First Adiabatic Invariants and Phase Space Densities for the Jovian Electron and Proton Radiation Belts—Galileo and GIRE3 Estimates

The fluxes and phase space densities for a fixed first adiabatic invariant for high‐energy electrons and protons provide important inputs for various scientific studies for determining the physics of particle diffusion and energization. This study provides estimates of the first adiabatic invariant and phase space density based on the complete and large data base available from the Energetic Particle Detector (EPD) on Galileo for the Jovian environment. To be specific, 10 min averages of the high‐energy electron and proton data are used to compute differential flux spectra versus energy between L = 8 and 25 over the mission. These spectra provide estimates of the differential fluxes and phase space density for constant first adiabatic invariants between 102 and 105 MeV/G. As would be expected, the electron and proton fluxes and phase space densities generally trend lower as the planet is approached. The results indicate that, whereas the overall trends for each orbit are consistent, detailed orbit to orbit variations can be observed. Galileo orbit C22 is presented as an example of deviations from the mean downward trend. To validate the Galileo results and extend the findings into L = 3, the GIRE3 model was also used to compute the fluxes and phase space densities for constant first adiabatic invariant versus L‐shell. Comparison between GIRE3 and EPD demonstrates that the model adequately reproduces the EPD data trends and they consistently show additional variations near Io. This provides proof that the GIRE3 is a useful starting point for diffusion analyses and similar studies.


First Adiabatic Invariants and Phase Space Densities for the Jovian Electron and Proton Radiation
• The long-term dynamics of particle trapping in the Jovian magnetosphere is investigated using high-energy electron and proton data • First adiabatic invariant and phase space densities are computed using the Energetic Particle Detector data as well as using the GIRE3 model • Both electrons and protons show a clear downward trend in flux and phase space density at constant first adiabatic invariant as the planet is approached To be specific, 10-min averages of the EPD electron data channels are averaged to provide omni-directional differential fluxes at 0.238, 0.416, 0.706, 1.5, 2.0, 11.0, and 31 MeV (the latter energy based on Pioneer 10 and 11 measurements) and between 3.2 and 10.1 MeV for protons between L = 8 and 25 for the 34 Galileo orbits. These allow determination of spectra which provide estimates for the differential fluxes and for the PSD for constant first adiabatic invariants between 10 2 and 10 5 MeV/G along Jupiter's magnetic equator.
The results permit studies of long-term overall trends and orbit to orbit variations of these parameters. To illustrate the latter, the Galileo orbit C22 event is studied and provides valuable information on short period time variability.
An important additional tool in the analysis of the radiation environment at Jupiter is the GIRE family of plasma and high-energy particle models (e.g., de Soria-Santacruz et al., 2016Divine & Garrett, 1983;Garrett et al., 2003Garrett et al., , 2005Garrett et al., , 2012Garrett et al., , 2015Garrett et al., , 2016Jun et al., 2019). The latest version of the GIRE3 model  is an amalgam of synchrotron measurements and Pioneer, Voyager, and Galileo in situ data. GIRE3 provides a definition of the electrons, protons, and various heavy ions between 2 Rj and 50 Rj and for energies of a few electron volts to several 100 MeV/nucleon. Here GIRE3 will be used to compute the first adiabatic invariant and PSD versus L over the same range as the EPD data. GIRE3 also permits estimates of these key parameters into 3 Rj and of their variations near Io. Though the agreement between the EPD data and model is not unexpected as GIRE3 is based in part on the EPD data, it provides further proof that GIRE3 is a useful tool for diffusion analyses and similar studies and for evaluating the latest models of losses and sources in the critical inner radiation belts (see, e.g., Woodfield et al., 2014).
This study is divided into two parts. First will be the computation of the 10 min electron and proton fluxes and the PSD for fixed values of the first adiabatic invariant between L = 8 and 25 for the Galileo data. These were broken out by orbit to study temporal variations-an example of which, orbit C22, will be presented.
In the second part, we will carry out a similar analysis using the GIRE3 model between L = 3 and 25. The electron and proton flux and phase space density contours versus energy are also computed to identify the applicable range of the analysis (i.e., between ∼100 keV and 100 MeV for the electrons and ∼600 keV and 100 MeV for the protons). Finally, the GIRE3 model and EPD variations with L will be compared and the results summarized.

Galileo EPD
The primary data source for this analysis is the Galileo APL/JHU EPD Low-Energy Magnetospheric Measurement System (LEMMS) which measures the high-energy electrons and protons from Jupiter orbit insertion in 1995 to the end of the mission in 2005 (Williams et al., 1992). Specifically, the steps undertaken to analyze Jupiter's trapped radiation in the Jovian magnetic equatorial plane in the range 8-25 Jupiter radii (1 Jovian radius = 71,400 km) using the Galileo EPD data are described in this section. First, the 10-min averages of the high-energy particle count rate data were combined with data on the location and magnetic field at the spacecraft-specifically, the position of the Galileo spacecraft and the magnetic field vector as modeled by the VIP4 magnetic field model (Connerney et al., 1998) from L = 8 to 25 (L-shell, rather than Rj, was used in this study as the data are better ordered in terms of the magnetic field). Of the 32 LEMMS channels, the most important ones for radiation modeling are the electron channels B1 ( The constants J 0 , A, B, and E 0 were computed for each 10 min interval using EPD and Pioneer data from the PDS. As will be discussed below, the Equation 1 constants then define a flux spectrum at each position that yields the electron first adiabatic invariant fluxes and the PSD functions. Whereas computing the high-energy electron flux spectra for the EPD data as just described was straightforward, computing the proton flux spectra was more involved. The primary reason is that the high-energy electrons inwards of 25 Rj were found to contaminate the high-energy EPD proton channels except for the B0 3.2-10.1 MeV channel (Jun et al., 2002). A proton flux spectrum in energy between 600 keV and 100 MeV is required to compute the fluxes and power spectral density for a specified first adiabatic invariant at a given point. Our method to do this assumes an appropriate proton spectrum scaled by the measured B0 flux at the point to the energy desired. Fortunately, APL has provided reference proton and heavy ion differential intensity spectra (Mauk et al., 2004) at 13 locations along the Galileo trajectory. These differential spectra, interpolated in L, are assumed to represent the shape of the proton flux distribution along the magnetic equator (Garrett et al., 2015). All the EPD ion data channels between ∼50 keV and ∼50 MeV were simultaneously fit by APL to differential intensity spectra of the form given in Equation 2 (Mauk et al., 2004): where: C, et, kT, γ 1 , γ 2 = parameters for the APL spectral fits to the EPD data at 13 locations (Mauk et al., 2004; As described in Garrett et al. (2015), the selected APL proton spectra were interpolated in L between L = 8 and 25. The resulting spectra were then scaled by the observed B0 flux at the desired point as derived from the PDS data. The estimated B0 fluxes (i.e., J(B0) PDS ) are plotted in Figure 1 and were computed as follows: where: J(B0) PDS = B0 channel isotropic differential proton flux based on observed count rates; (cm 2 -s-MeV) −1 B0(cts) = 10-min averages of the B0 channel available from the PDS; (counts per second) GF = average geometric factor for B0 channel; ∼0.0094 cm 2 -sr between 3.2 and 10.1 MeV (Jun et al., 2002) To scale the proton distribution at an arbitrary energy at a given B0 data location, the GIRE3 model was exploited. As the GIRE3 proton model is based on the APL spectra (Garrett et al., 2015), the spectra given by Equation 2 as interpolated in L are readily recovered at a given location. That is, the GIRE3 differential flux at the desired energy/location was divided by the corresponding GIRE3 B0 flux at 5.69 MeV (the geometric mean of the B0 channel 3.2-10.1 MeV). The result, the normalized flux versus energy at the B0 data location, was then multiplied by the observed B0 flux value to give the corresponding flux at the desired energy. The formula is: where: J ( The requirement to be able to determine the electron and proton spectra at a given location for a specific energy follows from the methods used to determine the first adiabatic invariant and power spectral density. The first adiabatic invariant I in terms of the relativistic momentum, P, and the magnetic field, B, is given by (e.g., McIlwain & Fillius, 1975;Roederer, 1970): The relativistic momentum can be shown to be given in terms of the particle kinetic energy E by: where: Assuming that the Galileo data are close to the magnetic equator and nearly isotropic (Garrett et al., 2012(Garrett et al., , 2015, α is 90°, and B eq is the magnetic field at the equator,   2 eq / 2 o I P m B . Deviations from this approximation for Galileo are expected to be on the order of a factor of 2 or less as a result (McIlwain & Fillius, 1975). Equations 5 and 6 can then be inverted to give: The procedure is to first define a value for I (e.g., 10 2 , …10 5 MeV/G) for the electrons or protons. Equation 7 then gives E for I and B eq where B eq is a function of L-shell and is given for each EPD data point. The differential flux for the electrons or protons at L is next computed for the value of E for the corresponding I and B eq . L-shell and B eq are computed using the VIP4 magnetic field model (Connerney et al., 1998). Results are plotted in Figures 2 and 3 for electrons and protons respectively for all Galileo orbits (note: the EPD data are plotted as dots, the solid and dashed lines are for the GIRE3 model and will be discussed in the next section). For reference, vertical black lines at L = 10 and 22 in Figures 2 and 3 indicate the uncertainty (assumed to be ± one standard deviation of the mean of the log of the fluxes in a ΔL = 2 interval) in the first adiabatic invariant (see also Jun et al., 2005 for a statistical error analysis of the EPD high-energy electron data). This uncertainty varies from a factor of x2 inside L = 12 to a factor of x6 to x10 at L = 24.
The Galileo results are plotted for L values between L-shells of 8-25. The limit at L = 25 is imposed as a result of how the Jovian plasma disc is modeled. While it is straightforward to compute values beyond L∼25, the B eq values in particular are very dependent on the magnetic field model assumed and L loses its relevance. Here the VIP4 model (Connerney et al., 1998) is assumed as opposed to the plasma sheet models of Khurana et al. (2005) which are used in the outer magnetosphere. The magnetic field models agree well inside L = 20 but deviate significantly beyond L∼25 because of the complexities brought about by the Jovian plasma disk.
The EPD data energy range of validity is also bounded as the flux contours for constant I indicate. That is, the EPD electron detectors are most reliable from ∼170 keV (the EPD F1 channel lower energy) to ∼30 MeV (the Pioneer 10 and 11). Even though the proton spectra provided by Mauk et al. (2004) were fit between ∼50 keV and ∼50 MeV, the validity of EPD low energy proton limit is estimated to be ∼0.6 MeV (Garrett et al., 2015). At the high-energy end, we have extrapolated both ranges up to 100 MeV, however, to allow for comparisons with the GIRE3 model inside L = 10.
While the general trend of both the electrons and proton fluxes for the EPD constant first adiabatic invariants is toward lower values as the planet is approached, the electrons decrease by only 1 order of magnitude or less between L = 8 and 25 for 10 4 -10 5 MeV/G while the protons decrease by 2 orders of magnitude. Indeed, the EPD 10 4 and 10 5 MeV/G electron first adiabatic constant fluxes are almost flat over this region and even rise slightly inside L = 10.
The second parameter of interest is the PSD. The PSD for relativistic particles is given explicitly (for constant first adiabatic constant I) by (e.g., McIwain & Fillius, 1975;Roederer, 1970): where: J' = Isotropic differential flux as a function of E; (cm 2 -s-sr-MeV) −1 P' = Relativistic momentum, assumed to be  2 P here; P ⊥ is in units of MeV/c The PSD is very important for determining the sources and losses in the diffusion equation (Woodfield et al., 2014). For Jupiter, it is assumed that the main source of the trapped high-energy particles is the inward diffusion and energization of lower energy particles (assumed to be the high-energy tail of the plasma particles streaming outward in the equatorial plane) from outside 25 Rj. The evidence for this inward diffusion in Figures 4 and 5 is the steady decrease of the PSD as one approaches the planet for both the electrons and protons. This has been reported by many authors for Jupiter such as McIlwain and Fillius (1975), Baker and Goertz (1976), Mogro-Campero and Fillius (1976), Thomsen et al. (1977), Cheng et al. (1983Cheng et al. ( , 1985, and Woodfield et al. (2014). While the electron PSD falls off smoothly with L-shell, a slight flattening of the fall-off in the proton curves is visible where the PSD data have a small inflection between L = 12 and 17 (particularly in the 10 5 MeV/G contour). This is possibly associated with Ganymede near L = 15 and, if the particles are infusing inward, may indicate it is a possible source of particles. There also appears to be a more rapid drop off at L∼9 that may represent absorption of inwardly diffusing particles by Europa. As for Figures 2 and 3, vertical black lines at L = 10 and 22 in Figures 4 and 5 indicate the uncertainty (assumed to be ± one standard deviation of the mean of the log of the fluxes in a ΔL = 2 interval) in the first adiabatic invariant. This uncertainty varies from a factor of x2 inside L = 12 to a factor of x6-x10 at L = 24.
The Galileo data can also be analyzed orbit by orbit. Although each of the orbits studied exhibit unique variations, orbit C22 (e.g., the 22nd orbit of Galileo which was targeted for a Callisto flyby) is the most unusual as a major intensification of the high-energy electrons was observed on Day 223 of 1999 as Galileo approached Jupiter. To illustrate the time evolution of the C22 observations, several of the EPD raw channels used in this study are plotted in Figure 6. The approximate duration of the C22 event is marked. Unfortunately, the EPD was not turned on sufficiently in advance to determine when the event actually started and it may have been in progress well before the data collection began. It is likewise not clear what the source of the event was (e.g., although we suspect it is unlikely, given the lack of earlier data Ganymede can't be ruled out as a source). Also shown in Figure 6 are data from the Galileo Star Scanner which fill in the gaps in the EPD data (note: the Star Scanner also apparently did not record the beginning of the event). The Star Scanner, which was based on a photomultiplier tube, served as an inertial attitude reference for Galileo. Although specifically shielded to limit radiation effects, it was found that, within about 20 Rj, the Star Scanner reliably responded to high-energy particle fluxes. When available, the Star Scanner data typically parallel the DC3 count rates as it is apparently sensitive to high-energy electrons (Fieseler et al., 2002). The Star Scanner data in Figure 6 imply that there were no other "unusual" impulses following the initial one. Finally, the periodic oscillations in the count rates are the results of the oscillations of the Jovian magnetic field-plotting the data in terms of L-shell largely removes these variations.  apparently peak near L∼9 and may be stochastically scattering and accelerating electrons from 0.1 to 1 MeV in this region. Another possibility is that an outwardly propagating enhancement near Io may be shadowed by Europa-there is, however, insufficient data inside L = 9 to decide.

GIRE3 Model Results
In this section, the GIRE3 model will be exploited to evaluate the electron and proton fluxes and phase space densities for constant first adiabatic invariant with the intent of comparing the model and EPD results. The GIRE family of jovian models has been used for some time (de Soria-Santacruz et al., 2016Divine & Garrett, 1983;Garrett et al., 2003Garrett et al., , 2005Garrett et al., , 2012Garrett et al., , 2015Garrett et al., , 2016Jun et al., 2019) to evaluate the radiation environment at Jupiter. The current model estimates the fluxes for the electrons, protons, and various heavy ions between 2 Rj and 50 Rj and for energies from a few electron volts to several 100 MeV/nucleon. Here the model is used to compute differential fluxes between 0.1 and 100 MeV for the electrons and 0.6 and 100 MeV for protons at selected energies versus L. The results in terms of constant first adiabatic invariant are converted to fluxes and PSDs for comparison with the EPD results. These are plotted as overlays in Figures 2-5 as solid lines. As would be anticipated since the GIRE3 model is based in part on the EPD data, there is agreement between L = 8 and 25-the model appears to trace the mean of the EPD 10 min electron and proton constant first adiabatic invariants. The comparisons provide proof that GIRE3 is a useful reference for diffusion analyses and for evaluating the latest models of losses and sources in the critical inner radiation belts.
To investigate the role of the Jovian moons Io and Europa on the first adiabatic invariant flux and the PSD as modeled by GIRE3, the GIRE3 model was run into L = 3. The results are shown as overlays in Figures 2-5 as solid lines. While for electrons the 10 2 , 10 3 , and 10 4 MeV/G first adiabatic invariant fluxes appear to show structure near L-shells associated with Io (L = 5-6) in Figure 2, there is minimal evidence for it in the 10 5 MeV/G ( Figure 2) and the PSD plots ( Figure 4). Indeed the GIRE3 contour plots imply that the PSDs drop off fairly smoothly inside the orbit of Europa into L = 3. In contrast, the protons show definite structure inside L = 8. Both in Figures 3 and 5, there appears to be an initial increase around L = 7 followed by a drop into L = 5-6 followed by another increase and then a fall off inside Io's orbit. There is indeed a notable enhancement in the GIRE3 proton PSD in Figure 5 at L = 5. Io may be acting as a block for the inward diffusing protons and/or as a source for outward diffusing protons. These latter variations, however, may be suspect for portions of the 10 4 and 10 5 MeV/G first adiabatic invariant proton contours as the GIRE3 predictions occur at energies above the 100 MeV proton energy contour (i.e., the dark red/purple dashed line which indicates the approximate upper energy bound of 100 MeV of the GIRE3 model).

Summary and Conclusion
The objective of this study was to investigate the variations in the flux and PSD at constant first adiabatic invariant between L = 8 and 25 in the Jovian equatorial plane for the high-energy electrons and ions over the Galileo mission using the JHU/APL EPD data. Thus, the results reported in this study represents a general long-term trend of particle trapping in the Jovian radiation belts. In addition, the results were compared with the GIRE3 Jovian particle model. The latter allowed the analysis to be extended into L = ∼3. As illustrated in Figures 2-5, the general trend of the electron and proton fluxes and PSDs for a given first adiabatic invariant was to decrease from L = 25 to L = 3. This is generally assumed to indicate the inward diffusion and energization of the electrons and protons and is consistent with current observations and understanding of the sources of the Jovian radiation belts. A new finding in this study is the more rapid fall off GARRETT AND JUN 10.1029/2020JA028593 7 of 10 (factor of 100) in the fluxes or the PSDs of the protons with L as compared to the electrons. While the electrons and protons show gradual changes in slope between Ganymede and Europa, the protons show much higher order variations between L = 10 and 3 (i.e., near the locations of Europa and Io).
The Galileo data set also allows the study of individual orbits. One example, the iconic C22 orbit presented here, shows detail in the first adiabatic invariant contours that is not visible in the overall mission set and in the GIRE3 predictions. As discussed, the proton fluxes and PSDs show little evidence of the initial impulse and, as the planet is approached, appear to be depressed although there is a peak at about L = 12. The electrons, on the other hand, clearly show the impulsive event at L = 15 which drops off rapidly as the planet is approached followed by another impulse at the orbit of Europa.
To conclude, the Galileo EPD high-energy electron and proton data and the GIRE3 Jupiter environmental model both provide useful and consistent information on the variations of the Jovian fluxes and PSDs for constant first adiabatic invariants. While both electrons and protons show a clear downward trend in flux and PSD at constant first adiabatic invariant as the planet is approached, the protons appear to fall off much more rapidly. This may indicate that the high-energy protons are not diffusing inwards as fast as the electrons. While the GIRE3 model predictions provide "mean" predictions of the electrons and protons from L = 25 well into L = 3, the Galileo data permit the study of orbit by orbit variations as exemplified by orbit C22. Finally, the Galileo data and the GIRE3 model potentially provide valuable inputs for studies of inward particle diffusion and energization at Jupiter.