Analytical view of diffusive and convective cosmic ray transport in elliptical galaxies

0.68 MB·English

by  T. Hein

Astronomy&Astrophysicsmanuscriptno.transport˙arxiv c ESO2008
(cid:13)
February2,2008
Analytical view of diffusive and convective cosmic
ray transport in elliptical galaxies
8 T.HeinandF.Spanier
0
0
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Lehrstuhlfu¨rAstronomie,UniversityofWu¨rzburg,AmHubland,D-97074Wu¨rzburg
n
a
J

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Received10.09.07,accepted07.11.07
1

]
h
p ABSTRACT
-
o
r
t Context.Ananalyticalsolutionofthegeneralizeddiffusiveandconvectivetransportequationis
s
a derivedtoexplainthetransportofcosmicrayprotonswithinellipticalgalaxies.
[
Aims.Cosmicraytransportwithinellipticalgalaxiesisaninterestingelementinunderstanding

1
theoriginofhighenergeticparticlesmeasuredonEarth.Asprobablesourcesofthosehighener-
v
9 geticparticles,ellipticalgalaxiesshowadenseinterstellarmediumasaconsequenceofactivity
2
inthegalacticnucleusormergingeventsbetweengalaxies.Thusitisnecessaryforanappropri-
6
1 atedescriptionofcosmicraytransporttotakethediffusiveandconvectiveprocessesinadense
.
1 interstellarenvironment intoaccount.Hereweshow thatthetransportequationscanbesolved
0
analyticallywithrespecttothegivengeometryandboundaryconditionsinpositionspace,aswell
8
0 asinmomentumspace.
:
v Methods.FromtherelativisticVlasovequation, whichisthemostfundamental equation for a
i
X kinetic description of charged particles within the interstellar medium in galaxies, one finds a
r generalized diffusion-convection equation in quasilinear theory. This has the form of a ‘leaky
a
box’ equation, meaning particlesareabletoescapetheconfinement regionbydiffusingout of
thegalaxy.Weapplyherethe‘diffusionapproximation’,meaningthatdiffusioningyrophaseand
pitchanglearethefastestparticle-waveinteractionprocesses.Ananalyticalsolutioncanbeob-
tainedusingthe‘scatteringtimemethod’,i.e.separationofthespatialandmomentumproblems.
Results.Thespatialsolutionisshownusingageneralizedsourceofcosmicrays.Additionally,
thespecialcaseofajet-likesourceisillustrated.Wepresentthesolutioninmomentumspacewith
respecttoanescapetermforcosmicrayprotonsdependingonthespatialshapeofthegalaxy.For
adelta-shapeinjectionfunction,themomentumsolutionisobtainedanalytically.Wefindthatthe
spectralindexmeasuredonEarthcanbeobtainedbyappropriatelychoosingofthestrengthof
FermiIandFermiIIprocesses.Fromtheseresultswecalculatethegamma-rayfluxfrompion
decayduetoproton-protoninteractiontogiveconnectiontoobservations.Additionallywedeter-
minetheescape-spectrumofcosmicrays.Theresultsshowthatbothspectraareharderthanthe
intrinsicpower-lawspectrumforcosmicraysinellipticalgalaxies.
Keywords.cosmicrays:general–transportmodel–diffusiveprocesses–convectiveprocesses
2 T.HeinandF.Spanier:Cosmicraytransportinellipticalgalaxies
1. Introduction
Since theirdiscoveryby ViktorHessin 1912,cosmicrayshavebeenone ofthe biggestfields of
interestinastrophysics,andyettheoriginoftheseparticlesisstillanopenquestion.Fullyionized
atomic nuclei reach the Earth coming from outside the solar system with very high energies up
to1020 eV.Mostofthemwithenergies< 1017 eV seemtooriginateintheMilkyWay,whilethe
highestenergeticonesare consideredto haveanextragalacticorigin(fora reviewsee Hoerandel
2007).
The most acceptedmodelfor the originof ultrahighenergycosmic rays(UHECRs) is accel-
erationinshockfrontsduetoFermiprocesses(Fermi1949);hence,themainsourcesaregamma
raybursts(GRBs),activegalaxieslikeactivegalacticnuclei(AGN)(Tavecchio2005),orcolliding
galaxies. The last two are specially interesting for two reasons. First UHECRs can be generated
in elliptical galaxies. Second the increase in the interstellar medium density in objects of these
typesinfluencescosmicraytransport(see Bekki&Shioya1998).Butsinceallellipticalgalaxies
haveaninterstellarmediumduetostarwinds,weconcludethatthestudyoftransportprocessesis
interestingingeneral(seeKnapp1999).
Inparticular,asaresultoftheGZK-effect,verycloseAGNarethemostprobablecandidates
forUHECRsources(seeBiermann1995).OneofthesenearbyAGNisthegiantellipticalgalaxy
M87. It is proposed that this galaxy is responsible for acceleration of cosmic ray protonsdue to
Fermi I processes (Blandford&Ostriker 1978; Riegeretal. 2007) in shockfronts within the jet
(Reimeretal. 2004). For an overviewof protonaccelerationin jets (see Mannheim1993). Since
particleaccelerationinajetislocatedwithinactivegalaxiessurroundedbyaninterstellarmedium,
thehighenergeticprotonsundergophysicaltransportprocessesbeforetheyescapeoutofthegalaxy
and reach the detectors on Earth, making a model for cosmic ray transport in elliptical galaxies
inevitable.Thishelpsgiveananswertothequestionabouttheoriginofcosmicrays.
Progress has been made in the field of modeling cosmic ray transport with the numerical
description of transportprocesses by Owens&Jokipii (1977) and Strong&Moskalenko (1998).
NeverthelesswefollowthebasicideaspresentedintheunderlyingpapersofLerche&Schlickeiser
(1985), Wang&Schlickeiser (1987), and Lerche&Schlickeiser (1988), who used an analytical
description of cosmic ray transport. In relation to our work such, a treatment has the following
advantages: the model is adequate for cosmic ray transport within any kind of elliptical galaxy
includingarbitrarycosmicraysourcesandthe physicalparametersinvolvedinourmodelcanbe
easily fitted to measurements. After all, our analytical model can serve as a test case for more
profoundnumericalmodels.
Inthispaperwesolvethecosmicraytransportequationanalyticallywithrespecttoa kinetic
descriptionof the interstellar plasma in elliptical galaxies. Specialattention is paid to the spatial
transportofchargednuclei.Inaddition,thesolutionofthemomentumequationisderivedto ex-
plain generalpropertiesof this model.As a result, we presentillustrative examplesof spatial, as
wellasmomentum,cosmicraytransportforgivensourcesofchargednuclei.Toshowtheconnec-
tiontoobservations,wecalculatethegamma-rayfluxfromneutralpiondecay.Thesemesonsare
producedby inelastic scattering processes between cosmic ray protons. The resulting power-law
Sendoffprintrequeststo:F.Spanier,
e-mail:[email protected]
T.HeinandF.Spanier:Cosmicraytransportinellipticalgalaxies 3
spectrum is slightly harder than the intrinsic one for cosmic rays. Finally, we presentthe escape
spectrumofchargedparticlesleavingellipticalgalaxies.Similartothegamma-rayflux,thisspec-
trumisflatterthantheintrinsicone.
2. Basicequations
To describe the propagation of cosmic ray nuclei within elliptical galaxies, we follow
Lerche&Schlickeiser(1985), Lerche&Schlickeiser(1988)andSchlickeiser(2002).Forthe de-
scriptionoftransportprocessestheyusethe‘diffusionapproximation’,whichmeansthatthefastest
particle-plasmawave interactionprocessesare diffusionin gyrophaseandpitch angle.Thusfol-
lowingJokipii(1966),Hasselmann&Wibberenz(1968),andSkilling(1975),wetakeanisotrope
particledistributionfunctioninmomentumspace.Hereweidealisetheinterstellarmediumasaho-
mogeneousvolumecontainingprimarycosmicraysbeingacceleratedfromthethermalbackground
mediumandsecondariesresultingfromfragmentationofprimarieshavinganegligibleabundance
inthebackgroundmedium(cf.Hayakawa1969;Cowsik1980).
The transport of these particles at large momenta (p > 10 GeV c 1 nucleus 1) is described
− −
bythesteady-statetransportequation(e.g.Schlickeiser1983).Suchatreatmentissuitabletoshort
timescalesofdiffusiveandconvectiveprocessescomparedtothedynamicaltimescaleofthegalaxy
(t 109years).Thisistrueinthecaseofhighenergeticparticles.Weassumeaspatialdiffusion
dyn
≈
coefficientK(r)of1029 cm2 s 1 at p =1GeV,whichisslightlylargerthanthevaluemeasuredin
−
theMilkyWay(K(r)=1028cm2,(cf.Schlickeiser2002)becauseofdiffusiveprocessesbeingless
effective in elliptical galaxies. Consequently we get for protons with TeV-energy a timescale of
108 years.Furthermore,thedynamicalageofthegalaxyhastobegreaterthanthetimescaleof
≈
sourcevariabilitytoobtainanappropriatedescription.Thisisusuallygiven,sincethesize ofthe
accretionregionontothe centralblackhole isof the orderof a few light-daysso thata maximal
variabilitytimescaleofsomedaysisassumed.
At large momenta, spatial diffusion in turbulent magnetic fields dominates convectionin the
galactic wind, so that we find a transport equation for the phase space density f(r,p) in spatial
coordinatesrandinthemomentumcoordinatep:
f + f +S(r,p)=0. (1)
r p
L L
Thespatialoperator isdefinedby
r
L
(r,p) K(r,p) (2)
r
L ≡∇ ∇
(cid:2) (cid:3)
containing spatial diffusion with the spatial diffusion coefficient K(r,p) = K(r)κ(p), where κ(p)
denotesthedimensionlessdependenceonthemomentumvariablepwithoutlossofgenerality.The
momentumoperator
∂ ∂ 1
(r,p) p 2 p2D(p) p2p˙ p2p˙ (3)
Lp ≡ − ∂p ∂p − gain− loss − τ
" # c
describes momentumdiffusion by second-orderFermi processes (D(p)), energy gain due to first
order Fermiprocesses (p˙ ), as well as continuous(p˙ ) and catastrophic (τ ) momentumloss
gain loss c
processes.AsshowninLerche&Schlickeiser(1985),fully-ionizedparticlesheavierthanprotons
havethesameFermiaccelerationratesasprotons,sohereinaftermomentummeansmomentumper
4 T.HeinandF.Spanier:Cosmicraytransportinellipticalgalaxies
nucleon.We areinterestedprimarilyinthebehaviourofthecosmicrayprimaryspectrum,sowe
donottakesecondaryparticlesduetofragmentationofprimariesintoaccount.Ontheotherhand,
weallowfragmentationofprimariesasagenerallossprocess.
Alinkbetweenspatialandmomentumdiffusionprocessescanbeseenintherelationbetweenthe
twodiffusioncoefficients
C p2
D(r,p)= 1 , (4)
K(r,p)
whereC standsfortheproportionalityfactorbeingindependentofrand p.Thiscloseconnection
1
arisesfromthe samebasic physicalprocessbehindspatialandmomentumdiffusion:Protonsare
scatteredinpitchangleduetothemagneticfieldsofMHDplasmawavescausingspatialdiffusion
alongorderedmagneticfieldlines,whereascyclotrondampingoftheelectricfieldassociatedwith
MHD waves affects diffusion in momentum space. For that reason it is necessary to solve this
modelinspatialcoordinatesaswellasinmomentumcoordinatestogetanacceptabledescription
oftransportprocessesinellipticalgalaxies.
3. ‘Scatteringtime’method
Weusethe‘scatteringtime’methodproposedbySunyaev&Titarchuk(1980)togetanimportant
classof exactanalyticalsolutionsofEq.(1)followingWang&Schlickeiser(1987). Thisimplies,
thatthespatialandmomentumoperatorscanbeseparatedas
(r,p)=h(p) (r) , (r,p)=g(r) (p). (5)
r r p p
L O L O
Foreaseofexposition,thesourcefunctionS(r,p)isalsoaproductoftwoseparablefunctions,i.
e.,
S(r,p)=q(r)Q(p). (6)
AsanasidewenotethattherequirementEq.(5)istriviallyfulfilled,ifK(r)isconstant(K(r)=K )
0
and p˙ , p˙ , as well as τ , are all independentof spatialvariables. Thuswe use the following
gain loss e
modelcontainingconstantfactorsa withn =1,2,3,4fortheproportionalityfactorsindependent
n
ofspatialandmomentumcoordinates.FirstwetakeD(p)= a p2/κ(p)todescribethemomentum
2
diffusioncoefficientsolelyinmomentumspace.Furthermoreweassume
p˙ =a p/κ(p) (7)
gain 1
tellingthatfirst-orderFermiaccelerationisrelatedtothe(momentumdependend)spatialdiffusion
coefficientdueto MHD plasma-wave-scatteringinteractionswithinthe accelerationprocess.The
continuouslossterm p˙ isindependentofspatialcoordinatesleadingto
loss
p˙ (p)= a ρ(p). (8)
loss 3
−
Similarly,weset
τ (p)=a 1θ(p) (9)
c −4
forthecatastrophiclosstimeduetofragmentation.
Undertheseconditions,inadditiontoEq.(5)and(6),wecanfindtheformalmathematicalsolution
ofEq.(1)asaconvolutionofthespatialandmomentumsolutionfunctionsT(r)andM(p)following
deFreitasPacheco(1971):
f(r,p)= ∞duT(r,u)M(p,u), (10)
Z0
T.HeinandF.Spanier:Cosmicraytransportinellipticalgalaxies 5
whereT(r,u)hastosatisfythegivenspatialboundaryconditions,and
∂T 1
= T (11)
∂u g(r)Or
with
= [K ] K ∆ , g(r)=1 (12)
r 0 0
O ∇ ∇ ≡
andtheconditions
T(r,u=0)=q(r)/g(r)=q(r) (13)
T(r,u= )=0. (14)
∞
Here,M(p,u)hastosatisfythegivenspatialboundaryconditions,and
∂M 1
= M (15)
∂u h(p)Op
with
1 ∂ a p4 ∂ a p3 a
= 2 1 +a p2ρ(p) 4 ; h(p)=κ(p) (16)
Op p2∂p κ(p) ∂p − κ(p) 3 − θ (p)
" # 4
andtheconditions
M(p,u=0)= Q(p)/h(p)= Q(p)/κ(p) (17)
M(p,u= )=0. (18)
∞
Asaconsequenceoftheformalmathematicalsolution,wenotethatwehavetosolvetwopartial
differentialEqs.(11)and(15)insteadofthemuchmorecomplicateddifferentialEq.(1).
4. Results
4.1.Spatialsolution
ThemostconvenientwaytofindtheformalsolutionofEqs.(11)and(15)istostartwiththespatial
problem. As can be seen from Eq.(12), the spatial operator is of Sturm-Liouville type (cf.
r
O
Arfken&Weber2005)andthereforehasacompleteeigenfunctionsystemE(r).Asaconsequence,
i
thesolutionfunctionT(r,u)canbeexpandedinthisorthonormalsystemas
T(r,u)= Ai(r)e−λ2iu. (19)
i
X
HeretheA(r)aredefinedby
i
A(r)=αE(r), (20)
i i i
implyingthatthecoefficientsα weighteacheigenfunctionE(r).Theλ denotetheeigenvaluesof
i i i
implyingthespecialspatialgeometry.
r
O
The shape of elliptical galaxies is adjusted to the cosmic ray transport Eq.(12) using prolate
spheroidalcoordinatesastheyaredefinedbyAbramowitz&Stegun(1972):
r +r r r
ξ = 1 2 , η= 1− 2 (21)
2f 2f
AscanbeseenfromFig.(1)r andr arethedistancestothefocioftheconfocalellipse,where2f
1 2
denotesthe distancebetweenthe two foci F and F . Additionally,we use the variableφ forthe
1 2
usualazimuthaldependencelikeinsphericalcoordinates.Thefollowingrelationsgivetherelation
betweenthesecoordinatesandthesemi-majoraxisaandthesemi-minoraxisb,respectively:
a= fξ , b= f ξ2 1. (22)
−
q
6 T.HeinandF.Spanier:Cosmicraytransportinellipticalgalaxies
Fig.1.Schematicalviewofanellipsewithitsfundamentalproperties.
Thenumericalexcentricityehasadirectrelationshiptothecoordinateξvia
e f/a=1/ξ. (23)
≡
Thevariableηisdefinedasη = cos θwithθtheanglebetweenthelineonwhichthefocilie,and
anarbitrarypointontheellipse.Asaresultthevariablesaredefinedintherange
ξ [1; [ , η [ 1;1] , φ [0;2π]. (24)
∈ ∞ ∈ − ∈
Fig.2. Left: Gray-shadedplane cuts ellipse leading to a cut view like the graph on the left-hand
side.
Right:Directionofunitvectorsofthevariablesξ,η,andφinprolatespheroidalcoordinates.
FromWeisstein(1999).
Figure (2) shows an illustration of the definition of the three spatial variables ξ, η, and φ.
Becauseofthesedefinitions,wecanwriteEq.(12)as
∂T K ∂ ∂T ∂ ∂T
= 0 (ξ2 1) + (1 η2) +
∂u f2(ξ2 η2) × ∂ξ − ∂ξ ∂η − ∂η
− ( " # " #
ξ2 η2 ∂2T
+ − . (25)
(ξ2 1)(1 η2) ∂φ2
− − )
T.HeinandF.Spanier:Cosmicraytransportinellipticalgalaxies 7
HereweusedtheLaplacianinprolatespheroidalcoordinates.Thegeneralsolutioncanbeobtained
byconsecutiveseparationofvariables(seeAppendixA):
T(ξ,η,φ,u) = R(1)(c,ξ) S(1)(c,η) cos(mφ) exp( k2u)=
mn × mn × × −
m,n
X
1
= ∞′ (2m+r)!dmn − ξ2−1 m/2
Xm,n∞′rX=(02,1m+rr)!!dmn r π J× (cξξ2) ! ×
× r! r 2cξ n+21 ×
r=0,1 r
X
dˆmn(c)Pm (η) cos(mφ) exp( k2u) . (26)
× r m+r × × −
o
Here the Pm (η) denotes associated Legendre functions of the first kind, order m + r, and the
m+r
J (cξ)denoteBesselfunctionsofthefirstkindandofordern+1.Thesumisextendedovereven
n+1 2
2
valuesofrasindicatedbythemark .Thefactorcisgivenbyc= fk .
′ √K0
Todefinereasonableboundaryconditions,weassumea‘leakybox’model.Cosmicrayparticles
aretrappedbydisorderedmagneticfieldswithintheconfinementregionofanellipticalgalaxy.In
thistheyundergodiffusiveandconvectivemovements.Attheedgeofthebox,leakageoutofthe
confinementareaispossible.
As an illustrative example for spatial boundary conditions, we show the solution depending
on a constant source function over the elliptical galaxy in Appendix B. To be more specific, we
take a jet-like source function here. The jet points in the direction η (represented by a Dirac
inj
delta function) with a length scale chosen to be f for any choice of ξ being smaller than an
max
arbitrarymaximumvalueoftheconfinementregionξ .Particlesleakoutattheedgeofthisregion.
c
Such a boundaryconditionis knownin the literature as a ‘free-escape’condition.For a realistic
assumptionwedecidetoletthejetendsmoothly(seethe‘Fermi’functioninEq.(29)).Weneglect
anydependenceonφforanadequateillustration.Theseconditionsaretakenintoaccountby
T(ξ=ξ ,η,φ,u)=0, (27)
c
byaperiodicalboundaryconditioninη
T(ξ,η=1,φ,u)=T(ξ,η=+1,φ,u), (28)
andby
δ(ξ ξ )δ(η η)
c inj
T(ξ,η,u=0)=q − − . (29)
0 exp(4(f f )+1
max
−
FinallyaftersomecalculationsinwhichEq.(26)hastomatchtheboundaryconditionsEq.(27-29),
(cid:2) (cid:3)
wederivethegeneralsolution(cf.AppendixB):
E (r) T(ξ,u)= ∞ ∞′ α 1 J y ξ P0(η)exp K0y2iru . (30)
i,r ≡ i=1 r=0 ir √ξ r+12 irξc! r −ξc2f2 
Theweightingfactorsare XX  
α = q0P0r(ηinj) 1ξcξ3/2Jr+12 yirξξc dξ. (31)
ir exp 4(f − fmax) +1 × R ξ2c2 Jr+3((cid:16)yir) 2(cid:17)
2
(cid:2) (cid:3) h i
Herethey standsforzerosofJ (x).Thesumoverrisextendedoverevenvaluesofthisparam-
ir r+1
2
eter.WeseethatthegeneralsolutionofEq.(25)withrespecttospatialboundaryconditionsindeed
8 T.HeinandF.Spanier:Cosmicraytransportinellipticalgalaxies
Fig.3.GraphicdemonstrationofthecosmicrayparticledensitygivenbythesolutionEq.(30)with
the weighting factors (31). The cosmic ray particle density is normalised and given in arbitrary
units.Asaneffectofchosencoordinatesonly‘onehalf’ofthegalaxyisvisible.Forillustrationwe
plottedT(f,η,u=0)withaconstantξ 1.2specifyinganE4ellipticalgalaxy.
≡
hasthe formoftheeigenfunctionexpansionEq.(19)with Eq.(20). ThissolutionisshowninFig.
(3)withrespectto100spatialeigenfunctions(i = 10,r = 18).Thejetisresponsibleforthe
max max
cosmicrayparticlesdistributedoverthewholegalaxydueto diffusiveprocesses.Itis broadened
withalargerdistancetothecentre.Thiseffectiscausedbythechosengeometry,whereastheloss
ofmagneticcollimationinrealastrophysicalsourcescanprovidethis.Toillustratetheshapeofan
ellipticalgalaxyweplottedT(f,η,u= 0)withaconstantξ 1.2specifyinganE4-galaxy.Using
≡
a more complicated source function, a better physical description of particle distribution within
ellipticalgalaxiescanbeobtained.
4.2.Consistencychecksoftheformalspatialsolution
Togetabetterunderstandingofourmodel,weprovetheformalmathematicalsolution(Eq.(26))
ofthespatialcosmicraytransportequation.Ourdiscussionisrelatedtothesolutionfoundinthe
illustrativeexampleinAppendixB,butcansimilarlydonewithEq.(26):
T(ξ,u)= ∞ α 1 J y ξ exp K0y2i u (32)
i=1 i √ξ 12 iξc! −ξc2f2 
withtheweightingfactors X  
q ξgξ3/2J y ξ dξ
α = 0 1 21 iξc . (33)
i R ξc2 J (y(cid:16)) 2 (cid:17)
2 3 i
2
Hereweusedperiodicalboundaryconditionsforhηandiφ
T(ξ,η= 1,φ,u)=T(ξ,η=+1,φ,u), (34)
−
T(ξ,η,φ=2π,u)=T(ξ,η,φ=0,u), (35)
and
T(ξ,η,φ,u=0)=q Θ(ξ ξ). (36)
0 g
−
T.HeinandF.Spanier:Cosmicraytransportinellipticalgalaxies 9
First,thissolutionhastoaccomplishthegivenspatialboundaryconditions.AssuggestedinEq.(36)
we useda constantsourcefunctionoverthe whole size of the galaxyin orderto lose all angular
dependencies. This behaviour is caused by the source function not having any dependence in η
and φ, but also due to the periodic boundary conditions chosen for these variables. Such an ef-
fectcanalso be seen ina sphericalor disc geometry.To the sphericalgeometryas performedby
Schlickeiseretal.(1987),weaddedangulardependenciesandaconstantsourcefunctionoverthe
galaxyasanexample.Theweightingfactorsofthegeneralsolutionfunctionsturnouttodisappear
inanycaseexceptfortheangularseparationconstantsbeingzero(theangularsolutionfunctions
aresphericalharmonicsinthiscase),whichmeansthatthereisnodependenceonthesevariables
asitisexpected.
Fig.4.Normalisedweightingfactorsα dependentonthenumberofzerosy inEq.(32)
i i
Second,weshowthatthesumofsolutionfunctions(cf.Eq.(26)),togetherwiththeweighting
factorsinadditiontothegivenspatialboundaryconditionsconverges.Fortheillustrativeexample
inAppendixB,thenormalisedexpansioncoefficientsα areshowninFig.(4).Toobtainapplicable
i
resultsit is essentialto includeonlythe firstfew onesdependingon the requestedaccuracy.The
resulting solution function is shown in Fig.(5). In the special case of a constantsource function,
the solution reproducesthe boundarycondition θ(ξ ξ) for the variable ξ. We used ξ = 4 for
g g
−
illustration.Ifwetakemanyeigenfunctionsintoaccount,weseeatthediscontinuitypointsξ = 0
andξ =ξ =4anoscillatoryphenomenon(Gibbsphenomenon).Butinthiscaseaclosersolution
g
forthegivenboundaryconditionisobtained.
Inanalyticalcalculations,itisacommonassumptiontoincludeonlythefirstspatialeigenvalue
λ . This one is associated with the longest escape timescale being the most important one for
1
modellingescapeofparticlesoutofthegalaxy.Fornumericalpurposes,thecomputingtimegives
anupperlimittothepossiblenumberofeigenvalues.Furthermore,thespatialsolutionasperformed
in this paper has to match the formalsolution for a spherical geometrywithin the limit of small
ellipticity(e 0).Schlickeiseretal.(1987)foundasthespatialsolution
→
T(R,u)= ∞ c 1 J y R exp K02y2mu , (37)
m=1 m √R 12 mR1! − R21 
X  
10 T.HeinandF.Spanier:Cosmicraytransportinellipticalgalaxies
Fig.5.SpatialsolutionEq.(32)inadditionwith weightingfactorsα (Eq.(20))asafunctionofξ.
i
As an illustration, we putξ = 4. For the solution representedby the solid line we included200
g
eigenvalues,whereasinthedashedsolutiononly10eigenvaluesaretakenintoaccount.
whichalreadyshowsaffinityto oursolutionEq.(32).They denotesthe zerosof J (x) (herewe
m 1
2
corrected Schlickeiseretal. (1987)) and R describes the edge of the galaxy. Generally we can
1
writeforbothsolutions:
T(X,u)= ∞ αZ(cX)exp( λ2u) (38)
i i i − i
i=1
X
TheasymptoticbehaviourofthesolutionEq.(32)iswithZ(cX) ξ 1/2J (cξ)
i − 1 i
≡ 2
Z(cξ) ciξ→∞ 1 cos cξ 1π . (39)
i −−−−−→ cξ i − 2
i !
Thelimitcξ satisfiesasphericalsymmetrywithanumericalexcentricityequaltozero(cf.
i
→ ∞
Eq.(23)).Furthermorewiththislimit,thesemi-majoraxiscξgoestoinfinity.Performingthesame
i
limittoEq.(37)withZ(cX) R 1/2J (cR),weget
i − 1 i
≡ 2
Z(cR) ciR→∞ 1 cos cR 1π , (40)
i i
−−−−−→ cR − 2
i !
which is equal to Eq.(39). As a consequencewe showed that our solution functionsembodythe
sphericalgeometrywithinthelimitofnumericalexcentricityequaltozero.Itisalsostraightforward
toshowthattheweightingfactorshavethesamestructureasthoseinSchlickeiseretal.(1987)so
thatthegeneralsolutioniscorrect.
4.3.Momentumsolution
Fortheformalmomentumsolution,itisnecessarytohaveacloserlookatthespatialsolution.As
notedabove,theeigenfunctionexpansionEq.(19),inadditiontoEq.(20),isindeedthebestwayto
solve the spatial transportEq.(11). Inserting this expansioninto the convolutionEq.(10), we can
write
f(r,p)= A(r)R(p) (41)
i i
i
X
with
Ri(p) ∞duM(p,u)e−λ2iu. (42)
≡
Z0
ConsequentlythemomentumsolutionR(p)obeystheordinarydifferentialequation
i
R(p) λ2g(p)R(p)= Q(p), (43)
Op i − i i −