Particle acceleration and entropy considerations

9 Pages·0.11 MB·English

by  John C Brown

Particle acceleration and entropy considerations
∗
JohnC Brown†, GregoryBeekman†, NormanGray†
AlexanderLMacKinnon‡
5
9
12 January1995
9
1

n
Abstract
a Maxwellian distribution might place a theoretical thermody-
J
namic constraint on the fraction of magnetic energy release

3 Possibleentropyconstraintsonparticleaccelerationspectraare likely to go into such acceleration. Brown (1993) has raised
2
discussed. thisissueagainandillustratedthepointwithasimpleexample

ofthethermal/nonthermalentropydifference.
1 Solarflaremodelsinvokeavarietyofinitialdistributionsof
v theprimaryenergyreleaseovertheparticlesoftheflareplasma Inthispaperweputthisquestiononamorequantitativebasis
3
–ie.,thepartitionoftheenergybetweenthermalandnonther- byexaminingtheentropyofmoregeneralparticledistribution
8
mal components. It is suggested that, while this partition can functions. Essentially the idea is that, solely on energy con-
0
1 take any value as far as energyis concerned, the entropyof a servationgrounds(firstlawofthermodynamics),depositionof
0 particledistributionmayprovideausefulmeasureofthelikeli- energyE amongN coldparticlescouldequallywellresultin
5 hoodofitsbeingproducedforaprescribedtotalenergy. (for example) a Maxwellian of temperature 2E/3Nk, a sin-
9
The Gibbs’ entropy is calculated for several nonthermal gleparticleofenergyE withtherestcold,ora deltafunction
/
h isotropicdistributionfunctionsf,forasingleparticlespecies, distribution of all the particles at particle energy E/N. In a
p andcomparedwiththatofaMaxwellian,alldistributionshav- statistical sense, however, it is obvious that the last two are
-
o ing the same total number and energy of particles. Specula- much less probable than the first – ie., they have much lower
r tions are made on the relevance of some of the results to the entropy. (Evolution of a plasma distribution function f from
t
s cosmic ray power-law spectrum, on their relation to the ob- a cold Maxwellian to a nonthermal form with high mean en-
a
served frequency distribution of nonthermal flare hard X-ray ergydoesofcourseinvolvea particleentropyincreaseand so
:
v spectrum parameters and on the additional energy release re- isnotprecludedbythesecondlawofthermodynamics. How-
i
X quiredtoachievelowerentropyfs. ever,whilenotimpossiblesuchanoutcomeisstatisticallymuch
r Keywords:entropy;acceleration;flares;cosmicrays. less probable than production of a hot Maxwellian f since it
a haslowerentropy). Quantifyinghowmuchlower the entropy
(probability)isshouldhelpconstrain,inastatisticalsense,the
1 Introduction likely forms of particle distributions produced by prescribed
energy release conditions or conversely constrain the energy
releaseconditionsrequiredtoyieldaprescribedparticledistri-
Models in which energetic particles are invoked as a major
bution.
component of primary energy release in the impulsive phase
of solar flares have been both popular and controversial for We recognise at the outset that the present calculations are
exploratory and rest on simplifying assumptions which need
decades(see eg, reviews(Brown, 1991; Simnett, 1991)). De-
furtherstudy. Inparticularweassume thatentropyconsidera-
spite everimprovingdata andtheoreticalmodelling,however,
itremainsunresolvedhowimportantsuchparticlesareinflare tionsarerelevant–thisraisesissuesregardingequilibriumand
closureofthesystem,forexample,whichhavebeenaddressed
energy transport. Brown and Smith (1980) suggested that
inbroadtermsbyTolman(1959,pp.560-564)–butwefeelthat
the lower entropy of accelerated particles compared to a pure
this will be so provided we deal only with newly-accelerated
∗ToappearinA&A.Preprintno.Astro-95/02,andastro-ph/9501083. particles outside the (presumably small) accelerating volume.
†Dept. ofPhysics and Astronomy, University ofGlasgow, Glasgow G12
Secondly, we compare here only the entropies of alternative
8QQ,U.K.
‡Dept.ofAdultandContinuingEducation,UniversityofGlasgow,Glasgow formsoftheparticledistributionfunctionwhichcouldbegen-
G128QQ,UK eratedbydepositionofenergyE amongN particlesin a pre-
1
scribed volumeV. We neglect, forthe moment, entropycon- = kN0 g(u)lng(u)d3u
tributionsfromothersystemdegreesoffreedomand,inpartic- − Zu
ular,fromplasmawaves,andfromthechangingmagneticfield 1
presumedresponsibleforgeneratingtheflareparticledistribu- +kN0(cid:26)lnv03− N0 Zrn(r)lnn(r)d3r(cid:27) (4)
tion. The possible importance of including plasma waves as
additionaldegreesoffreedominconsideringtheevaluationof where
plasmaentropyduringparticleaccelerationhasbeendiscussed N0 = n(r)d3r (5)
in (Grognard, 1983). Our neglect of magnetic field entropy Zr
is based on the heuristic argument that the field is associated isthetotalnumberofparticles. Ifweconsiderdifferentdistri-
solelywithanorderedzeroentropyflowcomponentofthepar- butionfunctionsf withthesamespatialdistributionn(r)and
ticles–theGibbsentropy flnfd3vofanyparticledis- totalvolume(ie., same N0) and use the same velocityunitv0
tribution function f(v) is∼unRcRhRanged by adding a systematic throughoutthenwecanmeasuretheentropydifferencesbythe
driftv0 (ie.,v v+v0). ‘scaledentropy’
→
1 1
2 Definitions Σ= S kN0 lnv03 n(r)lnn(r)d3r
kN0 (cid:18) − (cid:26) − N0 Zr (cid:27)(cid:19)
(6)
InthispaperwewilldealsolelywiththeGibbsentropySofthe sothatby(5),Σcanbeexpressedsolelyintermsofg(u),via
3-D (velocity space) distribution function f(r,v) for a single
speciesofnonrelativisticparticleofmassm,definedby Σ g = g(u)lng(u)d3u (7)
{ } −Zu
S = k flnfd3rd3v (1) andwenote,intermsofg,normalisation(3)is
− ZvZr
where k is Boltzmann’sconstant. For simplicity, we consider g(u)d3u=1. (8)
only distributions in which the velocity v distribution is sep- Zu
arable from the configuration space r dependence. With this
We donotconsiderfurtherherethespatialstructurecontri-
restriction,wemaywrite
butiontotheentropy(lasttermin(6))–i.e.,weonlyconsider
f(r,v)=n(r)G(v) (2) entropiesofdifferentg(u)butthesamen(r).Also,throughout
the rest of this paper we consider only isotropic f(v) so that
wheren(r)isthetotalparticlespacedensityatrand ud3u→ 0∞4πu2du. Wewillbecomparingtherelativeen-
tRropiesofvRariousdistributionsgwiththatofapureMaxwellian
G(v)d3v=1 (3) ofthesameN0andofthesametotalenergyE(ie.,wearecom-
Zv paringidenticalplasmas with the same total energydeposited
in them). Itis thereforeconvenientto use as velocityunitv0,
Because (1) is not linear in f, the absolute value of S de- thethermalspeedintheMaxwellianplasma
pendsonthevelocityandcoordinateunitsused.Here,however,
we will only be concerned with entropy differences between v0 =(2kT0/m)1/2 (9)
differentG(v). Ifwetakesomecharacteristicspeedv0 asve-
locityunitandwrite v = v0uandG(v) g(u)/v03 then(1) whereT0 = 2E/3N0k isthetemperatureoftheplasmawhen
and(3)yield → all of E is deposited as heat in a pure Maxwellian. With
this choice of v0 and with g isotropic the normalisation con-
S = k n(r)g(u)ln[n(r)g(u)/v3]d3rd3u dition (8) ensuring the same number of particles is that g(u)
0
− ZuZr shouldsatisfy
∞
1
= k n(r)lnn(r)d3r g(u)d3u g(u)u2du= (10)
− Zr Zu Z0 4π
k n(r)d3r g(u)lng(u)d3u whiletheconstraintensuringthesametotalenergybecomes
− Zr Zu ∞
3
4
+klnv03Zrn(r)d3rZug(u)d3u Z0 g(u)u du= 8π. (11)
2
3 Scaled entropies of specific distribu- 3.3 Pure powerlaw gPL(u)
tion functions
Acommonlyoccurringformofacceleratedparticlespectrumin
solarflaresandelsewhereinastrophysicsisthepowerlawv−α.
3.1 PureMaxwelliangM(u) Suchf(v)divergesunlessflattenedortruncatedatsmallv, so
wefirstconsideratruncatedpurepowerlawcomprisingallthe
Expressed in terms of g, a pure Maxwellian satisfying con-
plasma particles and again with the same total energy as the
straints(10)and(11)is
pureMaxwellian3.1.Withnumberconstraint(10),thishasthe
e−u2 form
gM(u)= π3/2 (12) 0 −α u<u1
g =α 3 u (17)
which, in (7), integrates straightforwardly to give the scaled PL  4π−u31 (cid:18)u1(cid:19) u≥u1
entropy
3 wherethedimensionlesslowenergycut-offspeedu1hastobe
Σ = (1+lnπ). (13)
M 2 relatedtoαtosatisfytotalenergyconstraint(11),viz.
3.2 Maxwellianwithbump intailgBIT(u) 3α 5 1/2
u1 = − (18)
(cid:18)2α 3(cid:19)
One of the forms of distribution function with an accelerated −
component is a Maxwellian with a ‘bump in tail’. Here we
and we note that α > 5 is required for finite total energy.
consideronlyasimplecasetoallowanalytictreatment,namely
We note that the relation between α and the spectral index δ
abumpintailcentredonspeedu1,andofnarrowwidth∆u normally used (in solar flare physics) to describe the non-
≪
1 over which the bump contribution to g is a constant added relativisticparticlefluxspectrumdifferentialinkineticenergy
to the Maxwellian componentwhich describes the rest of the
(Brown,1971)isα=2δ+1.
plasma(cfBrown(1993)). Becauseofenergyconstraint(11),
Substitutionof(17)and(18)in(7)leadstothescaledentropy
thetemperatureofthisMaxwelliancomponentisreducedbya
factorτ <1relativetothepureMaxwellianofcase3.1,bythe δ+ 1 27π2(δ 2)3
energyresidentinthebump. Ifthefractionoftheparticlesby ΣPL(δ)= δ 12 + 12ln(cid:20) 2 (δ−1)5(cid:21) (19)
numberinthebumpandin theMaxwellianarerespectivelyφ − −
and(1 φ)(tosatisfythenumberconstraint(10))then
− 3.4 Maxwellianwitha powerlaw tail
φ
g (u)= 1−φ e−u2/τ + 4πu2∆u uin∆u The conventionalwisdom regardingthe behaviour of the real
BIT (πτ)3/2  01 otherwise distributionfuntionforthebulkoftheelectronsinasolarflare
(14) isthatitisroughlyMaxwellianatlowvelocities(wherecolli-

with sionsdominate)butofpowerlaw(orothernonthermalform)at
1 2φu2 high velocities (where mean free paths become long and run-
τ = − 3 1 (15)
1 φ awaymayoccur). Sucha particlemodelspectrumfindssome
−
and we note that the condition u2 3/2φ has to be met supportinrecentinversionsofhighresolutionbremsstrahlung
1
≤ hard x-ray spectra (Johns and Lin, 1992; Thompson et al.,
since τ 0, equality holding when the Maxwellian compo-
≥ 1992;Piana,1994)thoughtheseinferencesareambiguousbe-
nentof(1 φ)N0 remainscold.
− cause of the effects of noise (Craig and Brown, 1986) and of
Substituting (14) and (15) in (7) and approximatingthe in-
tegrals on the assumptionthat ∆u 1 and φ/(4πu2∆u) averaging over an inhomogeneous source (Brown, 1971). It
1
1 φe−u21/τ/(πτ)3/2 (ie, the bump≪is locally ‘large’)we o≫b- isthereforeinstructivetoconsidersuchadistributionfromthe
− viewpointofentropy. Aconvenientfunctionalformwhichex-
tainforthescaledentropy
hibitstheseasymptoticbehavioursatlowandhighv,andwhich
ΣBIT(φ,u1,∆u)= circumvents most of the analytic messiness associated with a
piecewisedescriptionisf(v) (1+v2/v2)−α/2 (Robinson,
φ 3 1
∼
−φln(cid:18)4πu2∆u(cid:19)+ 2(1−φ) 1993). The high velocity spectral index is α, as in 3.3, and
1 the temperature T defining the shape at small v is fixed by α
1+lnπ 5ln(1 φ)+ln 1 2φu21 (.16) andv1 suchthat 21mv12 =αkT. Similarlytocases3.2and3.3,
×(cid:20) − 3 − (cid:18) − 3 (cid:19)(cid:21) α and v1 have to be interrelated to satisfy constraints (10)
3
and (11), namely u1 = v1/v0 = (α 5)/2. Taking these
−
constraintsintoaccount,theresultinpgg(u)is
2 3/2 Γ α 2 −α/2
2 2
g (u)= 1+ u
MPL (cid:20)π(α 5)(cid:21) Γ (cid:0)α−3(cid:1) (cid:18) α 5 (cid:19)
− 2 −
(cid:0) (cid:1) (20)
whereΓ(x)isthegammafunction.
Substitutionof(20)in(7)gives,forthiscase(withα=2δ+
1asinSect.3.3)
Figure1isinfilefig1.ps
Γ(δ+ 1)
Σ (δ) = ln 2
MPL − (cid:20)π3/2(δ 2)3/2Γ(δ 1)(cid:21)
− −
(δ+ 1)[ψ(δ 1) ψ(δ+ 1)](21)
− 2 − − 2
wherethepsifunctionis
d
ψ(x) [lnΓ(x)]. (22)
≡ dx
4 Entropy comparison of nonthermal
Figure 1: Scaled entropy ∆Σ Σ Σ, relative to the
M
≡ −
distributions with a pure Maxwellian Maxwellian for (a) a pure truncated power law and (b) a
Maxwellianwithapower-lawtail.
We now compare the entropies of nonthermal distributions
(16),(19),and(21)withthatofthepureMaxwellian(13).
4.2 Pure powerlaw
Infigure1a,weshowthequantity∆Σ (δ)=Σ Σ as
4.1 Bump-in-tail PL M − PL
a functionof δ forthe purepower-lawcase. Thekey features
ofthiscurveare
This case was already briefly discussed by Brown (1993). Its
mostimportantfeaturesare:
1. Thereisaminimumin∆ΣPL whichoccursatδ = δ0
≡
(3 + √7)/2 2.82. This is the value of δ at which a
1. Asφ 0,thepureMaxwellianentropy(13)isrecovered. ≈
→ sharplytruncatedpowerlawmostcloselyapproachesthe
2. Foranyφ,u1(ie,nomatterhowfewparticlesareacceler- entropyoftheMaxwellian.
atedortohowlowaspeed),therelativeentropy
→ −∞ 2. As δ , Σ and ∆Σ + . This is
as∆u 0–ie,asafinitenumberofparticlesinf(u)is → ∞ PL → −∞ PL → ∞
→ because as δ , more and more of the particles are
crowdedintoanarbitrarilynarrowvelocityrange.Consid- → ∞
eringtherelativeentropyasthenegativeofthelogofthe concentrated at a single velocity u1 = 3/2 (by (18))
and the distribution is more and more lipke a delta func-
probabilityP,thisquantifiesthequalitativeimprobability
tion,withassociatedlowprobability(cf.,4.1). Thesame
statementinSect.1concerningtherelativeimprobability
limitationapplieshereasinpoint2ofSect.4.1.
of nonthermal distributions. We note, however, that the
divergence of Σ to is an artefact of taking the ap- 3. As δ 2, ∆Σ + logarithmically. This is be-
proximatecontinuum−f∞unctionexpression(1)forentropy cause→thedistribuPtLion→func∞tionthenbecomesveryflatand
toapointwherethesmoothnessassumptionmadebreaks widewithadecreasingprobabilityoftheincreasingnum-
down(seeAppendix). berof particlesneededatu to satisfy thetotalen-
→ ∞
ergyconstraint(11).
3. Nomatterhowsmallthefractionφofparticlesaccelerated
is,asu21 → 32φ,ΣBIT →−∞becausethefractionofthe AnotherwaytoexpresstheentropycomparisonbetweengM
total energyin fast particles 1 atthatpoint, which is and g is to consider the temperature T (or total energy)
→ PL δ
arbitrarilyimprobable.Thesameremarkasin2applies. whichapureMaxwellianwouldneedtohaveinordertohave
4
as low an entropy as gPL with the same total number N0 of in ∆ΣMPL which tends to + as δ 2 and falls as δ in-
∞ →
particles. The generalisation of (12) to a Maxwellian of tem- creases. This is essentially because, for a prescribedtotal N0
peratureτ = T/T0 is gM(u,τ) = exp( u2/τ)/(πτ)3/2 and andE,thesteeppowerlawtailmoreandmorecloselyresem-
−
of(13)is bles the Maxwellian for large δ. On the other hand as δ de-
Σ (τ)= 3[1+ln(πτ)]. (23) creases,theentropybecomessmallerandsmallerasdeviation
M 2
fromtheMaxwellianincreases. Ifthisresultisinterpretedsta-
Equating(23)with(19)wecanfindthedimensionlesstem-
tisticallyin termsoftherelativelikelihoodofatailofgivenδ
perature τ(δ)=Tδ/T0requiredfortheMaxwellianentropyto occurring, as comparedto pure heating, it means that smaller
beaslowasthepowerlaw,viz. lnτ =2[Σ Σ ]/3,or
PL PL− M andsmallerδshouldoccurlessandlessfrequently.
4 δ 3 δ 2 Observational inference of δ in solar flares is achieved
lnτPL(δ)= 3(δ− 1) +ln(cid:20)(2π)1/3(δ −1)5/3(cid:21) (24) from the spectral index γ of bremsstrahlung hard X-rays (eg,
− − Brown(1971)). Intermsoftheelectronaccelerationspectrum
whichalsohasamaximumatthevalueofδ =δ0. theeventintegratedγisrelatedtoδbyδ =γ+1(forcollision
Theaboveresultsallowsomequantitativeinterpretation,al- dominatedthicktargetenergylosses)andby δ =γ 1forcol-
−
beit somewhat speculative. First, if we consider transient en- lisionally thin targets. Kane (1974)has reported the statistics
ergyreleaseevents,thentheentropiesofdistributionfunctions of observed occurrence of different γ values. The qualitative
ofdifferentδshouldbemeasuresoftherelativeprobabilitiesof trend of these data is similar to that predictedfrom the above
theseδsbeingrealisedif alltheenergywentintoapurepower- entropyargument–ie,asharpdeclineofnumbersofeventsof
law. Ina largeset ofsuch eventsthese shouldreflectin some smallspectralindexandafrequencydistributionflatteningoff
waytherelativefrequencyofoccurrenceofdifferentδs. Inpar- atlargerindices(thoughthedataareinstrumentallylimitedto
ticular,valuesofδnearδ0,whereΣPLhasamaximum,should indicesγ <6).
be commonest (This is certainly not the case in the observed Inview∼ofthesimplicationsmadeinthepresenttheoretical
frequencydistribution of flare δ values inferred from hard X- analysis(homogeneoussourcevolume,neglectofwavesandof
ray bursts, but nor are flare spectra those of pure power laws anisotropy)the trend of the theoreticaland observed distribu-
withnoMaxwelliancomponent!–cf.Sect.4.3). tionsisremarkablysimilar,andworthyoffurtherinvestigation.
Second, we note thatthe abovepower-lawentropyanalysis
alsoappliestotherelativisticregimewhenvisreplacedbythe
5 Conclusions
momemtump andthemaximumentropypower-lawspectrum
againhasindexδ0. Thisδ0 isveryclosetotheobservedmean
spectralindexofcosmicrayparticlesoveraverywideenergy The lower entropy of a hot plasma with an accelerated com-
range(Longair,1981)–afactpointedouttousbyBell(1994). ponent, as compared to a pure Maxwellian of the same total
Thatis,ifanunspecifiedmechanismoperatestoforceapower- particlenumberandenergy,quantifiestheextentto whichthe
lawformonthedistributionfunction,underconditionsofcon- hardest non-thermalcomponentsare further from equilibrium
stanttotalparticlenumberandenergy,thenthehighestentropy andsoshouldoccurlessfrequently.Calculationoftheentropy
(most likely) state is that with a power-law index close to the difference in the case of a Maxwellian with a power-law tail
observedcosmic-rayone. Thisfactshouldbeexploredfurther. of indexδ leadsto a predictedfrequencydistributionofspec-
tral indices in qualitative agreementwith the general trend of
the distributioninferredfromthe statistics of flare hard X-ray
4.3 Maxwellianwithpower-law tail
bursts.
In figure 1b we show the variation with δ of ∆Σ (δ) AlldistributionsexcepttheMaxwellianarenon-equilibrium
MPL
≡
Σ Σ (equivalenttoτ )forthiscase,definedsimilarly ones, which means that we cannot define a temperature for
M MPL δ
−
tothosein4.2,basedonequations(13)and(21),viz. themandso,inturn,thatwecannotuseourexpressionforthe
entropies of these distributions in any non-trivial purely ther-
3 Γ(δ+ 1)
∆Σ (δ) = +ln 2 modynamical argument. We can, however, make a cautious
MPL 2 (cid:20)(δ 2)3/2Γ(δ 1)(cid:21) statistical argument,aswe havedonein Sects. 4.2and 4.3, to
− −
+(δ+ 1)[ψ(δ 1) ψ(δ+ 1)](25) theextentthat, givena processwhichproducesa certainnon-
2 − − 2
(26) thermalparticledistribution,weexpectsomecharacteristicsof
that distribution to be more likely than others. We have no
Inthiscase,whichisexpectedtobemuchclosertothesolar doubtthattherearemore, andmoredetailed,argumentstobe
flare situation than the pure power law, there is no extremum madeusingtheseresults.
5
The close coincidence of the observed cosmic-ray spectral G(u )δ3u particles in the volume centred on u . Conse-
i i
indexwiththatofamaximumentropypurepower-lawislike- quently,thetotalnumberofwaysofdistributingtheN particles
wisetantalising. amongstthecellswillbethemultinomialΩ = N!/ N !. If
i i
we take δ3u to be large enough that each of the NQi is large,
thenwecanapproximatelnΩusingStirling’sformulalnN!
Acknowledgements ≈
NlnN N,toobtain
−
GB gratefully acknowledges the support of a Royal Society lnΩ N g(u )lng(u )δ3u. (28)
i i
≈−
of Edinburgh Cormack Vacation Research Scholarship. JCB, Xi
NG,andALMgratefullyacknowledgethesupportofaPPARC
wherewehaveusedthenormalisation
grant, an EEC Contract, and of the Research Centre for The-
oreticalAstrophysics,UniversityofSydney,wherepartofthis N = G(u )δ3u=N, (29)
i i
workwascarriedout.Thepaperhasbenefitedfromdiscussions
Xi Xi
oftheentropyproblemwithnumerouscolleaguesandinpartic-
ularwithDBMelrose,PRobinson,A.M.Thompson,andAR andthenusedthelargenessofG(u )δ3utoignorelnδ3u,com-
i
Bell. pared with lnG(u ), before substituting G(u) Ng(u). If,
i
finally,thescaleδ3uissmallenoughthatweca≡ntakethedis-
tributionfunctiong(u)tobeconstantoverit, thenwe canap-
Appendix: the limitations of Eq. (7)
proximatethis final estimate for the distributionfunction(28)
byanintegral,andwritetheentropy(27)as
As pointed out in Sect. 4.1, the divergence of the ‘scaled en-
tropy’ Σ to −∞ is an artefact of the approximationsmade in Σ[g] S[g]/Nk= g(u)lng(u)d3u, (30)
deriving(1).Tomakethisclear,wepresenthereabriefderiva- ≡ −Z
tionof(7)whichconcentratesonthestepsbeforeEq.(1)rather
wherewehavereplacedtheapproximationsignbyanequality.
than,asinSect.2,onthephysicsbehindthedefinitionofΣ.
Compare(7). Crucially,thevalidityoftheapproximation(30)
Asabove,wecanwrite(2)
dependsonthedistributionfunctionG(u)beingsuchthatthere
f(r,v)=n(r)G(v), canbeinfactascaleδ3uwhichsimultaneouslyhassufficiently
large N and sufficiently constant g(u). If this is not so, we
i
where n(r) and G(v) are position and velocity distribution generatetheartefactsdescribedinSects.4.1and4.2.
functions,respectively. Sincetheseareindependent,thestatis-
tical weightfor the distributionf(r,v) – the numberof ways
References
of realising it – is simply Ω[f] = Ω[n(r)] Ω[G(v)], so
that the entropyof the distributionf(n,v) wil×l be the sum of
Bell,A.R.,1994, personalcommunication
the entropies in position and velocity spaces separately; this
means that we can immediately ignore the constant contribu- Brown,J.C.,1971, Sol.Phys.18,489
tionofn(r)totheentropy,andinsteadconcentrateontheen-
tropyofthevelocitydistributionalone.Intheseterms,thisis Brown, J.C., 1991, in(CulhaneandJordan,1991),pp413–
424
S[G]=klnΩ[G], (27)
Brown,J.C.,1993, in(Uchidaetal.,1993)
where k is Boltzmann’s constant, and Ω[G] is the statistical
Brown,J.C.andSmith,D.F.,1980, Rep.Prog.Phys.43,125
weightofthedistribution,orthenumberofwaysinwhichthe
distributioncanberealised. Notethat,sinceΩ[G]isapositive Craig, I. J. D. and Brown, J. C., 1986, Inverse Problems in
integer,theentropyS[G]hasaminimumofzero(whichoccurs Astronomy, AdamHilger(Bristol)
whenG(v)isadeltafunction–thereisonlyonewayofgiving
Culhane, J. L. and Jordan, C. (eds.), 1991, The Physics of
alltheparticlesthesamevelocity).
First definea dimensionless‘velocity’u v/v0, forsome SolarFlares,Vol.336AofPhil.Trans.Roy.Soc.London
≡
arbitraryspeedv0. Imaginedividingtheaccessible volumeof Grognard,R.J.-M.,1983, Phys.Lett.95A(8),432
velocityspaceintoafinitenumberν ofsmall,butnotinfinites-
imal, volumes, size δ3u, at u , so that there will be N Johns,C.M.andLin,R.P.,1992, Sol.Phys.137(1),121
i i
≡
6
Kane, S. R., 1974, IAU Symposiumno.57, pp105–141,D.
Reidel
Longair,M.S.,1981, HighEnergyAstrophysics, Cambridge
UniversityPress
Piana,M.,1994, A&A288(3),949
Robinson,P.,1993, personalcommunication
Simnett,G.M.,1991,in(CulhaneandJordan,1991),pp439–
450
Thompson, A. M., Brown, J. C., Craig, I. J. D., and Fulber,
C.,1992, A&A265(1),278
Tolman,R.C.,1959, PrinciplesofStatisticalMechanics, In-
ternationalSeriesofMonographsonPhysics,OxfordUniver-
sityPress
Uchida, Y., Watanabe, T., Shibata, K., and Hudson, H. S.
(eds.), 1993, X-Ray Solar Physics from Yohkoh, No. 10 in
FrontiersScience,UniversalAcademicPress,Inc
7
This figure "fig1-1.png" is available in "png"(cid:10) format from:
http://arXiv.org/ps/astro-ph/9501083v1
(a)
3
2
L
P
Σ
-
M
Σ 1
0
2 4 6 8 10
δ
(b)
4
3
PL 2
M
Σ
- M 1
Σ
0
−1
2 4 6 8 10
δ