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Home All issues Volume 671 (March 2023) A&A, 671 (2023) A113 Full HTML
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A&A
Volume 671, March 2023
Article Number A113
Number of page(s) 12
Section Planets and planetary systems
DOI https://doi.org/10.1051/0004-6361/202245474
Published online 14 March 2023

© The Authors 2023

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1 Introduction

In addition to the prominent ring system of Saturn, rings have also been discovered around Jupiter (Smith et al. 1979), Uranus (Elliot et al. 1977; Smith et al. 1986), and Neptune (Hubbard et al. 1986; Smith et al. 1989). Therefore, it is reasonable to assume that extrasolar planets may show the same phenomena. The detection and characterization of circumplanetary rings of extrasolar planets potentially provides insights into the formation of planets and rings and the obliquity of the planet (Schlichting & Chang 2011), as well as into the formation of satellites and extrasolar moons (Mamajek et al. 2012). However, these so-called exorings have not been detected for any of the over 5000 confirmed extrasolar planets (as of November 2022; The Extrasolar Planets Encyclopaedia1, Schneider et al. 2011; The NASA Exoplanet Archive2, Akeson et al. 2013).

Nevertheless, several authors have investigated the detectability of rings of transiting extrasolar planets and predicted their photometric and spectroscopic signatures (e.g., Barnes & Fortney 2004; Ohta et al. 2009; Tusnski & Valio 2011; Zuluaga et al. 2015; Sucerquia et al. 2017; Akinsanmi et al. 2018). In addition, observations of light curves have been compared with ring models, such as for 51 Peg b, but in this case the detected reflected light in principle cannot be explained by a ring system (Santos et al. 2015). However, ring-like structures are consistent with observations and have been proposed for the systems J1407 (Mamajek et al. 2012) and PDS 110 (Osborn et al. 2017). Furthermore, Piro & Vissapragada (2020) explored whether super-puffs, which are planets with very low mean densities due to their large radii of ~4 R⊕ to 10 R⊕ but relatively low masses of ~2 M⊕ to 6 M⊕ (Lee & Chiang 2016; Wang & Dai 2019; Libby-Roberts et al. 2020), can be explained as ringed exoplanets. Finally, the scattered and reflected light for a ringed extrasolar planet has specific photometric signatures, making it detectable even for non-transiting systems (Arnold & Schneider 2004; Dyudina et al. 2005; Sucerquia et al. 2020; Zuluaga et al. 2022).

However, previous approaches to investigating reflected light have usually assumed simplified models, such as diffuse reflecting or transmitting rings, or planets approximated by Lambertian spheres. In addition, multiple scattering, a precise scattering phase function based on the ring particles, and polarization due to scattering were neglected. The radiation reflected by planets is usually polarized with an amplitude on the order of 10 ppm for unresolved hot Jupiters (Seager et al. 2000). In particular, in the case of HD 189733, Bott et al. (2016) reported an amplitude of about 29 ppm for the polarization variations. For the Sun, the net polarization is measured to be below 10 ppm (Kemp et al. 1987). For nearby FGK dwarfs, the polarization is on the order of 10 ppm (Bailey et al. 2010; Cotton et al. 2017). However, for active stars, the polarization can be on the order of about 20 ppm to 40 ppm (Cotton et al. 2017), while for M dwarfs the polarization reaches values of about 100 ppm (Bailey et al. 2010) and is, thus, significantly larger than the planetary contribution.

While polarimetry is used in our Solar System to characterize planetary atmospheres (Dollfus & Coffeen 1970; Dollfus 1975; Tomasko & Smith 1982; West et al. 1983a,b; Hansen & Hovenier 1974; Smith & Tomasko 1984; Tomasko & Doose 1984; Schmid et al. 2011; McLean et al. 2017) and circumplanetary rings (Kemp & Murphy 1973; Hall & Riley 1974; Dollfus 1979; Esposito et al. 1980; Johnson et al. 1980), it is also proposed for the search for Earth-like extrasolar planets (Bailey 2007; Stam 2008; Karalidi et al. 2011; Sterzik et al. 2012; Song & Qu 2017), to detect and characterize Jupiter-like extrasolar planets (Stam et al. 2004), and to characterize the cloud compositions of extrasolar planets (Lietzow & Wolf 2022). Lietzow et al. (2021) simulated the total reflected and polarized flux of a gas giant with a circumplanetary ring, assuming multiple scattering, Rayleigh scattering by molecular hydrogen in the atmosphere, and Mie scattering by small water ice particles in the ring. The authors find that the reflected polarized flux is also affected by an additional circumplanetary ring. Thus, polarization measurements are potentially a useful tool for detecting and characterizing not only extrasolar planets, but also the planetary environment, such as circumplanetary rings.

The aim of this study is to investigate the impact of circumplanetary rings consisting of spherical micrometer-sized particles on the net scattered light polarization of extrasolar gas giants. For this purpose, we studied the reflected flux and its polarization as a function of planetary phase angle at optical to near-infrared wavelengths. In Sect. 2, we describe the numerical methods that are used to calculate the scattered polarized radiation. In Sect. 3, we introduce our model setup, which includes the planetary atmosphere as well as the geometric and physical properties of the circumplanetary ring. Our results are presented in Sect. 4. We mainly focus on close-in gas giants in which Rayleigh scattering is dominating. A ringed cloudy Jupiter-like planet is also considered in Sect. 4.6. Finally, in Sect. 5 we summarize our study.

2 Methods

The scattered polarized radiation was calculated using the publicly available three-dimensional Monte Carlo radiative transfer code POLARIS3 (Reissl et al. 2016). Recently, it has been extended to calculate the radiation scattered in a planetary atmosphere and the planetary environment as well (Lietzow et al. 2021).

To describe the state and degree of polarization of the radiation, we use the Stokes formalism (e.g., Bohren & Huffman 1983). Here, the radiation is characterized by its total flux, I, the components Q and U, which describe the linear polarization, and the component V for the circular polarization. These parameters define the vector S = (I, Q, U, V)T. It is multiplied by a scattering matrix (Müller matrix), F(ϑ), at each scattering event to account for a change in polarization, S′∝F(ϑ)⋅S,$ {\bf{S'}} \propto {\bf{\rm F}}\left( \vartheta \right) \cdot {\bf{S}}, $(1)

where S′ is the Stokes vector after the scattering event and ϑ is the scattering angle. The linear polarized intensity and the degree of linear polarization are given by PI=Q2+U2, P=Q2+U2I,$ PI = \sqrt {{Q^2} + {U^2}} ,\quad P = {{\sqrt {{Q^2} + {U^2}} } \over I}, $(2)

respectively. The orientation of linear polarization is defined as tan(2χ)=UQ.$ \tan \left( {{2_\chi }} \right) = {U \over Q}. $(3)

For radiation that is polarized parallel to the scattering plane, that is, the plane containing the star, planet, and observer, it is χ = 0°. The wavelength- and phase-angle-dependent flux reaching the observer, Iλ, is given in W m−2 m−1 and, unless stated otherwise, is normalized to the planetary flux at a planetary phase angle of α = 0° at a wavelength of 0.55 μm. Here, the phase angle, α, is the angle between observer-planet and planet-star; thus, at α = 0°, the planet is at full phase. For single scattered radiation, it is α = 180° − ϑ.

3 Model setup

3.1 General parameters

The planet with radius, rp, was illuminated by a Sun-like star with a spectrum that is approximated by Planck's law for a temperature of 6000 K. We assumed that the emitted stellar radiation is unpolarized. The setup of the planet-ring model is shown in Fig. 1. The size of the ring is defined by its inner radius rin and outer radius rout. The vectors norb, nr, and nobs define the normal of the planetary orbital plane, the normal of the ring plane, and the direction toward the observer, respectively. The corresponding angles are given by cos i=norb⋅nr,cos ir=nr⋅nobs,cos iorb=norb⋅nobs.$\matrix{ {\cos \,i = {{\bf{n}}_{{\rm{orb}}}} \cdot {{\bf{n}}_{\rm{r}}},} \hfill \cr {\cos \,{i_{\rm{r}}} = {{\bf{n}}_{\rm{r}}} \cdot {{\bf{n}}_{{\rm{obs}}}},} \hfill \cr {\cos \,{i_{{\rm{orb}}}} = {{\bf{n}}_{{\rm{orb}}}} \cdot {{\bf{n}}_{{\rm{obs}}}}.} \hfill \cr } $

Throughout this study, we assumed an observer who is located in the orbital plane, that is, iorb = 90°.

3.2 Atmospheric model

We considered an atmosphere with properties dominated by gas. The gaseous component consists of molecular hydrogen to account for Rayleigh scattering, as it is the most abundant gas in a gas giant. The refractive index, the scattering cross section, and the scattering matrix of molecular hydrogen were calculated using the formulas given by Keady & Kilcrease (2000), Sneep & Ubachs (2005), and Hansen & Travis (1974), respectively. The pressure ranges from 10−5 bar at the top of the atmosphere to 100 bar, which defines the lower boundary. Below this gaseous layer, we assumed a thick cloud layer that absorbs all incident radiation. At a wavelength of 0.55 μm and assuming a Jupiter-like gravity, the scattering optical depth of the gas is approximately 12.

For this setup, the resulting geometric albedo is somewhat large, with a value of about 0.72 (see Sect. 4.1). For a semi-infinite Rayleigh scattering atmosphere, the geometric albedo is 0.7975 (Prather 1974). In contrast, hot gas giants (≳900 K) are expected to have low geometric albedos due to various opacity sources and missing reflecting clouds (Marley et al. 1999; Sudarsky et al. 2000, 2005; Cahoy et al. 2010, 2017; Cowan & Agol 2011). In particular, sodium and potassium are very important opacity sources in the visible wavelength range, whereas methane and water are important in the near-infrared (Tsuji et al. 1999; Burrows et al. 2000; Sudarsky et al. 2000). For example, very low albedos (<0.2) were found for CoRoT-1b (Alonso et al. 2009a; Snellen et al. 2009), CoRoT-2b (Alonso et al. 2009b; Snellen et al. 2010), HAT-P-7b (Welsh et al. 2010), HD 189733b (Evans et al. 2013), HD 209458b (Rowe et al. 2006, 2008; Brandeker et al. 2022), and TrES-2b (Kipping & Spiegel 2011), as well as for some confirmed Kepler giant planets (Angerhausen et al. 2015). For temperatures ≲350 K, water or ammonia condense and form reflecting clouds in the upper atmospheric layers, so the geometric albedo increases. Therefore, gas giants in our Solar System have high geometric albedos in the visible wavelength range, such as 0.52 or 0.47 for Jupiter or Saturn (e.g., Tholen et al. 2000), respectively.

Although cool Jupiters, which are gas giants with orbital periods longer than 100 d, are much more common than close-in hot Jupiters (Wittenmyer et al. 2020), we mainly focused on close-in gas giants since these planets are currently the best targets for the observation of reflected stellar radiation. Instead of modeling a real planetary atmosphere, the simple non-absorbing Rayleigh scattering atmosphere served as a reference model to set an upper limit for the reflected flux. Furthermore, the case of pure Rayleigh scattering is often used for radiative transfer calculations in planetary atmospheres and has been extensively studied (e.g., Chandrasekhar 1960; Coulson et al. 1960; Kattawar & Adams 1971; Prather 1974; van de Hulst 1980; Sromovsky 2005; Buenzli & Schmid 2009; Natraj et al. 2009). It has also been reported that some hot Jupiters have atmospheres, in which the scattering is dominated by Rayleigh scattering of small particles, such as HD 189733b (Pont et al. 2008, 2013). However, this is potentially due to a haze containing sub-micrometer-sized particles. In addition to a purely scattering atmosphere, we reduced the single scattering albedo of the gaseous particles in Sect. 4.1 to investigate the case of a low geometric albedo of the planet as well.

In Sect. 4.6, we considered a cool gas giant, similar to Saturn or Jupiter, for which additional cloud and haze particles are present in the atmosphere (see, e.g., Sato & Hansen 1979; Smith 1986; West et al. 1986; Karkoschka & Tomasko 1992, 1993; Ortiz et al. 1996; West et al. 2004). For this purpose, we assumed small spherical particles with a standard size distribution (Hansen 1971), n(s)∝s(1−3υeff)/υeffexp[ −s/(seffυeff) ],$ n\left( s \right) \propto {s^{{{\left( {1 - 3{\upsilon _{{\rm{eff}}}}} \right)} \mathord{\left/ {\vphantom {{\left( {1 - 3{\upsilon _{{\rm{eff}}}}} \right)} {{\upsilon _{{\rm{eff}}}}}}} \right. \kern-\nulldelimiterspace} {{\upsilon _{{\rm{eff}}}}}}}}\exp \left[ { - {s \mathord{\left/ {\vphantom {s {\left( {{s_{{\rm{eff}}}}{\upsilon _{{\rm{eff}}}}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {{s_{{\rm{eff}}}}{\upsilon _{{\rm{eff}}}}} \right)}}} \right], $(4)

where seff, and υeff are the mean effective radius and effective variance, respectively. We used Jupiter-like atmospheric properties similar to the model atmosphere by McLean et al. (2017) that best fits their polarimetric observations of Jupiter. The model considers an optically thin haze layer in the upper atmospheric layers above an optically thick cloud layer in the deeper atmosphere. The haze layer is located between 133 mbar and 100 mbar with an optical depth of 0.2. For the haze particles, we assumed an effective radius of 0.2 μm and an effective variance of 0.01. The refractive index is 1.5 + 10−3i and represents the optical properties of benzene and polycyclic aromatic hydrocarbons, which are proposed to be present in the polar hazes (Friedson et al. 2002). The cloud layer has a top pressure of 1 bar and an optical depth of 50. For the cloud particles, we assumed an effective radius of 0.5 μm and an effective variance of 0.05. The cloud particles have a refractive index of 1.42 + 1.5 × 10−2i and represent the optical properties of an ammonia-like composition. The scattering and absorption cross sections as well as the scattering matrix were calculated using the code miex (Wolf & Voshchinnikov 2004), which is based on the Mie scattering theory (Mie 1908). The general model parameters are summarized in Table 1.

Table 1

General model parameters.

thumbnail Fig. 1

Illustration of the planet-ring model setup. See Sect. 3 for details.

3.3 Ring model

For a star-planet separation below (L⋆16πσTsub4)1/2≈2.7(L⋆L⊙)1/2au,$ {\left( {{{{L_ \star }} \over {16\pi \sigma T_{{\rm{sub}}}^4}}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} \approx 2.7{\left( {{{{L_ \star }} \over {{L_ \odot }}}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}{\rm{au,}} $(5)

where L⋆ is the stellar luminosity, σ is the Stefan-Boltzmann constant, and Tsub ≈ 170 K is the sublimation temperature of water ice, circumplanetary rings cannot be composed of water ice (Gaudi et al. 2003). Thus, circumplanetary rings of close-in planets are potentially composed of rocky materials, such as silicates or graphites. This is consistent with the ring system of Saturn, where mainly icy material is assumed (e.g., Cook et al. 1973; Cuzzi et al. 1984, 2009). For the Jovian ring, Neugebauer et al. (1981) reported that its spectrum is consistent with reflection from rocky material. Thus, unless stated otherwise, we assumed ring particles that are composed of a silicate mixture that represents the optical properties of interstellar dust (Draine 2003).

The size distribution n(s) of the ring particles between a minimum grain size smin and a maximum grain size smax is described by a simple power law with a power law index, q, n(s)∝sq$ n\left( s \right) \propto {s^q} $(6)

that represents the outcome of an ideal collisional cascade (Dohnanyi 1969). For the main ring of Saturn, the power law index is about −3 (Cuzzi & Pollack 1978), while a power law index of about −4.6 was found in Saturn's F ring (Showalter et al. 1992). For the rings of Jupiter, a power law index of about −3.5 is consistent with observations (Tyler et al. 1981). For our default ring model, unless stated otherwise, we assumed small spherical micrometer-sized particles (0.1 μm ≤ s ≤ 100 μm, q = −3). Micrometer-sized particles have also been proposed for the Jovian ring system (Showalter et al. 1987), whereas the main ring system of Saturn mainly consists of centimeter and meter-sized particles (Cuzzi & Pollack 1978; French & Nicholson 2000).

As in the case of the clouds and haze particles (see Sect. 3.2), the scattering and absorption cross sections as well as the scattering matrix of the ring particles were calculated with the code miex (Wolf & Voshchinnikov 2004). Although dust particles in planetary rings are not spherical, optical properties of nonspherical particles can be approximated with equivalent spherical particles (Grenfell & Warren 1999; Neshyba et al. 2003). In contrast, the scattering phase function of large irregularly shaped particles differs from that of spheres (Asano & Sato 1980; Pollack & Cuzzi 1980; Mishchenko & Travis 1994). One approach is to approximate the scattering by the Henyey-Greenstein function (Henyey & Greenstein 1941), as for example in the case of debris disks observations (e.g., McCabe et al. 2002; Schneider et al. 2006, 2009; Stark et al. 2014; Milli et al. 2017; Lovell et al. 2022). Hedman & Stark (2015) reported that a linear combination of three Henyey-Greenstein functions is a good approximation for the particles inside the G and D ring of Saturn. However, the phase curves of these rings were also fitted equally well using the Mie theory (Hedman & Stark 2015). In addition, polarization characteristics such as the rainbow at large scattering angles are less pronounced or absent for nonspherical particles compared to spherical particles (Perry et al. 1978; Asano & Sato 1980; Cai & Liou 1982; Takano & Jayaweera 1985; Hess 1998; Goloub et al. 2000).

The default ring extends from 1.2rp to 2.3rp, similar to Saturn's main ring (e.g., Tholen et al. 2000). The ring has an opening angle of 2″, resulting in a maximum vertical height of approximately 134 m that is in agreement with the observed upper limit of the thickness of the rings of Saturn (Lane et al. 1982). Furthermore, planetary rings in our Solar System show various optical depths. Cuzzi et al. (1980) found a maximum optical depth in the rings of Saturn of about 1.5 in the optical wavelength range. Esposito et al. (1983) found a trimodal distribution of optical depth in the rings of Saturn with modes at ~0.08, ~0.5, and ≳2.5 in the ultraviolet during stellar occultation. In contrast, Showalter et al. (1987) found very low optical depths on the order of 10−6 in the ring system of Jupiter. Lane et al. (1986) measured mean optical depths of ≲0.6 of most of the Uranian rings, and mean optical depths in the range of 1.3 to 2.3 for the γ and ϵ ring in the ultraviolet wavelengths. The vertical optical depth of our ring model was set to a moderate value of 1 at a wavelength of 0.55 μm.

The size of a circumplanetary ring is constrained by the Hill sphere, in which the gravitational force is dominated by the planet, and the Roche limit. For radii larger than the Roche limit, particles potentially gather to form satellites. The radius of the Hill sphere and the (fluid) Roche limit are given by (e.g., Goldreich et al. 2004; Schlichting & Sari 2008) rH≈a(mp3M⋆)1/3$ {r_{\rm{H}}} \approx a{\left( {{{{m_{\rm{p}}}} \over {3{M_ \star }}}} \right)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}} $(7)

and (e.g., Murray & Dermott 1999; de Pater & Lissauer 2001) rR≈2.46rp(ρpρr)1/3≈1.53(mpρr)1/3.$ {r_{\rm{R}}} \approx 2.46{r_{\rm{p}}}{\left( {{{{\rho _{\rm{p}}}} \over {{\rho _{\rm{r}}}}}} \right)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}} \approx 1.53{\left( {{{{m_{\rm{p}}}} \over {{\rho _{\rm{r}}}}}} \right)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}}. $(8)

In Eq. (7), a is the semimajor axis, mp is the planetary and M★ is the stellar mass, and in Eq. (8), ρp and ρr are the density of the planet and its ring, respectively. Since rH ∝ a, rR∝ρr−1/3${r_{\rm{R}}} \propto \rho _{\rm{r}}^{ - {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}$, and rings of close-in extrasolar planets probably consist of rocky material (see Eq. (5)), the probability of ring formation decreases with decreasing semimajor axis. For example, a close-in Saturnlike planet with a semimajor axis of 0.1 au, the Hill radius is approximately 11rp. With a ring density of 3.5 g cm−3, assuming rocky, that is, dense silicate material, the Roche limit is about 1.4rp. However, Schlichting & Chang (2011) found that most of the currently known extrasolar planets have the potential to harbor circumplanetary rings. In addition, the lifetime of close-in extrasolar rings are strongly limited by the radiation drag or Poynting-Robertson force (Poynting 1904; Robertson 1937). The decay time of a ring particle in an optically thin ring is proportional to its size (Burns et al. 1979; Goldreich & Tremaine 1982), while for an optically thick ring, the decay timescales with the surface density of the disk (Schlichting & Chang 2011). Thus, smaller particles can survive longer in an optically thick ring, with lifetimes ranging from 106 years to 109 years (Schlichting & Chang 2011).

3.4 Single scattering properties

The ratio −F12/F11 describes the degree and orientation (see Eq. (3)) of polarization for incoming unpolarized radiation that is scattered once at a single particle. For −F12/F11 < 0, the radiation is polarized parallel to the scattering plane. Figure 2 shows the normalized matrix element F11 and the ratio −F12/F11 of the scattering matrix (see Eq. (1)) as a function of the scattering angle ϑ at a wavelength of 0.55 μm for various considered particles. These include molecular hydrogen, Jupiter-like haze and cloud particles (McLean et al. 2017), silicate particles with refractive indices from Draine (2003), water ice particles with refractive indices from Warren & Brandt (2008), and particles consisting of a mixture of 50 % silicate and 50 % water ice. For the haze and cloud particles, the size distribution described by Eq. (4) with seff = 0.2 μm and υeff = 0.01 as well as seff = 0.5 μm and υeff = 0.05 was used, respectively. For the ring particles, the size distribution described by Eq. (6) with q = −3 was used.

Since the matrix elements are a function of the particle size, they were averaged over the size distribution as follows: Fij=∫n(s)Fij(s)d s∫n(s)d s.$ {F_{ij}} = {{\int {n\left( s \right){F_{ij}}\left( s \right){\rm{d}}\,s} } \over {\int {n\left( s \right){\rm{d}}\,s} }}. $(9)

Furthermore, the element F11 was normalized, such that ∮ F11 dΩ = 1. Both the silicate and the water ice particles, as well as the mixture of both, show strong forward scattering. In addition, particles containing water ice show a maximum in the linear polarization at a scattering angle of about 140°. However, this polarization feature may be absent for nonspherical particles (see Sect. 3.3). Particles consisting of silicates have a polarization maximum at a scattering angle of about 150° with an opposite sign. For silicate particles, the radiation is polarized parallel to the scattering plane at scattering angles ≳100°. For icy material, it is polarized parallel for scattering angles ≲70° and ≳160°.

thumbnail Fig. 2

Normalized scattering matrix element F11 and ratio −F12/F11 as a function of the scattering angle, ϑ, for the considered chemical compositions of the ring and atmospheric particles at a wavelength of 0.55 μm. Forward scattering is at ϑ = 0°. For single scattered radiation, it is α = 180° − ϑ. See Sect. 3.4 for details.

4 Results

4.1 Impact of planetary rings on the net polarization: General remarks

First, we investigated the impact of the ring geometry. For this purpose, we considered: (i) a planet with a radius of 7 × 107 m and a simple cloud-free Rayleigh scattering atmosphere, (ii) a single ring with an obliquity or inclination of 30° without a planet, and (iii) a ringed planet, which is case 1 and 2 combined.

Figure 3 shows the spatially resolved total reflected and polarized flux maps of the planet with a circumplanetary ring at a wavelength of 0.55 μm. The fluxes are normalized to the maximum total flux at a phase angle of 0°. Depending on the planetary phase angle, shadows occur on the circumplanetary disk or the planet, decreasing the total reflected and polarized flux. Furthermore, the lower part of the planet is covered by the ring, decreasing the total reflected and polarized flux of the planet as well.

For a planetary phase angle of 0°, the orientation of linear polarization is pointing in radial direction toward the planet. For increasing phase angles, the orientation of the linear polarization flips. At a phase angle of 60°, the orientation of the linear polarization of the disk is parallel with respect to the scattering plane, while for the planet, it is perpendicular. The orientation of linear polarization of the circumplanetary ring flips, being oriented perpendicular to the scattering plane at larger phase angles of 120° or 150°. At these phase angles, the observed flux is dominated by radiation scattered by the dust particles in the ring. In addition, the stellar radiation that is reflected by the ring onto the night side of the planet (ringshine) is also visible. In contrast, stellar radiation that is reflected by the planetary atmosphere onto the ring (planetshine) is not visible because the directly scattered light dominates the net flux.

Figure 4 shows the total reflected flux, polarized flux, and the degree of polarization as a function of the planetary phase angle of the three models. The step width of the phase angle is 5°. The fluxes are normalized to the total reflected flux of a ringless planet at a phase angle of 0° at a wavelength of 0.55 μm. Following Chernova et al. (1993), the degree of polarization is multiplied by (− cos 2χ). Thus, the sign of the resulting polarization describes the orientation of polarization, so that if the resulting polarization value is positive or negative, the orientation is perpendicular or parallel to the scattering plane, respectively. In addition, for the degree of polarization, only the scattered radiation but not the unpolarized stellar flux is considered.

At a wavelength of 0.55 μm, the planet has a somewhat high geometric albedo of about 0.72, because of the non-absorbing atmosphere. Consequently, the polarized flux of the planet provides an upper limit and is expected to be lower for hot Jupiters (see Sect. 3.2), so the contrast of the planet and the ring increases. For this reason, a planetary atmosphere with a single scattering albedo of 0.6 for H2, resulting in a geometric albedo of about 0.17, is also shown in Fig. 4.

If no circumplanetary ring is present, the total reflected flux is largest at a phase angle of 0°, and decreases with increasing phase angle. The reflected polarized flux has its maximum at a phase angle of about 65° and decreases with increasing as well as decreasing phase angle. The degree of polarization has a maximum value of about 0.33, centered around a phase angle of about 95°. In addition, for a lower single scattering albedo, the reflected flux decreases whereas the degree of polarization increases due to the absorption and reduced multiple scattering. Along the planetary orbit, the radiation is polarized perpendicular to the scattering plane. These results are consistent with previous studies by, for example, Buenzli & Schmid (2009) or Bailey et al. (2018).

For the second model, solely a ring with an inclination of 30° was considered. Here, the total reflected and polarized flux are usually below the total and polarized planetary flux. First, at phase angles of ≳120°, the total reflected flux of the ring exceeds the total reflected planetary flux. This is due to the small dust particles that preferentially scatter the radiation in the forward direction. At phase angles of ≳130°, the polarized flux of the ring dominates the net polarization. The total reflected flux decreases at a phase angle of 90° because the surface area being illuminated is smallest, that is, at the equinox of the planet where the ring is facing the star edge-on. The degree of polarization is largest at a phase angle of about 30° with a value of about 0.2, and decreases to 0 at phase angles of ~85°. For smaller phase angles, the radiation is polarized parallel to the scattering plane, while it is polarized perpendicular to the scattering plane, for larger planetary phase angles.

Finally, we considered the combination of the previous models, that is, a planet with an additional circumplanetary ring. For a purely scattering atmosphere, the behavior of the total reflected flux is comparable to the flux of a planet without a ring at phase angles ≲110°. In contrast, for an absorbing atmosphere, the total flux shows a significant contribution of the ring at small phase angles as well. Although, the total reflected flux strongly increase due to scattering by the ring particles at large phase angles, shadows on the circumplanetary ring and on the planet are slightly decreasing the total flux. The polarized flux of the planet with a ring is somewhat lower compared to that of the planet without a ring because the polarized flux of the ring particles is small around a phase angle of 90° and it is polarized parallel to the scattering plane for phase angles ≲90°. The degree of polarization is largest at a phase angle of about 90°. In contrast to a planet without a ring, the polarization distribution of the ringed planet shows a skewness. In particular, at phase angles >90°, the degree of polarization decreases stronger compared to the degree of polarization of a planet without a ring. The impact of this effect increases if the single scattering albedo of the atmospheric particles decreases. Finally, at phase angles of ≳140°, the behavior of the degree of polarization mirrors the single scattering properties of the individual particles, that is, −F12/F11, at scattering angles ≲40° (see Fig. 2).

The increase in the scattered flux at large phase angles (≳120°) applies also for the total reflected flux of extrasolar planets covered by oceans (Williams & Gaidos 2008; Robinson et al. 2010; Zugger et al. 2010, 2011; Trees & Stam 2019). However, the total reflected flux of the Sun glint is wavelength-dependent due to the atmosphere above. In particular, the Sun glint disappears in the total reflected flux at short wavelengths (~0.4 μm) where the atmosphere becomes thicker, whereas the strength of the glint increases in the total reflected flux with increasing wavelength. In contrast, the increased flux of the ring only depends on the ring size and its particle properties.

thumbnail Fig. 3

Total reflected flux, Iλ (top), and polarized flux, PIλ (bottom), at four different planetary phase angles, a, at a wavelength of 0.55 μm. The fluxes are normalized to the maximum total flux at a phase angle of 0°. Here, it is i = 30° and ir = 60° (see Fig. 1). See Sect. 4.1 for details.

4.2 Impact of the ring size

As shown in the previous section, a circumplanetary ring with a radius of 2.3rp has a significant impact on the total reflected and polarized radiation at large phase angles. Since the total reflected radiation depends on the size of the circumplanetary ring, we compared rings with different outer radii of 2.3rp, 5rp, and 10rp. Figure 5 shows the total reflected flux, polarized flux, and the degree of polarization as a function of the planetary phase angle for the different considered outer radii of the ring at a wavelength of 0.55 μm. For comparison, the total reflected flux of a planet without a disk is shown as well.

With increasing ring radius, the total flux increases as well. In particular, at a phase angle of 0°, for a ring radius of 5rp and 10rp, the flux is about 1.7 and 4 times larger, respectively. Especially at large phase angles of ≳150°, the total reflected flux increases by a factor of ~10. In addition, the polarized radiation increases as well, and the polarized flux is dominated by radiation scattered by dust in the circumplanetary ring. Subsequently, the net radiation is polarized parallel to the scattering plane at phase angles ≲35° and ≲60° for a ring radius of 5rp and 10rp, respectively. For an outer ring radius of 10rp, the degree of polarization has a local maximum at a phase angle of 30°, which can also be seen in the single scattering properties of the silicate particles at a scattering angle of 150° (see Fig. 2). In contrast to a planet without a circumplanetary ring, the distribution of the polarization degree becomes sharper at a phase angle of 90°. In addition, there is zero polarization and a change in the orientation of polarization at phase angles of about 35° or 60°, depending on the ring radius.

4.3 Impact of the ring composition

Besides geometrical and orbit parameters, the total reflected and polarized flux also depends on the optical properties of the dust particles inside the ring. In this section we investigated the impact of the ring composition. For this purpose, we also assumed a ring composed of water ice particles, and a mixture of 50 % silicate and 50 % water ice.

Figure 6 shows the total reflected flux, polarized flux, and the degree of polarization as a function of the planetary phase angle for different ring compositions at a wavelength of 0.55 μm. For the circumplanetary ring containing icy material, the total reflected flux is somewhat larger compared to a ring containing silicate and rocky material. In particular, at phase angles ≳110°, the total reflected flux increases strongly if icy material is present in the circumplanetary ring. Furthermore, at a phase angles of about 135°, the polarization becomes zero and changes its orientation; thus, for phase angles ≳135°, the radiation is polarized parallel to the scattering plane for an icy ring. This is because of the opposite sign of the term −F12/F11 of water ice compared to silicate grains (see Fig. 2). While the polarized radiation increases for a silicate ring at phase angles ≳140°, the polarized radiation has a local maximum at a phase angle of about 155° if there are ice particles in the ring. Finally, for icy particles, a small maximum at a phase angle of 40° is visible in the polarized flux as well as in the degree of polarization. However, this polarization feature may be absent for nonspherical particles (see Sect. 3.3).

thumbnail Fig. 4

Total reflected flux, Iλ (top), polarized flux, PIλ (middle), and degree of polarization, P (bottom), as a function of the planetary phase angle, α, of a planet, a circumplanetary ring, and the combination of the two. The fluxes are normalized to the total reflected flux of a ring-less planet at a phase angle of 0° at a wavelength of 0.55 μm. The line styles correspond to different single scattering albedos, ϖ, of the gaseous particles (except for the ring model). For a single scattering albedo of 1.0 and 0.6, the planet has a geometric albedo of about 0.72 and 0.17, respectively. See Sect. 4.1 for details.

4.4 Impact of the grain size

The scattering-angle-dependent reflected radiation and polarization degree strongly depend on the particle size, more specifically on the size parameter x = 2πs/λ. To investigate the impact of the particle size, we assumed the same size distribution as defined by Eq. (6) with q = −3, but varied the minimum and maximum grain size. Figure 7 shows the normalized matrix element F11 and the ratio −F12/F11 of the scattering matrix as a function of the scattering angle ϑ at a wavelength of 0.55 μm for various grain size distributions of the ring particles. In addition to the previous broad size distribution with grain sizes ranging from 0.1 μm to 100 μm, we considered narrow size distributions ranging from 0.1 μm to 1 μm, 1 μm to 10 μm, and 10 μm to 100 μm. For increasing particles sizes, the magnitude of forward scattering increases, whereas back scattering decreases. Furthermore, the degree of polarization strongly increases at a scattering angle of 60° up to ~100% for the largest considered particles. For the smallest considered particles, the degree of polarization has a maximum of ~42 % at a scattering angle of about 150°, but the radiation is polarized parallel to the scattering plane.

Figure 8 shows the total reflected flux, polarized flux, and the degree of polarization as a function of the planetary phase angle for various grain sizes. The distribution with the smallest particles (0.1 μm to 1 μm) shows the highest reflected flux at all phase angles. In contrast, the polarized flux is the lowest compared to the other grain sizes. As a result, the degree of polarization is small as well. In particular, at phase angles of ≳140°, the degree of polarization is almost zero. For increasing grain size, the reflected flux decreases, especially at large phase angles (~150°). However, the polarized flux increases, resulting in a maximum degree of polarization at a phase angle of about 150°. Since the size distribution strongly decreases with increasing grain size (q = −3), these features of large particles are lost in the broad distribution (see Fig. 7).

thumbnail Fig. 5

Total reflected flux, Iλ (top), polarized flux, PIλ (middle), and degree of polarization, P (bottom), as a function of the planetary phase angle, α, of a circumplanetary ring with different outer radii. For comparison, the flux of a planet without a ring is also shown. The fluxes are normalized to the total reflected flux of a ringless planet at a phase angle of 0° at a wavelength of 0.55 μm. See Sect. 4.2 for details.
thumbnail Fig. 6

Same as Fig. 5, but for various ring compositions. See Sect. 4.3 for details.
thumbnail Fig. 7

Same as Fig. 2, but for various grain size distributions of the ring particles. See Sect. 4.4 for details.
thumbnail Fig. 8

Same as Fig. 5, but for various grain sizes. See Sect. 4.4 for details.

4.5 Impact of the ring inclination and orientation

Figure 9 shows the spatially resolved total reflected and polarized flux for different phase angles, with a ring inclination of 30°. In contrast to Fig. 3, the longitude or orientation of the ring is rotated by an angle of 30° (an orientation of 90° corresponds to an edge-on view). Thus, the angle, ir, between observer and ring normal amounts to ~64.3° (see Fig. 1). However, for single scattered radiation, and thus the fraction with the typically highest contribution to the net scattered flux, the scattering angle is still ϑ = 180° − α. Thus, at a phase angle of 60°, the radiation scattered in the ring has an orientation of linear polarization parallel to the scattering plane, while it is perpendicular for the presented cases for phase angles 120° and 150°. Although, the scattering angle for single scattered radiation does not change, varying the inclination and orientation of the ring alters the observable ring area, and the observed shadowing on the planet and its ring. Similar to Fig. 3, the night side of the planet is partially illuminated by ringshine.

Figure 10 shows the total reflected flux, polarized flux, and the degree of polarization as a function of the planetary phase angle for various ring inclinations. Here, the results are shown for an entire planetary orbit, that is, for phase angles from 0° to 360°. For ir = 64.3° and 69.3°, the ring inclination is i = 30°, and the orientation is rotated by 30° and 45°, respectively. For ir = 77.1° and 79.5°, the ring inclination is i = 15°, and the orientation is rotated by 30° and 45°, respectively.

The total reflected flux has a maximum at a phase angle of about 180° due to the forward scattering particles in the circumplanetary ring. In addition, depending on the orientation of the ring, there is a minimum in the total flux at phase angles between 200° to 240°, because the illuminated surface of the ring decreases strongly at these angles, that is, at the equinox of the planet. This minimum is only at phase angles greater than 180°, because at these phase angles, the ring is facing edge-on the star with a small planetary contribution at the same time. The polarized flux shows maxima at a phase angle of about 70° and, depending on the orientation, at 290° to 300°. The maximum degree of polarization at these phase angles also varies with the ring orientation. In addition, at phase angles of about 180°, the polarized flux is largest for ir ≈ 64.3°, whereas it decreases for increasing ir.

thumbnail Fig. 9

Same as Fig. 3, but for a ring orientation of 30° (instead of 0°), resulting in an angle of ir ≈ 64.3° between the observer and ring normal (see Fig. 1). See Sect. 4.5 for details.
thumbnail Fig. 10

Same as Fig. 5, but for various ring inclinations. See Sect. 4.5 for details.

4.6 Spectropolarimetric signatures

In this section, we investigated the wavelength dependence of the total reflected and polarized flux. For this purpose, we assumed that methane is present in the atmosphere to account for a wavelength-dependent absorption. The absorption cross section of methane was taken from Karkoschka (1994) with a volume mixing ratio of 1.8 × 10−3, similar to what was found in Jupiter (Sato & Hansen 1979). The refractive index and the scattering cross section of methane were calculated using the formulas given by Sneep & Ubachs (2005). Furthermore, cloud and haze particles are present in the atmosphere to represent the atmospheric structure and composition of Jupiter (see Sect. 3.2). The cloud and haze properties are listed in Table 1.

Figure 11 shows the total reflected flux, polarized flux, and the degree of polarization as a function of the planetary phase angle at various wavelengths for a cloudy planet with and without a circumplanetary ring containing silicate particles. There are methane bands at the considered wavelengths of 0.62 μm, 0.73 μm, and 0.89 μm. Since the strength of the absorption increases with increasing wavelength, the total reflected flux decreases with increasing wavelength. At a wavelength of 0.55 μm, 0.62 μm, 0.73 μm, and 0.89 μm, the planet has a geometric albedo of about 0.27, 0.14, 0.04, and 0.005. However, with increasing wavelength, the total reflected flux is dominated by the scattered radiation of the circumplanetary ring, because most radiation entering the atmosphere is absorbed. In particular, at a wavelength of 0.73 μm and 0.89 μm, the reflected flux is dominated by radiation scattered in the ring.

The degree of polarization is largest for the planet without a ring at a wavelength of 0.89 μm, because most of the radiation is absorbed, and only radiation scattered by the top gaseous layers with a high degree of polarization reaches the observer. For a planet with a circumplanetary ring, the distribution of the degree of polarization at a wavelength of 0.73 μm and 0.89 μm is about similar to the case of an increased ring radius (see Sect. 4.2). However, the maximum value at a phase angle of 90° is somewhat larger compared to an increased ring radius. For a wavelength of 0.89 μm, there is a maximum in the polarization at a phase angle of about 30° that is polarized parallel to the scattering plane, zero polarization at about 55°, and a sharp maximum at a phase angle of 90°.

Figure 12 shows spectra of the total reflected flux, polarized flux and degree of polarization for two models at three different planetary phase angles. Here, we considered a planet without a ring and a planet with an additional circumplanetary ring. The step width of the wavelength is 5 nm. The fluxes are normalized to the total reflected flux of a ringless planet at a phase angle of 0° at a wavelength of 0.55 μm. In addition to the total reflected flux, the single scattering albedo of CH4 is shown.

At a phase angle of 30°, the scattered radiation decreases with increasing wavelength and shows characteristic methane absorption features. The total reflected fluxes in both cases, a planet only and a ringed planet, are similar outside the methane band because they are dominated by the reflected planetary flux at this phase angle. At a phase angle of 135° or 150°, the total reflected flux of the ringless planet decreases strongly. In contrast, the total reflected flux of the planet with a ring is significant larger compared to a ringless planet, and it does not show methane absorption features in the spectrum.

For a planet without a ring, the degree of polarization increases in the methane bands, similar to what has been observed by Schmid et al. (2011) or McLean et al. (2017). The radiation is polarized perpendicular to the scattering plane. However, for a ringed planet, the radiation is polarized parallel to the scattering plane at wavelengths ~0.6 μm due to the dust grains in the ring at a phase angle of 30°. In addition, at phase angles of 135° or 150°, no significant methane absorption features are visible in the polarization spectrum. Thus, the degree of polarization is almost constant, with a value of about 0.05 and perpendicular to the scattering plane in the considered wavelength range.

thumbnail Fig. 11

Same as Fig. 5, but for a cloudy planet at various wave-lengths. At a wavelength of 0.55 μm, 0.62 μm, 0.73 μm, and 0.89 μm. the planet has a geometric albedo of about 0.27, 0.14, 0.04, and 0.005, respectively. See Sect. 4.6 for details.
thumbnail Fig. 12

Total reflected flux, Iλ (top), polarized flux, PIλ (middle), and degree of polarization, P (bottom), as a function of wavelength, λ, for a cloudy atmosphere. The fluxes are normalized to the total reflected flux of a ringless planet at a phase angle of 0° at a wavelength of 0.55 μm. At this wavelength, the planet has a geometric albedo of about 0.27. In addition, the single scattering albedo of CH4 is shown in the top panel. See Sect. 4.6 for details.

5 Conclusions

In this study, we have investigated the total reflected and polarized flux of a circumplanetary ring of an extrasolar planet and its impact on the net scattered light polarization of the planet-disk system. The circumplanetary ring consisted of spherical micrometer-sized particles. We compared the total reflected and polarized fluxes of a planet without a ring, a planet with a single ring, and a planet with a circumplanetary ring (see Sect. 4.1). Subsequently, we studied various parameters that have an impact on the reflected radiation. They include the ring size (Sect. 4.2), the ring composition (Sect. 4.3), the size of the ring particles (Sect. 4.4), and the ring inclination and orientation (Sect. 4.5). Finally, we also included a cloudy planetary model with methane to account for absorption in the atmosphere (see Sect. 4.6).

The dust particles in the circumplanetary ring show strong forward scattering for the considered grain size distribution at the considered wavelengths. As a result, for phase angles ≲120°, the impact of the circumplanetary ring on the total flux is insignificant because it is dominated by the total reflected and polarized flux of the planet. However, for phase angles ≳120°, the total scattered flux of the ring exceeds the total reflected planetary flux. For an increasing outer radius of the ring, the total reflected flux of the ring shows an increasingly significant contribution at small phase angles. In particular, the orientation of polarization is parallel to the scattering plane at small phase angles fora ring containing silicate particles. In contrast, for a ring containing water ice particles, the orientation of polarization is parallel to the scattering plane at large phase angles. For different ring inclinations and orientations, the total reflected and polarized flux show minima and maxima at specific phase angles. In addition, the maximum degree of polarization varies depending on the ring orientation. For smaller ring particles (0.1 μm to 1 μm), the reflected flux is usually larger at all phase angles compared to larger particles. In contrast, the degree of polarization is largest for large ring particles (10 μm to 100 μm).

For a Jupiter-like atmosphere with methane and aerosols, a planet without a ring, or at small phase angles (~30°) in general, methane absorption features are found in the spectrum of the total reflected and polarized flux. In contrast, for a ringed planet at large phase angles (~135° or ~150°), the reflected and polarized flux does not show any absorption features and the degree of polarization is constant across the considered wavelength range. Furthermore, at small phase angles, the orientation of polarization of a ringed planet changes for increasing wavelength due to the different orientation of polarization of the particles in the ring at a phase angle of 30°.

As shown in Fig. 4, the reflected flux of the considered circumplanetary ring is usually smaller compared to the reflected flux of the planet. In particular, for a planet with a geometrical albedo of about 0.17, the reflected flux as a fraction of the total stellar flux is about 94 ppm at a phase angle of 0° and a semimajor axis of 0.02 au. For the circumplanetary ring, the reflected flux as a fraction of the total stellar flux is about 50 ppm at the same distance. In addition, the polarized flux decreases to about 16 ppm at a phase angle of 70° with an additional circumplanetary ring. At a phase angle of 150°, where the polarized flux of the ring exceeds the polarized flux of the planet, the polarized flux of the ring as a fraction of the total stellar flux is about 8 ppm. Thus, the polarized flux has to be measured at a parts-per-million level, which is currently at the limit of modern polarimeters, such as the High-Precision Polarimetric Instrument-2 (HIPPI-2; Bailey et al. 2020), and below the parts-per-million level needed for an increasing semimajor axis to detect the scattered polarized flux of a circumplanetary ring. Consequently, only close-in planets, provided that rings exist at these distances, are suitable targets for current polarimetric instruments.

In this study, the observer was located in the plane of the planetary orbit, but reflected light polarimetry is not limited to transiting planets. In particular, inclined or even face-on orbits are also expected to show characteristic variations in the polarized radiation due to the ring geometry. However, this is beyond the scope of the current study.

In summary, we find unique features in the phase-angle- and wavelength-dependent total reflected and polarized flux due to scattering of the stellar radiation by dust in circumplanetary rings of extrasolar planets. Thus, exoplanet polarimetry not only provides the means to study the planetary atmosphere and surface, but also to identify the existence and constrain the properties of exoplanetary rings.

Acknowledgements

The authors thank the anonymous referee for useful comments and suggestions. This research made use of NASA's Astrophysics Data System (https://ui.adsabs.harvard.edu), Astropy (https://www.astropy.org), a community-developed core Python package for Astronomy (Astropy Collaboration et al. 2013, 2018), Matplotlib (https://matplotlib.org) (Hunter 2007), and Numpy (https://numpy.org) (Harris et al. 2020).

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All Tables

Table 1

General model parameters.

In the text

All Figures

thumbnail Fig. 1

Illustration of the planet-ring model setup. See Sect. 3 for details.
In the text
thumbnail Fig. 2

Normalized scattering matrix element F11 and ratio −F12/F11 as a function of the scattering angle, ϑ, for the considered chemical compositions of the ring and atmospheric particles at a wavelength of 0.55 μm. Forward scattering is at ϑ = 0°. For single scattered radiation, it is α = 180° − ϑ. See Sect. 3.4 for details.
In the text
thumbnail Fig. 3

Total reflected flux, Iλ (top), and polarized flux, PIλ (bottom), at four different planetary phase angles, a, at a wavelength of 0.55 μm. The fluxes are normalized to the maximum total flux at a phase angle of 0°. Here, it is i = 30° and ir = 60° (see Fig. 1). See Sect. 4.1 for details.
In the text
thumbnail Fig. 4

Total reflected flux, Iλ (top), polarized flux, PIλ (middle), and degree of polarization, P (bottom), as a function of the planetary phase angle, α, of a planet, a circumplanetary ring, and the combination of the two. The fluxes are normalized to the total reflected flux of a ring-less planet at a phase angle of 0° at a wavelength of 0.55 μm. The line styles correspond to different single scattering albedos, ϖ, of the gaseous particles (except for the ring model). For a single scattering albedo of 1.0 and 0.6, the planet has a geometric albedo of about 0.72 and 0.17, respectively. See Sect. 4.1 for details.
In the text
thumbnail Fig. 5

Total reflected flux, Iλ (top), polarized flux, PIλ (middle), and degree of polarization, P (bottom), as a function of the planetary phase angle, α, of a circumplanetary ring with different outer radii. For comparison, the flux of a planet without a ring is also shown. The fluxes are normalized to the total reflected flux of a ringless planet at a phase angle of 0° at a wavelength of 0.55 μm. See Sect. 4.2 for details.
In the text
thumbnail Fig. 6

Same as Fig. 5, but for various ring compositions. See Sect. 4.3 for details.
In the text
thumbnail Fig. 7

Same as Fig. 2, but for various grain size distributions of the ring particles. See Sect. 4.4 for details.
In the text
thumbnail Fig. 8

Same as Fig. 5, but for various grain sizes. See Sect. 4.4 for details.
In the text
thumbnail Fig. 9

Same as Fig. 3, but for a ring orientation of 30° (instead of 0°), resulting in an angle of ir ≈ 64.3° between the observer and ring normal (see Fig. 1). See Sect. 4.5 for details.
In the text
thumbnail Fig. 10

Same as Fig. 5, but for various ring inclinations. See Sect. 4.5 for details.
In the text
thumbnail Fig. 11

Same as Fig. 5, but for a cloudy planet at various wave-lengths. At a wavelength of 0.55 μm, 0.62 μm, 0.73 μm, and 0.89 μm. the planet has a geometric albedo of about 0.27, 0.14, 0.04, and 0.005, respectively. See Sect. 4.6 for details.
In the text
thumbnail Fig. 12

Total reflected flux, Iλ (top), polarized flux, PIλ (middle), and degree of polarization, P (bottom), as a function of wavelength, λ, for a cloudy atmosphere. The fluxes are normalized to the total reflected flux of a ringless planet at a phase angle of 0° at a wavelength of 0.55 μm. At this wavelength, the planet has a geometric albedo of about 0.27. In addition, the single scattering albedo of CH4 is shown in the top panel. See Sect. 4.6 for details.
In the text

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