A CLOUDINESS INDEX FOR TRANSITING EXOPLANETS BASED ON THE SODIUM AND POTASSIUM LINES: TENTATIVE EVIDENCE FOR HOTTER ATMOSPHERES BEING LESS CLOUDY AT VISIBLE WAVELENGTHS

Our cloudiness index is meaningfully defined only if there is a detection of the sodium or potassium line. We first wish to establish that when the line is undetected, it is most probably not due to it being dominated by Rayleigh scattering associated with hydrogen molecules. Rather, the absence of a sodium or potassium line is either due to a vanishingly low abundance of sodium or potassium or a very cloudy atmosphere. Because there is no way to distinguish between these scenarios without additional information, we focus on applying our cloudiness index only to objects with reported sodium/potassium line detections.

For the sodium or potassium line center to be unaffected by Rayleigh scattering associated with hydrogen molecules, the relative abundance of sodium or potassium (to H₂), by number, has to greatly exceed a threshold value,

$$\begin{eqnarray}&&{f}_{{\rm{Na/K}}}\gg \displaystyle \frac{{\sigma }_{{\rm{scat}}}}{{\sigma }_{0}}.\end{eqnarray} \tag{ 7 }$$

The cross-section for Rayleigh scattering by molecules is (Sneep & Ubachs 2005)

$$\begin{eqnarray}&&{\sigma }_{{\rm{scat}}}=\displaystyle \frac{24{\pi }^{3}}{{n}_{{\rm{ref}}}^{2}{\lambda }^{4}}{\left(\displaystyle \frac{{n}_{r}^{2}-1}{{n}_{r}^{2}+2}\right)}^{2}{K}_{\lambda },\end{eqnarray} \tag{ 8 }$$

where n_ref is a reference number density, n_r is the real part of the index of refraction, and K_λ is the King factor.

For molecular hydrogen, we set K_λ = 1 and n_ref = 2.68678 × 10¹⁹ cm⁻³ and take the refractive index to be (Cox 2000)

$$\begin{eqnarray}&&{n}_{r}=1.358\times {10}^{-4}\left[1+7.52\times {10}^{-3}\,{\left(\displaystyle \frac{\lambda }{1\mu {\rm{m}}}\right)}^{-2}\right]+1.\end{eqnarray} \tag{ 9 }$$

For the sodium/potassium cross sections at line center (σ₀), we take Equation (14) of Heng et al. (2015). The condition in Equation (7) becomes f_Na/K ≫ (T/1000 K)^1/2 10⁻¹⁶ for both the sodium and potassium doublets. Given that the elemental abundances of sodium and potassium in the solar photosphere are 2.0 × 10⁻⁶ and 1.3–10⁻⁷ (Lodders 2003), respectively, this condition is expected to possess some generality. For atmospheres in which this condition holds, the sodium/potassium line center (and the wavelengths near it in the line wings) is unaffected by Rayleigh scattering.

Other caveats include the assumption of an isothermal atmosphere in hydrostatic equilibrium and our neglect of non-local thermodynamic equilibrium (non-LTE) effects (Fortney et al. 2003).

For cloud-free atmospheres, we may directly infer the isothermal pressure scale height using Equation (5). For cloudy atmospheres, this approach is invalid. Equation (5) is robust in the sense that it is based on measuring differences in the transit radii and wavelengths, but this also makes it blind to whether Rayleigh scattering is associated with molecules or aerosols.

In a cloud-free atmosphere, the difference in transit radii, between the line center and wing of the sodium or potassium line, should take on a specific value. We denote this quantity by ΔR. Using Equations (16) and (17) of Heng et al. (2015), we obtain

$$\begin{eqnarray}&&{\rm{\Delta }}R=H\,\mathrm{ln}[{\lambda }_{0}{{\rm{\Phi }}}^{-1}{(2\pi {Hg})}^{-1/2}],\end{eqnarray} \tag{ 10 }$$

based on the reasoning that the chord optical depths associated with the transits at line center and wing have the same value. The line-center wavelength is denoted by λ₀. The line profile in the line wings is well approximated by a Lorentz profile (Heng et al. 2015),

$$\begin{eqnarray}&&{\rm{\Phi }}=\displaystyle \frac{{A}_{21}{\lambda }^{2}{\lambda }_{0}^{2}}{4\pi {c}^{2}{(\lambda -{\lambda }_{0})}^{2}},\end{eqnarray} \tag{ 11 }$$

where the Einstein A-coefficient is A₂₁ and the speed of light is c. For the sodium D₁ and D₂ lines, we have A₂₁ = 6.137 × 10⁷ s⁻¹ and 6.159 × 10⁷ s⁻¹, respectively, corresponding to λ₀ = 0.5897558 and 0.5891582 μm (Draine 2011). For the potassium D₁ and D₂ lines, we have A₂₁ = 3.824 × 10⁷ s⁻¹ and 3.869 × 10⁷ s⁻¹, respectively, corresponding to λ₀ = 0.770108 and 0.766701 μm (Draine 2011). For both the sodium and potassium doublets, we estimate that ΔR/H ≈ 20 with a gentle dependence on temperature. It implies that the isothermal pressure scale height need not be known precisely to compute ΔR/H. In other words, the uncertainty on ΔR is linearly proportional to the uncertainty on H.

A possible concern is that we have not taken pressure broadening of the line into account, which may render our calculations inaccurate. By using the expression for the chord optical depth (Fortney 2005; Heng et al. 2015) and the ideal gas law, one may obtain an expression for the pressure probed during a transit,

$$\begin{eqnarray}&&P\approx \displaystyle \frac{g}{\kappa }{\left(\displaystyle \frac{H}{2\pi R}\right)}^{1/2}.\end{eqnarray} \tag{ 12 }$$

The preceding expression is essentially the photospheric pressure with a correction term for transit geometry. Here, κ is interpreted as being the mean opacity of the atmosphere. Freedman et al. (2014) have shown that the Rosseland and Planck mean opacities are ∼0.01 cm² g⁻¹ and ∼1 cm² g⁻¹, respectively. If we adopt typical numbers (g = 10³ cm s⁻², H = 1000 km and R = R_J, where R_J = 7.1492 × 10⁹ cm is the radius of Jupiter), then we obtain P ≈ 0.05–5 mbar. We expect the sodium and potassium lines to probe pressures that are similar. Allard et al. (2012) have previously shown that pressure broadening of the sodium lines is only important for P ≳ 1 bar. Therefore, we conclude that pressure broadening is not a concern.

By measuring the actual difference in transit radii between the line center and wing (${\rm{\Delta }}{R}_{{\rm{obs}}}\equiv {R}_{0}-R$), we construct an index for the degree of cloudiness in the atmosphere,

$$\begin{eqnarray}&&C\equiv \displaystyle \frac{{\rm{\Delta }}R}{{\rm{\Delta }}{R}_{{\rm{obs}}}}.\end{eqnarray} \tag{ 13 }$$

Completely cloud-free atmospheres have C = 1. Atmospheres remain nearly cloud-free when C ∼ 1. Cloudy atmospheres have C ≫ 1.

We estimate the uncertainty associated with C, denoted by δ_C, using

$$\begin{eqnarray}&&\displaystyle \frac{{\delta }_{C}}{C}=\sqrt{{\left(\displaystyle \frac{{\delta }_{H}}{H}\right)}^{2}+{\left(\displaystyle \frac{{\delta }_{{R}_{0}}}{{R}_{0}-R}\right)}^{2}+{\left(\displaystyle \frac{{\delta }_{R}}{{R}_{0}-R}\right)}^{2}},\end{eqnarray} \tag{ 14 }$$

where the uncertainties associated with the transit radii at line center (R₀) and wing (R) are denoted by ${\delta }_{{R}_{0}}$ and δ_R, respectively. We do not include the uncertainties on the stellar radius.

As a bonus, one may directly infer the absolute abundance of sodium or potassium associated with the line center (Heng et al. 2015),

$$\begin{eqnarray}&&{n}_{{\rm{Na/K}}}\approx \sqrt{\displaystyle \frac{g}{{R}_{0}}}\displaystyle \frac{{m}_{e}c}{\pi {e}^{2}{f}_{{\rm{osc}}}{\lambda }_{0}},\end{eqnarray} \tag{ 15 }$$

where R₀ is the transit radius at line center, m_e is the mass of the electron, e is the elementary charge, and f_osc is the oscillator strength. For the sodium D₁ and D₂ lines, we have f_osc = 0.32 and 0.641, respectively (Draine 2011). For the potassium D₁ and D₂ lines, we have f_osc = 0.34 and 0.682, respectively (Draine 2011). The preceding expression assumes a cloud-free atmosphere. In a cloudy atmosphere, the expression would yield an upper limit for the abundance of sodium or potassium.

To begin our analysis requires that we have knowledge of the isothermal pressure scale height. But as we have already demonstrated, this cannot be reliably inferred in a cloudy atmosphere from measuring the spectral slope. Instead, we begin by computing the isothermal pressure scale height assuming that the temperature is the equilibrium temperature (T = T_eq),

$$\begin{eqnarray}&&{H}_{{\rm{eq}}}\equiv \displaystyle \frac{{k}_{{\rm{B}}}{T}_{{\rm{eq}}}}{{mg}},\end{eqnarray} \tag{ 16 }$$

which is then used to compute ΔR and C. We assume m = 2.4 m_H, where m_H is the mass of the hydrogen atom. In other words, we are assuming that H = H_eq, following Sing et al. (2016) and Stevenson (2016). We set the error or uncertainty associated with this assumption to be δ_H/H = 0.17 (see Section 4.2).

Table 2 shows our estimates for C, n_Na, and n_K. We subscript C with either "Na" or "K," depending on whether it was derived using the sodium or potassium lines; generally, there is consistency between them. We took the average of the values of A₂₁ and f_osc. Following Sing et al. (2016), we omit WASP-12b, WASP-19b and WASP-31b from the analysis for sodium. We also exclude HD 209458b, as there are no data points that precisely align with the sodium line centers. However, we include WASP-6b and HAT-P-12b. For the potassium line, we analyze the data of WASP-6b, WASP-31b, WASP-39b, and HAT-P-12b following Sing et al. (2016). We also use the ground-based data of Wilson et al. (2015) for HAT-P-1b.

We consistently obtain n_Na, n_K ∼ 10² cm⁻³. Only for the nearly cloud-free WASP-17b, WASP-31b, and HAT-P-1b are the values of n_Na and n_K actual estimates for the absolute abundances of sodium and potassium. For the other objects, they are upper limits as their atmospheres are cloudy. Note that we cannot estimate the mixing ratios (relative abundances) based on analyzing the sodium or potassium lines alone because we do not have an independent estimate of the total pressure being sensed. If the atmosphere is hydrogen-dominated (which is our assumption), then this would be the pressure associated with molecular hydrogen.

Generally, the temperature being sensed is not the equilibrium temperature, implying that there is an error associated with assuming H = H_eq. We can estimate what this error is by calculating the true pressure scale height (H) for the nearly cloud-free objects using Equation (5) and comparing it with H_eq. Specifically, we may perform this analysis for WASP-17b, WASP-31b, and HAT-P-1b, as they have C ∼ 1. We note that the claim for HAT-P-1b being cloud free is consistent with the work of Montalto et al. (2015).

By specializing to α = −4, Equation (5) tells us that

$$\begin{eqnarray}&&H=-\displaystyle \frac{1}{4}\displaystyle \frac{\partial R}{\partial (\mathrm{ln}\lambda )}.\end{eqnarray} \tag{ 17 }$$

We may directly estimate the value of $\partial R/\partial (\mathrm{ln}\lambda )$ by performing a linear fit to the spectral slopes, in the visible range of wavelengths, measured by Sing et al. (2016) for WASP-17b, WASP-31b, and HAT-P-1b. We use the data points from the bluest available wavelength up to the data point just blueward of the peak of the sodium line.

For WASP-17b, we have H_eq = 1662 km. Our linear fit and use of Equation (17) yields H = 1896 km, which translates into an error of δ_H/H = 12.3%. For WASP-31b, we obtain H_eq = 1181 km, H = 1390 km and δ_H/H = 15.0%. For HAT-P-1b, we obtain H_eq = 605 km, H = 485 km and δ_H/H = 24.6%. If we take the average of these three values, we obtain δ_H/H ≈ 0.17. This is the value we assume when using Equation (14).

Figure 1 plots C versus the equilibrium temperature, surface gravity, and mass (M) of the exoplanets. Curiously, there seems to be a tentative trend of increasing C with decreasing T_eq, which is a proxy for the incident stellar flux. The uncertainties associated with C are larger for cooler objects, because of the larger uncertainties on the transit radii.

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By contrast, it has previously been shown that the measured geometric albedos of hot Jupiters exhibit no trend with equilibrium temperature, surface gravity or stellar metallicity, with Kepler-7b being the oddball of having an unusually high geometric albedo (Heng & Demory 2013; Angerhausen et al. 2015). Physically, if this trend is real, it implies that clouds consisting of sub-micron-sized particles become less prevalent as the atmosphere becomes more irradiated. Any trend of C with g or M is less apparent.

For five out of seven objects, we may directly compare our values of C with the values of the near-infrared index of Stevenson (2016), which is denoted by (H₂O−J)/H_eq. Stevenson (2016) regards index values of (H₂O−J)/H_eq > 2 to correspond to cloud-free atmospheres, while index values between 1 and 2 correspond to partially cloudy to nearly cloud-free atmospheres. Cloudy atmospheres have (H₂O−J)/H_eq < 1. One would expect that atmospheres with C ∼ 1 would also have (H₂O−J)/H_eq ≥ 1. Within the uncertainties associated with both indices, HAT-P-1b fulfills the designation of being nearly cloud-free. Based on both indices, HAT-P-12b is cloudy. However, the comparison becomes less clear for WASP-17b, WASP-31b, and HD 189733b. It is conceivable that the visible and near-infrared wavelengths are probing different atmospheric layers and one layer being cloud-free does not preclude the other being cloudy. Future work with a larger sample, where both indices may be calculated, is needed to shed light on these discrepancies.