Journal of Geophysical Research: Space Physics Statistical Study of the Energetic Proton Environment at Titan’s Orbit From the Cassini Spacecraft

A statistical study of the energetic proton environment at Titan’s orbit as captured by the MIMI/LEMMS and MIMI/CHEMS instruments is performed. The data analyzed cover all the dedicated ﬂybys of Titan by Cassini as well as the orbit crossings that happen far from the moon. The energetic environment is found to be highly variable on timescales comparable to that of the duration of a ﬂyby. Analysis of H + ion ﬂuxes reveals a weak asymmetry in Saturn local time with the highest ﬂuxes occurring in the premidnight sector of the magnetosphere. A correlation between the energetic ion ﬂuxes and the location of Cassini in the magnetosphere with respect to the center of the current sheet can be observed. Finally, an empirical model of proton spectra for energies above 20 keV is derived based on ﬁts to Kappa distribution functions. This model can be used to better understand the interaction of Titan with the magnetosphere and the energy deposition by energetic particles below the main ionospheric peak.


Introduction
The interaction of Titan with its surrounding environment is arguably one of the most complex interactions of its kind in the solar system. The mean orbital distance of the moon to Saturn is 20.3 Saturn radii (R S ), which locates it close to the magnetopause stand-off distance, described using a bimodal model with mean distances of 22 and 27 R S by Achilleos et al. (2008), when crossing the subsolar point. As a consequence, while Titan spends most of the time in the outer region of the Saturnian magnetosphere, under strong solar wind conditions it can be located within the magnetosheath Edberg et al., 2013) or in the unshocked solar wind .
Titan's orbit lies almost at the equator but due to the fact that Saturn's current sheet can move up and down into a bowl shape due to solar wind activity , the moon can be located inside or outside the plasma sheet at any given time. This flapping motion of the current sheet, affected by magnetospheric dynamics, can be faster than the transit time of Cassini during dedicated flybys. This, together with the different trajectory geometries for different flybys, makes it possible for the plasma and field instruments to detect different plasma environments during the inbound and outbound parts of the trajectory (Simon et al., 2013).
Since Titan does not possess an internal magnetic field (Backes et al., 2005;Wei et al., 2010), its atmosphere and ionosphere interact directly with the different environments just described, creating a unique moon-magnetosphere interaction when located inside the magnetosphere. When located upstream of the Saturnian bow shock, the interaction is very similar to that of Mars (e.g., Brain et al., 2010;Brecht & Ledvina, 2006) or Venus (e.g., Russell et al., 2006;Slavin et al., 1980) with the solar wind . When located in the magnetosheath, Titan has been observed to retain signatures of the Saturnian magnetic field in the conducting ionosphere, something that has been referred to in the literature as fossil fields .
During the Cassini era, it became clear that the north-south orientation of the magnetic field observed during the Voyager 1 flyby (Neubauer et al., 1984) was not common, with only a single flyby reported in the literature, namely, T70 (Simon et al., 2013), having occurred during such conditions. For this reason, different efforts to Using electron data from the Electron Spectrometer (ELS), part of the Cassini Plasma Spectrometer (CAPS), and from the Low Energy Magnetospheric Measurement System (LEMMS), part of the Magnetospheric Imaging Instrument (MIMI), Rymer et al. (2009) identified four different regions according to the characteristic thermal electron environment. These regions are plasma sheet (high energy and density), lobe-like (high energy, low density), magnetosheath (low energy and high density), and bimodal (two superimposed populations). This classification was extended by Smith and Rymer (2014) to include all the data gathered at Titan's orbit (with and without Titan present) by the CAPS/ELS instrument until it was switched off in 2012 after 83 flybys.
A similar classification was made using ion data for the first 54 flybys from the CAPS/IMS instrument by Németh et al. (2011). They also looked at ion composition, finding short events with enhanced heavy ion densities occurring when Cassini crossed the narrow central sheet .
Being located in the outer magnetosphere, the planetary magnetic field configuration at Titan's orbit differs significantly from that of a dipolar configuration. Due to the fast rotation of the planet and the presence of heavy ions, the centrifugal force causes the partially corotating plasma to be confined near the equatorial regions, creating a current sheet that is present at all local times, although with varying thickness . All of this means that the magnetic field can also be used to estimate the location of Titan with respect to the magnetic equator. Using data from the Magnetometer (MAG) instrument, Simon et al. (2010) classified the magnetic environment during the TA to T62 flybys. The classification was later extended to include the data until the T85 flyby (Simon et al., 2013).
More recently, Kabanovic et al. (2017) provided an empirical model of the magnetic environment at the moon's orbit using magnetic field data obtained by Cassini. They found a perturbed field configuration close to noon local time, regardless of the season, while a dependence on the season is present in the nightside due to the change in the orientation of bowl-shaped current sheet.
The energetic plasma environment, specifically protons with energies from 27 to 255 keV, was studied by Garnier et al. (2010) for all the flybys and orbit crossings from Saturn Orbit Insertion (SOI) until January 2008, almost at the end of Cassini's Prime Mission. The then total number of crossings accounted for 39 dedicated flybys and 33 crossings far from Titan. In their work, they analyzed data from the MIMI suite, concentrating on the LEMMS and the Ion Neutral Camera (INCA) instruments. For the former, they looked at the energetic proton data (with energies between 27 and 255 keV) far from the region where the disturbances introduced by Titan are appreciable. By looking at mean fluxes for the channels analyzed and studying their correlation with Saturn local time (SLT), they found larger fluxes in the postmidnight to dawnside.
It is evident from the descriptions provided above that classification of Titan's upstream environment is the least developed in terms of energetic charged particles. The study of Garnier et al. (2010) provides some insights but covers only a small time period and is limited in energy coverage. It provides no detailed information about the shape and intensity of energetic ion spectra. With the other classifications in mind (by the CAPS and MAG instruments) and with the Cassini mission completed, we have the opportunity to update and extend the results of Garnier et al. (2010) providing a more detailed description of Titan's energetic particle environment. The availability of an empirical model of energetic particles is necessary to complement the study of how Titan interacts with the magnetospheric environment and how the energy deposition into the atmosphere changes with upstream conditions. Cassini performed the first dedicated flyby of Titan on 26 October 2004. During this flyby, known as the TA flyby, Cassini traveled, with an altitude at closest approach (CA) of 1,174 km, below the main ionospheric peak, located at around 1,200 km. Cravens et al. (2005) compared the results obtained by the Radio and Plasma Wave Science instrument during this flyby with those obtained using a photochemical model along the track of the spacecraft. They showed that by just considering photoionization (without the inclusion of electron impact ionization), the production rate predicted by the model was significantly lower than what the data showed, making it necessary to consider the magnetospheric input as well. Edberg et al. (2015) looked at the electron densities in Titan's ionosphere at different local times. After filtering out the ionization by solar extreme ultraviolet using a photochemical model, they found higher densities during flybys that occurred around the midnight sector of the magnetosphere and lower densities around noon. They suggested that this difference could be due to ionization from magnetospheric sources.
Using data obtained by the MIMI/LEMMS instrument during the T5 flyby, Cravens et al. (2008) calculated production rates from precipitating H + and O + from the magnetosphere of Saturn. They found that the these ions can contribute to the ionization of the atmosphere at altitudes between 500 km and 1,000 km, below the main ionospheric peak, which is mainly produced by solar extreme ultraviolet radiation, a result confirmed through test particle simulations by Regoli et al. (2016). Gronoff et al. (2009) also analyzed the role of energetic electrons in the ionization of Titan's atmosphere during the T5 flyby and concluded that the geometry of the draped field lines has a significant influence on the local electron fluxes. Similar results were obtained by Smith et al. (2009) by looking at energetic neutral atom (ENA) emissions produced by precipitating protons and detected by the MIMI/INCA instrument.
In terms of neutral particles, Brandt et al. (2012) studied the exosphere of Titan using ENA measurements from the INCA instrument. They found an H 2 exosphere that extends to about 50,000 km and also estimated the precipitation of ENAs to be comparable to that of energetic ions. This led to the suggestion that for any study analyzing the energy budget at Titan's atmosphere, ENAs need to be accounted for.
In this work we analyze the energetic environment at Titan's orbit, with a focus on energetic protons. Using data from the LEMMS and Charge Energy Spectrogram (CHEMS) instrument, we develop a method to characterize the proton fluxes and distribution encountered at Titan's orbit at different locations in the magnetosphere, as well as an empirical model of the energetic proton environment. The results of the present work are particularly interesting for the ionization of Titan's atmosphere and for understanding ENA emissions from Titan's exosphere, as observed by Cassini's INCA detector.

Instrumentation and Data Set
Cassini had a series of instruments devoted to the study of charged particles at different energy levels. Among those, CAPS and MIMI were designed to perform in situ measurements of the fluxes of charged particles with different energies in Saturn's magnetosphere.
CAPS (Young et al., 2004) was composed of three instruments, namely, ELS, the Ion Mass Spectrometer (IMS), and the Ion Beam Spectrometer. Among the three of them, ELS and IMS are of special interest when it comes to characterizing the low-energy environment of the Saturnian magnetosphere. ELS was used to characterize the plasma environment at which Titan was encountered at each of the flybys for which CAPS data are available (Rymer et al., 2009). Due to an electrical failure of the spacecraft, CAPS was switched off shortly after the T83 flyby that took place in May 2012. CAPS covered the low-energy part of the spectrum, with ELS reaching energies of up to 28 keV and IMS reaching energies of up to 50 keV. This was neatly complemented by MIMI (Krimigis et al., 2004), composed by LEMMS, CHEMS, and INCA.
The data used in the present study were collected by the LEMMS and CHEMS instruments. LEMMS was mounted on a rotating platform intended to provide 360 ∘ coverage on the spacecraft's x-z plane. The platform, however, stopped working at the beginning of 2005, leaving the instrument looking into a fixed direction (Krupp et al., 2012). This limited the pitch angle coverage of the instrument, although at Titan's orbit, as reported by Garnier et al. (2010), the ion distribution is quasi-isotropic.
LEMMS consisted of a double-ended telescope. The two ends of the instrument measured different energy ranges, one being labeled as low energy and one as high energy. The low-energy end was able to measure ions with energies between 27 keV and 4 MeV and electrons from 18 to 832 keV, while the high-energy end could measure ions with energies from 1.4 to 160 MeV/N and electrons from 0.1 to several tens of MeV .
For this study, only ion data gathered by the low-energy telescope are used. The corresponding channels on the LEMMS instrument are labeled A0 to A7 and the energy range for each one of these channels is presented in Table 1. For most of the time when Cassini was in the outer magnetosphere near Titan's orbit, the highest-energy channels of the instrument do not measure any fluxes above the detection threshold. For this reason, some of the analyses presented here will focus on the lowest-energy channels.
CHEMS was a mass spectrometer capable of distinguishing between ion species and their charge state, most importantly H + and W + ions, the two major magnetospheric species present at Titan's orbit (e.g., Arridge   three telescopes, each one containing an electrostatic analyzer to filter particles based on their energy/charge followed by a time-of-flight analysis to determine the mass of the detected ion. The central telescope was fairly well aligned with the low-energy telescope of LEMMS. In terms of energies, CHEMS covered the range between 2.8 keV and 220 keV. For all the analyses presented in the following sections of the paper, data collected between SOI and the beginning of 2017 are used. Since the focus of the paper is on the energetic H + environment at Titan's orbit, the data are filtered to cover the L shell range between 19 and 21 R S and the latitude is also limited by only including data collected within 1 R S of the equatorial plane. In general, higher latitudes can be considered (from where fluxes can be mapped to Titan's orbital plane); but due to the ambiguity of magnetic field models for Saturn's magnetosphere at Titan's distance, we restrict our study near the equatorial plane. For some of the analyses, subsets of the described data set are used, as indicated at the beginning of the corresponding section.
When analyzing data collected during dedicated flybys, the interaction region (defined as the region of the magnetosphere affected by the presence of the moon) is removed from the data to ensure that only the magnetospheric environment is taken into account.
In terms of accumulation time, two different sets were used for the study. When analyzing fluxes (sections 3 to 5), 10-min averaged data were used, while for the Kappa distribution analysis (section 6) 30-min averaged data were used. These accumulation times provide enough smoothing of the data while still capturing the possible spatial variations.
Taking into account the spatial constraints just mentioned, it took about 4 hr for Cassini to fly through the region of interest. This translates into approximately 24 data points available for each of the flybys and orbit crossings for the analysis for fluxes and about eight spectra for the Kappa distribution fits.

Dependence of the Energetic Ion Fluxes on the Local Plasma Environment
In order to correctly interpret any results involving the fluxes of energetic ions, it is important to determine whether these fluxes are affected in any way by the location in the magnetosphere at which Cassini was at the time the data were obtained (as is the case for lo-energy ions and electrons). For this, the classification introduced by Rymer et al. (2009) is used, where six different categories were defined, namely, plasma sheet (hereafter referred to as category 1), lobe-like (2), magnetosheath (3), bimodal (4), mixed (5), and unclassified (6). On 1 December 2013, Titan was encountered by Cassini while in the unshocked solar wind  so an extra category, namely, solar wind (7), will be included as well.
The first three categories are, as their names imply, related to the specific locations inside Saturn's magnetosphere. Bimodal spectra contain two different electron populations which were linked by Rymer et al. (2009) to an enhanced pickup ion environment. Mixed means that Cassini crossed more than one environment during the flyby or pass. Finally, unclassified means that the observed spectrum does not fit in any of the first four categories, but at the same time it is not uniform enough as to justify the creation of a new category. Here we look at the ion channel with the lowest energies from LEMMS (namely, A0, with energies from 27 to 35 keV) in order to determine whether a correlation exists between the observed average fluxes and the category in which each of the flybys occurred. We only use the lowest-energy channel since we are interested here in global trends in the magnetosphere. An analysis containing the full distribution of high-energy particles is presented in section 6.
The top panel of Figure 1 shows a plot of the median fluxes of the data from the A0 channel obtained during each of the first 83 flybys plus T85 and T96, and a subset of the orbit crossings for which the environment classification has been published (Smith & Rymer, 2014). Apart from all the flybys and orbit crossings with classification from the CAPS instrument, T85 is included because it was another magnetosheath flyby (Edberg et al., 2013) and T96 because it was the only solar wind flyby recorded by Cassini . The bottom panel shows the number of data points available at each environment. Table 2 shows the median values plotted in Figure 1 together with the number of data points available for each case. The data points correspond to all the spectra collected for the analysis and not to individual events.
Without taking into account the mixed and unclassified data points, from the median fluxes presented in Figure 1 and Table 2, it can be seen that the A0 channel fluxes are indeed influenced by the moon's location, with the solar wind fluxes being the most strongly influenced followed by the plasma sheet ones. Since there is only one flyby occurring in the solar wind, the statistical significance of this result cannot be asserted, although  the fact that the magnetopause and the bow shock boundaries were pushed beyond the orbit of Titan directly implies large solar wind fluxes.
In contrast to what is observed in the thermal plasma data from the CAPS instrument, the fluxes in the magnetosheath are lower than those encountered in the plasma sheet. At the same time, they are slightly higher than those in the lobes, although an important overlap between both classifications exists. From the number of available data points, it is also evident that the quality of the statistics from different environments is different, with the plasma sheet and the lobe regions inside the magnetosphere being the better sampled.
Another way of looking at the data while considering all the data points available is to make a rough distinction between plasma sheet and lobes based on the magnetic field data. This can be achieved by plotting the fluxes versus the ratio between the radial component of the magnetic field and the magnitude of the field. When this ratio is close to zero, it means the magnetic field is almost perpendicular to the orbit plane, meaning that the data were collected close to the center of the current sheet. Figure 2 shows a scatter of all the A0 channel measurements included in the study plotted against B r ∕|B|, together with a second-degree polynomial fit that shows the trend toward higher fluxes closer to the center of the current sheet and lower fluxes at the lobes.
In general, although with some overlap between the lobe-like and magnetosheath regions, the data follow a trend that is expected, with the fluxes in the plasma sheet being higher than those in the lobes. Using a single channel, though, provides an incomplete description of the environment.
In addition to the difference in kinetic energy, the overall shape of the distribution is another factor that makes it possible to use the thermal plasma data for classification but not the energetic data. The Rymer et al. (2009) and Németh et al. (2011) classifications rely on two main factors to define the different categories, namely, flux and energy of the peak of the distribution. While the fluxes can vary for energetic data as well, the peak of the distribution is always located at the lowest-energy channels of the MIMI instrument. It can be seen that no clear seasonal dependence is present, and overall, the fluxes are highly variable. As representative of higher or lower fluxes than the general trend, five specific periods are highlighted in the plot. These correspond to the times listed in Table 3 together with some relevant parameters.

Ion Fluxes Over Time
The three highlighted time periods with high fluxes coincide with two dedicated flybys, namely T32, T78 and T96. T32 was the first flyby of the mission that occurred with Titan inside the magnetosheath. The fact that Titan was located outside the magnetosphere means that the solar wind was energized and, even after being slowed down past the bow shock, the distribution of ions detected by Cassini was energized enough that the LEMMS instrument was able to capture an enhanced flux of particles.
The case of T78 is quite different. While T32 occurred at the noon sector of the magnetosphere, specifically at 13.56 SLT, T78 occurred in the afternoon sector, at 17.55 SLT. Furthermore, the flyby was classified using MAG data by Simon et al. (2013) as having taken place under plasma sheet conditions. At the time of the flyby, CAPS was switched off, so no classification from the low-energy plasma point of view is available (Smith & Rymer, 2014).
The third highlighted point corresponds to the T96 flyby, that took place at a time when the magnetosphere was compressed by the arrival of an ICME, and it corresponds to the only flyby during the Cassini mission when Titan was observed in the solar wind.
High fluxes are also visible for the very first data set which corresponds to the TA flyby. Being the first flyby of the mission, TA was studied in detail using different data sets and simulations and it was found that the fluxes were relatively high, comparable to those observed by Voyager 1 (e.g., Backes et al., 2005;Cravens et al., 2005;Hartle et al., 2006).
For the case of low fluxes, two times are highlighted in Figure 3. Both of them correspond to crossings of Titan's orbit, with the first data point having been collected at 2.16 SLT and the second one at 15.9 SLT. The selection of these data points was solely based on their relatively low fluxes as shown in Figure 3 and, at first glance, no noticeable features that could explain the low fluxes are present.
Given their lack of interaction with the solar wind boundaries, solar energetic particles (SEP) can penetrate the magnetosphere and thus serve as a proxy for the solar wind activity when Cassini was inside the magnetosphere of Saturn. A series of these events was identified by Roussos et al. (2017). The data points for the A0 and A5 channels that fall within the period of time when these SEP events were detected are highlighted in orange in Figure 4.
In general, although SEP events increase the fluxes in the highest-energy channels of the LEMMS instrument (Roussos et al., 2017), the effect does not seem to be as strong in the lowest-energy channels. This is not completely unexpected, given that the ability of a particle to penetrate the bow shock depends on the particle's energy, and those detected by the A0 channel might simply not have the energy needed to do so. This is apparent when comparing the fluxes of the SEP events in both channels shown in Figure 4. While those from the A0 channel show no special trend, the ones detected by the A5 channel do have a bias toward higher fluxes.

Average Fluxes
In this section, the local time dependence of the energetic ion fluxes is analyzed. For this, all the data available from the lowest-energy ion channel (A0) between SOI and the beginning of 2017 are taken into account. Figure 5 shows a plot of the median ion fluxes versus SLT divided into 2-hr bins for the two main magnetospheric environments, namely, current sheet (top) and lobe-like (bottom). Since the data are being divided by magnetospheric environment, only those times listed in Rymer et al. (2009) and Smith and Rymer (2014) are used. Both plots show the number of data points available at each SLT bin. The sampling is rather low for most bins, and for some of them there are no points available at all. The dayside/nightside asymmetry reported by Garnier et al. (2010) is not observable, although their analysis did not discern between magnetospheric environments.  When the environment restriction is removed, the plot shown in Figure 6 is obtained. In this plot an SLT asymmetry seems to be present, with the largest fluxes present between the afternoon and premidnight sector of the magnetosphere. However, the sampling bias observed in Figure 5 shows that many of the measurements taken in the nightside (especially in the premidnight sector, where fluxes seem to be higher) were made when Titan was in the current sheet, a region that, as shown in Figure 1, presents larger fluxes of energetic particles.
Part of the asymmetry in SLT observed in the electron densities in the ionosphere, as reported by Edberg et al. (2015) can be due to this sampling bias. While energetic particles are expected to penetrate below the altitude of Cassini's CA (Cravens et al., 2008;Regoli et al., 2016), part of the incident population could have grazing angles large enough that they will indeed deposit their energy at higher altitudes.

Analysis of Energetic H + Spectra using Kappa Distributions
Kappa distributions were first introduced by Vasyliunas (1968) to describe the spectra of plasma populations that could not be described with Maxwellian distributions in the Earth's magnetosphere and have since then been used to describe populations in other planetary magnetospheres (Dialynas et al., 2017) and plasma environments (e.g., Livadiotis, 2015;Livadiotis & McComas, 2013). In this study we use a simplified version of the original distribution that was described in Dialynas et al. (2009), based on a function introduced by Mauk et al. (2004) for the study of energetic ions in the Jovian magnetosphere.
Due to the energy distribution of particles in the magnetosphere, the MIMI/LEMMS and MIMI/CHEMS instruments capture the high-energy tail of a Kappa distribution centered at low energies that is detected by the CAPS instrument (Dialynas et al., 2009;Young et al., 2005). The high-energy fluxes have been organized by the value of the parameters of a Kappa distribution for the equatorial magnetosphere of Saturn, but no specific analysis for Titan's orbit has been made thus far, apart from the fact that the Dialynas et al. (2009) study included data collected by Cassini only until July 2007. In this section, a modified Kappa distribution function [equation (1)] is used with the aim of providing an empirical model of the fluxes of energetic particles at Titan's orbit. Figure 7 shows two examples of the combined LEMMS and CHEMS spectra used in this study for two different times, together with the corresponding result of the fitting process. As visible from both spectra shown, the Kappa distribution function provides a good description of the plasma population with energies above 20 keV, whereas below that threshold, the distribution seems to be closer to an inverse power law, which is most likely representative of a second Kappa distribution that peaks at lower energies (Dialynas et al., 2009). For instance, Young et al. (2005) showed broad peaks of ions at local corotation speeds. This change in the distribution is particularly visible in the right-hand panel of Figure 7 and it implies that the results presented in this section in terms of an empirical model of the fluxes are only valid for energies above 20 keV.
Due to the already described variabilities of the outer magnetosphere, when analyzing the results of the fitting process the different possible sources of this variability need to be considered. These include location where the data were collected with respect to Saturn (SLT), location with respect to the current sheet (magnetic latitude), and whether the data were collected inside or outside the magnetosphere (only relevant for flybys or passes that occurred close to the subsolar point, or close to 12 SLT).  Since most plasma sources in Saturn's magnetosphere are located in the inner region (with the main source, Enceladus, at a radial distance of 4 R S ), by the time the plasma is transported outward, any SLT asymmetries that could be originally present are smoothed out. This was shown by Dialynas et al. (2013) where they analyzed ion distributions from the MIMI instrument at radial distances between 5 and 20 R S and showed that the inner magnetosphere presents significant structure (especially on the dayside) that decreases with increasing L-shell.   Figure 9 shows the same parameters from the fits, this time organized by plasma environment using, once again, the classification from Rymer et al. (2009).
For the four well-defined environments all the three parameters show a similar trend, with the lowest values present for solar wind conditions, followed by plasma sheet, lobe-like, and magnetosheath conditions, where the maximum values are observed. There is, however, a significant spread in the data, another reflection of the variability of the energetic environment. Table 4 presents the obtained median values of the three parameters. The parameter indicates the shape of the distribution, with a larger value meaning a more thermalized population. For this reason, it is interesting to analyze the value obtained for each environment.
The classification used was obtained using thermal electron data, and thus the link to energetic ions might be weak. This makes it difficult to interpret the results obtained, especially for cases like the bimodal distribution, which is based on an extra source of thermal electrons that might not be linked at all to energetic particles. For similar reasons, the results for the mixed and unclassified environments are only useful in a statistical sense and a physical interpretation is difficult to be derived.
For the other four environments, while the link between the thermal electron data and the energetic particles might be, at best, weak, some physical interpretation can be given to the results, keeping in mind that the statistical uncertainty of the values obtained is significant when compared to the difference between the  values at specific environments. In general, the value of obtained for the solar wind is consistent with results reported in the literature (see, for instance, Table 1.1 in Livadiotis, , Livadiotis, ).
The fact that, on average, the value of at the magnetosheath is larger than that at the plasma sheet and the lobes could be related to a thermalization of the solar wind plasma. However, the shape of the distribution will also depend on external parameters such as the Alfvén mach number and the plasma beta (e.g., Thomsen et al., 2018) at any given point and these values will necessarily be smoothed by the averaging performed in the analysis presented here.
In order to simplify the environment description, we have also looked for correlation between the three spectrum coefficients, so that we can reduce the free parameters of the spectrum from three to one. In terms of interdependence of the parameters, the plots from Figure 10 show the (top panel) and kT (bottom panel) parameters plotted against C.
The black points in both panels show the respective values obtained from the fits described above together with the error bars that are obtained from the quality of the fits. In addition, the two orange curves represent fits to the data.
Both plots show a power law dependence [equations (2) and (3), respectively], with a spread present for large values of C, where most of the points are concentrated (about half of the fits throw a value for C that is larger than 5 × 10 12 ). The goodness of the fit in terms of R 2 for the versus C case is 0.8757 and for the kT versus C case is 0.3672. This difference is already visible in Figure 10 from the larger spread of kT for low values of C.
= −9.915 ⋅ C −0.03768 + 8.421 (2) The combination of the values of C provided for each magnetospheric environment together with the relations from equations (2) and 3 provide a description of the energetic particle fluxes at different environments.

Discussion and Conclusions
During the time orbiting Saturn, Cassini sampled different regions of the magnetosphere gathering enough data to understand part of the dynamics of the plasma circulation. Still, due to the size of the magnetosphere, the coverage is far from perfect. If we add this to the fact of only having a single spacecraft to analyze the dynamic processes that take place in the outer regions of the magnetosphere, characterizing the magnetospheric environment at Titan's orbit is a very challenging task.
Given that about 50% of the particle pressure in the magnetosphere of Saturn is contributed by the energetic particles (Sergis et al., 2009), understanding the fluxes from a statistical point of view is important to provide a 10.1029/2018JA025442 comprehensive model of the interaction of Titan with the Saturnian magnetosphere. In addition, an empirical model of the ion fluxes can help constrain the energy deposition by energetic particles that affect mainly the ionospheric densities below the main ionospheric peak.
While some progress has been made with the magnetic field and low-energy particle data, few studies have focused on the high-energy plasma. In this paper, we analyzed data from the MIMI/LEMMS and MIMI/CHEMS instruments in order to study, from a statistical point of view, the behavior of energetic H + ions. Toward this, we looked at the mean fluxes detected by LEMMS and also at the energy distribution by fitting Kappa distribution functions to the data.
Based on the analyses detailed on the paper, we conclude that the energetic environment is extremely variable. As expected based on prior studies and on this one, when looking at ions in the keV and MeV scale, we are only looking at the high-energy part of the total particle distribution and these particles, due to their high kinetic energy, are not as cleanly organized by their location with respect to the center of the plasma sheet as thermal particles are.
Fluxes in the higher-energy channels are notably increased during SEP events. This increase in the fluxes can be related to acceleration processes arising from the arrival of the SEPs or to the fact that solar wind ions with very high energies can penetrate the magnetosphere, thus reaching the moon even when located downstream of the magnetopause. In both cases, this implies that the energy deposition in the atmosphere might be affected during periods of enhanced solar activity, something that needs to be further analyzed to evaluate how the ionosphere below the main peak is affected.
With the aim of providing an empirical model of the energetic particle environment, instead of looking at the fluxes of a single energy channel, a better approach is to look at the distribution of particles. At the energies analyzed in this study, we see the high-energy tail of a Kappa distribution that peaks at low energies, and a Kappa distribution of an energized population, peaking at approximately 20 keV. In this sense, we derived correlations between the parameters describing this high-energy population as a Kappa distribution function.
While no local time dependance was found, there is a correlation of the Kappa distribution parameters and the magnetospheric environment and also between the different parameters of the distribution. More than half of the periods analyzed have values of C, kT, and within specific ranges. Apart from this, there is a strong power law correlation between and C and between kT and C.