ◦ ]010203040506070 P ‘ [ % ] L t = ◦ L t = ◦ L t = ◦ L t = ◦ Fig. 3.
Similar to Fig. 2 except for polar cusps clouds for di ff erent valuesof the threshold latitude L t . ate 300 independent, random cloud patterns. The curves shownin subsequent figures for a given F c are the averages of these 300patterns. This allows us to explore the range of possible valuesof the disk–integrated polarization due to di ff erent locations ofthe cloud patches on the planet. This variability is not directlyrelated to temporal variations, because the 300 patterns are in-dependent: they do not depend on the rotation of the planet, theposition of the patches is purely random and not bound to a re-alistic climate model.Figure 4 shows P (cid:96) for di ff erent combinations of F c andcloud-top pressures p c . The change of the strength of the rain-bow and the peak around α = ◦ is similar to what was seen inFig. 3. For a given value of F c , a larger value of p c is related tosmaller values of P (cid:96) , in particular around α = ◦ . This is due tothe di ff erent amount of gas above the clouds: a larger cloud-toppressure (i.e., a lower cloud-top altitude) leaves more gas abovethe clouds and thus more relatively highly polarized Rayleighscattered light (with increasing gas optical thickness above theclouds, the polarization would reach a maximum value beforestarting to decrease due to the increase of multiple scattering). Phase angle [ ◦ ] P ‘ [ % ] p c = F c = p c = F c = p c = F c = p c = F c = p c = F c = p c = F c = p c = F c = p c = F c = p c = F c = p c = F c = p c = F c = p c = F c = Fig. 4. P (cid:96) at λ =
500 nm for di ff erent cloud-top pressures p c (600, 700,and 800 mb) and cloud coverages F c (0.3, 0.5, 0.7, and 0.9) for planetswith patchy clouds. Di ff erent line–styles indicate di ff erent values of F c and di ff erent colors indicate di ff erent values of p c . Each curve is the av-erage of 300 curves pertaining to 300 randomly generated patchy cloudpatterns for the given values of F c and p c . The di ff erences in P (cid:96) due to di ff erences in cloud-top pressure p c should be regarded with care: Fig. 5 shows the average polar- Article number, page 5 of 14 & A proofs: manuscript no. paper ization curves for F c = . ff erent cloud-top pressures thatwere also shown in Fig. 4, except here the shaded area representsthe variability of the 300 curves computed for p c =
600 mb. Wehave considered the variability within the ± σ interval, unlessstated otherwise, where σ is the (absolute) standard deviation ofthe distribution of values for P (cid:96) obtained from the 300 generatedcloud patterns, at a given α . We used the ± σ interval in orderto limit the influence of outliers. As can be seen in Fig. 5, thevariability due to di ff erent cloud patterns for a given value of F c is larger than the di ff erences in P (cid:96) due to varying p c . Withpatchy clouds, it is thus di ffi cult to accurately retrieve p c frommeasurements of P (cid:96) in a single wavelength region. Interestingly,the variability in the rainbow due to di ff erent cloud-top pressuresappears to be negligible for every F c . Phase angle [ ◦ ] P ‘ [ % ] p c = F c = p c = F c = p c = F c = Fig. 5.
Similar to Fig. 4, except only for F c = .
5. The shaded areaindicates the 2 σ variability for the p c =
600 mb curve.
The variability in P (cid:96) depends on F c , as shown in Fig. 6: thesmaller F c , the larger the variability, because cloudy pixels havemany more possible locations on the planet (the comparison be-tween di ff erent cloud coverage types in this figure will be dis-cussed in Sect. 4.3). With a small cloud coverage, the proba-bility that the visible part of the planetary disk is completelycloud–free also increases, in particular at larger phase angles.Comparing Figs. 6 and 7, the e ff ect of the wavelength λ on thevariability can be seen: at longer wavelengths (Fig. 6), the di ff er-ence between the contribution of a cloudy and a cloud–free pixelis large. In the blue ( λ =
300 nm, Fig. 7), however, the gas abovethe clouds scatters more e ffi ciently and P (cid:96) is less sensitive to thecloud distribution, resulting in less variability.The dependence of the variability on the amount of Rayleighscattering also implies that with increasing cloud-top pressure p c (i.e., lower cloud-top altitude), the variability for a given value of F c decreases. This shows from the 1 σ variability of P (cid:96) of plan-ets with patchy clouds as a function of p c (Fig. 8) in the blue( λ =
300 nm). The 1 σ variability of the flux is insensitive to p c .The variability in P (cid:96) is not a direct proxy for cloud-top pressuresand hence altitudes, because the measured variability will alsobe determined by instrumental e ff ects and observational con-straints. Also, cloud patterns might not be as randomly locatedon a planetary disk as in our model, and cloud-top pressures willvary across the planet. Nevertheless, our results imply that thevariability of the polarization could be a source of information.Clearly, cloud-top pressure and cloud fraction both influence P (cid:96) and this could lead to ambiguous retrievals. As an example,Fig. 9 shows P (cid:96) and its variability in the visible ( λ =
500 nm)for patchy clouds with F c = . p c = . F c = . p c = . ff erences are much larger: as the cloud cover of case B islarger, this planet reflects more light than the planet of case A.More importantly, the variabilities of the two cases are mutuallyexclusive, especially for α < ◦ . Therefore, although the caseshave similar polarization signals, they could be distinguished us-ing their reflected flux, assuming the radius of the planet and / orits distances to its star and the observer are known accuratelyenough. For example, to distinguish case A from case B usingthe planet’s reflected flux, the planet radius should be knownwell within 10 %, assuming the distances are accurately knownand the albedo of the surface below the clouds can be assumedto be similar (see Sect. 4.4).We note that the strength of the rainbow is, again, nearlyidentical for both cases, both in flux (compared to the flux atslightly smaller or larger α ) and in polarization. This strengthensthe application of the rainbow feature for cloud particle charac-terization (Karalidi et al. 2012; Bailey 2007). We now compare the signatures of di ff erent types of cloud coverfor the same values of F c . From Fig. 6 ( λ =
500 nm), it seemsthat sub–solar clouds should be easiest to identify from measur-ing P (cid:96) across a range of phase angles, because P (cid:96) will follow theRayleigh polarization curve once the cloud has disappeared overthe limb. The Rayleigh polarization curve is very distinct fromthe cloud polarization curve, at least for liquid water clouds. Forexample, in Fig. 6, the cloud coverage for F c = .
10 correspondsto a sub–solar cloud with σ c = ◦ (see Table 2), which disap-pears completely from view around α = ◦ . The cloud cover-age for F c = . σ c = ◦ which disappears onlyaround α = ◦ .The polar and patchy cloud types are more di ffi cult to tellapart as they have a similar phase angle behavior. Polar cloudswould, however, provide a more stable signal than patchy clouds,especially for small values of F c , and one could presumably usethe variability as a proxy to determine the cloud type and to es-timate the cloud cover patchiness.It is also worth noting that the three types of cloud coverall show a distinct rainbow in a phase angle region where thevariability in P c due to cloud patchiness is small. This is impor-tant as the rainbow has been proposed as a tool to identify liquidwater clouds, and to characterize their micro-physical properties(Karalidi et al. 2011; Bailey 2007). The rainbow can apparentlynot be used to retrieve the type of cloud coverage, although thedi ff erence between P (cid:96) in the rainbow and in the continuum in-creases with increasing F c , for small ( < .
6) values of F c .Figure 7 is similar to Fig. 6, except for λ =
300 nm. At suchshort wavelengths, the scattering by the gas above the cloudsobliterates any di ff erences between the cloud coverage types.The maximum value of P (cid:96) (around α = ◦ ) decreases some-what with increasing F c for all cloud coverage types. The dif-ference between the maximum P (cid:96) obtainable for Rayleigh scat-tering, and the maximum observed could thus help estimating F c . Article number, page 6 of 14. Rossi and D. M. Stam: Using polarimetry to retrieve the cloud coverage of Earth-like exoplanets P ‘ [ % ] F c = F c = Phase angle [ ◦ ] P ‘ [ % ] F c = Phase angle [ ◦ ] F c = Patchy cloudspolar cuspssubsolar cloud
Fig. 6.
Comparison of P (cid:96) at λ =
500 nm for di ff erent types of cloud cover for F c = .
1, 0.4, 0.6, and 0.8. The cloud-top pressure p c is 800 mb. Thesolid line shows the average of 300 patchy cloud patterns. The shaded area shows the variability of the 300 curves. − P ‘ [ % ] F c = F c = Phase angle [ ◦ ] − P ‘ [ % ] F c = Phase angle [ ◦ ] F c = Patchy cloudspolar cuspssubsolar cloud
Fig. 7.
Similar to Fig. 6, except at λ =
300 nm.
In the simulations discussed so far, we have only consideredblack surfaces, thus ignoring any reflection and (de)polarizationthat could be induced by light reflected o ff the surface. On rockyplanets, however, the surface could be covered by various typesof rocks, sand, liquids, soil, and even vegetation, thus with di ff er-ent surface albedos and / or bidirectional reflection functions. Theinfluence of the albedo of Lambertian, thus completely depolar-izing, reflecting surfaces on planetary phase curves has been in-vestigated before (see, e.g., Stam 2008, and references therein).The main e ff ect of such a reflecting surface is that it adds un- polarized light to (the bottom of) the atmosphere, and thus usu-ally increases the flux and decreases the degree of polarizationof the light that emerges from the top of the atmosphere. Theunpolarized surface light might change the angular location ofthe maximum of polarization due to Rayleigh scattering, but notchange the general phase angle variation (Stam 2008). As notedby Karalidi et al. (2012), unpolarized surface reflection mightalso decrease the strength of the rainbow feature. But in all cases,increasing the cloud coverage reduces the e ff ect of the surface onthe phase curves in flux and polarization. Article number, page 7 of 14 & A proofs: manuscript no. paper σ p [ % ] F c = F c = Phase angle [ ◦ ] σ p [ % ] F c = Phase angle [ ◦ ] F c = P c = P c = P c = Fig. 8.
The 1- σ variability on P (cid:96) at λ =
300 nm as a function of α fordi ff erent cloud coverages F c and cloud-top pressures p c . Phase angle [ ◦ ] F l u x P c = F c = P c = F c = Phase angle [ ◦ ] − P ‘ [ % ] P c = F c = P c = F c = Fig. 9.
The flux (top) and P (cid:96) (bottom) at λ =
500 nm from two di ff erentpatchy cloud configurations with p c =
600 mb, F c = . p c =
800 mb, F c = . σ variability for 300 cloud patterns. The case of an ocean surface is less straightforward, becauseFresnel reflection is both anisotropic and polarizing. In particu-lar, Fresnel reflection produces the so–called sun-glint: the sharpreflection when the reflection angle equals the incident angle(and φ − φ = ◦ ). To investigate the e ff ect of Fresnel reflec-tion, we have performed similar computation as in the previoussections, except with cloud–free pixels in which the black sur-face is replaced by a Fresnel reflecting surface above a black wa-ter body, as also used by Stam (2008). We consider a calm, flatocean to obtain the largest e ff ect of the glint as waves random- ize and thus reduce the maximum of polarization due to Fresnelreflection (Zugger et al. 2010).The e ff ects of the Fresnel reflection are very small, as canbe seen in Figs. 10–11, where we show the di ff erences in thereflected flux and polarization at λ =
300 nm and 700 nm.The reflected flux is generally larger when Fresnel reflection isincluded across the whole phase angle range. With increasingcloud coverage F c , the di ff erences in flux between the Fresnelreflecting surface and the black surface decrease, as expected. At λ =
300 nm, the di ff erences in flux are larger than at λ =
700 nm;because the di ff use skylight is brighter at shorter wavelengths,and the surface receives and reflects light from and in all direc-tions. Also, with increasing F c , the di ff erences between the cloudcoverage types increase, with in particular, the flux di ff erence forthe subsolar clouds being lower than for the other types at α = ◦ and higher around α = ◦ , but the di ff erences are very small(about 1% at 300 nm) to start with. At 700 nm, the flux phasecurve for the subsolar cloud case is clearly di ff erent from theother curves for F c > . ff erencesare very small). At large phase angles, the subsolar cloud disap-pears completely from sight (even for F c = . α . Indeed, because at such short wavelengths, the surface is il-luminated from all directions by the di ff use skylight and reflectsback in all directions, the surface light decreases P (cid:96) of the lightemerging from the top of the atmosphere. The e ff ect is largest forthe subsolar cloud. At longer wavelengths ( λ =
700 nm), wherethe Rayleigh scattering hardly contributes, the e ff ect of the Fres-nel reflection is larger, and the Fresnel reflecting planet is morepolarized than the black planet at almost all α , except when thereare subsolar clouds. The polarization di ff erences for the plan-ets with subsolar clouds are actually very similar to those of theother planets up to the phase angle where the subsolar clouds dis-appear across the limb. While at moderate phase angles, the po-larization of the planet with Fresnel reflection is still influencedby the Rayleigh scattering (see Fig. 10), at larger phase angles,the polarization of the glint (that is perpendicular to the refer-ence plane) becomes significant. The variability of patchy cov-ers is barely a ff ected by the presence of the glint, with changesin values of σ on a similar scale as those due to the cloud-toppressure.
5. Observational strategy
Below, we propose an observational strategy that could allowus to retrieve information about the cloud cover and the cloudproperties.
Firstly , in the blue ( λ <
400 nm), the polarization is quiteinsensitive to the clouds because of the e ffi ciency of Rayleighscattering. This is true for all types of cloud coverage (Fig. 7).Short wavelengths could thus be used to retrieve information ona planet that would be otherwise remain entangled with cloudproperties and cloud coverage variations.An example of such information is an estimation of a planet’sorbital parameters, as this requires assumptions on the planet’spolarization (Fluri & Berdyugina 2010), which in the visible isstrongly influenced by for example, clouds. Changes in StokesQ and U as a function of orbital phase, usually under the form Article number, page 8 of 14. Rossi and D. M. Stam: Using polarimetry to retrieve the cloud coverage of Earth-like exoplanets − F F c = F c = Phase angle [ ◦ ] − F F c = Phase angle [ ◦ ] F c = Patchy cloudspolar cuspssubsolar cloud − − − − P ‘ [ % ] F c = F c = Phase angle [ ◦ ] − − − − P ‘ [ % ] F c = Phase angle [ ◦ ] F c = Patchy cloudspolar cuspssubsolar cloud
Fig. 10. Di ff erences F glint − F black (left) and P glint (cid:96) − P black (cid:96) (right) at λ =
300 nm, as functions of the phase angle α for di ff erent cloud coverage typesand values of F c . − − F F c = F c = Phase angle [ ◦ ] − − F F c = Phase angle [ ◦ ] F c = Patchy cloudspolar cuspssubsolar cloud − − − P ‘ [ % ] F c = F c = Phase angle [ ◦ ] − − − P ‘ [ % ] F c = Phase angle [ ◦ ] F c = Patchy cloudspolar cuspssubsolar cloud
Fig. 11.
Similar to Fig. 10, except for λ =
700 nm. of so–called QU –diagrams, that show linearly polarized flux Q as a function of linearly polarized flux U along the planetary or-bit, have been shown (Brown et al. 1978; Wiktorowicz & Stam2015) to be very helpful for the estimation of orbital parameters.Because clouds are still detectable in the blue as they do influ-ence the maximum value of P (cid:96) compared to that of a cloud–freeplanet (see Fig. 7), we have computed Q and U as a function ofthe planetary phase angle for some of the patchy cloud cases, fordi ff erent values of the orbital inclination angle i and the longi-tude of the ascending node Ω . Angle Ω is defined as the anglebetween the observer’s upward direction and the line through thetwo points of greatest elongation of a planet (for the descriptionof the computation of the QU –diagrams, see Appendix A).Figure 12 shows Q / F and U / F as a function of planetaryphase angle (i.e., the phase angle of the planet when seen in anedge-on configuration, see Appendix A for its relation with thephase angle at other inclinations) at 300 nm for two di ff erent val-ues of i and for di ff erent cloud fractions F c . At this short wave-length, the amplitude and orientation of the curves are insensitive to F c , indicating the use of measuring (part of) this pattern for or-bit parameter determination, without knowledge on the presenceand / or distribution of clouds. Figure 13 is similar to Fig. 12, ex-cept at 700 nm, where Rayleigh scattering is less e ffi cient. Here,the influence of light that has been scattered by cloud particlesresults in a strong dependence of the curves on F c , hence pre-venting the use of the determination of the orbital parameters.There is a caveat here: high altitude cirrus clouds have notbeen considered in this study. Although cirrus clouds can reachquite high altitudes, they are often optically thin (Dupont et al.2010) and their average coverage on Earth is quite small (lessthan 20%, according to Rossow & Schi ff er 1999). So it seemsreasonable to assume that the no-clouds approximation wouldbe valid in most cases for an Earth-like atmosphere.Other methods such as transits (Seager & Mallén-Ornelas2003), transit time variations (TTVs), astrometry (Chauvin et al.2012), analysis of reflectance phase curves (Kane & Gelino2011) and radial velocities could also provide estimates of theorbital parameters. Article number, page 9 of 14 & A proofs: manuscript no. paper − − − Q / F [ % ] i = Ω = i = Ω = F c = F c = F c = F c = F c = − − −
50 0 50 100 150
Planet phase angle − − − U / F [ % ] i = Ω = − − −
50 0 50 100 150
Planet phase angle i = Ω = Fig. 12. Q / F (upper half) and U / F (lower half) as a function of theplanetary phase angle at λ =
300 nm for planets in circular orbits withpatchy clouds and di ff erent cloud fractions F c . − − Q / F [ % ] i = Ω = i = Ω = F c = F c = F c = F c = F c = − − −
50 0 50 100 150
Planet phase angle − − U / F [ % ] i = Ω = − − −
50 0 50 100 150
Planet phase angle i = Ω = Fig. 13.
Similar to Fig. 12, except at 700 nm.
Secondly , at any wavelength (but particularly in the visible)one can quickly infer whether the cloud cover is stable or notby measuring the variability of P (cid:96) with time. Because P (cid:96) is arelative measure, this would require less stability of an instru-ment than when measuring variability in the reflected flux. If thevariations of P (cid:96) at a given phase angle are consistently greaterthan the measurement precision, and not periodic, one couldassume that the planet is covered with patchy clouds (althoughcloud coverage could partly be related to the distribution ofcontinents on a planet, and thus partly periodic). Thirdly , the detection of optical phenomena such as rainbowsand glories would give information about the microphysicalproperties of the clouds (Karalidi et al. 2011, 2012; Rossi et al.2015; Bailey 2007), although the glory would be di ffi cult todetect on an exoplanet, because it occurs in backscattered lightand thus requires phase angles less than 10 ◦ , (see García Muñozet al. 2014). The rainbow feature will be present for varioustypes of (water) cloud cover, and a large range of cloud coveragefractions F c and cloud-top pressures p c . Fourthly , when dealing with patchy clouds, it would bepossible to use the reflected flux of the planet to constrain the cloud coverage fraction and hence to retrieve unambigu-ously the cloud-top altitude. In order to do this, the planetaryflux should be well–calibrated over time, and the size of theplanet and the distances to the star and the observer should beaccurately known. We note that the cloud-top pressure couldalso be retrieved by studying the variability of the cloud coverin P (cid:96) , which shows some dependence on p c in the blue (cf.Fig. 8). For Earth–remote sensing, methods such as measuringthe reflected flux and / or polarization inside and outside of agaseous absorption band are routinely used to retrieve cloud–topaltitudes, and such methods could also be applied to exoplanets.However, applying those methods requires knowledge on thevertical distribution of the absorbing gas (see Fauchez et al.2017, and references therein for the application of this methodto Earth-like exoplanets). Fifthly , once the type of coverage and the micro-physicalproperties of the clouds have been found, further fits of ob-servations to models of the polarization could allow for adetermination of the cloud fraction F c within 10 % precision.Once the micro-physical properties of the clouds and thecloud fraction are obtained, these values could be used to makea refined estimation of the orbital parameters. Such an iterativeprocess would hopefully converge toward a best–fit solution forboth orbital parameters and cloud properties.All computations have been made assuming phase anglesfrom 0 ◦ to 180 ◦ are accessible for observations. This range, how-ever, depends both on the orbital inclination angle and on theinner working angle (IWA) of the telescope with the instrument(assuming observations that spatially resolve the planet from itsstar). Indeed, for an orbit that is observed under an inclinationangle i , the range of phase angles the exoplanet goes throughalong its orbit is given by 90 ◦ − i ≤ α ≤ ◦ + i (along an or-bit with, for example, i = ◦ , the smallest phase angle that theplanet attains is thus 60 ◦ and the largest 120 ◦ ). The IWA limitsthe actual phase angle range at which an exoplanet can be sepa-rated from its star, cutting o ff access to the smallest and largestphase angles, where the planet is too close to its star to be re-solved. We note that the IWA will usually depend on the wave-length.In Fig. 14, we show the phase angle range at which a planetat a distance of 1 AU from its star (i.e., the habitable zone arounda solar-type star) can be observed as a function of the distancebetween the observer to the star for IWAs of 5, 10, 20, and 40mas, ignoring the limitations of a planet’s orbital inclination an-gle i . For example, with an IWA of 40 mas, a planet at 1 AUfrom a star at 200 pc, can only be observed at phase angles largerthan about 20 ◦ . Whether or not the planet will actually presentitself at this phase angle at some point in time, will depend onthe planet’s orbital inclination angle. With an IWA of 20 mas, aplanet at 1 AU from its star cannot be spatially resolved if thesystem is at larger distances than about 160 light-years (50 par-secs). Also shown in the figure is the phase angle of 40 ◦ aroundwhich the primary rainbow, indicative for light scattered in liquidwater cloud droplets (Karalidi et al. 2012; Bailey 2007) would bevisible. For example, with an IWA of 20 mas, the rainbow wouldbe observable for systems that are closer than about 100 light-years, of course provided i > ◦ . For systems that are furtherout, the planet will too close to its star at the rainbow phase angleto be spatially resolved from its star with this IWA. Article number, page 10 of 14. Rossi and D. M. Stam: Using polarimetry to retrieve the cloud coverage of Earth-like exoplanets
Distance to star (in light-years) P ha s e ang l e ( i n deg r ee s ) Distance to star (in parsec) rainbow-angle
Fig. 14.
Observable phase angles for a planet at 1 AU of its star asa function of the observer’s distance to the star. Di ff erent lines showresults for di ff erent inner working angles (IWA). The solid line at α = ◦ corresponds to the angle at which the rainbow appears, assumingwater droplets. For the phase angle range 90 ◦ - 180 ◦ , the graph mirrorsthis one over α = ◦ .
6. Conclusion
Identifying and characterizing clouds on exoplanets is crucial forretrieving their atmospheric properties and for getting insight intheir climate and habitability. We have shown that polarimetrycould enable observers to derive information on the type andfraction of cloud coverage. In our modeling, we have concen-trated on Earth–like exoplanets with black surfaces with andwithout a Fresnel reflecting interface. The latter would be rep-resentative for a flat ocean surface. While similar conclusionscan be expected for gaseous planets, the e ff ects of deeper cloudlayers, with a possible strong vertical variation in cloud particlemicro-physical properties, remain to be studied.Polarimetry allows us to distinguish sub-solar clouds frompatchy and polar clouds, because at a phase angle that dependson the cloud’s spatial extension, sub-solar clouds will disappear(and reappear) over the limb of the planet, leaving (and remov-ing) the characteristic polarization signature of Rayleigh scatter-ing gas. Secondly, the variability of the polarization signatureof patchy clouds should allow us to distinguish them from polarclouds, as the latter exhibit less variability at all phase angles.Measurements of the variability of the polarization combinedwith accurate measurements of the planet’s reflected flux (whichrequires knowledge about the size of the planet and its distanceto the parent star) would provide a tool to reduce ambiguitiesbetween the fraction of patchy clouds and the cloud-top pres-sure, as the polarization variability due to the varying patchinessappears to be larger than that due to the cloud-top pressure.Finally, measurements at short wavelengths ( <
400 nm)would allow the observer to mostly ignore the e ff ect of the cloudson the planet’s polarization signal and would therefore allow usto characterize the gaseous atmosphere of the planet (down tothe cloud tops). At these wavelengths, a pure Rayleigh scatteringatmosphere approximation could also be used to derive orbitalparameters without too much interference of the clouds. Longerwavelengths could be used to estimate the cloud coverage, and,depending on the phase angle and the IWA of the telescope + in-strumentation, to derive micro-physical properties from observa-tions of the primary rainbow, provided the rainbow phase angle( α = ◦ ) is within reach for the planetary system under study. Acknowledgements.
LR thanks Emmanuel Marcq and Arnaud Beth for their use-ful comments regarding an earlier version of this work. We also thank the re-viewer who greatly helped improving the paper. LR acknowledges the support ofthe Dutch Scientific Organization (NWO) through the PEPSci network of plane-tary and exoplanetary science.
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Appendix A: Influence of computation parameters
Appendix A.1: The number of pixels n pix The number of pixels across the planet directly determines thespatial resolution on the planetary disk, and thus the size of thecloudy pixels on the disk. A large number of pixels allows us totake smaller spatial features into account but also leads to longcomputation times.To measure the influence of n pix on the computed degreeof polarization P (cid:96) , we have run simulations with a 50% patchycloud cover ( F c = .
50) for 100 cloud patterns and di ff erent val-ues of n pix . The result is shown in Fig. A.1, in which we indicatethe 2 σ variability of P (cid:96) . D e g r ee o f li n e a r p o l a r i za t i o n n pix =15 n pix =30 n pix =45 n pix =60 Fig. A.1.
Degree of linear polarization P (cid:96) as a function of the phaseangle α for a 50% patchy cloud coverage. The solid curves representthe average of P (cid:96) over 100 cloud patterns. The shaded areas correspondto the 2 σ variability. It can be seen that the main e ff ect of increasing n pix is thereduction of the variability, especially when increasing from n pix =
15 to n pix =
30. Overall, using a smaller value for n pix ,and thus larger pixels on the planetary disk, leads to more abruptpixel type di ff erences across the disk and therefore to more vari-ability in the polarization. The decrease of the variability withincreasing value of n pix , and thus smaller pixels, seems to haveconverged with n pix =
45 in the figure. As a compromise be-tween a good enough accuracy and a reasonable computationtime, we decided to pursue the calculations for this paper with n pix = Appendix A.2: The number of cloud patterns n pattern For patchy clouds and a given value of the cloud coverage F c ,the number of cloud patterns, n pattern , might influence the com-puted average value of P (cid:96) and its variability, so it is necessaryto find the minimum number of patterns required for accurateresults. Figure A.2 shows P (cid:96) for F c = .
50 and n pix =
40 fordi ff erent values of n pattern . As can be seen in the figure, n pattern has only a small influence on P (cid:96) and its variability: increasing n pattern smoothens the average curve and the variability. We haveperformed our computations with n pattern =
300 to ensure rep-resentative results without being burdened by long computingtimes. D e g r ee o f li n e a r p o l a r i za t i o n n iter =50 n iter =100 n iter =200 n iter =500 Fig. A.2.
Similar to Fig. A.1, except for n pix =
40 and di ff erent numbersof cloud patterns n pattern .Article number, page 12 of 14. Rossi and D. M. Stam: Using polarimetry to retrieve the cloud coverage of Earth-like exoplanets Appendix A: Rotating polarization reference planes
Appendix A.1: From local to disk–integrated Stokes vectors
We compute F planet , the disk–integrated Stokes vector of a modelplanet, by summing up Stokes vectors computed for N locationson the illuminated and visible planetary disk, as follows F planet = Σ Ni = F i a i , (A.1)where F i is the locally reflected Stokes vector, and a i the surfacearea of pixel i on the two dimensional planetary disk. The centerof pixel i , projected parallel to the line of sight onto the threedimensional planet indicates the location from where F i has beenreflected.Assuming that the planet presents a circular disk to the ob-server with a radius equal to 1.0, the following should hold Σ Ni = a i = π. (A.2)We divided the planetary disk in areas that are equal in size.Equation A.1 thus transforms into F planet = π Σ Ni = F i . (A.3)Before evaluating this summation, we had to make sure that alllocally reflected Stokes vectors are defined with respect to thesame reference plane.Our radiative transfer algorithm provides parameters Q i and U i of each locally reflected Stokes vector as defined with re-spect to the local meridian plane : the plane through the localdirections toward the zenith and the observer. We note that thislocal meridian plane is independent of the direction toward theparent star. The natural reference plane for the disk–integratedStokes vector F planet is the planetary scattering plane : the planethrough the center of the planet, the sun and the observer. Theadvantage of the planetary scattering plane is that when theplanet is mirror–symmetric with respect to this reference plane,disk–integrated Stokes parameter U planet will equal zero (disk–integrated circular polarization Stokes parameter V planet will thenalso equal zero).To rotate from one reference plane to another, we used rota-tion matrix L , which is given by (see Hovenier & van der Mee1983) L ( β ) = β sin 2 β − sin 2 β cos 2 β
00 0 0 1 , (A.4)with β the angle between the two reference planes, measuredrotating in the clockwise direction from the old to the new ref-erence plane when looking toward the planet (0 ◦ ≤ β < ◦ )(Hovenier & van der Mee (1983) write that β is measured ro-tating in the anti–clockwise direction when looking toward theobserver, which of course yields the same angle).Rotation angle β i for a given pixel i depends on its locationwith respect to the planetary scattering plane. In the following,we use a Cartesian xy -coordinate system, with the origin in thecenter of the disk and the x –axis horizontal through the disk (seeFig. A.1). The radius of the disk equals 1. We can distinguishthe following cases given pixel center coordinates ( x i , y i ): x i y i ≥ β i = arctan ( y i / x i ) x i y i < β i = ◦ + arctan ( y i / x i ) x -axis y -axis ( x i , y i ) planetary scattering plane β l o c a l m e r i d i a n p l a n e Fig. A.1.
The definition of rotation angle β for a pixel with center coor-dinates ( x i , y i ) on the planetary disk. Appendix A.2: From planetary scattering plane to detectorplane
The orientation of the planetary scattering plane with respect tothe observer depends on the inclination angle i of the planetaryorbit, on the angle κ between the observer’s upward directionand the projection of the normal on the planetary orbital plane onthe sky, and on the position angle ψ of the planet along its orbit(see Fig. A.2). The longitude of the ascending node Ω equals90 ◦ - κ . Below, we derive how the Stokes vector F planet that hasbeen computed for the planet as a whole and with respect tothe planetary scattering plane, can be rotated to an observer’sreference plane. For the latter we have used a horizontal plane,that we refer to as the detector plane.The orbital inclination angle i is defined as the angle betweenthe normal on the planetary orbit and the direction toward the ob-server. The inclination angle has a value between 0 ◦ (for a ’face–on’ orbit) and 90 ◦ (for an ’edge–on’ orbit). In the following, weuse κ = ◦ , assuming that the observer’s telescope and detectorare rotated to accomodate this (see Fig. A.2). The normal on theplanetary orbital plane thus falls in the plane that is perpendic-ular to the detector plane and that contains the direction towardthe observer.The planet’s orbital position angle ψ is measured from theposition where the planet is closest to the center of the stellardisk as seen by the observer. Thus, for i = ◦ , ψ = ◦ in themiddle of the primary transit, and ψ = ◦ in the middle of thesecondary transit. For i = ◦ , ψ = ◦ is undefined. Angle ψ ismeasured rotating from ψ = ◦ in the counter–clockwise direc-tion (for a planet orbiting in the clock–wise direction as seen bythe observer, ψ will thus decrease in time).For completeness, given the orbital inclination angle i andthe orbital position angle ψ , the planetary phase angle is givenby α = arccos ( − cos ψ sin i ) . (A.5)Angle β to rotate the planetary Stokes vector from theplanetary scattering plane to the detector plane depends on theorbital position angle ψ and the orbital inclination angle i :tan ψ ≥ . β = ◦ − arctan (cos i / tan ψ ) Article number, page 13 of 14 & A proofs: manuscript no. paper observer’s north detector plane o r b i t a l p l a n e κ p l a n e t a r y sc a tt e r i n g p l a n e ψ Fig. A.2.
An inclined, circular planetary orbit with the planet’s orbitalposition angle indicated by angle ψ (0 ◦ ≤ ψ ≤ ◦ ). The rotation ofthe normal on the planetary orbit as projected on the sky with respect tothe observer’s north is indicated by angle κ ( − ◦ ≤ κ ≤ + ◦ ). tan ψ < . β = − arctan (cos i / tan ψ )Applying this to the Stokes vector F planet , we thus obtained thefollowing expressions for parameters Q orbit and U orbit , as definedwith respect to the orbital plane: Q orbit = cos 2 β Q planet + sin 2 β U planet U orbit = − sin 2 β Q planet + cos 2 β U planet These equations hold both for circular and elliptical orbits, be-cause the ellipticity does not change the values of angle ψ , onlythe change of ψ in time.In case of a non–zero value of κ , and additional rotation overthe angle between the orbital plane used above and the actualreference plane should be performed. In particular, a rotationfrom the orbital plane and a reference plane that we’ll refer toas the detector plane , perpendicular to the direction toward theobserver’s north, would be described by the following equations:0 ◦ ≤ κ ≤ ◦ : Q detector = cos 2 κ Q orbit + sin 2 κ U orbit U detector = − sin 2 κ Q orbit + cos 2 κ U orbit − ◦ ≤ κ ≤ ◦ : Q detector = cos 2 κ Q orbit − sin 2 κ U orbit U detector = sin 2 κ Q orbit + cos 2 κ U orbitorbit