energetics of explosive events observed with sumer

THE ASTROPHYSICAL JOURNAL, 565 :1298È1311, 2002 February 1( 2002. The American Astronomical Society. All rights reserved. Printed in U.S.A.

ENERGETICS OF EXPLOSIVE EVENTS OBSERVED WITH SUMER

AMY R. WINEBARGER,1,2 A. GORDON EMSLIE,2 JOHN T. MARISKA,3 AND HARRY P. WARREN1Received 2001 June 12 ; accepted 2001 October 3

ABSTRACTObservations of solar chromosphere-corona transition region plasma show evidence of small-scale,

short-lived dynamic phenomena characterized by signiÐcant nonthermal broadening and asymmetry inthe wings of spectral line proÐles. These impulsive mass motions (explosive events) are thought to be theproduct of magnetic reconnection and to be similar in driving mechanism (though larger in size) to nano-Ñares, the small-scale events proposed to heat the corona. In this paper, we present a statistical analysisof the energetics of explosive events to address the viability of the nanoÑare heating theory. We considerhigh spectral, spatial, and temporal resolution spectra of the C III j977, N IV j765, O VI j1032,and Ne VIII j770 lines observed with the Solar Ultraviolet Measurements of Emitted Radiation(SUMER) telescope and spectrometer. Each line proÐle exhibiting explosive event characteristics wasanalyzed using the velocity di†erential emission measure (VDEM) technique. A VDEM is a measure ofthe emitting power of the plasma as a function of its line-of-sight velocity and hence provides a methodof accurately measuring the energy Ñux associated with an explosive event. We Ðnd that these eventsglobally release D4 ] 104 ergs cm~2 s~1 toward both the corona and chromosphere. This implies thatexplosive events themselves are not energetically signiÐcant to the solar atmosphere. However, the dis-tribution of these explosive events as a function of their energy has a power-law spectral index ofa \ 2.9^ 0.1 for the energy range 1022.7È1025.1 ergs. Since a is greater than 2, the energy content isdominated by the smallest events. Hence, if this distribution is representative of the size distributiondown to lower energy ranges (D1022 ergs), such small and (currently) undetectable events would releaseenough energy to heat the solar atmosphere.Subject headings : Sun: transition region È Sun: UV radiation

1. INTRODUCTION

One of the most important questions driving solarphysics today concerns the heating of the solar atmosphere.In the quiet Sun, approximately 3] 105 ergs cm~2 s~1 arerequired to heat the corona and 4 ] 106 ergs cm~2 s~1 toheat the chromosphere (Withbroe & Noyes 1977). Oneheating theory is that energy is released through dissipationof small current sheets that form between intertwined mag-netic Ðlaments (Parker 1988). Parker estimated that each ofthese reconnection events released 1024 ergs into the solarcorona. Since this is D10~9 times the energy of a typicallarge Ñare, he called these events nanoÑares. Current instru-mentation and line-of-sight integration limitations do notpermit resolution of these Ðlaments, and therefore nano-Ñares cannot be observed and their importance to theglobal heating of the solar atmosphere cannot be directlyassessed. The Sun, however, exhibits a variety of pheno-mena that are probably driven by magnetic reconnectionand are observable. These phenomena span a range of ener-gies and include solar Ñares (1028È1032 ergs event~1) andmicroÑares (1024È1027 ergs event~1). Through studyingthese larger events, it is possible to determine the impor-tance of reconnection events as a function of the energyreleased into the solar atmosphere and hence assess indi-rectly the viability of the nanoÑare theory.

The energetic signiÐcance of reconnection events dependson their rate of occurrence as a function of their energy. It

1 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, MS58, Cambridge, MA 02138 ; awinebarger=cfa.harvard.edu.

2 Department of Physics, University of Alabama in Huntsville, Hunts-ville, AL 35899.

3 Code 7673, E. O. Hulburt Center for Space Research, Naval ResearchLaboratory, Washington, DC 20375.

has been found for the hard X-ray bursts associated withsolar Ñares (e.g., Dennis 1985) that the di†erential numberof events as a function of their energetic content, E, falls o†as a power law with index, a, i.e., dN/dED E~a, where E(ergs event~1) is the energy released into the atmosphere byan event and N (events s~1) is the number of eventsoccurring globally per unit time. For a \ 2, the total power,/ E(dN/dE)dE, released by all events is dominated by theenergy of the largest events, but for a [ 2, the total power isdominated by the energy of the smallest events (see dis-cussion in Hudson 1991). For the dissipation process associ-ated with ParkerÏs nanoÑares to release the energynecessary to account for atmospheric heating, a must begreater than 2 in the nanoÑare energy range (Hudson 1991 ;Porter, Fontenla, & Simnett 1995 ; Haisch, Strong, &Rodono 1991).

The index, a, associated with solar Ñare hard X-ray burstshas been determined to be approximately 1.8 throughnumerous analyses of bursts with energies greater than 1028ergs event~1 (for a review, see Crosby, Aschwanden, &Denis 1993). This index has also been determined forsmaller events such as transient brightenings observed withnarrowband Ðltergrams on EUV Imaging Telescope (EIT)and T ransition Region and Coronal Explorer (T RACE), butwith ambiguous results. For these brightenings, the indexwas found to be between 2.3 and 2.4 for events with energiesin the range 1024.9È1026.2 ergs event~1 (Krucker & Benz1998), between 2.0 and 2.6 for events with energies in therange 1023È1026 ergs event~1 (Parnell & Jupp 2000), and1.8 for events with energies in the range 1024È1026 ergsevent~1 (Aschwanden et al. 2000). (Though these transientbrightenings were referred to as ““ microÑares ÏÏ by Krucker& Benz [1998] and ““ nanoÑares ÏÏ by Parnell & Jupp [2000]and Aschwanden et al. [2000], they have similar spatial

1298

ENERGETICS OF EXPLOSIVE EVENTS 1299

scales [D103 km] and overlapping energy ranges. Further-more, they are all larger than the spatial scale of Ðlamentsand hence are not ParkerÏs putative nanoÑares. To avoidconfusion, the transient brightening events will hereafter bereferred to as microÑares and ParkerÏs Ðne-scale reconnec-tion events will be called nanoÑares.) The microÑares weredetected by an enhanced intensity signature over the sur-rounding background emission, and their energies werederived by interpreting this increase in emission as heating.The discrepancies in the indexes for microÑares have beenattributed to the di†erences in selection techniques and inmodel assumptions used to calculate the electron densities(Aschwanden et al. 2000).

From previous analyses of microÑares, it is not clearwhether the index of the distribution of reconnection eventsis greater or less than 2 for moderate energy events. Fur-thermore, all the above results depend heavily upon theassumed efficiency of the process to convert the releasedenergy into either hard X-ray emission or thermal energy.This efficiency may be (waveband) sensitive to the type ofenergy transfer involved and may dramatically inÑuence thedetermination of the distribution function and its index.Porter et al. (1995) attempted to correct the distribution byconsidering an efficiency function that varied across thespectrum from UV to X-rays but unfortunately found thatthis distribution (and its associated index) could not unam-biguously be obtained.

To build upon these previous analyses, we present here astatistical analysis of the energies of explosive events anddetermine, for the Ðrst time, the distribution of the numberof explosive events as a function of their energies. Explosiveevents were Ðrst observed by the Naval Research Labor-atoryÏs High Resolution Telescope and Spectrograph(HRTS) instrument as small-scale (D1500 km), short-lived(D60 s), moderate-energy (D1024 ergs event~1) events inthe chromosphere-corona transition region spectral lines(Brueckner & Bartoe 1983). These events were characterizedby enhanced nonthermal broadening and/or asymmetry inthe wings of the line proÐles ; they had a global birthrate of600 events s~1 (Dere et al. 1984, 1989). Explosive eventsmost often occurred in areas of mixed polarity near mag-netic neutral lines during times of magnetic cancellation(Chae, Yun, & Poland1998). Furthermore, the spectral sig-nature of explosive events showed enhancements in boththe red and blue wings of the proÐle, implying high-velocityÑows similar to a bidirectional jet associated with magneticreconnection (Innes et al. 1997). Hence, explosive eventswere thought to be the products of magnetic reconnection,speciÐcally magnetic cancellation of the photospheric Ðeldswith either emerging Ñux (Dere et al. 1991) or dissipatingactive region Ñux (Porter & Dere 1991). A large volume ofliterature has been published detailing the characteristics ofexplosive events (Brueckner & Bartoe 1983 ; Dere et al.1986, 1989, 1991 ; Cook et al. 1988 ; Dere 1994 ; Innes et al.1997 ; Chae et al. 1998 ; et al. 1999 ; & DoylePe� rez Pe� rez2000).

The relationship between microÑares (transient bright-enings observed in narrowband Ðltergram instruments) andexplosive events (high-velocity events observed with high-resolution spectrometers) remains unclear. Because of theirsimilarity in spatial scale, lifetime, energy range, and drivingmechanism, it was originally thought that an explosiveevent was simply the spectral signature of a microÑare(Porter et al. 1987). Yet, it is not currently clear whether

these two phenomena are the same process observed withdi†erent instruments or two related yet di†erent events (seethe discussion in Porter & Dere 1991). Nevertheless, bothexplosive events and microÑares can be used to determinethe power-law distribution of reconnection events withenergies around 1024 ergs events~1. Unlike microÑares,however, explosive events have a distinct spectral signature(extremely broadened or skewed line proÐles) and hence donot rely on detection techniques that require a signiÐcantbrightness compared to the surrounding area. This allowsus to detect smaller energy events that are not signiÐcantlybrighter than their surroundings and to improve our under-standing of the distribution of events at lower energies.

In this paper, we address the questions (1) of whetherexplosive events themselves are energetically signiÐcant tothe chromosphere and corona and (2) whether the distribu-tion of the number of explosive events as a function of theirenergy implies that smaller, undetectable events (ParkerÏsnanoÑares) may have an e†ect on the energy balance of thesolar atmosphere. We measure the energy Ñux associatedwith explosive events using the velocity di†erential emissionmeasure (VDEM) technique. This technique was thor-oughly discussed by Winebarger et al. (1999) but is brieÑyreviewed in ° 2. This technique is applied to data from theSolar Ultraviolet Measurements of Emitted Radiation(SUMER) instrument Ñown aboard the Solar and Helio-spheric Observatory (SOHO). We Ðnd that explosive eventsglobally release D4 ] 104 ergs cm~2 s~1 toward both thechromosphere and corona, approximately 10% of coronalheating requirements and 1% of chromospheric heatingrequirements. We Ðnd the average index of the distributionsof events as a function of their energy released toward thecorona and toward the chromosphere is 2.9 ^ 0.1 for theenergy range D1023È1025 ergs event~1. The implications ofthese results will be discussed in ° 7.

2. THEORY

An explosive event is characterized by a broadenedand/or skewed line proÐle. This Doppler-shifted emissioncontains information on the bulk Ñows present in thesource. From the bulk Ñows, we can directly measure theenergy Ñows associated with the reconnection event (asopposed to simply interpreting emission enhancements asheating). Furthermore, we can assess and separate theenergy Ñowing toward the observer or away from the obser-ver (i.e., toward the corona or toward the chromosphere,respectively, for observations made near disk center).

The average energy Ñux toward the observer (]) or awayfrom the observer ([) carried by a plasma conÐned to movealong a single ““ parallel ÏÏ direction (such as a bidirectionaljet associated with reconnection) can be written as

jEB

\ 52cnkT vAB

] 12co(2m2vAB

] vA3

B) , (1)

where k (ergs K~1) is BoltzmannÏs constant, n (cm~3) is thetotal number density, o (g cm~3) is the mass density, and T(K) is the temperature of the Ñowing plasma. In equation (1)a volumetric Ðlling factor, c, has been added to account forthe Ðlamentary structure of the transition region and a non-thermal rms velocity term, m (km s~1), has been included toaccount for the typical turbulent, nonthermal velocity thatis observed at transition region temperatures (e.g., Mariska1992). In equation (1) (km s~1) is the bulk velocity alongv

Athe parallel direction and the average represented by the baris over the parallel bulk velocity distribution, of thef

A,

1300 WINEBARGER ET AL. Vol. 565

plasma, i.e.,

vA`k \

P0

`=vAk f

A(v

A)dv

A,

vA~k \

P~=

0vAk f

A(v

A)dv

A. (2)

We deÐne the bulk velocity distribution as a function thatdescribes the fraction of plasma moving with a given bulkvelocity along the direction of motion. (For a derivation ofthe energy-Ñux equation and further discussion on theparallel bulk velocity distribution, see Winebarger et al.1999.)

To approximate the bulk velocity distribution, we use theVDEM function (Newton, Emslie, & Mariska 1995).VDEM [photons cm~2 s~1 (cm s~1)~1] is a measure of theemitting power of a portion of the atmosphere travelingwith a given line-of-sight velocity. Formally, it is deÐned by

VDEM(vlos)\ ne2G(T )ds

dvlos, (3)

where is the electron density, G(T ) is the emissivity func-netion associated with the observed spectral line, s is the dis-

tance along the line of sight, and is the velocity along thevlosline of sight. VDEM is related to the observed intensity ofthe emitted line through

I(j)\ Ac4nD2

PVDEM(vlos)K(j, vlos)dvlos , (4)

where A is the area under observation, D is the Sun-satellitedistance, and K(j, is a kernel function. For the tran-vlos)sition region species analyzed in this paper, the kernel func-tion is well-represented by a thermally broadened Gaussianfunction centered on the wavelength corresponding to theline-of-sight velocity in question, i.e.,

K(j, vlos)\1

J2npK2

expA[ Mj[ j0[1[ (vlos/c)]N2

2pK2

B(5)

(Newton et al. 1995), where

pK

\ pth\ j0cSkT

mi

(6)

c is the speed of light, is the rest wavelength and is thej0 mimass of the emitting ion. The width of the kernel function

can also be expanded to include the e†ects of instrumentalbroadening, i.e., where is thep

K\ (pth2 ] pinstr2 )1@2, pinstrwidth of the instrumental function (see Winebarger et al.

1999 for further details).Deconvolving the kernel from an observed line proÐle

produces the corresponding VDEM function. Figure 1a dis-plays a sample spectrum of the O VI j1032 line. The arrowhighlights a spatial pixel exhibiting nonthermal broadeningand a strong blue-wing asymmetry. The intensity at thisspatial pixel is shown in Figure 1b, while Figure 1c showsthe VDEM returned from deconvolving the line proÐleusing the discretized inversion procedure described byNewton et al. (1995).

If the line of sight coincides with the direction of motionand the temperature and density are uniform throughoutthe volume emitting the spectral line, then a normalizedVDEM function is proportional to the parallel bulk veloc-

ity distribution (Winebarger et al. 1999). It is assumed,therefore, that the parallel moments of the velocity can becalculated using the appropriate weighted means of theVDEM function :

vA`k \ /0`= v

Ak VDEM(v

A)dv

A/~=`= VDEM(v

A)dv

A

,

vA~k \ /~=0 v

Ak VDEM(v

A)dv

A/~=`= VDEM(v

A)dv

A

. (7)

(For a complete discussion on this assumption, see Wine-barger et al. 1999.) Uncertainties in the measured velocitymoments are derived from uncertainties in the intensity ofthe line, as well as any uncertainties associated with thewavelength calibration of the original line proÐle.

Several possible systematic errors were addressed beforeapplying VDEM analysis to spectral data. For instance,VDEM returns information only about the velocity dis-tribution along the line of sight. The assumption that theline of sight coincides with the direction of motion should ingeneral introduce a deprojection factor (which wouldalways be greater than 1) in the resulting energy Ñux. As aresult, the energy Ñux thus determined is a lower limit to thetrue value. Similarly, we would like to interpret motiontoward the observer as toward the corona and motion awayfrom the observer as toward the chromosphere. Thus, welimit ourselves to observations within 10@ of disk center.

Two other potentially signiÐcant systematic errors wereaddressed through simulations of hypothetical event lineproÐles. The Ðrst possible error stems from the assumptionthat the observed emission proÐle (which su†ers fromuncertainties due to Poisson statistics) well represents theemitting plasma. These statistical uncertainties are mostimportant when deconvolving a VDEM function from alow count-rate proÐle. The second possible error is due tothe assumption that the plasma is at a constant, uniformdensity and temperature. This assumption is surely naive inthe presence of a dynamic reconnection event, yet it must bemade to calculate the width of the kernel function (see eq.[5]) and recover the VDEM from an event line proÐle.

The magnitudes of these two systematic errors wereaddressed through Monte Carlo simulations of event lineproÐles. We began with a hypothetical density, temperature,and bulk velocity distribution that could represent an eventplasma on the Sun and calculated the energy Ñuxes associ-ated with them. By weighting the bulk velocity distributionwith and an emissivity function representative of a tran-ne2sition region spectral line, we recover a hypothetical VDEMfunction. With this VDEM function, we simulate count-rateproÐles (see eq. [4]) using Monte Carlo techniques. We thendeconvolve a VDEM function from the simulated lineproÐle using a kernel function derived from assumed con-stant temperature and density. Finally, we calculate theenergy Ñuxes from this recovered VDEM and comparethem to the true energy Ñuxes. Di†erences in these two setsof values indicate systematic errors from either the decon-volution process and/or inaccurate assumptions of tem-perature and density.

To assess the number of counts required to minimize thee†ect of Poisson uncertainties on the recovered VDEM, weassumed that the temperature and density of the plasmawere constant and uniform. We varied the number of totalcounts in a series of simulated proÐles and compared themeasured energy Ñuxes to the true energy Ñuxes. We found

No. 2, 2002 ENERGETICS OF EXPLOSIVE EVENTS 1301

FIG. 1.È(a) SUMER spectrum of O VI j1032 line taken on 1996 May 10 at Sun center. The horizontal axis is the dispersion pixel number, and the verticalaxis is the spatial pixel along the slit. The arrow points out spatial pixel 26, whcih exhibits explosive event characteristics. The line proÐle associated with thispixel is shown in (b). The VDEM function recovered from this line proÐle is shown in (c). Note the velocity axis of the VDEM function has been reversed sothat positive (upward) Ñows correspond to the blueshifts observed in the line proÐle in the plot above it.

that the systematic errors in the energy Ñux were less thanerrors derived from uncertainties from count-rate statisticsfor proÐles containing more than 1000 counts.

To investigate the e†ects of temperature (and possibleassociated density) gradients in the source on the measuredVDEM functions, we simulated line proÐles with severaltemperature and density scenarios, recovered VDEM func-tions from the simulated proÐles, and determined theassociated energy Ñuxes. Regardless of the scenario, theenergy Ñuxes were always less than the true energy Ñuxes.This result is due to the narrow emissivity function, G(T ), ofalmost all transition region ions. The emissivity functiondrops rapidly when the plasma is heated or cooled from itspeak temperature. In a constant density plasma, the changein the intensity is proportional to the change in emissivityfunction ; hence, the intensity of the hotter or cooler plasmais insigniÐcant compared to the emission from the plasma atthe peak temperature. In the case of a constant pressureplasma, an increase or decrease in temperature results in andecrease or increase in density, respectively. Because theobserved intensity is proportional to we might expectn

e2,

that a cooler, denser plasma to produce more emission. Theemissivity function, G(T ), however, falls o† more steeply onthe low-temperature side of the peak than the square of thedensity. Hence, even for a constant pressure plasma, a

change in temperature produces a drop in emission : anincrease in temperature results in a decrease in both thedensity and emissivity function, while a decrease in the tem-perature results in a drop in the emissivity function thatmore than o†sets the increase in density.

Because of the strongly peaked form of G(T ), it is reason-able to assume that any emission that is observed originatesin plasma at the peak temperature of the emissivity func-tion. The measured energy Ñux will then be representativeof the plasma at that temperature. Any plasma at hotter orcooler temperatures will not contribute signiÐcant emissionto the line in question, and its associated energy Ñux willtherefore not be measured. Hence the measured energyÑuxes will always be the lower limit to their true values ifthe density or temperature gradients are present.

3. INSTRUMENT AND DATA

The SUMER experiment is a high-resolution, normal-incidence spectrometer capable of imaging the Sun at wave-lengths between 660 and 1600 SUMERÏs Ðrst-orderA� .spectral resolution is about 45 and is only weaklymA�dependent on wavelength. The spectrometer is stigmaticwith a spatial resolution close to Exposures from1A.5.SUMERÏs detectors (referred to as A and B) are limited to asingle wavelength range approximately 45 wide, and onlyA�


TABLE 1

CHARACTERISTICS OF THE DATA SETS USED FOR THIS STUDY

Parameter C III O VI N IV and Ne VIII

Date(s) observed . . . . . . . . . . . . . 1996 May 10 1996 May 10 1996 Oct 18ÈNov 2Slit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0A.3 ] 120A 1A ] 120A 1A ] 120ADetector . . . . . . . . . . . . . . . . . . . . . . . A A BAverage solar position . . . . . . . 200A west, 150A north 200A west, 150A north 0A, 0ASpatial resolution (km) . . . . . . 740 740 2170Temporal resolution (s) . . . . . . 48 36 120Number of proÐles . . . . . . . . . . . 17,520 39,360 43,560Chromospheric lines . . . . . . . . . Yes Yes No

one detector may be operated at a time. The central 20 ofA�the detectors are coated with KBr and, at most wave-lengths, are more sensitive than the bare microchannelplates. There are also 10] attenuators covering the ends ofthe detectors that allow for observations of Lya. The pre-launch radiometric calibration of the instrument was estab-lished using plasma discharge lamps referenced to asynchrotron radiation source (Hollandt et al. 1996) withrelative uncertainties of ^15% between 540 and 1250 A�and ^30% at longer and shorter wavelengths (Wilhelm etal. 1997). The contribution of dark counts, o†-band light,

and scattered light to SUMER disk observations is verysmall (Wilhelm et al. 1997) and can be ignored. A detaileddescription of the instrument is given in Wilhelm et al.(1995), and its initial in-Ñight performance is reviewed byWilhelm et al. (1997) and Lemaire et al. (1997).

Explosive events are small-scale, short-lived phenomena ;hence the data used in this study must be of high spatial andtemporal resolution. Furthermore, to minimize uncer-tainties in the measured energy Ñux, the wavelengths associ-ated with each proÐle must be well deÐned fromreference-line calibration. The SUMER data archives were

FIG. 2.ÈSpectrum around the O VI j1032 line along with the corresponding line proÐle averaged along the slit


FIG. 3.ÈSpectrum around the C III j977 line along with the corresponding line proÐle averaged along the slit

scanned for data that closely met our criteria. Only obser-vations of the quiet Sun were considered. The character-istics of the selected data sets are discussed below and givenin Table 1. The reduction of the raw SUMER data includeddead-time correction, local-gain correction, Ñat-Ðeld sub-traction, and correction of the geometric distortion.

The Ðrst data set was taken over 5 hr on 1996 May 10 asa set of rasters centered at approximately 200A west and150A north of Sun center. During the Ðrst 3 hr of obser-vations, the O VI j1032 line (formed around 105.45 K) wasobserved with the 1@@] 120@@ slit, and during the Ðnal 2 hr of

TABLE 2

THE SPECTRAL LINES USED FOR WAVELENGTH CALIBRATION FOR THE

N IV AND Ne VIII SPECTRA

Laboratory SolarWavelength Temperature Velocity Wavelength

Ion (A� ) (K) (km s~1) (A� )

N III . . . . . . 763.340 104.85 [7.7 763.360N III . . . . . . 764.357 104.85 [7.7 764.377S V . . . . . . . 786.47 105.2 [11.1 786.50O IV . . . . . . 787.741 105.24 [11.1 787.740

observations, the C III j977 line (formed around 104.8 K)was observed with the slit. Both observing0A.3@@] 120@@sequences used detector A. Throughout the observations,6 s exposures of 50 pixels (D2 around the main lines wereA� )recorded, with the exception of a single long (100 s) obser-vation of 512 pixels (D20 taken in the middle of eachA� )sequence. These long exposures of the O VI and C III wave-length ranges, shown in Figures 2 and 3 respectively,contain observations of O I lines, which we use as referencelines to provide wavelength calibration. The resultinguncertainty in the wavelength calibration is approximately^5 km s~1.

The second data set included 36 hr of observations takenover 9 days in the period between 1996 October 18 and1996 November 2 at or near Sun center. The N IV j765 line(formed at 105.15 K) and Ne VIII j770 line (formed at 105.9K) were simultaneously observed with the 1@@] 120@@ slitprojected onto detector B. The primary observationsinclude 20 s exposures of 50 pixels around the N IV andNe VIII lines, with one full detector (1024 pixels or 45 A� )exposure taken for context approximately every 40 minutes(see the example in Fig. 4). The wavelength range of thesefull detector observations do not include any chromo-spheric spectral lines ; thus we use four transition region


FIG. 4.ÈExample spectrum around the N IV j765 line and Ne VIII j770 line along with the corresponding line proÐle averaged along the slit

spectral lines (given in Table 2) as reference lines to deter-mine the wavelength calibration for the spectra. To correctfor the typical redshifts associated with transition regionspectral lines, the temperature of formation of each emittingion is used to estimate the an average redshift velocity fromChae et al. (1998). The resulting uncertainty in the wave-length calibration is approximately ^6 km s~1.

ProÐles must contain 1000 counts in order to well modelthe energy Ñux with VDEM analysis (see ° 2). However, theaverage number of counts in each O VI proÐle is approx-imately 250, in each C III proÐle approximately 200, and ineach N IV and Ne VIII proÐles approximately 80. To allowfor analysis of the majority of the proÐles, we increase thisaverage value to 1500 counts per proÐle by summing everysix consecutive O VI spectra and eight consecutive C III

spectra. For the N IV and Ne VIII data, we sum every sixconsecutive spectra and every three spatial pixels along theslit. This results in a total of 39,360 O VI line proÐles with atemporal resolution of 36 s and spatial resolution of 740 km,17,520 C III line proÐles with a temporal resolution of 48 sand spatial resolution of 740 km, and 43,560 N IV andNe VIII line proÐles with a temporal resolution of 120 sand spatial resolution of 2170 km.

4. EVENT SELECTION AND CHARACTERISTICS

The data considered in this analysis includes 144,000 pro-Ðles, but only a fraction of these proÐles display explosiveevent characteristics (i.e., excessive broadening and/orskewness in the wings of the proÐle). Previous techniquesfor selecting event proÐles include direct analysis of thespectra (e.g., Cook et al. 1988) and Ðtting each proÐle with aGaussian function to see if it is excessively broadened (e.g.,

Chae et al. 1998). The Ðrst method is biased toward larger,more easily distinguishable events, while the second methodrequires Ðtting a proÐle that is intrinsically non-Gaussianwith a Gaussian function. We Ðnd the Ðtting procedureoften fails and the widths that are retrieved are poor repre-sentations of the proÐleÏs true width.

To select the event proÐles for this analysis, we measurethe width, and the skewness, of each(j [ j)2, (j[ j)3,proÐle directly from the line intensity using

(j [ j)n \ / I(j)(j [ j)ndj/ I(j)dj

, (8)

where is the average Doppler shift of the proÐle. If ajproÐle is to be considered an explosive event, the measuredwidth and/or skewness must be ““ large,ÏÏ at least larger thanthe measured width or skewness of a normal, noneventproÐle. To determine the critical values for the width andskewness that separate normal, nonevent measurementsfrom ““ large ÏÏ event measurements, we consider the widthsand skewnesses of all the proÐles in the data sets. Figures 5and 6 show histograms (solid lines) of the measured widthsand skewnesses of the line proÐles observed in (a) the C III

j977 line, (b) the N IV j765 line, (c) the O VI j1032 line, and(d) the Ne VIII j770 line.

Each distribution shown in Figures 5 and 6 is really thecombination of two separate distributions : the distributionassociated with nonevent proÐles and the distributionassociated with event proÐles. Because most of the proÐlesdo not exhibit explosive event characteristics, nonevent pro-Ðles dominate the combined distributions near their modes.Conversely, event proÐles dominate the combined distribu-tion at large widths and skewnesses.


FIG. 5.ÈHistograms (solid lines) of widths of (a) the C III proÐles, (b) the N IV proÐles, (c) the O VI proÐles, and (d) the Ne VIII proÐles. The portion aroundthe mode of the histogram is Ðtted with a Gaussian function (dashed lines) and the number of excess events (dotted lines) above this Gaussian is found. Thecritical width values (dash-dotted lines) for each line is then taken to be the width where the number of events was larger than the number of nonevents.

We now describe our procedure for determining whetheror not a given proÐle qualiÐes as an event. First, we estimatethe distributions of normal, nonevent widths and skewnessby Ðtting Gaussian functions to the portion of the combineddistributions around the modes. These Gaussian Ðts areshown in Figure 5 and 6 as dashed lines. We then Ðnd theevent distributions (dotted lines) by subtracting the Gauss-ian curves from the original histograms. We deÐne the criti-cal width, as the width above which the number ofkwid (A� 2),events contributing to the distribution becomes larger thanthe number of nonevents. The critical width values foundfrom each ionÏs distribution are given in Table 3 and areshown as the vertical lines in Figure 5. Similarly, the criticalskewnesses (Fig. 6, vertical lines) are deÐned to be the skew-nesses at which the number of events becomes larger thanthe number of nonevents. Because the center of the skew-ness distributions are o†set to negative values, the criticalvalues are di†erent for proÐles with negative or positiveskewnesses. Both values, denoted with for thekskew`

(A� 3)positive and for the negative, are given for eachkskew~ (A� 3)ion in Table 3.

If a proÐle exhibits either width or skewness larger thanthe corresponding critical value, the proÐle is selected as anevent. The number of proÐles that meet these criteria fromeach data set are given in Table 4. Approximately 12% of allproÐles qualify as events. Of these selected proÐles, many(D24%) exhibit both statistically signiÐcant width andskewness. Of the proÐles exhibiting signiÐcant skewness, anaverage of 71% have dominant blue wings, i.e., (j [ j)3\ 0.This dominance is most evident in the C III and O VI pro-Ðles, while the N IV and Ne VIII data have roughly equalnumbers of events with dominant blue and red wings.

There are two ways to proceed with the analysis after theevent proÐles have been selected. Much of the analysisdepends on the average size of the event, i.e., the eventÏs areaand lifetime. It becomes important, then, to consider howthe proÐles are grouped into events. One option is to groupproÐles together as a single event if they occur in adjacentspatial pixels in a single spectrum or in the same spatialpixel in consecutive spectra. The area of the event is then theaverage length along the slit squared and the eventÏs lifetimeis the number of consecutive spectra multiplied by the expo-

TABLE 3

CRITICAL WIDTH AND SKEWNESS VALUES

Parameter C III N IV O VI Ne VIII

kwid (A� 2) . . . . . . . . . 7.8 ] 10~3 1.2 ] 10~2 1.2] 10~2 1.4 ] 10~2kskew`

(A� 3) . . . . . . 2.7 ] 10~4 2.8 ] 10~4 3.7] 10~4 3.2 ] 10~4kskew~ (A� 3) . . . . . . [2.7 ] 10~4 5.3 ] 10~4 [4.4] 10~4 [5.6 ] 10~4


FIG. 6.ÈHistograms (solid lines) of the skewnesses for (a) the C III proÐles, (b) the N IV proÐles, (c) the O VI proÐles, and (d) the Ne VIII proÐles. The portionaround the mode of the histogram is Ðt with a Gaussian function (dashed lines) and the number of excess events (dotted lines) above this Gaussian is found.The critical skewness values (dash-dotted lines) is then taken to be the value where the number of events was larger than the number of nonevents.

sure time of each spectrum. Previous observations of explo-sive events show, however, that di†erent events tend tooccur in the same area of the disk over time and that indi-vidual events may not be resolved spatially along the slit(e.g., Dere 1994). Because the resolution of the spectra usedin this study is comparable to or worse than that of theprevious analyses, we instead assume that each individualproÐle exhibiting event characteristics is a single event witharea equal to the width of the pixel squared and lifetimeequal to the exposure time of the spectrum. This treatmentimplies that each proÐle is a ““ snapshot ÏÏ of the dynamicsoccurring in the observed volume of plasma and is notphysically connected to the surrounding volume elements.

The e†ects of this assumption on the results are discussed in° 7.

The global birthrate of explosive events, R (events s~1), isfound from

R\N0AsunqAslit

, (9)

where is the total number of events (from Table 4),N0 Asun(cm2) is the solar surface area, q (s) is the total observationtime of the data, and (cm2) is the projected area of theAslitslit onto the solar surface. The steady state number ofevents that are occurring globally, is calculated byNglobal,

TABLE 4

CHARACTERISTICS OF EXPLOSIVE EVENT PROFILES


Number of event proÐles . . . . . . . . . . . . . . . . . . . . . . . 3403 2505 5531 2907Percentage of total proÐles . . . . . . . . . . . . . . . . . . . . 20 6.4 14 7.5Percentage of event proÐles :

Exhibiting enhanced width . . . . . . . . . . . . . . . . . . 97 79 96 70Exhibing enhanced skewness . . . . . . . . . . . . . . . . 29 45 28 51Exhibing both . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 24 25 21

Percentage of skewed event proÐles :With dominant blue-wing asymmetry . . . . . . 96 57 81 51With dominant red-wing asymmetry . . . . . . . 4 43 19 49

Global birthrate, R (events s~1) . . . . . . . . . . . . . . . 4.5 (4) 1.9 (3) 4.3 (4) 2.2 (3)Steady state number, Nglobal (events) . . . . . . . . . . 2.2 (6) 2.2 (5) 1.5 (6) 2.6 (5)

NOTE.È Values in parentheses are powers of 10, i.e., 2(5)\ 2 ] 105.


multiplying the global birthrate by the average lifetime ofthe events, or The global birthrates andt

E, Nglobal \Rt

E.

steady state numbers for the events observed in each spec-tral line are given in Table 4. The global birthrate is between103 and 104 events s~1, while the steady state number is105È106 events. These numbers are 1È2 orders of magnitudelarger than the birthrate and steady state numbers found inprevious analyses (e.g., Dere 1994).

5. EVENT ANALYSIS

To measure the energy Ñuxes (see eq. [1]), we Ðrstmeasure or assume values for temperature, density, massdensity, Ðlling factor, and average nonthermal velocity forthe source, as well as calculate the moments of the velocityby recovering a VDEM function from the event line proÐle.We assume that the temperature of the emitting plasmaobserved in each spectral line is the peak temperature ofeach lineÏs (fairly narrow) emissivity function. Furthermore,we assume the electron pressure in the transition region isconstant at 1015 K cm~3 (e.g., Mariska 1992), so the elec-tron density, (cm~3), is simply the pressure divided by then

etemperature in Kelvin. The number density, n (cm~3), andmass density, o (g cm~3), are found from the electrondensity and the abundances given in Mariska (1992). Thepeak temperature of each ionÏs emissivity function, thenumber density, and the mass density are given in Table 5.

The Ðlling factor is found for every event proÐle by takingthe ratio of the volume implied by the number density,temperature, and total intensity of the proÐle to the area ofthe pixel times the scale height of the atmosphere. Theaverage Ðlling factors for the event proÐles are also given inTable 5. The Ðlling factor varies signiÐcantly in each lineobserved, with an average of 23%.

The nonthermal rms velocity, m, is found from the averagewidth of the nonevent proÐles in each data set. The resultsare given for each ion in Table 5. The values are slightlygreater than the values generally reported in the literature.

Before recovering a VDEM function, each line proÐlethat meets the event criteria is calibrated to units of Ñux(photons cm~2 s~1 and background subtracted. TheA� ~1)width of the kernel is calculated from the above numberdensity and temperature and the instrumental width of theSUMER detectors, 37.8 for detector A and 57.0 formA� mA�detector B (Chae et al. 1998). Using this kernel, we Ðnd theVDEM function and moments of velocity associated witheach event line proÐle (eq. [7]). Finally, the energy Ñuxesboth toward and away from the observer are evaluated foreach event proÐle using equation (1). The averages of thesevalues are given in Table 5. The average energy Ñux towardthe observer over all event proÐles is 2.3] 105 ergs cm~2

s~1. The average energy Ñux away from the observer is2.5] 105 ergs cm~2 s~1.

6. THE GLOBAL ENERGETICS OF EXPLOSIVE EVENTS

The global contribution of energy Ñux toward thecorona, (ergs cm~2 s~1), or toward the chromo-jglobal`sphere, (ergs cm~2 s~1), from explosive events can bejglobal~constructed from the average of the total energy releasedtoward the corona or chromosphere by a single event, E

B(ergs event~1), times the global birthrate of the events, R(events s~1), divided by the surface area of the Sun, Asun(cm2), or

jglobalB\EB

RAsun

. (10)

The global birthrate has already been found in ° 4, and thusthe only unknown in equation (10) is the average energy perevent released toward the corona or chromosphere. Thisenergy is given by

EB

\ jEB

AtE

, (11)

where A is the area and is the lifetime of the event. ThetEaverage energy released toward the corona or chromo-

sphere per event is then

EB

\ 1N0

;i/0

N0jEBi

AitEi

, (12)

where is the total number of events.N0By substituting equations (9) and (12) into equation (10),the global energy Ñux can be written as

jglobalB\ ;i/0

N0jEBi

A Ai

Aslit

BAtEiqB

, (13)

which is simply the sum of the measured energy Ñuxesweighted by a spatial ““ Ðlling factor ÏÏ and a tempo-(A

i/Aslit)ral ““ duty cycle ÏÏ The standard error in the global(t

Ei/q).

energy Ñux is due to the uncertainties in the energy Ñux ofeach event, as well as the uncertainty in the event area (12Ai

)and lifetime resulting from the spatial and temporal(12tEi)resolution of the data and is proportional to N0~1@2.

Table 6 summarizes the global energy Ñux toward thecorona and chromosphere derived from the events observedin the four spectral lines. The average global energy Ñuxcontribution toward the corona is 3.6 ] 104 ergs cm~2 s~1,and the average global energy Ñux contribution toward thechromosphere is 4.4 ] 104 ergs cm~2 s~1. Both valuesimply that the global energy Ñux associated with explosiveevents is in itself too small to be signiÐcant in atmosphericheating. However, the variation of number of events versusthe energetics of the event may point to a signiÐcant role for

TABLE 5

VALUES USED TO MEASURE THE ENERGY FLUX


T (K) . . . . . . . . . . . . . . . . . . . . . . 104.8 105.15 105.45 105.9n (cm~3) . . . . . . . . . . . . . . . . . . . 1010.48 1010.13 109.83 109.38o (g cm~3) . . . . . . . . . . . . . . . . 10~13.51 10~13.86 10~14.16 10~14.61c (%) . . . . . . . . . . . . . . . . . . . . . . . 30 9 52 1m (km s~1) . . . . . . . . . . . . . . . . . 21.1 29.0 24.1 30.3jE`

(ergs cm~2 s~1) . . . . . . 2.30^ 0.05 (5) 9.1 ^ 0.2 (4) 5.69^ 0.06 (5) 1.27 ^ 0.02 (4)jE~ (ergs cm~2 s~1) . . . . . . [4.41^ 0.05 (5) [2.01] 0.04 (5) [3.54^ 0.05 (5) [1.21^ 0.02 (4)



TABLE 6

CHARACTERISTICS OF THE GLOBAL ENERGY FLUX CONTRIBUTION OF EXPLOSIVE EVENTS

Flows C III N IV O VI Ne VIII

Coronal

jglobal` (ergs cm~2 s~1) . . . . . . 4.46 ^ 0.08 (4) 1.64^ 0.04 (4) 7.9 ^ 0.1 (4) 2.67 ^ 0.05 (3)a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 ^ 0.1 2.8^ 0.1 3.3 ^ 0.4 2.8 ^ 0.5log10 C . . . . . . . . . . . . . . . . . . . . . . . . 45.1 ^ 0.1 46.2^ 0.1 57.7 ^ 0.4 45.5 ^ 0.5Applicable range (ergs) . . . . . . 1022.7È1024.2 1023.8È1025.1 1023.1È1024.3 1022.8È1024.1NanoÑare energy (ergs) . . . . . . 1021.3 1021.8 1022.3 1020.1R

n(events s~1) . . . . . . . . . . . . . . . . 106.6 106.7 105.9 109.0

Chromospheric

jglobal~ (ergs cm~2 s~1) . . . . . . [8.6 ^ 0.1 (4) [3.63^ 0.07 (4) [4.96 ^ 0.07 (4) [2.54 ^ 0.04 (3)a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.0 ^ 0.2 2.5^ 0.3 2.7 ^ 0.2 3.1 ^ 0.2log10 C . . . . . . . . . . . . . . . . . . . . . . . . 50.7 ^ 0.2 39.6^ 0.3 43.3 ^ 0.2 51.6 ^ 0.2Applicable range (ergs) . . . . . . 1023.0È1024.0 1023.9È1025.3 1022.7È1024.0 1022.7È1024.0NanoÑare energy (ergs) . . . . . . 1021.4 1020.3 1020.1 1019.9R

n(events s~1) . . . . . . . . . . . . . . . . 107.6 109.0 109.0 109.5


very small events, below the SUMER detectability thresh-old, in coronal heating. This is discussed in the next, andconcluding, section.

7. DISCUSSION

In this paper, we have presented a statistical analysis ofthe energetics of four sets of explosive events observed indi†erent spectral lines formed over a range of temperatures.Using new criteria for selecting events, we Ðnd that theyhave a global birthrate of 2 ] 103È5 ] 104 events s~1 and asteady state number of 2 ] 105È2 ] 106 events, both 1È2orders of magnitude greater than the global birthrate andsteady state number derived from the previous analyses(e.g., Dere 1994).

The measured contribution of explosive events to theglobal energy Ñux released toward the corona (D3.6] 104ergs cm~2 s~1) and to the global energy Ñux releasedtoward the chromosphere (D4.4] 104 ergs cm~2 s~1)implies that the events themselves are not signiÐcant to theenergy balance in the solar atmosphere. The values reportedhere are approximately 10 and 100 times smaller than thevalues needed to heat the corona or chromosphere, 3] 105and 4 ] 106 ergs cm~2 s~1, respectively.

However, in assessing the contribution of explosiveevents to coronal heating, we must be cognizant of thepossibility that the overall energy content may be domi-nated by a large number of low-energy events too weak tobe detected. Clues to the presence of such low-energy eventscan be found in the di†erential distribution of explosiveevent energies. To determine this distribution, we generatedhistograms of the number of events with energies between

and where d is 0.1. The di†erentiallog10 EB

(log10 EB

) ] d,number distribution of events occurring globally per unittime per unit energy is then found by multiplying thenumber of observed events in each bin, by theNobs(EB

),solar surface area and dividing by the observation time,area of the slit, and width of the energy bin, i.e.,

dNdE

B

\ AsunAslit q

Nobs(EB)

*EB

, (14)

where These distributions are shown*EB

\EB(10d[ 1).

in Figure 7 (toward the corona) and in Figure 8 (toward thechromosphere).

If the distributions follow a power-law form, i.e.,then plotting it on a logarithmic scaledN/dE

B\CE

B~a,

should result in a straight line with slope of [a and y-intercept of log C, i.e.,

logdNdE

B

\ log C[ a log EB

. (15)

Yet, the curves shown in Figure 7 and Figure 8 do notimmediately appear to be straight lines. The shape of thedistributions at high energies does appear to be linear, butat low energies the curves level o† and begin decreasing. Ifthis turnover is real (rather than an artifact of, for example,the event selection process), then the total energy content ofthe distribution of events as a whole will be determined to alarge extent by the energy at which the distribution Ñattensout.

To determine empirically the low-energy cuto† for eachdistribution, an initial slope, a, and standard error, ispa,found by considering only the highest energy points in thedistribution. Then, a single additional point at the imme-diate lower energy to those points considered is included todetermine its inÑuence on the Ðt. The energy at which the Ðtdeteriorates by more than 2 away from a with the addi-pation of the next point in the distribution is deÐned to be theempirical cuto† value for the distribution ; it represents theenergy at which the distribution breaks from the power lawand is shown for each distribution in Figures 7 and 8 as adash-dotted line. The slopes, standard errors in the slopes,y-intercepts, and applicable energy ranges associated withthese empirical cuto† values are given in Table 6. Theaverage slope of these distributions is a \ 2.9^ 0.1. Sincethis value is greater than 2, the energy content of the power-law part of the distribution is dominated by the lowestenergy events in that part of the distribution.

Because of the dominance of low-energy events in deter-mining the total energy content in the entire distribution ofevents, it is of paramount importance to determine whetherthe downward break from a power law at low event ener-gies in Figure 7 is an intrinsic property of reconnectionevents or an artifact of our event selection criteria. Recallthat the event selection criteria used in this paper are thatproÐles must contain 1000 counts and have a statistically


FIG. 7.ÈDistributions of events observed in the (a) C III line, (b)N IV line, (c) O VI line, and (d)Ne VIII line as a function of their energy released toward thecorona. The dash-dotted lines are the empirical energy cuto†, while the dashed lines are the nominal energy cuto†. The Ðts of these distributions, shown withthe solid line, are to the points above the empirical energy cuto†. The results are given in Table 6.

signiÐcant width and/or skewness. Determining the eventenergy corresponding to these threshold values is non-trivial. For example, there is no exact correlation betweenthe number of counts in a proÐle and the correspondingevent energy (e.g., a proÐle with a small number of countscan have a high energy Ñux if it is sufficiently shifted and/orskewed, while a proÐle with a high number of counts couldhave a negligible shift and/or skewness and hence a smallenergy Ñux). We can, however, identify the event energiescorresponding to 1000-count proÐles with typical (mean)width and skewness. These ““ nominal ÏÏ energies are 1022.3,1023.3, 1022.7, and 1022.6 ergs for the C III, N IV, O VI, andNe VIII lines, respectively, and are shown as dotted lines inFigure 7. For comparison, the empirical cuto† values deter-mined for the distribution functions of the explosive eventsobserved in these lines are, respectively, 1022.7, 1023.8,1023.1, and 1022.8 ergs. Further inspection of Figure 7reveals that the distribution curves are in all cases fairly Ñatfrom the ““ empirical ÏÏ cuto† energy down to the ““ nominal ÏÏcuto† energy and fall o† only sharply below the nominalenergy. For example, the largest di†erence between thenominal energy and the empirical energy is for O VI (Fig.7c), and this distribution has the Ñattest maximum. Con-versely, the smallest di†erence between the nominal energyand the empirical energy is for Ne VIII (Fig. 7d), and thatdistribution has the sharpest peak.

This observation suggests that perhaps our criterion thata proÐle contain 1000 counts (° 4) is causing us to reject asigniÐcant number of actual events, thereby artiÐciallydriving down the number of events at the low energies at

and below our nominal threshold. It is therefore importantto verify to what extent the observed fallo† may be due tosystematic errors associated with the selection of events. Toinvestigate this, we analyzed 15 proÐles that containedD1000 counts and had energies corresponding to thenominal energy cuto†. We obtained average values forthe mean-square and mean-cube velocities from(v2) (v3)these low-energy, low count-rate proÐles. We then con-structed bulk velocity distributions (Manheimer 1977) withvalues of and that were typical of the 1000-countv2 v3proÐles. Next we simulated a series of 1000-count proÐlesfrom this parent distribution function using a Monte Carlotechnique and determined what fraction of these proÐlesmet our criteria for selection as an event (° 4). The results ofthis experiment revealed that only 22% of the simulatedproÐles failed to meet one or more of the criteria for selec-tion as an event. This event rejection rate therefore causesonly a small deviation (a factor of D1.3 at the nominalcuto† energy) between the ““ true ÏÏ and ““ observed ÏÏ eventdistributions, a factor that is not large enough to accountfor the typical order-of-magnitude di†erence between theobserved distribution and the extrapolated power law at thenominal energies (except possibly for the case of Ne VIII, forwhich the di†erence is small). Hence the paucity of events atsuch energies relative to the extrapolated power laws ofFigure 7 is not primarily due to erroneous rejection ofactual events.

There remains the possibility that the deviations from thepower-law distributions in Figure 7 are due to other sys-tematic errors in our analysis. For example, we derived the


FIG. 8.ÈDistributions of the events observed in the (a) C III line, (b) N IV line, (c) O VI line, and (d) Ne VIII line as a function of their energy released towardthe chromosphere. The dash-dot lines are the empirical energy cuto†. The Ðts of these distributions, shown with the solid lines, are to the points above theempirical energy cuto†. The results are given in Table 6.

density of the source from a constant pressure assumptionand assumed that the density was uniform in all eventsobserved in the same spectral line. Furthermore, weassumed that the area and lifetime of each explosive eventwas the spatial and temporal resolution of the spectral data.If these assumptions were systematically wrong, e.g., thearea of all the events was actually twice the spatialresolution of the data, the distributions of the events as afunction of energy would systematically shift to a higher orlower energy scale. The expected nanoÑare energy andbirthrate would both be changed, but the slope of the dis-tribution would not change. If, however, the area, lifetime,and/or density are correlated or anticorrelated with theenergy of the events, the slope would be a†ected. Such acorrelation has been found in previous studies of micro-Ñares observed with EIT; Berghmans, McKenzie, & Clette(2001) found the size of the event was directly proportionalto its peak intensity and hence total energy release. If thesame were true for explosive events, the distribution wouldÑatten and the slope would be reduced, possibly to lessthan 2.

In an attempt to address how the area of the event a†ectsthe slope of the distribution, we revisited our decision toassume that every proÐle is an individual event (see ° 4). Wegrouped the event proÐles together if they occurred in adja-cent spatial pixels in the same spectrum or in the samespatial pixel in consecutive spectra. We found the eventsgroups would simply begin to outline the network bound-aries. The area of the groups did not appear to be correlatedwith event energy ; both large and small events could occur

in groups with large areas. We believe such ““ blind ÏÏ group-ing techniques are extremely unreliable and should not betrusted. To address fully the true size, lifetime, and densityof explosive events, it is necessary to have high-resolution,high-cadence images of the solar transition region, as wellas information on the local magnetic Ðeld motions and thehigh-quality spectral data. Until then, the e†ect of the areaand lifetime of events on the distribution of events as afunction of energy cannot be addressed.

Recognizing that the deviation from a power law couldstill be due to such e†ects, it is therefore still of interest todetermine to what extent the power-law distribution mustbe extrapolated downward to create a total energy Ñux inexplosive events consistent with that required for coronalheating. Using such an extrapolated power-law Ðt, we cancalculate the required nanoÑare energy, fromE minB,

PTB

\PEminB

EmaxBEB

dNdE

B

dEB

\ CPEminB

EmaxBE

B~a`1 dE

B

\ C[a ] 2

(EmaxB~a`2 [ EminB~a`2)

BC

a [ 2EminB~a`2 , (16)

where a and C are the values from Table 6 and are thePTBpowers required to heat the corona and chromosphere,


respectively. Similarly, we can predict the birthrate of nano-Ñares of the required energy :

RnB

BC

a [ 1EminB~a`1 . (17)

The resulting values of and are given in TableEmin BR

nB6. They show that the power-law distribution of number ofevents versus energy must extend downward to D1021.9ergs for the resulting energy content to be sufficient to heatthe corona. The corresponding global birthrate of suchevents would be 108.4 events s~1. A similar analysis fordownward Ñux events (Fig. 8) reveals that the power-law

distribution would have to extend down to an energy of1020.8 ergs, with a corresponding global birthrate of 109.1events s~1. Future observations are necessary to determinewhether such a production rate of low-energy events indeedexists.

The SUMER project is Ðnancially supported by DLR,CNES, NASA, and the ESA PRODEX program (Swisscontribution). The authors are grateful to the NASA GSRPprogram and the Sun-Earth Connection Guest Investigatorprogram.

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