On the Pulse Shape of Ground Level Enhancements
R.D. Strauss, O. Ogunjobi, H. Moraal, K.G. McCracken, R.A. Caballero-Lopez
aa r X i v : . [ phy s i c s . s p ace - ph ] M a r Solar PhysicsDOI: 10.1007/ ••••• - ••• - ••• - •••• - • On the Pulse Shape of Ground Level Enhancements
R.D. Strauss · O. Ogunjobi · H. Moraal · K.G. McCracken · R.A. Caballero-Lopez c (cid:13) Springer ••••
Abstract
We study the temporal intensity profile, or pulse shape, of cosmic ray groundlevel enhancements (GLEs) by calculating the rise ( τ r ) and decay ( τ d ) timesfor a small subset of all available events. Although these quantities show verylarge inter-event variability, a linear dependence of τ d ≈ . τ r is found. Weinterpret these observational findings in terms of an interplanetary transportmodel, thereby including the effects of scattering (in pitch-angle) as these parti-cles propagate from (near) the Sun to Earth. It is shown that such a model canaccount for the observed trends in the pulse shape, illustrating that interplane-tary transport must be taken into account when studying GLE events, especiallytheir temporal profiles. Furthermore, depending on the model parameters, thepulse shape of GLEs may be determined entirely by interplanetary scattering,obscuring all information regarding the initial acceleration process, and hencemaking a classification between impulsive and gradual events, as is traditionallydone, superfluous. Keywords:
Cosmic Rays: Solar; Energetic Particles: Propagation; Magneticfields: Interplanetary
1. Introduction
Ground level enhancements (GLEs) are sudden increases in the cosmic ray (CR)intensity as observed at Earth’s surface, in recent times mostly through neu-tron monitors (NMs). Since the first observation of a GLE on 28 February1942, such events have been observed seventy-one times (see the review byMcCracken et al. , 2012). A database of all GLE events is available at ftp://cr0.izmiran.rssi.ru/COSRAY!/FTP GLE/ (hosted by E. Eroshenko), https://gle.oulu.fi/ , as well as http://usuarios.geofisica.unam.mx/GLE Data Base/ with details of the latter described by Moraal and Caballero-Lopez (2014). It is Centre for Space Research, North-West University,Potchefstroom, South Africa.Email: [email protected] Institute for Physical Science and Technology, University ofMaryland, College Park, MD, 20742, U.S.A. Ciencias Espaciales, Instituto de Geof´ısica, UniversidadNacional Aut´onoma de M´exico, 04510 M´exico D.F., M´exico.
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Figure 1.
The temporal profile, or pulse shape, of GLE 45 (blue curves) and GLE 69 (redcurves) in order to illustrate the classification of so-called gradual (such as GLE 45) andmore impulsive (such as GLE 69) events. For each event, the different lines correspond tomeasurements by different NMs. generally believed that transient solar eruptive episodes, such as coronal massejections (CMEs) and solar flares, are responsible for producing the enhanced CRflux detected during a GLE event, and hence these particles can be classified assolar energetic particles (SEPs). If the primary SEPs reach Earth with kineticenergies ≥
500 MeV per unit charge, or rigidity ≥ et al. (2012), andCaballero-Lopez and Moraal (2016).Despite the large amount of SEP observations by both ground and spacebased CR detectors over the years, the relationship between flares and CMEs,and their role in accelerating particles to relativistic energies during major solarevents remains an exigent scientific challenge. Most studies, e.g. Reames (1999),classify SEPs into two distinct classes, impulsive and gradual events. Particlesfrom these events are believed to be accelerated by different processes in differenttransient structures; impulsive SEPs are associated with magnetic reconnectionin solar flares, while gradual SEPs are thought to be accelerated mainly bydiffusive shock acceleration in the shock fronts of large CMEs. Having an accu-
SOLA: GLE_2017_with_editor.tex; 30 September 2018; 6:45; p. 2 n the Pulse Shape of Ground Level Enhancements rate classification of SEP events may therefore give insight into where and howthese particles were accelerated ( e.g.
Reames, 2013). The usual classification,based on the shape of the observed intensity time profile, the so-called pulseshape, is furthermore motivated by differences in the composition and chargestate of the SEPs ( e.g.
Mewaldt et al. , 2012), for example, impulsive events havehigher ionization states. For ground-based observations of GLE, however, boththe composition and change state information is lost and the GLEs are classifiedexclusively on their pulse shape. In Figure 1, as an example, we show the pulseshape of GLE 45 (blue curves) and GLE 69 (red curves) for all available NM sta-tions with a cut-off rigidity below 1 GV. It is clear that GLE 45 was observed tobe more gradual than the more impulsive GLE 69, but it is impossible to extractinformation solely from this temporal profile about the acceleration mechanismsresponsible for each particle event without taking interplanetary transport intoaccount. Once the SEPs are accelerated, presumably near the Sun, they stillhave to propagate (mainly) along the turbulent interplanetary magnetic fieldto reach Earth. Depending on the level and nature of the underlying turbulentfluctuations and how efficiently they can disrupt ( i.e. scatter) the particle tra-jectories, the SEP temporal profile reaching Earth may be significantly differentfrom that near the acceleration region.In this study we will show that the interplanetary transport conditions mayalter the pulse shape in such a way as to remove any source information by thetime it reaches Earth, possibly making the classification of GLEs into differentclasses superfluous.
2. Characterising the Pulse Shape
The data base, which contains all available NMs observation, of all 71 GLEssince 1942, has been decsribed by McCracken et al. (2012). In this study, weselect only 14 GLEs, based on the following 5 criteria: i) in order to removeany energy effects, we only use NMs with cut-off rigidities below 1 GV, as allthese NMs are sensitive to the same particle energy range. As we will show laterin this paper, particle propagation is very dependent on the transport coeffi-cients, which are believed to be energy dependent, making a direct comparisonof the pulse shape of different energy particles (i.e. NMs with different cut-offrigidities) impossible. ii) We select only GLEs that were observed simultaneouslyby more that 10 NM stations. It is well-known that NMs are sensitive to thearrival direction of particles due to deflection by the Earth’s magnetosphere( e.g.
Shea and Smart, 1990). For an anisotropic event, such as a GLE whereparticles are predominantly streaming towards Earth, different stations at thesame cut-off rigidity may respond differently to the same SEP pulse. Note, forexample, the difference in peak intensity of GLE 69 (shown in Figure 1) asobserved by different NMs due to this geographical effect of essentially viewingparticles with different pitch-angles. By using more NM stations, we try, as faras possible, to remove this observational bias. In Figure 2 we show the calculated
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Table 1.
A summary of the GLE events selected for this study. The approximate solar longitude of the transientevent and the maximum increase is taken from McCracken et al. (2012).GLE no. Sol. long. a Max. increase b Rise time Decay time NM stations c ( ◦ W) (%) (min) (min) ( < a Ideal magnetic connection between the source of the SEPs and Earth would correspond to asource located at ≈ ◦ W in terms of solar longitude. b The maximum increase of the omni-directional counting rate. c ALTR – Alert, APTY – Apatity, BRBY – Barentzburg, CALG – Calgary, CAPS – CapeSchmidt, DPRV – Deep River, FSMT – Fort Smith, GSBY – Goose Bay, INVK – Inuvik,MCMD – McMurdo, MRNY – Mirny, MWSN – Mawson, NRLK – Norilsk, OTWA – Ottawa,PWNK – Peawanuk, TERA – Terra Adelie, THUL – Thule, TXBY – Tixie Bay
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Figure 2.
Geographical locations of a selection of the contributing NMs with geomagneticcut-off rigidity below 1 GV (large markers) and the corresponding asymptotic viewing di-rections (small markers). For each NM, the asymptotic viewing direction is calculated forrigidities of 0.7, 1.4, 2.1, 2.9, 3.6, 4.3 and 5.0 GV, with the highest rigidity position closest toeach station. asymptotic viewing directions for a number of NM stations used in this study.Note the nearly uniform longitudinal coverage near the equatorial regions. iii)The selected GLEs have amplitudes more than 10% and iv) consists of bothsmall and large events v) showing both gradual and impulsive characteristics.Table 1 summarizes all the events that were selected and used in this study.We consider this selection as a good subset of all available 71 events. Lastly,before quantifying the pulse shape, we average the time profiles of the separateNMs to remove the observational pitch-angle bias and refer to this as the omni-directional time profile. This is also done in order to facilitate a later comparisonto modelled solutions.In Figure 3, we demonstrate how the pulse shape is characterized in this study:for each event, the time profile of the omni-directional intensity is calculated andthe time of maximum intensity, t max , recorded. We then search for times whenthe intensity is half of its maximum value and note the corresponding times t a / and t b / . The rise time is now defined as τ r := t max − t a / and the decay time as τ d := t b / − t max . Here we deviate from Moraal, McCracken, and Caballero-Lopez(2016), and usual SEP studies such as Dresing et al. (2014), by not calculatingthe so-called onset time (defined as the time when the observed intensity breachesthe background level) and defining the rise time in terms of this quantity. Ob-servationally, the onset-times are biased towards larger events, or events withfast rising intensities ( i.e. for impulsive events), that breach the backgroundquicker ( e.g. Xie et al. , 2016), while, from a modelling point-of-view any chosenbackground level in a model is somewhat arbitrary ( e.g.
Wang and Qin, 2015).
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Figure 3.
Illustrating how we characterize the pulse shape of a GLE through the rise anddecay times.
Our re-definition of the rise time removes this bias, while sufficiently quantifyingthe temporal slope of the increasing phase of an SEP event.Figure 4 shows our calculations of the decay and rise times for the 14 selectedGLE events as the data points. Although there is still a significant spread in theseobserved values, a linear trend is discernible. By fitting a linear curve (throughthe origin) to the data, we find τ d = (3 . ± . τ r , where the error is simplytaken to be a standard deviation. The regression curve, with its error, is shownon the figure as the red line and shaded region respectively. All the data-pointsare fairly well described by the linear fit, with the possible exception of thatcorresponding to GLE 48 (the circled data-point) which has an exceedingly longdecay time. A closer inspection of GLE 48’s temporal profile shows a second peak(which might be a second event) some ≈ i.e. τ d ≈ τ r , although they used a different definition of τ r , but similarly alsofound a linear relationship between τ r and τ d . In the next sections, we interpretthis seemingly universal dependence in terms of a particle transport model. SOLA: GLE_2017_with_editor.tex; 30 September 2018; 6:45; p. 6 n the Pulse Shape of Ground Level Enhancements
Figure 4.
The datapoints show the rise and decay times as calculated for our selection of14 GLEs. The red line shows a linear fit to the observations, while the red band indicates a1 σ (standard deviation) error on the fitted slope. The data-point corresponding to GLE 48 iscircled.
3. Comparison with Theory ∂f ( z, µ, t ) ∂t = − µv ∂f∂z − − µ L v ∂f∂µ + ∂∂µ (cid:18) D µµ ( z, µ ) ∂f∂µ (cid:19) , (1)which is valid for motion along a magnetic field directed along e z . In this ex-pression, µ is the cosine of the particle’s pitch angle, so that the parallel speedbecomes v || = vµ , and focussing due to the diverging Parker (1958) magneticfield is given in terms of the focussing length L − ( z ) = 1 B ( z ) ∂B ( z ) ∂z (2)and pitch-angle diffusion is included via the pitch-angle diffusion coefficient D µµ .Although some analytical solutions of Equation (1) do exist (see e.g. Effenberger and Litvinenko
SOLA: GLE_2017_with_editor.tex; 30 September 2018; 6:45; p. 7 .D. Strauss et al. (2014)), these are notoriously inaccurate during the rising phase of a SEP event,and as such, we opt to solve the TPE numerically as outlined in Appendix A.Usually, all transport quantities are specified, not in terms of z , but rather interms of spherical coordinates, and as such, we need to convert between thesetwo coordinate systems, z ( r, φ ). The unit vector e z is defined to point alongthe magnetic field, B , so that e z ≡ B B = cos Ψ e r − sin Ψ e φ (3)in terms of spherical coordinates with Ψ being the magnetic field spiral angle, i.e. the angle between the magnetic field and the radial direction. For a Parker(1958) magnetic field, B = B h r r i ( e r − tan Ψ e φ ) , (4)where B is some reference values at r , we havetan Ψ = − B φ B r = Ω r sin θV , (5)which, at Earth in the equatorial plane ( θ = π/ ≈ ◦ when usingobserved values of the solar wind speed V ≈
400 km.s − and the solar rotationrate Ω ≈
25 days. As the magnitude of z is now defined as the length along thespiral field, we may calculate z ( r ) = Z r d z ( r ) = Z r d r p Ψ (6)where we have used d φ = − Ω V d r. (7)For ease, Equation (6) is integrated numerically in the present model. Dueto the curvature of the magnetic field, we find z ( r = 1AU) ≈ . z ( r = 3AU) ≈ . et al. (2010), we parametrize the pitch-angle diffusion coef-ficient as D µµ ( z, µ ) = D ( z ) (cid:0) − µ (cid:1) n | µ | q − + H o (8)where q = 5 / H = 0 .
05 is a parameter to account for non-linear or dynamicalprocesses which scatter particles though µ = 0 (see e.g. Dr¨oge et al. , 2010). Here D ( z ) is a function which we will use to obtain the required magnitude of D µµ . SOLA: GLE_2017_with_editor.tex; 30 September 2018; 6:45; p. 8 n the Pulse Shape of Ground Level Enhancements
In order to compare our results to analytical approximations, and to get a moreintuitive feeling for the level of particle scattering, we specify the parallel meanfree path λ || in the model rather than D µµ . These two quantities are relatedthrough (Hasselmann and Wibberenz, 1968) λ || ( z ) = 3 v Z +1 − (cid:0) − µ (cid:1) D µµ ( z, µ ) d µ, (9)while λ || is also related to the effective radial mean free path λ rr (in the absenceof perpendicular diffusion) through λ rr = λ || cos Ψ . (10)We parametrize λ rr as λ rr = λ (cid:20) rr (cid:21) α , (11)where λ is a reference value at r = 1 AU. Note that the same parametrizationis used in Appendix B, keeping in mind that κ rr = v/ λ rr , in order to comparewith the analytical approximations. By specifying λ and α , we can then followEquations (9) - (11) in reverse to estimate D ( z ) which is the input to the model.The magnitude of λ can already give us an indication as to the behaviour ofSEP intensities: at Earth, L ≈ z ≈ . λ ≪ L, z ≈ i.e. λ ≈ f ( z, µ, t ) we cancalculate the omni-directional intensity by simply averaging over pitch-angle: F ( z, t ) = 12 Z +1 − f ( z, µ, t )d µ, (12)and compare the modelled time profile of F ( z = 1 . , t ) to that derived fromGLE observations. See also the recent modelling, using a similar approach, byB¨utikofer et al. (2016).3.2. General Modelling ResultsTo be compatible with the GLE observations, we solve Equation (1), for 2 GVprotons, by specifying isotropic impulsive injection at the inner boundary of themodel, f ( z = 0 . , µ, t ) = δ ( t ). Examples of the resulting time profiles of F ,at Earth, are shown in the left panel of Figure 5 for a fixed value of α = 0 anddifferent values of λ . As expected, the level of particle scattering (quantifiedthrough the magnitude of λ ) has a large influence on the temporal profiles. SOLA: GLE_2017_with_editor.tex; 30 September 2018; 6:45; p. 9 .D. Strauss et al.
Figure 5.
The left panel shows model solutions of the omni-directional intensity for differentvalues of λ when α = 0 is kept constant. The right panel shows the distribution function, for λ = 0 .
05 AU and α = 0, at different pitch-angles. All profiles are shown at a radial positionof 1 AU. For large values of λ (weak scattering), we note a very fast rise in intensities,followed by a sharp decrease. For smaller values of λ (stronger scattering),the intensities rise more gradually. We are therefore able to control the temporalprofile of the distribution at Earth by adjusting the amount of scattering that theparticles experience. Hence, the injection profile (here, simply a delta-function)is not the only controlling factor of the pulse shape of GLEs; interplanetarytransport conditions can also have a large influence. In the right panel of Figure5, we keep α = 0 and λ = 0 .
05 AU constant and show the distribution functionat different pitch-angles. Of course, the particles that stream along the magneticfield, i.e. these with µ = 1, reach Earth first, and their profile shows a veryquick rise ( i.e. a short rise time). Particles mostly gyrating perpendicular to thefield ( µ = 0), take longer to reach Earth but also have a longer rise time. Thisis an additional motivation for us to use only the omni-directional intensity toquantify the temporal profile (for both the model and the observations) of theGLE events, thereby removing the dependence of the rise time on pitch-angle, or,in the observational sense, the dependence of the rise time on the geographicalposition of the NMs.Following our standard definition, we calculate the rise time from the modelledsolutions, and present these in Figure 6. As alluded to in Figure 5, larger valuesof λ lead to shorter rise times. What is however very interesting, is the linear SOLA: GLE_2017_with_editor.tex; 30 September 2018; 6:45; p. 10 n the Pulse Shape of Ground Level Enhancements
Figure 6.
The calculated rise time, as a function of λ , for different values of α . dependence of τ r for small values of λ ; when λ is less than ≈ . τ r decreases linearly with time, with the slope of the linearity dependent on thevalue of α . These small values of λ must result in fairly isotropic distributionsand, as we will show in the next section, this linear dependence is characteristicof the isotropic scenario where pitch-angle scattering is sufficiently strong. Asshown in the next section, a similar linear dependence is observed for τ d in thislimit.3.3. Comparison between Theory and ObservationsWe show the modelled relationship between τ r and τ d in Figure 7, as the symbols,for different choices of α . As already discussed, by decreasing λ in the model,both τ r and τ d increase, and in the limit of λ ≪ τ r >
10 min, a linearrelationship is evident between τ r and τ d with the slope dependent on the value of α . This linear dependence is indicative of isotropy, and this idea can be tested bycomparing the modelled solution to the analytical approximations discussed inAppendix B, which are derived for an isotropic particle distribution, and shownon Figure 7 as the different dashed lines. Indeed, at larger values of τ r boththe model and the approximations follows the same linear trend. Our resultsthus indicate that for GLEs with τ r >
10 min, we may approximate the particledistributions to be isotropic and use the approximations derived in Appendix B.
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Figure 7.
The symbols show the modelled relationship between τ r and τ d for different valuesof α , while the dashed lines are the corresponding analytical approximations discussed inAppendix B. The red band illustrates the relationship as derived from observations in Section2. This linear trend of course breaks down below τ r <
10 min, as these events arehighly anisotropic with λ ≈ τ r and τ d as derived form GLE observations in Section 2. We can now explain thislinear dependence in terms of particle transport in the isotropic limit. Moreover,a comparison between modelled and observed results indicates the value α ≈ λ rr independent of radial distance) is consistent with the observations; thepossible implications thereof are discussed in the next section.
4. Summary and Conclusions
By calculating the rise and decay times for a small subset of all GLE events, wefind a linear relationship between these two quantities, conveniently summarizedas τ d ≈ . τ r , i.e. the decay phase lasts, on average, 3 to 4 times longer thanthe rising phase of the event. This relationship seems to hold for a large rangeof τ d and τ r values (almost two orders of magnitude), suggesting that GLEsdo not fall into two distinct classes of either impulsive or gradual events, butfollow a continuous distribution of impulsive-like or gradual-like events. It is SOLA: GLE_2017_with_editor.tex; 30 September 2018; 6:45; p. 12 n the Pulse Shape of Ground Level Enhancements difficult to imagine how different acceleration mechanisms can lead to sucha “universal” linear relationship, and hence we interpret this result as beingindicative of interplanetary transport. We show that interplanetary scatteringcan significantly affect the temporal profile of GLEs as observed at Earth, ina sense obscuring the initial acceleration profile. In the limit of very effectiveparticle scattering, i.e. when the resulting distribution is nearly isotropic, boththe numerical and analytical solutions reproduce the observed linear trend. Wethus conclude that interplanetary transport may have an extremely large effecton the observed pulse shape of GLEs and should not be completely ignored asis mostly done.There are very large inter-event variations in the values of τ d and τ r , althoughthe values follow the same general trend. These large variations are most likelyrelated to changes in the level of interplanetary scattering between events (see e.g. Dr¨oge (2000)) and can be accounted for in our model by varying, for example, λ . A comparison between the observations and especially the analytical approx-imations, suggests, however, that a constant radial dependence of λ rr ∝ r (i.e. λ rr being independent of radial position) best reproduces the observations forall events. Perhaps rather fortuitously, the same assumption is frequently usedin many transport models (see the discussion by Dr¨oge (2000)), giving us someconfidence in our modelling approach and conclusions.In future, we plan to extend our analyses to include more events, as our14 selected events only represent ≈
20% of the total recorded GLEs. However,based on the findings of Moraal, McCracken, and Caballero-Lopez (2015), whoincluded more events, the qualitative conclusions presented here are not expectedto change much. Of particular interest will also be to perform an equivalent studyfor SEP events that are observed in-situ by spacecraft and that can be classifiedmore rigorously into impulsive (flare accelerated) and gradual (CME accelerated)events.
AppendixA. Notes on the Numerical Model
Following Strauss and Fichtner (2015), we integrate Equation (1) numerically byfirst transforming it into a set of three, 1D convection and diffusion equations,using a first-order operator splitting method, to obtain13 ∂f∂t ′ + µv ∂f∂z = 0 (13)13 ∂f∂t ′ + 1 − µ L v ∂f∂µ = 0 (14)13 ∂f∂t ′ = ∂∂µ (cid:18) D µµ ∂f∂µ (cid:19) , (15) SOLA: GLE_2017_with_editor.tex; 30 September 2018; 6:45; p. 13 .D. Strauss et al.
Figure 8.
A comparison between our numerical solver (FD model; indicated by the redlines) and that of Dr¨oge et al. (2010) (SDE model; indicated by the blue symbols). The leftpanel compares the calculated omni-directional intensity and the right panel the correspondingfirst-order anisotropy. where d t ′ = d t/
3. The diffusion equation (Equation (15)) is solved by an ex-plicit Euler method, while the convection/advection equations (Equations (13)and (14)) are solved by a flux limiter corrected upwind scheme. For all equa-tions, flux conserving boundary conditions are applied (see the discussion byStrauss and Fichtner (2015)). To validate this modelling approach, we compareour results to those of Dr¨oge et al. (2010) in this section. These simulations, usingexactly the same transport parameters as given by Dr¨oge et al. (2010), are for4 MeV protons with the results shown as a function of time at Earth’s position.In Figure 8 our model results are labelled as the finite-difference (FD) solutions,while the Dr¨oge et al. (2010) results, computed by making use of stochasticdifferential equations (SDEs) are labelled accordingly. The left panel comparesthe omni-directional intensity and the right the first-order anisotropy, defined as A = 3 R − µf d µ R − f d µ . (16)The excellent agreement between the two independent models validates ourmodelling approach while giving us confidence is our calculated temporal profiles. SOLA: GLE_2017_with_editor.tex; 30 September 2018; 6:45; p. 14 n the Pulse Shape of Ground Level Enhancements
Figure 9.
Finding the roots of Equation (23) for different values of α . B. (Semi-) Analytical Approximations
If the particle distribution can be approximated to be nearly-isotropic, we canfollow Moraal, McCracken, and Caballero-Lopez (2016) and obtain f by solvingthe spherical symmetric diffusion equation ∂f∂t = 1 r ∂∂r (cid:18) r κ rr ∂f∂r (cid:19) . (17)Assuming that the effective radial diffusion coefficient, κ rr , can be parametrizedas κ rr = κ (cid:20) rr (cid:21) α , (18)the solution of f for impulsive injection at r = 0 and t = 0 is (Duggal, 1979) f ∝ t / ( α − exp (cid:20) − r ( r /r ) α (2 − α ) κ t (cid:21) . (19)The time of peak, or maximum, intensity is evaluated as SOLA: GLE_2017_with_editor.tex; 30 September 2018; 6:45; p. 15 .D. Strauss et al. t max = r ( r /r ) α − α ) κ , (20)with a corresponding peak intensity of f max ∝ (exp (1) · t max ) / ( α − , (21)so that Equation (19) may be rewritten, in terms of these quantities, as ff max = (cid:20) tt max exp (cid:18) t max t − (cid:19)(cid:21) / ( α − . (22)This analytical approximation is shown, for a variety of different parameters,and discussed in more detail in Moraal, McCracken, and Caballero-Lopez (2015,2016). Here, however, we are only interested in times where the distributionobtains a half of its maximum value, i.e. we are interested in finding t a / and t b / as defined in Figure 3. By setting 2 f = f max in Equation (22), we obtainthe required transcendental equation t / t max = 2 (2 − α ) / exp (cid:18) − t max t / (cid:19) (23)that must be solved numerically to obtain the two values of t a / < t max and t b / > t max . Note also that this expression is independent of the magnitude ofthe diffusion coefficient and only on its radial dependence through α . Figure 9illustrates how to find the roots (values of x where y = 0) of Equation (23) fordifferent values of α by setting x := t / t max , (24) y ( x, α ) := x − (2 − α ) / exp (cid:18) − x (cid:19) . (25)For example, α = 0 gives t a / ≈ . t max and t b / ≈ . t max , which, substitutedinto our definitions of the rise and decay times (see again Section 2) leads to τ d ≈ . τ r . Our results are summarized as α = − τ d ≈ . τ r (26) α = − τ d ≈ . τ r (27) α = 0 : τ d ≈ . τ r (28) α = 1 : τ d ≈ . τ r (29)with the special case of τ d = τ r for α = 2 describing a free-streaming scenario.Note that, for all values of α , we obtain a linear relationship between τ d and τ r . SOLA: GLE_2017_with_editor.tex; 30 September 2018; 6:45; p. 16 n the Pulse Shape of Ground Level Enhancements
These resulting estimates, referred to as the isotropic solutions, are included inFigure 7 as the dashed lines.
Acknowledgements
This work is based on the research supported in part by the NationalResearch Foundation (NRF) of South Africa (grant no. 106049). Opinions expressed andconclusions arrived at are those of the authors and are not necessarily to be attributed tothe NRF. OO acknowledges the support of the post-doctoral programme of the North-WestUniversity in South Africa.
This paper is dedicated to the memory of the late Harm Moraal.
Disclosure of Potential Conflicts of Interest
The authors declare that they have noconflicts of interest.
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