Empirical Photochemical Modeling of Saturn’s Ionization Balance Including Grain Charging

We present a semianalytical photochemical model of Saturn’s near-equatorial ionosphere and adapt it to two regions (∼2200 and ∼1700 km above the 1 bar level) probed during the inbound portion of Cassini’s orbit 292 (2017 September 9). The model uses as input the measured concentrations of molecular hydrogen, hydrogen ion species, and free electrons, as well as the measured electron temperature. The output includes upper limits, or constraints, on the mixing ratios of two families of molecules, on ion concentrations, and on the attachment rates of electrons and ions onto dust grains. The model suggests mixing ratios of the two molecular families that, particularly near ∼1700 km, differ notably from what independent measurements by the Ion Neutral Mass Spectrometer suggest. Possibly connected to this, the model suggests an electron-depleted plasma with a level of electron depletion of around 50%. This is in qualitative agreement with interpretations of Radio Plasma Wave Science/Langmuir Probe measurements, but an additional conundrum arises in the fact that a coherent photochemical equilibrium scenario then relies on a dust component with typical grain radii smaller than 3 Å.


Introduction
Saturn's upper atmosphere was probed in situ during the proximal orbits of the Cassini mission in 2017. During all proximal orbits, the Radio Plasma Wave Science/Langmuir Probe (RPWS/LP) was operating, offering different means to extract the electron number density (e.g., Wahlund et al. 2018;Morooka et al. 2019;Persoon et al. 2019). Likewise, during all passages, the Ion Neutral Mass Spectrometer (INMS) operated such that the number density of neutral species, most importantly H 2 , could be determined (e.g., Waite et al. 2018;Miller et al. 2020). Cassini's orbits 288 (perigee on 2017 August 14) and 292 (perigee on 2017 September 9) were special among orbits reaching below 2000 km above the 1 bar level, in the sense that they included measurements of light (<8 amu) ion species by the INMS (e.g., Waite et al. 2018). These orbits reached ∼1700 km above the 1 bar level at their closest approaches, which occurred just south of the equator.
A couple of studies have focused on orbits 288 and/or 292 and combined RPWS/LP and INMS ion and neutral data to test our understanding of Saturn's photochemistry or fill in gaps where existing measurements do not provide direct or definite answers. For instance, the ion measurements of INMS ) are restricted to species with mass-to-charge ratios <8 Da, which essentially covers only H + , H 2 + , H 3 + , and He + (and isotopic variants). For altitudes near 1700 km, the measured total number densities of these species are markedly lower than the electron and total ion number densities inferred from RPWS/LP measurements (Morooka et al. 2019), which leads to the obvious question of what species actually dominates the ion composition in this region. Dreyer et al. (2021) combined RPWS/LP ion (number density) and electron (number density and temperature) data with INMS H 2 and H 2 + (number density) data to estimate an upper limit of the effective recombination coefficient at a reference electron temperature of 300 K. Comparing their derived upper limits with recombination rate constants measured in laboratories, they argued for a dominance of ion species characterized by comparatively low rate constants and raised HCO + as a prime suspect. This is at odds with predictions of H 3 O + dominance from earlier model works (e.g., Moore et al. 2018). Considering the low proton affinity of CO in comparison to species like H 2 O and NH 3 and the associated efficiency of proton transfer reactions like HCO + + H 2 O → H 3 O + + CO and HCO + + NH 3 → NH 4 + + CO, it is actually hard to imagine HCO + dominance in said region of the ionosphere unless the mixing ratio of CO markedly exceeds (an order of magnitude or so) those of H 2 O, NH 3 , and other species characterized by higher proton affinities than CO. On the one hand, the INMS analysis by Miller et al. (2020) of mixing ratios of different species in Saturn's upper atmosphere gives no obvious support for the idea of CO being more abundant than species like H 2 O. On the other hand, Cravens et al. (2019) presented independent results, suggesting that the mixing ratio of CO (or CO-like molecules) exceeds that of, e.g., H 2 O in Saturn's near-equatorial ionosphere. We return to the details of their study below and note merely, for the time being, that the work supports, at least indirectly, the idea of an ionosphere at least regionally dominated by HCO + , as proposed by Dreyer et al. (2021). However, the HCO + dominance proposed by Dreyer et al. (2021) is not carved in stone. To estimate the upper limit of the effective recombination coefficient, the total ion number density, as measured by the LP, enters the denominator in the formalism of Dreyer et al. (2021). This implies that an overestimated value of the ion number density would bring about too low of an estimate of the effective recombination coefficient. Around an altitude of ∼1700 km above the 1 bar level, the reported total ion number density for all flybys reaching such depths is markedly higher than the number density of free electrons (see Figure 6 of Morooka et al. 2019), a feature suggesting significant electron depletion due to grain charging. While grains falling into Saturn's upper atmosphere have been suggested from model calculations to originate mainly from the B and C rings (Hsu et al. 2018), the flow into the near-equatorial upper atmosphere also has important contributions from the D ring . As noted by Dreyer et al. (2021), earlier Saturn ionospheric models did not work with the assumption that dust was ubiquitous enough to have any notable influence on the ionization balance. Even in some recent works (e.g., Moore et al. 2018;Cravens et al. 2019), authors have utilized the assumption that the total number density of gas-phase positive ions equals the number density of free electrons. Dreyer et al. (2021) merely noted that if, in their analysis, they had forced the ion number density to be equal to the measured number density of free electrons, the resulting effective recombination coefficient would be more compatible with an H 3 O + -dominated ionosphere.
One may speculate that secondary electron emission (from "ram neutrals" smashing into the LP at >30 km s −1 ) could contribute to the current on the ion side of the LP voltage sweeps and-when not accounted for-give the impression of higher ion number densities than actually present. For instance, a very strong secondary electron emission was reported for Giotto (and Vega I and Vega II) when flying through the waterdominated coma of comet 1P/Halley at ∼70 km s −1 (Grard et al. 1987). Studies with the aim to estimate the actual contribution of secondary electron emission to the current on the ion side of the LP bias voltage sweeps performed in Saturn's ionosphere are therefore ongoing in parallel. In the present work, the total ion number density is, as explained below, merely an output parameter of a model.
To briefly return to and explain the work of Cravens et al. (2019), they drew awareness to the fact that in the deep ionosphere, the measured number density of H + locally exceeds that of H 3 + , despite the latter ion being produced at a rate ∼10 times higher. This is not possible to explain solely ( Figure 4 in Cravens et al. (2019), it is seen that near the equatorial crossing of orbit 288, the empirically derived mixing ratio of R-type molecules exceeds 10 −3 , while that of M-type molecules is only a few times 10 −5 . This can again be contrasted with the results reported in Miller et al. (2020), who reported an average mixing ratio of a few times 10 −4 for H 2 O and overall higher average mixing ratios of M-type than of R-type molecules if translated into the nomenclature of Cravens et al. (2019).
It should be stressed that the method of Cravens et al. (2019) results in mixing ratios of R-and M-type molecules that vary along Cassini's trajectory. It is particularly at near-equatorial latitudes along the Cassini trajectory where the mixing ratios of R-type molecules grow significantly in comparison to M-type molecules. An interesting speculation by the authors is that the R-type molecules might be generated by ablation of grains entering the atmosphere. This is also touched upon by Yelle et al. (2018), who further noted that molecules liberated from micrometeorite bombardment are prone to recondense more or less effectively on particles of ring origin. They argued that cosmically abundant molecules with high vapor pressures, such as CO, can be expected to flow into Saturn's equatorial atmosphere.
In the present work, we make similar calculations as Cravens et al. (2019) but extend their photochemical reaction network in ways that allow predictions of additional parameters (most importantly, the total ion number density and the number density of specific ion types) while retaining the possibility of solving the system semianalytically. In contrast to Cravens et al. (2019), we do not enforce quasi-neutrality in the classical sense of n e = n i . Instead, we couple our ion chemistry model to the grain-charging formalism of Draine & Sutin (1987), which allows us to calculate the typical grain size needed for realizing a coherent photochemical equilibrium scenario that respects overall quasi-neutrality.
The outline of our paper is as follows. In Section 2, we present our model. In Section 3, we apply the model to two locations encountered during the inbound part of Cassini's orbit 292 and present both input and output parameters of the model. We proceed by discussing the findings with an emphasis on persisting conundrums. A summary with concluding remarks is given in Section 4.

Brief Description
The inputs into the model are the electron number density n e , the electron temperature T e , and the concentrations of H 2 , H + , H 2 + , and H 3 + . The ion temperature is set equal to T e by default but can be shifted. The model first calculates an upper limit for the concentration of M-type molecules (molecules like H 2 O, NH 3 , and CH 4 being reactive with both H + and H 3 + ). The value of [M] is then stepped from close to zero to the upper limit, and for each fixed value of [M], the model calculates a range of parameters. These include, e.g., the concentration of R-type molecules (molecules like CO and N 2 being reactive with H 3 + but not H + ) and the concentrations of ion species MH + and RH + (protonated versions of the M-and R-type molecules, e.g., H 3 O + and HCO + , respectively). Another output is the electron loss rate due to dust attachment. The model thus allows for solutions with the total gas-phase ion number density n i exceeding n e . As a final step in the model, the graincharging theory of Draine & Sutin (1987) is incorporated to calculate the typical grain size needed to allow for a selfconsistent solution of the ionization balance within the framework of photochemical equilibrium. Because a model run renders output parameters over ranges (rather than unique fixed values), we refrain from presenting along Cassini trajectory altitude profiles but rather focus on two separate regions encountered during the inbound part of Cassini's orbit 292 (see Sections 3.1 and 3.2).

Ion-electron Pair Production
Ion-electron pair production is conceived as driven by photoionization and electron-impact ionization of H 2 (e. , where k pt = 2.0 × 10 −9 cm 3 s −1 is the rate constant for the reaction. The above expression also represents the production rate of H 3 + , which we denote by p 3 . The production rate of the H + + e − pair, p 1 , is taken as a fraction γ of p 3 . We adapt γ = 0.20 as a default value. This is set rather crudely and subject to further discussion below. In summary, the production rates of protons, H 3 + , and free electrons are calculated as

Ion-neutral and Ion-electron Reactions
The following ion-neutral and recombination reactions are taken into consideration: Importantly, we assume that the M + produced in Equation (2a) is rapidly (on the order of a second) converted to MH + through a reaction with H 2 . Guided by Cravens et al. (2019), we adapt k ct = 5.0 × 10 −9 and k M = 3.5 × 10 −9 cm 3 s −1 . Lead by rate constants listed on the UMIST database for astrochemistry (McElroy et al. 2013; see also udfa.ajmarkwick.net) for reactions of H 3 + with CO or N 2 , we utilize k R = 2.0 × 10 −9 cm 3 s −1 . Similar  Table 1 in Dreyer et al. 2021) and, to some extent, by the modeling results of Moore et al. (2018).

Grain Attachment and Mass-corrected Ion Number Density
Our model does not enforce quasi-neutrality in the classical sense of n e = n i , where n i denotes the total positive ion number density. Instead, ions, as well as electrons, are allowed to attach to dust particles. We assume the dust component to be in current balance, collecting electrons and ions at equal rates, and, as at ∼10 au, we ignore the photoelectric effect. Letting s e and s 1 denote the inverse lifetimes of free electrons and protons against dust attachment, we enforce [ ] + denote the rate at which H 3 + attaches to dust grains, but from the grain-charging formalism of Draine & Sutin (1987), it is clear that s 3 can be expressed as s s 3 3 1 = . By Equation (3), we have also introduced θ C , a mass-corrected ion number density useful (as shall be seen later) when modeling the grain charging in detail. Note that we assume the MH + and RH + ions to have characteristic masses of 19 and 29 amu, respectively. The mass selected for RH + ions should be fine, as thus far, we have only identified CO and N 2 as probable R-type molecules. Using 19 amu as characteristic mass for the MH + ions assumes this population to be dominated by H 3 O + , but we note that the masses of other candidate molecules, like CH 5 + and NH 4 + (17 and 18 amu, respectively), are not far off.
We discuss in Section 3.3.1 the effect on the model results if we instead adapt a mass of several hundred amu for the MH + ions, which would be consistent with them being dominated by complex organic ions.

Procedure of Calculations
Balancing production and loss of H + gives rise to the equation where p 1 is given by Equation (1a), [M] denotes the concentration of M-type molecules, and s 1 is the inverse lifetime for a proton against dust attachment. Already from Equations (1a) and (4) while the mass-corrected ion number density θ C is calculated according to Equation (3). Equation (3) also gives the associated electron loss rate due to dust attachment, i.e., s e n e ; this is a key quantity for connecting our ion chemistry scheme to the grain-charging formalism of Draine & Sutin (1987). Before describing that step, let us just note that another quantity of interest (for comparison with observations) is the harmonic mean ion mass, which can be calculated as where m p is the proton mass. We consider spherical dust grains of a single size that can be at most singly positively charged, while we set no strict limit on the maximum negative charge. We let E + , E 0 , E −1 , etc. denote the inverse lifetimes against electron attachment of dust grains in charge states +1, 0, −1, etc. Likewise, we let P 0 , P −1 , P −2 , etc. denote the inverse lifetimes against ion attachment of dust grains (we enforce P + = 0). Letting n + , n 0 , n −1 , etc. denote the number density of grains in charge state +1, 0, −1, etc., we are facing an equation system P n E n , 11a 0 0 ( ) = + + P n E n E P n , 11b while n 0 = n + E + /P 0 . While the inverse lifetimes are given from the formalism of Draine & Sutin (1987) as described below, the value of n + needs to be adjusted so that the solution respects overall quasi-neutrality. This is taken care of via where the sum goes over negative charge states (e.g., j = 3 is associated with grains in charge state −3), n i is from Equation (9), and f j (dimensionless) is given by We have included an upper limit of j = m for the sum in the denominator, mainly to emphasize that for practical consideration, it may in some cases not be necessary to consider particularly high values of j (for instance, for subnanometer grains, f j becomes very small already for j = 2 or 3). For clarification, notice that Equations (12) and (14) can be combined such that n + f j = n −j . It then follows that Equation (13) is equivalent to a simple charge balance equation, where the equality between −n + + n −1 + 2n −2 + ... and n i − n e represents overall charge neutrality. From the formalism of Draine & Sutin (1987), we calculate inverse lifetimes (the "E:s" and "P:s") according to where n x is either n e or the mass-corrected ion number density θ C , k B is Boltzmann's constant, T x is T e or the ion temperature, m x is the electron or proton mass, r is the grain radius, and J(ν, τ x ) is a function of ν (the ratio of the charge of the grain and the charge of the attaching species) and τ x . The latter is a dimensionless parameter referred to as the reduced temperature and given by  We step the grain size radius by 0.1 Å starting at r = 1 Å (which is below the limit where the formalism is applied in Draine & Sutin 1987)  matches s e n e , as given by Equation (3). The corresponding grain radius is the one needed for the entire system to be in ionization balance within our theoretical framework (for the results presented in Section 3, we have verified the uniqueness of the solutions).

Results and Discussion
The nature of the model output (as can be understood from the figures presented in this work) is such that it is not suitable for being presented in a traditional altitude profile style. We refer readers to Figure ( Figure 1. The mixing ratio of M-type molecules is allowed to be several hundred ppm, and the calculated mixing ratio of R-type molecules also amounts to several hundred ppm. Both of these values are in qualitative agreement with the average mixing ratios reported in Table 2 of Miller et al. (2020), a work discussed further in Section 3.3.1. The total ion number density of ∼3000 cm −3 is consistent with RPWS/LP measurements (Morooka et al. 2019), and a dust population with grains of ∼0.5-1 nm radii can explain the electron depletion, at least while the mixing ratio f M < 150 ppm. For f M > 200, the required grain radius quickly drops to values below 3 Å. To this point, it should be noted that Draine & Sutin (1987) applied their grain-charging theory to grains with radii 3 Å. The calculated number density of grains (not shown) remains close to the difference between the calculated ion number density and the electron number density, meaning that the bulk of the grains are singly negatively charged. Another observation to make from Figure 1 is that the calculated m i,harm is in the range of about 4-5 m p , which is roughly twice the value inferred from RPWS/LP measurements at the corresponding location (see Figure 2 of Morooka et al. 2019).

Results When Applied to ∼1700 km above the 1 Bar Level during the Inbound Part of Orbit 292
We have also used the model for conditions encountered near closest approach of the inbound part of Cassini's orbit 292. This corresponds to ∼1700 km above Saturn's 1 bar level and a latitude range of ∼3°-4°S.   Similar to Figure 1, the dashed horizontal lines in Figure 2 represent fixed model input parameters, while the solid lines are model output parameters. The model output suggests a significant level of electron depletion, which is in qualitative agreement with RPWS/LP measurements (Morooka et al. 2019). Again, the dust population is dominated by singly negatively charged particles, and the total dust number density (not shown) is thus close to the difference between the calculated ion number density and the electron number density. The model also predicts the dominance of RH + ions, which is in line with the work of Dreyer et al. (2021), who proposed that the deep ionosphere is dominated by ion species with comparably low recombination coefficients. However, to this point, it should be noted that the model does not include reactions where MH + ions are produced at the expense of RH + ions; see Section 4. The computed m i,harm of ∼10 amu is, again, roughly twice the value of ∼5-6 amu derived from LP sweep analysis in Morooka et al. (2019). At the moment, we refrain from speculating on the cause for the seemingly systematic discrepancy. Instead, we discuss in Section 3.3 two other conundrums: first, in Section 3.3.1, that the mixing ratios of R-and M-type molecules depart significantly from the INMS averages (Miller et al. 2020) and second, in Section 3.3.2, that the grain radius is already forced below 2 Å for M-type mixing ratios exceeding 20 ppm.

Conundrums Associated with the Results from the Model
When Applied to ∼1700 km above the 1 Bar Level during the Inbound Part of Orbit 292 Looking at the variations in T e and of the input concentrations over a 70 km interval from closest approach, we notice notable departures from the utilized rounded values mainly for H [ ] + and H 3 [ ] + (see Figure 3 in Moore et al. 2018). We have inspected the model sensitivity to these variations and concluded that the central qualitative conclusions and identified main conundrums persist. However, it should be stressed that we target the inbound of orbit 292 specifically, so the stated input should not be viewed as necessarily typical at nearequatorial latitudes, ∼1700 km above the 1 bar level in Saturn's sunlit atmosphere.

Mixing Ratio Conundrum
We note from Figure 2 that our model suggests an upper limit of the mixing ratio of M-type molecules of ∼40 ppm. The upper limit can be conceived as even lower, since, for f M > 28 ppm, the grain size required to complete the photochemical equilibrium scenario drops below 1 Å. The mixing ratio of R-type molecules is calculated as stable around 2000 ppm. While these mixing ratios seem roughly consistent with the empirical estimates presented for orbit 288 by Cravens et al. (2019; see their Figure 4), they conflict with the average mixing ratio reported from closed-source neutral-mode measurements by the INMS Waite et al. 2018;Yelle et al. 2018;Miller et al. 2020). In Table 2 of Miller et al. (2020), H 2 O, NH 3 , and CH 4 (all M-type molecules) are specified to have average mixing ratios within the range 200-400 ppm, and, specifically for CH 4 , a mixing ratio of a few hundred ppm seems to be fairly stable with varying altitude (see Perry et al. 2018;Yelle et al. 2018). For the R-type molecules, average mixing ratios of ∼200 ppm are stated for CO and N 2 in Table 2 of Miller et al. (2020). While it is questionable to compare with average mixing ratios, it seems as if the model generates a mixing ratio of M-and R-type molecules that is roughly an order of magnitude too small and too large, respectively. It can be added that adjusting the assumed mass of MH + ions from 19 to 200 amu in the model (connecting back to discussions in Section 2.4) has no notable influence on the model output. Reducing the parameter γ from 0.20 to 0.10 (connecting back to discussions in Section 2.2) mainly has the effect of further reducing the calculated upper limit of M-type molecules down to ∼20 ppm.
This mixing ratio conundrum has been highlighted before, although from a slightly different viewpoint. Moore et al. (2018) made use of average mixing ratios of several species (both R-and M-type) as input in their Saturn ionospheric model. The equivalent problem then faced in the output (see their Figure 3)  is endoergic by ∼4.3 eV and requires significant internal excitation in both reactants to be feasible. We have inspected a simplified chemical scheme, neglecting dust attachment and recombination but adding the CID reaction, and found that an effective rate coefficient of k CID ∼ 3 × 10 −12 cm 3 s −1 yields f M ≈ f R ≈ 200 ppm. There are several concerns with such a solution on the mixing ratio conundrum that are so severe that we lean toward ruling out its feasibility. Not only is the required internal excitation indeed substantial, but also, if H 2 is in the fourth vibrational state or higher, it opens up for H + loss via , a process that has historically received a lot of attention in discussions of Saturn's ionization balance (see Moore et al. 2017 and references therein) and that, with reactants in the ground state, is endoergic only by ∼1.8 eV.

Grain Size Conundrum
The red solid line in Figure 2 shows the calculated grain size radius that is required to complete a photochemical equilibrium scenario for the ionization balance. Note that the unit is given in pm, and so the resulting radii for f M > 20 ppm remain smaller than 2 Å. Recall that Draine & Sutin (1987) only applied their grain-charging theory to grains with radii 3 Å. We note that a reduction of the ion temperature to values closer to the neutral temperature makes the situation even more problematic. We feel more or less forced to abandon the idea of compact and conducting spherical dust particles acting as the main negative charge carrier in Saturn's deep ionosphere unless drastic flaws prevail in our model input. We intend to study in detail the effect on the model results of introducing sizedependent sticking coefficients for electron-and ion attachment onto grains (at the moment these are set to unity). Another way out may be offered by complex-shaped grains, potentially capable of charging up more negatively than spherical grains of the same size, but we see no simple way of testing this.
The surprisingly rich organic chemistry of Saturn's ionosphere Waite et al. 2018) combined with the apparent requirement of "molecular-sized" negative charge carriers motivates us to also explore the option of certain carbon-rich molecules acting as electron attachment sites. The species in question may, for instance, be conceived as a mixture of carbon chains and small polycyclic aromatic hydrocarbons (PAHs), but we are careful not to pinpoint their nature and refer to the hypothesized species collectively as "Cparticles." We refer to the number densities of positively charged, neutral, and negatively charged C-particles as n C+ , n C , and n C− , respectively. We set rate constants for various reactions based on a survey of the UMIST database for astrochemistry, inspecting the reactivity of species like C 6 H and C 6 H 6 and a range of other hydrocarbons. We adapt a rate constant of k ptC = 2 × 10 −9 cm 3 s −1 for proton transfer from any ion to neutral C-particles, an electron recombination coefficient of α C+ = 5 × 10 −7 cm 3 s −1 for positively charged C-particles, a rate constant of k aC = 1 × 10 −7 cm 3 s −1 for electron attachment to neutral C-particles, and a rate constant of k MN = 7 × 10 −8 cm 3 s −1 for the mutual neutralization of negatively charged C-particles with any ion. These (effective) rate constants may very well be off by 50% or even more. We neglect interactions between the C-particles themselves and other processes not mentioned above. Balancing the production and loss rates of positive and negative C-particles and respecting quasi-neutrality renders, respectively, k n n n n , 18a Approximating n i ≈ 2n e = 10,000 cm −3 (under the guidance of Figure 2) gives 2k R n C = α C n C+ , k a n C = 2k MN n C− , n C− = n e + n C+ , and, after some algebra, a solution with n C+ = 57, n C = 7080, and n C− = 5057 cm −3 . This gives a total concentration of C-particles of ∼12,000 cm −3 , corresponding to a mixing ratio of 2 ppm. Besides the rather loose notion of C-particles and the uncertainty in effective rate constants, there are several caveats to the crude concentration calculations just made. For instance, the radiative electron attachment to neutral C-particles may possess nonnegligible activation energy barriers, as is suggested to be the case for the electron attachment to C 60 and possibly also many PAHs (Petrie & Bohme 2000). Also, at least for carbon chain anions, with or without H or N inclusion, a potentially highly competitive loss pathway is through associative electron detachment with atomic hydrogen. Measured rate coefficients for these reactions are on the order of (5-10) × 10 −10 cm 3 s −1 (Barckholtz et al. 2001;Yang et al. 2010). Considering a mixing ratio of atomic hydrogen possibly as high as ∼1% at the pressure level of interest (e.g., Müller-Wodarg et al. 2012;Kim et al. 2014), the loss of negative C-particles through this process may commence up to an ∼50 times higher rate than the loss through mutual neutralization, essentially pushing the "required" mixing ratio of the C-particles up to ∼100 ppm. On the one hand, such high mixing ratios of organic material are not in conflict with reports from INMS measurements (e.g., Waite et al. 2018;Miller et al. 2020). On the other hand, from a literature/database survey, we have reason to conceive the C-particles as M-type molecules. A mixing ratio of several tens of ppm or more would thus make it even harder to explain the high H + concentration in Saturn's deep ionosphere and further reduce the already "too low" upper limit for the mixing ratios of molecules like H 2 O and CH 4 . This poses a problem as long as a viable mechanism for enhanced H + production (lacking in the model) has not been identified. Theoretical predictions (Moore et al. 2008;Sakai & Watanabe 2016) suggest that the electron temperature is closer to the neutral temperature at the pressure level of interest. It can be noted that a model run utilizing an electron temperature of T e = 370 K (≈neutral temperature; see, e.g., Yelle et al. 2018) instead of 1500 K causes a reduction in the calculated n i from >10,000 cm −3 to ∼6000 cm −3 , which is a consequence of the fact that dissociative recombination is more efficient at lower T e . A reduced level of electron depletion relaxes the requirements on the grain size to complete a self-consistent solution for the ionization balance. However, setting T e = 370 K conflicts with the standard interpretation of the LP sweep characteristics; to this point can be added the fact that the inferred T e is not particularly sensitive to the fraction of the apparent ion current that is assumed to be caused by impactinduced secondary electron emission.

Summary and Concluding Remarks
When applied to conditions encountered near an altitude of ∼1700 km above the 1 bar level during the inbound part of orbit 292, our Saturn ionospheric model, which builds on the empirical approach of Cravens et al. (2019), suggests in comparison to INMS measurements Waite et al. 2018;Yelle et al. 2018;Miller et al. 2020) too-low mixing ratios of M-type molecules and too-high mixing ratios of R-type molecules. As a reminder, using the terminology of Cravens et al. (2019), M-type molecules (e.g., H 2 O, CH 4 , and NH 3 ) are reactive with H + and H 3 + , while R-type molecules (CO and N 2 ) are reactive with H 3 + but not H + . Our mixing ratio results are in accord with Cravens et al. (2019), who focused on orbit 288, and the identified conundrum is essentially equivalent to the problem of reproducing the H 3 + and H + number densities in Saturn's deep ionosphere as highlighted in Moore et al. (2018). We considered in Section 3.3.1 the CID of H 3 + , with H 2 as a way of enhancing H + production at the expense of H 3 + . An effective rate constant of k CID ∼ 3 × 10 −12 cm 3 s −1 suffices for the calculated mixing ratios of Mand R-type molecules to better match the INMS averages, but we also raised several concerns with this hypothetical solution.
Our model suggests that the total ion number density exceeds the electron number density by a factor of ∼2 (∼1700 km above the 1 bar level during the inbound part of orbit 292). This proposes an n i /n e ratio of ∼50%, which, while low, is higher than the ratio of ∼20% reported earlier (see Figure 6 of Morooka et al. 2019). A conundrum arises in that the calculated typical grain radius required for a self-consistent solution becomes smaller than 3 Å, making it difficult to separate them from semicomplex molecules and questionable whether the grain-charging formalism of Draine & Sutin (1987) is even applicable. Ways out of the grain size conundrum were speculated on in Section 3.3.2, but at this stage, we are not in a position to favor any particular explanation.
Drastic modifications made to (artificially) enhance [M] and reduce [R] bring notable changes to the total ion concentration. This is not surprising, since MH + ions are characterized by roughly four times higher electron recombination rate constants than RH + ions. With substantial concentrations of both M-and R-type molecules, the total ion number density is foreseen to be more sensitive to the molecular species that actually dominate the respective population. The maximum total ion number density is realized in a scenario wherein the M-type molecules are heavily dominated by CH 4 and the R-type molecules are dominated by CO. This facilitates effective proton transfer from CH 5 + to CO and a plasma dominated by HCO + , which has a low recombination constant (bringing this in line with the work of Dreyer et al. 2021). At the other extreme, a high concentration of H 2 O and/or NH 3 will render a heavy ion population dominated by H 3 O + and/or NH 4 + ions, characterized by markedly higher recombination rate constants than HCO + . Extending the proposed model with a more sophisticated chemical reaction scheme (e.g., Moore et al. 2018) seems a natural way forward.