STATISTICAL STUDY OF THE CORRELATION OF 

HARD X-RAY AND TYPE I11 RADIO BURSTS 

IN SOLAR FLARES 

Russell J. Hamilton and VahC Petrosian 

Center for Space Science and Astrophysics 

Stanford University 

Stanford, California q430x 

CSSA-ASTRO-89-03 

/1/AZ& ywa May 1989 

Submitted to The Astrophysical Journal 

National Aeronautics and Space Administration Grant NSG-7092 

National Science Foundation Grant ATM 8705084 

Also Department of Physics, Stanford University 

Also Department of Applied Physics, Stanford University 
Abstract 

We have analyzed a large number of hard X-ray events which were recorded by the 

Hard X-Ray Burst Spectrometer (HXRBS) on the Solar Maximum Mission (SMM) during 

the maximum of the 2lSt solar cycle (circa 1980) in order to study their statistical correlation with type I11 bursts. We confirm qualitatively the earlier finding by Kane (1981) that 

flares with stronger hard X-ray emission, especially those with harder spectra, are more 

likely to produce a type I11 burst. The observed distribution of hard X-ray and type I11 

events and their correlations are shown to be satisfactorily described by a bivariate distribution consistent with the assumption of statistical linear dependence of X-ray and radio 

burst intensities. From this analysis we determine the distribution of the ratio of X-ray 

intensity (in counts/s) to type I11 intensity (in solar flux units) which has a wide range 

and a typical value for this ratio of about 10. The implications of our results for impulsive 

phase models are discussed. 

Subject headings: Sun: flares - Sun: radio radiation - Sun: x-rays 

2 
I. Iiitroduction 

It is generally accepted that the electrons which produce the impulsive hard X-ray 

bursts and those which cause the type I11 radio emissions during a solar flare have a 

common origin. The hard X-rays are a result of bremsstrahlung radiation of electrons 

which penetrate to the lower corona and chromosphere while type I11 radiation is produced 

by those electrons which escape on open field lines generating plasma turbulence as they 

travel away from the sun. The correlation of hard X-ray emission with type I11 emission has 

been studied both statistically (Kane 1981) and by detailed analysis of individual events 

(see e.g. Kane, Pick, and Raoult 1980, Icane, Benz, and Treumann 1982, Dennis et aZ.1984, 

or Raoult et aZ.1985). There is a wide variation in the relative strengths of the hard X-ray 

and type I11 bursts. In fact, the majority of hard X-ray events are not accompanied by an 

identifiable type I11 burst and vice versa. From this data one can estimate the ratio of the 

number of electrons involved in the production of hard X-rays to those producing type I11 

radiation. For example, Ftamaty et aL(1980) estimates an average value of lo2 to lo4 for 

this ratio. However, the range of this ratio is very large. In some hard X-ray events with 

no detectable type I11 burst this number must exceed lo7 (Simnett and Benz 1986). 

This kind of information is not very useful in the theoretical understanding of the 

phenomena. A knowledge of the mean value and dispersion (or in general the distribution) 

of such a ratio would be much more useful. To obtain such information from data with 

large dispersion a statistical approach to study the relationship between these two emission mechanisms is essential. A statistical study can establish the general trends in the 

data (such as mean values and dispersion) and therefore be helpful in the understanding 

of the general behavior of flares, such as the directionality of the accelerated electrons 

and the magnetic field configuration surrounding the acceleration region. Kane’s (1981) 

investigation dealt with the correlation of the X-ray and type I11 emissions observed during the solar maximum of the 20th solar cycle (circa 1969). He found that the correlation 

between X-ray and type I11 events increased with the intensity and starting frequency of 

3 
the type I11 burst, and with the peak energy flux and spectral hardness of the hard X-ray 

burst. He concluded that the observations suggest that the acceleration region is located 

in the corona and that the electrons responsible for both emissions are accelerated in a 

single process. The observed correlations were shown to be consistent with a bivariate 

distribution function of X-ray and type I11 burst intensities by Petrosian and Leach (1983, 

hereafter PL). Although the distribution could not be determined from Kane’s data, a 

procedure was outlined for the determination of the distribution given a larger data set 

with better resolution of radio burst intensities. Therefore, we decided to look at a large 

sample of flares which had either a hard X-ray event or a type I11 burst for the most recent 

solar maximum and to apply the method of PL to determine the bivariate distribution 

function and discuss the resulting implications. 

--* 

The paper is organized as follows. In the next section, we describe the data set that 

we analyzed. In $111, we use the data to find the resulting bivariate distribution function 

and discuss its properties. A discussion of our results is given in §IV and a brief summary 

in §V. 

11. Data 

The hard X-ray events were obtained from the tables in the Hard X-Ray Burst Spectrometer Event Listing 1980 to 1987 (Dennis et aZ.1988) excluding events which were 

interrupted or were nonsolar in origin. The radio data consisted of observations made 

at Bleien (BLEN) and at the Osservatorio Astronomico Di Trieste (Trieste) tabulated in 

their quarterly report Osservazioni Solari. These data were accumulated and kindly made 

available to us by Dr. A.O. Benz. The BLEN data does not give a precise quantitative 

measure of the strength of the type I11 bursts such as the flux density. Consequently, we 

use the Trieste data which gives the peak fluxes, the start time, the peak time, and the 

duration of the burst at 237, 327, and 408 MHz for our analysis. 

For our study, a hard X-ray event and a type I11 burst are considered to be associated 

4 
if the peak in the radio flux occurred during the duration of the hard X-ray burst or the 

- peak in the count rate of the HXRBS detector occurred during the duration of the radio 

burst. We studied bursts which occurred during the maximum of the 2lSt solar cycle 

from the beginning of the HXRBS data record 1980 February 19 until 1981 September 

30. During this period 3490 HXRBS events were recorded which are included in our study 

and we have X-ray spectral information for 241 of these bursts which was kindly made 

available to us by Dr. B.R. Dennis. There were 944 recorded bursts at 237 MHz during 

this same period of time with 134 of these bursts correlated with a hard X-ray burst. 

Table 1 summarizes the statistics of the hard X-ray and the 237 MHz radio data used in 

our analysis below. 

l 

I 111. Analysis of the Correlation 

a) Distribution of Burst Intensities 

I For the purpose of description of the distribution of bursts, we choose the readily 

available peak count rate of the hard X-ray bursts and peak flux density at 237 MHz (in 

solar flux units which is equal to 10-22Wm-2Hz-1) for the radio bursts which we denote 

by X and R, respectively. The choice of the 237 MHz channel is somewhat arbitrary, but 

similar results are found for all three radio channels. 

Figure 1 shows the distribution of the 134 correlated events (those with both X-ray 

and radio fluxes) in the R-X plane. This figure shows the wide range of the fluxes in both 

energy bands with no apparent clear correlation between X and R. Instrumental cut-off 

seems to be at 20 or 30 counts per second for X-rays and about 50 to 100 solar flux units 

at 237 MHz. We let the number of bursts with intensities in the range X to X+dX and 

R to R+dR be Af(X, R)dXdR, where f(X, R) is the distribution function of the bursts 

(normalized to unity) and A is the total rate of occurrence of all bursts. The moments 

of this function are useful for demonstrating the existence of a correlation between the 

occurrence of a hard X-ray burst of intensity X and the occurrence of a type I11 burst of 

5 
intensity R. 

-- From the bivariate distribution f(X, R), the number of hard X-ray bursts with intensity between X and X+dX with radio intensity greater than R, is 

Similarly, the number of type 111 bursts with intensity between R and R+dR with X-ray 

intensity greater than X, is 

For R, = 0, equation (1) gives the distribution N,(X) of peak count rates of all X-ray 

bursts. Similarly, N,(R) gives the distribution of radio fluxes of all type I11 bursts. If the 

radio and the hard X-ray intensities are stochastically independent of each other then the 

distribution is separable; f(X, R) = A$(X)$(R). For this situation equations (1) and (2) 

become 

and roo 

respectively. In this case the effect of changing the sensitivity R, (or X,) is only to 

change the normalization of the distributions, but the functional form is identical to the 

distributions of all bursts N,(X) (or N,(R) ). Therefore, we can immediately test for 

statistical independence by comparing the distribution NR, (X) or Nx,(R) for different 

sensitivities R, or X, with N,(X) or N,(R). Figures 2 and 3 show such a comparison 

for the data described in $11. The error bars given are proportional to the square root of 

the number of bursts in each bin (Note that the data is probably incomplete at the low 

end). It is evident that the curves are not parallel and that the magnitude of the slope of 

the distributions NR,(X) shown in Figure 2 decrease with increasing radio sensitivity R,. 

G 
This means that larger radio bursts are more likely to be accompanied by a hard X-ray 

burst. Alternatively, with the statistically independent assumption, one would expect the 

function G(X/R,) = NR,(X)/N,(X), which is the fraction of hard X-ray bursts which are 

associated with a type I11 burst of intensity greater than R,, to be constant for all values 

of R,. However, as shown in Figure 4, this fraction increases with X indicating that the 

larger hard X-ray bursts are more likely to be accompanied by a type I11 burst. 

As a qualitative measure of this behavior, we list the best fit power-law indices (excluding the incomplete bins) to the curves in Figures 2 and 3 in Table 2. The power-law 

index for N,(X) agrees with the well-known result for the HXRBS data (Dennis 1985 Fig. 

3) and the power-law indices systematically decrease with decreasing sensitivity (increasing R, or X,), both for the hard X-ray and type I11 distributions. We conclude, therefore, 

that the data is inconsistent with the stochastic independence assumed in equations (3) 

and (4). 

The question that arises is, what kind of correlation exists between these two types of 

emissions or parameters? We have yet to determine the functional form of the distribution 

f(X,R) which gives rise to the observed distributions NR,(X) and Nx,(R). In general, 

there are a multitude of functional forms which can describe a given set of observations. 

Without any loss of generality, we can write 

where Y is an arbitrary function of X and R. 

Y = X/R discussed by PL. This form for the distribution function gives 

Here we first analyze the simple form 

where 

G(Y) = g(Y’)dY’ with G(w) = 1 (7) .6’ 

is the fraction of bursts with hard X-ray intensity X which have radio bursts of strength 

R > X/Y. Note that N,(X) = A$(X). This is a useful and general way of describing 

7 
the correlation between X and R. If X and R are perfectly correlated (X/R = Y, for 

all events) then g(Y) oc 6(Y - Yo). For complete independence g(Y) is featureless (e.g. a 

simple power-law). In other words, the “width”of the distribution g(Y) provides a measure 

of the degree of stochastic dependence of the quantities X and R. 

This distribution assumes stochastic independence of the X and Y parameters and 

therefore can be subjected to a similar analysis described above for the X - R distribution. 

The distribution of the correlated events in the Y-X plane are shown in Figure 5. Also 

plotted is the effective instrumental cut-off at R, = 75. The absence of data points in the 

upper left region, Y 5 lo-’ , X 2 lo3 of this figure, as well as the absence of data below 

the cut-off line, make these quantities appear to be strongly correlated. However, based on 

the occurrence rates for all flares one only expects one or two points to be in the first of the 

above mentioned regions so that the absence of observed flares there is not unreasonable. 

Due to the presence of a cut-off, the construction of the distributions g(Y) or +(X) is 

not straightforward. If X and Y are stochastically independent then the non-parametric 

method described by Petrosian (1986) is one way to construct these distributions. This 

method gives the cumulative functions @(X) = JF $(X‘) dX’ and G(Y). In Figure 6 we 

show the cumulative function @(X) for the Y parameter in the ranges 0 < Y < 0.4 and 

0.4 < Y < 6.4. The fact that these distributions are fairly similar is an indication of the 

independence of X and Y. The distribution Ny,(X) = @(X)G(Yo) versus X is fitted to a 

power-law function from which we calculate the index dln+/dlnX for various values of 

Yo. The indices, given in Table 2, show no systematic variation with Yo and within the 

errors agree with the power-law “index”found for all bursts (Yo -+ 00). This again shows 

that the separation of the distribution given by equation (5) is consistent with the data. 

Using the same non-parametric method we can evaluate the cumulative function G(Y) 

[see Figure 7). The functions ?b[-Y) = N,(X)/A and g(Y) = dG(Y)/dY then give a 

complete description of the distribution. This procedure, however, requires an malytic 

form for G(Y). Except for the case of perfect correlation, there is no unique form for 

8 
g(Y) or G(Y). We chose a few simple functional forms with two free parameters and 

found the values of these parameters that best fit the data. The forms tried were gaussian, 

exponential, and simple power-law like forms. No satisfactory fit was obtained for the 

gaussian or exponential forms. The resulting best fit for the power-law like form is 

G(Y) = 1/(1+ (Yo/y)b) , g(Y) = bYL1(Y/Yo)b-l/(l + (Y/Yo)b)2 , (8) 

with b = 0.82f0.07 and Yo = 8 f2. This fit is shown in Figure 7. To check our method for 

internal consistency, we use the analytic functions g(Y) and $(X) 0: X-1.82 to evaluate 

the expected number of bursts NR,(X) and compare this distribution with the data. The 

circles in Figure 2 show the results of this comparison for two values of R,. Similarly, we 

can compute the expected distributions N-y,(R) of type 111 events, 

?) (X )g (X/ R)XdX/R2 . J,. Nx,(R) = A (9) 

We see in Figure 3 that the calculated points (circles) agree with the observed histograms 

here as well (The lowest point doesn't fit due to the incompleteness of the radio data at low 

flux levels.). Therefore, the form for the bivariate distribution function of hard X-ray and 

type 111 burst intensities given by equation (5) with g(Y) given by equation (8) describes 

the observed distribution of bursts adequately. 

In order to obtain a more quantitative measure of the degree of correlation between 

the X-ray and type 111 radiations, we have repeated the same procedure with two other 

simple functional forms for Y: namely, Y' = a/R, and Y" = X2/R. We find that the 

power-law index of Nyt(X) changes systematically from 1.40 (with 25 points) to 1.63 (with 

132 points) as Y' is increased from 0.01 to 1.0. Furthermore, the index of 1.63 does not 

agree with the expected result of 1.82 for all X-ray bursts (cf. Table 2). For the choice 

Y', the power-law index of N~J~(X) is too large with values of 2.44 (28 points), 2.74 (70 

points), and 2.55 (116 points), for Y" = 5, 50, and lo4, respectively. Therefore, neither 

Y' nor Yrr appear to be stochastically independent of X and our choice of Y = X/R 
provides a better description of the data. More elaborate functions might be chosen, such 

as Y = (X - Xth)/R, where Xth may be due to another process contributing to the Xray flux (e.g. thermal bremsstrahlung) but not to the type I11 emission. In this case, the 

distribution in Xth would also have to be unfolded. We feel that the data does not warrant 

such detailed analysis and that the thermal contribution only adds more dispersion to the 

data (see §IV below). 

c 

We note that our choice for the independent parameters is slightly arbitrary. Instead 

of X and Y we could have chosen R and Y. In fact, if X (or R) and Y were strictly statistically independent, then one could choose any function of X (or R) and Y as independent 

parameters. This, however, would not alter observable quantities such as the distributions 

NR,(X), Nx,(R) or the values of the moments of these functions. We have repeated the 

calculations leading to Figures 5 through 7 using R instead of X, and Y as our parameters 

and have fitted the data represented in this way to the function described by equation 

(8) times $(X) oc X-1.82 = (RY)-1.82 and find b = 0.83 and Yo = 7. These values are 

in agreement with the values b = 0.82 f 0.07 and Y, = 8 f 2 found above. It is clear 

from this analysis that the distributions No(X) and N,(R) should be identical. As shown 

in Table 2, the power-law indices (1.82 and 1.75) only differ by slightly more than one 

standard deviation. Therefore, the observed No(R) distribution could be a random sample 

of the N,(X) distribution. Finally, we note that the ratio of X-ray to radio limiting flux 

fiim = Xlim/Rlim N 25/75 = 1/3 is less than the representative value Yo of the Y parameter determined by either method. Ideal samples should have xim N Yo which means that 

we need an improvement of a factor of 24 (Rljm N 3 solar flux units) in the radio data if 

Y, - 8. Further lowering of Rlim will not increase the size of the data set significantly, or 

improve it, unless accompanied by proportional decreases in Xlim. 

b) Dependence on Spectral Index 

As summarized in Table 1, we have the values of the X-ray spectral index, y, from 

power-law fits to the X-ray spectra for 241 of the bursts, 34 of which have been observed to 

10 
be associated with a type I11 burst. In this subsection, we see to what extent the spectral 

index effects the correlation between the hard X-ray and type I11 emissions. *- 

The histogram in Figure 8 shows the fraction of hard X-ray bursts which have detected 

type I11 fluxes versus the spectral index (solid line) for R 2 50. For comparison we also 

show (dashed line) the data from Figure 10 of Kane (1981). Both histograms show that 

a larger fraction of harder (smaller y) X-ray bursts produce type I11 bursts. This effect is 

much less pronounced in the new data. In fact, a X2-test gives x2 = 4.6 (with 7 degrees of 

freedom) and therefore a probability of 0.70 that the x2 found from a parent distribution 

which is uniform would be this large or larger. Hence, we cannot exclude the possibility 

that there is not a correlation. This presumably is not true for Kane’s (1981) results and 

we do not know the reason for this difference. To determine whether or not the spectral 

index dependence that we do see is due to selection effects, such as correlation of X-ray 

intensity with spectral index, we have preformed the following test. We binned all X-ray 

bursts with known spectral index (241 of them) by spectral index. Since the probability 

that an X-ray event of strength X is associated with a type I11 burst is known (cf. Figure 

2), we can compute the expected number of type I11 associated events for each bin. This 

is found by summing the probability of type I11 association for each individual event over 

all X-ray events in the bin. The result of this calculation is shown in Figure 9 along 

with the observed number of hard X-ray bursts (with known spectral indices) associated 

with a type I11 burst. We see that for y > 4.5 the number of associated bursts is equal 

to that expected from the total distribution while for y < 4.5 the number of associated 

bursts is twice that expected. In this case, we find x2 = 3.0 (8 degrees of freedom) which 

gives a higher probability (0.93) that this x2 could have come from a distribution which 

is uniform in spectral index. This indicates that some sampling bias exists. In order to 

conclude that the increased correlation between hard X-rays and type I11 emission for 

harder X-ray spectra is a real effect, a larger sample is needed. 

11 
IV. Discussion 

We have shown that the observed distribution of the type I11 and hard X-ray emission 

from solar flares can be described by a bivariate distribution with independent parameters 

the X-ray intensity and the ratio of the X-ray to type I11 intensities. This implies a 

correlation between these two types of emission. However, this correlation is stochastic 

as demonstrated by the large range (or “width”) of the distribution g(Y) in equation (8). 

Strictly speaking, both the average value and mean dispersion of Y for this distribution 

diverges. This, however, is an artifact of the simple distribution chosen here and can be 

remedied by the introduction of additional parameters. The important conclusion we draw 

from this is that a typical value for the ratio Y is the quantity Yo - 8 and the range of Y 

is even greater than Yo and can be as large as 102Yo N lo3. We now discuss the meaning 

of these results and the results from consideration of the spectral index dependence in the 

frame work of models for the impulsive phase of the flare and for the two types of radiation 

under consideration here. 

a) Nature of the Correlation 

We first consider the correlation we have found between the X-ray and type I11 emissions and in particular the appropriateness of the choice Y = X/R as a stochastic parameter. 

The X-rays of energies 20 - 100 keV are produced mainly by bremsstrahlung of 

electrons with comparable energies. Three distinct models have been advanced to explain 

the solar hard X-ray emission by electrons, the thick-target, thin-target, and thermal 

models (trap plus precipitation models are classified as thick-target here because the Xray production is via thick-target bremsstrahlung of the precipitating electrons). In both 

the thick and thin target models, nonthermal electrons produce the impulsive X-ray bursts 

and the strength of the burst is proportional to, n,, the number of nonthermal electrons. 

In thermal models, the level of X-rays produced is proportional to n:. Since the corona is 

12 
optically thin to photons in this energy range, the observed radiation is also proportional to 

n, (nonthermal) or n: (thermal). Although there is often some contribution from thermal 

emission, the impulsive X-ray bursts are most likely due to nonthermal electrons. 

The relationship between the number of type I11 producing electrons, 12111, and the 

intensity of type I11 bursts is more complicated. Type 111 bursts are produced by nonthermal electrons with energies E - 30 keV (/? - 1/3) which develop a positive slope in their 

velocity distribution as they propagate. This situation is unstable to the generation of longitudinal plasma waves (Langmuir waves) with frequency near the local plasma frequency. 

These waves don’t propagate or escape the plasma, so that a mechanism for conversion of 

these waves to escaping radiation is needed. There are two processes which are believed to 

occur. The coalescence of two longitudinal waves to form a transverse wave at twice the 

plasma frequency (second harmonic emission) and the scattering of a longitudinal wave 

off ion-acoustic turbulence into a transverse wave with frequency near the local plasma 

frequency (fundamental emission). To find the relation between nIII and the intensity of 

type I11 radiation we must first relate nzzz to the level of plasma turbulence generated, 

Wl, and then relate this to the intensity, R, of type I11 radiation. 

The first of these relations is simple Wr oc n111. The relationship between the turbulence level and the intensity of type I11 radiation was discussed by Smith and Spicer 

(1979). For second harmonic emission they showed that R oc Wl when optically thick and 

R 0: W: for optically thin sources. Therefore, R oc nzzz or R oc n:Iz for second harmonic 

emission from an optically thick or optically thin source, respectively. They also showed 

that R 0: Wl for optically thin fundamental emission. For Optically thick sources the 

fundamental emission depends on the ratio of Langmuir turbulence to ion-acoustic turbulence. However, the second harmonic emission is usually dominant for type I11 bursts and 

for solar type 111 bursts the source is generally optically thick. So that for most type 111 

bursts R 0: MJl 0: nIII. 

Combining the results for X-ray and type I11 emission, we see that for the most 

13 
likely situation where impulsive hard X-rays are produced by nonthermal electrons and 

the type I11 burst is second harmonic emission from an optically thick source the quantity 

Y = X/R oc n,/nIII. Therefore, if this ratio is independent of X or R then Y is a good 

variable with which to characterize the correlation of X-ray and type I11 bursts. Other 

situations are of course possible and would lead to other parameterizations. For example, 

for nonthermal models (X oc n,) with optically thin second harmonic type I11 emission 

(R 0: n2rII) parameterization in terms of Y” = X2/R 0: (n,/nIII)2 would be more natural. 

Or, if X-rays are produced by thermal bremsstrahlung (X 0; nz) and type I11 radiation 

by optically thick second harmonic emission, then the parameter Y’ = O/R oc nX/n11I 

would be the natural parameter if nIII oc n,. For these cases, however, there may not be 

any correlation between n, and 72111 so that one would expect no correlation between X 

and R and a distribution function as described in connection with equations (3) and (4). 

As shown in the previous section, all of these other possibilities are less likely than the 

parameterization in terms of X and Y = X/R, corresponding to nonthermal X-rays and 

optically thick second harmonic type I11 emission. 

cThus, we can conclude that the ratio Y = X/R is proportional to the ratio Ye = 

n,/nIII of the number of X-ray to type I11 producing electrons; Y = qYe. Part of the 

large (three or four orders of magnitude) observed dispersion in the value of Y is due to 

the dispersion of the proportionality “constant”q which depends on the properties of the 

flare plasma, especially those determining the level of plasma turbulence Wl. We estimate 

q to be 10-1 to based on the relation between n, and X which is straightforward 

(cf. Hoyng, Brown, and Van Beek 1976) and on the relation between nIII and R which 

is very model dependent. We have used the relations given by Smith and Spicer (1979) 

and assumed a velocity of c/3 for the electrons, a range for plasma temperature of lo6 

to IO7, and efficiency of plasma wave generation of 0.1 to 1. Equally important in the 

dispersion of Y is the variation of the ratio Ye from flare to flare. The ratio 1; depends 

on a number of factors related to the acceleration mechanism and the transport of the 

14 
accelerated electrons in the solar atmosphere. The most important among these factors 

--- is the ratio of the number of accelerated electrons which end up on open field lines (lines 

extending to the upper corona and/or the interplanetary medium) to those on closed field 

lines, which we denote by E. 

All electrons on the closed field lines and a fraction fz of those on the open field lines 

contribute to the X-rays while the rest contribute to the type I11 emission. (We assume 

that X-ray emitting electrons have a negligible contribution to the type I11 emission and 

vice versa). The ratio fz in turn depends 

on two factors. The first of which is the fraction of the electrons on the open field Iines 

directed outward from the sun, fup, which is determined by the pitch angle distribution 

of the accelerated electrons (e.g. fup = 1/2 for isotropic pitch angle distribution). Due to 

pitch angle diffusion and field convergence a fraction jref of the down going electrons are 

reflected back so that fz = (1 - frej)(l - fup) and 

Thus, the ratio Ye = (1 + efZ)/e(l - fz). 

which is the same as equation (17) of PL with different notation. 

First we note that based on our estimated range of values for 7, a typical observed 

ratio Y = Yo requires the value Ye0 - Yo/7 - lo3*’. The important point here is that 

despite the large uncertainty we require Ye >> 1 even for those flares with minimal X-ray 

emission having Y - 1. According to equation (lo), this can be the case if either E << 1 or 

(fup + fief) << 1 or both. It can be shown that within a factor of two this expression can 

be simplified as 

which should be sufficient for our discussion considering the large uncertainties in the other 

parameters. The large dispersion in the value of I7 can then be easily accommodated by 

the dispersion in the three parameters in equation (11). Of these, E and fup depend on 

15 
the acceleration process and the geometry of the magnetic field (0 < E < m, 0 < fup < 1) 

.- while fief depends on the transport effects and is expected to be small (fief << 1). 

Another contribution to outgoing type I11 producing electrons may come from possible 

diffusion across the magnetic field lines (from closed to open field lines) as in the model 

proposed by Emslie and Vlahos (19S0, hereafter EV). In the general version of the model, 

Ye is given by equation (10) or (11) with the addition in the denominator of another term 

fdiff which is the fraction of electrons in the closed loop that diffuse to the open field lines 

with pitch angles so that they stream outward from the sun. In the version of this model 

proposed by EV, all particles are on closed field lines so that E = 0 and 

In general, fdjff is expected to be small so that Ye >> 1 as required, but the distribution 

of fdiff (the values of the expected average and rms values) is not explicitly specified by 

the model and difficult to predict. 

Finally, let us consider a third model for the production of type I11 radiation proposed by Sprangle and Vlahos (1983, hereafter SV). In this model, the primary electrons 

which are accelerated in a closed loop with a converging magnetic field reflect to acquire 

a loss cone distribution. This electron distribution is unstable and results in cyclotron 

maser emission. The emitted radiation then escapes the closed field lines of the loop and 

accelerates secondary electrons on open field lines. This model is able to account for short 

time delays between peaks in the type I11 emission relative to peaks in the X-ray emission 

(Dennis et aZ.1984). SV show that the number of accelerated electrons is proportional to 

the magnitude of the loss cone, and therefore to the number of reflected electrons, times 

factors related to the efficiency of the secondary acceleration process. Since the number of 

reflected electrons is proportional to nz, this model also agrees with the Y parameterization that we used and predicts a value of Ye given by equation (10) or (11) with addition 

of another term 0: fief in the denominator for the contribution to nllI by the secondary 

16 
process. Of course, as in the EV model, this model is of interest when E = 0. So that in 

the simplest form of the model Ye oc frl'f. As we shall see below fTef << 1 and Ye >> 1 as 

expected. However, again determination of the average or rms value of Ye is difficult in 

this model. 

b) Dependence on Spectral Index 

The discussion above shows that most model predictions agree with the observed 

I 

distribution and its parameterization in terms of X and Y = X/R. We now consider the 

dependence of the X-ray and type I11 correlation on the X-ray spectral index. In general, 

for nonthermal models there is a simple relation between the X-ray spectral index y and 

the electron spectral index S. In particular, for the thick target models considered here in 

their simplest form, y = S - 1. This relation is not exact and depends on the parameters 

of the model (see Leach and Petrosian 1983, or Leach 1984), but for our purposes here it is 

sufficient to note that as 6 increases y increases so that the observed correlation between 

the distribution and y (if any) reflects correlation between S and Ye. 

As shown in the previous section, the new data does not show the strong dependence 

on spectral index found by Kane (1981), so that the degree of the correlation is not well 

determined. However, we believe that bursts with harder X-ray spectra are more likely to 

also have type I11 emission. This can be described mathematically by making Yo (or b) to 

be a function of spectral index. If we take b to be independent of spectral index then for 

each spectral index bin the fraction of bursts with type I11 emission is 

For $(X) 0: X-" and G(Y) given by equation (S) this reduces to 

00 

f(y) = Z;-l/(a - 1) s,. dZ 2*-y1+ 2*) , (14) 

ivhere 2, = Y,;,/Yo(y). As noted above I?i, - 1/3 << Yo so that 2, << 1, in which 

case, to within some slowly varying logarithmic factor, the variation with spectral index 

17 
of this fraction can be approximated as f(y) x [l'jim/Yo(y)]b. Our results and those by 

Kane (1981) shown in Figure 8 indicate f(y) x y-l and respectively. If we assume 

f(y) M y-2 and neglect the slight difference between the exponent b and unity the data 

would require Yo(y) oc y2. This is a very approximate relation that implies Ye oc S2, a 

requirement that the models must satisfy. If instead of Yo we assume b to be a function 

of y then the above approximation is still valid and lnf(y) = b(y)ln(Kim/Yo). Thus, an 

observed f(y) c( y-a can be described by b = bo - a ln(y/yo), where bo is the value of b at 

yo. In what follows we consider the case that b is constant and Yo = Yo(y). 

-- 

In the simple nonthermal model it is evident from equation (11) that we require e # 0 

otherwise there will not be any type I11 emission. Furthermore, in its simplest form this 

model does not require or predict dependence on the spectral index 6 for this parameter 

or for fup which is a moment of the pitch angle distribution of the accelerated electrons. 

Then, as proposed by PL, the only dependence on 6 will arise from the variation of fief 

with 6. As also shown by PL, there is considerable variation of frep with S especially 

for models with electrons beamed primarily downward fup + 0 so that Ye 0: fr:'f. From 

Table I of PL we see that for this case (their parameter a: < 1) frep increases more rapidly 

than linearly, so that the trend agrees roughly with the above requirement. 

On the other hand, if the pitch angle distribution of the accelerated electrons is 

isotropic fup = 1/2, then we require E << 1 so that Ye oc (1 + 2fr,f)-l. Since, in general frep << 1, this predicts a slower variation. In addition, for an isotropic pitch angle 

distribution the above mentioned Table of PL shows a slower variation of frer with 6. 

Consequently, we expect very little variation of Ye with S and Yo (or f(y) ) with y. Note 

that for both of these cases the dependence on 6 is weakened with increasing magnetic 

field convergence. 

For a quantitative comparison of these models with the data we assume that the 

spectral indices are related as in the simple thick-target model, y = 6 - 1, and choose E/V 

so that Yo = 8 for the median spectral index y = 4. This means that qYe(6 = 5) = 8. 

18 
The fraction f(7) is computed using equation (13) with Y,(y) given by equation (IO). The 

-- model parameters are listed in Table 3 and the corresponding f(7)’s are shown in Figure 

10 for comparison with the data. We find that the calculated fractions agree with the 

above qualitative discussion. The isotropic models show very little variation with spectral 

index while the beam models fit the data well, suggesting that the pitch angle distribution 

of accelerated electrons is beamed toward the sun. 

In the EV model (with E = 0) Ye c( &if. It is not clear why fdiff would be dependent 

on spectral index 6. However, since higher energy particles penetrate to higher depths 

below the transition region and if the diffusion rate across the field lines also increases 

with depth then bursts with harder spectra (lower 6) will have larger fdiff, a lower Ye or 

Yo, and therefore a larger fraction f(7) associated with type I11 emission. 

In the SV model, the strength of the loss cone determines the strength of the type 111 

emission. Since the bulk of the strength of the loss cone comes from lower energy electrons 

and since the loss cone is produced by mirroring before much collisional energy loss occurs, 

we expect little dependence on the spectral index. For this model the field convergence is 

strong so that fief approaches unity with very weak dependence on 6. 

Clearly we require more spectral index information before such observations can be 

used to distinguish between the various possibilities described above. 

V. Summary 

We have analyzed a large number of hard X-ray events which occurred during the 

maximum of the 21’‘ solar cycle in order to study their correlation with type I11 bursts. 

From our analysis, we found that the distribution of occurrences of hard X-ray bursts which 

are correlated with a type 111 radio burst is significantly different than the distribution of 

all hard X-ray bursts, which is consistent with results found by previous authors. We 

were able to show that this result is consistent with the assumption that the hard X-ray 

and type I11 intensities are somewhat correlated. A bivariate distribution function of the 

19 
burst intensities was fit to the data. From this distribution we determined that the typical 

ratio of X-ray intensity to type I11 intensity is Y - 8 and that the ratio of the number 

of X-ray producing electrons to type I11 producing electrons is in the range Y, N lo2 to 

lo4. However, there is a large dispersion in the ratio of peak X-ray intensity to type I11 

intensity. 

.- 

Three models that have been proposed to explain the relation between hard X-ray 

and type I11 producing electrons were examined in the context of these observations. All 

models predict that Y c( Ye and that it is independent of both X-ray and type I11 intensity 

which agrees with the data. The important differences in the models are whether or 

not both electron populations are accelerated simultaneously and whether the correlation 

between hard X-rays and type I11 radiation increases for harder X-ray bursts. The data 

indicate that this correlation increases slightly with decreasing X-ray spectral index. The 

increased correlation of X-ray and type I11 events with decreasing photon spectral index 

is explained in the model of PL by the increase in the fraction of reflected electrons for 

harder electron spectra suggesting that reflected electrons play a role in type I11 bursts. 

The observed variation with spectral index is best fit by models in which the accelerated 

electrons, especially those on open field lines, are beamed toward the sun. More data is 

needed to further quantify this correlation so that the predictions of the impulsive phase 

models can be further tested. However, this also requires more detailed acceleration models 

so that the spectral index dependence of the ratio Ye can be determined. 
Acknowledgments 

-- 

We would like to thank Dr. B.R. Dennis for providing the HXRBS data and Dr. A. 0. 

Benz for providing the radio data. We are grateful to Dave Caditz for making some 

statistical analysis subroutines available, to Jim McTiernan for useful discussions, and 

to Katrina Lane who aided us in the initial gathering and analysis of the data. We 

acknowledge support of this work by National Aeronautics and Space Administration grant 

NSG-7092 and National Science Foundation grant ATM 8705084. 

21 
Table 1 

Summary of hard X-ray, type 111 data, and the correlated events. 

Hard X-Ray Bursts 

All events: 

Total number of bursts 3490 

Number of bursts with known spectral index 241 

Correlated events: 

Total number of bursts 134 

Number of bursts with known spectral index 34 

Total number of bursts 944 

Type I11 Bursts 

Number of correlated bursts 134 

22 
I Table 2 

Yd Power-law fit to the functions N R , ( X ) , N x , ( R ) , and Ny,(X). 

Ro n CY Acy Xo n cy Acy YO n cy Acy 

0 3490 1.82 0.02 0 944 1.75 0.04 00 3490 1.82 0.02 

I 75 126 1.39 0.06 20 132 1.54 0.08 6.4 113 1.82 0.07 

150 93 1.33 0.06 40 101 1.48 0.09 1.6 100 1.78 0.08 

300 63 1.30 0.07 SO 76 1.44 0.10 0.4 59 1.98 0.17 

I 600 38 1.27 0.09 320 36 1.37 0.13 0.1 24 1.95 0.25 

n is the number of points in the distribution. 

I 

CY is the power-law index such that f(z) cx z-~. 

23 
Table 3 

PL Model Parameters 

B5 0 5 0.37 

I1 

I5 

1 

2 

- 

1 

2 

- 

1 

5 

-0.62 

-0.67 

T, is the factor by which the magnetic field increases from the injection point to the 

transition region. 

24 
References 

I- Dennis, B.R. 1985, Solar Phys., 100, 465. 

Dennis, B.R., Benz, A.O., Ranieri, M., and Simnett, G.M. 1984, Solar Phys., 90, 383. 

Dennis, B.R. et a1.1988, The Hard X-Ray Burst Spectrometer Event Listing 1980-1987, 

NASA Technical Memorandum 4036. 

Emslie, A.G. and Vlahos, L. 1980, Ap. J., 242, 359. 

Hoyng, P., Brown, J.C., and Van Beek, H.F. 1976, Solar Phys., 48, 197. 

Icane, S.R. 1981, Ap. J., 247, 1113. 

Kane, S.R., Pick, M., and Raoult, A. 1980, Ap. J. (Letters), 241, L113. 

Kane, S.R., Benz, A.O., and Treumann, R.A. 1982, Ap. J., 263, 423. 

Iiane, S.R. et al.1980, in P.-4. Sturrock (ed.), Solar Flares, (Boulder: Colorado Assoc. 

Univ. Press), p.187. 

Leach, J. 1984, “The Impulsive Hard X-rays from Solar Flares ”, Stanford University (Ph.D. 

Thesis). 

I Leach, J. and Petrosian, V. 1983, Ap. J., 269, 715. 

Petrosian, V. and Leach, J. 1983, Solar Phys., 87, 165. 

Petrosian, V. 1986, in G. Giuricin (ed.), Structure and Evolution of Active Galactic Nuclei, 

I (Dordrecht [Netherlands]: Reidel), p. 353. 

Ramaty, R. et al.1980, in P.A. Sturrock (ed.), Solar Flares, (Boulder: Colorado Assoc. 

Univ. Press), p.117. 

Raoult, A., Pick, M., Dennis, B.R., and Icane, S.R. 1985, Ap. J., 299, 1027. 

Simnett, G.M. and Benz, A.O. 1986, Astr. Ap., 165, 227. 

Smith, D.F. and Spicer, D.S. 1979, Solar Phys., 62, 359. 

Sprangle, P., and Vlahos, L. 1983, Ap. J. (Letters), 273, L95. 
Figure Captions 

1- 

Fig. 1. Distribution of the correlated events in the R-X plane. 

Fig. 2. Distribution of all HXRBS events N,(X) and NR,(X) for R, = 75 and 300 with the 

R, = 300 histogram reduced by a factor of ten for clarity. The solid lines show the 

best fit power-law for each of the histograms. The circles are the values calculated 

using the best fit to the function G(Y). 

Fig. 3. Same as Figure 2 except for type I11 distributions N,(R) and Nx,(R) for X, = 40 

and 320 with the X, = 320 histogram reduced by a factor of ten for clarity. 

Fig. 4. Fraction of X-ray events correlated with a type I11 event for minimum radio 

intensities of R, = 75 and 300 solar flux units. The R, = 300 histogram is reduced 

by a factor of ten for clarity. 

Fig. 5. Distribution of the correlated events in the Y-X plane. Also shown is the instrumental cut-off at R=75. 

Fig. 6. Cumulative distribution @(X) for 0 < Y < 0.4 and 0.4 < Y < 6.4 obtained using 

the non-parametric method with the 0 < Y < 0.4 curve reduced by a factor of two 

for clarity. Also shown is the best fit power-law for each of the distributions, 

Fig. 7. The cumulative distribution, G(Y), obtained using the non-parametric method 

along with the best fit curve equation (8). 

Fig. 8. Ratio of correlated events to total HXRBS events as a function of photon spectral 

index (solid). Also depicted are the results from Figure 10 of Kane (1981) (dashed). 

The power-law functions f 0: y-l and yF3 are also shown. 

Fig. 9. The number of observed correlated bursts (solid) and the expected number of 

correlated bursts (dashed) as a function of X-ray spectral index. 

Fig. 10. The observed fraction of correlated events as a function of spectral index f(y) is 

shown (histogram) along with the predictions of the PL models of Table 3. Models 

B1 and I1 are shown as solid lines while B5 and I5 are dashed. 

26 
Addresses 

.- 

Russell J. Hamilton 

323 ERL 

Center for Space Science and Astrophysics 

Stanford University 

Stanford, CA 94305 

Vahk Petrosian 

304 ERL 

Center for Space Science and Astrophysics 

Stanford University 

Stanford, CA 94305 

27 
h 

I 05 

cn 

C 

-e, 

.- 

C 

103 0 

>. 

a 

U 

I 

X 

1 0’ 

1 

++ 

Peak Radio Flux 

Fig. 1. Distribution of the correlated events in the R-X plane. 
Peak X-Ray Counting Rate 

Fig. 2. Distribution of all HXRBS events N,(X) and Nn,(X) for R, = 75 and 300 with the 

R, = 300 histogram reduced by a factor of ten for clarity. The solid lines show the 

best fit power-law for each of the histograms. The circles are the values calculated 

using the best fit to the function G(Y). 
Icn C 

aJ 

L 

a 

LL 

4 

rc 

0 

1 IZI0 

10-’ 

I 0-5 

10’ 

t 

I 05 

Peak Radio Flux 

Fig. 3. Same as Figure 2 except for type I11 distributions N,(R) and N,yo(R) for X, = 40 

and 320 with the X, = 320 histogram reduced by a factor of ten for clarity. 
Peak X-Ray Counting Rate 

Fig. 4. Fraction of X-ray events correlated with a type I11 event for minimum radio 

intensities of R, = 75 and 300 solar flux units. The R, = 300 histogram is reduced 

by a factor of ten for clarity. 
+ 

Y 

Fig. 5. Distribution of the correlated events in the Y-X plane. Also shown is the instrurnerital cut-off at R=75. 
I IZ1° 

m 

a, 

L 

a 

C 

0 

-e, 

3 

.- 

a 

.- 10-’ 

L 

m 

0 

i-, 

.- 

a, 

> 

i-, 

a 

1 0-2 

1 

Peak X-ray Counting Rate 

Fig. 6. Cumulative distribution 9(X) for 0 < E’ < 0.4 and 0.4 < Y < 6.4 obtained using 

the non-parametric method with the 0 < Y < 0.4 curve reduced by a factor of two 

for clarity. Also shown is the best fit power-law for each of the distributions. 
Y 

Fig. 7. The cumulative distribution, G(Y), obtained using the non-parametric method 

along with the best fit curve equation (8). 
-U 

a, 

a 

Q) 

L 

L 

0 

0 

C 

0 

-e, 

.- 

4 

W 

a 

L 

LL 

Photon Spectral Index 

Fig. 8. Ratio of correlated events to total HXRBS events as a function of photon spectral 

index (solid). Also depicted are the rcsults from Figure 10 of Kane (1981) (dashed). 

The power-law functions f oc y-' and y-3 are also shown. 

8 

5 
0 

m 

L 

3 

-c, 

m 

4- 

0 

L 

a, 

a 

E 

3 

7 

la, 

I- 

- -. I 

I 

Ill 

I 

L---I I 

I 

rI 'I L 

Photon Spectral Index 

Fig. 9. The number of observed correlated bursts (solid) and the expected nunibcr of 

correlated bursts (dashed) as a function of X-ray spectral index. 
C 

0 

.- 

-P 

u 

a 

L 

L.lPhoton Spectral Index 

Fig. 10. Tlie observed fraction of correlated events as a function of spectral iridcx f(7) is 

shown (histogram) along with the predictions of the PL models of Table 3. Moclcls 

131 and I1 are shown as solid lines while 135 and I5 are dashed.