ELECTRON DENSITIES AND ALKALI ATOMS IN EXOPLANET ATMOSPHERES

Due to their different ionization potentials, ionization of each atom is initiated at a different pressure level in the atmosphere. In the upper atmosphere we consider ionization of H, He, C, N, and O, the ion-chemistry of which we have already discussed in Koskinen et al. (2013a). These elements have large ionization potentials and they do not significantly affect the ionization of the alkali atoms deeper in the atmosphere. In order to demonstrate the dominance of the alkali atoms in the ionization of the lower atmosphere, here we only discuss the effects of their photoionization and electron recombination. Additional processes that could affect the profiles of the alkali atoms are discussed in Section 4.

Potassium has the most extended photoionization cross section reaching to ∼2900 Å. Photons at these wavelengths are not substantially absorbed by the other atmospheric gaseous compounds resulting in ionization of K to pressures greater than 1 bar (Figure 4). For Na, photons available for ionization with λ ⩽ 2412 Å are also absorbed by other gaseous species, such as H₂O and H₂S (as identified also by Fortney et al. 2003) and photoionization is efficient down to ∼10⁻² bar. At low pressures Na⁺ and K⁺ recombine radiatively with electrons. This process is slow, resulting in large electron densities. At deeper pressures, greater than 10⁻³ bar, Na⁺ and K⁺ are dominantly lost through three-body recombination, which is much more rapid. Hence, most of the ionized alkali atoms return to their neutral state and the electron densities decrease (Figure 5). At even higher pressures, larger than 0.1 bar, thermal ionization becomes efficient and the electron densities start to increase again. The three-body recombination also becomes more efficient with increased pressure though, forcing the majority of the alkali atoms to remain in their neutral state.

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The variable ionization regions of these atomic elements have their signature on the resulting electron densities (Figure 6). For pressures lower than 10⁻⁸ bar, electrons are produced from the ionization of atomic hydrogen, with secondary contributions by He and C photoionization. At higher pressures the majority of the electrons is produced by the photo-ionization of Na and K, with the former dominating over the latter up to pressures of ∼10⁻² bar. Below the 1 bar level, photo-ionization rates are small and most of the electrons are produced by thermal ionization of K. Although thermal ionization produces large electron densities below the 1 bar level, it is important to note that photo-ionization dominates at lower pressures. The evaluation of the thermal electron population based on the Saha equation considering all atomic elements used in the kinetic model shows that the thermal electron density at 10⁻² bar should be ∼4 × 10⁶ cm⁻³ (green line in Figure 6). The photon-induced electron density at the same level is almost 100 times larger. Such high electron densities so deep in the atmosphere are not common among the giant planets of our solar system, and for HD 209458b result from the combination of strong stellar fluxes for this close-in exoplanet, and high atmospheric temperatures that allow Na and K to remain in atomic form. Thus, we can conclude that photo-ionization in the atmospheres of exoplanets similar to HD 209458b will produce large electron densities in the middle atmosphere between 10 μbar and 1 bar.

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We now turn our attention to the comparison of our model results with the available observations of alkali metals in HD 209458b. Our opacities for the transit depth calculation include H₂ Rayleigh scattering, absorption by Na and K, collision induced absorption by H₂–H₂ and H₂–He collisions, and molecular absorption by H₂O (Figure 7). We use the atomic line properties from the NIST database and we calculate the line profiles for the different pressure-temperature conditions of the model atmosphere assuming Voigt line profiles. For the alkali atoms we use the line parameters and broadening coefficients from Iro et al. (2005). We are aware of the divergence of the alkali line wings from the Voigt description at the far wings (Allard et al. 2003), but the conclusions we draw below are not affected by this issue. For the temperature profile we assume the average terminator profile (see Figure 2), while for the chemical abundances we use the densities derived by the day side conditions since we expect that the strong zonal transport suggested by the GCMs will efficiently redistribute the abundances from the disk to the terminator (Showman et al. 2009). We discuss further below possible implications for the differences between the dawn and dusk terminators.

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The transit depth at the sodium line appears to be consistent with the sodium wings observed by Sing et al. (2008a) in the medium-resolution Space Telescope Imaging Spectrograph (STIS) observations (Figure 7). The analysis of the same observations by Knutson et al. (2007) and Deming et al. (2013) at larger spectral bins suggests the presence of a less broad Na wing, but our model is consistent with these observations as well, given the larger uncertainty in the mid-resolution observations. The presence though or not of a broad wing at the sodium line has important implications for the interpretation of the observations and specifically the question of whether or not the atmosphere is clear. A broad atomic line wing implies a clear atmosphere, while a shallow atomic core would indicate the presence of clouds (or haze) at the probed altitudes.

A closer look at the Na line double core (Figure 8) shows that our model generates a differential transit depth that is larger than the original observations by Charbonneau et al. (2002) and the subsequent observations by Sing et al. (2008a). The very low differential transit depth at the sodium line core was originally interpreted as an indication of a cloud deck at the probed pressure levels around the sodium core that reduced the contrast between the line center and the wings (Charbonneau et al. 2002). We find that such an interpretation is not consistent with the subsequent observations of Rayleigh scattering at shorter wavelengths, and in general with the low-resolution spectra. This argument becomes obvious if we assume a spectrally flat cloud deck around the Na core that would be consistent with the observed differential transit depth values. Our calculations suggest that this would occur for a cloud transit depth close to 0.00153 (see Figure 8), which results in a differential transit depth that is consistent with both the magnitude and the spectral dependence of the observations. However, such a cloud would be inconsistent with the observed transit depth at shorter wavelengths since it has a larger absorption than the strongest absorption observed at 0.3 μm (see dashed line in Figure 7). In other words, the cloud properties implied by the high-resolution differential transit depth are not consistent with the Na wing detection and the observed slope at shorter wavelengths.³ We further discuss the implications of this result below.

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At longer wavelengths, absorption by H₂O alone provides a consistent picture with the HST observations by Deming et al. (2013) between 1.1 and 1.6 μm, and the Spitzer observations at longer wavelengths by Beaulieu et al. (2010) and Crossfield et al. (2012), although the model seems to lack some opacity at 5.8 and 8 μm.

The most striking feature of the comparison of the model with the observations is the large transit depth at the potassium line in the model relative to practically no detection in the observations (Knutson et al. 2007; Sing et al. 2008a; Jensen et al. 2011). It is commonly anticipated that the lack of potassium could be explained by the rapid photo-ionization of this element due to its very low ionization potential. Our transit depth calculations imply that for the assumed solar elemental abundances, the K profile would have to be depleted to very deep levels in the atmosphere (close to 10 mbar) in order to be consistent with the observations. Our photochemical calculations presented above demonstrate that this is not likely. Before considering other options to explain this discrepancy between observations and modeling, we decided to further investigate the alkali chemistry in order to verify that other photochemical processes cannot help reduce the abundances of Na and K. These studies are described in Section 4.

We have also considered the possibility of other atoms in the atmospheres of hot extrasolar giant planets. For example, Mg, Si, Ca, Al, and Fe combine a low-ionization potential with a high elemental abundance relative to other heavy atoms (Table 1) and could contribute to the electron densities. Most of these heavy atoms should condense in the low atmosphere for the temperature conditions assumed (see Figure 2), in which case they would not significantly contribute to the ionosphere. The observations of HD 209458b seem to favor an atmosphere in which Si, at least, survives in the upper atmosphere. Specifically, the signature of Si iii in the transit depth observations (Linsky et al. 2010) implies a large abundance of atomic silicon in the thermosphere, which should not be possible if condensates such as forsterite or enstatite form in the lower atmosphere (Koskinen et al. 2013a, 2013b). Based on these observations and interpretation, we performed calculations including the above atoms in order to evaluate their potential impact on the electron densities and the transit depth.

We used ionization cross sections for all new atoms from Verner et al. (1996). We could not find recombination rates for all elements considered, thus certain estimations had to be made based on the electronic structure of each element. For Mg and Si we used recombination rates from Aldrovandi & Pequignot (1973) and for Ca we assumed the same rate as for Mg. For Al we estimated the recombination rate with that of Si iii from Aldrovandi & Pequignot (1973), while for Fe we used the recombination rate from Nahar et al. (1997). Three-body rates were estimated based on the rate for sodium. Based on the NASA thermochemistry model (Burcat & Ruscic 2005), for the atmospheric conditions of HD 209458b, the most abundant form of the heavy elements is the atomic, with the exception of Si and Al. Aluminum at the lower boundary is dominantly in the form of AlOH and AlH, which we follow kinetically in the model through the reactions

$$\begin{eqnarray*} &&{\rm Al + H_2O \leftrightarrow AlOH + H} \\ &&{\rm Al + H + M \leftrightarrow AlH + M} \end{eqnarray*}$$

with rates for the forward reactions taken from McClean et al. (1993), Sharipov et al. (2011), and Swihart et al. (2003), respectively. Silicon, like aluminum, is present in molecular compounds at the lower boundary with SiO the most abundant species. For this case, though, there is a large number of other silicon species that can be formed. In our calculations we have followed kinetically the transition between SiO and SiH₄ through different intermediate species, as well as the interaction of silicon with other elements to form SiS, SiN, and SiC. The SiO-SiH₄ equilibrium is described kinetically with a scheme similar to the CO-CH₄ equilibrium (Moses et al. 2011), but modified by the thermochemical parameters of the silicon-based species. For the most part of the atmosphere SiO and SiH₄ remain close to their equilibrium abundances. The details of these calculations are beyond the scope of the current study and will be presented in a future study.

According to our calculations, the two aluminum molecular compounds have similar abundances at the lower boundary but diverge at lower pressures with AlOH becoming the most abundant species close to 10² bar (Figure 9). At pressures below 10⁻² bar, reaction with atomic hydrogen gradually returns aluminum to its atomic form before becoming mainly ionized at pressures lower than 10⁻³ bar. Calcium has a similar ionization pressure region since its ionization limit is close to that of aluminum. The remaining elements, Mg, Fe, and Si, have all significantly higher ionization potentials compared to the previous elements, and their corresponding ions become important only above 10⁻⁶ bar. Mg, Fe, and Si photo-ionization contributes to the local increase of the electron density close to 10⁻⁶ bar, while the signature of Ca and Al photo-ionization is clear close to 10⁻⁴ bar. The increased electron abundance also affects the Na and K profiles close to 10⁻⁴ due to the enhanced recombination rates.

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Inclusion of these new atomic profiles to the transit depth calculations produces numerous narrow absorption lines at short wavelengths (see also Barman 2007). Although most of these lines are too narrow to be detectable with the current instrumentation, some of them, originating mainly from Ca (∼423 nm) and Fe (all other broad absorption features between 0.3 and 0.4 μm), are wide enough to be observed. The medium-resolution HST/STIS observations (Sing et al. 2008a) demonstrate some structure in this part of the spectrum, but they suffer from large uncertainties that do not allow for a clear assignment of absorption signatures. On the other hand, the low-resolution analysis of the same observations (Knutson et al. 2007; Deming et al. 2013) do not indicate the presence of such absorption features, which would affect the observed transit depth according to our calculations (Figure 10). The residuals between the model and the broadband observations are about 6σ and 20σ at the major Fe and Ca absorption signatures, respectively, while close to the calcium line the residuals relative to the medium-resolution observations are more than 3σ. These results indicate that the abundances of Ca and Fe, at least, are decreased at the probed pressure levels of the HD 209458b relative to the solar elemental abundances, a behavior anticipated due to condensation. Therefore, we decided to consider atomic opacities of Na and K only for the remainder of this study.

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For the temperatures considered here, alkali atoms can form molecular structures. We follow the populations of NaH, NaOH, KOH, and KH, which are the thermochemically most abundant molecular structures at the thermal conditions assumed. We follow these populations kinetically so that we can identify any potential increases of their densities above the (otherwise low) thermochemical levels, due to quenching by photochemistry and dynamics. The chemical processes we consider for the description of these species are similar for Na and K (although with different rates for each element). For example, reaction of Na with water leads to NaOH:

$$\begin{eqnarray*} &&{\rm Na + H_2O \leftrightarrow NaOH + H}, \end{eqnarray*}$$

while reaction with hydrogen forms NaH:

$$\begin{eqnarray*} &&{\rm Na + H_2 \leftrightarrow NaH + H}. \end{eqnarray*}$$

These molecules can thermally decompose back to sodium and the corresponding radicals:

$$\begin{eqnarray*} &&{\rm NaOH + M \leftrightarrow Na + OH + M} \\ &&{\rm NaH + M \leftrightarrow Na + H + M}, \end{eqnarray*}$$

with both forward and reverse pathways important depending on the assumed temperature conditions. For the cases studied here, the formation of these molecules is favored only deep in the atmosphere and their resulting abundance is always smaller than the atomic (Figure 11).

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We also track the populations of NaCl and KCl, which require the inclusion of Cl and HCl in the study. Deep in the atmosphere chlorine is dominantly in the form of HCl in a balance with atomic Cl that is controlled through the reaction

$$\begin{eqnarray*} &&{\rm HCl + H \leftrightarrow Cl + H_2}, \end{eqnarray*}$$

while at lower pressures reaction with alkali atoms leads to

$$\begin{eqnarray*} &&{\rm Na,K + HCl \leftrightarrow NaCl, KCl + H} \end{eqnarray*}$$

which affects the HCl population to a small degree. We also consider the reaction

$$\begin{eqnarray*} &&{\rm HCl + OH \leftrightarrow H_2O + Cl}, \end{eqnarray*}$$

but found that it has a third role in the overall chlorine species profiles, after alkali atoms. Photolysis of HCl has a minor impact on its abundance and we included it with cross sections from Brion et al. (2005). The results show that close to the temperature minimum the abundances of both NaCl and KCl increase. KCl becomes large enough close to 10 mbar that induces a small decrease in the atomic potassium abundance (Figure 11). This decrease, though, is too small to reduce the transit depth of the potassium line to the level of the observations. Hence, molecular chemistry cannot explain the low alkali abundances. Another conclusion we can derive from these results is that, notwithstanding the small differences in the profiles of the different species involved, Na and K have a very similar chemical behavior, as anticipated.

Due to the large photon fluxes at visible wavelengths and the large oscillator strengths for the alkali resonant transitions, the photo-excitation rates of these states are orders of magnitude larger than the corresponding photo-ionization rates (Figure 4). Therefore, a large population of excited alkali atoms can potentially exist in the exoplanet atmospheres. Processes forbidden or weak for ground-state atoms could be efficient for the excited states and in this way drive the system further away from the thermochemical equilibrium. Moreover, if the resulting atomic state population of Na and K is much different from the LTE abundances, further modifications in the transit depth would be expected (Barman et al. 2002). Thus, a detailed description of these processes is necessary in order to evaluate their role in the abundances of alkali atoms.

Two electronic states of Na and K can be efficiently excited from their grounds states directly. These are 3(²P) and 4(²P) for Na and 4(²P) and 5(²P) for K (Figure 12). For simplicity we reduce the nomenclature of these states to Na[3P], Na[4P], K[4P], and K[5P]. Each of these states is practically a doublet, but we are treating them as a single state in the photochemical calculations, i.e., we assume they have the same reactivity and radiative lifetimes. Although different reactivities between the two terms have been observed for higher laying states, such effects are small and they will not affect our conclusions regarding the importance of the excited states chemistry on the overall abundance of the alkalis. Higher excited states of ²P symmetry can also be excited from the ground state, but the required shorter energy photons will be absorbed by other species higher up in the atmosphere and the oscillator strengths for these transitions decrease with increasing energy level. Further interaction of the first excited states with photons leads to the excitations of the [4S] and [3D] states of Na and the [5S] and [3D] states of K, which in their turn excite to the Na[4P] and K[5P] states, respectively (Figure 12). These are the four states we consider for each atom. We only consider excited states of the neutral alkali atoms, since excited states of the ions are separated by tens of eVs from the corresponding ground ion states.

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We investigate in our calculations the different chemical reactivity of the excited states toward H₂, H₂O, and HCl (Table 3). These reactions can lead to the formation of molecular complexes that could reduce the atomic abundances. To avoid interrupting the flow of the text we moved the details of this study to the Appendix. Our results demonstrate that these processes cannot significantly affect the atomic abundances of Na and K. In addition, the simulated non-LTE populations are not large enough to affect the transit depth observations. Our assumed atmospheric conditions and composition are different from those used by Barman et al. (2002), which is probably the reason for the different conclusions derived between the two studies about the efficiency of the non-LTE processes. Nevertheless, we note that the overall effect of the extended chemistry processes we include (see below) can affect the upper atmosphere abundances of the alkali atoms.

Excited alkali metals can further contribute to the ionization of the atmosphere. Direct photoionization from the radiation field

$$\begin{eqnarray*} &&{\rm Na}^* + {h}{\rm v} \rightarrow {\rm Na}^+ + {\rm e}^- \end{eqnarray*}$$

is more efficient for the excited states since they are closer to the ionization potential, i.e., the ionization cross sections of the excited states extend to longer wavelengths than those of the ground state. Therefore, their ionization rates can be significant since the stellar fluxes increase with increasing wavelength in this part of the spectrum. We calculated the photo-ionization cross section of the excited states following the analytical description from Zhang & Shneider (2010) that is consistent with available measurements (Wippel et al. 2001; Amin et al. 2005).

Energy pooling reactions can also populate higher excited levels that can photo-ionize faster. For example, Huennekens & Gallagher (1983b) have measured reaction rates for the process

$$\begin{eqnarray*} &&{\rm 2 Na^* \rightarrow Na}{[nl]}{\rm \ + \ Na} \end{eqnarray*}$$

and found cross sections of 23 and 16 Å² for production of the [4D] and [5S] states, respectively, from collision of two Na[3P] atoms at 600 K. Theoretical calculations that are consistent with these measurements show that at higher temperatures the cross section for the formation of the [4S] state should increase to ∼40 Å² at 1000 K, while that for [5S] should decrease, but less steeply (Geltman 1989). Energy pooling reactions for potassium were measured for collisions between [4P] states to produce [5P], [6S], and [4D] states (Namiotka et al. 1997). According to our calculations, the loss rate of Na* states due to energy pooling is small compared to the other processes we consider and since we did not include any higher states than the Na[4P], we did not include this mechanism. For potassium we included the energy pooling rate to the K[5P] state, but it has a minor role.

Other processes of direct ionization without a secondary photon involvement are those of Penning ionization:

$$\begin{eqnarray*} &&{\rm 2 Na^* \rightarrow Na^+ + Na + e^-} \end{eqnarray*}$$

and associative ionization:

$$\begin{eqnarray*} &&{\rm 2 Na^* \rightarrow Na_2^+ + e^-}. \end{eqnarray*}$$

These mechanisms occur for both sodium and potassium, but also for combinations of the two. Associative ionization for Na has been observed for the [3P] states (Huennekens & Gallagher 1983a) as well as from collisions between [3P] and [3D] states (Babenko et al. 1995), while theoretical studies have demonstrated that the collision cross section for the associative ionization of two [3P] atoms at 1000 K is more than an order of magnitude smaller than the corresponding energy pooling cross section (Geltman 1988). Associative ionization for potassium was measured by Klucharev et al. (1977) for two [4P] states, while Kimura & Saikan (1981) measured the penning and associative ionization of excited Na atoms with ground state K atoms and found that production of K⁺ and NaK⁺ is possible but occurs only for sodium energy levels above the [6S] state. Thus, associative ionization cannot have an important contribution to the ionization of the Na and K for the conditions we simulate, as demonstrated also by the resulting profiles of the dimer ions (Figures 20 and 21).

Penning ionization is possible for collisions between the two resonant states of sodium and potassium:

$$\begin{eqnarray*} &&{{\rm Na}[3P] + {\rm Na}[4P] \rightarrow {\rm Na} + {\rm Na}^+ + e^-} \\ &&{{\rm K}[4P] + {\rm K}[5P] \rightarrow {\rm K} + {\rm K}^+ + e^-}, \end{eqnarray*}$$

but also for the cross collision processes:

$$\begin{eqnarray*} &&{ {\rm Na}[4P] + {\rm K}[4P] \rightarrow {\rm Na} + {\rm K}^+ + e^-} \\ &&{{\rm Na}[3P] + {\rm K}[5P] \rightarrow {\rm Na} + {\rm K}^+ + e^-} \\ &&{{\rm Na}[4P] + {\rm K}[5P] \rightarrow {\rm Na} + {\rm K}^+ + e^-}. \end{eqnarray*}$$

These processes have large reaction cross sections and we used an estimated reaction rate of 10⁻⁹ cm³ s⁻¹ based on the measurements for Na[3P] + Na[5S] collisions from Carré et al. (1984). Similar processes for the other states are also possible, but the populations of the resonant states are much larger and have the highest contribution to the alkali ionization through this pathway.

Inclusion of these processes does not affect the densities of Na and K below 1 mbar and therefore does not change our transit depth results. Any change in the profiles occurs only in the upper atmosphere and could be only be observationally detected by looking at the cores of the Na and K lines that are probing this region of the atmosphere. Our calculations show that inclusion of the extended chemistry reduces the differential transit depth at the Na line by ∼15% at the smallest bands widths around the line cores.

We finally add a few notes on the ground state ion chemistry. Radiative recombination of atomic ions usually leaves the neutral atom in an excited state from which it subsequently de-excites. We have not found any state specific information for the recombination of Na and K ions, thus we assumed that the produced neutrals are at the first excited states (Na[3P] and K[4P]). Assuming recombination to the Na[4P] and K[5P] states did not change significantly our results. In the three-body recombination the excess energy is captured by the collision partner and the produced atoms should be on their ground state. We note that the released energy from the recombination of sodium (5.14 eV) is larger than the binding energy of the H₂ molecule (4.52 eV), thus the three-body recombination could lead to the dissociation of H₂:

$$\begin{eqnarray*} &&{\rm Na^+ + e^- + H_2 \rightarrow Na + 2 H}, \end{eqnarray*}$$

or at least to significant vibrational excitation. This process occurs deep enough in the atmosphere for the atoms to rapidly recombine to H₂. Thus, the H₂ profile is not affected by the three-body recombination. For potassium the ionization potential (4.32 eV) is less than (although comparable with) the H₂ binding energy and therefore the recombination will not dissociate the molecule although it will probably leave it in an excited vibrational level.

For the high-pressure limit of three-body recombination we assumed a rate of 10⁻⁷ cm³ s⁻¹ due to lack of information. Typically, recombination rates depend inversely on the square root of T, which means that the high-pressure limit of the recombination rate close to the temperature minimum could be ∼50% smaller than deeper in the atmosphere. Our sensitivity test on this parameter show that we need to assume a high-pressure limit of 10⁻¹¹ cm³ s⁻¹ before the change in the total recombination rate becomes significant to affect the profiles of Na and K. Thus, modifications to the assumed recombination rate do not seem capable to reduce the alkali abundances.

Due to their lower ionization potentials, charge transfer reactions of alkali atoms with other species are not efficient. Thus, the only charge transfer reactions to consider is

$$\begin{eqnarray*} &&{\rm Na^+ + K \leftrightarrow Na + K^+}, \end{eqnarray*}$$

with the forward rate in this case favored by 0.82 eV. Another ionization mechanism is that of the ion-pair formation:

$$\begin{eqnarray*} &&{\rm Na + X \rightarrow Na^+ + X^-}, \end{eqnarray*}$$

where X is a species (atom or molecule) with a large enough electron affinity to make this reaction efficient at high temperatures. The best candidates in our case (in terms of abundance and electron affinity) are H, O, and OH with electron affinities of 0.75, 1.46, and 1.83 eV, respectively, which means that the barriers for the ion-pairs formation will be 4.39, 3.68, and 3.31eV. The overall effect of these processes on the alkali atoms is small. It is interesting to note that Na and K also form stable negative ions with electron affinities of 0.55 and 0.50 eV, respectively. In this case the corresponding ion pair reactions should be limited among the alkali atoms:

$$\begin{eqnarray*} &&{\rm Na + Na \rightarrow Na^+ + Na^-} \\ &&{\rm K + K \rightarrow K^+ + K^-} \\ &&{\rm Na + K \rightarrow Na^+ + K^-} \\ &&{\rm Na + K \rightarrow Na^- + K^+}. \end{eqnarray*}$$

The reduction in the involved energy barriers for these reactions is not as large as for the ion-pair formation processes with other species and thus these mechanisms are not efficient. In any case, the ion-pair recombination (reverse of the above processes) is a barrierless reaction, therefore the net contribution of the ion-pair mechanism will be important only at very high temperatures and does not affect our conclusions for HD 209458b.

Associative ionization of sodium atoms with water molecules, although important in the atmospheres of our solar system, is not significant for the high-temperature exoplanet atmosphere as the bond energy is small (∼1 eV) and the formed ligands would not survive in the atmosphere in significant abundances (Gilligan et al. 2001). This effect is already demonstrated for the alkali dimers described above. The same conclusion also applies to the formation of clusters of alkali ions with H₂O or H₂, which can reduce the ion abundances due to their faster recombination rates. The high atmospheric temperatures would inhibit a significant population of the clusters, which even if they were possible would only make the agreement with the observations worse since the population of neutral atoms would increase. Therefore, we do not explicitly follow these processes in our calculations.

Finally, we investigated the role of the atmospheric mixing on the profiles of Na and K. Specifically, we assumed a eddy mixing profile that is 10 times smaller than the nominal profile we use. The resulting mixing ratios are not significantly different from the previous results in the region probed by the transit observations. As a result, our attempts to match the Na differential transit depth and explain the non-detection of K with our photochemical model are not successful.

One of the possible explanations proposed for the reduced Na transit depth and the lack of K is a sharp decrease in the temperature profile of HD 209458b at pressures below ∼3 mbar (Sing et al. 2008b) that would reduce the abundance of both elements due to their capture into condensates (Na₂S) or molecular structures (KCl). Below this pressure level the temperature would have to remain high in order to reproduce the observed Na wing by Sing et al. (2008a). The vapor pressure curves for these conditions (see Figure 2) indicate that temperatures lower by ∼300 K from the atmospheric skin temperature would be required for the alkali atoms to be significantly reduced. Such low temperatures, however, are not supported by the observations.

At this point we need to make the distinction between day and night conditions. On one hand, the current circulation models for HD 209458b from which our temperature profiles for day-time disk and limb conditions are taken, suggest that strong zonal winds efficiently redistribute the heat flux from the day side to the night side (Showman et al. 2009). Thus, temperatures much lower than the skin temperature are not anticipated. On the other hand even if the night side is much colder than anticipated and the alkalis are captured in the form of condensates or molecules then the reduction in the observed transit depth would be only a factor of two, since the observations probe both the dusk and dawn terminators. Thus, a reduced temperature scenario does not seem able to explain the observed transit depth.

An argument against globally lower temperatures arises from the thermal emission of HD 209458b (Figure 15). With the assumed disk temperature profile and the calculated H₂O and CO abundances from the photochemical model, we derived a thermal emission spectrum that is consistent with the available observations from Spitzer. The latter include the initial observations by Knutson et al. (2008) between 3.6 and 8 μm, and the more recent observations at these wavelengths by Diamond-Lowe et al. (2014) that suggest a lower flux ratio, as well as the 24 μm observations by Crossfield et al. (2012). The calculated spectrum indicates that a slightly warmer atmosphere would bring the thermal emission even closer to the observations. Hence, a colder atmosphere in order to the reduce the abundances of the alkali atoms is not supported by these observations.

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If we assume that the short wavelength slope in the transit depth observations is due to H₂ Rayleigh scattering (des Etangs et al. 2008), we can derive an estimate of the atmospheric temperature. Assuming different isothermal conditions we can investigate the variation of the simulated slope of the transit depth at short wavelengths against the observations (Figure 16). Changing the atmospheric temperature, shifts the whole transit depth curve due to the changes in the atmospheric scale-height. Thus, in order to focus only on the variations of the Rayleigh slope, we normalized our simulated transit spectra for different temperatures to 0.485 μm. For comparison with the HST/STIS observations (which have small enough error bars to allow for this test) we scaled the simulated spectra so that they would result to the same transit depth at 0.485 μm. This comparison of the simulated slope with the observed slope demonstrates that temperatures close to 1000 K generate transit depth slopes that are in better agreement with the observed slope compared to spectra generated for temperatures of 500 or 1500 K. A best-fit search assuming only H₂ results in an isothermal profile of ∼1090 K, which is very close to the isothermal part of the limb temperature profile we use (Figure 2). Note that this conclusion does not depend on which data set of the broad band observations we use (Knutson et al. 2007; Deming et al. 2013), since we only compare the relative absorption depths at different wavelengths that are approximately the same for the two studies. Therefore, if H₂ is responsible for the short wavelength slope of the observations, the temperature cannot be low enough for Na and K to condense or be captured in molecular structures.

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We note at this point that Vidal-Madjar et al. (2011) derived from the differential transit depth observations a cold enough temperature profile for HD 209458b to allow for the loss of Na due to condensation. Their retrieval was based on the assumption of a constant with altitude Na profile, though, which our calculations show is not realistic above ∼1 mbar (see Figure 5). For a constant profile the temperature retrieval does not depend on the Na abundance, but this does not hold for a variable profile for which the differential transit depth has a different slope and magnitude (Figure 17). Thus, a more reliable temperature retrieval based on these observations needs to consider the variation of the Na profile with altitude due to the photochemistry. Given the apparent inconsistency between the mid- and low-resolution transit depth observations, we are reluctant to perform such an analysis before a better understanding of the observations is possible.

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A much warmer temperature scenario does not provide a solution either; even if more exotic metals become available in the atmosphere such as TiO or VO (Fortney et al. 2003; Desert et al. 2008), their spectral signature is not large enough to cover the broad wings of the potassium line. For example, assuming a mixing ratio of 10⁻¹⁰ for TiO, the transit depth at the sodium line center is larger than the observations, while the K wings are not affected (Figure 18). Assuming a larger abundance (10⁻⁹), the transit depth at wavelengths between 0.4 and 0.75 μm increases, but at shorter wavelengths it remains the same, as the TiO cross section sharply decreases for λ < 0.4 μm. Thus, a large TiO abundance is inconsistent with the observed broadband slope, if the latter is due to H₂ Rayleigh scattering. If the slope is due to a different source of opacity, then a much higher temperature would be required in order to allow for large TiO abundance, enough to cover the K line and obscure the Na line. Such high temperatures would contradict the disk emissions, with which the temperature profile we use is consistent.

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Another potential solution is that of a heterogenous opacity. In this scenario, the signatures of Na and K are reduced because a cloud or a photochemical haze is providing a spectrally homogeneous opacity that masks the alkali signature. As we demonstrated earlier, this scenario, originally suggested as an explanation of the reduced differential transit depth around the Na doublet (Charbonneau et al. 2002), is not consistent with the subsequent observations (Ballester et al. 2007; Knutson et al. 2007; Sing et al. 2008a). On the other hand a reduced differential transit depth was retrieved in the subsequent observations from the high-resolution spectra (Sing et al. 2008b) around the sodium core. One way to reconcile the two different conclusions derived from the same observations is to assume that the spectral region around the Na core is affected by another gaseous absorber with a relatively flat absorption spectrum. The problem with this scenario is that the Na doublet is one of the strongest absorption lines and in order to obscure its wings with another gaseous absorber, the abundance of the later would have to be too large. For example, TiO, which has numerous lines around the Na core, would have to be present at a mixing ratio of 10⁻⁸ before it substantially affects the differential transit depth. In that case numerous absorption lines at shorter wavelengths would have an observable signature in the transit spectrum, and generate a different spectral slope than observed. Therefore, it is clear that the analysis of the low- and high-resolution observations from HST/STIS provide inconsistent results. The same problem appears in the comparison of the low-resolution analysis results (Knutson et al. 2007; Deming et al. 2013) with the medium-resolution spectra (Sing et al. 2008a), with the later demonstrating a more pronounced Na wing compared to the former. All these differences demonstrate the difficulties inherent in the analysis of these observations.

Nevertheless, if we momentarily neglect the differential transit observations (that are subject to higher uncertainty relative to the broadband results), a heterogeneous opacity can still be invoked to help understand the observations, provided that the haze or cloud structure can provide a short wavelength slope that is consistent with the observations. The probed pressures around the alkali lines indicate that this heterogenous opacity needs to be located close to the 10 mbar pressure level. It is important to keep in mind that we are observing the limb of the atmosphere. In this way the opacity of this cloud/haze does not have to be large in the vertical direction as long as the tangent opacity is consistent with the observations. Moreover, the consistency of the theoretical thermal emission spectrum with the observations (Figure 15) implies that heterogeneous opacity is small and/or that the particle size is much smaller than ∼1 μm in the probed region. Otherwise, scattering by the particles would modify the emitted radiation.

Heterogeneous opacities can also have an active photochemical role. For example, particles can accumulate free electrons in the atmosphere, and thereby decrease the recombination rates (e.g., Lavvas et al. 2013). This way the sodium and potassium abundances could be reduced deeper in the atmosphere. For the high temperatures of HD 209458b, the large collision rates of the ions with the charged particles would limit the decrease in the recombination rate, though. Heterogenous reactions on the surfaces of the particles, however, could still affect the abundances of the alkali metals. All these processes require a dedicated study that will be presented in a future publication.

We should also note here that the case of heterogeneous opacity could also be consistent with the cases where only potassium is observed. In this case, a low-pressure opacity would be required (i.e., a photochemical haze rather than a cloud) in order to mask the Na line and with the appropriate slope in order to allow for the K line to be detected. The magnitude of the required slope could constrain the particle properties and be used to evaluate the feasibility of the haze scenario against other options. Moreover, the presence or not of line wings in the observations is a limiting factor for these scenarios, therefore high-resolution observations, able to resolve these features are a requirement for future measurements.

Until now we assumed solar elemental abundances in all of our calculations. The simplest method to reduce the Na and K transit depths is to assume that both elements have much different abundances in the atmosphere of HD 209458b. This difference can be absolute or relative. If the ratio of the two elements is the same as solar but the absolute abundance is smaller, then we would have to reduce the abundances by a factor of 10 relative to the solar values in order to make the transit depth at the potassium line consistent with the observations (Figure 19). In this case, though, the corresponding reduction of the Na abundance would make the sodium wings much more narrow than what is suggested by the STIS observations analyzed by Sing et al. (2008b), and the overall transit depth would be much lower than the broadband absorption retrieved by Knutson et al. (2007) or Deming et al. (2013). Thus, a homogeneous decrease of the elemental abundances relative to the solar values is not consistent with the observations.

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Is there though a reason to anticipate that the Na/K ratio in HD 209458b is similar to the solar ratio? Before answering this question we need to evaluate the degree at which we need to modify the assumed elemental abundances to bring the simulations in agreement with the observations. As discussed above, the solar abundance of K needs to be reduced by a factor of 10 to make its signature consistent with the transit depth spectrum from the Deming et al. (2013). Explaining the mid-resolution HST/STIS observations, as presented in Desert et al. (2008), requires decreasing the solar K abundance by a factor of ∼50 to bring the transit depth at the line core (the bin including the K line center in Desert et al. 2008) within 3σ of the observations. On the other hand, the solar Na abundance provides an acceptable fit to the sodium wing transit depth derived from the HST/STIS observations (neglecting the differential transit depth), implying that no major change is required for this element. These results imply that Na/K ≃150–750 in the atmosphere of HD 209458b, which is a substantial modification to the corresponding solar ratio, (Na/K)_☉ ≃15. Thus, we can reformulate the above question as: is such a large abundance ratio realistic?

There are two approaches to answer this question. The first involves the actual spectrum of HD 209458, which demonstrates the presence of both Na and K as seen from ground-based observations (e.g., Jensen et al. 2011). Although an evaluation of the abundance of these elements in the star's atmosphere is not available, the structure and magnitude of the spectral lines of the two elements imply that the abundance of K is not two orders of magnitude smaller than that of Na. Although not conclusive, this is the only constraint available for this system currently.

From a wider perspective we can address the question of the relative abundances of Na and K by observing the statistics of their ratio in different stars. Observations of very low metallicity massive stars, which are representative of the first stars formed in the universe, can provide us with information about the initial elemental abundance of heavy elements in the interstellar medium. Measurements of the sodium and potassium abundance ratios relative to magnesium in such stars of our galaxy give mean values of [Na/Mg] = −0.8 dex and [K/Mg] = −0.3 dex, with scatter of 0.14 and 0.09, respectively (Andrievsky et al. 2010). Although these values suggest lower elemental abundances of the alkalis relative to the solar values (as anticipated for the early stages of the universe) they demonstrate that the Na/K ratio ranges between 0.16 and 0.62 relative to the solar ratio. Observations of stars with similar properties as HD 209458b and the sun show that Na and K abundances are close to the solar values (Shimansky et al. 2003; Adibekyan et al. 2012), while investigations of high metallicity A stars show that the variability of their alkali abundance is consistent with Na/K < 100 (Takeda et al. 2012). Thus, a consistent picture arises from these observations that the large Na/K values required to explain the observations are not likely, at least in stellar atmospheres. This conclusion though concerns only the stellar atmosphere rather than the planet.

Planetary atmospheres do not have to be of the same elemental abundance as the parent stars, as we well know from the solar system. It is important to note though that even if the elemental abundances in the atmosphere of HD 209458b have been modified after the formation of the planet (e.g., due to migration in the stellar system followed by condensation and/or chemical modification), then we would expect Na and K to undergo the same changes, since they have very similar chemical behavior, as demonstrated above. In other words, we would not expect a preferential decrease of the potassium abundance over that of sodium during the evolution of the planetary atmosphere due to chemical processes only. Thus, any potential changes in the Na/K ratio have to be traced back to the planet formation from the protoplanetary disk.

If the Na/K ratio varied across the protoplanetary disk that gave rise to the planetary system, then modifications of the atmospheric ratio relative to the stellar value are possible. Variations of the abundances of different elements on the disk can occur as a result of turbulence, chemistry, and radiation pressure. Due to the large oscillator strengths of the resonant lines, Na and K are subject to strong radiative acceleration. This effect will be important only at disk regions that are optically thin and where the alkali metals are not ionized, and therefore will only affect a small part of the disk's mass. Moreover, studies of this effect in debris disks demonstrate that, for low viscosity conditions, the Na/K ratio (at 100 AU) decreases relative to the solar value, contrary to the required modification in our case (Xie et al. 2013).

On the other hand, models coupling transport with photochemistry in protoplanetary disks (between 10 and 800 AU) demonstrate that the impact of turbulence varies among different gaseous species and ices, with the effect being more prominent on the latter component (Semenov & Wiebe 2011). This occurs as a result of the transport of ices/grains to regions of the disk where heterogeneous chemistry is more active (for the homogenous chemistry alone, no modifications were observed). The dominant condensates of alkalis in protoplanetary disks can be different from the condensates in planetary atmospheres, as the lack of cloud formation and stratification due to gravity, allows for the modification of condensates through interaction with other molecules. Thus, under thermochemical equilibrium Na and K should condense in the form of feldspar (NaAlSi₃O₈, KAlSi₃O₈) in protoplanetary disks compared to Na₂S and KCl in planetary atmospheres (Lodders 2010a). Due to the limited life time of protoplanetary disks (<10 Myr) and the variable lifetimes required for thermochemical equilibrium at different locations of the disk (due to variations in pressure and temperature), the abundance of the different components needs to be evaluated with detailed models that take into account the multiple processes involved. Unfortunately, a detailed study on the variation of Na and K over the protoplanetary disks is currently lacking. Any insight from observations is also limited. Spatially resolved spectra from the β-Pictoris debris disk demonstrate that Na is distributed over a large part for the disk, from 13 to 323 AU (Brandeker et al. 2004) but the corresponding distribution for K is unknown.

Nevertheless, we can evaluate the protoplanetary disk conditions required to explain the HD 209458b observations. The necessary Na/K ratio retrieved from the observations implies that HD 209458b should have emerged from a region of the disk that was deprived of K. This means that not only does K need to be lost from the gas phase but also its condensates or molecules need to be redistributed at a different radial location of the disk from where the planet is formed. Otherwise, gravitational heating (and the possible migration of the planet closer to the star) would have released the condensed alkalis back into the atmosphere.⁴ At the same time Na would have to remain unaltered by the processes affecting the K abundance. The current understanding of giant planet formation suggests that HD 209458b was formed further away from the star (>1 AU) and subsequently migrated to its current location. Hence, both Na and K should be present in condensed form at the planet formation region, and would be subject to similar differentiation processes. Therefore, Na and K condensates should not have very different responses to the differentiation mechanisms. Moreover any differentiation process will have to occur in a short timescale, before the disk is accreted to the star, in order to allow for the migration of the planet at its current short orbital distance. Hence it is difficult to explain the observations of HD 209458b based on chemical and physical processes in the protoplanetary disk at the time of planet formation.

Therefore, based on the expected primordial elemental abundances and alkali chemistry either in the protoplanetary disk or the atmosphere of the planet, HD 209458b should not have a Na/K ratio that is substantially different from the solar ratio. While we cannot rule this possibility out unequivocally, it appears unlikely based on the above arguments.

Finally, there is one more aspect of the observations that requires some attention. The broad band observation on the red side of the K line demonstrate a significantly larger absorption than the model can produce. Absorption by H₂O would be consistent with these observations if the temperature was higher than the profile we assume, and in this case the simulated spectrum would be in better agreement with the Spitzer/Infrared Array Camera (IRAC) observations at 5.8 and 8 μm. In this case though the model would provide too much absorption for the other observational constraints from HST/Wide Field Channel (WFC) and Spitzer. These differences among the different observations could potentially be related to differences in the analysis applied, but could also be related to temporal changes in the properties of the star. Similar effects could be possible at shorter wavelengths and could affect the relative levels of the alkali lines relative to the slope at shorter wavelengths. Simultaneous and repeated observations of the whole spectrum are required in order to clarify all these observational issues.