Titan brighter at twilight than in daylight

TL;DR

Titan's brightness at twilight surpasses its daylight brightness across various wavelengths due to its extended atmosphere and haze scattering, revealing unique atmospheric properties and potential insights for exoplanet studies.

Contribution

This study presents comprehensive brightness measurements of Titan across multiple wavelengths, demonstrating its unusual twilight brightness and linking it to atmospheric scattering effects.

Findings

Titan's twilight outshines daylight at various wavelengths

Extended atmosphere causes efficient forward scattering of sunlight

Implications for atmospheric characterization of hazy exoplanets

Abstract

Investigating the overall brightness of planets (and moons) provides insight into their envelopes and energy budgets [1, 2, 3, 4]. Titan phase curves (a representation of overall brightness vs. Sun-object-observer phase angle) have been published over a limited range of phase angles and spectral passbands [5, 6]. Such information has been key to the study of the stratification, microphysics and aggregate nature of Titan's atmospheric haze [7, 8], and has complemented the spatially-resolved observations first showing that the haze scatters efficiently in the forward direction [7, 9]. Here we present Cassini Imaging Science Subsystem whole-disk brightness measurements of Titan from ultraviolet to near-infrared wavelengths. The observations reveal that Titan's twilight (loosely defined as the view when the phase angle 150deg) outshines its daylight at various wavelengths. From the match between measurements and models, we show that at even larger phase angles the back-illuminated moon will appear much brighter than when fully illuminated. This behavior is unique to Titan in our solar system, and is caused by its extended atmosphere and the efficient forward scattering of sunlight by its atmospheric haze. We infer a solar energy deposition rate (for a solar constant of 14.9 Wm-2) of (2.84+/-0.11)x10^14 W, consistent to within 1-2 standard deviations with Titan's time-varying thermal emission spanning 2007- 2013 [10, 11]. We propose that a forward scattering signature may also occur at large phase angles in the brightness of exoplanets with extended hazy atmospheres, and that this signature has valuable diagnostic potential for atmospheric characterization.

Equations

$$\mathcal{F}=\sum _j(I_j/F)S_{\mathrm{pixel}}/{\ell }^2,$$

$$\mathcal{F}=\pi R^2/{\ell }^2{A}_{\mathrm{g}}\Phi (\alpha )$$

$$P_{\mathrm{dep}}=P_{\mathrm{inc}}-P_{\mathrm{sca}}.$$

$$P_{\mathrm{inc}}={\int }_{200\mathrm{nm}}^{5000\mathrm{nm}}\pi R_{\mathrm{opt}}^2(\lambda )F_{\odot }(\lambda )d\lambda .$$

$$P_{\mathrm{sca}}={\int }_{200\mathrm{nm}}^{2500\mathrm{nm}}\pi R^2{A}_{\mathrm{g}}(\lambda )q(\lambda )F_{\odot }(\lambda )d\lambda ,$$

$$\delta P_{\mathrm{inc}}^{R_{\mathrm{opt}}}=\int 2\pi R_{\mathrm{opt}}(\lambda )\delta R_{\mathrm{opt}}(\lambda )F_{\odot }(\lambda )d\lambda$$

$$\delta P_{\mathrm{sca}}={\left[(\delta P_{\mathrm{sca}}^{A_{\mathrm{g}}}{)}^2+(\delta P_{\mathrm{sca}}^q{)}^2\right]}^{1/2}$$

$$\delta P_{\mathrm{sca}}^{A_{\mathrm{g}}}=\int \pi R_{\mathrm{p}}^2\delta A_{\mathrm{g}}(\lambda )q(\lambda )F_{\odot }(\lambda )d\lambda$$

$$\delta P_{\mathrm{sca}}^q=\int \pi R_{\mathrm{p}}^2{A}_{\mathrm{g}}(\lambda )\delta q(\lambda )F_{\odot }(\lambda )d\lambda .$$

Full text

Titan brighter at twilight than in daylight

A. García Muñoz 111Corresponding author: garciamunoz@astro.physik.tu-berlin.de; tonhingm@gmail.com

Zentrum für Astronomie und Astrophysik, Technische Universität Berlin, D-10623 Berlin, Germany

P. Lavvas

Groupe de Spectrométrie Moléculaire et Atmosphérique, UMR 7331, CNRS, Université de Reims Champagne-Ardenne, Reims 51687, France

R.A. West

Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California 91109, USA

Pre-print of a manuscript published in

NATURE ASTRONOMY 1, 0114 (2017)

DOI: 10.1038/s41550-017-0114

www.nature.com/nastronomy

Investigating the overall brightness of planets (and moons) provides insight into their envelopes and energy budgets [1, 2, 3, 4]. Titan phase curves (a representation of overall brightness vs. Sun-object-observer phase angle) have been published over a limited range of phase angles and spectral passbands [5, 6]. Such information has been key to the study of the stratification, microphysics and aggregate nature of Titan’s atmospheric haze [7, 8], and has complemented the spatially-resolved observations first showing that the haze scatters efficiently in the forward direction [7, 9]. Here we present Cassini Imaging Science Subsystem whole-disk brightness measurements of Titan from ultraviolet to near-infrared wavelengths. The observations reveal that Titan’s twilight (loosely defined as the view when the phase angle $\gtrsim$150∘) outshines its daylight at various wavelengths. From the match between measurements and models, we show that at even larger phase angles the back-illuminated moon will appear much brighter than when fully illuminated. This behavior is unique to Titan in our solar system, and is caused by its extended atmosphere and the efficient forward scattering of sunlight by its atmospheric haze. We infer a solar energy deposition rate (for a solar constant of 14.9 Wm*-2*) of (2.84$\pm$0.11)$\times$1014 W, consistent to within 1-2 standard deviations with Titan’s time-varying thermal emission spanning 2007-2013 [10, 11]. We propose that a forward scattering signature may also occur at large phase angles in the brightness of exoplanets with extended hazy atmospheres, and that this signature has valuable diagnostic potential for atmospheric characterization. **

We produced Titan phase curves (Fig. 1) from calibrated, whole-disk images taken with the Narrow Angle Camera of the Cassini Imaging Science Subsystem (ISS) [12, 13]. The timespan of the images (2004–2015), phase angle coverage (\alpha$$\leq166*∘*; see Supplementary Fig. 1 for sketch of the viewing geometry), sampling ($\sim$400 datapoints/curve on average), and variety of filters (15, effective wavelengths $\lambda_{\rm{eff}}$=300–940 nm) in this work significantly expand on previous treatments [5, 6]. The phase curves are presented in the size-normalized way $A_{\rm{g}}\Phi$($\alpha$), adopting Titan’s solid radius of 2575 km for the normalization. $A_{\rm{g}}$ is the geometric albedo and $\Phi$($\alpha$=0)$\equiv$1 by definition.

From full illumination ($\alpha$=0) to \alpha$$\sim90*∘, the curves describe Titan’s progressive dimming as less of its dayside appears visible to the observer. Our Cassini/ISS data confirm the Pioneer 11 measurements in blue and red passbands for \alpha$$\leq96∘* [5] (Fig. 1) and ground-based spectroscopic data at \alpha$$\sim5.7*∘* [14] (Fig. 2). The incomplete sampling near full illumination does not permit the confirmation of proposed strong backscattering at very small phase angles and short wavelengths [15, 16], a task that would require nearly continuous sampling for \alpha$$\leq5*∘* in the passbands with $\lambda_{\rm{eff}}$=441 and 455 nm (Fig. 1). We note though that the data for $\lambda_{\rm{eff}}$=343 nm do not show a discernible enhancement in the antisolar direction, which suggests that backscattering in the ultraviolet is less strong than recently proposed. The curve for the narrowband filter with $\lambda_{\rm{eff}}$=938 nm is particularly well sampled with nearly 2000 datapoints. In this passband, Titan’s surface contributes to the emergent radiation (Fig. 3). The dispersion in the $A_{\rm{g}}\Phi$($\alpha$) values for this passband at small phase angles (probing deeper in the atmosphere) is tentatively attributed to rotational effects as different surface patches enter and exit the field of view [17, 18].

For larger phase angles up to $\sim$140*∘, the diminishing size of Titan’s visible dayside is compensated by efficient forward scattering of its upper-atmosphere haze, and the curves exhibit plateaus or mild increases in brightness. As Titan nears back-illumination (\alpha$$\geq150∘), the twilight brightness rises steeply and at \alpha$$\sim160–166∘* it is comparable to or higher than at full illumination. This is particularly the case for wavelengths at which absorption by haze ($\lesssim$600 nm) or atmospheric methane (strong bands occur at e.g. 619, 727, and 889 nm; Fig. 2) make Titan’s dayside darker. The brightness surge is unrelated to the central flash caused by atmospheric refraction during stellar occultations [19].

We have investigated the empirical phase curves with a Monte Carlo radiative transfer model that solves the multiple-scattering problem of reflected sunlight in stratified, spherical-shell atmospheres [20] (‘Methods’). Our implementation of aerosol optical properties (extinction coefficients, single scattering albedos, scattering phase functions) is based on the latest interpretation of the in-situ measurements made by the Huygens Descent Imager/Spectral Radiometer (DISR) [16]. The DISR observations were made from 150 km altitude down to the surface, but they were sensitive to aerosols above 150 km, as confirmed by this study.

The model phase curves based on the DISR aerosol implementation (solid red curves, Fig. 1) reproduce well the Cassini/ISS data for wavelengths $\geq$440 nm unaffected by methane absorption. The DISR implementation is poorly constrained shortwards of 490 nm though [15, 16]. To reproduce the observations in the passbands with $\lambda_{\rm{eff}}$=306 and 343 nm, we slightly modified the aerosol absorption above 150 km, which led to better fits (dashed red curves) (‘Methods’). In the passbands affected by methane, an adjustable amount of methane absorption at all altitudes provided the required attenuation to reproduce the observations (dashed red curves, \lambda_{\rm{eff}}$$\geq619 nm). The reported best fits represented by the dashed red curves were the result of minimizing the relative measurement-model error.

The grey areas of Fig. 1 quantify the amount of energy that Titan scatters in all possible three-dimensional directions with a phase angle $\alpha$. This is mathematically expressed by $\Phi$($\alpha$)$\sin$($\alpha$) [1]. The integrated area (properly normalized) is the phase integral $q$, and depends on wavelength (‘Methods’). Thus, our inferred phase integrals are passband-averaged values. According to our best-fitting models, phase angles \alpha$$\geq166*∘* contribute to $q$ by 13% ($\lambda_{\rm{eff}}$=343 nm), 5% (649 nm) and 7% (928 nm). These non-negligible contributions originate from layers above 150 km (Fig. 3) and substantiate the role of the upper-atmosphere haze in Titan’s energy balance by scattering part of the incident sunlight. The phase integrals calculated here ($q$=1.9–2.9) are notably larger than earlier estimates ($q$=1.3–1.7) based on incomplete phase angle coverage [5, 21].

At small phase angles, Titan’s brightness is dictated by solar photons that scatter tens of times before exiting the atmosphere. Our model shows that the photons emerging from the $\alpha$=0 configuration scatter preferentially at altitudes of 150–300 km (455-nm photons) and 50–250 km (938-nm photons) (Fig. 3). In contrast, the brightness for large phase angles is caused by photons undergoing only a few collisions. For 455- and 938-nm photons and $\alpha$=166*∘*, the preferential range of scattering altitudes is 250–350 km and 200–350 km, respectively. As a rule, the larger phase angles are more sensitive to higher altitudes.

Atmospheric stratification is key in the interpretation of the twilight brightness, and the product $p_{\rm{a}}$(\theta$$\rightarrow0)$H_{\rm{a}}/R$ of the particles scattering phase function ($\theta$ is the scattering angle between the incident and exiting photon directions), $p_{\rm{a}}$($\theta$), times the ratio of the aerosol scale height to Titan’s radius, $H_{\rm{a}}/R$, becomes a key factor. Titan’s haze particles are fractal aggregates, each comprising thousands of small ($\leq$0.05 $\mu$m) spherical monomers [8, 22]. To the effects of radiative transfer modeling, the haze particles behave with their own aggregate-averaged properties, which may significantly differ from those of the monomers. The large effective size of the aggregates (equal-projected-area radii of 2–3 $\mu$m [15, 22, 23]) causes the prominent forward lobe in the particles scattering phase function that has been known since the times of the Voyager spacecraft [7, 9] and that translates into a large $p_{\rm{a}}$(\theta$$\rightarrow0). The DISR measurements, some of them obtained while looking less than 10*∘* away from the Sun, have shown that the forward lobe had been severely underestimated for decades [15, 16]. Titan’s extended atmosphere results in H_{\rm{a}}/R$$\sim45/3000$\sim$1.5$\times$10*-2*, much larger than e.g. $\sim$4/6150$\sim$6.5$\times$10*-4* and $\sim$27/70000$\sim$3.8$\times$10*-4* for Venus and Jupiter, respectively. This difference has an impact on their corresponding forward scattering components [20, 24].

The large value of $p_{\rm{a}}$(\theta$$\rightarrow0)$H_{\rm{a}}/R$ in Titan’s upper atmosphere is ultimately responsible for the whole-disk brightness surge at large phase angles. Based on our model’s capacity to reproduce the measured phase curves for \alpha$$\leq166*∘, we predict that Titan’s brightness for \alpha$$\rightarrow180∘* exceeds the full-illumination brightness by an order of magnitude or more, depending on the observation wavelength. The predicted A_{\rm{g}}$$\Phi(\alpha$$\rightarrow180*∘*) are quoted in Fig. 1. The diminishment in particle sizes (and in the strength of their forward scattering lobe) above 400 km [22] has no impact on the brightness surge because the atmosphere at these altitudes is optically thin at the wavelengths investigated here.

The Titan aerosols participate in vertical and horizontal structures, and in transient behaviors on diverse timescales [25, 26]. This complexity results from the strong ties of aerosol formation with super-rotating winds, seasonal heating, and both neutral and ion photochemistry. The match between the Cassini/ISS phase curves and models over all measured phase angles supports the DISR implementation [16] as a functional representation of the globally-averaged optical properties of Titan’s aerosols. This conclusion was unanticipated because Titan’s brightness at the larger phase angles is dictated by the atmosphere above 150 km (Fig. 3), and also because in principle the DISR conclusions apply principally to the Huygens descent conditions.

We have used the inferred whole-disk scattering properties to calculate the rate of solar energy deposited into the Titan atmosphere as the difference between the incident and scattered rates, $P_{\rm{dep}}$=P_{\rm{inc}}$$-$$P_{\rm{sca}} (‘Methods’). For the incident contribution, scaled to a solar constant $S_{\odot}$=14.9 Wm*-2* specific to Saturn’s semi-major axis of 9.58 AU, we obtain $P_{\rm{inc}}$=(3.87$\pm$0.07)$\times$1014 W. For the rate of energy scattered by Titan, we obtain $P_{\rm{sca}}$=(1.03$\pm$0.08)$\times$1014 W, its uncertainty being comparable to that for $P_{\rm{inc}}$. Figure 4 and Supplementary Table 1 summarize the partial contributions to both $P_{\rm{inc}}$ and $P_{\rm{sca}}$. Finally, we infer $P_{\rm{dep}}$=(2.84$\pm$0.11)$\times$1014 W and, from the definition of Bond albedo, $A_{\rm{B}}$=$P_{\rm{sca}}$/$P_{\rm{in}}$=0.27$\pm$0.02. This is strikingly similar to previous estimates of the Bond albedo [5, 21, 27] even though some of the intermediate quantities used in these works (including the phase integrals) were poorly constrained.

Titan has been estimated to emit thermally at rates of $P_{\rm{emi}}$=(2.86$\pm$0.01)$\times$1014 W in 2007 (S_{\odot}$$\sim16 Wm*-2*) and (2.79$\pm$0.01)$\times$1014 W in 2013 (S_{\odot}$$\sim14 Wm*-2*) [10, 11], suggesting that the emitted energy dropped less than the solar irradiation during that period. Scaling $P_{\rm{dep}}$ from our time-averaged treatment of $P_{\rm{inc}}$ and $P_{\rm{sca}}$ by the relevant solar constants, we obtain $P_{\rm{dep}}$=(3.05$\pm$0.11)$\times$1014 W in 2007 and (2.67$\pm$0.11)$\times$1014 W in 2013. Thus, $P_{\rm{dep}}$ and $P_{\rm{emi}}$ are consistent to within 1–2 standard deviations during 2007-2013. Our study cannot rule out an energy imbalance, but it sets strict limits based on contemporaneous measurements of scattered sunlight and Titan’s thermal emission. We note the order-of-magnitude difference in the uncertainties quoted for $P_{\rm{dep}}$ and $P_{\rm{emi}}$, and the difficulty of further constraining a putative imbalance if either the optical radius (a measure of Titan’s sunlight-intercepting cross section, ‘Methods’) or the wavelength-dependent reflectance change over time.

Titan’s brightness surge has direct implications on the characterization of exoplanets with extended and hazy atmospheres, two oft-cited properties amongst known exoplanets [28, 29, 30, 31]. For illustration, we estimate (assuming a hydrogen-helium bulk composition and equilibrium temperature of 930 K) the background scale height of the low-gravity hot sub-Neptune CoRoT-24b to be $H$/R$$\sim3.5$\times$10*-2*, and therefore larger than Titan’s $H_{\rm{a}}$/$R$. Whether the haze on these exoplanets (provided it exists) produces significant forward scattering is difficult to anticipate, as current haze formation models [32, 33] have limited capacities to predict particle sizes and the corresponding $p_{\rm{a}}$(\theta$$\rightarrow0).

Therefore, it remains plausible that some exoplanets will exhibit brightness surges at large phase angles such as that experienced by Titan, which may in addition affect the measured transit radius [34]. Indeed, for a typical hot Jupiter on an edge-on orbit around a Sun-like star, phase angles of $\sim$175*∘* are probed immediately before and after transit. In that viewing configuration, an extended atmosphere bearing Titan-like haze will appear a few times brighter than indicated by its geometric albedo. The eventual detection of this phenomenon will help constrain the scale height and particle size of their atmospheric haze. Future theoretical work must therefore assess whether forward-scattering haze forms in exoplanet atmospheres. Also, characterization efforts with existent data from the CoRoT and Kepler missions, and with data from upcoming spacecraft such as CHEOPS, JWST, PLATO and TESS, should study the larger phase angles, as they offer diagnostic possibilities complementary to those usually explored with primary and secondary eclipses.