A M U L T I T H E R M A L A N A L Y S I S O F S O L A R X - R A Y E M I S S I O N

K E N N E T H P. DERE, D O N A L D M. HORAN, and ROBERT W. K R E P L I N

Naval Research Laboratory, Washington, D.C. 20375, U.S.A.

(Received 2 April, 1974)

Abstract. The NRL SOLRAD 10 satellite carries six ionization chambers to measure solar X-radia- tion in the 0.5 to 60 ~ wavelength band. The X-ray emission spectrum in this range is determined by the derivative of the coronal emission measure ( [Ne2d V) with respect to temperature when the thermal processes of bremsstrahlung, radiative recombination and line radiation are considered. If a simple model for this differential emission measure is used and detector responses to the calculated spectra are fitted to the SOLRAD data by a least squares method, the differential emission measure can be obtained for temperatures between 2 • 106 K and 64 • 106 K. Data during quiet and flaring periods are analyzed and the general behavior of the differential emission measure during flares is presented. This analysis is based on experimental measurements of the efficiencies of the SOLRAD detectors.

1. Introduction

Solar X-ray emission in the wavelength band between 1 A and 100 A is generally re-

cognized to be the result o f thermal processes in the corona. Analysis o f the radiat ion in different parts of this 1 A to 100 A band yields temperatures as low as about 1.5x 106K and up to at least 30x 106K during some flares. Calculations by Kahler

et al. (1970) and Doschek (1972) show that deviations f rom thermal equilibrium in the

electron gas or f rom ionization equilibrium should disappear quickly at the tempera- tures ( ~ 10 x 106K) and electron densities (,-~101~ cm -3) found in flares (e.g.,

Cowan and Widing, 1973; Phillips et al., 1973; Culhane et al., 1970). There does not

seem to be any evidence for nonthermal X-ray emission f rom the nonflaring corona.

The appropriate model for analyzing solar X-ray data covering an extended wave-

length range is then a multi thermal coronal plasma. Batstone et al. (1970), Parkinson (1973), Chambe (1971) and Walker (1972) have

analyzed X-ray and E U V line emission f rom the nonflaring solar corona in terms of a

multi thermal model. They have fit their data with differential emission measures that decrease with increasing temperature for temperatures above 2 x 10 6 K and that can have significant components at temperatures as high as 8 x 10 6 K. Analysis o f solar

flare X-ray data have often employed, either explicity or implicitly, a two temperature model. With the implicit two temperature model as used by Horan (1971), one

temperature is considered to be constant but unknown and accounts for the preflare

emission. Values of the X-ray emission during the flare, greater than the preflare values,

are attributed to an isothermal solar flare plasma whose effective electron temperature

and emission measure are derived. Herr ing and Craig (1973) have used an explicit two

temperature model to fit X-ray flare emission in the 2.6 to 10 A range. With this mode,

values of the temperature and emission measure are obtained for both component re-

gions. Recently, Meekins (1973) has analyzed solar X-ray cont inuum emission detected

Solar Physics 36 (1974) 459472. All Rights Reserved Copyright �9 1974 by D. ReideI Publishing Company, Dordrecht-Holland

460 K E N N E T H P. DERE ET AL.

by a Bragg crystal spectrometer and found that plasma with temperatures extending up to about 50 • 106K was required to explain the emission from an active corona. For the work presented here, solar X-ray emission detected by ionization chambers in the 0.5 to 60 A range is analyzed in terms of a model that considers a continuous distribution of material at temperatures greater than 2 • 10 6 K.

2. The Multithermal Analysis of S O L R A D Data

The data used in this analysis was obtained by ionization chambers on the NRL SOLRAD 10 satellite. The relevant detectors were sensitive in the following wave- length bands: 0.5 to 3A, 1 to 5A, 1 to 8A, 8 to 16A, 1 to 2 0 A and 44 to 60A. These detectors have been discussed in detail by Horan and Kreplin (1972). Recently, the quantum efficiencies of these detectors have been measured experimentally and significant differences have been found between the measured and the theoretical efficiencies used in previous calculations (Meekins et aL, 1973). The SOLRAD 10 data are available in two different formats: (1) Satellite memory data which provide one minute time resolution for most of the above detectors (three minute time resolu- tion for the 44 to 60 A and 1 to 20 A_ detectors) at all times when the satellite is in sunlight and not in the Earth's particle belts; (2) Real time pulse code modulated (PCM) telemetry provides data for each experiment at 1.6 s time intervals (0.8 s time intervals for the 0.5 to 3/k and 1 to 8 A detectors) for periods of about ten minutes when the satellite is over the Blossom Point, Md. tracking station. Analog to digital conversion of the memory data uses 5 bits, and conversion of the PCM data uses 8 bits so that the PCM words are inherently 8 times more precise. Since the data for some of the experiments produced memory values of only 3 or 4 counts during quiet periods, the more accurate PCM data was used in this initial analysis in spite of the limited duration of transmission.

Solar X-ray emission is predominantly produced by bremsstrahlung, recombination, and line emission in the corona. With an accuracy sufficient for our purposes, this emission is completely determined by specifying the differential emission measure as a function of the electron temperature, i.e.

dB _ d f N ~ d V , (1) d in T d i n T

where dB/d l nT is the logarithmic differential emission measure we use in this study. This is related to the normal differential emission measure dB/d T by

dB _ T d B _ T d f N ~ d V , (2) d In T dT dT

where N e is the coronal electron density, dV is a differential volume element in the corona and T is the electron temperature. Henceforth, we will refer to both dB/dT and dB/d in T as the differential emission measure and mathematical expressions will indicate which formulation is employed.

A M U L T I T H E R M A L A N A L Y S I S O F S O L A R X - R A Y EMISSION 461

The current Is. induced by the coronal X-ray emission in ionization chamberj is then

f fdE( ,~ , T) dB d l n T d s (3) I s=ecos.A s a s(2) dB d l n ~ '

where e is the electron charge, co s is the number of ion pairs produced in the detector fill gas per unit absorbed photon energy, Aj is the detector window area, as. (2) is the detector quantum efficiency and E (2, T) is the energy emission spectrum (energy flux/wavelength interval) of a coronal plasma with electron temperature T. Both cos. and as (2) have been obtained from fits to experimental data (Meekins et al., 1973).

The procedure for finding dB (T)/d In T starts with the assumption that it can be adequately represented by a simple function of several free parameters. For this analysis, we have represented dB (T)/d In T by

dB(T) - C1 exp ( - C 2 xC3), (4)

d i n T where

X = l n ( T / 2 x 10 OK) (5)

and the Ci's are free parameters and T is the electron temperature in deg K. Previous analyses by Chambe (1971) and Walker (1972) have shown that dB/d Thas a maximum at about 2 x 10OK and then decreases with increasing temperature in the absence of flares. With C: constrained to be greater than zero, this function will have these characteristics while maintaining considerable freedom in the possible shape of the curve. The parameters Ci are then determined by finding those values which yield the minimum value for (#,

(p = Z (l i (measured) -- I s (expected))2/A 2 (6) j detectors

with A s. the estimated experimental error for detector j. This is accomplished by first searching a predetermined array of values for each Ci until those values giving a minimum value for ~o are found. The value of q~ is then further minimized by varying the parameters Ci in an iterative process until the minimum is reached.

Calculations of the coronal emission spectrum E (2, T) included continuum and line emission. The formulas for continuum emission by free-free and free-bound transitions were taken from the work of Culhane (1969) and were applied as in Horan (1970) with, however, the following differences: (1) the ionization equilibria for the higher stages of Fe as calculated by Jordan (1970) were included: (2) the value of the free-free Gaunt factor remained a function of both electron temperature and photon energy and was obtained from the calculation of Karzas and Latter (1961); (3) the free-bound Gaunt factor was equal to unity unless [ log(E/Z z Ry)[ ~> 1, (Eis the photon energy, Z, the nuclear charge and Ry the energy of one Rydberg) when the asymptotic values of Karzas and Latter (1961) were used: (4) important recombination edges were included. For the line emission spectrum, fits to the calculations of Tucker and Koren (1971) were used.

462 K E N N E T H P. D E RE ET AL.

There are several limitations inherent in the nature of the data and this analysis which must be recognized before interpreting the results. The most probable shape of the differential emission measure is determined by finding the values of the para- meters C, which yield the minimum value for q0. As a function of the parameters Ci, qo usually has several local minima and there is no guarantee that the minimum value for cp that is found is indeed the absolute minimum. In practice, it has been found that the problem of several minima can be avoided by choosing a sufficiently small initial grid size when first searching for the approximate solution. The function used to re- present the differential emission measure is not the most general but yields better fits than several others that have been tried. For example, use of two isothermal regions as in the analysis by Herring and Craig (1973) did not fit our data as well as the function finally used and required four free parameters in contrast to the three needed in our analysis. Another limitation of the chosen model is seen when the spectrum is hardest and minimizing ~o requires C 3 ~ 1. In this case, it is probable that the calculated values of the differential emission measure at 2 x 106 K are much higher than they actually are. These higher values result from constraints on the high temperature portions of the function. Part of this is due to the fact that the uncertainties in the detector efficiencies are lowest for the short wavelength, beryllium window detectors (0.5 to 3/k, 1 to 5/~, and 1 to 8 A). Although the calculations took into account regions where the electron temperature was above 2 x 106 K, the results are usually reliable only for temperatures above about 2.5 x 106 K. The choice of the model also precludes finding a maximum in the differential emission measure at any typical flare temperature or finding a sharp drop off at some maximum temperature.

In the rest of this paper, we present the results of this multithermal analysis as applied to SOLRAD 10 PCM data obtained during quiet periods and during portions of three X-ray flares.

3. Analysis of Nonflaring Periods

Out of the available PCM data, two passes were chosen to illustrate the typical results obtained for nonflaring periods during the months of July, August and September 1971. X-ray activity was low to moderate during that period. In all cases, the 0.5-3 ~ flux was below detector threshold and not included as a data point. In Figure 1, the analysis as performed for two extreme cases in this time period is shown. The differential emission measure is plotted only for those temperatures which con- tribute significantly to the current in any of the detectors. On 10 August 1971 (710810) at 1029 UT, the X-ray emission was quite low and the X-ray telescope maps published in Solar-Geophysical Data show that there are no active regions with any significant X-ray emission. The multithermal analysis indicates that the response of the SOLRAD detectors, in particular the 1-5 A detector, was due to radiation generated in regions with temperatures less than about 5 x 106K. Nonflare X-ray emission on 29 August 1971 (710829) at 1901 UT was significantly higher than average for this time period. X-ray telescope data in Solar-Geophysical Data indicate that McMath region 11482

f

LLI n"

U3

LLI

Z 0 CO U")

Ld

.._1

l-- Z hl

W U- LL

C~

r

x~

1051.

,o oL 1049

1048

10 47

10461 2

Fig. 1.

A M U L T I T H E R M A L ANALYSIS OF SOLAR X - R A Y EM[SS[ON

710829 1901UT

710810 1029 UT

I I 4 8

ELECTRON TEMPERATURE (I06K)

The differential emission measure for two nonflaring periods.

463

16

and perhaps region 11484 produced enhanced emission at this time. Significant radiation is produced at temperatures ~ 8 x 106 K. Data for several other periods were analyzeci and these differential emission measure curves fall between the two extreme cases presented in Figure 1. For temperatures >3 x 106 K, these results fall between the low activity ( f t ) and high activity (f2) models of Chambe (1971), but at lower temper- atures our results are generally higher than either of his models.

4. Analysis of Portions of Three X-Ray Flares in September 1971

Data on the X-ray, visible and radio observations of the three flares to be discussed are presented in Table I. The X-ray data are obtained from detectors on SOLRAD 9 and SOLRAD 10 and the radio and visible flare data are taken from Solar-Geophysical Data. 'Graybody flux' refers to the flux obtained by multiplying the value of the current in an ionization chamber by a conversion constant calculated with the assumption that the solar emission spectrum is a graybody spectrum. In Figures 2, 4,

464 KENNETH P. DERE ET AL.

T A B L E I

Flare da ta

Date 710912 710916 710918

S O L R A D 1-8 A graybody flux M a x i m u m flux (ergs cm 2 s-Z) 4.3 x 10 -e 1.3 • 10 -:~ 8.7 x 10 3 Begin t ime 1615 U T 1408 U T 1331 U T Peak t ime 1623 U T 141245 U T 133930 U T End t ime 1641 U T 1445 U T 1415 U T

H ~

Class -- F - - F -- B Begin t ime 1617 U T 1407 U T 1330 U T Peak t ime 1640 U T 1413 U T 1338 U T End t ime 1653 U T 1432 U T 1411 U T M c M a t h region 11516 11516 11515

2700 M H z Peak intensity (10 -22 W m 2 Hz- i ) 7.9 6.6 40.1 Begin t ime 1618.6 U T 1408.5 U T 1327,4 U T Peak t ime 1620.5 U T 1409.7 U T 1336.3 U T End time 1623.4 U T 1420.7 U T 1339.0 U T

Metric type IlI bursts Start t ime 1616 U T End t ime 1622 U T

SOLRAD I0

I0 o

7o iO-i

7

~ 10_2

x D i0 -3

g Io-%

10-5

10-6/ 1610

SEPTEMBER 12, 1971

/ 5-2

, I / ,c PCM DATA ~'1~"" / I A ~ ~ i 1620 1630 1640

UNIVERSAL TIME 1650

Fig. 2. S O L R A D 10 graybody flux values for a flare on 12 September 1971 obta ined f rom satellite m e m o r y and real t ime data.

A M U L T I T H E R M A L ANALYSIS OF SOLAR X-RAY EMISSION 465

1050!

4-" 'E r

w [ 0 4 9 8g

u3

W

Z 0 1 0 4 8 (.13 1.0

W

d

10 4 7 z w

uJ u_ u_

1 0 4 6 c

q m x3

IO 4 5

SEPTEMBER 12, 1971 I 162401 UT 2 162700 LIT 3 163000 UT 4 163555 UT

1 0 4 4 l I I I I I 2 4 8 16 3 2 64

ELECTRON TEMPERATURE (I06K)

Fig. 3. The differential emission measure at four instants during the decay of the flare on 12 September 1971.

and 6, the graybody fluxes in the 0.5 to 3A, 1 to 5 •, 1 to 8 A, 8 to 16 It and 44 to 60 A detectors are plotted for the three flares. The plotted X-ray data includes both SOLRAD 10 memory and realtime PCM data although only the PCM data was used in this analysis. The e-folding rise and fall times for the three flares are presented in Table II.

The X-ray emission time histories of all three flares have the usual profiles with total rise times on the order of minutes and a total decay time on the order of tens of minutes. Only the September 12 flare showed evidence of an early 'nonthermal' component like those studied by Kahler and Kreplin (1971). Between 1615 UT and 1619 UT, simultaneous with a type III radio burst, a short enhancement that is most evident at shorter wavelengths is seen superimposed over an increasing 'thermal' component.

The analysis of this flare uses data during the decay phase of the 'thermal' event, by

466 KENNETH P.DERE ET AL.

i00

T ~, IO -J ct~

cxl I E o i0_ 2

g

X 1 0 - 5 / i,

>- 0 0 m 1 0 - 4

rY {9

10-5

Fig. 4.

10-6 1405

SOLRAD I0 SEPTEMBER 16, 1971

PCM DATA I -~-

4 4 - 60 J

~8-16 A__

/

i i i I i i i

1415

\ \ l - B ~,

I - 5 ~ , \

\

0.5-:5

\ , I T i ~ , i i I ~ , , ,

1425 1455 1445

UNIVERSAL TIME

SOLRAD 10 graybody flux values for a flare on 16 September 1971 obtained from satellite memory and real time data.

TABLE II

e-folding times 1 to 8.~i

Date Rise time Decay time

710912 60 s 250 s 710916 100 s 1250 s 710918 120 s 670s

which t ime the ' non the rma l ' c o m p o n e n t should have comple te ly d isappeared . In

Figures 3, 5, and 7 the differential emission measure as ob ta ined by this mul t i the rmal

analysis is p lo t ted at representa t ive t imes in the progress o f the flares. The er ror bars

are ob ta ined by examining those solut ions, for which ~o is less than or equal to twice

the m i n i m u m value o f ~o.

General ly , the deviat ions o f the observed ion iza t ion chamber currents f rom the fitted

values were well within the es t imated exper imenta l errors. These exper imenta l errors

were de te rmined f rom the precis ion with which the efficiencies o f ioniza t ion chambers

were measured. The possible errors involved in calculat ing the expected solar emission

A MULTITHERMAL ANALYSIS O1: SOLAR X-RAY EMISSION 467

4" E o v w n~

w

Z o

w

-3

I-- Z W

w h

C

m "U

,o5, I SEPTEMBER 16, 1971

1 0 5 0 ~

1049 ' ~ L

1048 ~ ~ i ~ ' ~ ~

1046

1045

I 140830 UT 2 140950 UT :3 141315 UT

[0 4 4 2 4 8 16 32 6 4

ELECTRON T E M P E R A T U R E (106 K)

F i g . 5. The differential emission measure at three instants during the beginning o f a flare on 16

September 1971. A t 2 x 106 K the identification of the curves is as fol lows: top - 140950 U T ; m i d d l e -

1 4 0 8 3 0 U T : b o t t o m - 141315 U T .

spectrum were not included. The uncertainties here, which could be considerable, are due to the uncertainties of the element abundances and of the various atomic parameters needed in the numerous calculations of ionization, recombination, ex- citation rates, etc. For the flare on September 12, values of ~o obtained were much higher than those of the other two flares. On examination, it was found that the large values of ? were due to a poor fit to the 1 to 20 A detector. The observed 1 to 20 ,& flux values were consistently lower than the expected values and the harder the spectrum, the larger this discrepancy. The analysis of this flare was then repeated without the 1 to 20 A data and the values of ~o then obtained were lower than those found in the

other two flares - in line with the reduced number of degrees of freedom. Excluding the

468 K E N N E T H P . D E R E E T A L .

T

g x..

i0-~ v

x

10 -4 >-

o m 10-5

10-6: 1320'

toOL SOLRAD I0 SEPTEMBER 18, 1971

/ ~ j ~ t - - - - - 44 - 60 ,~

~ . .. 8 -16 \

1350 1540 1550 1400

UN;VERSAL TiME

Fig. 6.

i i p ,

1410 1420

SOLRAD 10 graybody flux values for a flare on 18 September 1971 obtained from satellite memory and real time data.

1 to 20 A data had only a small effect on the calculated shape of the differential emission measure curve. The differential emission measures for the September 12 event plotted in Figure 7 were determined by excluding the 1 to 20 A detector data. A possible explanation for the errors in the 1 to 20 A detector data is that the pressure of the detector fill gas is lower than assumed. This could account for the close to expect- ed long-wavelength response and the lower than expected short-wavelength response. During the calibration of the various ionization chambers, some of the greatest variations of the efficiency between supposedly identical detectors were found in the 1 to 20 A detectors.

Since a considerable amount of analysis of X-ray flare emission has been performed with an isothermal model, it would be useful to compare the results of the isothermal and multithermal approaches. This isothermal temperature can be calculated by taking the ratio of the graybody flux values of the 0.5 to 3 A and 1 to 8 A detectors, as in the previous study by Horan (1971). For example, this ratio technique yields a peak temperature of 8.5 x 106K for the September 16 flare at 140950 UT. The multi- thermal analysis shows that the maximum response of the 0.5 to 3/k detector at that time is to regions at about 16 x 106 K. Other flares that have been examined show the same result; there are significant amounts of material at temperatures considerably higher than those determined by the isothermal techniques.

5. Discussion

Based on the analyses performed on the September 16 and 18 flares and on several others, a general pattern for the behavior of the differential emission measure with time suggests itself. In Figure 8, typical differential emission measure curves are

i E u

klA [12

<~ q,i

Z O U3

W

d

I--

Z w n~ bJ b_ b_

c

co

Fig. 7.

A M U L T I T H E R M A L A N A L Y S I S O F S O L A R X - R A Y E M I S S I O N

1051 i SEPTEMBER 18, 1971 I 133812 UT

L 2 135100 UT

1050

\

1048 -

1047 --

[046 --

1045

1044 I I I I I 2 4 8 16 32 64

ELECTRON TEMPERATURE ( I06K)

The differential emission measure at two instants during the decay of the flare on 18 September 1971.

469

plotted for four periods during a typical X-ray flare. Curve number 1 is typical of the differential emission measure during the early rising portion of the X-ray flare but before the peak temperature, as obtained by detector response ratios, is reached. When this peak temperature is attained, the differential emission measure is similar to curve number 2, where the differential emission measure has increased at all tempera- tures from the previous values in curve number 1. Curve number 3 is representative of the differential emission measure near times of peak flux in the detectors covering the 0.5 to 8/~ band. Here it is seen that the differential emission measure is still increasing at lower and intermediate temperatures but at higher temperature the increase is very small and the highest temperature components are already decaying. Later, during the decay stage of the X-ray flux values, curve number 4 is typical and shows the differ-

ential emission measure decaying at all temperatures but fastest at the highest

470

~-" 1050[

i

E l.b

W I 0 4 9 cr

W

Z 0 1048 if) (,9 :s W d ~ 1 0 4 7 - Z W n-" W I.L Ix_

1_.2 1 0 4 6 - c

-(j

rrl

KENNETH P. DERE ET AL.

10451 I I I I I 2 4 8 16 3 2 6 4

ELECTRON T E M P E R A T U R E ( I 0 6 K )

Fig. 8. Typical differential emission curves at four instants during the evolution of the typical 'thermal' flare. The curves are normalized to give the same value of the differential emission

measure at 2 x 106K.

temperatures. The behavior outlined above is typical for the behavior of six flares studied with high resolution PCM data.

The only exception to this behavior was exhibited by the September 12 flare which was a limb flare and was the most impulsive of the events studied. This flare produced a considerable enhancement of the differential emission measure at temperatures between 2 x 106K and 4 x 106K. This low temperature enhancement then decayed along with the higher temperature components until it reached typical preflare values. In all other flares which we have analyzed it is only the higher temperature regions which grow and then decay. In this case, the low temperature values are probably real and not forced by the requirements made on the high temperature portions of the model differential emission measure.

A possible explanation for the different behavior of the September 12 flare is that it occurred lower in the corona than the others. This may imply that this X-ray flare plasma had a greater electron density. This would produce the faster decay rate of the September 12 flare, assuming that radiation losses, which vary directly with the elec- tron density, are responsible for a significant part of the energy loss. When the flare

A MULTITHERMAL ANALYSIS OF SOLAR X-RAY EMISSION 471

plasma extends low into the corona it will be in proximity to material at high electron densities and low electron temperatures (~< 2 x 106 K). It is possible that, when parts of the flare plasma are heated to temperatures of 20 x l 0 6 K and higher, some of this lower temperature material is simultaneously heated to temperatures above 2 x 106K, causing the enhancement of the differential emission measure at those temperatures. This is similar to the model Noyes (1973) has used to explain EUV flares. However, on the basis of the correlation of Ha flares with EUV flares (Wood et al., 1972), one might then expect the September 12 flare to produce a brighter H~ flare than the - F observed.

6. Conclusion

It has been demonstrated here that it is possible to account for solar X-ray emission from the nonflaring corona and for at least the 0.5 to 60 A radiation from X-ray flares by purely thermal processes. This is in contrast to the results of Kahler et al.

(1970) who have interpreted their 3 to 32 keV proportional counter data from the decay portion of an X-ray flare as due to simultaneous thermal and nonthermal processes. The temperatures they obtained are comparable to temperatures obtained from the ratio of the NRL 0.5 to 3 A and 1 to 8 A ionization chambers. We have found that a large part of the 0.5 to 3 A flux is actually produced in regions with temperatures considerably higher than the ratio determined temperatures and so it is possible that their data throughout the 3 to 32 keV range could have been accounted for by a suitable multithermal plasma. As shown by Kahler and Kreplin (1971), there are inherent difficulties in distinguishing thermal from nonthermal continuum radiation in the 2 to 8 keV range. However, Phillips et al. (1973) have found good agreement between a thermal interpretation of the continuum between 4 and 8 keV and iron line emission around 1.9 A. They fit their continuum data with a single time-varying temperature that at one time reached 30 x 106K and used the resulting temperature and emission measure to calculate the expected emission at --, 1.9 A from Fe xx~v and Fe xxv. Meekins (1973) also found a reasonable agreement between measured line radiation values and values predicted by a multithermal analysis of solar continuum radiation. Similar results at other wavelengths have been reported previously by Meekins et al. (1970) and Doschek et al. (1972).

Based on an analysis of solar X-ray line emission, Neupert et al. (1973) found that the differential emission measure during a flare had a local maximum at ,-~ 10 x 10 6 K. The two temperature analysis by Herring and Craig (1973) could also be interpreted to indicate the same situation. Although our model for the differential emission mea- sure does not allow any local maximum at temperatures greater than 2 x 106 K, it can be argued that, on the basis of the good fits obtained, this local maximum does not exist for most flares.

Acknowledgements

We wish to thank Drs J. F. Meekins and G. A. Doschek for reading and commenting on tiffs manuscript.

472 K E N N E T H P. DERE ET AL.

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