On the residence-time of Jovian electrons in the inner heliosphere
A. Vogt, N. E. Engelbrecht, R. D. Strauss, B. Heber, A. Kopp, K. Herbst
AAstronomy & Astrophysics manuscript no. aanda c (cid:13)
ESO 2020July 1, 2020
On the residence-time of Jovian electrons in the inner heliosphere
A. Vogt , N. E. Engelbrecht , R. D. Strauss , B. Heber , A. Kopp , , and K. Herbst Institut für Experimentelle und Angewandte Physik, Christian-Albrechts Universität zu Kiel, Leibnizstraße 11, D-24118 Kiel,Germanye-mail: [email protected] Centre for Space Research, North-West University, 2520 Potchefstroom, South Africa Theoretische Physik IV, Ruhr-Universität Bochum, Universitätsstr. 150, 44801 Bochum, Germany
ABSTRACT
Context.
Jovian electrons serve as an important test-particle distribution in the inner heliosphere and have been used extensively inthe past to study the (di ff usive) transport of cosmic rays in the inner heliosphere. With new limits on the Jovian source function (i.e.the particle intensity just outside the Jovian magnetosphere), and a new set of in-situ observations at 1 AU for both cases of good andpoor magnetic connection between the source and observer, we revisit some of these earlier simulations. Aims.
We aim to find the optimal numerical set-up that can be used to simulate the propagation of 6 MeV Jovian electrons in the innerheliosphere. Using such a set-up, we further aim to study the residence (propagation) times of these particles for di ff erent levels ofmagnetic connection between Jupiter and an observer at Earth (1 AU). Methods.
Using an advanced Jovian electron propagation model based on the stochastic di ff erential equation (SDE) approach, wecalculate the Jovian electron intensity for di ff erent model parameters. A comparison with observations leads to an optimal numericalset-up, which is then used to calculate the so-called residence (propagation) times of these particles. Results.
Comparing to in-situ observations, we are able to derive transport parameters that are appropriate to study the propagation of6 MeV Jovian electrons in the inner heliosphere. Moreover, using these values, we show that the method of calculating the residencetime applied in former literature is not suited to being interpreted as the propagation time of physical particles. This is due to anincorrect weighting of the probability distribution. We propose and apply a new method, where the results from each pseudo-particleare weighted by its resulting phase-space density (i.e. the number of physical particles that it represents). Thereby we obtain morereliable estimates for the propagation time.
Key words. keyword – keyword – keyword
1. Introduction
During the last decade, solving particle transport equations bymeans of Stochastic Di ff erential Equations (SDEs) became anincreasingly popular tool due to increasing computational power.Whereas many of these studies are focused on Galactic CosmicRays (GCRs), this work builds upon recent research regardingJovian electrons (see Vogt et al. 2018; Nndanganeni & Potgi-eter 2018, and references therein). Since Jupiter is essentiallya “steady state point source” of MeV electrons, Jovian elec-trons present an unique opportunity to study particle transport inthe inner heliosphere. Detailed computational parameter studieshave also become more feasible due to the enhanced computa-tional capabilities acquired by utilizing Nvidia’s Compute Uni-fied Device Architecture (CUDA) as described by Dunzla ff et al.(2015). Here we study the residence time of energetic Jovians inthe heliosphere (see e.g. McKibben et al. 2005, and referencestherein). Since Jupiter is assumed to release electrons contin-uously, these times are not directly measurable, and thereforehave to be derived from theory / modelling. In order to developa reliable estimation of the residence or propagation time, it istherefore necessary to1. determine if the computational setup is realistic and to2. validate the transport parameters against spacecraft data.This work will address both topics.As for modulation studies of GCRs, an essential constraintfor such an investigation is the knowledge of the source spec- trum. The Jovian electron source spectrum has been recently de-termined by Vogt et al. (2018) and is shown in Fig. 1. Knowledgeof this spectrum allows to estimate the e ff ects of various physi-cal parameters on computed Jovian intensities, which can also becompared to spacecraft observations of the same at Earth takenduring periods of good and bad magnetic connection with theJovian source. These comparisons can then provide insight as tothe behaviour of quantities such as the low-energy electron dif-fusion coe ffi cients parallel and perpendicular to the heliosphericmagnetic field, as well as an optimised, realistic parameter set forfurther model computations as proposed in Tab. 2, which is usedto study the residence times of Jovian electrons. We focused oursimulations on 6 MeV electrons during quiet-time conditions.This choice is in agreement with most prior investigations onJovian electrons (see e.g. Kissmann et al. 2004; Nndanganeni& Potgieter 2018, and references therein) as it covers the de-tection range of several particle detection instruments such asUlysses / KET, Voyager 1 / TET, ISEE 3 / ICE, IMP-8 / CRNC andSOHO / EPHIN.In order to estimate the residence times we propose a similarformalism as used to transform the probability densities result-ing from the Transport Equation (TPE) into di ff erential intensi-ties. As this is done by a convolution with the source spectrum(see e .g. Strauss & E ff enberger 2017, and references therein),an equivalent convolution is applied to the simulation times pro-vided by the SDEs code. It is demonstrated that the methodof calculating residence times employed by, e.g., Florinski & Article number, page 1 of 10 a r X i v : . [ phy s i c s . s p ace - ph ] J un & A proofs: manuscript no. aanda
Fig. 1: The Jovian source spectrum according to Vogt et al.(2018).The upper panel shows thesource spectrum as fitted tothe
Pioneer 10
CPI and
Ulysses
COSPIN / KET data. The
Voy-ager 1
TET data added in this plot seems to be in agreement aswell. The lower panel complementary shows the relative devia-tion of the spacecraft data from the fit. The shaded area coversthe ± σ uncertainty.Pogorelov (2009) and Strauss et al. (2013), cannot be interpretedas the propagation time of physical particles. A novel approachis proposed, where the results from each pseudo-particle areweighted by its resulting phase-space density (i.e. the number ofphysical particles that it represents), to obtain estimates for thepropagation time that are more consistent with the limited obser-vational constraints, but also more representative of the propaga-tion time of the physical particles themselves.
2. Scientific Background
The scientific discussion of Jovian electrons dates back to theearly 1970s when McDonald et al. (1972) proposed the exis-tence of a dominant Jovian source by addressing the correlationof the ∼
13 month periodicity in low-MeV electron countingrate measurements at Earth orbit with Jupiter’s synodic period.After the Jupiter flyby of
Pioneer 10 , Teegarden et al. (1974)were able to confirm this hypothesis based on data obtained bythe Charged Particle Instrument (CPI). Pyle & Simpson (1977)showed, by analysing the electron fluxes as function of distanceand their dependence on the occurrence of Corotating Interac-tion Regions (CIRs), that the Jovian source is not only quasi-continuous but also point-like. This refers to the observation thatno electron emission could be detected from Jupiter’s magneto-tail which extends up to over 1 AU into the heliosphere.As the dominant particle population from a few to severaltens of MeV in the inner heliosphere, Jovian electrons soon be-came the subject of charged particle transport modelling (Conlon1978; Zhang et al. 2017). Due to Jupiter’s decentral position, themagnetic connection by means of the Parker spirals determines whether the electrons reach the observer primarily via motionalong the field or by di ff usion perpendicular to it. Thus, Jovianelectrons are ideal candidates to probe the electron di ff usion co-e ffi cients in the inner heliosphere.Jovian electrons were used as test particles to model thecharged-particle transport computationally (see e. g. Chenetteet al. 1977; Conlon 1978; Fichtner et al. 2000; Zhang et al.2007, and references therein) to ascertain the di ff usion coef-ficients parallel and perpendicular to the Heliospheric Mag-netic Field (HMF). This was usually done by comparing com-puted with measured electron intensities at Earth during peri-ods of good / bad magnetic connection. Furthermore, given thedemonstrated sensitivity of computed low-energy galactic elec-tron intensities to various turbulence quantities (see Engelbrecht& Burger 2010, 2013; Engelbrecht 2019), it may be possibleto draw conclusions from Jovian electrons as to the behaviourof those quantities in regions of the heliosphere where space-craft observations of the samedo not exist (see, e.g., Engel-brecht 2017). Since these transport parameters and the di ff u-sion coe ffi cientsdepend on are highly spatially dependent, thetime that particles reside in a certain part of the heliospheremay yield significant insights to the modulation of GCRs aswell. Florinski & Pogorelov (2009) showed this dependency forGCR protons, investigating the time they spend in the helio-tail, the heliosheath and in the solar wind within the terminationshock, respectively. Utilizing both galactic electrons and pro-tons, Strauss et al. (2011b)focused on the connection betweenthe total propagation time and energy losses. They find a sig-nificant non-linear dependency on the total propagation times,which is strong enough to influence also the observations of Jo-vian electrons. These energy losses are entirely caused by adia-batic e ff ects as other possible influences such as particle-particleinteractions are negligible in the TPE due to a lack of signif-icance in the interplanetary medium. As the adiabatic energychanges dE / dt ∼ − / · Eu S W / r are connected to the radial posi-tion, the corresponding energy loss rate per step only depend onthe temporal step size ∆ s and the radial position after the step.The radial direction of the step thereby is irrelevant. This leadsto particles spending more simulation time at small radii losingmore energy and implicitly to a statistical connection betweenthe average energy losses and the particle’s mean free paths. Parallel to the e ff orts of studying Jovian electron transport, theJovian source itself has been investigated intensively. The mainunknowns are the energy spectrum of the source and how theparticles are accelerated. Source energy spectrum:
Although widely considered to bedominant in its energy range, the shape and exact strength ofthe Jovian source remains a topic of debate. A first suggestionwas published as soon as
Pioneer 10 confirmed the existence ofthe source by Teegarden et al. (1974), but the limited amount offlyby data and the general di ffi culties inherent to measuring elec-trons, especially electron spectra, made it di ffi cult to further con-strain both the magnitude and the shape. Suggestions were pub-lished by Baker & van Allen (1976), Eraker (1982), Haasbroeket al. (1997) and Ferreira et al. (2001) based on both Pioneer10
CPI flyby and Earth orbit data. The two Voyagers and Ulyssesare the only spacecraft equipped with particle instruments thatcould resolve the electron spectra above a few MeV (see Heberet al. 2005; Vogt et al. 2018; Nndanganeni & Potgieter 2018, andreferences therein). The latter two published source energy spec-
Article number, page 2 of 10. Vogt et al.: On the residence-time of Jovian electrons in the inner heliosphere tra j jov ( E ) shown in Fig.1 on the base of these flyby data. Vogtet al. (2018) proposed j jov ( E ) = . · m s sr MeV (cid:32) EE (cid:33) − . e − E / E b (1)with E , and E the kinetic and rest energy of the electron, and E b = . α = − .
63 is very much in agreement with the findings ofFerreira et al. (2001) and Baker & van Allen (1976), the shapeproposed in Eqn. (1) is more similar to the suggestions by Tee-garden et al. (1974) and Eraker (1982) and acceleration theoryin itself as Ferreira et al. (2001) proposed a combination of twospectra to fit the spectral break.The
Voyager 1 flyby spectrumas published by Nndanganeni & Potgieter (2018) is included inFig. 1 and supports these results. An overview of the electronspectral data utilized for this study, obtained both during flybysand at Earth orbit, is listed in Tab. 1. Previous studies derived thespectra by solving the particle transport equation and fitting theresults to measurements close to Earth (see for example Moses1987, and references therein).
Acceleration processes:
Since the findings of Bolton et al.(1989) it is widely accepted that the Jovian electrons originate inthe solar wind, are picked up by the Jovian magnetosphere anddi ff use inwards. In situ data has been obtained by spacecraft suchas Pioneer 10 , Galileo and
Cassini and found to suggest acceler-ation via wave-particle interaction within the radiation belts, asdiscussed by Horne et al. (2008), alongside adiabatic processessuggested by de Pater & Goertz (1990). Because the existenceof the Jovian source seems to be linked to the planet’s extendedand strong magnetic field, the question as to whether these pro-cesses also apply close to Saturn is also relevant (see e.g Lange& Fichtner 2008). Recently,
Cassini measurements (as reportedby Palmaerts et al. 2016; Roussos et al. 2016) revived this discus-sion and support, together with
Galileo data, a dominant role ofadiabatic processes within both planets’ magnetospheres in or-der to accelerate MeV electrons (see Kollmann et al. 2018, andreferences therein).Due to the recent measurements of electrons outside the he-liosphere (Stone et al. 2013), the question of how to distinguishthe Jovian population from the Galactic background, based onsuggestions for the very local interstellar spectrum (VLIS) is dis-cussed by Bisscho ff et al. (2019, and references therein). Apartfrom its implications regarding the modulation of GCRs, Nndan-ganeni & Potgieter (2018) find the Jovian population dominatingthe spectrum up to energies of about E ∼
25 MeV, followed byan energy range of 25 MeV ≤ E ≤
40 MeV where the spectrumconsists of a mixture of Jovian and Galactic electrons varied bythe co-longitudinal dependency of the Jovian intensities. Regard-ing the radial dependency, Jovian electrons were found to be thedominant populations up to radial distances of R ∼
15 AU, basedon simulations of the 6 MeV Jovian and Galactic electron inten-sities.
3. Calculating Differential Intensities
For the purpose of this study, the SDE code as discussed by Dun-zla ff et al. (2009) was utilized, which is based on previous codesby e.g., Strauss et al. (2011a,b). This SDE solver is written inCUDA to optimise on performance time, and therefore incorpo-rates a simplified analytical approach for both the solar wind ve-locity as well as for the HMF. The solar wind velocity is chosen Fig. 2: Sketch of the simulation setup as detailed in Dunzla ff et al. (2015). Color coded the four di ff erent exit possibilities areshown: The Sun (yellow), the Jovian magnetosphere (orange),the Heliopause (blue) as well as the option the phase space tra-jectory is terminatedby reaching a pre defined end time (green).Note that the figure is not to scale.to be u S W =
400 km / s and directed radially outwards. The HMFis assumed to be Parker-like and included geometrically withinthe di ff usion tensor according to Burger et al. (2000). Thereforethe results of this study are applicable to solar minimum condi-tions.It has been established, from a modelling (e.g. Potgieter1996; Burger et al. 2000; Ferreira & Potgieter 2002) as well asfrom a theoretical perspective (e.g. Bieber et al. 1994; Engel-brecht & Burger 2010), that drift e ff ects can be neglected whenstudying the transport of electrons with energies of a few MeV.Inorder to optimize the performance time of the code, the energy-independentapproach of Dunzla ff et al. (2015) and Strauss et al.(2011a) is used to include the parallel and perpendicular meanfree paths: λ (cid:107) is normalized to a value of the parallel mean freepath λ at 1 AU, such that λ (cid:107) ( r ) = λ (cid:32) + rr (cid:33) . (2)This value of the parallel mean free path, together withthe particle speed ν , scales the parallel di ff usion coe ffi cient as κ (cid:107) ( r ) = νλ (cid:107) ( r ) /
3, and the perpendicular di ff usion coe ffi cient viathe proportional factor χ such that κ ⊥ ( r ) = χκ (cid:107) ( r ) . (3)Although the above expressions represent an essentially adhoc approach to modelling di ff usion parameters, the radial de-pendencyof the parallel di ff usion coe ffi cient does reproduce thecorresponding behaviour of the quasi-linear theory electron par-allel di ff usion coe ffi cient employed by Engelbrecht & Burger(2013), between 1 and 5 AU. Perpendicular mean free path ex-pressions from theory, however, can behave in a manner quitedi ff erent from what is assumed in this study (see, e.g., Shalchi2009; Engelbrecht & Burger 2015; Gammon & Shalchi 2017). Article number, page 3 of 10 & A proofs: manuscript no. aanda
Table 1: Overview of the electron spectra used in this study.Location of observation Spacecraft mission / Instrument SourceFlyby Pioneer 10 / CPI Teegarden et al. (1974)Ulyssses / KET Heber et al. (2005)Voyager 1 / TET Nndanganeni & Potgieter (2018)Earth orbit SOHO / EPHIN Kühl et al. (2013)(well connected) Ulyssses / KET Heber et al. (2005)ISEE 3 / ICE Moses (1987)Voyager 1 / TET Nndanganeni & Potgieter (2018)Earth orbit ISEE 3 / ICE Moses (1987)(badly connected)For a preliminary study, however, the above expression for κ ⊥ should provide a reasonable approximation that meets the re-quirements for the scientific tasks investigated herein. This as-sumption is confirmed later within this study as the approachleads to reasonable results for the limited energy range consid-ered.As described in great detail by e. g. Kloeden & Platen (2011),the stochastic nature of di ff usion is treated within the SDEmethod as a Wiener process dW t = ζ √ dt with ζ being a vector ofGaussian distributed random numbers. The resulting set of fourintegral equations, as they are derived by Strauss et al. (2011a),is solved iteratively by applying the Euler-Maruyama scheme.This leads to a random walk type solution which is terminatedif a spatial or temporal boundary is reached, as shown in Fig. 2for various possible exit positions. A time-backward phase-spacetrajectory can either terminate at the assumed position of the he-liopause (blue line), the Jovian magnetosphere (orange line), orafter a pre-specified number of steps have been made withoutan encounter with either of the formerly mentioned structures(green line). As discussed in Sec. 2.2, in the low-MeV energyrange Jovian electrons dominate the spectrum. The solar popula-tion therefore can be assumed to be negligible during quiet times,and thus trajectories exiting at the Sun are discarded. Galacticelectrons on the other hand are considered via the local interstel-lar spectrum according to Potgieter & Vos (2017), although thetransport of the Galactic population into the inner heliosphereis much more sensitive to drift e ff ect, which are not consideredhere.Instead of solving Parker’s TPE directly to obtaina time de-pendent distribution function f ( r , t ) covering the whole phasespace, the SDE method provides a chain of point-like solutionsboth in space and time. The solutions must be sampled at a largenumber of phase-space points to approximate a spatial solutioncovering the phase space of interest. In order for the computa-tional results to be comparable with spacecraft data, the time-backward setup is solved as derived and discussed by e.g. Koppet al. (2012). A comparison between the the time-forward andtime-backward setup, made using a simpler 1D model, can befound in the review by Strauss & E ff enberger (2017).The left panel of Fig. 5 shows histograms of the so calledexit energy (i.e. the energy of which pseudo-particles leave thecomputational domain)at times of good magnetic connection be-tween the Jovian magnetosphere and the observational point, incontrast to Fig. 5b, illustrating the energy distribution for an ob-servational point in opposition to the Jovian source. The distribu- Note that the term exit energy refers to a time-backward simulationsetup. Therefor the exit energy describes the energy the particle wouldhave been emitted with at its source - which is in the case of this studythe Jovian magnetosphere. tions in Fig. 5b are much broader. This of course is in agreementwith Fig. 5a being dominated by the much more e ffi cient par-allel di ff usion along the nominal Parker spirals, while pseudo-particles reaching the Jovian magnetosphere in Fig. 5b must doso via the more ine ffi cient perpendicular di ff usion process. Theseparticles su ff er much more scattering, leading to a more di ff usedistribution.Taking the integral distributions (dashed lines) into accountit shows that these extreme particle trajectories correspondingto high exit energiescontribute very little to the total di ff eren-tial intensity. In case of of good connection Fig. 5a suggest thatthe pseudo-particles with exit energies below 10 MeV contributeabout 90 % to the total di ff erential intensity whereas Fig. 5bshows that in case of bad connection they still make up about80 %. Therefore, we conclude that although the trajectory ofeach pseudo-particle is (mathematically) equally likely, they donot contribute equally towards the di ff erential intensity. This isequivalent to stating that di ff erent pseudo-particles represent dif-ferent amounts of physical particles corresponding to their phys-ical significance.The question now becomes: If each pseudo-particle repre-sents a di ff erent number of real particles, why should we weighthe pseudo-particles equally when calculating physical quanti-ties , such as the residence time,from their distribution? A major aspect of modelling physical processes is to simplify thecomputational setup in order to save resources and time withouta ff ecting the physical validity of the results. Therefore the vari-ability of our results was tested against various model set-ups inorder to find the most optimal modelling scenario.The choice of time increment ∆ s has the most obvious in-fluence on the simulation results due to its association with theWiener process and hence with the mathematical representa-tion of di ff usion. Both the e ff ects of the size of the time incre-mentsand the have been tested. For ∆ s a high sensitivity to themagnetic connection was observed as expected since the timeincrement (which is related to the adiabatic energy changes) in-fluences the duration of the random walk. Thereby it is importantto note that the e ff ect is more significant for observation pointsof good magnetic connections. As the trajectories for bad con-nections are much more di ff usion dominated the e ff ect of thesize of the time increment is more likely to be cancelled outby the number of timesteps. For good connections, however, thetrajectories are much more dominated by the geometry of theunderlying HMF. Therefor larger time increments cause largerabbreviations and subsequently longer simulation times. It wasfound that the simulated di ff erential intensity converges for val- Article number, page 4 of 10. Vogt et al.: On the residence-time of Jovian electrons in the inner heliosphere N o r m a li z e d D i ff . I n t e n s i t y Corotating Jupiter
Init. Energy Static Jupiter E init =6.1 MeV E init =7.4 MeV E init =9.0 MeV E init =11.0 MeV E init =13.5 MeV E init =16.5 MeV E init =20.1 MeV Longitude φ / ◦ j c o r o t a t i n g / j s t a t i c Fig. 3: Di ff erential intensities of Jovian electrons for di ff erentinitial energies E init . The top panel shows the longitudinal vari-ation due to varying connection for a co-rotating Jovian source,while the middle panel shows the case of a static Jupiter. The bot-tom panel shows the ratio of the results of the two approaches.ues of ∆ s ≤ .
001 in program units (about 0 .
004 days). Thenumerical dependence on the maximal duration of simulation-appears to be very weak (above a certain limit), with a requireddurationof about 700 to 800 program unitsin order to avoid devi-ations of more than 20 . This leads to a total time of simulationsof (cid:80) ∆ s ≥
300 days. As the numerical step sizes of the pseudo-particles are ultimately dependent on the choice of the energytransport parameters (which influences the di ff usive step size),there is a slight energy dependence on the optimal size of thetimestep.Due to the fact that Jovian electrons originate inand populatemainlythe inner heliosphere, it seems natural to raise the ques-tion as to whether a model heliosphere could be restricted in sizeso that calculation results remain una ff ected. It was found thatresults become insensitive to the size of the model heliosphereif R HP ≥
80 AU is used. For smaller radii the resulting di ff er-ential intensities increase exponentially, regardless of magneticconnection. Therefore we use a value of R HP =
120 AU in or-der to assure convergence, and, as reported by e. g. Gurnett et al.(2013), this isthe radial distance at which
Voyager V φ = Ω S un · r .Fig. 3 shows the calculated Jovian electronintensities, at Earth, for di ff erent energies as a function of longi-tude. Simulations were performed for both a co-rotating coordi-nate system(top panel) and for the case of a static Jupiter (bottompanel). The bottom panel shows the ratio of these two scenariosand proves that the ratio between the di ff erential intensities fora co-rotating and a static approach appears to be significant. Weconclude that Jovian co-rotation must be included in the modelto produce reasonable results. We concluded that the co-rotation of the Jovian source is an important e ff ect to incorporate into thenumerical model. Fig. 5 shows that only lower-energy pseudo particles (with exitenergies near the initial energy) contribute significantly to par-ticle intensity at Earth. As this energy range only covers an or-der of magnitude, the energy dependence of the mean free pathsboth parallel and perpendicular can be neglected for the meansof this study. Note, however, that the resulting di ff usion coe ffi -cients, which are the parameters used in the model, might stillhave some energy dependence due to their dependence on theparticle speed ν .Thus, to estimate λ , the Jovian spectral data at Earth orbitas listed in Tab. 1 were compared to the results of a parame-ter study covering the parameter space in question, namely λ (cid:107) and λ ⊥ . As already discussed in Sec. 2.1, the decentral positionof the Jovian source allows one to estimate the e ff ectiveness ofparallel and perpendicular transport dependent on the observa-tional point’s magnetic connection with Jupiter. In order to en-sure consistency, λ (cid:107) was determined when investigating the caseof best magnetic connection between the observational point andthe source, before determining χ which scales λ ⊥ with respect to λ (cid:107) according to Eqn. (3) and therefore dominates the results incase of magnetic opposition.The upper panel of Fig. 4a shows the available Jovian elec-tron spectral data at Earth orbit during times of good magneticconnection. For a more detailed discussion with regards to theJovian source spectrum see Vogt et al. (2018) or the individualpublications listed in Tab. 1. The black dashed line marks theshape of the Jovian source as fitted to the Earth orbit data in or-der to have a measure to define the deviation. Color coded in allthree panels are the results of the simulations with varying paral-lel mean free paths covering the range of λ (cid:107) = [0 .
05 AU , . .
05 AU, and a value of χ = .
01 in agreementwith the suggestions by Palmer (1982) as well as Bieber et al.(1994) and succesfully implemented in previous studies by (e.g.Strauss et al. 2011b; Vogt et al. 2018, amongst others) is used.The results are shown in the two upper panels of Fig. 4a; thetop panel depicting the simulated intensities whereas the middlepanel shows their relative deviation from the nominal spectrumprovided by the fit. The grey area marks the range of a ± . λ (cid:107) , leading to simulation results within this ± . λ (cid:107) / ⊥ .The energy rangeis chosen in order to cover the range of exit energies contributingto the total di ff erential intensity for an inititial energy of 6 MeVaccording to Fig. 5. As it appears a value of λ (cid:107) = .
15 AU seemsto match the observations best over the whole energy range in-vestigated herein. This is an additional motivation for our choiceof an energy-independent mean free path.Fig. 4b shows the relative deviation of simulations withvarying χ = [0 . , .
04] and for two values for λ (cid:107) = [0 . , .
15] AU. The uppermost panel of Fig 4b shows the spec-tral data at Earth orbit according to Tab. 1, specifically obtainedduring times of bad magnetic connection between the spacecraftand the Jovian magnetosphere. Similar to Fig. 4a, the dashed line
Article number, page 5 of 10 & A proofs: manuscript no. aanda -2 -1 D i ff . E l e c t r o n I n t e n s i t y / ( m − s − s r − M e V − ) Model Parameters ∆ t = . min Part. = v SW = . km/sData Fit by I ( E )= I · ( E/E ) γ exp( − E/E break ) Ulysses
COSPIN-KET, 1991
ISEE , 1979-1984SOHO-EPHIN
Voyager BSe, 1977 R e l a t i v e D e v i a t i o n λ =0.05 λ =0.075 λ =0.1 λ =0.125 λ =0.15 λ =0.175 λ =0.2 λ =0.225 λ =0.25 λ =0.275 λ =0.3 λ =0.325 λ =0.35 λ =0.375 λ =0.4 Energy E / MeV M e a n F r ee P a t h λ / / A U ± . Relative Deviation (a) Simulated Jovian electron spectra at the longitudinal point of best ef-fective magnetic connection for di ff erent values of λ (cid:107) = [0 . , .
4] AU.The top panel shows a data fit (dashed line) along with the simulationsresults and spacecraft data obtained during corresponding time periods.The second panel displays the simulations’ relative deviation from thefit. In order to estimate the agreement, the ± . λ (cid:107) within the margin of lessthan ± . -2 -1 D i ff . E l e c t r o n I n t e n s i t y / ( m − s − s r − M e V − ) Data Fit by I ( E )= I · ( E/E ) γ exp( − E/E break ) ISEE , 1979-1984SOHO-EPHIN R e l a t i v e D e v i a t i o n χ =0.005 χ =0.0075 χ =0.01 χ =0.0125 χ =0.015 χ =0.0175 χ =0.02 χ =0.0225 χ =0.025 χ =0.0275 χ =0.03 χ =0.0325 χ =0.035 χ =0.0375 χ =0.04 χ = λ / λ λ (1 AU )=0 . AU ± . Relative Deviation R e l a t i v e D e v i a t i o n Energy E / MeV χ = λ / λ λ (1 AU )=0 . AU (b) Simulated Jovian electron spectra at the longitudinal point of leaste ff ective magnetic connection for di ff erent values of χ = [0 . , . ff erent values for λ (cid:107) . The fit (dashed line) was performed utilizing spacecraft data by ISEE3 , the only spectral data for magnetically poorly connected observationtimes as given by Tab. 1. The second and fourth panels are equivalentto the second of Fig. 4a, whereas the third and fifth panel correspond tothe third of Fig. 4a showing the possible ranges for χ . Fig. 4: Simulated Jovian electron spectra used to optimize the values of λ (cid:107) and χ . The lower panels show the corresponding relativedeviations from the fit (dashed lines)and the resulting best fit behaviour for both quantities, giving an estimation of the energydependency of λ (cid:107) / ⊥ .indicates the spectral shape of the source when fitted to the Earthorbit data. In this case the data are presumably dominated by thee ff ects of perpendicular di ff usion. As already discussed above,larger values of λ (cid:107) lead to more e ff ective di ff usion and thereforeto increased di ff erential intensities at the observational point.This relation is also present in the results depicted by Fig. 4b,as higher values of λ (cid:107) demand smaller values of χ in order to fitthe data. Although the estimation of parallel and perpendicularmean free paths based on turbulence theory is still an ongoingtopic of research, previous studies suggest values between 0 . . λ (cid:107) (Palmer 1982; Bieber et al. 1994) which weretested successfully for electrons in the inner heliosphere such bystudies such as (e. g. Ferreira et al. 2003; Ferreira 2005; Strausset al. 2011a; Dröge et al. 2016, amongst others).Although val-ues of λ (cid:107) = .
175 AU themselves would fit well within thisrange, theywould demand values of χ out of a realistic rangePalmer (see e.g. 1982); Bieber et al. (see e.g. 1994); Strauss et al. Table 2: The computational and physical parameters used for thecode (when not stated otherwise).Computational Parameters ∆ t T End R HP
120 AU u S W
400 km / s λ (cid:107) (1 AU) 0 .
15 AU χ = λ ⊥ /λ (cid:107) . E init Article number, page 6 of 10. Vogt et al.: On the residence-time of Jovian electrons in the inner heliosphere
Taking the best fitting χ for λ (cid:107) = .
15 AU (as the best fit-ting value for λ (cid:107) according to Fig. 4a)into account, a value of χ = . χ is somewhat smaller than the range expected fromPalmer (1982), where 0 . ≤ χ ≤ .
08, but well within the rangeof values commonly used in numerical modulation studies (see,e.g., Ferreira et al. 2001; Engelbrecht & Burger 2013; Nndanga-neni & Potgieter 2018). Therefore, unless otherwise indicated,these parameters were utilized throughout this study as summa-rized in Tab. 2.
As already pointed out above, the phase space trajectories ob-tained by the SDE solver are mathematical solutions and, in con-trast to a physical interpretation, each phase space trajectory isequally probable. However, not each phase space trajectory rep-resent the same number of physical particles. Although they areoften referred to as pseudo-particles, this term is slightly mis-leading because the phase space trajectories represent the evolu-tion of the particle density distribution f along a curve throughthe heliosphere and has no connection with the trajectory of ac-tual charged particles in a turbulent plasma.The TPE is solved by integrating the SDEs via the time-backward Euler-Maruyama scheme which leads to an increaseof the pseudo-particle’s energy E i due to the inverse adiabaticprocesses (adiabatic energy losses treated in a time-backwardsfashion). Applying therefore the source spectrum j jov ( E ) as theboundary weight (see, e. g., Strauss et al. 2011a; Kopp et al.2012) leads to the expression j ( r , E ) = (cid:80) Ni = j jov ( E exiti ) N (4)with the distribution function f as the solution of the TPErelated to the di ff erential intensity j = P f , with P = pc / q be-ing the particle rigidity, depending on the momentum p and thecharge q .Eqn. (4) can also be derived by calculating the Green’s func-tion G ( x i , s = T ) for any given boundary or initial condition inorder to solve the convolution (see, e.g., Pei et al. 2010) f ( x , T ) = (cid:90) T (cid:90) x G ( x (cid:48) , t ) f b ( x (cid:48) , t ) dx (cid:48) dt (5)with f b ( x (cid:48) , t ) denoting the boundary value. As discussed byStrauss & E ff enberger (2017), neglecting the initial conditionsimplifies Eqn. (5) to f ( x , t ) = (cid:90) t (cid:90) x ∈ Ω b f b ( x , t ) ρ ( x , t ) d Ω dt (6)with x = ( r , E ), f b ( x , t ) representing the boundary condition(i.e. the particle population’s source distribution), ρ ( x , t ) beingthe conditional probability density and f ( x , t ) the distributionat the observational point. Focusing on steady state solutionswhere t → ∞ ⇒ ρ ( x i , t ) : → ρ ( x i ), as well as calculating f i ( x , t )for each phase space trajectory ρ i individually reducing the spa-tial boundary of the integration domain Ω b to the exit position r exit as well as the range of the energy coordinate to E exit , thenleads to the expression as given by Eqn. (4). Fig. 5 shows the probability density (binned distributionof pseudo-particles, N i / N ; red) and the elements of Eqn. (4)( j jov ( E exiti ); blue) for an initial energy of E = ff erentlyto the total di ff erential intensity; from the figure it is evidentthat pseudo-particles may reach the Jovian magnetosphere withseveral tens of MeV (which implies significant energy loss inthe normal time-forward scenario). However, the blue histogramhints that only pseudo-particles with up to 10 MeV contributesignificantlyto the total di ff erential intensity. The integral dis-tributions (dashed line)for both cases further support this as-sumption as they converge roughly at the exit energies for whichthe fractional contribution of the di ff erential intensity gets lowerthan the fractional contribution of pseudo-particles.Looking atjust the distribution of the pseudo-particles may therefore bemisleading if we are interested in deriving physical quantitiesfrom the SDE method. It is important to note that, even if somepseudo-particles contribute very little to the di ff erential inten-sity, they cannot be disregardedin the calculations as they stillcontribute to the denominator of Eqn.( 4).
4. Residence Times
Because the Euler-Maruyama-Scheme (as well as alternativemethods) utilizes a well-defined time increment ds in order tosolve the system of integral equations, it is possible to derive ameasure τ as the time that the phase space trajectory takes toconnect the source to the observational point. Thereby τ is cal-culated as the number of time steps needed for the phase spacetrajectory n i multiplied by the time increment ds , if the lattertime step is assumed to be constant. Generally τ can be givenas τ i = | s exiti − s i | , with s exiti = n i · ds and s i often being simplythe start time of the calculation. Although τ i is often referred toas the propagation time or as the residence time too, the moreaccurate description would be to call τ i the trajectory duration orsimulation time.This estimation raises the question of how to interpret the in-dividual phase space trajectories. As discussed in Sec. 3, phasespace trajectories are equally mathematically possible but notequally physically significant. If we now want to derive physicalquantities from these trajectories, how should we weigh the re-sulting distributions? We aim to address this question in the nextparagraphs. Traditionally, the residence or propagation time is assumed to beequivalent to the expectation value of the time, as weighed by aprobability density ρ , τ = (cid:82) ρ ( x , t ) tdt (cid:82) ρ ( x , t ) dt . (7)Usually, in previous work, ρ is constructed from the SDE so-lutions as the (normalized) distribution of the pseudo-particles’exit time (see e. g. Florinski & Pogorelov 2009; Strauss et al.2013). For 6 MeV Jovian electrons, these distributions are shownin Fig. 6, as the red histogram, for the case of good (left) and bad(right) magnetic connection to the source. From the red distribu-tion, a propagation time of ∼ −
600 days is calculated, andfrom the figure itself, we note that most pseudo-particles onlyreach the observer within ∼ − Article number, page 7 of 10 & A proofs: manuscript no. aanda
10 206
Exit Energy E exit / MeV F r a c t i o n O f J o v . D i ff . I n t e n s i t y ( b l u e ) P h a s e Sp a c e T r a j e c t o r i e s ( r e d ) X Phase Space Trajectories =
Diff. Intensity j Jovian = . m − s − sr − MeV − N o r m . I n t e g r a l D i s t r . O f J o v . D i ff . I n t e n s i t y (a) Distribution of exit energies E exiti , simulated with an obser-vational point at Earth orbit, magnetically well connected to theJovian source
10 206
Exit Energy E exit / MeV F r a c t i o n O f J o v . D i ff . I n t e n s i t y ( b l u e ) P h a s e Sp a c e T r a j e c t o r i e s ( r e d ) X Phase Space Trajectories =
Diff. Intensity j Jovian = . m − s − sr − MeV − N o r m . I n t e g r a l D i s t r . O f J o v . D i ff . I n t e n s i t y (b) Distribution of exit energies E exiti simulated with an observa-tional point at Earth orbit magnetically poorly connected to theJovian sourceFig. 5: Binned distributions of the exit energies E exiti (red) and the binned contribution of each energy point to the total di ff erentialintensity, j jov ( E exiti ) (blue), for an initial energy of E initi = ff erential intensity as shown in blue. Exit Time s exit / Days F r a c t . C o n t r . O f J o v . P r o p . T i m e ( b l u e ) M e a n S i m u l a t i o n T i m e ( r e d ) Averaged Sim. Time X s exiti /N = . DaysResidence Time τ = . Days N o r m . I n t e g r a l D i s t r . O f J o v . R e s i d e n c e T i m e (a) Distribution of exit times of pseudo-particles for a good mag-netic connection between the observational point and the Joviansource. Exit Time s exit / Days F r a c t . C o n t r . O f J o v . P r o p . T i m e ( b l u e ) M e a n S i m u l a t i o n T i m e ( r e d ) Averaged Sim. Time X s exiti /N = . DaysResidence Time τ = . Days N o r m . I n t e g r a l D i s t r . O f J o v . R e s i d e n c e T i m e (b) Similar to the left panel, but now for the case of poor mag-netic connection between the observational point and the Joviansource.Fig. 6: Binned distributions of the Jovian electron propagation time using the normal (red) and new (blue) way of defining theprobability distribution. The blue distribution thereby shows the weighted exit times, which we refer to as the propagation (residence)time of Jovian electrons. Again the integral distribution of the pseudo-trajectories contribution to τ is shown by the dashed line. Theresidence times are obtained via Eqn. (10).long for relativistic electrons di ff using only a radial distance of ∼ ff usion barrier between Earth and Jupiter these authors findthat it takes ≈ ff usion barrier has passed Jupiterfor the Jovian electrons to be detected again.Thenumericaldefinition of τ according to Eqn. (7)essentiallyweighs each pseudo-particle with the same probability. How- ever, we have already seen earlier in this paper that each pseudo-particle does not represent the same number of physical parti-cles. We therefore propose to rather use the distribution of parti-cle density to calculate the propagation time by specifying ρ ( x , t ) = f ( x , t ) f ( x ) (8)which is normalized by the total phase-space density Article number, page 8 of 10. Vogt et al.: On the residence-time of Jovian electrons in the inner heliosphere f ( x ) = (cid:90) f ( x , t ) dt . (9)Thereby f ( x , t ) represents the solution of the TPE and is cal-culatedat the exit position x exit of the random walk. Note that f ( x exit ) cancels in the calculation of τ , which in discrete formreads as τ ( r , E ) = (cid:80) Ni = s ( E exiti ) · f ( E exiti ) (cid:80) Ni = f ( E exiti ) , (10)and s is the exit (integration) time of the pseudo-particles.Using this new definition, their weighted contribution to thepropagation times are shown in Fig. 6 as the blue distribution.The parameters used for this simulation are listed in Tab. 2 asdiscussed in Sec. 3.2.These weighted propagation times are gen-erally much shorter than those obtained by weighing the pseudo-particles equally. Similar values as the ones found by this studywere only obtained using larger mean free paths (compare e.g. Strauss et al. 2013) which would turn out to be unrealisticas discussed in Sec. 3.2 as our findings are in agreement withprior studies on electron mean free paths such as (Palmer 1982;Bieber et al. 1994; Tautz & Shalchi 2013, amonst others) andmodelling approaches by (e g. Potgieter & Ferreira 2002; Dröge2005; Strauss et al. 2011a, amongst others).By integrating theblue histograms, we find propagation times of ∼ −
11 days,almost two orders of magnitude shorter than the traditional cal-culation. Again, the comparison with the red histogram showingthe pseudo-particle trajectories’ exit times makes this di ff erencecomprehensible. The integral distribution of the blue histograms(dashed lines)illustrate that the maximum of the exit times distri-bution (red) is almost a magnitude higher than the range of con-vergence. Normalized to the value of the residence time, the in-tegral distributions ( like for the di ff erential intensities as shownin Figs. 3) illustrate how little long exit times corresponding tolarge exit energies influence the residence times if calculated viaEqn. (10).Thus leading to the disparity between the averaged exitor simulation times and our estimations of the residence timeequivalent to the di ff erences between the two distributions.As we have shown and discussed in Sec. 3.3, this is causedbythe diverging physical significance of the phase space trajecto-ries.As Eqn. (10) addresses and solves this problem,we considerournew calculation as being more representative of the propa-gation time of a physical particle. This is supported by the factthat Eqn. (10) considers the exit or simulation times according totheir representation of physical particles and therefore providesa measure consistent with the total di ff erential intensity.
5. Discussion and conclusions
In this paper we have discussed a Jovian electron transport modelthat was used to study the propagation times (residence times) of6 MeV electrons.We have discussed the optimal numerical set-up, includingthe choice of time step, integration times, and the required sizeof the numerical domain. We have also shown that taking into ac-count the co-rotation of the Jovian source leads to non-negligiblee ff ects. This is an important e ff ect due to its influence on the fi-nite propagation time of Jovian particles from the observer (as-sumed at Earth, but at di ff erent longitudes) to the source. Fig. 7shows how the calculation of, for example, the propagation time, R e s i d e n c e T i m e τ J ov i a n / D a y s Corotating Jupiter
Init. Energy E init =6.1 MeV E init =7.4 MeV E init =9.0 MeV E init =11.0 MeV E init =13.5 MeV E init =16.5 MeV E init =20.1 MeV Longitude φ / ◦ τ c o r o t a t i n g / τ s t a t i c Fig. 7: Similar to Fig. 3, the residence times ( τ ) of Jovian elec-trons are now shown in order to illustrate the influence of co-rotation of the Jovian source as compared to a static source. Thebottom panel shows the ratio of the solutions. Di ff erent coloursindicate di ff erent energies.changes due to the assumption of either a static or a co-rotatingsource.We have also used the unique magnetic geometry of the Jo-vian propagation problem to quantify the appropriate mean freepaths. By reproducing the 1 AU intensity spectrum during timesof good magnetic connection, we were able to show that λ || ≈ . χ ≈ .
01 isappropriate. Both these values fall within the range of what isexpected from previous studies.Using the optimal numerical set-up, we calculated the exittimes of 6 MeV electrons using the traditional SDE formalismwhere each pseudo-particle contributes equally to the final re-sults. A value of ∼
600 days was found, which, although for-mally correct, does not seem physically consistent with relativis-tic particle propagating across a distance of ∼ ∼ ff erent.Especially regarding CIRs a more realistic measure of Jovianresidence times could improve our understanding. As Jovianelectron simulations successfully have been applied to this topicby (Kissmann et al. 2003, 2004) the numerical set-up providedby this study provides the opportunity to revisit this approachwith further insight. Simulations of residence times could alsohelp to determine how much time Jovian electrons (and otherparticle populations) spend within structures like CIRs or mag-netic flux tubes. More recently (Daibog et al. 2013) highlightedthe role of Jovian electrons as test particles within this matteragain, suggesting that deviations from their quiet time variationscould serve as probes for the inner heliosphere’s structure. Acknowledgements.
This work is based on the research supported in part bythe National Research Foundation of South Africa (Grant Number 111731) and
Article number, page 9 of 10 & A proofs: manuscript no. aanda the Deutsche Forschungsgemeinschaft (Grant Number FI 706 / References
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