Escaping dynamics o ela i is ic p o ons in he Ea h’s magne osphe e
´
Al a o Mesegue ,1, ∗Juan C. Vallejo,2Jes´
us M.
Seoane,2F ancisco Ma qu´
es,1and Miguel A.F. Sanju´
an2
1Depa amen de F´
ısica, Uni e sidad Poli `
ecnica de Ca alunya, Ba celona 08034, Spain
2Nonlinea Dynamics, Chaos and Complex Sys ems G oup,
Depa amen o de F´
ısica, Uni e sidad Rey Juan Ca los,
Tulip´
an s/n, 28933 M´
os oles, Mad id, Spain
(Da ed: Ap il 3, 2025)
1
Abs ac
In his wo k, we s udy he mo ion o p o ons unde he in luence o he Ea h’s magne ic ield. We
in es iga e he dynamics and opology o in a ian spa ial egions in wo magne ic ield models, namely,
he dipola and Luhmann ail con igu a ions. In bo h cases, we analyze he mo ion o he p o ons in he
ela i is ic egime, wi h kine ic ene gy in e als in he ange 10 < E < 100 MeV. Speci ically, we
examine si ua ions in which he p o ons unde go chao ic sca e ing due o hei in e ac ion wi h he Ea h’s
magne ic ield. Addi ionally, we conside scena ios whe e he p o ons, du ing hei in e ac ion, all on o
he Ea h’s su ace. We compa e he dipole and Luhmann ail cases by compu ing he dis ibu ions o
escape and esidence imes, he exi basins, and he ac al dimensions o hei bounda ies. We p esen
a obus symplec ic nume ical scheme which is sui able o analysing he impac o he nonlinea e ec s
and he p esence o s ong sensi i i y o ini ial condi ions when he model goes beyond he pu e dipola
ield. Su p isingly, we unco e a scaling law be ween he coe icien o he decay law and he ene gy o he
p o ons. Fu he mo e, ou compu a ion o he ac al dimension o he exi basin bounda ies D e sus he
dis ance be ween he p o ons and he Ea h z0 e eals a dec ease in Das z0inc eases. We expec his wo k
o be use ul o a be e unde s anding o he beha io o p o ons unde he in luence o he Ea h’s magne ic
ield when hey a e in he chao ic sca e ing egime.
I. INTRODUCTION
The elec odynamics o Ea h’s magne osphe e is e y complex, and i has a ac ed he in e es
o many physicis s and ma hema icians o o e a cen u y. Fo a ho ough e iew o his opic, in-
cluding a comp ehensi e his o ical o e iew and he mos ele an and up- o-da e s udies, we e e
he eade o he s anda d monog aph by Russell, Luhmann, and S angeway [1], and e e ences
he ein. Figu e 1 shows he main cha ac e is ics o Ea h’s magne osphe e.
Close o he Ea h, he magne ic ield is e y simila o a magne ic dipole, gene a ed by he
cu en s in he Ea h’s me allic co e. Howe e , he sola wind d ama ically al e s he dipola
s uc u e o he magne ic ield a a ew adii om he Ea h. On he Sun- acing side, a bow shock
is o med when he sola wind pa icles a e de lec ed by Ea h’s magne ic ield. On he opposi e
side o he Ea h, away om he Sun, he so-called magne o ail is o med, gene a ed mainly by a
plasma shee nea he equa o ial plane.
∗al [email p o ec ed]
2
FIG. 1. A simple schema ic iew o he Ea h magne osphe e and i s main componen s. The magne ic ield
lines in ed, he Sun on he le .
As he sola wind in e ac s wi h he Ea h’s magne osphe e, high-ene gy pa icles a e apped
and held a ound he Ea h, o ming he so-called Van Allen adia ion bel s, whe e he popula ion
o cha ged pa icles is much la ge han a e age [2, 3]. Two bel s a e usually desc ibed. The inne
bel mainly consis s o ene ge ic p o ons, wi h kine ic ene gy alues anging om 10 MeV o
100 MeV, gene a ed h ough he so-called Cosmic Ray Albedo Neu on Decay (CRAND) [4, 5].
No e ha , om now on, we deno e by E he kine ic ene gy o he p o ons. This inne bel is
p ima ily he p oduc o cosmic ay collisions in he uppe a mosphe e. Some o he collision
by-p oduc s can en e he a mosphe e, bu when he agmen is a ee neu on, i can decay in o
a p o on (which cap u es mos o he ene gy), an elec on, and a neu ino. Al hough he neu on
is e y as , i can s a is ically decay while s ill close o Ea h. Hence, he esul ing ene ge ic
p o on will emain apped by he magne osphe e. This esul s in a bel wi h a iable bounda ies,
ypically ex ending om 0.2RL o 3.0RL, whe e RLis he Ea h adius.
Al hough he inne bel can also con ain high concen a ions o elec ons wi h ene gies wi hin
he ange o hund eds o keV, see o ins ance [6], i is he ou e bel which consis s mainly o high-
ene gy elec ons. Ou wo k will ocus on he analysis o he dynamics o he cha ged pa icles
o ming he inne bel as hey a e injec ed in o he Ea h’s magne osphe e. This analysis will
use he ools o nonlinea dynamics om he pe spec i e o an open chao ic sca e ing p oblem,
aiming o ge insigh in o he esidence imes o he injec ed pa icles, he opology o he in a ian
apping egions whe e hese pa icles emain cap u ed, as well as he li e imes o hese egions.
Sca e ing p oblems a e de ined as he in e ac ion be ween an inciden pa icle and a po en ial
3
egion o massi e objec ha sca e s i . In his con ex , we say ha he sca e ing is chao ic when
he inal s a e o he pa icle has sensi i e dependence on ini ial condi ions (see Re . [7]). The
equa ions o mo ion o he es pa icles ha model hese in e ac ions a e nonlinea , and we can
ind s ong dependence on ini ial condi ions and chao ic dynamics. One de ines he sca e ing
egion as he domain whe e he pa icle is a ec ed by he po en ial ield while, ou side his zone,
he in luence o he po en ial o e he pa icle can be neglec ed. The sca e ing sys ems a e labeled
as open when he es pa icles may escape once hey en e ed in he sca e ing egion a e bouncing
back and o h o a while. In hese open cases, chao ic sca e ing is ega ded as a ansien chaos
dynamics, and, al hough he es pa icle can emain bounded, he esul ing dynamics is no a
simple one.
Rega ding he dynamics in a magne osphe ic model, complex s uc u es can be ound e en
in he mos simple case o a magne ic dipola ield. This is he so called S ¨
o me p oblem (see
Re . [8–10]). He e, chaos is ypically p esen a high ene gies and has been obse ed o be in-
dependen o he ini ial eloci y [11–13]. When conside ing he eal Ea h’s magne osphe e, he
magne ic dipole componen s ill p edomina es, bu he dynamics o he pa icles and he opology
o he bel s a e s ongly dependen on he p esence o he magne o ail, as shown in Fig. 2. One
o he main goals o his s udy is he e o e o analyse he e ec s o he p esence o absence o he
magne o ail ega ding esidence imes (li e imes) o he pa icles. Besides, we s udy how he p es-
ence o absence o he magne o ail in luences he opological changes o he apping egions by
s udying he ac ali y o he exi basins. Acco dingly, he explo a ions epo ed he e will conside
he pu e dipola model and will compa e i when adding a magne o ail. In bo h cases, we aim
a de e mining how bo h models di e in he esidence imes o he injec ed pa icles and which
alues o he pa ame e s d i e he con igu a ion and li e imes o he egions whe e hese pa icles
emain apped. We will also s udy he coexis ence o escape and apping egions o he wo
models, and hei dependence as a unc ion he ene gy and ini ial condi ions (day-side o nigh -
side launches). Because o he b oad se o pa ame e s modeling he magne osphe ic s uc u es,
his s udy will ocus on he dynamical s udy o p o ons. Howe e , he ma hema ical o mula ion
and nume ical me hodologies de eloped he e a e comple ely gene al and can be applied o o he
amilies o cha ged pa icles.
When modeling pa icles in he magne osphe e, he guiding cen e app oxima ion has been
o en used in he pas because i educes he equi ed compu a ional ime, see [14] and e e ences
he ein. Howe e , he e, we need o sol e he ull dynamics o he pa icles, modeling i s h ee
4
main dynamical componen s, gy a ion, bounce and azimu hal d i , because we aim o analyse he
impac o he nonlinea e ec s and he p esence o s ong sensi i i y on ini ial condi ions when he
model goes beyond he pu e dipola ield and bo h high and low ene gies a e possible. The e o e,
we will p esen a obus nume ical scheme wi h educed nume ical dissipa ion, which is sui able
o his analysis.
The pape is o ganized as ollows. In Sec. II, we in oduce he dipola and dipola - ail magne ic
models, illus a ing he main ea u es o he esul ing p o on dynamics. In Sec. III we p o ide some
analy ical esul s ega ding he dipola model, add ess he symme ies o he esul ing dynamical
sys em, and in oduce he symplec ic nume ical me hodologies used o he ene gy-p ese ing
ime in eg a ion o he p o on’s mo ion. Sec ion IV is de o ed o cha ac e ize he opology o
he space egions acco ding o hei associa ed esidence imes (i.e., he o e all ime spen by he
p o ons in hose egions be o e an e en ual escape). The s udy o he decay law o he pa icles
in he sca e ing egion is ca ied ou in Sec. V. Sec ion VI p esen s bo h he exi basins and he
basins o a ac ions and hei ac al na u e. Finally, he main conclusions and a discussion o he
esul s a e p esen ed in Sec. VII.
II. MODEL DESCRIPTION
The ela i is ic dynamics o a cha ged pa icle o mass mand cha ge qin he p esence o
elec ic Eand magne ic B ields is desc ibed by he Lo en z o ce
md(γ )
d =q(E+ ×B), γ = (1 − 2/c2)−1/2.(1)
This equa ion can be ob ained om he ela i is ic Lag angian and Hamil onian
L=−mc2p1− 2/c2−q(V− ·A),(2a)
H=p(p−qA)2c2+m2c4+qV, p=mγ +qA.(2b)
whe e Vand Aa e he elec ic and magne ic po en ials, espec i ely. In he absence o elec ic
ield (E= 0) om (1) we see ha he magni ude o he eloci y is cons an , and he γ ac o mo es
ou side he ime de i a i e:
mγ d
d =q ×B(3)
This equa ion is almos iden ical o he New onian case, excep o he p esence o he γ ac o ,
which is a cons an p esc ibed by he ini ial condi ions o he pa icle. As has been obse ed by
5
o he au ho s [15, 16], in he absence o elec ic ields we can use he modi ied Lag angian and
Hamil onian gi en by
L=1
2mγ 2+q ·A,H=1
2mγ (p−qA)2,p=mγ +qA,(4)
whe e we assume γis cons an in he de i a ion o he equa ions o mo ion, and Ais he magne ic
ec o po en ial de ined by B=∇×A. These modi ied Lag angian and Hamil onian, which
esemble he New onian case excep o he p esence o he γ ac o , ep oduce he ela i is ic
equa ions o mo ion (3), and we will use hem because hey a e simple han (2), and acili a e he
discussion o symme ies and conse ed quan i ies.
In o de o model he magne ic ield o he magne osphe e, we will use a supe posi ion o wo
di e en magne ic ields: he Ea h dipole magne ic ield, Bd, which is dominan close o he Ea h,
and a magne ic ield modeling he magne o ail, B , using he Luhmann model [17]. Acco dingly,
he ma hema ical exp essions o he magne ic ec o po en ial and he magne ic ields ead as
ollows:
A=Ad+A ,Ad=µˆ
z×
3, µ =µ0M
4π,A =−B δln cosh(z/δ)ˆ
y,(5)
B=Bd+B ,Bd=µ3z
5−ˆ
z
3,B =B anh(z/δ)ˆ
x,(6)
whe e Mis he Ea h magne ic dipole momen poin ing owa ds he zdi ec ion; ˆ
x,ˆ
y,ˆ
za e he
uni ec o s in he x,yand zdi ec ions, espec i ely. A easonable alue o M, ha we will use
in he pape , is M=−8×1022 A m2, he minus sign accoun ing o he ac ha he Geog aphic
No h Pole is app oxima ely he magne ic sou h pole. This gi es a alue µ=µ0M/(4π) =
−8×1015 T m3. Hence o h, ˆ
xand ˆ
ya e he uni ec o s in he Ca esian xand ydi ec ions,
espec i ely, whe e he xaxis poin s in he di ec ion om he Sun o he Ea h (so ha he posi i e
xdi ec ion poin s away om he Sun).
In Eq. (6), B measu es he s eng h o he magne o ail magne ic ield, and he alue we a e
using in his s udy is B =−10−8T, consis en wi h o me models [17], whe eas δis a con enien
leng h measu ing he hickness o he densi y cu en associa ed wi h B . The densi y cu en
associa ed wi h a p esc ibed magne ic ield is gi en by he Amp`
e e-Maxwell equa ion ∇×B=
µoj. Fo he magne ic dipole he cu en is loca ed a he o igin (in p ac ice in he Ea h’s co e)
because ∇×Bd= 0.
Fo he magne o ail, we ge a densi y cu en a he equa o ial plane in he ˆ
ydi ec ion:
j =B
µ0δcosh−2(z/δ)ˆ
y≈4B
µ0δe−2|z|/δ ˆ
y.(7)
6
Because 95% o he cu en is concen a ed in a laye o wid h 3δa ound he equa o ial plane, in
his s udy we will use δ= 3RL. Mode a e a ia ions o he alue o δdid no esul in ema kable
changes in he dynamics obse ed.
(a) (b)
-20 0 20 40 60 80
x
-30
-15
0
15
30
z
FIG. 2. Magne ic ield lines o a magne ic dipole Bdincluding a magne o ail B . (a) Me idional plane. (b)
3D pe spec i e iew. Red do s a e he poin s whe e B= 0, and he ed lines a e he ield lines h ough hese
poin s. B =−10−8Tand δ= 3RL. Leng hs a e gi en as mul iples o Ea h adius. No ice ha he day
side is he egion on he le o he Ea h (co esponding o nega i e alues o x) while he nigh side is he
egion on he igh o he Ea h (co esponding o posi i e alues o x).
Figu e 2 shows magne ic ield lines in he me idional plane y= 0 o B =−10−8Tand
δ= 3RL. The magne ic ield is usually ep esen ed by plo ing he magne ic ield lines, ha a e
he solu ion o d /dτ =B, whe e τpa ame izes he magne ic lines. The alue B= 0 is ound
a he wo poin s ma ked in ed in he igu e, wi h coo dina es gi en by =z0(−√2,0,±1),
z0= 9.442 77RL. The uns able mani old o he uppe B= 0 poin and he s able mani old o
he lowe B= 0 poin de ine he magne osphe e and magne o ail o he Ea h. They a e plo ed
in ed in Fig. 2. The magne opause on he noonside is limi ed by he bow shock. The limi o he
magne opause on he noonside in he equa o ial plane is e y close o x0=−13.3704RL.
I is con enien o ende he p oblem dimensionless using he Ea h adius RL= 6.371 ×
106m, he eloci y o ligh cand he Ea h magne ic ield a he equa o Beq =µ/R3
L=
0.309 362 T, as uni s o leng h, eloci y and magne ic ield. The esul ing dimensionless go -
7
e ning equa ions a e
d
d =b ×B,L=1
2 2+b ·A, b =qµ
mγ ,(8)
A=ˆ
z
× 3−b δln cosh(z/δ)ˆ
y,B=3z
5−ˆ
z
3+b anh(z/δ)ˆ
x,(9)
whe e band b a e he dimensionless e sions o qµ/(mγ)and B /µ espec i ely. Pa ame e b
depends on γand he a io q/m, and he e o e on he ini ial condi ions and na u e o he pa icle
conside ed, while b = 3.232 457 ×10−4is ixed, measu ing he dimensionless s eng h o he
magne o ail ield, in good ag eemen wi h he a o emen ioned Luhmann model [17].
III. THEORETICAL ANALYSIS AND NUMERICAL INTEGRATION OF THE MODEL
In his sec ion, we desc ibe he Lag angian s uc u e o he equa ions o mo ion, he conse ed
quan i ies de i ed om he Lag angian o mula ion, and he nume ical echniques used. To s eam-
line he main ex , he de ails o he analysis ha e been con ined o he Appendix A, and we p esen
he e a summa y o he ele an conclusions.
Since he model does no explici ly depend on ime, he Hamil onian, ha coincides wi h he
kine ic ene gy o he pa icles, is conse ed. In he dipola model wi hou a magne o ail, i.e., when
b = 0, he e is an addi ional conse ed quan i y, ela ed o he angula momen um o he pa icle
ℓ=ρ2˙
θ+bρ2
3,(10)
whe e uis he modulus o he pa icle eloci y (conse ed), and ρ=px2+y2is he dis ance o
he dipole axis; see equa ion (A2) in Appendix A. The conse a ion o ℓand u esul s in a bounded
quan i y, g(ρ, z), depending only on he spa ial coo dina es:
g(ρ, z) = a
ρ−ρ
3,|g(ρ, z)| ≤ u
|b|, a =ℓ/b. (11)
In he case a > 0 he e exis bounded egions, o |g| ≤ 1/4and ρ≤2/a, whe e he pa icles
emain apped, and ha e a lunula shape simila o he Van Allen bel s, as shown in Fig. 3. Pa icles
apped in hese egions can pene a e along he edges o he lunulas in o he a mosphe e and e en
each he Ea h’s su ace, esul ing in he au o as (bo ealis and aus alis).
The shape and size o he apping egions depends on he alue o a=ℓ/b, i.e. depends on he
ini ial condi ions o he pa icle and he na u e o he pa icle (p o ons, elec ons, e c.). The e o e
he Van Allen bel s a e no ixed in space, bu depend on he kind o pa icles coming om he
8
0 1 2 3
a;
-1
-0.5
0
0.5
1
az
-20
-10
0
1
FIG. 3. Con ou s o he unc ion g o a=ℓ/b > 0. The ed lines co espond o g=±1/4, he g een line o
g= 0, and he black cu es o g=±0.15. The g ay egion is he lunula bounded by he cu es g=±0.15.
sola wind, ha end up popula ing he bel s. In he ail model, he e is jus one conse ed quan i y,
namely he kine ic ene gy o he pa icle. The analysis done abou apping egions in he dipole
case, can no be ex ended in he p esence o he ail. Howe e , conside ing he weakness o he
ail magne ic ield nea he Ea h compa ed wi h he dipole s eng h, i is easonable o assume ha
some o he cha ac e is ics o he apping egions will s ill apply. In any case, he ajec o ies a e
e y complex, o en chao ic, and can only be compu ed nume ically.
T ajec o ies con ined on he equa o ial plane a e exac solu ions o he equa ions o mo ion,
e en including he magne o ail ield. In ac he magne o ail ield is ze o on he equa o ial plane,
he e o e bo h models coincide in his plane.
S anda d ime in eg a o s o non-s i ODEs a e gene ally non-symplec ic and he e o e do no
p ese e ene gy du ing he ime e olu ion o he sys em. Howe e , e en symplec ic in eg a o s
may ail o p ese e addi ional conse ed quan i ies, such as he ℓmagni ude in he dipole case.
In his s udy, we ha e used and adap ed he Runge-Ku a 9(8) ime-s eppe [23], modi ied wi h
a p ojec ion me hod [24–26] o ensu e he p ese a ion o conse ed quan i ies. We ha e chosen
a high-o de me hod wi h a a iable ime s ep o allow easonable s ep sizes ha educe compu-
a ional cos while main aining high accu acy in he esul s; see de ails in Sec. 4 o Appendix
A.
Figu e 4 shows he ajec o y o a p o on in he magne o ail ield, ob ained h ough nume i-
cal in eg a ion using he a o emen ioned symplec ic-p ojec ion me hod. The ajec o y appea s
chao ic and is con ined wi hin a lunula-shaped egion.
9
V. ESCAPE TIMES AND DECAY LAWS
Ou sys em can be iewed as a sca e ing p oblem. Chao ic sca e ing e e s o he in e ac ion
be ween an inciden pa icle and a po en ial egion o massi e objec , which sca e s he pa icle
[28]. The e o e, he analysis o escape imes also p o ides insigh s in o he dynamical p ope ies
o he sys em.
Decay laws desc ibe he e olu ion o popula ions o injec ed pa icles o e ime, as well as
hei likelihood o emaining bound as ime p og esses. In his wo k, we analyze he dynamical
decay law o p o ons, which de ines he ime dependence o he pa icles’ su i al p obabili y
wi hin he sca e ing egion. Pa icle mo ion can be ei he bounded o unbounded, ollowing eg-
ula o chao ic ajec o ies. In conse a i e sys ems, egula mo ion is ypically associa ed wi h
Kolmogo o –A nold–Mose (KAM) o i. In hype bolic chao ic sca e ing, pe iodic o bi s a e un-
s able, and KAM o i do no exis in he phase space. In his case, he dynamical decay law o he
pa icles ollows an exponen ial o m. Howe e , in he nonhype bolic egime, KAM o i coexis
wi h chao ic saddles, leading o algeb aic decays in he pa icles’ su i al p obabili y [29].
We no e ha we will examine how p o ons escape he in luence o he magne ic ield o ene -
gies in he ange o [10,100] MeV, which co esponds o he ela i is ic egime. In he ela i is ic
case [30], i has been shown ha , gene ally, when he pa icle eloci y is su icien ly la ge, he
decay law o he pa icles becomes exponen ial.
A. Decay laws
He e, we aim o in es iga e whe he he injec ed p o ons ollow he cha ac e is ic exponen ial
decay law ypically associa ed wi h ela i is ic sys ems. To do so, we conside pa icles wi h
ene gies in he ange o [10,100] MeV, ixing he ini ial injec ion heigh a z0= 2.0RL. Figu e 7
shows he a io Ro bound pa icles o bo h he dipole and magne o ail models, wi h he la e
including day-side and nigh -side injec ions. The ime ho izons and ene gies ange wi hin he
in e als ∈[0,30] s and E∈[10,100] MeV, espec i ely.
Unexpec edly, Fig. 7 shows ha he magne o ail seems o ha e a apping e ec on he pa icles,
albei mode a e, bu clea ly no iceable. Compa ed wi h he dipola model, Fig. 7(a), Fig. 7(b) and
Fig. 7(c) show ha he su i al a io sligh ly inc eases when he magne o ail is p esen , al hough
he pa icles a e eleased om he side opposi e o he mos ac i e egion o he magne o ail.
16
FIG. 7. E olu ion o he a io o bound p o ons, R, wi h ime, o p o ons injec ed a z0= 2RL, wi h
he colo code indica ing ene gy le els. (a) Dipole ield. (b) Magne o ail Luhman’s D- ail model (day-side
injec ions) (c) Magne o ail Luhman’s N- ail model (nigh -side injec ions).
The a io Ro bound pa icles is ela ed o he likelihood o a p o on o emain bound o e ime.
This, in u n, can be linked o he e olu ion o he ene gy spec um, o he dis ibu ion unc ion
o he numbe o p o ons as a unc ion o ene gy, which cha ac e izes he pa icle popula ion a a
gi en ime. The e o e, Fig. 8 shows he e olu ion o hese ene gy spec a o bo h he nigh -side
and day-side popula ions. F om a quali a i e pe spec i e, we obse e ha bo h dis ibu ions a e
qui e simila , al hough he nigh -side popula ion seems o se le sligh ly ea lie
A = 0, he popula ion s a s wi h a la dis ibu ion, co esponding o all pa icles being
injec ed om he nigh side. The igu e illus a es ha he mos ene ge ic p o ons, wi h E=
100 MeV, a e he i s o escape. As ime p og esses, lowe -ene gy p o ons begin o escape as
well. A ound T≈15 s, he dis ibu ion s abilizes, wi h no u he escape o pa icles occu ing a
a gi en ene gy.
17
FIG. 8. E olu ion o he ene gy spec um o he popula ions o p o ons injec ed a z0= 2.0RL, bo h om
he day side, D- ail model (le ) and nigh side, N- ail model ( igh ). La ge ene gies escape ea lie , bu a e
oughly T≈15 s he spec um emains s able.
The panels o Fig. 7 show a ansien beha io o Ra ound T≈15 s, obse ed in bo h he
dipole and magne o ail cases. Howe e , he ansien is mo e clea ly seen in he magne o ail cases.
To analyze his beha io , we now plo he e olu ion o he ac ion Ro unbound pa icles o
di e en injec ion heigh s z0, ixing he ene gy a E= 25 MeV and E= 100 MeV.
Figu e 9 shows he inc ease in he ac ion o escaping pa icles, R, wi h ene gy and ini ial
injec ion heigh , z0. I also illus a es he dependence o he asymp o ic bound popula ion on di -
e en injec ion heigh s, speci ically z0= 0.5,1,2,5RL. A la ge z0 alues, almos all injec ed
pa icles escape, whe eas a di e en beha io is obse ed a smalle z0. Typically, an ini ial ac-
ion o pa icles escapes immedia ely a e injec ion, while ano he g oup emains bound o some
ime be o e e en ually escaping. In he asymp o ic egime, only he pa icles ha emain bound
h oughou he en i e in eg a ion ime pe sis . The igh mos panels o Fig. 9 p o ide a zoomed-in
iew o he e olu ion o his ansien popula ion. He e, wo dis inc slopes in he decay law cu es
a e isible, wi h he slopes being clea ly di e en o he lowe ene gies (con inuous line). A he
la ges z0, his asymp o ically bound popula ion disappea s, as nea ly all injec ed pa icles e en u-
ally escape, and no long- e m bound popula ion emains, wi h he decay con inuing inde ini ely.
18
FIG. 9. Dynamical decay law, o log −plo e olu ion o he ac ion o pa icles ha emain bound as he
ajec o y p og esses. Top panels co espond o day-side injec ions, bo om panels o nigh -side injec ions.
Righ mos panels de ail he e olu ion a ea ly imes o he e olu ion up o = 15 ime-uni s.
B. Exponen ial beha io s a ela i is ic egimes
In a gene al classical scena io, he ac ion o emaining pa icles in he sca e ing egion de-
c eases linea ly wi h ene gy, as shown in Re . [31]. Howe e , in he ela i is ic domain, he decay
law o he pa icles ollows an exponen ial beha io gi en by R∼e−α , whe e 1/α ep esen s
he cha ac e is ic decay ime due o sca e ing p ocesses. We conside p o ons wi h ene gies in he
ela i is ic egime, whe e he eloci y o he pa icles exceeds β > 0.1. A his poin , nea ly all
ajec o ies depa om he non- ela i is ic egime [30].
To s udy he po en ial exponen ial beha io o Ra di e en ene gies, we ha e again chosen
ene gy alues in he ange E∈[10,100] MeV and ixed z0= 2RL. Figu e 10 shows he end
19
FIG. 10. Plo s o he coe icien o he exponen ial law α e sus ln Eco esponding o p o ons launched
om z0= 2RL. Dipole ( op), magne o ail day-side injec ions (bo om le ) and magne o ail nigh -side
injec ions (bo om igh ). We can obse e simila linea ends in all cases. The p o ons a e escaping
as e in he dipole case compa ed wi h he magne o ail day-side injec ion when he ene gy is e y la ge,
since he e is no e ec o a magne o ail. Howe e , he p o ons escape as e in he case o he nigh -side
magne o ail.
ac oss he h ee di e en scena ios conside ed. We obse e ha he expec ed exponen ial beha io
holds in all h ee models. The a ionale behind his linea end is in ui i ely unde s andable. As
demons a ed in ou p e ious wo k [31], he ac ion o apped pa icles dec eases wi h ene gy.
Since his ac ion is desc ibed by he exp ession e−α , i is easonable o expec ha αscales
linea ly wi h ln E.
The op panel plo s he exponen o he decay law, α, agains ln E o he dipole model on
he day side. The end closely ollows a s aigh -line i ( 2= 0.995). The middle panel o
Fig. 10 shows he Luhmann ail model o day-side injec ions. He e, he plo exhibi s a simila
linea end wi h 2= 0.994. Howe e , i is no iceable ha p o ons escape sligh ly as e in he
20
-2 -1 0 1 2
Log z0
-0.2
-0.15
-0.1
-0.05
0
α
FIG. 11. Plo s o he coe icien o he exponen ial law α e sus log z0 o he nigh side in which he p o ons
ha e E= 25 MeV and we a y he injec ion heigh z0. A ough exponen ial decay can be obse ed whe e
he p o ons escape as e inso a he dis ance zis inc easing.
dipole model. To quan i y his di e ence, we de ine he ac ion =∆α
∆ ln Eas an indica o o he
escape eloci y. Speci ically, o he dipole case, ≈0.022, while o he Luhmann ail model,
≈0.027, sugges ing ha he magne o ail e ains pa icles o a sho e du a ion be o e hei
escape.
The bo om panel depic s o he Luhmann ail model, nigh -side injec ions. The linea end
be ween αand ln Eis also e iden , wi h an 2= 0.995. He e, ≈0.02, indica ing ha p o ons
ake longe o escape he in luence o he magne ic ield compa ed o p e ious cases, owing o he
e ec o he ail. Mo eo e , hey ake mo e ime han on he day-side injec ions case. One can
conclude ha he in luence o he magne o ail is s onge o pa icles injec ed om he nigh -side.
This di e ence migh a ise because p o ons a e close o he ail on he nigh side.
We also in es iga ed he e ec o a ying he injec ion heigh on he escape ime, while keeping
he ene gy o he p o ons ixed. Fo his, we se E= 25 MeV and a y z0∈[0.25,5]RLon he
nigh side. Figu e 11 shows he exponen o he decay law, α, plo ed agains ln z0. An exponen ial-
like beha io is obse ed, wi h p o ons escaping mo e apidly o smalle alues o z0.
VI. EXIT BASINS AND FRACTAL DIMENSION
In he p e ious sec ions, we ha e analyzed he empo al beha io o he p o ons as hey we e
injec ed. This sec ion will nume ically explo e he complexi y o he dynamics o he exi basins.
21
The goal is o in es iga e how hese basins e ol e wi h he con ol pa ame e s o he model, com-
pa ing he beha io s o day-side and nigh -side pa icles, and ela ing he apping s uc u es o he
empo al beha io s, esidence imes, and escape ime dis ibu ions discussed ea lie .
In dissipa i e sys ems, a basin o a ac ion is de ined as he se o poin s ha , when aken as
ini ial condi ions, a e a ac ed o a speci ic a ac o . When wo dis inc a ac o s exis wi hin a
ce ain egion o phase space, wo basins a e o med, sepa a ed by a basin bounda y. This basin
bounda y may be a smoo h cu e o , in some cases, a ac al cu e. In ou s udy, we ha e obse ed
ha he injec ed pa icles can ei he emain bounded, all o Ea h, o escape a e some ime,
exi ing he sys em h ough a speci ic exi egion. The e o e, in analogy o basins o a ac ion, we
de ine exi basins (o escape basins) as he se o ini ial condi ions ha lead o a pa icula exi
[7]. When wo o mo e escape egions a e possible, ac al bounda ies can eme ge, e lec ing he
sys em’s chao ic dynamics. In such cases, he bounda y sepa a ing one basin om ano he is no
clea ly de ined, leading o unp edic able beha io .
Figu e 12 compa es he s uc u e o he exi basins in he dipole and Luhmann ail models. The
escaping pa icles a e p o ons wi h ene gies o 25 MeV and 100 MeV, and hei ini ial posi ions
(x0, y0, z0)lie on cons an z=z0planes, as indica ed. The egion (x0, y0)∈[−10,0]RL×
[−10,10]RLis di ided in 100 ×200 small cells, wi h pa icles launched om he cen e o each
cell owa d Ea h. The ajec o ies a e compu ed up o a ime o Te= 30 s, bu he ime in eg a ion
s ops when < 1(Ea h all, shown in ed in he igu e) o > 80 (escape om Ea h). When a
pa icle escapes, we dis inguish be ween z(Te)>0(No h escape, shown in blue) and z(Te)<0
(Sou h escape, shown in g een). The emaining ini ial condi ions, co esponding o pa icles ha
emain apped nea he Ea h, a e plo ed in whi e. The ime spen be o e escaping o alling is
indica ed by he g ading be ween whi e and he co esponding colo .
The No h exi , Sou h exi , and Ea h- alling basins exhibi ac al cha ac e is ics. The a ea o
hese ac al egions inc eases wi h he dis ance om he equa o ial plane o he injec ion poin
(i.e., wi h inc easing z0) and wi h wi h he ene gy o he pa icle. The p esence o he magne o ail
signi ican ly enhances he ac ali y o he basins. Addi ionally, he ime spen by he pa icle
be o e alling o Ea h, indica ed by he in ensi y o he ed do s wi hin he whi e egions, also
displays ac al beha io .
In o de o quan i y his e olu ion o he ac ali y o he basins, we ha e compu ed he box-
coun ing dimension Do he ini ial condi ions co esponding o he con ined p o ons, ha is, he
whi e egions o Fig. 12. Figu e 13 shows he a ia ion o he box-coun ing dimension Dwi h
22
Dipole 25 MeV D-Tail 25 MeV N-Tail 25 MeV Dipole 100MeV D-Tail 100MeV N-Tail 100MeV
z0= 0.5RL
z0=RL
z0= 2RL
z0= 3RL
FIG. 12. Exi basins o he dipole and Luhmann models. Each panel ep esen s he ini ial condi ions plane,
x0−y0ini ial condi ions plane in which |x0| ∈ (0,10)RLand y0∈(−10,10)RL. G een co esponds
o unbounded ajec o ies escaping wi h z > 0, while blue co esponds o hose escaping wi h z < 0.
Red means pa icles alling o Ea h. Whi e means con ined pa icles. F om op o bo om, p o ons a e
injec ed a z0= 0.5,1.0,2.0,3.0RL. The i s h ee le mos columns co espond o 25 MeV, while he
h ee igh mos columns o 100 MeV.
23
he heigh o he injec ion poin z0. We ha e depic ed di e en physical si ua ions o in e es : he
nigh side o he ail model (deno ed in blue and da k blue), he day side o he ail model (deno ed
in ed and pu ple) and he dipole model (ligh g een and g een), whe e all o hem o 25 MeV
and 100 MeV, espec i ely. We can obse e ha he smalle he z0, he la ge he alue o he
dimension o he whi e a ea, meaning i almos co e s a su ace.
1 2 3 4 5
z
0
0.5
1.0
1.5
2.0
D
N-Tail 25 MeV
N-Tail 100 MeV
D-Tail 25 MeV
D-Tail 100 MeV
Dipole 25 MeV
Dipole 100 MeV
FIG. 13. Va ia ion o he ac al dimension Do he a eas co esponding o con ined pa icles, when
launched a di e en z0. He e, we show di e en physical si ua ions: he nigh side o he ail model
(deno ed in blue and da k blue), he day side o he ail model (deno ed in ed and pu ple), and he dipole
model (deno ed in ligh g een and g een), whe e 25 MeV and 100 MeV, espec i ely.
A dec easing end o Dwi h z0is clea ly obse ed bo h o p o ons wi h E= 25 MeV and
o hose wi h la ge ene gy alue, E= 100 MeV. The la ge he ene gy, he smalle ac ion o
pa icles ha emains bounded, hence, he smalle he box dimension. The e is a s eep dec ease o
he dimension when z0>2.0. Mo eo e , beyond z0≈3.5, he numbe o ini ial condi ions leading
o bounded ajec o ies is e y small, and one can obse e some luc ua ions on he compu ed
alues o he box dimension Da la ge z0. Rega ding he dipole da a, a he la ges z0, he e a e
no con ined mo ions, hence no da a o plo .
VII. CONCLUSIONS AND DISCUSSION
Ou wo k has been ocused on analyzing he dynamics o p o ons injec ed in o Ea h’s mag-
ne osphe e a a ying heigh s, z0, using ools om nonlinea dynamics and chaos heo y. These
p o ons a e he esul o neu on decay nea Ea h. Howe e , gi en he high eloci ies o he
24
neu ons, we can conside injec ions occu ing a g ea e dis ances om Ea h as well.
We ha e i s analyzed he p o on dynamics analy ically o he case o a pu e dipola magne ic
ield. Howe e , o explo e a mo e ealis ic scena io ha includes a magne ic ail, a nume ical
app oach is necessa y. Ou in es iga ion e ealed ha he chosen in eg a ion scheme mus be
highly accu a e o accoun o pa icles alling in o he Ea h’s poles, due o he s ong sensi i i y
o ini ial condi ions in his scena io.
When analyzing esidence ime plo s, e en conside ing ou ini ially limi ed se o ho izon al
launches o de ining he ini ial condi ions o he p o ons, we ha e obse ed ha he models al-
eady p oduce s uc u es esembling he inne bel . He e, he ole o he ail does no seem o be
e y c i ical in c ea ing hese s uc u es, which a e o med by bo h bound and unbound p o ons.
We ha e also analyzed he decay laws associa ed o hose popula ions. In he ela i is ic egime,
decay laws a e exponen ial, as expec ed. In e es ingly, he shapes o he adia ion bel s do no
change signi ican ly when compa ing he dipole and magne o ail models. Howe e , he magne o-
ail appea s o ha e a apping e ec on he esidence imes o he injec ed pa icles. This e ec ,
al hough mode a e, is clea ly no iceable. The in luence o he magne o ail is s onge o pa i-
cles injec ed om he nigh side, possibly because p o ons a e close o he ail on he nigh side.
An exponen ial-like beha io can be obse ed, wi h p o ons escaping om Ea h mo e apidly o
smalle alues o z0. The decay laws also show ha he escaping popula ions exhibi some an-
sien beha io . The highe he ene gy o he p o on, he sho e he du a ion o he ansien . Only
a e some ime pos -injec ion does he ene gy spec um o he popula ions s abilize o a gi en z0.
Because apped p o ons a e mo e likely o in e ac wi h he Ea h’s a mosphe e, hese low-ene gy
popula ions could be he mos p one o such in e ac ions [32].
Finally, we ha e also analyzed he exi basins. They show he ex en o which he esul ing
p o ons emain bound. The No h exi , Sou h exi , and Ea h- alling basins exhibi a ac al na u e,
wi h he a ea o he ac al egions inc easing wi h bo h he dis ance om he equa o ial plane o
he injec ion poin (inc easing z0) and he ene gy o he pa icle. The p esence o he magne o ail
signi ican ly inc eases he ac ali y o he basins. A p elimina y compu a ion o he ac al di-
mension o he basin bounda ies, D, as a unc ion o he dis ance z0, shows a dec ease in Dwi h
inc easing z0.
Ou Luhmann model is s ill e y simple, neglec ing he il o he dipole, day/nigh side asym-
me ies o he magne ic lobes, and ing cu en s. I is also a s a ic model, igno ing any high-
equency luc ua ions ha migh a ec he dynamics o he s udied pa icles. Howe e , despi e
25
sonably la ge ime s eps o abou 10−2, excep du ing pola pene a ions, which equi e a educ ion
o he ime s ep by wo o de s o magni ude.
To p ese e conse ed quan i ies, his ime-s eppe has been sui ably modi ied by adding a
p ojec ion o he p edic ed ajec o y o e he in a ian mani old, as desc ibed in [24–26]. In all
he cases explo ed in his wo k, such p ojec ion me hod led o phase space in eg a ion o bi s ha
nume ically p ese e ene gy (and angula momen um, in he dipola model) almos o machine
p ecision.
Figu e 4 shows he ajec o y o a p o on in he magne o ail ield, and i s ow o igu e 16
shows he ajec o y o a p o on wi h he same ini ial condi ions in he dipole ield. Bo h ajec o-
ies look chao ic, and show pene a ions along he wedges o he lunula egion whe e he p o ons
a e con ined. The di e ences a e clea ly seen in he second ow o Fig. 16. This igu e shows
empo al se ies o x,θand ℓin he wo cases (dipole and magne o ail). The mean angula eloci y
⟨˙
θ⟩is sligh ly la ge in he p esence o he magne o ail, as shown in he θ( )plo . Time s eppe s
p ese ing he conse ed quan i ies (ene gy, and ℓin he dipole case) ha e been used. The ime
se ies o ℓ( )shows ha in he dipole case ℓis p ese ed (e o less han 10−14), while in he
magne o ail case i shows empo al oscilla ions, ye e y iny.
10!10 10!5
Tol
10!2
10!1
E o h_
3i
10!15 10!10 10!5
Tol
10!14
10!12
10!10
10!8
"`
10!15 10!10 10!5
Tol
10!15
10!10
"u
FIG. 17. Figu e showing e o es ima es as a unc ion o esolu ion. We obse e ha a subs an ial educ ion
in e o equi es ole ances smalle han app oxima ely 10−12.
E o es ima es and con e gence es s o nume ical in eg a ions a e usually conduc ed by com-
pa ing esul s ob ained wi h di e en ime s eps. In ou case, since we use a iable ime s eps, we
assess nume ical accu acy based on he ole ance equi ed in he Runge-Ku a p ojec ion me hod.
Figu e 17 p esen s hese accu acy es s o he ajec o y illus a ed in Fig. 16. The second
32
and hi d panels show he maximum de ia ion o he conse ed quan i ies along he ajec o y o
di e en ole ances, εℓ=||ℓ−ℓo||∞and εu=||u−uo||∞. We obse e ha o ole ances less han
o equal o 10−10, he conse ed quan i ies emain cons an up o machine p ecision. The i s panel
displays he ela i e e o o he mean azimu hal angula eloci y o di e en ole ances. ⟨˙
θ⟩is a
highly sensi i e a iable, as i c i ically depends on he accu a e esolu ion o pola pene a ions,
especially o chao ic ajec o ies such as he one conside ed in Fig. 16. The e o is compu ed by
compa ing wi h he smalles ole ance used. We obse e ha a subs an ial educ ion ( h ee o de s
o magni ude) in he e o equi es ole ances smalle han app oxima ely 10−12. Fo his eason,
we ha e used a e y small ole ance o 3×10−14 in all compu a ions.
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