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Flux and polarisation spectra of water clouds on exoplanets

Karalidi T Stam DM Hovenier JW

Published inAstronomy amp Astrophysics

DOI1010510004-6361201116449

Link to publication

Citation for published version (APA)Karalidi T Stam D M amp Hovenier J W (2011) Flux and polarisation spectra of water clouds on exoplanetsAstronomy amp Astrophysics 530 DOI 1010510004-6361201116449

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Download date 07 May 2018

AampA 530 A69 (2011)DOI 1010510004-6361201116449ccopy ESO 2011

Astronomyamp

Astrophysics

Flux and polarisation spectra of waterclouds on exoplanets

T Karalidi13 D M Stam1 and J W Hovenier2

1 SRON ndash Netherlands Institute for Space Research Sorbonnelaan 2 3584 CA Utrecht The Netherlandse-mail TKaralidisronnl

2 Astronomical Institute ldquoAnton Pannekoekrdquo University of Amsterdam Science Park 904 1098 XH Amsterdam The Netherlands3 Astronomical Institute Utrecht University Princetonplein 5 3584 CC Utrecht The Netherlands

Received 6 January 2011 Accepted 7 April 2011

ABSTRACT

Context A crucial factor for a planetrsquos habitability is its climate Clouds play an important role in planetary climates Detecting andcharacterising clouds on an exoplanet is therefore crucial when addressing this planetrsquos habitabilityAims We present calculated flux and polarisation spectra of starlight that is reflected by planets covered by liquid water clouds withdifferent optical thicknesses altitudes and particle sizes as functions of the phase angle α We discuss the retrieval of these cloudproperties from observed flux and polarisation spectraMethods Our model planets have black surfaces and atmospheres with Earth-like temperature and pressure profiles We calculatethe spectra from 03 to 10 μm using an adding-doubling radiative transfer code with integration over the planetary disk The cloudparticlesrsquo scattering properties are calculated using a Mie-algorithmResults Both flux and polarisation spectra are sensitive to the cloud optical thickness altitude and particle sizes depending on thewavelength and phase angle αConclusions Reflected fluxes are sensitive to cloud optical thicknesses up to sim40 and the polarisation to thicknesses up to sim20The shapes of polarisation features as functions of α are relatively independent of the cloud optical thickness Instead they dependstrongly on the cloud particlesrsquo size and shape and can thus be used for particle characterisation In particular a rainbow stronglyindicates the presence of liquid water droplets Single scattering features such as rainbows which can be observed in polarisationare virtually unobservable in reflected fluxes and fluxes are thus less useful for cloud particle characterisation Fluxes are sensitive tocloud top altitudes mostly for α lt 60 and wavelengths lt04 μm and the polarisation for α around 90 and wavelengths between 04and 06 μm

Key words polarization ndash methods numerical ndash radiative transfer ndash planets and satellites atmospheres ndash Earth

1 Introduction

Since the discovery of the first exoplanet orbiting a solarndashtypestar more than a decade ago (Mayor amp Queloz 1995) the questfor signs of habitable exoplanets has started And while manyscientists intertwine habitability with the existence of liquid wa-ter on the planetary surface another key factor for habitabilityis the planetary climate (Kasting et al 1993) Clouds are amongthe major factors that affect a planetary climate

Regarding the Earth Goloub et al (2000) presented an ex-tended series of studies of the Earthrsquos climate both at an ob-servational as well as at a modelling level which clearly indi-cates the crucial and diverse roles of clouds In particular cloudsare responsible for the modulation of both the shortwave radi-ation (from the sun) as well as the long-wave radiation (fromthe planet) budgets of the Earth (Kim amp Ramanathan 2008Ramanathan et al 1989 Cess et al 1992 Malek 2007 and oth-ers) Additionally by modulating the solar radiation clouds af-fect the atmospheric photolysis rates which change the atmo-spheric photochemistry and chemical composition (Pour Biazaret al 2007) And clouds are responsible for the storage of atmo-spheric volatiles such as the organic volatiles that are indicatorsof the existence of bio and fossilndashmasses which can damage thesoil and groundwater and can react with sunlight to create tro-pospheric O3 (Klouda et al 1996) Because of their roles in the

climate clouds on Earth have been subjected to intense study forthe past five decades both from a theoreticalmodelling pointof view as well as from an observational point of view (fromthe ground as well as from space) The effects of clouds on theEarthrsquos climate have been shown to depend on the sizes andshapes (ie thermodynamical phase) of the cloud particles andon the optical thickness and vertical extension of a cloud

In this paper we present results of numerical simulations offlux and especially polarisation spectra of starlight that is re-flected by cloudy exoplanets and discuss the sensitivity of thesespectra to the size of the cloud particles the cloud optical thick-ness and the cloud top altitude In general the composition of acloud or cloud layer in a planetary atmosphere will depend onthe ambient chemical composition and the pressure and temper-ature profiles (the latter themselves will of course also be influ-enced by the presence of clouds) It is well known that cloudshave strong effects on flux spectra of planets in the Solar Systemand beyond Examples of simulated flux spectra of exoplanetswith water clouds can be found in Marley et al (1999) Tinettiet al (2006) Kaltenegger et al (2007) Far less work has beendone regarding polarisation spectra of cloudy planets Examplesof simulated polarisation spectra of exoplanets with liquid waterclouds can be found in Stam (2008) In this paper we extend thiswork to different types of clouds at various altitudes

Article published by EDP Sciences A69 page 1 of 14

AampA 530 A69 (2011)

Terrestrial liquid water clouds are in general comprised ofparticles with radii ranging from sim5 μm to sim30 μm (Han et al1994) and their optical thickness varies typically from b sim 1 tob sim 40 or more (van Deelen et al 2008) There appears to be acorrelation between the cloud particle sizes and the cloud opticalthickness (Han et al 1994 and references therein) that couldoriginate in the properties of the condensation nuclei that thecloud partices condense on but it is not a strong one (Stephenset al 2008) Knowing the sizes of cloud particles is importantfor understanding a cloudrsquos influence on a planetary climate be-cause it determines how a cloud particle scatters and absorbs in-cident light and thermal radiation (see eg Chapman et al 2009and references therein) How a particle scatters the light dependsin particular on the size parameter ie the ratio of 2π times theparticle radius to the wavelength (see Eq (9)) We will considersize parameters ranging from less than 1 up to sim120 cover-ing the so-called Rayleigh regime where the scattering particlesare much smaller than the wavelength to the geometric opticsregime where particles are large with respect to the wavelengthThe cloudrsquos optical thickness (which when assuming sphericalparticles is the product of the column number density and theextinction cross-section of the cloud particles) depends on thecloud particlesrsquo sizes shapes and composition and hence alsoon the wavelength of the radiation We will use cloud opticalthicknesses ranging from 05 to 60 (at 055 μm) and sphericalwater particles

Another cloud parameter that is not only important for the ra-diation field but also for the thermo- and hydro-dynamical pro-cesses that take place in planetary atmospheres is the altitude orpressure of the top of the cloud Cloud top altitudes and pressuresare routinely determined in Earth remote-sensing and regularlyin planetary observations (eg Wark amp Mercer 1965 Weigeltet al 2009 Peralta et al 2007 Garay et al 2008 Matcheva et al2005) In Earth and planetary observations knowledge of cloudtop altitudes is also essential for accurate derivations of mixingratios of atmospheric trace gases from the depths of gaseous ab-sorption bands in planetary spectra since clouds will change theband depths For example in Earth observation cloud top alti-tudes are used in the retrieval of the trace gas ozone (O3) Thesecloud top altitudes are usually derived from the depth of the so-called A absorption band of the wellndashmixed gas oxygen (the O2A band is located around 076 μm) (eg Yamamoto amp Wark 1961and Fischer amp Grassl 1991) Note that in case an absorbing gasis not well-mixed and its vertical distribution is not known itsabsorption bands cannot be used to provide absolute cloud tops

Clouds on Earth are found at a wide range of altitudes Mostclouds are located in the troposphere the lowest portion of theatmosphere where the temperature generally decreases with al-titude On average the top of the troposphere decreases with in-creasing latitude and with that the maximum cloud top altitudefrom about 20 km in the tropics to about 10 km in the polarregions The tops of the highest clouds will usually contain iceparticles The thin whisphy clouds commonly known as cirrusclouds are composed entirely of water ice crystals Since abovethe tropopause the atmosphere contains relatively little watervapour most types of clouds are confined to the troposphereThe few cloud types that can be found above the tropopause arethin ice clouds such as polar stratospheric clouds and noctilu-cent clouds

We will limit ourselves to clouds that are composed entirelyof liquid water droplets and hence limit the cloud top altitudes inour model atmospheres to about 4 km (where the temperaturesare still high enough to exclude the presence of ice particles)Our main reason to exclude clouds with ice particles is that the

single scattering properties of ice (crystal) particles are usuallyvery different from those of liquid (spherical) particles (see egGoloub et al 2000) and as a result their presence will influencethe light that is reflected by a cloud (in particular the polarisedsignal) Modelling and analysing this influence (which will de-pend on absolute and relative ice particle number densities iceparticle sizes and shapes and orientation) will be the subject offurther research

In this paper we present not only numerically simulatedflux spectra but especially polarisation spectra Polarimetry iemeasuring the direction and degree of polarisation of light isconsidered to be a powerful tool for the direct detection of ex-oplanets (Keller 2006 Keller et al 2010 Stam et al 2004)The reason for this is that when integrated over the stellar diskstarlight of solar type stars will be virtually unpolarised (Kempet al 1987) while starlight that has been reflected by an exo-planet will generally be polarised due to scattering and reflec-tion processes in the planetary atmosphere and on the surface (ifpresent) Polarimetry can thus enhance the contrast between aplanet and its star by a factor of sim104minussim105 (Keller et al 2010)and thus facilitate the direct detection of an exoplanet Anotheradvantage of polarimetry for exoplanet detection is that it en-ables the direct confirmation of a detection since the degree anddirection of polarisation of a detected object will exclude it beinga background star

The real strength of polarimetry for exoplanet research ishowever that it cannot only be used for the direct detection ofexoplanets but also for the characterisation of the physical prop-erties of these planets The reason is that the state of polarisa-tion of starlight that is reflected by a planet is very sensitive tothe composition and structure of the planetary atmosphere andsurface (if present) (see Hansen amp Travis 1974 Hovenier et al2004 Mishchenko et al 2010 and references therein) An earlyexample of this application of polarimetry is the derivation ofthe composition and size distribution of the droplets forming theupper Venusian clouds as well as the cloud top altitudes fromEarth-based disk-integrated Venus observations by Hansen ampHovenier (1974)

The application of polarisation for the detection and charac-terisation of exoplanets has been shown for gaseous exoplanetsby eg Seager et al (2000) Saar amp Seager (2003) Stam (2003)Stam et al (2004) and for terrestrial planets by Stam (2008)Note that in the first two papers planets are considered that aretoo close to their star to be spatially resolved The observabledegree of polarisation for these systems is thus the ratio of thepolarised flux of the planet to the total flux of the star (plus that ofthe planet) and consequently very small In the latter three pa-pers the planet is assumed to be spatially resolvable from its starIn that case the observable degree of polarisation is thus the de-gree of polarisation of the planet itself (apart from a contributionof unpolarized background starlight) which can be several tensof percents In this paper we will consider spatially resolvableplanets Our results can straightforwardly be applied to spatiallyunresolvable planets by scaling them with the stellar flux

The simulations we present in this paper are useful for thedesign development and optimisation of instruments for the di-rect detection of exoplanets Since the presence of water-cloudsis not restricted to terrestrial planets these can be instrumentsfor the detection of gaseous planets andor terrestrial planetsand both for ground- and space-based telescopes An exampleof such an instrument is SPHERE (Spectro-Polarimetric High-contrast Exoplanet Research) a second generation planetfinderinstrument for the European Southern Observatoryrsquos (ESO) VeryLarge Telescope (VLT) For SPHERE first light is expected in

A69 page 2 of 14

T Karalidi et al Spectra of water clouds on exoplanets

2012 SPHERE has broadband polarimetric capabilities in theI-band (06minus09 μm) EPICS (ExoPlanets Imaging Camera andSpectrograph) has been proposed as the planetfinder instrumentfor ESOrsquos Extremely Large Telescope (ELT) and will also havea polarimeter to detect and characterise exoplanets EPICS isstill in its design and optimisation phase and first light is ex-pected not earlier than 2020 An example of a space telescopeconcept for exoplanet research that would be ideally suited toobserve both the flux and the state of polarisation of starlightthat is reflected by exoplanets is the New Worlds Observer(NWO) that has been and will be proposed to NASA (Oakley ampCash 2009 Cash amp New Worlds Study Team 2010) An exam-ple of a space-telescope with polarimetric capabilities for exo-planet research that has been proposed to the European SpaceAgency in response to its Cosmic Vision 2015minus2020 call fora medium sized mission (M3) is Spectro-Polarimetric Imagingand Characterization of Exo-planetary Systems or SPICES

This paper is organised as follows In Sect 2 we give a gen-eral description of light including polarisation and present theradiative transfer algorithm we use for our numerical simulationsof starlight that is reflected by planets In Sect 3 we describeour model atmospheres and in Sect 4 the flux and degree of po-larisation of light that has been singly scattered by the modelcloud particles In Sects 51 and 52 we show the results ofour numerical simulations of the flux and degree of polarisa-tion of reflected starlight for different cloud particle microphysi-cal properties and in Sect 53 for different cloud top pressuresFinally in Sects 6 and 7 we summarise and discuss our resultsand future work

2 Description of starlight that is reflectedby an exoplanet

Light that has been reflected by an exoplanet can be fully de-scribed by a flux vector πF as follows

πF = π

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

FQUV

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (1)

where parameter πF is the total reflected flux parameters πQand πU describe the linearly polarised flux and parameter πVthe circularly polarised flux (see eg Hansen amp Travis 1974Hovenier et al 2004 Stam 2008) All four parameters arewavelength dependent and their dimensions are W mminus2 mminus1Parameters πQ and πU are defined with respect to the so-calledplanetary scattering plane ie the plane through the centers ofthe planet the host star and the observer (see Stam 2008)

The degree of polarisation P is defined as the ratio of thepolarised flux to the total flux as follows

P =

radicQ2 + U2 + V2

Fmiddot (2)

In case a planet is mirrorndashsymmetric with respect to the plane-tary scattering plane and for unpolarised incoming stellar lightthe disk integrated fluxes πU and πV of the reflected light willequal zero due to symmetry (see Hovenier 1970) and we can usethe following alternative definition of the degree of polarisationthat includes the direction of polarisation

Ps = minusQFmiddot (3)

For Ps gt 0 (ie Q lt 0) the light is polarised perpendicular tothe reference plane while for Ps lt 0 (ie Q gt 0) the light ispolarised parallel to the reference plane

The flux vector πF of stellar light that has been reflected bya spherical planet with radius r at a distance d from the observer(d r) is given by (Stam et al 2006)

πF(λ α) =14

r2

d2S(λ α)πF0(λ) (4)

Here λ is the wavelength of the light and α the planetary phaseangle ie the angle between the star and the observer as seenfrom the center of the planet Furthermore S is the 4times4 planetaryscattering matrix (Stam et al 2006) and πF0 is the flux vector ofthe incident stellar light For a solar type star the stellar flux canbe considered to be unpolarised when integrated over the stellardisk (Kemp et al 1987) Further assuming that the stellar light isunidirectional we can thus describe πF0 as (see Eq (1))

πF0(λ) = πF0(λ)1 (5)

with πF0 the flux of the stellar light that is incident on the planetmeasured perpendicular to the direction of incidence and 1 theunit column vector

For unpolarised incident stellar light (see Eq (5)) and fora planet that is mirrorndashsymmetric with respect to the planetaryscattering plane the degree of polarisation Ps (Eq (3)) of thelight that is reflected by the planet depends on only two elementsof the scattering matrix as follows

Ps(λ α) = minusb1(λ α)a1(λ α)

(6)

(for a derivation see Stam 2008) Since the degree of polarisa-tion is a relative measure it has no dependence on planetary andstellar radii distances nor on the incident stellar flux

In this paper we present numerically simulated flux and po-larisation spectra of planets that are completely covered by waterclouds as functions of the planetary phase angle α Even thoughsuch a model is probably not very realistic in this first study wewill use it in order to limit the number of parameters we need tostudy in our models and to get some first idea on the importantfeatures we need to investigate in planetary signals We assumethat the ratio of the planetary radius r and the distance to the ob-server d is equal to one and that the incident stellar flux πF0 isequal to 1 W mminus2 mminus1 The hence normalised flux πFn that isreflected by a planet is thus given by

πFn(λ α) =14

a1(λ α) (7)

(see Stam 2008) and corresponds to the planetrsquos geometricalbedo AG if α = 0 (AG is defined as the ratio of the total flux πFthat is reflected by the planet at α = 0 to the flux πFL that is re-flected by a Lambertian surface subtending the same solid angleon the sky) Our normalised fluxes πFn can straightforwardly bescaled for any given planetary system using Eq (4) and insertingthe appropriate values for r d and πF0 As mentioned above thedegree of polarisation Ps is independent of r d and πF0 and willthus not require any scaling

To calculate the planetary scattering matrix elements a1 andb1 of the reflected stellar light we employ the algorithm de-scribed in Stam et al (2004) which consists of an efficient andaccurate adding-doubling algorithm (de Haan et al 1987) incombination with a fast numerical disk integration algorithmto calculate the radiative transfer in the locally plane-parallelplanetary atmosphere and to integrate the reflected light acrossthe illuminated and visible part of the planetary disk

A69 page 3 of 14

AampA 530 A69 (2011)

Table 1 The altitude (in km) pressure (in bar) and temperature(in K) at the bottom and top of each layer of our model atmosphere(McClatchey et al 1972)

Altitude Pressure Temperature00 1013 294020 0802 290030 0710 279040 0628 27301000 30 times 10minus7 2100

3 Our model atmospheres

The model planetary atmospheres we use in our numerical sim-ulations are composed of stacks of horizontally homogeneousand locally plane-parallel layers which contain gas moleculesand optionally cloud particles We create vertically inhomoge-neous atmospheres simply by stacking different homogeneouslayers Each atmosphere is bounded below by a flat homoge-neous surface We base our atmospheric temperature and pres-sure profiles on representative ones of the Earthrsquos atmosphere(McClatchey et al 1972) Here we use only four atmosphericlayers (see Table 1 for the pressures and temperatures at thetops of these layers) and a black planetary surface The molecu-lar or Rayleigh scattering optical thickness of each atmosphericlayer and its spectral variation is calculated as described in Stam(2008) assuming an atmospheric gas composition of the modelatmospheres that is similar to that of the Earth For our simula-tions of the flux and polarisation spectra we focus on the con-tinuum and we ignore absorption by atmospheric gases (such asO2 H2O and O3) Even though we use pressure and temperatureprofiles and a gas composition that are typical for an Earth-likeplanet most of our results are also applicable to other planetseg gas giants with high altitude water clouds

We will use a clear model atmosphere ie without cloudsand cloudy model atmospheres In the latter one of the atmo-spheric layers contains cloud particles in addition to the gasmolecules To study the dependence of the flux and degree ofpolarisation of the starlight that is reflected by the model planeton the altitude of the top of the cloud layer we will use the fol-lowing three cases a low cloud layer with its top at an ambientpressure of 0802 bar (corresponding to an altitude of 2 km onEarth) a middle cloud layer with its top at 0710 bar (3 kmon Earth) and a high cloud layer with its top at 0628 bar (4 kmon Earth) Unless stated otherwise the geometrical thickness ofthe cloud layer is 1 km

Optical thicknesses of clouds on Earth show a huge varia-tion on daily and monthly timescales and from place to place Inparticular the optical thickness in completely cloudy cases canreach values up to 40 (van Deelen et al 2008) or more The op-tical thickness b of our cloud layers is chosen to range between05 and 60 (at λ = 055 μm) Our standard cloud layer has anoptical thickness of 10 (at λ = 055 μm) which appears to be anaverage value The spectral variation of the cloud layerrsquos opticalthickness depends on the microphysical properties of the cloudparticles such as their composition shape and size (see below)

Our model cloud layers are composed of liquid sphericalwater particles Water cloud observations on Earth show a largevariation in droplet sizes that is attributed to amongst others avariation in the number density and composition of cloud con-densation nuclei (Han et al 1994 Martin et al 1994 Segal ampKhain 2006) In particular typical cloud particle radii range fromabout 5 μm to 15 μm with a global mean value of about 85 μmabove continental areas and 118 μm above maritime areas

001 010 100 1000 10000particle size (μm)

00001

00010

00100

01000

10000

100000

1000000

n(r)

B part distr

A part distr

Fig 1 Particle size distributions (see Eq (8)) for model A particles(reff = 20 μm veff = 01 red dashed line) and model B particles(reff = 60 μm veff = 04 black solid line) Each size distribution hasbeen normalised such that the integral over all sizes equals 1

(Han et al 1994) On the lower limit sizes down to 2 μm havebeen reported from satellite measurements (Minnis et al 1992)and on the upper limit sizes up to 25 μm (Goloub et al 2000)

We describe the sizes of our cloud particles by a standardsize distribution (Hansen amp Travis 1974) as follows

n(r) = C r(1minus3veff )veff eminusrveffreff (8)

where C is a normalisation constant n(r)dr is the number ofparticles with radii between r and r+dr per unit volume and reffand veff are the effective radius and variance respectively (seeHansen amp Travis 1974) The units of reff are [μm] while veff isdimensionless

The scattering properties of particles often depend stronglyon the ratio of the radius of the particles to the wavelength ofthe light This so-called effective size parameter xeff that willbe used later in this paper is defined as

xeff =2πreff

λ(9)

(see Hansen amp Travis 1974) For xeff 1 the scattering is usu-ally referred to as Rayleigh scattering and for xeff 1 we getinto the regime where light scattering can be described with ge-ometrical optics

Our cloud layers are composed of either small model A par-ticles with reff = 20 μm and veff = 01 (Stam 2008) or largermodel B particles with reff = 60 μm and veff = 04 The lat-ter are similar to those used by van Diedenhoven et al (2007)who consider this to represent an average terrestrial water cloudFigure 1 shows the size distributions of the particles of the twostandard models used further in this paper (A and B particles)To study the influence of a size distributionrsquos effective varianceveff on the reflected light we will also use model A2 particleswith reff = 20 μm and veff = 04 and model B2 particles withreff = 60 μm and veff = 01 (see Table 2)

The real part of the refractive index of water in the wave-length region of our interest is slightly wavelength dependent Itvaries from 1344 at λ = 04 μm to 1320 at 10 μm (Daimon ampMasumura 2007 van Diedenhoven et al 2007) The imaginarypart of the refractive index is small but varies strongly (Pope ampFry 1997) from about 10minus8 at 03 μm to about 10minus5 at 10 μm

A69 page 4 of 14

T Karalidi et al Spectra of water clouds on exoplanets

2012 SPHERE has broadband polarimetric capabilities in theI-band (06minus09 μm) EPICS (ExoPlanets Imaging Camera andSpectrograph) has been proposed as the planetfinder instrumentfor ESOrsquos Extremely Large Telescope (ELT) and will also havea polarimeter to detect and characterise exoplanets EPICS isstill in its design and optimisation phase and first light is ex-pected not earlier than 2020 An example of a space telescopeconcept for exoplanet research that would be ideally suited toobserve both the flux and the state of polarisation of starlightthat is reflected by exoplanets is the New Worlds Observer(NWO) that has been and will be proposed to NASA (Oakley ampCash 2009 Cash amp New Worlds Study Team 2010) An exam-ple of a space-telescope with polarimetric capabilities for exo-planet research that has been proposed to the European SpaceAgency in response to its Cosmic Vision 2015minus2020 call fora medium sized mission (M3) is Spectro-Polarimetric Imagingand Characterization of Exo-planetary Systems or SPICES

This paper is organised as follows In Sect 2 we give a gen-eral description of light including polarisation and present theradiative transfer algorithm we use for our numerical simulationsof starlight that is reflected by planets In Sect 3 we describeour model atmospheres and in Sect 4 the flux and degree of po-larisation of light that has been singly scattered by the modelcloud particles In Sects 51 and 52 we show the results ofour numerical simulations of the flux and degree of polarisa-tion of reflected starlight for different cloud particle microphysi-cal properties and in Sect 53 for different cloud top pressuresFinally in Sects 6 and 7 we summarise and discuss our resultsand future work

2 Description of starlight that is reflectedby an exoplanet

Light that has been reflected by an exoplanet can be fully de-scribed by a flux vector πF as follows

πF = π

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

FQUV

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (1)

where parameter πF is the total reflected flux parameters πQand πU describe the linearly polarised flux and parameter πVthe circularly polarised flux (see eg Hansen amp Travis 1974Hovenier et al 2004 Stam 2008) All four parameters arewavelength dependent and their dimensions are W mminus2 mminus1Parameters πQ and πU are defined with respect to the so-calledplanetary scattering plane ie the plane through the centers ofthe planet the host star and the observer (see Stam 2008)

The degree of polarisation P is defined as the ratio of thepolarised flux to the total flux as follows

P =

radicQ2 + U2 + V2

Fmiddot (2)

In case a planet is mirrorndashsymmetric with respect to the plane-tary scattering plane and for unpolarised incoming stellar lightthe disk integrated fluxes πU and πV of the reflected light willequal zero due to symmetry (see Hovenier 1970) and we can usethe following alternative definition of the degree of polarisationthat includes the direction of polarisation

Ps = minusQFmiddot (3)

For Ps gt 0 (ie Q lt 0) the light is polarised perpendicular tothe reference plane while for Ps lt 0 (ie Q gt 0) the light ispolarised parallel to the reference plane

The flux vector πF of stellar light that has been reflected bya spherical planet with radius r at a distance d from the observer(d r) is given by (Stam et al 2006)

πF(λ α) =14

r2

d2S(λ α)πF0(λ) (4)

Here λ is the wavelength of the light and α the planetary phaseangle ie the angle between the star and the observer as seenfrom the center of the planet Furthermore S is the 4times4 planetaryscattering matrix (Stam et al 2006) and πF0 is the flux vector ofthe incident stellar light For a solar type star the stellar flux canbe considered to be unpolarised when integrated over the stellardisk (Kemp et al 1987) Further assuming that the stellar light isunidirectional we can thus describe πF0 as (see Eq (1))

πF0(λ) = πF0(λ)1 (5)

with πF0 the flux of the stellar light that is incident on the planetmeasured perpendicular to the direction of incidence and 1 theunit column vector

For unpolarised incident stellar light (see Eq (5)) and fora planet that is mirrorndashsymmetric with respect to the planetaryscattering plane the degree of polarisation Ps (Eq (3)) of thelight that is reflected by the planet depends on only two elementsof the scattering matrix as follows

Ps(λ α) = minusb1(λ α)a1(λ α)

(6)

(for a derivation see Stam 2008) Since the degree of polarisa-tion is a relative measure it has no dependence on planetary andstellar radii distances nor on the incident stellar flux

In this paper we present numerically simulated flux and po-larisation spectra of planets that are completely covered by waterclouds as functions of the planetary phase angle α Even thoughsuch a model is probably not very realistic in this first study wewill use it in order to limit the number of parameters we need tostudy in our models and to get some first idea on the importantfeatures we need to investigate in planetary signals We assumethat the ratio of the planetary radius r and the distance to the ob-server d is equal to one and that the incident stellar flux πF0 isequal to 1 W mminus2 mminus1 The hence normalised flux πFn that isreflected by a planet is thus given by

πFn(λ α) =14

a1(λ α) (7)

(see Stam 2008) and corresponds to the planetrsquos geometricalbedo AG if α = 0 (AG is defined as the ratio of the total flux πFthat is reflected by the planet at α = 0 to the flux πFL that is re-flected by a Lambertian surface subtending the same solid angleon the sky) Our normalised fluxes πFn can straightforwardly bescaled for any given planetary system using Eq (4) and insertingthe appropriate values for r d and πF0 As mentioned above thedegree of polarisation Ps is independent of r d and πF0 and willthus not require any scaling

To calculate the planetary scattering matrix elements a1 andb1 of the reflected stellar light we employ the algorithm de-scribed in Stam et al (2004) which consists of an efficient andaccurate adding-doubling algorithm (de Haan et al 1987) incombination with a fast numerical disk integration algorithmto calculate the radiative transfer in the locally plane-parallelplanetary atmosphere and to integrate the reflected light acrossthe illuminated and visible part of the planetary disk

A69 page 3 of 14

AampA 530 A69 (2011)

Table 1 The altitude (in km) pressure (in bar) and temperature(in K) at the bottom and top of each layer of our model atmosphere(McClatchey et al 1972)

Altitude Pressure Temperature00 1013 294020 0802 290030 0710 279040 0628 27301000 30 times 10minus7 2100

3 Our model atmospheres

The model planetary atmospheres we use in our numerical sim-ulations are composed of stacks of horizontally homogeneousand locally plane-parallel layers which contain gas moleculesand optionally cloud particles We create vertically inhomoge-neous atmospheres simply by stacking different homogeneouslayers Each atmosphere is bounded below by a flat homoge-neous surface We base our atmospheric temperature and pres-sure profiles on representative ones of the Earthrsquos atmosphere(McClatchey et al 1972) Here we use only four atmosphericlayers (see Table 1 for the pressures and temperatures at thetops of these layers) and a black planetary surface The molecu-lar or Rayleigh scattering optical thickness of each atmosphericlayer and its spectral variation is calculated as described in Stam(2008) assuming an atmospheric gas composition of the modelatmospheres that is similar to that of the Earth For our simula-tions of the flux and polarisation spectra we focus on the con-tinuum and we ignore absorption by atmospheric gases (such asO2 H2O and O3) Even though we use pressure and temperatureprofiles and a gas composition that are typical for an Earth-likeplanet most of our results are also applicable to other planetseg gas giants with high altitude water clouds

We will use a clear model atmosphere ie without cloudsand cloudy model atmospheres In the latter one of the atmo-spheric layers contains cloud particles in addition to the gasmolecules To study the dependence of the flux and degree ofpolarisation of the starlight that is reflected by the model planeton the altitude of the top of the cloud layer we will use the fol-lowing three cases a low cloud layer with its top at an ambientpressure of 0802 bar (corresponding to an altitude of 2 km onEarth) a middle cloud layer with its top at 0710 bar (3 kmon Earth) and a high cloud layer with its top at 0628 bar (4 kmon Earth) Unless stated otherwise the geometrical thickness ofthe cloud layer is 1 km

Optical thicknesses of clouds on Earth show a huge varia-tion on daily and monthly timescales and from place to place Inparticular the optical thickness in completely cloudy cases canreach values up to 40 (van Deelen et al 2008) or more The op-tical thickness b of our cloud layers is chosen to range between05 and 60 (at λ = 055 μm) Our standard cloud layer has anoptical thickness of 10 (at λ = 055 μm) which appears to be anaverage value The spectral variation of the cloud layerrsquos opticalthickness depends on the microphysical properties of the cloudparticles such as their composition shape and size (see below)

Our model cloud layers are composed of liquid sphericalwater particles Water cloud observations on Earth show a largevariation in droplet sizes that is attributed to amongst others avariation in the number density and composition of cloud con-densation nuclei (Han et al 1994 Martin et al 1994 Segal ampKhain 2006) In particular typical cloud particle radii range fromabout 5 μm to 15 μm with a global mean value of about 85 μmabove continental areas and 118 μm above maritime areas

001 010 100 1000 10000particle size (μm)

00001

00010

00100

01000

10000

100000

1000000

n(r)

B part distr

A part distr

Fig 1 Particle size distributions (see Eq (8)) for model A particles(reff = 20 μm veff = 01 red dashed line) and model B particles(reff = 60 μm veff = 04 black solid line) Each size distribution hasbeen normalised such that the integral over all sizes equals 1

(Han et al 1994) On the lower limit sizes down to 2 μm havebeen reported from satellite measurements (Minnis et al 1992)and on the upper limit sizes up to 25 μm (Goloub et al 2000)

We describe the sizes of our cloud particles by a standardsize distribution (Hansen amp Travis 1974) as follows

n(r) = C r(1minus3veff )veff eminusrveffreff (8)

where C is a normalisation constant n(r)dr is the number ofparticles with radii between r and r+dr per unit volume and reffand veff are the effective radius and variance respectively (seeHansen amp Travis 1974) The units of reff are [μm] while veff isdimensionless

The scattering properties of particles often depend stronglyon the ratio of the radius of the particles to the wavelength ofthe light This so-called effective size parameter xeff that willbe used later in this paper is defined as

xeff =2πreff

λ(9)

(see Hansen amp Travis 1974) For xeff 1 the scattering is usu-ally referred to as Rayleigh scattering and for xeff 1 we getinto the regime where light scattering can be described with ge-ometrical optics

Our cloud layers are composed of either small model A par-ticles with reff = 20 μm and veff = 01 (Stam 2008) or largermodel B particles with reff = 60 μm and veff = 04 The lat-ter are similar to those used by van Diedenhoven et al (2007)who consider this to represent an average terrestrial water cloudFigure 1 shows the size distributions of the particles of the twostandard models used further in this paper (A and B particles)To study the influence of a size distributionrsquos effective varianceveff on the reflected light we will also use model A2 particleswith reff = 20 μm and veff = 04 and model B2 particles withreff = 60 μm and veff = 01 (see Table 2)

The real part of the refractive index of water in the wave-length region of our interest is slightly wavelength dependent Itvaries from 1344 at λ = 04 μm to 1320 at 10 μm (Daimon ampMasumura 2007 van Diedenhoven et al 2007) The imaginarypart of the refractive index is small but varies strongly (Pope ampFry 1997) from about 10minus8 at 03 μm to about 10minus5 at 10 μm

A69 page 4 of 14

AampA 530 A69 (2011)

Table 1 The altitude (in km) pressure (in bar) and temperature(in K) at the bottom and top of each layer of our model atmosphere(McClatchey et al 1972)

Altitude Pressure Temperature00 1013 294020 0802 290030 0710 279040 0628 27301000 30 times 10minus7 2100

3 Our model atmospheres

The model planetary atmospheres we use in our numerical sim-ulations are composed of stacks of horizontally homogeneousand locally plane-parallel layers which contain gas moleculesand optionally cloud particles We create vertically inhomoge-neous atmospheres simply by stacking different homogeneouslayers Each atmosphere is bounded below by a flat homoge-neous surface We base our atmospheric temperature and pres-sure profiles on representative ones of the Earthrsquos atmosphere(McClatchey et al 1972) Here we use only four atmosphericlayers (see Table 1 for the pressures and temperatures at thetops of these layers) and a black planetary surface The molecu-lar or Rayleigh scattering optical thickness of each atmosphericlayer and its spectral variation is calculated as described in Stam(2008) assuming an atmospheric gas composition of the modelatmospheres that is similar to that of the Earth For our simula-tions of the flux and polarisation spectra we focus on the con-tinuum and we ignore absorption by atmospheric gases (such asO2 H2O and O3) Even though we use pressure and temperatureprofiles and a gas composition that are typical for an Earth-likeplanet most of our results are also applicable to other planetseg gas giants with high altitude water clouds

We will use a clear model atmosphere ie without cloudsand cloudy model atmospheres In the latter one of the atmo-spheric layers contains cloud particles in addition to the gasmolecules To study the dependence of the flux and degree ofpolarisation of the starlight that is reflected by the model planeton the altitude of the top of the cloud layer we will use the fol-lowing three cases a low cloud layer with its top at an ambientpressure of 0802 bar (corresponding to an altitude of 2 km onEarth) a middle cloud layer with its top at 0710 bar (3 kmon Earth) and a high cloud layer with its top at 0628 bar (4 kmon Earth) Unless stated otherwise the geometrical thickness ofthe cloud layer is 1 km

Optical thicknesses of clouds on Earth show a huge varia-tion on daily and monthly timescales and from place to place Inparticular the optical thickness in completely cloudy cases canreach values up to 40 (van Deelen et al 2008) or more The op-tical thickness b of our cloud layers is chosen to range between05 and 60 (at λ = 055 μm) Our standard cloud layer has anoptical thickness of 10 (at λ = 055 μm) which appears to be anaverage value The spectral variation of the cloud layerrsquos opticalthickness depends on the microphysical properties of the cloudparticles such as their composition shape and size (see below)

Our model cloud layers are composed of liquid sphericalwater particles Water cloud observations on Earth show a largevariation in droplet sizes that is attributed to amongst others avariation in the number density and composition of cloud con-densation nuclei (Han et al 1994 Martin et al 1994 Segal ampKhain 2006) In particular typical cloud particle radii range fromabout 5 μm to 15 μm with a global mean value of about 85 μmabove continental areas and 118 μm above maritime areas

001 010 100 1000 10000particle size (μm)

00001

00010

00100

01000

10000

100000

1000000

n(r)

B part distr

A part distr

Fig 1 Particle size distributions (see Eq (8)) for model A particles(reff = 20 μm veff = 01 red dashed line) and model B particles(reff = 60 μm veff = 04 black solid line) Each size distribution hasbeen normalised such that the integral over all sizes equals 1

(Han et al 1994) On the lower limit sizes down to 2 μm havebeen reported from satellite measurements (Minnis et al 1992)and on the upper limit sizes up to 25 μm (Goloub et al 2000)

We describe the sizes of our cloud particles by a standardsize distribution (Hansen amp Travis 1974) as follows

n(r) = C r(1minus3veff )veff eminusrveffreff (8)

where C is a normalisation constant n(r)dr is the number ofparticles with radii between r and r+dr per unit volume and reffand veff are the effective radius and variance respectively (seeHansen amp Travis 1974) The units of reff are [μm] while veff isdimensionless

The scattering properties of particles often depend stronglyon the ratio of the radius of the particles to the wavelength ofthe light This so-called effective size parameter xeff that willbe used later in this paper is defined as

xeff =2πreff

λ(9)

(see Hansen amp Travis 1974) For xeff 1 the scattering is usu-ally referred to as Rayleigh scattering and for xeff 1 we getinto the regime where light scattering can be described with ge-ometrical optics

Our cloud layers are composed of either small model A par-ticles with reff = 20 μm and veff = 01 (Stam 2008) or largermodel B particles with reff = 60 μm and veff = 04 The lat-ter are similar to those used by van Diedenhoven et al (2007)who consider this to represent an average terrestrial water cloudFigure 1 shows the size distributions of the particles of the twostandard models used further in this paper (A and B particles)To study the influence of a size distributionrsquos effective varianceveff on the reflected light we will also use model A2 particleswith reff = 20 μm and veff = 04 and model B2 particles withreff = 60 μm and veff = 01 (see Table 2)

The real part of the refractive index of water in the wave-length region of our interest is slightly wavelength dependent Itvaries from 1344 at λ = 04 μm to 1320 at 10 μm (Daimon ampMasumura 2007 van Diedenhoven et al 2007) The imaginarypart of the refractive index is small but varies strongly (Pope ampFry 1997) from about 10minus8 at 03 μm to about 10minus5 at 10 μm

A69 page 4 of 14

T Karalidi et al Spectra of water clouds on exoplanets

Table 2 The effective radius reff (in μm) and variance veff of the stan-dard size distributions (see Hansen amp Travis 1974 and Eq (15)) thatdescribe our model cloud particles

Particle reff veffA 20 01A2 20 04B 60 04B2 60 01

with a minimum of 8times10minus10 at 05 μm We use a constant refrac-tive index that is equal to 1335plusmn 000001i We checked that ourresults are virtually insensitive to the assumption of a constantvalue of the refractive index

Given the wavelength refractive index effective radius reffand variance veff we calculate the cloud particlesrsquo extinctioncross-section single scattering albedo and the expansion co-efficients of the single scattering matrix in generalised spher-ical functions (see Hovenier et al 2004) using Mie theory(van de Hulst 1957 de Rooij amp van der Stap 1984) With thehence obtained extinction cross-sections we calculate the cloudlayerrsquos optical thickness at wavelengths other than 055 μmFigure 2 shows the spectral variation of the absorption and scat-tering optical thicknesses of four cloud layers each with a totaloptical thickness of 20 at λ = 055 μm that are composed of themodel A A2 B and B2 particles respectively As can be seen inthe figure the effective variance veff plays only a minor role indetermining the cloudrsquos scattering and absorption optical thick-nesses

4 Single scattering properties of water cloudparticles

The degree of polarisation of the starlight that is reflected by aplanet is very sensitive to the single scattering properties of theatmospheric particles (see eg Hansen amp Travis 1974) The rea-son for this is that due to the generally low degree of polarisationof multiple-scattered light the main angular features observedin a polarised planetary signal will be due to single scatteredlight The contribution of the multiple scattered light to the sig-nal is mostly to decrease the overall degree of polarisation notto change the shape or angular distribution of the features Thusin order to understand the features observed in starlight that isreflected by a planet as presented in Sect 5 knowledge of thesingle scattered light is essential

In this section we therefore present and discuss the flux anddegree of polarisation of incident unpolarized light Ps that issingly scattered by our model liquid water cloud particles asfunctions of the single scattering angle Θ and the wavelengthλ The flux as a function of Θ is usually referred to as the (flux)phase function and we will refer to Ps as a function of Θ as thepolarised phase function

Figure 3 shows the flux and Ps of the four model cloud parti-cles A B A2 and B2 (see Table 2 for their sizes) as functions ofthe scattering angle Θ at λ = 055 μm For comparison we havealso added the phase function of the gas molecules (Rayleighscattering) using an Earth-like depolarization factor ie 0028(Bates 1984) The flux phase functions are normalised such thattheir average over all scattering directions equals unity they thusdo not include the particlesrsquo single scattering albedo

As can be seen in Fig 3 Ps of light that is singly scattered bythe gas molecules and by the model cloud particles equals zero atΘ = 0 (forward scattering) and 180 (backward scattering) The

03 04 05 06 07 08 09 10wavelength (μm)

00

01

02

03

04

05

b abs

A particlesB particlesA2 particlesB2 particles

03 04 05 06 07 08 09 10wavelength (μm)

15

16

17

18

19

20

21

22

b sca

Fig 2 Spectral variation of the absorption (upper panel) and scattering(lower panel) optical thicknesses babs and bsca of cloud layers com-posed of the following model particles (see Table 2) A (reff = 20 μmveff = 01 black solid line) A2 (reff = 20 μm veff = 04 blue dashedndashdotted line) B (reff = 60 μm veff = 04 red dashed line) and B2

(reff = 60 μm veff = 01 orange dashedndashtripplendashdotted line) Thetotal optical thickness (b = bsca + babs) of each cloud layer is 20 atλ = 055 μm

reason for this is that the incoming light is unpolarised and that atthese scattering angles the scattering process is symmetric withrespect to the incoming and the scattered light The phase func-tion and Ps of Rayleigh scattered light is smooth and symmetricaround Θ = 90 The maximum value of Ps of this light is 094(ie 94)

The flux phase functions of all four types of cloud particleshave a strong peak in the forward scattering direction due to lightthat is diffracted by the particles and the so-called glory in thebackscattering direction (cf Hansen amp Travis 1974) The polar-isation phase functions of the cloud particles have several inter-esting features In particular for each particle type Ps changessign (thus direction) several times between Θ = 0 and 180For Θ 50 the angular features of Ps appear to be insensitiveto the particle size while for larger scattering angles the featuresdepend clearly on reff and veff

Between Θ = 135 and 150 the phase functions of allfour types of cloud particles show a local maximum that isusually referred to as the (primary) rainbow which is formedby light that has been reflected once inside the particles (seevan de Hulst 1957 Hansen amp Travis 1974) These rainbows are

A69 page 5 of 14

AampA 530 A69 (2011)

0 30 60 90 120 150 1800scattering angle Θ (degrees)

-2

-1

0

1

2

3

4 A particlesB particlesA2 particles

Rayleigh scatteringB2 particles

phas

e fu

nctio

n

0 30 60 90 120 150 1800scattering angle Θ (degrees)

-04

-02

00

02

04

06

08

10

Deg

ree

of P

olar

isat

ion

Fig 3 Phase function on a logarithmic scale (upper panel) and degreeof polarisation Ps (lower panel) for incident unpolarised light at λ =055 μm that is singly scattered by our model water cloud particles (seeTable 2) as functions of the scattering angle Θ model A (black solidlines) model B (red dashed lines) model A2 (blue dashed-dotted lines)and model B2 (orange dashed-triple-dotted lines) For comparison thephase function and Ps are also shown for gas molecules (ie Rayleighscattering) (purple long-dashed lines)

clear indicators of the spherical shape of the scattering parti-cles and their angular position depends strongly on the compo-sition (refractive index) of the scattering particles (see Hansenamp Travis 1974) and slightly on reff of the particles (see below)While especially for the small cloud particles (models A andA2) the primary rainbow is hardly visible in the flux phase func-tion (see Fig 3) in Ps it is the strongest and most prominent an-gular feature for each of the four cloud particle types Note thatthe rainbows that are seen in the Earthrsquos sky during showers arenot formed in cloud particles but in raindrops On Earth rain-drops have radii on the order of millimeters corresponding toxeff 1000 at visible wavelengths

For reff = 2 μm (particles A and A2) the rainbow is locatedat Θ asymp 148 For reff = 6 μm (particles B and B2) the rainbowis more pronounced and shifted to slightly smaller scattering an-gles ie to Θ asymp 143 The strength of the primary rainbow in Psdepends on reff and on veff the smaller reff the smaller the max-imum value of Ps in the primary rainbow and the smaller veff the larger this maximum value In particular for reff = 20 μmand veff = 01 (model A) Ps of the primary rainbow is 058(58) while for reff = 20 μm and veff = 04 (A2) it is 050 For

log10F (veff=01)

0 30 60 90 120 150 1800

20

40

60

80

100

120

-05

905

-05

905

-05905

05656

17216

287

76

scattering angle (degrees)

x eff

-175 -117 -059 -001 057 114 172 230 288 346

Ps (veff=01)

0 30 60 90 120 150 1800

20

40

60

80

100

120 -030-030

-01

0-0

10

-010

-010-010

010

010010

010

010

010

030

030

030

030

030

scattering angle (degrees)

x eff

-050 -040 -030 -020 -010 000 010 020 030 040 050 060 070

Fig 4 Phase function on a logarithmic scale (upper panel) and degreeof polarisation Ps (lower panel) for incident unpolarised singly scat-tered light as functions of the scattering angle Θ and the effective sizeparameter xeff = 2πreffλ for model cloud particles with veff = 01 iemodels A and B2 For model A (with reff = 20 μm) the curves shown inFig 3 correspond to xeff = 23 while for model B2 (with reff = 60 μm)they correspond to xeff = 69

reff = 60 μm and veff = 01 (B) Ps of the primary rainbow is080 and for reff = 60 μm and veff = 04 (B2) Ps = 069 Thescattering angle of the primary rainbow (Θ where Ps is maxi-mum) depends mostly on reff at λ = 055 μm the angle is 150for reff = 20 μm (A and A2) and 143 for reff = 60 μm (B andB2)

For the large particles a secondary peak in the flux phasefunction is visible around Θ asymp 120 This peak is the secondaryrainbow which is formed by light that has been reflected twiceinside the particles While in the phase function this rainbowis hardly visible and only for the largest particles Ps clearlyshows the secondary rainbow for all four cloud particle typesfor reff = 60 μm (B and B2) around Θ = 120 and for thesmaller particles at slightly smaller angles Noteworthy is theminor peak around 155 in Ps of the large B2 particles this is asupernumerary arc (see eg Dave 1969) For the same particleswe see a peak around Θ sim 100 which could also be a supernu-merary arc Supernumerary arcs are interference features whichexplains their washing out with increasing veff

The effects of reff and veff on the singly scattered light areeven more clear from Figs 4 and 5 which show the scatteredflux and Ps as functions of Θ and the effective size parameter

A69 page 6 of 14

T Karalidi et al Spectra of water clouds on exoplanets

log10F (veff=04)

0 30 60 90 120 150 1800

20

40

60

80

100

120

-05

02

-0502

06540654

1811

296

7

scattering angle (degrees)

x eff

-166 -108 -050 008 065 123 181 239 297 355

Ps (veff=04)

0 30 60 90 120 150 1800

20

40

60

80

100

120

-01

0-0

10

-010

-010

-010

010

010

010

010

010

010

030

030

030 030030

scattering angle (degrees)

x eff

-050 -040 -030 -020 -010 000 010 020 030 040 050 060 070

Fig 5 Similar to Fig 4 except for particles with veff = 04 ie mod-els A2 and B For model A2 (with reff = 20 μm) the curves shown inFig 3 correspond to xeff = 23 while for model B (with reff = 60 μm)they correspond to xeff = 69

xeff (cf Eq (9)) for veff = 01 (A and B2) and veff = 04 (A2 andB) respectively Similar graphs for other size distributions andrefractive indices can be found in for example Hansen amp Travis(1974) and Hansen amp Hovenier (1974) The curves shown inFig 3 correspond to horizontal cuts in Figs 4 and 5 at xeff = 23(reff = 2 μm and λ = 055 μm particles A and A2) and xeff = 69(reff = 6 μm and λ = 055 μm particles B and B2) respectively

Comparing the graphs of the flux phase functions in Figs 4and 5 it is clear that the phase function depends mostly on xeff thus on the ratio 2πreffλ and that it is not very sensitive to veff which mostly determines the smoothness of the phase functionie the larger veff the more subdued the angular variation in thephase function (for a given reff) (see Hansen amp Travis 1974)Additionally we see that Ps appears to be somewhat more sen-sitive to veff than the flux phase function but the overall appear-ance of Ps is the same for the two values of veff In particularfor Θ lt 20 both figures show a peninsula with small values ofPs As can also be seen in Hansen amp Travis (1974) and Hansenamp Hovenier (1974) the shape of this peninsula is sensitive to vefffor small values of xeff

In both Figs 4 and 5 the primary rainbow (just belowΘ = 150) is only a slight crest in the flux phase functions butby far the strongest feature in the polarisation phase functionsThe strength of the polarised primary rainbow increases with in-creasing xeff In other words for particles with a given reff the

04 05 06 07 08 09wavelength (μm)

140

145

150

155

160

Rai

nbow

sca

tterin

g an

gle

(deg

rees

)

A particlesB2 particles

Fig 6 Scattering angle Θ where the primary rainbow in Ps occurs asa function of the wavelength for the model A (reff = 20 μm) and B2

(reff = 60 μm) particles

strength of the rainbow increases towards the blue andor at agiven wavelength the strength of the rainbow increases with in-creasing reff Interesting to note is that for small values of xeff (iele60) the primary rainbow shifts to larger values of Θ with de-creasing xeff ie for a given effective particle radius reff the rain-bow shifts to larger values of Θ towards the red Consequentlywhen observed with polarimetry the rainbow in visible light thatis scattered by small water cloud particles should be inverted incolour compared to the rainbow of light that is scattered by wa-ter raindrops (this can not be observed in the flux since the rain-bow is virtually unobservable in the flux of light scattered bysmall cloud particles)

This colour inversion is shown more clearly in Fig 6 wherethe scattering angle of the primary rainbow in Ps is plottedas a function of λ for the model A and B2 particles For thesmallest particles the rainbow angle increases from 147 atλ = 03 μm to 157 at λ = 10 μm From these and other nu-merical simulations (not shown) it appears that dΘdλ decreaseswith increasing reff until reff asymp 20 μm Indeed for particles withxeff sim 100 whitish rainbows (so-called fogbows) are observed influx (Adam 2002) For much larger particles such as rain dropsdΘdλ becomes negative and decreases further with increasingreff We checked that the behavior of dΘdλ that we calculatedis not affected by our choice of using a wavelength independentrefractive index nor by our choice of the particle size distribu-tion

Our single scattering results suggest that in particular observ-ing the dispersion of the primary rainbow in polarisation wouldbe a useful tool in the retrieval of cloud particle shapes and sizesin exoplanetary atmospheres To evaluate the usefulness of thistool numerical simulations of multiple scattered light are re-quired Results of such simulations are presented and discussedin Sect 5

5 Light that is reflected by planets

In the previous section we have presented the single scatteringphase functions of our model cloud particles Here we presentthe normalised flux πFn and the degree of polarisation Ps of (sin-gle and multiple scattered) starlight that is reflected by a planetas a whole thus integrated over the illuminated and visible partof the planetary disk We show the dependence of πFn and Ps on

A69 page 7 of 14

AampA 530 A69 (2011)

0 30 60 90 120 150 180000

02

04

06

08

planetary phase angle (degrees)

Nor

mal

ised

flux

no cloudb=05b=10b=100b=200b=400b=600

0 30 60 90 120 150 1800-02

00

02

04

06

08

10

planetary phase angle (degrees)

Ps

Fig 7 Normalised flux πFn (upper panel) and Ps (lower panel) at λ =055 μm as functions of the planetary phase angle α for model planetswith a cloudfree atmosphere (black solid line) and with atmospherescontaining a cloud layer ranging from 3 to 4 km composed of model Acloud particles with the following cloud optical thicknesses b (at λ =055 μm) 05 (red dashed lines) 10 (green dash-dotted lines) 100(blue dash-triple-dotted lines) 200 (orange long-dashed lines) 400(pink diamond lines) and 600 (purple cross lines)

the cloud optical thickness in Sect 51 on the size of the cloudparticles in Sect 52 and on the cloud top pressure in Sect 53We show πFn and Pn as functions of the planetary phase angleα which equals 180 minus Θ with Θ the single scattering angle

51 The effects of the cloud optical thickness

Figure 7 shows πFn and Ps of light reflected by cloudy plan-ets as functions of the phase angle α At α = 0 the planetrsquosilluminated side is fully visible and at α = 180 we see theplanetrsquos night side The cloud layer on each planet is composedof model A particles (see Table 2) and the cloud top pressureis 0628 bar (on Earth this would correspond to a cloud top alti-tude of 4 km see Table 1) At 055 μm the molecular scatteringoptical thickness of the whole model atmosphere is 0098 andthat of the gaseous atmosphere above the cloud 006 In the fig-ure the cloud optical thickness varies from 0 (ie no cloud atall) to 60 (at λ = 055 μm) Note that a cloud optical thicknessof 60 appears to be very large for a 1 km thick cloud We haveincluded this large value in our simulations for the purpose ofcomparison

As can be seen in Fig 7 the quasi-monochromatic geomet-ric albedo AG the cloudfree planet is as small as 0063 which isdue to the black surface and the small atmospheric (molecular)scattering optical thickness The normalised flux πFn decreasessmoothly with α The degree of polarisation Ps of the cloudfreeplanet is zero at α = 0 (due to symmetry) and 180 (due tofirst order scattering see Hovenier amp Stam 2007) (cf Fig 3)In between these phase angles Ps varies smoothly with α andis positive except for α 168 These negative values whichindicate that the light is polarised parallel instead of perpendic-ular to the reference plane (as it is at the smaller phase angles)are due to second order scattered light The maximum degree ofpolarisation of the cloudfree planet is 087 (87) and occurs atα = 90

Figure 7 also shows that πFn generally increases with in-creasing cloud optical thickness and converges rapidly for b 40 With such large optical thicknesses the cloud layer appearsto be semi-infinite making πFn insensitive to further increasesof b The sensitivity of πFn to b decreases with increasing phaseangle and in particular for α gt 150 πFn hardly changes withb since in this limb viewing geometry most of the reflectedstarlight has been scattered in the layers above the cloud layeror by the highest cloud particles and did not penetrate deep intothe layer

The normalised flux πFn shows a few angular features Atα sim 30 (Θ sim 150) the primary rainbow (cf Fig 3) is slightlyvisible and the local maximum in πFn at α sim 8 can be tracedback to the cloud particlesrsquo single scattering feature at Θ asymp 170in Fig 3

The degree of polarisation Ps of the cloudy planets is a mix-ture of the degree of polarisation of light that is scattered by theatmospheric gases and of light that is scattered by the cloud par-ticles (and includes of course light that has been scattered byboth) In particular for an atmosphere with a cloud with a totaloptical thickness b of only 05 (at λ = 055 μm) the maximum Psstill clearly shows the angular features of a Rayleigh scatteringatmosphere with a maximum of 046 around α asymp 78 At mostphase angles |Ps| decreases with increasing value of b due to theincreasing contribution of light with a low degree of polarisa-tion that has been multiple scattered within the cloud layer Thedegree of polarisation converges rapidly for b 20 It does notnecessarily converge to zero because even for the thickest cloudPs is determined by the degree of polarisation of light that hasbeen singly scattered within the upper parts of the cloud and thenearly unpolarised flux from the deeper parts which convergesfor large values of b

Indeed even for optically thick clouds Ps of the planetshows the traces of the degree of polarisation of the singly scat-tered light For example the negative values of Ps of the cloudyplanets around α asymp 10 can be traced back to the single scat-tering feature of the cloud particles at Θ asymp 170 (see Fig 3)The maximum in Ps around α asymp 30 is the primary rainbowIncreasing b decreases Ps of this rainbow because multiple scat-tering increases the unpolarised total flux for b = 05 Ps = 024(24) while for b = 20 Ps = 006 (6) The detection ofthe primary rainbow in starlight that is reflected by an exoplanetwould indicate that the cloud particles are made of liquid wa-ter (this was also pointed out by eg Hansen amp Travis 1974Liou amp Takano 2002 except for individual clouds not for wholeplanets) and if the phase angle of the rainbow could be deter-mined accurately it would hold information on the particle sizesFor cloudy exoplanets (with a disk integrated signal) this rain-bow was also discussed by Bailey (2007) (whose radiative trans-fer calculations do not include multiple scattering) and clearly

A69 page 8 of 14

T Karalidi et al Spectra of water clouds on exoplanets

π Fn

0 30 60 90 120 150 180010

09

08

07

06

05

04

030075

01500225

planetary phase angle (degrees)

λ (μ

m)

-000

003

008

012

017

022

028

Ps

0 30 60 90 120 150 180010

09

08

07

06

05

04

03

010

00

100

01000100

030

00

300

030003000

500

050

0

05000500

planetary phase angle (degrees)

λ (μ

m)

-050 -020 -010 000 010 020 030 040 050 060 070 080 090

Fig 8 Normalised flux πFn (upper panel) and Ps (lower panel) as func-tions of the planetary phase angle α and wavelength λ for the cloudfreemodel planet

shows up in the numerical simulations (that do include multi-ple scattering) by Stam (2008) The latter uses only the smallestof our cloud particle sizes (ie with reff = 2 μm) and a rela-tively thick cloud layer (b = 10 at λ = 055 μm) which resultsin a rather subdued rainbow (Ps = 010 at λ = 044 μm) Thesecondary rainbow that was seen in Fig 3 does not show up inFig 7 because for small cloud optical thicknesses b it vanishesin the contribution of the Rayleigh scattered light while for largevalues of b it is suppressed by nearly unpolarised multiple scat-tered light

The normalised flux and degree of polarisation of the re-flected starlight depend not only on phase angle α but also onwavelength λ In Fig 8 we show πFn and Ps of light that is re-flected by a cloudfree (b = 0) planet as functions of α and λThe atmospheric molecular scattering optical thickness rangesfrom 11 at λ = 03 μm to 0009 at λ = 10 μm Clearly πFnis largest at the smallest values of λ and α where the atmo-spheric optical thickness is largest and where most of the illumi-nated hemisphere of the planet is visible and decreases smoothlywith increasing λ and α The degree of polarisation Ps shows thestrong maximum around α = 90 that is due to Rayleigh scatter-ing The general increase of Ps with λ is due to the decrease ofthe atmospheric optical thickness and hence the multiple scat-tering which usually decreases Ps with λ The decrease of theatmospheric optical thickness also explains why the phase an-gle region where Ps is negative narrows with increasing λ the

π Fn

0 30 60 90 120 150 180010

09

08

07

06

05

04

03

0096

009

6

0192

019

2

028

8

0288

planetary phase angle (degrees)

λ (μ

m)

-000

003

010

016

022

029

035

Ps

0 30 60 90 120 150 180010

09

08

07

06

05

04

03

006

006

006

006

022038

planetary phase angle (degrees)

λ (μ

m)

-010 -002 006 014 022 030 038 046 054 062

Fig 9 Similar to Fig 8 except for a model planet with a cloud layerwith b = 100 (at λ = 055 μm) with its top at 0628 bar that is composedof model A particles (reff = 20 μm)

smaller the optical thickness the longer the path through the at-mosphere and hence the larger the phase angle required to haveenough second order scattered light to change the sign of Ps

Figure 9 is similar to Fig 8 except for an atmosphere thatcontains a cloud layer composed of model A particles withb = 100 (at λ = 055 μm) and with its top at 0628 bar Thecloud layer strongly increases πFn except at large values of αwhere the observed light has not penetrated deep enough into theatmosphere to encounter the cloud The features seen at α asymp 10were also seen in Fig 7 and trace back to the single scatteringfeatures at Θ asymp 170 in Fig 3 Like in Fig 7 the primary rain-bow is hardly visible in the fluxes shown in Fig 9

Adding a cloud layer to the model atmosphere strongly de-creases Ps especially at longer wavelengths as can be seenfrom comparing Figs 9 and 8 At the shortest wavelengths theRayleigh scattering maximum of Ps (around α = 90) is stillvisible because there the molecular scattering optical thicknessabove the cloud layer is still significant With increasing λ thisoptical thickness decreases and the contribution of light that isreflected by the cloud layer increases In particular the ridge inPs near α = 35 at 05 μm and extending towards α = 25 at10 μm is the primary rainbow The strength of this rainbow de-creases with increasing λ from about 015 (15) at λ = 03 μmto about 007 (7) at λ = 10 μm

The branch of negative values of Ps in Fig 9 that appearsfrom λ = 04 to 085 μm for α 20 corresponds to the branch

A69 page 9 of 14

AampA 530 A69 (2011)

0 30 60 90 120 150 180010

09

08

07

06

05

04

03

00725

007

25

01725

017

25

planetary phase angle (degrees)

λ (μ

m)

-001

002

007

012

017

022

027

Ps

0 30 60 90 120 150 180010

09

08

07

06

05

04

03

006

006

006

006

022

022

022

038

054

planetary phase angle (degrees)

λ (μ

m)

-010 -002 006 014 022 030 038 046 054 062 070

Fig 10 Similar to Fig 9 except for a cloud layer that is composed ofmodel B particles (reff = 60 μm)

of negative values of Ps for light that has been single scatteredby the model A cloud particles for 150 lt Θ lt 180 (see Fig 4)The negative values of Ps in Fig 9 that appear for λ gt 07 μmand for intermediate phase angles are related to the band of neg-ative values of the single scattering Ps for Θ lt 90 (see Fig 4)

52 The effects of the cloud particle sizes

To study the effects of the cloud particle sizes on πFn and Ps ofthe reflected starlight we replace the model A particles in ourcloud layer with model B particles while keeping the cloud toppressure at 0628 bar and its optical thickness b equal to 10 (atλ = 055 μm) The resulting πFn and Ps are shown in Fig 10For comparison Fig 11 shows a cross-section of Figs 9 and 10at a phase angle of 90 and includes lines for clouds composedof model A2 particles

Comparing Figs 10 and 9 we can see that the introductionof the larger cloud particles in our model atmosphere leaves cleartraces in the reflected πFn and Ps Figure 11 shows for α = 90that the effect of the cloud particle variance on πFn and Ps isnegligibly small Indeed the particle effective radius appears tobe the parameter that influences the reflected signals most

In Figs 10 and 9 normalised flux πFn is significantly lowerwith the larger particles except for the largest values of λ and αThe lower values of πFn are explained by the smaller single scat-tering albedo of the model B cloud particles (see Fig 2) Indeed

03 04 05 06 07 08 09 10λ (μm)

002

004

006

008

010

012

014

Nor

mal

ised

flux

A particlesB particlesA2 particlesα=90deg

03 04 05 06 07 08 09 10λ(μm)

-01

01

03

05

07

Ps

Fig 11 Cross-sections through Figs 9 and 10 at α = 90 to show thespectral effects of the particle size on the reflected normalised flux anddegree of polarisation model A particles (black solid line Fig 9)and model B particles (red dot-dashed line Fig 10) Also includedmodel A2 particles (blue triple-dot-dashed line)

while the albedo of the model A particles equals sim1 across thewavelength range under consideration the albedo of the model Bparticles varies from 084 at 03 μm to 090 at 055 μm and094 at 10 μm The differences between the reflected normalisedfluxes at the largest phase angles are relatively small becausethere most of the reflected starlight has been scattered in thelayers above the cloud

With the model B cloud particles πFn of the light that isreflected by the planet shows a somewhat stronger primary rain-bow than with the smaller model A particles (Fig 9) This isexplained by the difference in strength of the primary rainbowin the single scattering phase functions of both particle types(Fig 3)

With the model B cloud particles (Fig 10) Ps shows strongerangular features than with the smaller model A particles (Fig 9)In particular the maximum around α = 90 is higher which isdue to the lower single scattering albedo of the model B parti-cles because more light is absorbed within the cloud layer thereis less multiple scattering and less (little polarised) light is scat-tered upward by the cloud layer With increasing λ the maxi-mum in Ps shifts towards smaller phase angles because therethe Rayleigh scattering feature blends with the strong features inthe single scattering Ps of the model B particles that can be seenfor Θ gt 100 in Figs 3 and 5

A69 page 10 of 14

T Karalidi et al Spectra of water clouds on exoplanets

Furthermore as expected from the single scattering polar-isation phase function (Fig 3) and as in the case of πFn theprimary rainbow is a much more prominent feature in Ps ofthe reflected starlight with the model B particles than with themodel A particles In particular at λ = 055 μm with themodel A particles Ps in the rainbow is about 012 while withthe model B particles Ps in the rainbow reaches a value as highas 026 The latter value is comparable to the case in which theatmosphere contains a cloud layer of model A particles with anoptical thickness of only b = 05 (see Fig 7) In observationsthe two cases should be separable because with the model B par-ticles the primary rainbow peak is the global maximum acrossthe entire α regime while in the case of the thinner cloud madeout of model A particles it is just a local maximum since Ps inthe Rayleigh scattering peak is as large as 044

53 The effects of the cloud top pressure

To study the effects of the cloud top pressure on πFn and Pswe use a cloud composed of model A particles with b = 10(at λ = 055 μm) and place it with its top at three differ-ent pressures ptop in the model atmosphere The geometricalthickness of each cloud is 1 km We use a low cloud withptop = 0802 bar (corresponding to 2 km on Earth) a middlecloud with ptop = 0710 bar (3 km on Earth) and a high cloudwith ptop = 0628 bar (4 km on Earth) The corresponding cloudtop temperatures on Earth are 285 279 and 273 K respectively(see Table 1) These cloud top pressures are chosen to corre-spond with terrestrial liquid water clouds

Figure 12 shows πFn and Ps as functions of λ at α = 90for the three values of ptop At wavelengths shorter than about06 μm πFn increases with increasing ptop because of the in-creasing amount of gaseous molecules and hence molecularscattering optical thickness above of the cloud layer With in-creasing wavelength the sensitivity of πFn to ptop vanishes be-cause of the decreasing molecular scattering optical thicknessabove the cloud layer The increase of πFn with increasing λthat occurs for all three values of ptop is due to the correspond-ing increase of the scattering optical thickness of the cloud layerand the decrease of its absorption optical thickness (see Fig 2)

As can be seen in Fig 12 Ps increases with increasing ptopat almost all wavelengths because of the increasing amount ofmolecules which scatter light with a relatively high degree ofpolarisation above the cloud layer In Fig 12 the largest in-crease in Ps is 0044 at λ = 044 μm when ptop increases from0628 bar to 0802 bar The change of Ps with ptop vanishes atthe shortest and longest wavelengths When λ 032 μm Psactually decreases slightly with increasing ptop because here theincreasing amount of molecules leads to an increase of multiplescattered little polarised light At the longest wavelengths themolecular scattering optical thickness of the atmosphere abovethe cloud layer is too small for each of the three values of ptop tosignificantly influence Ps

The dependence of πFn and Ps on ptop varies not only with λbut also with α In Fig 9 we presented πFn and Ps as functionsof α and λ for a cloud layer with ptop = 0628 bar To get abetter view of the change of πFn and Ps with ptop Fig 13 showsπFn(ptop = 0802) minus πFn(ptop = 0628) and Ps(ptop = 0802) minusPs(ptop = 0628) as functions of α and λ As can be seen inFig 13 πFn increases with increasing ptop except for α 110and λ 032 μm The changes of πFn with ptop from 0802 to0628 bar are very small at maximum sim33 for λ = 03 μmand α sim 10 In particular at the phase angles around 90 wherean exoplanet will be easiest to observe directly because it will

03 04 05 06 07 08 09 10λ (μm)

006

008

010

012

Nor

mal

ised

flux

0802 bar0710 bar0628 bar

α=90deg

03 04 05 06 07 08 09 10λ (μm)

-01

00

01

02

03

04

05

06

Ps

Fig 12 Normalised flux πFn (upper panel) and Ps (lower panel) asfunctions of the wavelength λ for model planets with cloud layerswith cloud top pressures ptop equal to 0802 bar (solid line) 0710 bar(dashed line) and 0628 bar (dashed-dotted line) The cloud layer hasan optical thickness b = 10 (at 055 μm) and is composed of model Aparticles The planetary phase angle α is 90

be relatively far from its star the sensitivity of πFn to ptop isextremely small making the derivation of cloud top pressures inthis range using flux measurements practically impossible

Regarding the polarisation plot of Fig 13 increasing ptopyields the largest increases in Ps for λ 035 μm and α asymp 90As was also explained for Fig 12 Ps increases with increasingptop because of the increase of the amount of molecules abovethe cloud layer In particular the largest change in Ps is 005 atλ = 044 μm and α = 94

In Fig 14 finally we present πFn and Ps as functions of ptopfor a cloud layer with an optical thickness of 10 (at 055 μm)consisting of A or B particles for λ = 055 μm and α = 90Not surprisingly both πFn and Ps vary smoothly with ptop Thenormalised flux increases with ptop for both cloud particle typesIn particular for the model A (B) particles πFn increases fromabout 0072 (0032) at ptop = 0 bar to 0079 (0044) at ptop =1 bar At this wavelength the degree of polarization Ps increaseswith ptop as well for the model A particles Ps increases from-0015 at ptop = 0 bar to about 017 at ptop = 1 bar and forthe model B particles Ps increases from 0045 at ptop = 0 barto about 034 at ptop = 1 bar The negative values of Ps for themodel A particles and ptop lt 01 bar are explained by the singlescattering properties of these particles at Θ = 90 (see Fig 3)

A69 page 11 of 14

AampA 530 A69 (2011)

Δ (π Fn)

0 30 60 90 120 150 180010

09

08

07

06

05

04

03

00005

00005

000

05

00025

planetary phase angle (degrees)

λ (μ

m)

-0002 -0001 0000 0001 0002 0003 0004 0005 0006 0008 0008 0009 0010

Δ Ps

0 30 60 90 120 150 180010

09

08

07

06

05

04

03 -0005

0012

0012

0028

planetary phase angle (degrees)

λ (μ

m)

-001

000

001

002

003

004

Fig 13 Differences πFn(ptop = 0802) minus πFn(ptop = 0628) (upperpanel) and Ps(ptop = 0802) minus Ps(ptop = 0628) (lower panel) as func-tions of λ and α for a model planet with a cloud layer with b = 10 (atλ = 055 μm) that is composed of model A particles

6 Summary and discussion

We have presented numerically simulated normalised flux (πFn)and polarisation (Ps) spectra from 03 to 10 μm of exoplanetsthat are completely covered by liquid water clouds We studiedthe effects of the cloud optical thickness the size of the cloudparticles and the cloud top altitude on the spectra as functions ofthe planetary phase angle Knowing the microphysical propertiesof cloud particle on a planet is important for our understandingof the cloudrsquos influence on the planetary climate In particularfrom Earth studies we know that the cloud particle sizes stronglyinfluence the cloud radiative forcing (see eg Chapman et al2009 Kobayashi amp Adachi 2009 and references therein) sincethey change the way cloud particles scatter and absorb incidentsunlight and thermal radiation

Given the huge number of free parameters in systems likethis our aim was not to cover the whole parameter space butrather to explore the information content of in particular the de-gree of polarisation of starlight that is reflected by a planet andthe spectral and phase angle ranges that would provide this infor-mation Although we used atmospheric temperature and pressureprofiles that are typical for an Earth-like planet our results canalso be used to represent liquid water clouds on gaseous planetsexcept that the cloud top pressures are likely to be different

00 02 04 06 08 10Cloud top pressure (bar)

002

004

006

008

010

012

Nor

mal

ised

flux

A particles

B particles

00 02 04 06 08 10Cloud top pressure (bar)

-01

00

01

02

03

04

PS

Fig 14 Normalised flux πFn (upper panel) and Ps (lower panel) at λ =055 μm as functions of ptop for model cloud layers with b = 10 (atλ = 055 μm) that are composed of model A or model B particles Theplanetary phase angle α is 90

The cloudrsquos optical thickness b strongly influences the nor-malised flux and polarisation spectra of our model planets Inparticular up to about b = 40 an increase of b leads to anincrease of πFn For larger optical thicknesses the cloud layerappears to be semi-infinite and the normalised flux spectra nolonger change significantly In polarisation increasing b lowersthe (polarisation) continuum because it increases the amount ofmultiple scattered light with usually a low degree of polarisa-tion to the total amount of reflected light While multiple scat-tering subdues the angular features in the polarisation even forthe largest values of b Ps as a function of the planetary phaseangle still carries the angular features that are representative forlight that was singly scattered by the cloud particles in the upperlayers of the cloud In particular the locations of maxima andso-called zero-points (where Ps equals zero) are characteristicfor the particle size shape and composition

Changing the effective radius reff of the size distribution ofthe water cloud particles changes their single scattering proper-ties and hence both the normalised flux and polarisation spec-tra of the reflected starlight As an example increasing reff from2 μm to 6 μm (keeping veff constant) increases Ps by sim020 in theblue for a cloud with b = 10 a cloud top pressure of 0628 barand at α = 90 (Fig 11) The effective variance veff has an almostnegligible effect on the normalised flux and polarisation spectraand can thus not be derived from such observations

A69 page 12 of 14

T Karalidi et al Spectra of water clouds on exoplanets

Because Ps preserves the angular features of the singly scat-tered light the detection of the primary rainbow in Ps at plane-tary phase angles around 30 would be a clear indicator of thepresence of liquid water clouds This rainbow is present acrossa wide range of particle sizes but is only detectable in Ps notin πFn (rainbows seen lsquoin scattered fluxrsquo on the Earth originatein rain droplets which are much larger than cloud droplets)Interestingly our simulations show that for small water clouddroplets (1 μm reff 10 μm) the dispersion of the polarisedldquocloudrdquo rainbows is opposite to that found for ldquorainrdquo rainbowsat λ = 056 μm the maximum of the polarised rainbow is lo-cated at α sim 32 (corresponding to a single scattering angle Θof 148) while for λ = 10 μm the largest Ps is found at α sim 24(Θ = 156) For ldquorainrdquo rainbows Θ decreases with increasing λOur simulations show that the dispersion for the ldquocloudrdquo rain-bows decreases with increasing particle size Indeed in case ourmodel cloud consists of the larger model B particles the max-imum polarised rainbow is found at α sim 38 (Θ = 142) forλ = 056 μm and at α sim 36 (Θ = 144) for λ = 10 μm (seeFig 10)

The inverse dispersion of a polarised ldquocloudrdquo rainbow shouldbe observable for terrestrial clouds eg when measured lookingdown towards a cloud layer from an airplane As far as we knowsuch observations have not been done yet

In order to use the polarised rainbow feature for determin-ing the composition and shape of cloud particles an exoplanetshould be observed across the appropriate phase angle range(from about 30 to 40) and with an appropriate angular resolu-tion (about 10) To also derive the cloud particle size an angularresolution of a few degrees (2minus5) would be required across therainbow phase angle range We expect that if an exoplanet werefound that could have liquid water clouds and that would be di-rectly observable a dedicated observing campaign to search forthe rainbow would be conceivable

The strength of rainbows and other angular features will de-pend slightly on the cloud top altitude because the latter is re-lated to the scattering optical thickness of the gaseous atmo-sphere above the clouds Our simulations show that the lowerthe cloud (hence the larger the cloud top pressure) the largerthe reflected πFn (in the absence of gaseous absorption) The in-crease of πFn with decreasing cloud top altitude decreases withincreasing wavelength above about 06 μm πFn appears to beindependent of the cloud top altitude This is due to the decreaseof the molecular scattering cross-section hence the gaseous scat-tering optical thickness with increasing wavelength The sensi-tivity of πFn depends strongly on the planetary phase angle Inparticular around α = 90 the sensitivity is very small πFnchanges by at most sim1 for a cloud top pressure increase of theorder of sim017 bar (on Earth this corresponds to a cloud top al-titude drop of sim2 km) The sensitivity of πFn to the cloud topaltitude seems to increase slightly with decreasing phase angleHowever with decreasing phase angle the difficulty for measur-ing the normalised flux that is reflected by the planet increasesbecause of the interfering starlight

The degree of polarisation of the reflected starlight is mostsensitive to the cloud top altitude at wavelengths between 04 and05 μm and around α = 90 Around this phase angle and wave-length range Ps shows a maximum due to Rayleigh scatteredlight and increasing the cloud top pressure (lowering the cloud)leads to an increase of Ps because of the increase of light thathas been singly scattered by the gas molecules above the cloudIn this regime increasing the cloud top pressure by sim018 bartypically increases Ps by 005 (5) At shorter wavelengthsmultiple Rayleigh scattering is significant and increasing the

0 30 60 90 120 150 1800planetary phase angle (degrees)

000

005

010

015

020

025

030

Nor

mal

ised

flux

b1=10 b2=10

b1=20 b2=00

b1=00 b2=20

b1=15 b2=05

b1=05 b2=15

0 30 60 90 120 150 1800planetary phase angle (degrees)

-02

00

02

04

06

Ps

Fig 15 The normalised flux (πFn) and degree of polarisation (Ps) ofstarlight reflected by a planet covered by two cloud layers the lower(upper) one composed of model particles B (A) The top of the uppercloud is at 0628 bar and for these calculations the cloud layers them-selves do not contain any gas molecules The lines pertain to differ-ent combinations of cloud optical thicknesses (at 055 microns) uppercloud b2 = 05 lower cloud b1 = 15 (orange dashed-tripple-dottedline) upper cloud b2 = 10 lower cloud b1 = 10 (black solid line)upper cloud b2 = 15 lower cloud b1 = 05 (green long-dashed line)For comparison we also included the cases in which both cloud layerscontain particles B with total b1 = 20 (red dashed line) and particlesA with total b2 = 20 (blue dashed-dotted line)

cloud top pressure results in an increase of multiple scatteringand hence a (small) decrease of Ps At longer wavelengths thegaseous scattering optical thickness is too small to change Pssignificantly

7 Future work

The simulated signals that we showed in this paper all pertain toplanets with a homogeneous cloud layer ie there are no vari-ations in cloud particle composition andor size andor shape inthe clouds In real planetary atmospheres we expect variationsOn Earth for example particle sizes usually show some vari-ation with altitude within the cloud Such variations could in-fluence the retrieval of cloud particle properties and should beinvestigated As an example in Fig 15 we show πFn and Ps ofstarlight reflected by a model planet covered by two homoge-neous cloud layers on top of each other The lower cloud layercontains model B particles and has its top at 0710 bar and the

A69 page 13 of 14

AampA 530 A69 (2011)

upper cloud layer contains model A particles and its top is lo-cated at 0628 bar For comparison we also show πFn and Psfor the cases in which both layers contain particles B or A re-spectively The total optical thickness of the two cloud layers is20 (at λ = 055 μm) for each case but the ratio of the opticalthicknesses of the two layers varies

The curves in Fig 15 clearly show that even when the uppercloud has a relatively small optical thickness Ps of the planetis mainly determined by the properties of the upper cloud par-ticles This is due to the fact that the angular features of Ps aremostly due to singly scattered light which originates mostly inthe upper part of a cloud Combining polarisation observationsat a range of wavelengths eg from the UV to the near-infraredor even the infrared (where polarisation signatures would be dueto scattered thermal radiation) would probably help probing var-ious depths in the cloud layers since cloud optical thicknessesdepend on the wavelength

Another interesting extension to the work presented in thispaper would be to study the influence of variations in particleshape within the clouds In this paper we placed the clouds at al-titudes where the ambient temperatures ensure that the cloud par-ticles are liquid and hence spherical in shape With increasingcloud top altitude and decreasing ambient temperatures cloudswill contain more and more ice particles The single scatter-ing flux and polarisation phase functions of ice cloud particleswill differ strongly from those of liquid water particles becauseof the non-spherical possibly crystalline shape of the ice parti-cles We can thus expect that ice particles will significantly influ-ence the polarisation spectrum of a cloudy planet Using Earth-observation data Goloub et al (2000) showed that the strengthof the primary rainbow polarisation peak that is characteristic forliquid water cloud particles decreases when the column numberdensity of overlaying ice particles increases For a full retrievalof cloud parameters we will thus have to take into account thepolarisation characteristics of ice particles Knowledge of thephase (liquid or solid) of cloud particles gives valuable informa-tion on the ambient atmospheric temperatures especially whencombined with knowledge of cloud top altitudes Flux and po-larisation signatures of ice and mixed clouds will be the subjectof further study

Furthermore the model planets that we used in this paperhave horizontally homogeneous atmospheres We will adapt ournumerical radiative transfer and disk integration code to investi-gate the influence of horizontally inhomogeneities of cloud lay-ers (eg partial cloud coverage or Jupiter-like cloud belts andzones) on the flux and polarisation spectra and on the retrievedcloud parameters

ReferencesAdam J A 2002 Phys Rep 356 229Bailey J 2007 Astrobiology 7 320Bates D R 1984 Planet Space Sci 32 785Cash W amp New Worlds Study Team 2010 in ASP Conf Ser 430 ed V Coudeacute

Du Foresto D M Gelino amp I Ribas 353Cess R D Kwon T Y Harrison E F et al 1992 J Geophys Res 97 7613Chapman E G Gustafson Jr W I Easter R C et al 2009 Atm Chem

Phys 9 945Daimon M amp Masumura A 2007 Appl Opt 46 3811

Dave J V 1969 Appl Opt 8 155de Haan J F Bosma P B amp Hovenier J W 1987 AampA 183 371de Rooij W A amp van der Stap C C A H 1984 AampA 131 237Fischer J amp Grassl H 1991 J Appl Meteor 30 1245Garay M J de Szoeke S P amp Moroney C M 2008 J Geophys Res

(Atmospheres) 113 18204Goloub P Herman M Chepfer H et al 2000 J Geophys Res 105 14747Han Q Rossow W B amp Lacis A A 1994 J Clim 7 465Hansen J E amp Hovenier J W 1974 J Atm Sci 31 1137Hansen J E amp Travis L D 1974 Space Sci Rev 16 527Hovenier J W 1970 AampA 7 86Hovenier J W amp Stam D M 2007 J Quant Spect Rad Trans 107 83Hovenier J W Van der Mee C amp Domke H 2004 Transfer of polarized light

in planetary atmospheres basic concepts and practical methods AstrophysSpace Sci Lib 318

Kaltenegger L Traub W A amp Jucks K W 2007 ApJ 658 598Kasting J F Whitmire D P amp Reynolds R T 1993 Icarus 101 108Keller C U 2006 in SPIE Conf 6269Keller C U Schmid H M Venema L B et al 2010 in SPIE Conf 7735Kemp J C Henson G D Steiner C T amp Powell E R 1987 Nature 326

270Kim D amp Ramanathan V 2008 J Geophys Res (Atmospheres) 113 2203Klouda G A Lewis C W Rasmussen R A et al 1996 Env Sci Technol

30 1098Kobayashi T amp Adachi A 2009 EGU General Assembly 2009 held 19minus24

April in Vienna Austria 11 11842Liou K N amp Takano Y 2002 Geophys Res Lett 29 090000Malek E 2007 AGU Fall Meeting Abstracts A18Marley M S Gelino C Stephens D Lunine J I amp Freedman R 1999

ApJ 513 879Martin G M Johnson D W amp Spice A 1994 J Atm Sci 51 1823Matcheva K I Conrath B J Gierasch P J amp Flasar F M 2005 Icarus 179

432Mayor M amp Queloz D 1995 Nature 378 355McClatchey R A Fenn R Selby J E A Volz F amp Garing J S 1972

AFCRL-720497 (US Air Force Cambridge research Labs)Minnis P Heck P W Young D F Fairall C W amp Snider J B 1992 J

Appl Meteor 31 317Mishchenko M I Rosenbush V K Kiselev N N et al 2010[arXiv10101171]

Oakley P H H amp Cash W 2009 ApJ 700 1428Peralta J Hueso R amp Saacutenchez-Lavega A 2007 Icarus 190 469Pope R M amp Fry E S 1997 Appl Opt 36 8710Pour Biazar A McNider R T Doty K amp Cameron R 2007 AGU Fall

Meeting Abstracts D740Ramanathan V Barkstrom B R amp Harrison E F 1989 Phys Today 42 22Saar S H amp Seager S 2003 in Scientific Frontiers in Research on Extrasolar

Planets ed D Deming amp S Seager ASP Conf Ser 294 529Seager S Whitney B A amp Sasselov D D 2000 ApJ 540 504Segal Y amp Khain A 2006 J Geophys Res (Atmospheres) 111 15204Stam D M 2003 in Earths DARWINTPF and the Search for Extrasolar

Terrestrial Planets ed M Fridlund T Henning amp H Lacoste ESA SP 539615

Stam D M 2008 AampA 482 989Stam D M Hovenier J W amp Waters L B F M 2004 AampA 428 663Stam D M de Rooij W A Cornet G amp Hovenier J W 2006 AampA 452

669Stephens G L Vane D G Tanelli S et al 2008 J Geophys Res (Oceans)

113 0Tinetti G Rashby S amp Yung Y L 2006 ApJ 644 L129van de Hulst H C 1957 Light Scattering by Small Particles (New York John

Wiley amp Sons)van Deelen R Hasekamp O P van Diedenhoven B amp Landgraf J 2008 J

Geophys Res (Atmospheres) 113 12204van Diedenhoven B Hasekamp O P amp Landgraf J 2007 J Geophys Res

(Atmospheres) 112 15208Wark D Q amp Mercer D M 1965 Appl Opt 4 839Weigelt A Hermann M van Velthoven P F J et al 2009 J Geophys Res

(Atmospheres) 114 1204Yamamoto G amp Wark D Q 1961 J Geophys Res 66 3596

A69 page 14 of 14