LIGHT SCATTERING FROM EXOPLANET OCEANS AND ATMOSPHERES

Since 1992 (Wolszczan & Frail 1992), astronomers have discovered over 480 planets⁷ outside our solar system (exoplanets), primarily through the radial velocity technique (which measures red and blue shifting of the starlight caused by motion of the parent star as the planet orbits around it). Once the candidate planets are confirmed, the Kepler mission⁸ is expected to more than double this count by measuring small dips in stellar brightness due to planets passing in front of the star (the transit method). Recent papers have announced the direct imaging of multi-Jupiter-mass planets in orbits similar to or larger than that of Uranus (Kalas et al. 2008; Marois et al. 2008), but direct detection of Earth-sized planets in Earth-like orbits is not yet possible. However, space telescopes now under consideration, such as NASA's proposed Terrestrial Planet Finder–Coronagraph (TPF-C),⁹ will eventually allow astronomers to capture reflected light from such planets and separate it from that of the parent star.

In addition to simply locating Earth-sized planets, we wish to determine whether they might be habitable and whether any of them might harbor life. Detection of liquid water on a planet's surface would be a strong marker of potential habitability. Although the presence of liquid water does not necessitate the presence of life, it is considered to be one of the best indicators of habitability because (1) liquid water requires both a significant atmosphere and moderate surface temperatures and (2) the physical and chemical properties of liquid water make it an ideal medium for biochemical processes.

However, liquid water is not easily identified by remote detection. Various vibration–rotation bands of water vapor might be found in the near- and thermal-infrared of exoplanets (Des Marais et al. 2002; Tinetti et al. 2007), but exoplanet spectroscopy cannot easily distinguish between atmospheric and surface water. Long time series, multi-band photometry might potentially be used to identify land-ocean contrasts, as has been done for Earth using the EPOXI mission (Cowan et al. 2009); however, it is not clear that this technique will be feasible for the much fainter signals received from exoplanets. Here, we compute polarized and unpolarized light curves of water-covered planets, compare them with light curves expected from planets with Lambertian or dark surfaces, and determine the atmospheric and surface properties under which an ocean is detectable. We focus initially on idealized cases of planets with diffusely reflecting Lambertian surfaces, dark surfaces, or specular-reflecting water surfaces, beneath varying cloud fractions and Rayleigh atmospheres of different optical thicknesses. We then calculate light curves of a few example water Earths which include Rayleigh scattering, aerosols, absorption, and waves.

Oakley & Cash (2009) modeled orbital and diurnal light curves of Earth-like exoplanets, but concentrated on planets with Earth-like geography, and did not study polarization. Mallama (2009) generated radiometric light curves for the terrestrial planets, but did not consider other types of planets or model polarization. Williams & Gaidos (2008) demonstrated that large oceans could be detected on exoplanets using the amplitude and shape of polarized and unpolarized orbital light curves. However, the model considered only surface scattering and did not include atmospheric effects. Also, the model assumed isotropic rather than Lambertian reflectance for diffuse scattering from clouds and rough surfaces. McCullough (2006) also modeled polarization and included Rayleigh scattering, clouds, and different surfaces, but the work was unfortunately never published. Neither of the abovementioned polarization papers investigates the significant effects of absorption, aerosol scattering, scattering from within the water column, or varying degrees of ocean waviness, and neither paper compares the polarization signatures of ocean planets and dry planets. Dry planets with Rayleigh polarization signatures diluted by diffuse scattering might produce polarization signatures similar to those from ocean planets, resulting in false positives for the presence of oceans. Stam (2008) modeled Earth-like atmospheres over water surfaces, but she used a simple Fresnel model for oceans which does not include waves, sea foam, or scattering from within the water, and did not model light curves of different atmospheres over an ocean. We seek here to extend the efforts of these previous workers by simulating polarized and unpolarized orbital light curves over a larger variety of atmospheric and ocean parameters.

We have coupled an atmospheric and surface modeling program (6SV)¹⁰ with a planetary surface and orbital geometry program (Oceans). The 6SV atmospheric code was originally developed by the Laboratoire d'Optique Atmospherique and the European Centre for Medium Range Weather Forecast (Vermote et al. 2006) and was modified to compute polarization (Kotchenova & Vermote 2007; Kotchenova et al. 2006). The 6SV code calculates molecular scattering and absorption, aerosol scattering, and surface effects, including water surfaces with waves, and computes polarization effects in all of these scattering calculations. We have modified 6SV and written an IDL program which calls the modified 6SV and produces a lookup table of calculated scattering in both polarizations for thousands of combinations of stellar zenith angle, viewer zenith angle, and relative azimuth.

The Oceans code was developed originally by Williams and used to simulate scattering from planets without atmospheres (Williams & Gaidos 2008). We have modified this code to use lookup tables generated by 6SV as inputs. The Oceans code computes the three-dimensional geometry of exoplanet orbits, calculates the light scattered to the observer at both polarizations from each 2° × 2° grid area on the planet, sums all of these contributions over the illuminated surface of the planet at each orbital point, and generates light curves and graphics illustrating orbit parameters, for planets without atmospheres. As modified for this work, the Oceans code also rotates the polarization reference plane from a ground-referenced system used by 6SV to the scattering-plane reference before summation.

The 6SV code is an atmospheric and surface radiative transfer code, written in Fortran, which simulates multiple scattering using successive orders of scattering. Coupling between the surface and the atmosphere is also included. The code approximates the vertical structure of the atmosphere with 30 layers and solves the radiative transfer equation for each layer. An Earth-like pressure and temperature profile is assumed. The code performs numerical integration using decomposition in Fourier series for the azimuth angle and using Gaussian quadratures for the zenith angle. The 6SV code uses 48 Gaussian angles to simulate scattering without aerosols; for aerosol simulations, 83 angles are used, including 0°, 90°, and 180°. Polarization is incorporated by calculating the first three Stokes parameters (I, Q, and U); circular polarization (V) is ignored. The effects of wavelength are included by using 20 node wavelengths between 0.25 and 4.0 μm and interpolating between them. We include absorption in the water Earth models; this is computed by 6SV for O₃, H₂O, O₂, CO₂, CH₄, and N₂O using statistical band models with a resolution of 10 cm⁻¹. Earth-like altitude profiles are used for ozone and water vapor, and the other species are assumed to be well mixed.

We also include the effects of maritime aerosols in the water Earth models. Maritime aerosols are modeled after those found over Earth's oceans, and consist of a mixture of water droplets and crystals of sea salts. The parameters of maritime aerosols are as described in Levoni et al. (1997), and an exponential aerosol profile with a scale height of 2 km is assumed. The size distribution of the aerosols is then calculated by assuming a log-normal distribution, normalized so that the extinction coefficient at 550 nm corresponds to the visibility selected by the user; we use the standard visibilities of 5 km and 23 km, along with 80 km. The 6SV code then computes the aerosol scattering using the Lorenz–Mie solution.

The 6SV code has been used to calibrate the MODIS instruments on the Earth-observing satellites Aqua and Terra, and it was recently verified against other codes and actual data for some cases (Kotchenova & Vermote 2007; Kotchenova et al. 2006). It has some inherent limitations, however, so we modified it as follows.

• 1.  
  The code as written reports "apparent reflectance," which assumes a diffuse surface, and for ocean surfaces calculates a reflectance that increases without limit for large zenith angles. We modified the code so that it reports reflective Stokes parameters for ocean and diffuse Lambertian surfaces as values between 0 and 1, with a perfect mirror being assigned the value of unity. For ocean surfaces, we did this by replacing the wave tilt probability formula used in two subroutines¹¹ of 6SV and adding appropriate output statements to the main routine.
• 2.  
  The above change in the ocean surface model also made the model insensitive to wind direction, which should be viewed as beneficial, as this will generally be unknown and variable over time and space for an exoplanet.
• 3.  
  We modified the Rayleigh scattering algorithm to allow exact round-number values of optical depth to be used for figures.

We also developed an algorithm to use the Kasten & Young (1989) equation to partially compensate for the difference between the plane-parallel and spherical atmosphere assumptions, but the resulting maximum differences in the integrated light curves were less than 1%.

To model hypothetical planets we must of course make some assumptions; with regard to wavelength, we do this in three different ways. The three cases are as follows.

• 1.  
  Lambertian surfaces, Lambertian clouds, and dark surfaces are assumed to be gray, and thus these results are independent of wavelength given that the albedo modeled corresponds to the albedo of the surface or cloud in the wavelength band of interest.
• 2.  
  For the water Earth models, we assume an Earth-like Rayleigh scattering atmosphere, molecular absorption, and maritime aerosols and specify the baseline TPF waveband of 500–1000 nm.
• 3.  
  For other cases, we parameterize Rayleigh scattering by atmospheres of hypothetical planets by showing how the planetary light curve changes with the Rayleigh optical depth τ_R. When we parameterize by τ_R, each curve for a given τ_R can represent a range of combinations of wavelengths and atmospheric densities. In order to give a sense of scale, the captions for these figures include the equivalent wavelength for an Earth-like Rayleigh scattering atmosphere corresponding to each value of τ_R.

The caption for each figure includes the relevant wavelength range for cases (2) and (3), and for case (1) states that the curves are wavelength independent.

In keeping with our desire to present idealized examples, we consider exoplanets with a single surface type and atmosphere type, in circular, edge-on orbits. Circular orbits are assumed both for simplicity and because such systems are presumed to be more likely to have stable climates suitable for continuously liquid water and life. We also assume that planets have a horizontally homogeneous atmosphere and a single surface type. A planet with a homogeneous surface in an orbit that is face-on to the observer (inclination, i = 0) has little variation with orbital phase, and so is not of interest. The effects we seek are maximized for edge-on inclinations (i = 90), so this is the case we model. This restriction is not unduly limiting, because half of all exoplanets will have orbital inclinations in the range 60 < i < 120 because of geometrical considerations (Williams & Gaidos 2008). We will also assume an Earth-size planet at 1 AU from a Sun-like star.

The angle of polarization is defined relative to a chosen reference plane. In this case, we use the scattering plane, which is defined by the parent star, the planet, and the observer. For our edge-on geometry, the scattering plane is identical to the orbital plane. In Figure 1, it is also the plane of the paper. Although it appears at first glance that the scattering plane depends on what point on the star a particular light ray originates, and from where on the planet it is scattered, these effects are entirely negligible. One can understand this by noting that the apparent diameter of the Sun as seen from Earth is only about 05, and the size of the Earth as seen from the Sun is 1% of that, and of course both objects appear very much smaller from the distance of another star system.

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Light propagating from the planet to the observer can be divided into components in which the electric field is parallel or perpendicular to the scattering plane. The difference between the perpendicular and the parallel components, divided by the sum, is called the polarization fraction:

$$\begin{equation} {\rm PF} = \frac{{F_ \bot - F_{/\!/} }}{{F_ \bot + F_{/\!/} }}. \end{equation} \tag{ 1 }$$

Orbital longitude (OL) is defined such that OL = 0° at new phase, when the planet passes in front of the parent star as seen from Earth (transit), and full phase (OL = 180°) occurs when the planet passes behind the star and is fully illuminated. When the planet appears farthest from the parent star, OL is 90° or 270°, and the planet is said to be at quadrature. If the planet (or moon) is close enough to be resolved, as with Mercury and Venus, the planet will be in crescent phase in the "front" portion of the orbit between 270° and 90°, in the first or last quarter at 90° and 270°, and in gibbous phase in the "back" portion of the orbit, between 90° and 270°. We take advantage of orbital symmetry to simplify our curves by including only waxing phases (OL 0° to 180°) in our calculated light curves.

Three OLs are of particular interest: OL = 74°, where the peak polarization of a flat water surface occurs, OL = 90°, where the peak polarization of Rayleigh scattering occurs, and OL = 140°, where the rainbow peak for water aerosols occurs. Polarization from Rayleigh scattering peaks when the source, scattering volume, and observer form a 90° scattering angle, a fact that can be verified by observing a clear blue sky at varying angles to the Sun with polarized sunglasses. Physically, the parallel component is unable to propagate to the observer in this geometry because the electric field vector is pointing in the direction of propagation. Therefore, for Rayleigh scattering in a thin atmosphere over a dark planet surface, the polarization fraction approaches 100%. If the atmosphere is thick, multiple scattering occurs, and the polarization fraction decreases. However, Chandrasekhar & Elbert (1954) calculated a number of cases and found that polarization from a Rayleigh scattering atmosphere can exceed 90% before being limited by multiple scattering. In our case, the polarization fraction can be limited either by multiple scattering, by dilution from unpolarized or partially polarized light from the planet's surface, or both.

Simplistically, polarization from a flat water surface peaks at OL = 74°, as indicated in Figure 1. Light reflecting off a flat air/water interface at the Brewster angle, which for water is 531 to the normal, will also be polarized nearly 100%. Therefore, when the planet is at OL ∼ 180° − (2 × 53°) = 74°, the polarization fraction from a flat water ocean (neglecting sea foam and scattering within the water column) is greatest. It is the parallel component that is not well reflected because it is oriented in the direction of travel. (This description is approximate and based on a simple conceptual model, but provides a useful starting point.) Our model below includes the effects of waves, sea foam, scattering from within the ocean, multiple Rayleigh scattering, clouds, and three-dimensional geometry, and we compare various ocean planets against Lambertian land surfaces under Rayleigh atmospheres.

Returning to Figure 1, we see a third polarization peak at the rainbow angle OL = 140°, as described by Bailey (2007).¹² This peak results from light interacting with approximately spherical airborne water droplets, as in a rainbow; light striking along the side of a droplet is refracted as it enters the droplet, reflects off of the inside back surface of the droplet, and refracts again as it leaves the droplet on the other side. The resulting angle between the incoming and outgoing light rays is approximately 40°, resulting in a peak near OL = 140°.

• 1.  
  The Lambertian light curve closely matches the analytical result (Russell 1916; Sobolev 1975) and varies smoothly in an S-curve between zero flux and full flux as phase varies from new to full. At quadrature, the flux is 1/π ≅ 0.32 of that at full phase. The Lambertian planet is faint at small OLs because of the small illuminated surface fraction and the cosine weighting of the reflected flux.
• 2.  
  The ocean light curve shows the opposite behavior: it peaks at small OLs (near 30°) when the planet is in crescent phase. This is because reflection from (calm) water is largest (∼100%) near grazing incidence, and smallest (∼2%) at normal incidence. The calm ocean curve was generated assuming a light wind of 1.5 m s⁻¹, which roughens up the surface enough so that nearly every illuminated pixel reflects some light to the observer. At full phase, OL = 180°, the entire face of the planet is illuminated, but with little reflectance. As the planet moves from full phase through the gibbous phase toward quadrature, the illuminated fraction becomes smaller, but the loss is mostly compensated for by increased reflectivity. At OLs below 90°, the reflectance increases rapidly, much faster than the loss of illuminated surface, as the planet goes into crescent phase. The flux peaks near OL = 30°, where the incidence and reflection angles for specular (mirror-like) reflection are 75°, and the reflectance has increased tenfold to 20%. At OL < 30°, the loss in illuminated surface area dominates and the flux falls toward zero.
• 3.  
  At small OLs, the normalized Rayleigh flux is higher than the normalized Lambertian flux because the pathlength available for Rayleigh scattering becomes larger with increasing stellar zenith angles through the atmosphere. Both the normalized Rayleigh and Lambertian planet fluxes grow at high OLs because more of the observable planet surface is illuminated. (The Rayleigh flux curve here assumes an Earth-like atmospheric pressure profile; the shape of the curve would change somewhat with a different atmospheric pressure profile, but should remain distinct from the light curve of an ocean planet with a thin atmosphere and no clouds.)

Although it appears from Figure 2 that discriminating between three types of surface scattering is straightforward, it may be difficult to do in practice for several reasons. First, the planet can only be viewed when it is at sufficient separation from the star to be outside the inner working angle of the coronagraphic telescope. Exactly when that occurs will vary from one target to the next, but we can conservatively assume that most target planets will be observable only in the OL = 45°–135° window. Second, the contrast ratio between the planet and the star must be sufficient for the planet to be able to be observed. Dark planets may not be observable, even at quadrature. And, third, real planets are likely to represent a combination of our three end-member cases, and so the respective light curves must somehow be deconvolved. Here, we discuss the specific issue of contrast ratios.

It is first important to define what is meant by albedo. The Bond albedo, or planetary albedo, is the fraction of all electromagnetic energy from the parent star that is scattered back into space at all angles. This is the albedo commonly used in energy balance and greenhouse effect calculations. Also relevant here is the geometric albedo, which is the ratio between the light reflected by a planet and the light that would be reflected by a white Lambertian disk. For a Lambertian surface, with the planet at full phase (OL = 180°), the ratio between the geometric and Bond albedos is 2/3. Real planets have smaller or larger ratios between the geometric and Bond albedos depending on surface type. The Bond albedo of Earth is 0.29–0.31, but the geometric albedo is 0.367 (Seidelmann 1992). Mars is closer to being a Lambertian planet, with a Bond albedo of 0.25 and a geometric albedo of 0.15.¹³

The contrast ratio between an exoplanet and its parent star is an important factor in determining the observability of an exoplanet, and thus helps motivate the design of a planet-finding mission like TPF-C. The TPF Science and Technology Study Definition Team¹⁴ calculated the contrast ratio C₀ between an Earth-sized Lambertian planet at 1 AU and its parent solar-type star at quadrature (OL = 90°) as

$$\begin{eqnarray} C_0 (90) &=& \frac{2}{3}A_{{\rm Bond}} \frac{1}{\pi }\left({\frac{{r_{{\rm planet}} }}{{a_{1\,{\rm AU}} }}} \right)^2 = A_{{\rm geo}} \frac{1}{\pi }\left({\frac{{6371\;{\rm km}}}{{149.6 \times 10^6 \;{\rm km}}}} \right)^2 \nonumber\\ &=& 1.154 \times 10^{ - 10}. \end{eqnarray} \tag{ 2 }$$

The above calculation also assumes an Earth-like Bond albedo of 0.3. Table 1 lists example planets with Lambertian and water surfaces, and the corresponding contrast ratios at quadrature and full phase, relative to C₀(90). Also, we follow McCullough (2006) in comparing the ocean planet values to the analytical result for a spherical mirror with a reflectance of 0.02, which is the value for water at normal incidence.

Table 1. Contrast Ratios for Lambertian and Ocean Planets with Thin Atmospheres Relative to Results from Analytical Calculations^a

Surface	C(90°) Rel. to Lamb.	C(180°) Rel. to 2% Mirror	Comments
Lambertian, Abond= 0.3	1.000	⋅ ⋅ ⋅	From Sobolev (1975)/TPF report
Cloud Lambertian, A = 0.3	0.997	...	Cloud planet–verifies Oceans code
Surface Lambertian, ρ = 0.3	0.998	...	Lamb. surface planet–verifies 6SV/Oceans
Spherical  mirror, ${{\boldrho}}$ = 0.02	...	1.00	From Tousey (1957)
Ocean 1.5 m/s, 900-1000 nm	0.079	1.06	Rayleigh reduced using longer wavelengths
Ocean 1.5 m/s, 500-1000 nm	0.089	1.34	Significant Rayleigh scattering within water

Note. ^aRows in bold italics show results from analytical calculations for comparison.

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The first three rows in Table 1 show that both our Lambertian cloud and surface models match the analytical results closely. The bottom three rows show that the calm ocean model at full phase gives close agreement with the spherical mirror approximation. Note that, at full phase, the brightness of the ocean planet is only about 8% of the brightness of the Lambertian planet. The ocean planet is also much dimmer than Earth because Earth's albedo is dominated by clouds, with smaller contributions from Rayleigh scattering and from continental surface scattering.

Mallama (2009) derived light curves for Mercury, Venus, and Mars, compensated for distance, and compared them to light curves based on earthshine from the Moon. All four terrestrial planet light curves decrease nearly monotonically as the planet moves from full phase to new phase, with Venus appearing three to four times as bright as the others. All four curves resemble some combination of our Rayleigh and Lambertian curves, except that Mercury shows a marked increase in brightness near full phase, and Venus shows a small flux increase near OL = 10°.

In summary, the Lambertian, Rayleigh-dominated, and ocean planets have widely differing unpolarized light curves that appear readily distinguishable. However, it must be remembered that the ocean planet would be comparatively very dim and that observing at OL ≤ 45° may not be possible for many systems. Additionally, a realistic ocean planet would require a background atmosphere to hold onto its water, and it would almost certainly have clouds. It is nonetheless useful to simulate such idealized planets so that we understand the end-member cases. Also, the decreasing brightness of the water planet in the range OL = 45° to 90° should be a useful diagnostic. Because ambiguities will likely remain in the interpretation of radiometric (unpolarized) exoplanet light curves, we now address these by considering the polarization effects of various end-member planet types.

As discussed earlier, the results shown here are for a planet with a homogeneous surface in an edge-on orbit (i = 90°). For increasingly face-on orbits, the variation in both total flux and polarization fraction over the planet's orbit is expected to approach zero, as was shown explicitly in Figure 6 of Stam (2008). On the other hand, slightly inclined orbits could increase the fraction of the orbit in which the planet–star distance exceeds the minimum value that can be observed.

Our model is limited to waviness caused by sea-level wind speeds up to approximately 14 m s⁻¹, the highest wind speed investigated by Cox & Munk (1954) measured at 41' (12 m) above the sea surface. An ocean planet with no land to impede the winds might conceivably have high wind speeds and a foamy, bright Lambertian surface. Alternatively, such a planet could conceivably have sustained winds from the same direction, creating a reflectance pattern that is highly asymmetric in azimuth, resulting in a complicated light curve that may be asymmetric in OL. This source of confusion might be minimized by observing over all OLs.

Using the Deep Impact spacecraft to observe the Earth as if it were an exoplanet, the EPOXI team (Cowan et al. 2009) obtained light curves of Earth over seven 100 nm wide wavebands between 300 and 1000 nm. Principal component analysis showed that two "eigencolors" captured 98% of the diurnal color changes caused by Earth's rotation. These eigencolors are essentially spectra of filters which could be used to distinguish between land surfaces and water surfaces. The cutoffs are gradual, but the land filter passes the red and near-infrared wavelengths between about 700 and 1000 nm, while the water filter passes the green, blue, and violet wavelengths below about 550 nm. Using this method the team was able to map the longitudinal variation in land surface area, which peaks when Africa and Europe are being observed and which approaches zero in the mid-Pacific. The technique proved successful despite the confusion generated by 50% cloud cover. This method of detecting land-sea contrasts might also work for an exoplanet, provided that enough photons are available to resolve diurnal variations.

For planets with liquid droplets in the atmosphere, Bailey (2007) shows that at the "rainbow angle," total flux is higher and the polarization fraction can be as high as 0.2. For water droplets, the rainbow angle occurs at about OL = 140° and for methane droplets at OL = 131°. His calculations show that the effect is consistent for particle sizes from 10 to 100 μm at a wavelength of 400 nm and that it weakens for smaller particles. These results are scalable throughout the TPF-C wavelength band when the ratio of particle size to wavelength is held constant. The Bailey paper predicts this effect for water droplets in clouds, but our model shows that the effect is similar for water aerosols such as the maritime aerosols typically found over Earth's oceans. Bailey's model and the 6SV aerosol model both use Lorenz–Mie scattering, so our result is a confirmation both of Bailey's result and of the 6SV aerosol model.

The effect of the rainbow angle for an ocean planet is to add a third competing polarization peak near OL = 140° to those caused by the water surface (near OL = 74°) and by atmospheric Rayleigh scattering (near OL = 90°). In our cloud-free water Earth model, the rainbow peak from aerosols shifts the total polarization peak to about 99°. We model clouds as simple Lambertian reflectors. If our cloud model included Lorenz–Mie scattering and exhibited a rainbow angle enhancement, clouds would serve not only to dilute the Fresnel water polarization peak, but to further strengthen the competing rainbow polarization peak. Such a cloud model may be developed for future analysis; however, clouds, even considering only those on Earth, are varied and complex (see Section 6.5).

Since seven of the eight solar planets and many of their moons have clouds of some type, we can reasonably expect most exoplanets with atmospheres to also have clouds. Across the solar system the variety of cloud types is impressive: sulfuric acid clouds on Venus, ice clouds on Mars, ammonia clouds on Jupiter and Saturn, and methane clouds on Neptune and Uranus. Earth clouds alone include a wide variety of water clouds with varying droplet sizes and distributions, as well as ice clouds made of particles ranging from highly symmetric hexagonal crystals to irregular particles. Each type of cloud has different scattering properties, so the parameter space for scattering from cloudy planets is enormous.

As mentioned earlier, some liquid water clouds have shared the aerosol property of producing a polarized "cloudbow" near OL = 140°, which in some water-rich planets could reinforce the polarization signal from water aerosols. Ice clouds have different properties; work by Takano & Liou (1989, 1995) found that hexagonal and irregular ice clouds can exhibit a wide range of both local scattering peaks and polarization peaks depending on the particle size and shape distribution.

Rather than attempt to parameterize this range of characteristics of hypothetical exoplanet clouds, we have chosen to model clouds as Lambertian, a common practice in remote sensing of Earth, as pointed out by Acarreta et al. (2004). The Lambertian approximation is reasonable (especially for a broadband unpolarized stellar input) because, depending upon the variety of types of clouds present, the particle size range, and the amount of multiple scattering, the total cloud integrated signal can be nearly Lambertian. We discuss the cloudbow effect as one possible deviation from our model.

Zodiacal light in our solar system is caused by scattering of sunlight by dust particles concentrated in and near the plane of the ecliptic. First detected around β-Pictoris by Smith & Terrile (1984), exo-zodiacal light (exo-zodi) scattered by similar disks in exoplanet systems is now considered to be a significant concern for future exoplanet investigation, and will generally be partially polarized. Additional data from surveys are needed to determine the characteristics of exo-zodi in order to optimize both future planet finding telescopes and analysis software.

The presence of significant linear polarization in the light from a candidate planet could be used in showing that the object is indeed in orbit around the parent star, and is not a background object. Regardless of whether the polarization is caused by Rayleigh scattering, a liquid surface, or both, a reduction in the "parallel" polarization component—the component parallel to the plane defined by the parent star, the planet, and the observer—would indicate that the object is probably a planet that is being illuminated by the star it appears to orbit. To date, polarization from giant exoplanets has been difficult to detect in the combined light of the planet and star (Lucas et al. 2009), but our model predicts significant levels of polarization for many different types of terrestrial planets when the planets can be resolved from the parent star.

Likewise, polarization could assist in determining the inclination of the orbit of planets with surfaces and atmospheres that are approximately horizontally homogeneous. For edge-on orbits, the polarization fraction varies widely as the planet orbits, but the direction of the polarization vector does not; for face-on orbits, the polarization factor is constant but the direction rotates with the planet's orbital motion. A combination of these two parameters could be used to determine the most likely orbital inclination angle. This effect would complement data from radial velocity measurement and from the planet's position angle in an image, and thereby help to determine the orbital inclination of the planet being observed.

The variety of parameters which dilute, shift, or compete with the Fresnel polarization peak suggests that solving the inverse problem—determining from observations whether or not a planet has a large ocean—could be subject to numerous false positives and false negatives. Likewise, the end-member unpolarized light curves are distinctive, but an ocean surface with clouds, Rayleigh scattering, and aerosols can easily be dominated by any of the three atmospheric effects, or a combination of these and absorption. Still, some nearby ocean planets, if they exist, may have thin atmospheres and light cloud cover; other wet planets may be dominated by aerosols or water clouds, and detection of these would also be an indicator of potential habitability, whether or not they hide an ocean below.