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THE ASTROPHYSICAL JOURNAL, 529 :554È569, 2000 January 202000. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

STATISTICAL ANALYSIS OF THE ENERGY DISTRIBUTION OF NANOFLARES IN THE QUIET SUN

C. E. PARNELL AND P. E. JUPP

School of Mathematical and Computational Sciences, University of St. Andrews, North Haugh, St. Andrews, Fife, KY16 9SS, Scotland ;clare=msc.st-and.ac.uk

Received 1999 June 2 ; accepted 1999 September 13

ABSTRACTFor many years it has been debated whether the quiet solar corona is heated by nanoÑares and micro-

Ñares or by magnetic waves. In this paper TRACE data of events with energies in the range 1023È1026ergs are investigated. A new stable and objective statistical technique is proposed to determine the index,[c, of a power-law relation between the frequency of the events and their energy. We Ðnd that c ishighly dependent on the form of the line-of-sight depth assumed to determine the event energies. If aconstant line-of-sight depth is assumed, then c lies between 2.4 and 2.6 ; however, if a line-of-sight depthof the form is assumed, where is event area and k is a constant, then c lies between 2.0 and(A

e/k2)1@2 A

e2.1. In all cases the value of c is greater than 2 and therefore implies that the events with the lowestenergies dominate the heating of the quiet solar corona. Moreover, there are strong indications thatthere is insufficient energy from events with nanoÑare energies (i.e., energies in the range 1024È1027 ergs)to explain the total energy losses in the quiet corona. However, our results do not rule out the possi-bility that events with picoÑare energies (i.e., energies in the range 1021È1024 ergs) heat the quiet corona.From analysis of the spatial distribution of the events, we Ðnd that events are mainly conÐned to regionswith the brightest EUV emission, which are presumably the regions connected to the strongest magneticÐelds. Indeed, just 16% of the quiet corona possesses such events.Subject headings : MHD È Sun: corona È Sun: Ñares

1. INTRODUCTION

Twenty-Ðve years ago Levine (1974) proposed a ““ newtheory of coronal heating ÏÏ in which he suggested that amultitude of small reconnection events may be responsiblefor heating the solar corona. Over several years Parker(1981, 1983, 1988) built on these ideas and introduced theterm nanoÑare, which he deÐned as any impulsive energyrelease with energy less than 1027 ergs, the minimum energyof a conventional microÑare. He considered the averageenergy of a nanoÑare to be 1024 ergs. Parker referred toobservations that seemed to indicate that his ideas wereplausible. The rocket-borne observations of Lin et al. (1984)show many hard X-ray spikes with energies in the rangefrom 1024 up to 1027 ergs. The more numerous of thesespikes have energies at the lower end of the range, and allthe signs indicated that there were many more events witheven smaller energies. Furthermore, Porter et al. (Porter,Toomre, & Gebbie 1984 ; Porter et al. 1987) observed local-ized brightenings throughout the magnetic network. Theseevents, observed in C IV, have lifetimes from just a minute orso up to an hour and were interpreted by Parker to benanoÑare-like impulsive heating events.

In 1991, Hudson calculated that if the power needed toheat the corona is generated by events of varying sizes thenthe total power, P, per unit area is equal to the integral ofevent energies, E, times their frequency of occurrence perunit area and per unit time, f(E),

P\PEmn

Emxf (E)EdE , (1)

where f (E) has dimensions of ergs~1 cm~2 s~1 and thelimits and are the energies of the smallest andEmn Emxlargest events, respectively. If the frequency of events, f (E),follows a power law of the form

f (E)\ f0E~c , (2)

then, assuming that and c\ 2,Emx ?Emn

PBf0

2 [ cEmx2~c , (3)

implying that events with large energy provide the domi-nant contribution to the heating. If, however, c[ 2, then

PBf0

c[ 2Emn2~c , (4)

and small-scale events dominate the heating.Although the frequency of event energies is unlikely to be

one single power law over the entire energy spectrum,authors of various papers have tried to determine values forthe power-law index, [c, in particular energy ranges. Inmany of the early papersÈsee Crosby, Aschwanden, &Dennis (1993) for a good reviewÈthe frequency of regularÑares was estimated by assuming that the energy in a Ñare islinearly related to the peak Ñux or peak count rate of theÑare. Power-law relations derived from histograms of theseparameters give estimates for c of around 1.8. Recently,however, attempts have been made to estimate actualenergies. Crosby et al. (1993) calculated energies from Ñareelectrons observed using hard X-ray bremsstrahlungobservations. They found c to be 1.53^ 0.02 for events withenergies in the range 1028È1031 ergs. Shimizu (1995) studiedactive-region transient brightenings, small-scale events withenergies in the range 1027È1029 ergs, and they estimated cfor the frequency distribution of these events to be between1.5 and 1.6. From this result he calculated that active-regiontransient brightenings supply at most 20% of the energyrequired to heat active regions.

Clearly, all these results seem to indicate that nanoÑarescannot be the dominant heating mechanism in the solarcorona. This could be for two reasons : either nanoÑaresreally are unimportant, at least in the active corona, or thefrequency of events drastically increases as the energy of

554

DISTRIBUTION OF NANOFLARES IN QUIET SUN 555

events decreases. Cargill (1994), therefore, decided to inves-tigate the problem from a di†erent angle by trying topredict the signatures of observed nanoÑares, assuming aparticular heating model for loops in active regions.CargillÏs predictions do not contradict existing obser-vations, moreover, they suggest possible new observablesignatures which, he hoped, would be observed by EIT onSOHO.

On the other hand, one could consider competing heatingtheories. These generally involve the dissipation of waves inthe corona. Ten years ago it was thought that slow magne-toacoustic waves generated below or in the photosphere byturbulent convection would steepen and shock beforereaching the corona. Also, fast magnetoacoustic waves gen-erated in a similar way were again believed not to reach thecorona, because they would either steepen and shock likethe slow waves or be reÑected o† density gradients in thetransition region. waves, on the other hand, wereAlfve� npredicted to propagate easily into the corona. The bigproblem was how to dissipate them. Phase mixing appearedpromising in coronal holes, whereas resonant absorptionwas a possibility in long closed loops. Recently, however,there has been a series of new observations that have putsome of the above theories into doubt. Slow magneto-acoustic waves have been observed both in giant loops(Berghmans & Clette 1999) and in polar plumes (DeForest& Gurman 1998). Astonishingly, these waves appear to pro-pagate up from the photosphere or low chromosphere,which seems to be at odds with established theories. Fastmagnetoacoustic waves have also been observed, in associ-ation with Ñares, using EIT on SOHO. Fast propagatingwaves have been detected in the corona moving at velocitiesof 200È600 km s~1 away from sites of solar Ñares andcoronal mass ejections. These waves are known as coronalMoreton waves or Ñare waves (Thompson et al. 1999). Fur-thermore, Nakariakov et al. (1999) have observed trappedfast waves in postÑare loops using TRACE. These loopshave periods of 4È6 minutes and are observed to dampenover a period of 10È12 minutes. However, wavesAlfve� nhave, as yet, not been observed in the low corona, thoughthese waves must exist, since there are in situ measurementsof waves in the solar wind taken by UVCS onAlfve� nSOHO. All these measurements of waves look very prom-ising for wave heating. However, answers to the key ques-tions of (1) whether or not these waves have sufficient Ñux toexplain the energy losses from the corona and (2) how theirenergy is dissipated have still not been determined.

The EIT on SOHO has not only been used to establishthe existence of waves in the corona. It also provided anexcellent opportunity for nanoÑares and microÑares to bedetected. Krucker & Benz (1998, hereafter KB), analyzedquiet-Sun data from EIT and found that the frequency ofevents in the range 1025È1026 ergs has a c between 2.3 and2.6. They suggest that this power-law relation would haveto continue to about 3 ] 1023 ergs in order to match theobserved minimum heating requirement for the quietcorona. This is estimated to be at least 3] 105 ergs cm~2s~1 by Withbroe & Noyes (1977), although KB show that,from radiative losses alone, this Ðgure might be muchnearer 4.5] 105 ergs cm~2 s~1. Furthermore, Falconer etal. (1998) compared EIT Fe XII images and Kitt Peak mag-netograms and discovered a high coincidence betweenbrightenings in the EIT images, no matter how small, andtiny magnetic bipoles in magnetograms. This suggests that

many of the very small, 1 or 2 pixels, brightenings observedby EIT are real events. More evidence that many very smallevents exist comes from observations made by MDI onSOHO. Schrijver et al. (1998) investigated the evolution ofmagnetic Ðelds in the quite corona and found that the““ magnetic carpet ÏÏ has a turnover time of 40 hr. That is tosay, the amount of magnetic Ñux that emerges in a 40 hrperiod is equal to the total absolute Ñux on the quiet Sun.Since the total absolute Ñux remains more or less constant,this means that the cancellation rate must be equal to theemergence rate of Ñux. Moreover, they estimate that thesecancellations provide sufficient energy to explain the energylosses from the quiet corona.

Clearly, the result of KB is signiÐcantly di†erent fromother estimates determined for c. This may well be becausethe data analyzed by KB are of the quiet corona, whereasother estimates of the power law index have used activeregion data. It is, therefore, worth testing their result usingdi†erent data. The natural choice was, of course, to useTRACE data, since TRACE has a higher spatial and tem-poral resolution than EIT. Also, the Ðlters on the TRACEand EIT are very similar and, therefore, similar types ofevents can be investigated.

In this paper we use a technique similar to that of KB todetermine event energies in the range a few times 1023È1026ergs using quiet Sun TRACE data (° 3). However, weanalyze our energy results using a new technique (describedin ° 2) which gives a reliable and consistent estimate of thepower-law index for the event frequency function. The newtechnique avoids all problems associated with binning dataand takes into account the fact that the energy of events isgenerally under-reported due to the discrete nature of theobserved data. This means that a much more accurateestimate of the power-law index can be calculated (° 4).Furthermore, it gives a much better lower bound for thetotal power from these events (° 6).

2. STATISTICAL APPROACH

Following HudsonÏs example (Hudson 1991) and otherauthors (e.g., Shimizu 1995 and KB, etc.) initially a histo-gram approach to analyzing the data was chosen. Thelogarithm of the observed energies, was deÐned to beEobs,

y \ log Eobs/E1 ,

where the constant, is such thatE1, E1[minimum Eobs.The yÏs were then binned into bins of constant width, suchz6 ,that the frequency of occurrence of events in the bin cen-tered on z equals

h(z) \ h0M(z)

z, (5)

where M(z) is the number of events with log energies yin the range and the constantz[ z6 /2 [ y \ z] z6 /2,

with A and T the area and time period inh0\ 1/(AT E1)which the events occur.An equivalent graph to a histogram of the data can be

drawn by plotting log h(z) against z. Such a graph is shownin Figure 1a. One can then use a variety of methods to Ðt aline through the data and determine the index of the powerlaw from the gradient of the Ðtted line. However, problemsarise due to the subjective nature of such an approach.What bin size, should be used? What method should bez6 ,used to Ðt a line though this data? Should all data points be

556 PARNELL & JUPP Vol. 529

FIG. 1.ÈPlot of the binned data showing the frequency of occurrence ofevents vs. event energy. The data have been binned with a bin size of

and the two lines have been Ðtted using the least absolute devi-z6 \ 0.05ation method (dashed line) and minimum s2 (dot-dashed line).

used to Ðt the line? Clearly, there are no absolute right orwrong answers to these questions. Furthermore, how do wedetermine errors for the power-law indices? Simply usingthe errors calculated from Ðtting a line is not sufficient, sincethey do not take into account the true extent of the varia-tion possible due to di†erent bin sizes and line Ðttingmethods.

Furthermore, careful inspection of the histograms sug-gests that Ðtting a single power law to the data is inade-quate. This is because there are fewer events with the lowestobserved energies than the line that Ðts the bulk of the datasuggests. This short fall of events with low energies appearsalso in the data analyzed by KB. Why should this occur?

The most likely explanation for a short fall of events atlow energies is that the energies we observe are not the““ true ÏÏ event energies, but energies that are just less than thetrue energies. The drop in energy probably arises becauseour data are discrete. We assume that an observed energy isassociated with a change in emission measure, *EM,between a peak and its preceding minimum (explained morefully in ° 3). Even though the cadence of the TRACE data ishigher than that of any other data set used for this sort ofanalysis to date, it is still not sufficient to resolve an individ-ual event properly and therefore we observe a change inemission measure smaller than the true change (Fig. 2).Hence, the energies observed under-report the true energiesof events.

In order to circumvent the problems of Ðnding thepower-law index using the above log histogram method, wedecided to employ a more objective method. The methodwe chose is essentially the standard statistical procedure ofmaximum likelihood estimation. The data can be graphi-cally compared with the results from such a method byusing empirical frequency functions (see Appendix for anexplanation). The problem of under-reporting is taken intoaccount by using a skew-Laplace distribution instead of asingle power law. A skew-Laplace distribution is a distribu-tion that on the left-hand side is a power law with index, /say, and on the right-hand side is a power law with index,[c say, where / and c are both positive.

FIG. 2.ÈSketch of the variation of the true emission measure in 1 pixelvs. time (solid line). The stars denote the discrete observed emissionmeasures and the vertical lines indicate the change in emission measure,

for event i.*EMi,

An eventÏs ““ observed ÏÏ energy, will be related to itsEobs,true energy, E, by

Eobs \ uE , (6)

where u is an under-reporting factor which satisÐes 0 \Since, we do not know the exact form for the densityu [ 1.

function g(u), it is reasonable to assume that it is a powerlaw with index equal to the gradient of the left-hand side ofthe skew-Laplace distribution, since the left-hand side is aplot of under-reported energies. Thus, we shall assume thatu has density function

g(u) \ 45600(/] 1)uÕ 0 \ u [ 1 ;0 otherwise .

(7)

Furthermore, we shall assume that u and are indepen-Eobsdent.Now, if we assume that the ““ true ÏÏ energies E have a

power-law frequency function

pt(E) \ 4

5600

p0(c[ 1)(E/E0)~c/E0 EZ E0 ,0 E[ E0 ,

(8)

then it follows from a result of Hinkley & Revankar (1977)that the observed event energies, will have a frequencyEobs,function of the form

pobs(Eobs) \

4

5

6

00p0(/] 1)(c[ 1)(Eobs/E0)~c/[E0(/] c)]

E0[ Eobs\ O ;p0(/] 1)(c[ 1)(Eobs/E0)Õ/[E0(/] c)]

0 \ Eobs[ E0 .(9)

Note that the power-law index, [c, for the observedfrequency of energies greater than is equal to that forE0the true energy frequency. This frequency function can thenbe reformulated in terms of

y \ log Eobs , l\ log E0 ,

No. 1, 2000 DISTRIBUTION OF NANOFLARES IN QUIET SUN 557

to give a skew-Laplace frequency function for y, whichequals

q(y)\

4

5

6

00[p0(/] 1)(c[ 1)/(/] c)] exp ([(c[ 1) o y [ l o )

l[ y \ O ;[p0(/] 1)(c[ 1)/(/] c)] exp ([(/] 1) o y [ l o )

[ O \ y [ l .(10)

Next, we need to estimate the parameters c, /, and l. Agood method of doing this is the maximum likelihoodmethod. However, for large samples, computation ofmaximum likelihood estimates is tedious. A simplerapproach is to use the large-sample approximation to themaximum likelihood estimates derived in Hinkley &Revankar (1977). We deÐne

Sr\ ;

i/1

ry(i) , L

r\ 1 [ r/n

Sn/n [ S

r/r

,

where r \ 1, . . . , n and are they(1)[ . . . [ y(i)[ . . . [ y(n)ordered values of y.If is the value of r at which is maximum, then ther‘ L

restimates and of c, /, and l are given byc‘ , /‘ , l‘l‘ \ y(r‘`1) ,

c‘ \ 1Sn/n [ S

r‘/r‘

] 1 ,

/‘ \ c‘1 [ r‘ /n

r‘ /n[ 1 . (11)

Clearly, we need to know how accurate our estimates ofc, /, and l are, and so we must calculate conÐdence inter-vals for our estimates. The 95% large-sample conÐdenceintervals for l, c, and /, are, respectively,

l‘ ^ 1.96S 2

n(c‘ [ 1)(/‘ ] 1),

c‘ ^ 1.96S(c‘ [ 1)2

n [ r‘,

/‘ ^ 1.96S(/‘ ] 1)2

r‘. (12)

The goodness of Ðt of the model to the observed fre-quencies of events can be assessed graphically by plottingthe empirical frequency function, against on aDobs(i), Eobs(i)log-log plot, where andy(i)\ log Eobs(i)

Dobs(i)\

4

5

6

00

p0(/‘ ] 1)(c‘ [ 1)/(/‘ ] c‘ )

] [1 [ (i[ r‘ [ 1/2)/(n [ r‘ )]/Eobs(i)i\ r‘ ] 1, . . . n ;

p0(/‘ ] 1)(c‘ [ 1)/(/‘ ] c‘ )

] [(i[ 1/2)/r‘ ]/Eobs(i)i\ 1, . . . , r‘ .

(13)

We can compare the data with the model by plotting pobsagainst on the same graph using the values of c, /, andEobsestimated from the data to calculate If the two plotsE0 pobs.are close, then the model is a good Ðt to the data. For aderivation of the empirical frequency function seeDobs(i),Appendix.

3. OBSERVATIONS

We analyzed data taken by TRACE (the TransitionRegion and Coronal Explorer), which was launched in 1998April (Golub et al. 1999). The TRACE instrument has anangular resolution of 1A and an average cadence of 30 s. ItsCCD detector has pixels and an array of 1024 ] 10240A.5pixels. However, TRACE can observe only an area some-what less than 512 ] 512 arcsec2, because of the circularnature of its Ðlters. It has Ðlters with eight wavelength bandscapable of observing plasma with temperatures from 4000up to 4 ] 106 K.

The data set which we use was taken using 2 Ðlters : theFe IX/X Ðlter at 173 which observes plasma with tem-A� ,peratures between 1.6] 105 and 2 ] 106 K, and the Fe XII

Ðlter at 195 which observes plasma with temperaturesA� ,between 5] 105 and 2 ] 106 K. It was taken on 1998 June16 between 20 :29 and 21 :53 and consists of images takenalternately with each Ðlter at a rate of 115 s between pairs of195/173 images. In the two cases where the cadence variesfrom this rate, the cadence drops to 114 s. All the 173 A�images have exposure times of 27.6 s and all the 195 A�images have exposure times of 65.5 s. In total we have 26images, making 13 195/173 pairs. Just 13 pairs of obser-vations is not many, however, out of a week-long series ofobservations this is the longest set we could Ðnd that, whencleaned, had a cadence varying by at most 1 s and hadexposure times that were identical for all 173 images andidentical for all 195 images. This data set has considerablybetter spatial and temporal resolution than any data setused so far to consider the distribution of event energies. Itis also much more consistent in its exposure times andcadence variation. This is very important, because our deÐ-nition of an event depends on comparing changes betweenimages. Clearly, these changes will naturally increase ordecrease as the time gap between the images increases ordecreases.

The TRACE data were prepared by Ðrst cleaning toremove spikes from cosmic-ray hits, then a pedestal(including dark current), calculated individually for eachimage, was subtracted. Next the 173 images were resizedA�from 1024 ] 1024 pixels down to the size of the 195 A�images, 512] 512 pixels, giving an angular resolution of 1A.Finally, the images were paired and the ratios per pixel oftheir intensities determined. From these ratios we derivedaverage temperatures for each pixel using TRACE responsecurves. The TRACE response curves were determined usingthe CHIANTI atomic data, an assumed density of 109 cmand the TRACE Ðlter response curves. Average emissionmeasures were derived from the average temperatures andthe 173 data numbers for each pixel. For further informa-A�tion see Klimchuk & Gary (1995). Random errors for thetemperature and emission measure were calculated usingthe formulae given by Klimchuk & Gary (1995). We Ðndthat the modal and mean errors for temperature are 1.7%and 1.9% and for emission measure are 6.1% and 6.7%,respectively. The sizes of the systematic errors are notknown. However, Klimchuk & Gary (1995) suggest thatthey may be even larger than the random errors.

To estimate the energies of events, we use a methodsimilar to that of KB. That is to say, we assume that thesignature of an event is an enhancement in emissionmeasure. Such an enhancement could occur for a numberof reasons : evaporation of heated photospheric/


chromospheric plasma; in situ heating or cooling of existingplasma to the Ðlter temperatures ; and the movement ofplasma along loops. Of these possible mechanisms the mostlikely ones are evaporation of chromospheric material or insitu heating. The temperature of the plasma we observe is atthe higher end of the range for the quiet Sun. It is, therefore,unlikely that many of our enhancements are due to coolingplasma, however, we cannot rule out this possibility (this ofcourse would be quite di†erent if we were looking at theactive corona). Also we see little evidence of our eventsmoving, which we would expect them to do if our enhance-ments were triggered by wave motions along a loop. Ithas been suggested that it is hard to distinguish betweenresonant absorption heating and impulsive heating frommultiple nanoÑare-like events. A recent paper by Ofman,Klimchuk, & Davila (1998) shows that resonant absorptioncould give rise to many small bursty brightenings thatwould move along loops. We see little evidence of moving ofevents, however, this again does not rule out such a heatingmechanism.

Even with all these apparent draw backs with the methodused it is still important to investigate the frequency ofevents and the index of the power law. If the power-lawindex is still less than 2, even when we count all the events(nanoÑare, cooling or resonant absorption generated), itimplies that nanoÑare heating is rather unlikely to be thedominant heating mechanism in the quiet corona. On theother hand, if we Ðnd that the power-law index is greaterthan 2, then we can at least say that nanoÑare heating iscertainly a possibility and needs to be investigated further.

Clearly, if an event is due to evaporation, then the energyof the event must be enough not only to heat the plasma tothe observed temperature, but also to raise the plasma fromthe photosphere/chromosphere into the corona. Let us con-sider the ratio of the thermal energy and potential energy insuch events :

Thermal EnergyPotential Energy

\ (3/2)kB NTe

NmgH\ 75.3

(Te/106 K)

(H/108 cm),

where N is the total number of particles, is the Bolt-kBzmann constant, is the temperature of the plasma, g is theTegravitational acceleration at the surface of the Sun and H is

the distance the plasma has been raised. Finally, m is themean particle mass, taken here to be where is the0.6m

p, m

pmass of a proton.In general, in the quiet corona the plasma temperature is

between 106 and 2 ] 106 K and the events have loops withradii at most 1.5 ] 109 cm (half the diameter of a super-granule cell). More often, however, H will be considerablysmaller than this. We, therefore, Ðnd that the thermalenergy of these small events is at least 5 times the potentialenergy, and so here we merely consider the thermal energyof events.

After taking into account the e†ect of di†erential rota-tion, we identiÐed peaks in the emission measure (cm~5) ofeach pixel. An energy per pixel is associated with the di†er-ence, *EM, between a peak in emission measure and itspreceding minimum by

Eobs\ 3kB TeJ(*EM)qhA

p2 , (14)

where is the area of the pixel, h is the observed line-of-Apsight depth and q is the Ðlling factor, taken here to be 1.0.

Since h is unknown, we consider various cases. In the Ðrst, his assumed to be constant, and in the second, h \ (A

e/k2)1@2,

where is the area of the event and k2 is Ðxed at a suitableAeestimate of event length over event width. If no preceding

minimum of a peak in emission measure exists, then the Ðrstemission measure observed for that pixel is taken as theminimum. This means that in the Ðrst few frames, not onlyare far fewer events observed, but also the sizes of theseevents are, in general, smaller than in the later time steps.To determine a peak in emission measure, we merelycompare the emission measure in each pixel with the emis-sion measure in that pixel at the preceding and followingtimes. Furthermore, we assume that if 2 pixels are next toeach other and they peak in exactly the same time step, thenthey are counted as being part of the same event. A pixel isconsidered to be next to another pixel if it is any one of thesurrounding eight pixels.

Clearly, not every enhancement in emission measure iscaused by an event and so we set a lower limit, or threshold,above which an emission measure enhancement, *EM, iscounted as a genuine event. This avoids problems withpeaks due to errors in the emission measures. The limit ofemission measure enhancement is set at np. To calculate pwe suppose that the errors, of the observed emissione

i,

measures have a normal distribution centered at 0 and withstandard deviation, p. Using maximum likelihood thethreshold, p, is estimated to be

pü \S1

n;i/1

nei2 .

In this paper, we take p as and n to be either 2 or 3. Thispümeans that if events have emission measure enhancementsof at least 2 p, then their enhancements are greater than95% of all emission measure errors. However, if events haveemission measure enhancements of at least 3 p, then theirenhancements are greater than 99.8% of all emissionmeasure errors.

The threshold used here is di†erent from that used byKB. Their threshold was calculated from Ðtting a normaldistribution to just the peak of the emission measure errordensity function as opposed to the whole function (Benz &Krucker 1998).

4. RESULTS

The region studied in this paper is the central region ofthe SOHO/EIT image above (Fig. 3a). An enlargement ofthis region taken by TRACE is shown in Figure 3b. TheEIT pixel area is almost 7 times as large as the TRACEpixel size.

The bottom left hand corner of the TRACE image con-tains some extended loop structures. These appear, from theEIT image, to be emanating from an active region. Since, inthis paper, we are interested only in the heating of the quietcorona we will, for the most part, only consider the threequiet quarters of the TRACE region. In ° 4.2, we use sta-tistical arguments to explain further the reasons for this.

4.1. Frequency of EventsUsing the above method of preparing the data and deter-

mining event energies, we Ðnd that the number of goodpixels per 3/4 image (i.e., those pixels that are within theregion of both the 173 and 195 Ðlters for the entireA�observing period) is 175,372, and so we observe an areaequivalent to 420] 420 arcsec2. The range of temperaturesof the pixels lies within the diagnostic range of the Ðlters(i.e., between 8.0] 105 K and 1.85] 106 K), so we have no


FIG. 3.È(a) Full disc Fe XII image of the solar corona taken by SOHO/EIT on the 1998 June 16 at 20 :25. (b) TRACE Fe XII image of the observed regiontaken at 20 :40 on 1998 June 16.

problems with trying to interpret temperatures near theturning points of our response curves.

The total number of observed peaks in emission measure,no matter how small the enhancement, is 191,418. By deÐni-tion, a peak in emission measure in a particular pixel mustbe greater than the emission measure in that pixel at thepreceding and following time steps. This means that therecan be no peaks in the Ðrst or last time steps and that a peakcan occur, at most, every other time step. We Ðnd that forour data p \ 9.2] 1025 cm~5 and that the total numbers ofpixels which have peaks with enhancements, at least 2 p orat least 3 p are 37,872 and 11,712, respectively.

As mentioned in the previous section, an event may bemore than 1 pixel in size, and so the number of events isobviously less than the number of peak pixels. We Ðnd thatthere are 16,272 and 4497 events with enhancements of atleast 2 p and at least 3 p, respectively, in total over thewhole time period. This relates to about 1479 and 409events per time step for each case. However, when determin-ing an enhancement in emission measure, we consider thedi†erence between a peak and its preceding minimum, asalready described in ° 3. If no minimum exists, as is possiblein the Ðrst few time steps, then the Ðrst observed emission

measure is assumed to be a minimum. This means that inthe Ðrst few time steps there will be fewer events than in thelater time steps and that these events will have smaller eventenergies. For example, in the case where events haveenhancements greater than 2 p there are only 593 and 808events observed in the second and third time steps and 1691and 1987 observed in the second from last two time steps(there are no events in the Ðrst and last time steps). If wediscount the Ðrst four steps in each case, then the averagenumbers of events rise to 1636 and 480 events per time stepfor each case. To avoid any errors from this problem, wealways determine the frequency of events using just thoseevents that occur in the time steps 5 to 13. From now onwhen we refer to the whole data set we mean just thoseevents in time steps 5 to 13.

First, we consider events calculated with a constant line-of-sight depth, h \ 7.26] 107 cm, which is equal to thewidth of 1 pixel. Such a small line-of-sight depth is chosenbecause 78% of events have areas less than 3 pixels and97% have areas less than 10 pixels. The observed energies ofevents with enhancements at least 2 p range from 2.5 ] 1023up to 5.7] 1025 ergs. It is important to note that the exactvalue of the constant h does not change the values of the


FIG. 3b

power-law indices, c and /. It does, however, shift andstretch the energy range over which this power law holdsand, therefore, does e†ect the estimate of the total powerdue to these events. For instance, an increase in h by afactor k2\ 3 will produce an increase of in all theJ3energy estimates, implying an observed energy range from4.3] 1023 up to 9.9 ] 1025 ergs.

The Ðtted and empirical frequency functions of eventenergies, for events with constant h, are plotted in Figure 4a.Estimates, and for the power-law indices are detailed inc‘ /‘ ,the Ðrst two lines of Table 1 for events with emissionmeasure enhancements of at least 2 p and 3 p. This tablealso gives conÐdence intervals for c and estimates ofminimum ““ true ÏÏ energy, for each case.E� 0\ exp (l‘ ),

TABLE 1

ESTIMATES OF THE THREE PARAMETERS THAT DETERMINE THE FREQUENCY FUNCTIONS OF

EVENTS FOR EVENTS WITH AT LEAST s PIXELS AND EMISSION MEASURE

ENHANCEMENTS OF AT LEAST np

Minimumnp Number of E� 0

n (cm~5) Pixels, s c‘ 95% C. I. for c /‘ (ergs)

2 . . . . . . 1.83 ] 1026 1 2.56 (2.53, 2.59) 41.6 3.0 ] 10233 . . . . . . 2.74 ] 1026 1 2.42 (2.37, 2.47) 36.0 3.7 ] 10232 . . . . . . 1.83 ] 1026 2 2.44 (2.40, 2.48) 34.1 5.8 ] 10232 . . . . . . 1.83 ] 1026 3 2.42 (2.37, 2.47) 35.2 9.6 ] 10232 . . . . . . 1.83 ] 1026 4 2.46 (2.39, 2.53) 28.0 1.3 ] 1024

NOTE.ÈEnergies were calculated assuming a constant line-of-sight depth of 7.26] 107 cm


FIG. 4.ÈFrequency of events vs. event energy for (a) events of any sizeand emission measure enhancements at least 2 p and 3 p and (b) eventswith at least 1, 2, 3, and 4 pixels and emission measure enhancementsof at least 2 p. Energies used in these graphs were determined usingh \ 7.26] 107 cm. These plots show the observed data (solid line and dots)and the right-hand power law of the Ðtted skew-Laplace distribution(dashed line). The frequencies of events for the four cases in (b) are multi-plied by 1, 10~2, 10~4, and 10~6, respectively, so they can all be drawnwithout overlap on the same graph.

In the above cases we have counted events of all sizes. Letus consider, instead, events with at least 2, 3, and 4 pixelsand emission measure enhancements at least 2 p. In each ofthese cases, we have 5131, 2842, and 1861 events, respec-tively, occurring in the observed period. The frequencygraphs for these cases are plotted in Figure 4b and theestimates and are detailed in the last three lines ofc‘ , /‘ , E� 0Table 1. From this table estimates show that c is greaterthan 2 and lies between 2.37 and 2.59. The power-law index,/, on the other hand, is at least an order of magnitudelarger. This implies that event energies are under-reportedby only a small amount. Remember, our under-reportingfactor u and the observed energies are independent andEobsso too are their frequency functions. Furthermore, we onlyassume the general mathematical form of these frequencyfunctions and, hence, / and c are free parameters which aredetermined by the observed data. This means / could havebeen small or large. Indeed, where this statistical techniquehas been applied in other Ðelds the values of / have beeneither much smaller than c or roughly the same size. For

example, Hinkley & Revankar (1977) investigated theunder-reporting of peoples incomes for tax purposes andfound, may be unsurprisingly, a small value of /, of about0.5, in comparison to their c of about 3.3 ! Note, that as webecome more strict in our deÐnition of an event, demandingthat events have a certain minimum area, the power lawindex does not decrease monotonically as the minimumarea increases. This suggests that our estimates of thepower-law indices are not dependent on event area. Clearly,however, the minimum event energy is dependent on eventarea.

In the above calculations, we assumed that the line-of-sight depth, h, which is used to calculate event energies, isconstant. However, this is not necessarily a reasonableassumption. Events covering a small area are likely to havea much smaller depth, h, than events covering a larger area.Assuming that this is the case, one possible form for thedepth h is where is the area of an event and k2(A

e/k2)1@2, A

eis a constant that assumes that all events are loops that arek2 times long as they are wide. Hence, events are loopswhose width equals their depth, h. Estimates and arec‘ , /‘ E� 0detailed in Table 2 below for the same Ðve cases givenabove, using energies calculated assuming that k \ 1. Notsurprisingly, the estimates for c are lower than in the con-stant h case. Interestingly though, they are still greater than2, but only just. Furthermore, the range of energies whichthe events span has increased, so that now the observedevents have energies ranging from 2.5] 1023 up to6.1] 1026 ergs (Fig. 5).

Relaxing the assumption that k \ 1, so assuming that allevents are loops of length k times their width, does notchange the indices of the power law. This is because k issimply a constant, and so its e†ect is just to decrease orincrease all energies by the same factor. Thus, it will causechanges to the estimates of total energy from the events.

4.2. Assessing Spatial HomogeneityAs already mentioned, the bottom left corner of our

sample region contains loops that could extend from anactive region (Fig. 3b). There also appears to be anunusually large number of events along the edge of thiscorner. To test whether this region is really di†erent fromthe other 3/4 of the observed region, we divide our sampleregion up into quarters, to give four independent data setsand compare the results with those attained when analyzingthe whole sample region. In this analysis, we consider eventswith energies calculated using a constant line-of-sightdepth, h \ 7.26] 107 cm. Table 3 details the results, whichshow that the estimates of c vary from 2.44 to 2.56. Todetermine whether this spread of values for c is signiÐcant,we use the likelihood ratio test of homogeneity of c.

Looking at the results for the estimates of c in the fourquarters, we see that the value of for the bottom leftc‘quarter appears much smaller than the estimates for theother three quarters. We can test this statistically by usingthe likelihood ratio test. Let us assume that in each quarter,i, where i \ 1, . . . , l, there exist events and that we esti-n

imate the power-law index, of the frequency function by[ci,

using the statistical approach described in ° 2. Then thec‘i,

likelihood ratio statistic is

wl\ 1

(c‘ [ 1)2 ;i/1

lni(c‘

i[ c‘ )2 , (15)


TABLE 2

ESTIMATES OF THE THREE PARAMETERS THAT DETERMINE THE FREQUENCY FUNCTIONS OF

EVENTS FOR EVENTS WITH AT LEAST s PIXELS AND EMISSION MEASURE

ENHANCEMENTS OF AT LEAST np

Minimumnp Number of 95% C. I. E� 0

n (cm~5) pixels, s c‘ for c /‘ (ergs)

2 . . . . . . 1.83 ] 1026 1 2.13 (2.11, 2.15) 40.3 2.9 ] 10233 . . . . . . 2.74 ] 1026 1 2.02 (1.99, 2.05) 33.4 3.7 ] 10232 . . . . . . 1.83 ] 1026 2 2.04 (2.01, 2.07) 33.9 8.7 ] 10232 . . . . . . 1.83 ] 1026 3 2.02 (1.98, 2.06) 40.4 1.6 ] 10242 . . . . . . 1.83 ] 1026 4 2.04 (1.99, 2.09) 28.0 2.6 ] 1024

NOTE.ÈEnergies were calculated assuming a variable line-of-sight depth equal to Ae1@2 ,

where is the event areaAe

where is the estimate of the power-law index of the[ c‘frequency function using the whole data set. The variationbetween the is considered signiÐcant if is largec‘

i-values w

lcompared with a distribution. We Ðnd that for oursl~12

four quarters Since, the probability (called thew4\ 11.98.p-value) of getting a number at least 11.98 from a dis-s32tribution is less than 0.08, we conclude that i\ 1, . . . , 4c

i,

are not all equal. If, however, we just compare the estimatesfor the three quiet quarters, top left, bottom right and topright with the estimates of the parameters calculated for thecombination of these three quarters (see Table 1), then we

Ðnd Such a has a p-value of 0.48, so we canw3\ 1.53. w3accept at the 48% signiÐcance level that the forci-values

these three quarters are equal.The same statistical test can also be performed using the

energies derived using The results from suchh \ (Ae/k2)1@2.

an investigation do not show quite as convincingly that thebottom left corner has a signiÐcantly di†erent value of c.However, there are certainly far fewer events in the bottomleft quarter than in all the others and so to avoid any pos-sible misleading results we simply do not count events inthis quarter.

FIG. 5.ÈFrequency of events vs. event energy for (a) events of any size and emission measure enhancements at least 2 p and 3 p and (b) events with at least1, 2, 3, and 4 pixels and emission measure enhancements of at least 2 p. Energies used in these graphs were determined using These plots show theh \ A

e1@2.

observed data (solid line and dots) and the right-hand power law of the Ðtted skew-Laplace distribution (dashed line). The frequencies of events for the fourcases in (b) are multiplied by 1, 10~2, 10~4, and 10~6, respectively, so they can all be drawn without overlap on the same graph.

TABLE 3

RESULTS FROM THE ANALYSIS OF THE FOUR QUARTERS OF THE SAMPLE REGION IN COMPARISON WITH THE WHOLE DATA SET

Property Bottom Left Top Left Bottom Right Top Right Whole

Number of good pixels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52,463 61,100 57,015 57,257 227,835p cm~5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.22 ] 1026 9.75 ] 1025 8.50 ] 1025 9.10 ] 1025 9.94 ] 1025Percent good pixels with º2 p enhancement . . . . . . 8.9 15.5 15.2 17.2 13.3Number of events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2,503 4,385 4,248 4,504 14,632c‘i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.44 2.56 2.56 2.53 2.53

95% C. I. for ci. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.38, 2.50) (2.51, 2.61) (2.51, 2.61) (2.48, 2.58) (2.50, 2.56)

E0 (ergs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 ] 1023 3.2 ] 1023 3.0 ] 1023 3.0 ] 1023 3.8 ] 1023

NOTES.ÈResults from the analysis of the four quarters of the sample region in comparison with the whole data set, where the number ofevents, power-law index and the minimum ““ true ÏÏ energy all refer to events of any size with emission measure enhancements of at([ c

i) (E0)least 2 p. Energies were calculated assuming h \ 7.26] 107 cm


5. SPATIAL DISTRIBUTION AND AREA OF EVENTS

Not only is the frequency of occurrence of events of inter-est, but so too is their spatial distribution. Here, it does notmatter what the line-of-sight depth, h, is since it is the posi-tion of the event, not the actual energy of the event that isimportant. Events that are of any size and have emissionmeasure enhancements of at least 2 p or at least 3 p occur injust 16% and 6%, respectively, of the pixels out of our totalarray of 175,372 good pixels in 3/4 of the image over the lastnine time steps. These numbers are signiÐcantly lower thanthe numbers observed by KB, probably because our observ-ing period was just 15 minutes as opposed to 42 minutes fortheir data. There were 27,924 pixels that had an event withan emission measure enhancement of at least 2 p and con-tained one or more pixels. Of these pixels, 25,095 had justone event, 2683 had two events, 135 had three events and 11had four or more events in the whole time period. Thus,only 1.6% of the good pixels have at least two events withinthe 15 minute period. In Figure 6b all pixels that have atleast one event during the observing period are blacked out.

We show here events not only in the 3/4 of the image thatwe are studying, but also those events that occurred in thebottom left quarter for comparison. This image is comparedwith the sum of all Fe XII images taken during the observingperiod (Fig. 6a). It is clear that events occur where there isbright emission in the Fe XII image and so, one wouldassume, where there are relatively strong concentrations ofmagnetic Ðeld. This, of course, is exactly as one wouldexpect. Furthermore, there appear to be two di†erent typesof pointlike events : those that occur at the footpoints ofloops and outline the edges of supergranule cells (just tothe lower right of center) ; and those that occur higher in thecorona and appear along loops (visible not only in thebottom left quarter of the image, but also in the quietregions of the image).

Obviously events vary in size (area). We Ðnd that eventscan be as small as 1 pixel (1 arcsec2) and as large as 114 and85 arcsec2 if the events have, respectively, enhancements ofat least 2 p and 3 p. The mean areas of events with enhance-ments of at least 2 p and 3 p are 2.3 and 2.6 arcsec2. In allcases the modal area is 1 arcsec2. Over 96% of all events

FIG. 6.È(a) Sum of all Fe XII images of the region studied during observed period on 1998 June 16. (b) Spatial distribution of events with enhancements atleast 2 p in size and with 1 or more pixel.


FIG. 6b

with enhancements at least 2 p or at least 3 p have areas lessthan 10 arcsec2. Clearly, this implies that spatially we arenot resolving the bulk of the events that we observe in thequiet corona. A resolution of 0.25 arcsec2 would mean thataround 10 pixels would cover a mean event size of 2.5arcsec2. However, to resolve 1 arcsec2 events we wouldreally need data with pixels of 0.125 arcsec2, giving a cover-age of 8 pixels to every one TRACE pixel. As already men-tioned, to resolve events temporally we also need data witha higher cadence than used here. For 1 arcsec2 pixels, likethose on TRACE, a cadence of 25 s (5 times the cadence ofthe data used here) is probably sufficient, provided that therandom errors can be kept as low as those estimated here.For smaller pixels it is likely that data of even highercadence will be needed.

6. QUIET-SUN ENERGY ESTIMATES

To estimate the total power per unit area produced bythe events which we have observed, we must Ðrst calculatethe constant In equation (5) where N is thep0. p0\ N/(AT ),total number of events observed in the whole time period T

and A is the area observed. This assumes that we areobserving events with true energies in the range up to O.E0If, as in our data set, the range of true energies is from toE0where then is modiÐed by a factor, almostEmx, Emx ? E0, p0equal to 1, such that

p0\ NAT [1[ (E

mx/E0)1~c]

. (16)

We can now calculate, from equation (1) with f (E)\ pt(E)

and the power per unit area, P, pro-f0\ p0(c [ 1)/E0~c~1,duced by all the events using their true event energies,

P\ p0E0c[ 12 [ c

CAEmxE0

B2~c [ 1D

. (17)

Results for and P are calculated in Table 4 for the eventsp0with emission measure enhancements at least 2 p and of sizeat least 1 and 2 pixels, as well as for the events withenhancements at least 3 p and of size at least 1 pixel. In theÐrst three lines of the table events are determined using aconstant h \ 7.26] 107 cm and in the last three lines eventshave a line-of-sight depth where k \ 1.h \ (A

e/k2)1@2,


TABLE 4

TOTAL POWER PER UNIT AREA ESTIMATES FOR EVENTS WITH EMISSION MEASURE ENHANCEMENTS OF AT

LEAST np AND OF SIZE AT LEAST s PIXELS

Minimum PercentNumber of E0 Emx p0 P Total

np Pixels, s c (ergs) (ergs) (cm~2 s~1) (ergs cm~2 s~1) Power

2 p . . . . . . 1 2.56 3.0 ] 1023 5.7 ] 1025 1.8 ] 10~20 1.4 ] 104 52 p . . . . . . 2 2.44 5.8 ] 1023 5.7 ] 1025 6.5 ] 10~21 1.2 ] 104 43 p . . . . . . 1 2.42 3.7 ] 1023 4.7 ] 1025 5.2 ] 10~21 5.6 ] 103 22 p . . . . . . 1 2.13 3.0 ] 1023 6.1 ] 1026 1.8 ] 10~20 2.9 ] 104 102 p . . . . . . 2 2.04 8.7 ] 1023 6.1 ] 1026 6.5 ] 10~21 3.4 ] 104 113 p . . . . . . 1 2.02 3.7 ] 1023 1.6 ] 1026 5.2 ] 10~21 3.7 ] 104 4

NOTE.ÈEnergies were calculated assuming h \ 7.26] 107 cm in the Ðrst three rows and in the last threeh \ Ae1@2

rows

The total power per unit area from events determinedassuming a line-of-sight depth h \ 7.26] 107k2 cm can beeasily calculated by multiplying the P-values given in theÐrst three lines of Table 4 by the factor k. Hence, if a line-of-sight depth of h \ 2.2] 108 cm were used, then k \ J3and it is estimated that about 32% of the total powerneeded to heat the corona is produced from events withemission measure enhancements greater than 2 p. To esti-mate the total power per unit area from events determinedassuming a line-of-sight depth whereh \ (A

e/k2)1@2, k D 1

we need to multiply the Ps given in the last three lines ofTable 4 by a factor 1/k. Hence, if it is assumed that eventsare all loops that have lengths k2\ 4 times their width ordepth, then and it is estimated that just 5%h \ (A

e/4)1@2

of the total power needed to heat the corona is producedfrom events with emission measure enhancements greaterthan 2 p.

In each of the cases mentioned, the events produce only asmall fraction of the total power per unit area (assumed tobe 3 ] 105 ergs cm~2 s~1) needed to heat the quiet solarcorona. If, however, we assume that the frequency functionsfound here extend down to ergs and beyondEmn\ 1020and that they extend up to ergs, then the totalEmx \ 1027power, P, from all events in the range to ergsEmn Emxequals

P\ p0E0c[ 12 [ c

CAEmx

E0

B2~c [AEmn

E0

B2~cD. (18)

We can, therefore, estimate the minimum energy, ofEmn,events needed to give a total power, P equal to 3 ] 105 ergscm~2 s~1, the estimated total power loss from the quietcorona. The estimates for in each of the three constantEmnh cases discussed above are given in Table 5. If a di†erentconstant line-of-sight depth were used, say one equal to

h \ 7.26] 107k2 cm, then the minimum energy neededEmnto heat the quiet corona can be estimated by multiplying thegiven in Table 5 by k*(1~c)@(2~c)+. Hence, in theEmn-values

case with emission measure enhancements at least 2 p and aconstant line-of-sight depth of 2.2 ] 108 cm, the multi-plying factor is 4.6 and a minimum event energy of 6] 1021ergs will produce sufficient energy to explain the heat lossesfrom the quiet corona.

In the cases where the energies are estimated using aline-of-sight depth equal to the power lawh \ (A

e/k2)1@2

index, c is barely greater than 2. This implies that anunphysically small minimum energy for events is obtainedfrom the above calculations, and so, although small-scaleevents are dominant in the quiet corona, they do not supplysufficient energy to heat the quiet corona.

7. CONCLUSIONS

In this paper an objective and stable statistical approachis presented which determines the index of a power lawdistribution simply and self-consistently. In this approach, askew-Laplace distribution is Ðtted to the data by maximumlikelihood and the goodness of Ðt is assessed by plottingempirical distribution functions. Not only does theapproach give estimates for the index, but it also enables thecalculation of conÐdence intervals. It takes into account thefact that observed energies of events are, in general, under-reported and explains how the observed energy distributionis related to the true energy distribution. This technique,which avoids all the problems related with choosing arbi-trary bin sizes and di†erent line Ðtting methods, whichwould be faced if using a histogram method, can be appliednot only to energy distributions, but to any situation wherepower law distributions arise and the variable observed islikely to be under-reported : for example, the distribution offragment Ñuxes in the photosphere.

TABLE 5

ESTIMATES OF THE MINIMUM ENERGY OF EVENTS, NEEDED TO HEAT THE QUIET CORONA FOR EVENTS WITHEmn,EMISSION MEASURE ENHANCEMENTS OF AT LEAST np AND OF SIZE AT LEAST s PIXELS

Minimum PercentNumber of Emn Emx p0 P Total

np Pixels, s c (ergs) (ergs) (cm~2 s~1) (ergs cm~2 s~1) Power

2 p . . . . . . 1 2.56 1.3 ] 1021 1 ] 1027 1.8 ] 10~20 3.0 ] 105 1002 p . . . . . . 2 2.44 5.1 ] 1020 1 ] 1027 6.4 ] 10~21 3.0 ] 105 1003 p . . . . . . 1 2.42 3.8 ] 1019 1 ] 1027 5.2 ] 10~21 3.0 ] 105 100

NOTE.ÈEnergies were calculated assuming h \ 7.26] 107 cm.


We apply this approach to calculating the index of theenergy-frequency power law distribution for eventsobserved using TRACE. The range of energies observed isfrom a few times 1023 to 1026 ergs, where the energy of anevent is related to an enhancement in emission measure.This means that we observe events with energies in thenanoÑare range as deÐned by Parker (1988). The exactenergy for a particular event is dependent upon the line-of-sight depth, h, assumed. If we assume that the frequency ofevents follows a power-law relation with index, [c, thenthe value of c is dependent on the form assumed for h, notthe exact value for h. For instance, if h is assumed to be aconstant, as is assumed by KB, then we Ðnd that for eventsof any size with emission measure enhancements at least 2 pand at least 3 p, estimates for c are 2.56 and 2.42, respec-tively. These values are equivalent to those estimated byKB. Whereas, if we assume where is theh \ (A

e/k2)1@2, A

eevent energy and k is a constant, then estimates for c are2.13 and 2.02 for events of any size and emission measureenhancements at least 2 p and at least 3 p. These estimatesare about 0.4 lower than if a constant h had been assumedbut are about 0.4 greater than those estimated by Shimizu(1995) for microÑares in active regions. Our estimates for care all greater than 2, which implies that events with thelowest energies are not only more numerous than eventswith larger energies, but, since c is greater than 2, they arealso the dominant contributor to the heating of the quietcorona.

Although not detailed in this paper, it is worth mention-ing that the order of pairing of the images does not e†ect theestimates for c. For instance, in the results shown here wehave paired each 195 images with the next 171 images ;however, if we pair each 195 image with the preceding 171image, there is not real change in the results.

If we assume that these power-law relations extend downseveral orders of magnitude to 1021 ergs and lower, then wecan estimate the minimum energy of events needed tosupply the estimated 3 ] 105 ergs cm~2 s~1 required toheat the quiet corona (Withbroe & Noyes 1977). We Ðndthat for events with energies estimated using a constant

h \ 7.3] 107 cm and with enhancements of at least 2 p therange of event energies required to explain the heat losses inthe quiet corona is (1.3 ] 1021, 1027) ergs, whilst for eventsof any size and enhancements of at least 3 p the energyrange is (3.8 ] 1019, 1027) ergs. However, if h \ 2.2] 108cm then for events with emission measure enhancements atleast 2 p the heat required for the quiet corona can beprovided by events in the range (6] 1021, 1027) ergs. Forevents with energies determined assuming cm andh \ A

e1@2

events with emission measure enhancements at least 2 p theheat required for the quiet corona can be provided byevents in the range (1017, 1027) ergs.

Clearly, the above results suggest that ParkerÏs idea(Parker 1988) that nanoÑare-type events with energies inthe range 1024È1027 ergs do not explain the heat losses fromthe quiet corona. There is a possibility that picoÑares,events with energies in the range 1021È1024 ergs may heatthe quiet corona ; however, this looks unlikely. We, there-fore, need a data set that resolves events yet smaller still todetermine whether multiple picoÑare-type events heat thequiet corona. Although higher resolution is essential tosolving this problem, it is equally important that at theseresolutions, we still have high count rates so that errors and,therefore, the parameter p, are kept low. Furthermore, weneed data from density sensitive lines so the problem ofestimating the line-of-sight depth may be resolved.

C. E. Parnell would like to thank E. R. Priest for anintroduction to the literature on this topic and T. N. Neu-kirch for giving helpful suggestions and encouragement ; theTRACE team for the data and E. Deluca and R. McMullenfrom the CFA, and D. Brown from St. Andrews, for all theirhelp and support in preparing it ; and also the RoyalAstronomical SocietyÈthis work was done whilst C. E.Parnell was the RAS Sir Norman Lockyer Fellow. Bothauthors are very grateful to Peter Cargill, Arnold Benz, Sa� mKrucker, Karel Schrijver, and Markus Aschwanden forreading the manuscript critically and giving sage advice forits improvement.

APPENDIX

EMPIRICAL DISTRIBUTION FUNCTIONS

Empirical distribution functions are widely used by statisticians as a method of comparing graphically the goodness of Ðt ofa data set to a model.

The probability density function v(E) of a random variable E is the function such that the probability of E lying in theinterval (a,b) is

Pr(a \ E\ b) \Pa

bv(E)dE . (A1)

One particular probability density function (p.d.f.) that is useful here is

v(E)\ 45600(/] 1)(E/E0)Õ/E0 0 \ E[ E0 ;0 E0\ E\ O .

(A2)

where the probability that E lies in the range (0, O) equals the probability that E lies in the range (0, which equals 1.E0],Often, it is useful to know the probability that E is less than X. This is written

Vl(E[ X) \ V

l(X) \

Pa

Xv(E)dE , (A3)


and is known as the cumulative distribution function (c.d.f.). It is a function that increases monotonically from 0 to 1. We willcall it the left c.d.f., since it tells us the probability that E is to the left of (smaller than) some X.

Clearly, we can also deÐne a right cumulative distribution function which gives the probability that E is to the right ofVr(X),

(greater than) some X,

Vr(Eº X) \ V

r(X) \

PX

=v(E)dE . (A4)

Again, this function varies monotonically as X increases, but this time it decreases from 1 to 0 as X increases. Notsurprisingly, we Ðnd for obvious reasons.V

l(X) ] V

r(X)\ 1,

The left c.d.f. for the p.d.f. given in equation (A2) is

Vl(E[ X)\ V

l(X) \

P0

X /] 1E0

A EE0

BÕdE\

AXE0

BÕ`1. (A5)

Clearly, and as X ] 0,Vl(E0)\ 1 V

l(X)] 0.

The frequency function, in equation (9) for events with energies less than has a similar form to that of the p.d.f.pobs(E), E0v(E) in equation (A2) and, therefore, has a pseudo-c.d.f., equal to times a constant. The dashed line in Figure 7a is a plotVl(X)

of the left c.d.f., against X, where X represents energy of events less thanVl(X), E0.If we had a data set that had data values where and we believed that they had a p.d.f. of the form of v(E)E1, . . . , E

n, E

i[ E0,in equation (A2), then we could draw an empirical c.d.f. in the following manner. Let our data set containing n data values be

ordered such that Then, the empirical c.d.f. is deÐned as0 [ E(1)[ . . . [E(i)[ . . . [E(n) [E0\O.

V (i)* \ i [ 1/2n

, (A6)

where i\ 1, . . . , n. Clearly, if n is large and In between these limits, increases monotonically from 0 to 1V (1)* B 0 V (n)* B 1. V (i)*as i increases. In this case, a plot of the empirical c.d.f. against is shown as a solid/dotted line in Figure 7a.V (i)* E(i)It is useful to compare the empirical c.d.f. with the estimated model c.d.f.. To do this we could plot versus A plotVl(E(i)) V (i)* .

of the above c.d.f.Ïs is given in Figure 7b. If the data Ðtted the model c.d.f. exactly, we would expect to get a straight linethrough (0, 0) and (1, 1). However, as we already know for the data used here, that when E gets small our observations areunreliable, since we are near the observational limits of our instrument.

If, on the other hand, we had a data set that had a p.d.f. of the form

w(E)\ 45600

0 0 [ E\ E0 ,(c[ 1)(E/E0)~c/E0 E0 º E\ O ,

(A7)

then it would make more sense to consider a right c.d.f. For example, the frequency function, in equation (9) for eventspobs(E),with energies greater than has a similar form to this p.d.f.. The right c.d.f. for this p.d.f. would look likeE0

Wr(Eº X)\ W

r(X) \

PX

= c[ 1E0

A xE0

B~cdx \

AXE0

B1~c. (A8)

Then, clearly, and for large X, A plot of the right c.d.f. for events with energies at least is given inWr(E0)\ 1 W

r(X)] 0. E0Figure 8a.

FIG. 7.È(a) Plot of the c.d.f. for the left-hand power law of the skew-Laplace distribution against energy, X, where the dashed line is the left c.d.f., andVl,

the solid/dotted line is the empirical c.d.f., V *, for the data. (b) Plot of the left c.d.f., against the empirical c.d.f., V *, for the left-hand power law of theVl,

skew-Laplace distribution.


FIG. 8.È(a) Plot of the c.d.f. for the right-hand power law of the skew-Laplace distribution against energy where the dashed line is the right c.d.f., andWr,

the solid/dotted line is the empirical c.d.f., W *, for the data. (b) Plot of the right c.d.f., against the empirical c.d.f., W *, for the right-hand power law of theWr,

skew-Laplace distribution.

If the data set of events with energies at least contains n data values and we order them such thatE0 0 [ E0[ E(1)[ . . . [then the empirical c.d.f. has the formE(i)[ . . . [E(n)\ O,

W (i)* \ 1 [ i [ 1/2n

, (A9)

where i\ 1, . . . , n. Clearly, for large n, and In between these limits decreases monotonically from 1 toW (1)* B 1 W (n)* B 0. W (i)*0 as i increases. A graph of against is plotted in Figure 8a as a solid/dotted line.W (i)* E(i)Again, a graphical assessment of the goodness of Ðt of the model c.d.f. to the empirical c.d.f. is attained from plotting ofversus (Fig. 8b). As before, if the data Ðtted the c.d.f. exactly, we would expect to get a straight line through (0, 0)W

r(E(i)) W (i)*and (1, 1). However, the Ðt is bad when gets large and our data are sparse.E(i)In general, statisticians prefer to use plots of the model c.d.f. versus the empirical function to demonstrate the goodness of Ðt

of a model to the data. However, we felt that, in this paper, it would be helpful to the reader if we actually plottedenergy-frequency plots for the data.

As already mentioned, is related to v(E) and w(E). Indeed,pobs(E)

pobs(E)\ p0/] c

[(c[ 1)v(E) ] (/] 1)w(E)] .

So is related simply to the two p.d.f.Ïs discussed above and the integral of between 0 and O equals thepobs(E) pobs(E) p0,number of events per unit area per unit time, instead of 1. Furthermore, the function can be written in the formpobs(E)

pobs(E)\ 45600p0(/] 1)(c[ 1)/(/] c) ] (E/E0)Õ`1/E 0 \ E[ E0 ;p0(/] 1)(c[ 1)/(/] c) ] (E/E0)1~c/E E0[ E\ O .

(A10)

Clearly, to compare the data to our model we can replace the factor by the empirical form for events with energies(E/E0)Õ`1less than and we can replace the factor by the empirical form for events with energies at least However, weE0 (E/E0)1~c E0.need to know how many events are in each of these two data sets. From equation (10), the estimate of is deÐned asl\ log E0the element of the ordered observations Thus, Hence, since the total number of observations is n,(r‘ ] 1)th y(i). E� 0\ E(r‘`1).there are observations with energies less than and with energies greater. It follows from equation (A5) thatr‘ E� 0 n [ r‘

(E(i)/E0)Õ`1 B (i [ 1/2)/r‘ , i \ 1, . . . , r‘ ,

and from equation (A8) that

(E(i)/E0)1~c B 1 [ (i [ r‘ [ 1/2)/(n [ r‘ ), i \ r‘ ] 1, . . . , n .

Therefore, an empirical analogue of the frequency function, ispobs(x),

Dobs(i)\45600

p0(/] 1)(c[ 1)/(/] c)[(i [ 1/2)/r‘ ]/E(i) i \ 1, . . . , r‘ ;p0(/] 1)(c[ 1)/(/] c)[1 [ (i [ r‘ [ 1/2)/(n [ r‘ )]/E(i) i \ r‘ ] 1, . . . n .

(A11)

In Figures 4a and 4b and Figures 5a and 5b, we plot against on a log-log graph. These plots are forms ofDobs(i) Eobs(i)energy-frequency plots that avoid the use of arbitrary choices.


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