Hard X-ray Spectra and Positions of Solar Flares observed by RHESSI: photospheric albedo, directivity and electron spectra

0.24 MB·English

by  J. Kasparova

Astronomy&Astrophysicsmanuscriptno.kasparova07 (cid:13)c ESO2008
February5,2008
Hard X-ray Spectra and Positions of Solar Flares observed by
RHESSI: photospheric albedo, directivity and electron spectra
JanaKasˇparova´1,EduardP.Kontar2,andJohnC.Brown2
1 Astronomical Institute, Academy of Sciences of the Czech Republic, Fricˇova 298, 251 65 Ondˇrejov, Czech Republic e-mail:
[email protected]
7 2 DepartmentofPhysicsandAstronomy,UniversityofGlasgow,G128QQ,UK
0 e-mail:eduard,[email protected]
0
Received*,2006;accepted*,2006
2

n ABSTRACT
a
J Aims.WeinvestigatethesignatureofthephotosphericalbedocontributioninsolarflarehardX-rayspectra,theeffectoflowenergy

0 cutoffsinelectronspectra,andthedirectivityofhardX-rayemission.
3 Methods.UsingRamatyHighEnergySolarSpectroscopicImager(RHESSI)flaredataweperformastatisticalanalysisofspatially
integratedspectraandpositionsofsolarflares.

1 Results.Wedemonstrateclearcentre-to-limbvariationofphotonspectralindicesinthe15-20keVenergyrangeandaweakerdepen-
v dencyinthe20-50keVrangewhichisconsistentwithphotosphericalbedoasthecause.Theresultsalsosuggestthatlow-energycut-
1 offssometimesinferredinmeanelectronspectraareanartefactofalbedo.Wealsoderivetheanisotropy(ratioofdownward/observer
7 directedphotons)ofhardX-rayemissioninthe15-20keVrangeforvariousheliocentricangles.
8
Keywords.Sun:flares,Sun:X-rays,Sun:particleemission,Scattering,Techniques:spectroscopic
1
0
7
0 1. Introduction found, contrary to above mentioned works based on hard X-
/ ray occurrence, significant centre-to-limb spectral variation in
h The successful operation of the Ramaty High Energy Solar
p the 10- 100keV rangeandattributedit to anisotropyof X-ray
Spectroscopic Imager RHESSI (Linetal. 2002) since February
- emission. Then, Vestrandetal. (1987) demonstrated that Solar
o 2002 has providedus with a substantial database of solar hard MaximumMission(SMM)GRSdatainthe25-200keVrange
r X-rayflaresthatcanbeusedfordetailedspectroscopicanalysis.
andabove300keV,showsignificantspectralhardeningofevents
t
s Thehighaccuracyofthespectroscopicdata,incombinationwith nearthelimb.SimilarresultswerereportedbyBogovalovetal.
a thepositionsofsolarflares,allowsustodeterminethecentre-to-
(1985) from Venera13 data in the 50 - 100 keV range andby
:
v limbspectralvariationsinvariousenergyranges. McTiernan&Petrosian(1991)inthe0.3-1MeVrange.
i Variation of hard X-ray flare occurrence and spectral in-
X dex with solar position of the sources have been of great in- It has been known for a while that X-ray photons can be
r terestforseveraldecades.As suggestede.g. byElwert&Haug effectively backscattered by photosphere atoms and electrons
a (1971); Brown (1972); Petrosian (1973); Leach&Petrosian (Tomblin1972;Bai&Ramaty1978).Thus,atenergiesnotdom-
(1983), centre-to-limb variations could provide essential infor- inatedbyabsorptionthebackscatteredalbedofluxmustbeseen
mation on directivity of X-ray emission and hence acceler- virtuallyin everysolarflarespectrum,the degreeofthealbedo
ated electrons. Most earlier analyses concentrated on centre- contribution depending on the directivity of the primary X-ray
to-limb variation of flare occurrence. Ohki (1969) and Pinte´r flux (Kontaretal. 2006). The solar flare photonsbackscattered
(1969) reported significant centre-to-limb variation at energies by the solar photosphere can contribute significantly (the re-
above 10 keV although they did not correct the hard X-ray flectedfluxis50-90%oftheprimaryinthe30-50keVrange
flareoccurrenceforHαflarelongitudedistribution.Subsequent forisotropicsources)tothetotalobservedphotonspectrum.For
works(Drake1971;Catalano&vanAllen1973;Phillips1973; thesimplecaseofapower-law-likeprimarysolarflarespectrum
Datloweetal.1974;Kane1974)coveringvariousenergyranges (withoutalbedo),thephotonsreflectedbythephotospherepro-
from1-10keVclaimednocentre-to-limbvariationofhardX- duceabroad‘hump’component.Photosphericalbedomakesthe
rayoccurrence.Datloweetal.(1977)concludedthatanisotropy observedspectrum flatter below ∼ 35 keV and slightly steeper
in the 10 - 100 keV range was so low that it ruled out the above,incomparisonwiththeprimaryspectrum.Theamountof
downward-beamed thick target model due to lack of limb backscatteredphotonsisgivenbythescatteringcross-section
brightening predicted e.g. by (Brown 1972). Theoretical mod- and is roughly dependent on the projected area of the albedo
els (Petrosian 1973; Langer&Petrosian 1977; Bai&Ramaty patch, which makes the photospheric albedo stronger for so-
1978) predicted that at hard X-ray energies, above 15 keV, lar centre events. Photospheric albedo has been treated using
centre-to-limb variations of spectral properties should be sen- MonteCarlosimulationstocalculatetheobservedspectrumfor
sitive to electron directivity. Although such studies were not anassumedprimarypower-lawspectrum(Bai&Ramaty1978).
verycommon,Datloweetal.(1974)andRoy&Datlowe(1975) Kontaretal. (2006) developeda new approachto photospheric
albedobasedontheGreen’sfunctionsofMagdziarz&Zdziarski
Sendoffprintrequeststo:JanaKasˇparova´ (1995) that produces the correction for any solar flare X-ray
2 JanaKasˇparova´etal.:Photosphericalbedoanddirectivity
However,applyingthemodel-independentalbedocorrection
method of Kontaretal. (2006) to the August 20, 2002 flare,
Kasˇparova´etal. (2005) have clearly demonstrated that such a
high value of low-energy cutoff is removed by albedo correc-
tion for an isotropic source.Equally,the albedocorrectedpho-
ton spectra do not require a gap in the inverted electron spec-
trum(Kasˇparova´etal.2005;Kontar&Brown2006a).Usingthe
presentsurvey,wecanpotentiallyfindmanyflareswithsuchlow
spectralindicesthattheymighthaveclearlow-energycutoffap-
pearingasagapbetweenthermalandnon-thermaldistributions.
The spectrum-independent photospheric albedo correction
method is introducedin Section 2. Data selection and analysis
are discussed in Section 3. In Section 4, our surveyshows that
the centre-to-limb variations of spectral indices are consistent
with photospheric albedo. Moreover, we find limits on the X-
rayspectraldirectivityforvariousheliosphericangles.Section5
confirmsthepredictionthatthelow-energycutoffinF¯(E)isre-
movedwhenthespectraarealbedocorrectedforalleventswith
lowspectralindexfoundinthesurvey.Lastly,Section6reviews
theresultsobtainedandtheirconsequencesfortheangulardis-
tributionofX-rayemittingelectrons.
2. Photosphericalbedoandspectralindex
Thealbedocontributionproducessubstantialchangeto the ob-
served spectrum in the range between 10 and 100 keV. It be-
comes negligible for energies below 10 keV due to the photo-
electricabsorptionbyatomswhilephotonsabove100keVpen-
etratedeepintohighdensitylayersofthesolaratmosphereand
Fig.1. Photon spectrum I(ǫ) (upper panel) and spectrum in-
donotescapeback.Therefore,thereflectivityofthephotosphere
dexγ(ǫ)(lowerpanel)foramodelledprimaryphotonspectrum
hasabroadmaximuminthe30-50keVrange.Thetotalspec-
Ip(ǫ) ∼ ǫ−3 foranisotropicsourcelocatedatµ = 0.8.Primary trumintheobserverdirectionµ(µ=cosθ,whereθisthehelio-
I (ǫ) and γ (ǫ) are plotted with a dashed line and the observed
p p centricangle)canbeexpressedas
I (ǫ)andγ (ǫ)withasolidline.Horizontallinesindicatespec-
o o
tral indices γ , γ , and γ obtained by single power-law fits in I (ǫ,µ)=I (ǫ,µ)+I (ǫ,µ)
0 1 2 o p A
the15-20,20-35,and35-50keVenergyranges,respectively.
I (ǫ,µ)=G(ǫ,ǫ′,µ)I (ǫ′,µ)α(µ), (1)
A p
where I (ǫ,µ) is the primary photon spectrum of the flare,
spectrumandcanbeusedbothforforwardandinversespectrum p
I (ǫ,µ)isthespectrumreflectedintotheobserverdirection,Gis
calculations.Theinfluenceofalbedowasnottakenintoaccount A
theGreen’smatrix(Kontaretal.2006)accountingforCompton
forspectralcentre-to-limbvariationinearlieranalysesanditwas
backscattering and photoelectric absorption, and α(µ) is a pa-
wrongly dismissed (Datloweetal. 1974) as being unimportant
rameterofemissionanisotropy,roughlytheratioofthefluxto-
in the 20 - 50 keV range. However, Bai&Ramaty (1978) and
wardtheSunI (ǫ,µ < 0) = α(µ)I (ǫ,µ)andthefluxtowardthe
Vestrandetal.(1987)pointedoutthattheDatlowe’sresultsare p p
observer I (ǫ,µ). We simplify the problemhere by considering
consistent with an albedo contribution to the observed photon p
onlyα(µ)independentofǫ (cfAlexander&Brown2006).
spectrum. Using the Bai&Ramaty (1978) results, the impor-
Thephotonspectrum,I (ǫ,µ),backscatteredtowardtheob-
tanceofalbedoinelectronspectruminferencewasfirstincluded A
serverstronglydependsonthreeparameters:thespectralindex
by Johns&Lin (1992) and analysed by Alexander&Brown
of the primary spectrum, the location of the flare on the solar
(2002).
disc, andthedirectivityofX-rayemission(anisotropyparame-
On the other hand, it has been long believed that the
mean electron flux distribution F¯(E) defined by (Brown ter α(µ)). The contribution of the albedo to the observed spec-
trum is greater for flatter primaryspectra (low γ) and for solar
1971; Brownetal. 2003) can exhibit low-energy cutoff – e.g.
discflares(largeµ).Further,themoreX-raysarebeameddown-
Nittaetal. (1990) showed that photon spectra of the impulsive
wardsα(µ)>1,thelargeristhecontributionofthealbedotothe
componentflattentowardlowenergiesandsuggestedthatalow-
observedspectrum.
energycutoffofthe electronspectrumaround50keV couldbe
Theenergydependentspectralindex
responsible.Fa´rn´ıketal.(1997),usingYohkohdata,observeda
flarewithaflatspectrumbelow33keV(photonspectralindexas ǫ dI(ǫ) dln(I(ǫ))
low as 1.98).Morerecently,Kasˇparova´etal. (2005) reporteda γ(ǫ)≡− =− (2)
I(ǫ) dǫ dln(ǫ)
clearlow-energycutoffinferredfromtheobservedphotonspec-
tra of the August 20, 2002 flare, neglecting albedo. Using a for the primary spectrum γ (ǫ) substantially differs from that
p
regularised inversion technique, Kontar&Brown (2006a) re- (γ (ǫ))ofthetotalI (ǫ,µ)towardtheobserver(includingalbedo)
o o
portthreeflareswithasimilarfeature–anabsenceofelectrons –see Figure1forthecaseofγ = 3.Intherange10-25keV
p
around20- 30 keV betweenpossiblythermaland non-thermal thephotonspectrumbecomesflatter,γ <γ ,whiletheobserved
o p
component. spectralindexishigheratenergiesabove35keV,γ >γ .
o p
JanaKasˇparova´etal.:Photosphericalbedoanddirectivity 3
Using a single power-law fit in various energy ranges
(Figure 1), we can estimate the local spectral index assuming
itisconstantineachenergyrange.(Notethatgenerallyfora
photon spectrum I(ǫ) ∼ ǫ−c(ǫ), c(ǫ) = γ(ǫ) only for c = const
(Conwayetal.2003).)
Hence, as a result of heliocentric angle dependent albedo,
thespectralshapeofphotonspectrashouldvaryasafunctionof
theirpositiononthesolardisc,i.e.acentre-to-limbvariationof
spectralindicesisexpectedandthedegreeofthespectralindex
variationwithheliocentricanglecouldusefullyprovideuswith
the directivity of hard X-ray sources, i.e. the anisotropycoeffi-
cientα(µ).
3. Dataselectionandanalysis
In our analysis we used the whole RHESSI flare list (from
February 12, 2002 until April 18, 2006) to search for any evi-
denceofcentre-to-limbvariationinphotonspectra.
RHESSI has 9 detectors on board with FWHM resolution
about ∼ 1 keV. For X-ray spectroscopy analysis we use the
six front segments of the detectors ignoring detector 2, 5 and
7 due to either low resolution or the segmentation problem
(Smithetal.2002).
To handle such a large set of data, we set up an automatic
procedure which selects suitable events and retrieves the flare
Fig.2. Example of the flare spectrum for July 30, 2002 flare,
photonspectra.Tobechosenforfurtheranalysistheeventshad
17:37:36–17:37:48UT.Theupperpanelcorrespondstothereg-
tomeetthefollowingmajorcriteria:
ularisedphotonspectrum,thelowerpanelshowsthecountspec-
– Thespectraoftheeventsshouldbeobservedabove25keV. trum. Observed spectrum is plotted with thin crosses, back-
Thus,wehaveselectedflaresflaggedasobservedabovethat groundsubtractedflare spectrumwith thick crosses. The back-
energyfromthe RHESSI flare list andhavedeterminedthe groundis displayedasa histogram.Horizontalsizes of crosses
timeofpeakfluxinthe25-50keVrange.Theeventsflagged andwidthofthehistogrambinscorrespondtowidthsofenergy
asparticleeventswereignored.Thisledusto∼1500events. bands,while verticalsizesrepresent1σstatistical uncertainties
– Countratesexceedingabout2×103countspersecondperde- oftheflux.
tectorcausepulsepile-upinthedetectorsthatrequirestrou-
blesomecorrection(Smithetal.2002).Toavoidthepile-up
issues,wehavethrownawayeventswithcorrectedlivetime
counter < 90%. We disregarded also weak events (back- Theregularisedinversionofacountfluxspectrumgaveusanon-
groundsubtractedcountratelowerthan5countspersecond parametricphotonspectrumanditsuncertainties.
inthe25-50keVrange)andeventspossiblyaffectedbyX-
rayabsorptioninEarth’satmosphere(peaktimecloserthan
3.1.Photonspectraofselectedevents
60 s to RHESSI eclipse time). This reduces our survey to
∼800events. As discussed in Section 2, addition of the albedo contribution
– We have used CLEANed images in the range 10 - 20 and changes the shape of the primary spectra. To describe its ef-
20-50keVtocheckfortruesolareventsandtofindtheirpo- fects,theregularisedphotonspectrahavebeenfittedwithasin-
sitions.Finally,thetotalnumberoftheselectedflareevents glepower-lawin threeseparateenergyranges15-20,20- 35,
dropsto703. and 35 - 50 keV thus obtaining correspondingspectral indices
γ ,γ ,andγ ,respectively.
RHESSI isan instrumentwith highbackgroundthatmakes 0 1 2
background subtraction an important step in the data analysis Notethatspectrainthe15-20keVenergyrangemaybe
(Schwartzetal.2002).Thecodesearchesfortwotimeintervals, affectedbythermalcomponent(s)whichcouldbecompara-
one before and one after the flare peak, where count rates are ble to or dominate the non-thermal emission and masking
less than 0.01 of the peak count rate in 15 - 50 keV range and the albedo contribution. Considering a widely used isother-
0.1above50keV.Thebackgroundtimeintervalsaredetermined malapproximation,valueofγ0insuchflareswouldbehigher
for several energy bands separately and the background at the in comparison with events without strong thermal compo-
peak is estimated by a linear fit (Schwartzetal. 2002). See the nent.
flareexampleinFigure2. Since the countflux decreases with energywhile the back-
Thecountspectra[countssec−1 keV−1 cm−2]wereaccumu- ground level does not (Figure 2), naturally not all 703 events
latedin12stimeintervalswithaquasi-logarithmicenergybin- have significant flux above background at all three energy
ning from 10 to 100 keV to improve the signal-to-noise ratio ranges. For γ and γ analysis we accepted only those events
0 1
(SNR) at higher energies (Figure 2). After background sub- withcountfluxsignal-to-noiseratio(SNR)>3atallthreeen-
traction, theresultingpeakcountflux(onespectrumperflare) ergyranges.Furthermore,theGreen’smatricesandthereforethe
and the detector response matrix (DRM) for the instrument at correctedDRMarenon-diagonalatγ energies,thephotonflux
2
thistimeintervalwereusedasaninputfortheregularisedinver- at this energyrangedependingon the signalat higherenergies
sion routine (Scullionetal. inpreparation; Kontaretal. 2004). andmakingγ unreliableforflareswithlowSNRabove50keV.
2
4 JanaKasˇparova´etal.:Photosphericalbedoanddirectivity
Thus, forγ analysisadditionalconditionSNR > 3 in the 50 -
2
66keVenergyrangewasrequired.
Hence, the aboveconditionsselect only eventswith signif-
icant flux above the roughly estimated backgroundin energies
higherthan theenergyrangewherethe spectralindicesarede-
termined.Thisreducedoursetto398eventsforγ andγ analy-
0 1
sisandto123eventsforγ analysis–seeFigure3.Notethatthe
2
eventsarenotuniformlydistributedoverµ. Themainreasonis
thatactiveregionsandflaresarelocatedinarathersmallrange
oflatitudes.Thusfewereventsperunitµare observedclose to
the limb than at the disc centre. This also puts a constraint on
thewidthofµbinsforanisotropyanalysis–seeSection4.1and
Table2.
We also apply an isotropic (α(µ) = 1) albedo correction
using Equation (1) and the approach discussed in Kontaretal.
(2006)todeterminespectralindicesofalbedocorrectedspectra.
The relative occurrenceof spectral indices of observed spectra
andofthosecorrectedforisotropicalbedoisshowninFigure3.
Notetheexcessofeventswithγ <2whichisremovedafterthe
0
isotropicalbedocorrection.Theexcessofγ >7correspondsto
0
eventswithverysteepspectra.
4. Centre-to-limbvariation
The resulting spectral indices γ , γ , and γ of the flares are
0 1 2
shown Figure 4. There is no clear upper bound on γ val-
ues (primary and observed spectral index can be arbitrar-
ily large) andthe albedo contributionto the observed spec-
trum becomes negligible for steep primary spectrum - see
Section 2. On the contrary, the primary spectral index for
bremsstrahlungemissioncannotbelowerthan1.Therefore,
fordisplaypurposesonlyvaluesupto3(3.5forγ )areshown;
2
eventswithlargerspectralindicesfillthewholerangeofµ.
Thespectralindexγ showssignificantcentre-to-limbvaria-
0
tion(seeFigure4).Lowervaluesofγ tendtobelocatedcloser
0
to disc centre,formingan edge in the γ (µ) distribution.There
0
arenoflareswithγ < 2locatedatµ < 0.5(θ > 60o),allsuch
0
beingatµ≥0.5.Wedonotseesimilarpatternathigherenergies
(spectralindicesγ andγ )andthe Figure4 showsnocorrela-
1 2
tionwiththepositionofflaresintheseenergyranges.
As one can see from our albedo model with α , α(ǫ)
(Figure 1), the strongest centre-to-limbvariation due to photo-
sphericalbedoisexpectedatlowerenergies,i.e.forγ ,andal-
0
mostnodependencyisexpectedonflarepositionathigherener-
gies.Thecorrespondingtheoreticalmodelforsinglepower-law
primaryspectraisshownbyvariouslinesinFigure4.
Figure 4 also shows that anisotropyplays a significantrole
incentre-to-limbspectralvariationonlyatγ energies.Although
0
wehavenoinformationabouttheprimaryspectraanddistribu-
tion of primary spectral indices of analysed events, it is clear
fromthefigurethatthealbedomodelpredictions,assumingsin-
gle power-law primary spectra, are consistent with the general
variationsoftheobservedspectralindiceswithµ.Inordertoput
further constraint on the model, particularly at energies above
20 keV, spectral indices need to be determinedwith uncertain-
ties smaller than the predicted centre-to-limb variation – see
Fig.3. Top: Positions of selected flares. Crosses indicate 398
Figure4.
flares used for the centre-to-limb analysis of γ and γ , small
Following previous results (Vestrandetal. 1987; 0 1
circlescorrespondto 123flaresforthe γ analysis(seetextfor
Datloweetal. 1974, 1977), and to assess the change of 2
details).Largecirclesindicateregionsofµ = 0.2,0.4,0.6,and
spectralindices with µ, we divide the sample into two subsets;
µ<0.5,andµ≥0.5andforeachintervalwecalculatethemean 0.8. Middle and bottom: Relative occurrence of γ0 (solid), γ1
(dashed),and γ (dotted)from observedspectra and those cor-
values of γ , γ , and γ of observed and isotropic albedo cor- 2
0 1 2
rectedforisotropicalbedo,respectively.
rectedphotonspectra.WethenapplytheKolmogorov-Smirnov
(K-S) test (Pressetal. 1992) to calculate the probability that
JanaKasˇparova´etal.:Photosphericalbedoanddirectivity 5
the distributions of spectral indices at µ < 0.5 and µ ≥ 0.5
are drawn from the same parent distribution. The mean values
and the probabilities(see Table 1) show the trend expected for
isotropic albedo. The mean γ¯ value of γ increases after the
0 0
albedo correction is applied as one would expect. The mean
spectral indices γ¯, γ¯ with and without albedo correction do
1 2
not differ significantly – i.e. are within the uncertainties of the
mean values. The K-S probabilities show a similar trend, the
distributions with the albedo component removed show larger
probabilities of being drawn from the same dataset. However,
theprobabilityvaluesforobservedindicesaretoolargetoreject
the null hypothesis, partly because the albedo contribution is
smearedovertoobroadarangeofµ,∆µ=0.5.
Better results can be achievedby taking two more extreme
distinct sets of flares; one close to the limb with µ < 0.1 (θ >
84◦),wherethealbedoisnegligible,anddisccentreeventswith
µ ≥ 0.9 (θ ≤ 25◦) where the albedo should be strongest. The
valueofK-Sprobabilityfortheγ distributionsdropsto 0.9%,
0
allowingustorejectthenullhypothesisat0.05and0.01sig-
nificantlevelsandconcludethattheγ distributionsare‘sig-
0
nificantlydifferent’.
After applying the isotropic albedo correction, the γ dis-
0
tributions can no longer be considered as being different – see
Table1andtheγ distributionsinFigure5.However,thesame
0
conclusion cannot be reached for γ and γ – see Table 1. It
1 2
shouldbepointedoutthatthenumberofeventsforγ analysis
2
for those µ bins is rather low – see Table 1 – so a larger set of
eventswillbeneededtoreachanyconclusion.
4.1.DirectivityofX-rayemission
Theintensityofphotonsbackscatteredfromthephotosphereis
determinedbythedownwarddirectedprimaryphotonspectrum.
The largerthe downwarddirectivityof the source,the stronger
isthealbedocontributiontotheobservedspectrum(seethepre-
dictions for α = 1,2,4 and the model of a single power-law
primaryspectruminFigure4).We describethedownwardver-
sustowardobserverfluxratiobytheanisotropyparameterα(µ),
seeEquation1(whichwetaketobeindependentofenergy).
InordertoassesstherangeofprimaryX-raydirectivitycon-
sistent with the data, we have assumed limb spectra, I (ǫ,µ <
o
0.1), to be true primary spectra toward the observer, I (ǫ,µ) ≡
p
I (ǫ,µ<0.1),andaddedtothesethealbedocomponentforvar-
o
ious α(µ). Using Equation (1), the modelled spectra I in the
M
observerdirectionµtaketheform
I (ǫ,µ)=(cid:2)1+α(µ)G(ǫ,ǫ′,µ)(cid:3)I (ǫ′,µ<0.1). (3)
M o
Thisapproachdoesnotdecreasethesignal-to-noiseratioof
photon flux as in the case of albedo subtraction which reduces
the photon flux but keeps the corresponding uncertainty at the
samelevel(uncertaintyonthealbedocontributionissettozero.).
The K-S test was then applied to distributions of spectral Fig.4. Spectralindicesγ , γ , andγ versuscosine µ of helio-
0 1 2
indices of the modelled I and observed I spectra in several centricangleθ.Verticalerrorbarsindicateaverageuncertainties
M o
rangesofµ. The widthofthe µbinswaschosento containap- on the values as determined from single power-law fits. Lines
proximatelythesamenumberofevents.Duetothenon-uniform show the predicted dependency for single power-law primary
distributionofeventswithµ,seeFigure3,thewidthoftheµbins spectra with γ = 2.0,2.5and for α(µ) = 1,2,4(solid, dashed,
p
increasestowardsthelimb(smallerµ).Weregardthedirectivity anddottedlinesrespectively),seealsoSection4.1.Highvalues
α(µ) as consistent with the data if the K-S probabilityis larger of spectral indices are not shown to emphasize their lower
than0.05or0.01significancelevel,i.e.whenthenullhypothesis bounds. Crosses represent flares for which F¯(E) was deter-
that two distributions of observed and modelled spectral index mined fromthe totalspectra. Flares with a dip in F¯(E) are
for given α(µ) are drawn from the same distribution cannot be denotedasstars(seeSection5).
rejected.Intheγ energyrangewefindtherangesofallowable
0
directivitytobe0.5<α(µ)<4and0.2<α(µ)<5,respectively
6 JanaKasˇparova´etal.:Photosphericalbedoanddirectivity
albedocorr. µ γ¯ (N) γ¯ (N) γ¯ (N)
0 1 2
none µ<0.5 4.6±0.1(157) 4.2±0.1(157) 3.61±0.09(45)
none µ≥0.5 4.3±0.1(241) 4.07±0.06(241) 3.88±0.09(78)
K-Sprob. 0.2 0.2 0.2
α(µ)=1 µ<0.5 4.8±0.1(157) 4.3±0.1(157) 3.5±0.1(45)
α(µ)=1 µ≥0.5 4.8±0.1(241) 4.12±0.06(241) 3.6±0.1(78)
K-Sprob. 1.0 0.2 0.5
none µ<0.1 4.7±0.2(49) 4.4±0.2(49) 3.5±0.2(13)
none µ>0.9 4.2±0.2(81) 4.1±0.1(81) 3.9±0.1(29)
K-Sprob. 0.009 0.4 0.4
α(µ)=1 µ<0.1 4.8±0.2(49) 4.4±0.2(49) 3.4±0.2(13)
α(µ)=1 µ>0.9 4.8±0.2(81) 4.2±0.1(81) 3.6±0.1(29)
K-Sprob. 0.4 0.4 0.8
Table1.Meanspectralindicesγ¯,γ¯ andγ¯ forallselectedeventsatselectedrangesofµ.K-Sprob.correspondstoprobabilitythat
0 1 2
apairofdistributionscomesfromthesameparentdistribution.Thenumberofeventsineachµbinisindicatedinbrackets.
Fig.5. Distributions of γ for ‘centre’ (µ > 0.9, dashed line) and ‘limb’(µ < 0.1, solid line) flares. The left plot correspondsto
0
observedγ ,therightplotisforspectracorrectedforisotropicalbedo,i.e.withdirectivityα=1.
0
– see Table 2 and Figure 6. The results are also consistent the
hypothesis that α(µ) does not vary significantly with µ for the
0.05and0.01significancelevels.
In the energy range of γ no conclusions about anisotropy
1
canbedrawnbecausehighvaluesoftheK-Sprobabilitiesforthe
distributionofγ do notpermitusto distinguishthem,though,
1
theresultsarenotinconsistentwiththealbedomodelwhichdoes
notpredictsignificantvariationofγ withµandαinthisenergy
1
range–seeFigure4.
Finally, the small number of suitable events at high energy
prevented us from analysing γ . On the basis of the albedo
2
model,seeFigure4,wedonotexpecttofindasignificantinflu-
enceoftheanisotropyintheγ energyrange,butthisprediction
2
isnottestedbythepresenteddata.
5. Low-energycutoffin F¯(E)demandedbythedata Fig.6. Directivity at different µ determined from the distribu-
tions of γ at the limb and a given range of µ. Cross-hatched
0
Thesurveypresentedinthethispaperisalsosuitableforanalysis areasshowthedirectivityforwhichthehypothesisthattheob-
offlareswherethehardX-rayspectrummayindicatesuspicious serveddistributionatagivenrangeofµ,andthedistributionI
featuresinF¯(E). atthelimbwithaddedalbedocontribution,aredrawnfromthMe
Usingaregularisedinversiontechnique(Kontaretal.2004; same parent distribution cannot be rejected at the 0.05 signif-
Scullionetal.inpreparation),wedeterminedthemeanelectron icance level; crossed areas correspond to the 0.01 significance
fluxdistributionF¯(E)fromcountspectraofseveraleventswhich level.SeealsoTable2.Thedashedlineshowstheisotropiccase,
had either flat photon spectra or a large change in local spec- i.e.α(µ)=1.
tral index with energy.Such spectral behaviour is expected for
JanaKasˇparova´etal.:Photosphericalbedoanddirectivity 7
K-Sprob.forα(µ)
µ N 10 5 4 2 1 0.5 0.25 0.2
0.1−0.4 80 3×10−3 0.03 0.07 0.2 0.07 0.02 8×10−3 6×10−3
0.4−0.7 88 9×10−6 4×10−4 1×10−3 0.02 0.4 0.1 0.03 0.02
0.7−0.9 100 7×10−6 2×10−4 4×10−4 0.02 0.3 0.04 7×10−3 7×10−3
0.9−1.0 81 2×10−3 9×10−3 0.04 0.2 0.5 0.06 0.01 4×10−3
Table2.K-Sprob.forseveralvaluesofdirectivitiesatdifferentrangesofµ(Nisthenumberofeventsinthatrange).Thevalues
correspondtoprobabilitythattheobserveddistributionofγ atagivenrangeofµandthemodelleddistributionI ofγ determined
0 M 0
from the eventsat the limb (µ < 0.1) with added albedo contributionfor the given µ and directivity α are drawn from the same
parentdistribution.
photon spectra either affected by the albedo – see Figure 1 – (whichappliesfora primarypower-lawphotonspectrum).The
or producedby F¯(E) with an absence of electrons in same en- inferredanisotropyratio,0.2 < α(µ) < 5,(Figure6)inthe15-
ergyrange.We foundseveralnew cases which exhibita statis- 20 keV range, and especially its large spread, allows no clear
tically significant dip in F¯(E) – e.g. Figure 7 (left). The posi- conclusion about the anisotropy. Moreover, at these energies
tionsandspectralindicesofeventswithsuchadipareindicated anisotropyofelectronbeamemissioncanbemaskedbyisotropic
inFigure4.However,whentheisotropicalbedocorrectionwas thermalemission, thus making the determinationof anisotropy
applied (i.e. α(µ) = 1), all such dips in F¯(E) were removed - fromspatiallyintegratedspectradifficult.
e.g.Figure7(right).OursetofeventsincludestheSep17,2002 The value of anisotropy ratio obtained is consistent with
(Kontaretal. 2006) and Apr 25, 2002 flares (Kontar&Brown the predictions of hard X-ray emission of downward colli-
2006a) for which a statistically significantdip in F¯(E) at ener- matedbeamandcannotrejectevenquitestronglybeamedcases.
giesbelow30keVhasbeenreported,neglectingalbedo. For example, the detailed model of Leach&Petrosian (1983,
The albedo contribution in events close to the limb should Fig.4) predicts α <∼ 3 at 22 keV for disc centre events. On
benegligible.Therefore,featuresinF¯(E)foreventstherecould the other hand, the determined anisotropy α(µ) is also con-
notbeascribedtothedistortionofthephotonspectrabyalbedo. sistent with isotropic X-ray emission and lends support to
F¯(E) was determined for a number of events close to the limb quite isotropic electron distribution when puttogether with the
which showed flattening of the photon spectra. None of these Kontar&Brown(2006b)analysisofdetailedspectraofindivid-
events close to the solar limb (µ < 0.5) was found to have a ualevents.Theanisotropyparameterα(µ)wasalsofoundnotto
statisticallysignificantdipinF¯(E),butrevealedonlysomeflat- needsignificantdependenceonµ–seeFigure6–indicatingthat
teningofF¯(E).AlleventswithdeterminedF¯(E)areindicated the beam X-rayemission neednotstronglyvaryin the upward
inFigure4. hemisphere(differentvaluesofµprobedifferentviewingangles
of upward emission). We stress that with α(µ) we measure the
anisotropyofdownwarddirectedphotonsratherthanupwarddi-
6. Discussionandconclusions rectedphotonsatdifferentanglesaswasdoneinmostprevious
studiesofsolarflaredirectivities.
ThehighenergyresolutionofRHESSIdownto∼ 1keVanda
largedatabasehaveallowedus to studystatistically 398events Thedataanalysedinthepaper,aswellaspreviouspublished
notaffectedby pulse pile-upbuthavingsufficientcountrate to results (Nittaetal. 1990; Fa´rn´ıketal. 1997; Kasˇparova´etal.
measure local spectral indices in three different energy bands. 2005; Kontar&Brown 2006a) describe events with observed
Combiningthesewithflarepositionmeasurementsweanalysed photon spectra which are flat at low energies. Such behaviour
centre-to-limb variation of hard X-ray flare spectra. The data suggests a low-energycutoff in the mean electron spectrum as
clearly show a distinct variation of low energyspectral indices derived from such observed photon spectra. We show that all
with heliocentric angle. This dependencyis strong for spectral such flares were located close to the solar disc centre (µ > 0.5
index in the range 15 - 20 keV but very weak for higher ener- (θ<60◦)),wherethealbedocomponentmustbeproperlytaken
gies. The mean values of low energy index γ are decreasing intoaccount.Nittaetal.(1990)reportedthattwoflaresobserved
0
for larger µ (smaller heliocentric angles). We have compared with SMM on 15 October 1981 at 4:43 UT and on 29 March
centre-to-limbspectralvariationswiththepredictionsofalbedo 1980at9:18UThadγ∼2andγ=1.9,respectively,bothflares
induced spectral index change (Kontaretal. 2006) for primary being located not far from disc centre (µ ≥ 0.6). Fa´rn´ıketal.
spectra assumed to be power-law and found the behaviour of (1997)observedafewflareswithveryflatobservedhardX-ray
meanspectralindexesγ¯ ,γ¯ ,andγ¯ (energyranges15-20keV, photonspectrafromthesameactiveregionnearthedisccentre
0 1 2
20-35keV,and35-50keV)tobeconsistentwiththespectral (µ=1.0).WenotethatZhang&Huang(2003,2004)studiedthe
changeduetoComptonbackscatteringalbedo.Forγ andγ at combinedeffectofthelow-energycutoffandalbedocontribution
1 2
higherenergiesthere is verylittle constraintinformationin the toexplainveryflatphotonspectraobservedbyYohkoh/HXTand
data. concludedthatbotheffectcouldcauseflatteningofphotonspec-
Theintensityofbackscatteredphotonsisdeterminedbyin- tra.However,toaccountforalbedotheyusedintegralreflectivity
tensityofdownwardbeamedphotons,sowecanestimatethera- (angle-averaged)whichisnotapplicabletocentre-to-limbstud-
tio of downward/observerdirected photons. Although our data ies.
set contains large number of events of photon spectra with Wehaveappliedanisotropicalbedocorrectiontotheflares
high energy resolution, it can be only used to test the hard X- inoursurveyandhavefoundthismakestheallegedlow-energy
ray anisotropy below 20 keV. Even for an anisotropic primary cutoff (or a ‘dip’) in electron spectra statistically insignificant.
source, the albedo induced spectral change and centre-to-limb Thissurveystronglysuggeststhehypothesisthatthelow-energy
variationaresignificantonlybelow20keV–seeFigure1and4 cutoffinferredinthemeanelectronfluxdistributionisanartefact
8 JanaKasˇparova´etal.:Photosphericalbedoanddirectivity
Fig.7.F¯(E)correspondingtothetotal(left)andprimary(right)flarephotonspectrumforaneventclosetothedisccentre(19Jul
2004).Verticalerrorbarscorrespondto3σuncertainties,horizontallinesshowtheFWHMofenergyresolutiondeterminedfrom
regularisedinversion.
of the photosphericalbedo.Hence, we can confidentlyconfirm Kontar,E.P.,MacKinnon,A.L.,Schwartz,R.A.,&Brown,J.C.2006,A&A,
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Sol.Phys.,225,293
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