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A statistical study of high coronal densities from X-ray line-ratios of Mg XI
Adv. Space Res. Vol. 11,No. l,pp. (l)147—(l)150, 1991 Printed in GreatBritain. All rights reserved.
0273—1177/91 $0.00 + .50 Copyright © 1991 COSPAR
A STATISTICAL STUDY OF HIGH CORONAL DENSITIES FROM X-RAY LINE-RATIOS OF Mg XI G. A. Linford, J. R. Lemen and K. T. Strong Lockheed Palo Alto Research Laboratory, Department 91-30, Building 256, 3251 Hanover Street, Palo Alto, CA 94304, U.S.A.
ABSTRACT We applied an X-ray line-ratio density diagnostic to 50 Mg XI spectra of flaring active regions on the Sun recorded by the Flat Crystal Spectrometer on the Solar Maximum Mission. The plasma density is derived from R, the flux ratio of the forbidden to intercombination lines of the He-like ion, Mg XI. The R ratio for Mg XI is only density sensitive when the electron density exceeds a critical value (about 1012 cm3), the low-density limit (LDL). This theoretical value of the low-density limit is uncertain as it depends on complex atomic theory. Reported coronal densities above 1012 cm3 are uncommon. In this study, we estimate the distribution of R ratio values about the LDL and derive the empirical values for the 1st and 2nd moments of this distribution from 50 Mg XI spectra. From these derived parameters we establish the percentage of our observations which indicated densities above this limit and discuss other aspects of these observations such as the types of events and the time duting which spectra were obtained (e.g. rise phase or decay). INTRODUCTION The importance of reliable electron density measurements of the solar flaring plasma cannot be overstated: the electron density is necessary to ascertain the flare energetics, thermodynamic properties of the loop, heating and cooling timescales of the plasma, and evolution of source volumes. Yet in most coronal studies, the electron density is rarely measured directly. Direct measurements of the coronal density have been made (e.g. /1,2,3,4,5/) from the X-ray line ratio, R /6/, which is the flux ratio of the forbidden (J) to intercombination (i) lines of the helium-like ions, e.g., 0 VII, Ne IX, and Mg XI. The most recent R-matrix calculations of f and i excitation rates for Mg XI find an LDL R ratio value of R 0=3.0, at
=
5.5 MK /7/. The relation between ~e and R, is given by (17/), 71e (cm3)
=
7.94 x
x (R 0IR
—
1);
(1)
3. The theoretical value of the LDL increases with for R0 = 3.0, the LDL corresponds ~e =1012as cm the atomic number of the ion and istouncertain it depends on complex atomic theory. It is very important to check this value empirically, firstly, to verify the atomic theory, and secondly, to establish that spectra whose values of R are significantly differently from R 0 can be positively identified as coming from high-density plasmas, and not due to an uncertainty in the atomic theory. This3.latter point applies particularly to studies of Mg XI spectra, since the LDL corresponds to n~= 1012 cm OBSERVATIONS
G. A. Linford eta!.
(1)148
studying the helium-like Mg XI (9.169 A) lines. For our analysis we have selected the 50 best spectra from 18 solar flares which varied in X-ray (GOES classification) magnitude from Cl to M5. These events varied from short impulsive C flares (lasting a few minutes) to long duration M flares (lasting over 10 hours) and several of these events may have been multiple flares. Many of the observations were not optimized for the collection of Mg XI spectra, so as a result the actual collection times of these spectra were primarily during the decay phase. Thus, for these data the electron distributions are expected to be Maxwellian. SYNTHETIC SPECTRAL FITS AND THE R RATIO DISTRIBUTION The synthesis of the soft X-ray spectrum from 9.1 to 9.35 A used to fit the observed spectra includes only Mg X-XI lines. We have extended the method described by Linford et al. /8/ to fit the widths of the lines independently of electron temperature. To compute the synthetic spectra for dielectronic satellites with n= 2, 3, and 4, we used the atomic data from Steenman-Clark et al. j9/ and the atomic data necessary to compute the contributions of inner-shell excitation of Mg X, the direct excitation of Mg XI, and the radiative recombination of Mg XII were taken from Faucher et al. /10/. The density dependence of the intercombination and forbidden line rates are estimated from the R and G ratio functions given by Keenan et a!. /7/. Voigt profiles are calculated for all the lines in the synthesized spectrum by convolving a Lorent.zian componentequal to the crystal rocking curve of the ADP crystal (FCS channel 3) with a Gaussian component equal to the thermal Doppler broadening. In order to fit an observed spectrum we computed many synthetic spectra covering a grid of electron temperatures (1 to 15 MK), R ratios which corresponded to densities (10~to 1013 cm ), and Doppler temperatures (1 to 20 MK). For each fit we minimized x2 by adjusting the emission measure and the background. The best fit parameters were determined by the fit which gave the smallest X2 statistic (X2min)~ Since for the purpose of this work we were primarily interested in a single quantity, namely, the R ratio, the uncertainty of the fitted ratio was derived from the confidence interval corresponding to x ~ + 1. These values are shown in Table I. In Figure 1 the fits for the electron density expressed in terms of the R ratio are shown as a frequency distribution for all 50 spectra. Note that density increases to the left on this plot (see eqn. [1]). Since many of the spectra were obtained from a late flare decay phase, most points cluster near R=3. The width of the bins (0.15) for the R ratio histogram was chosen so that several measurements are contained in each bin. From this distribution it is possible to obtain an empirical measurement of R 0. If we initially assume that all the measurements of R were from plasmas with low densities, and if we assume that there is a normal distribution of values about the true mean, then the first and second moments of the distribution can be estimated by fitting a Gaussian function. When all 13 bins are included in the fit, we obtain a Gaussian with a mean value of 2.94 and a l/e width of20.65. of 8.8.However, On the the acceptance of the thisoriginal fit is estimated to be 21 percent a measured x basis of this, welevel reject hypothesis thatonly all spectra werefrom emitted from low-density plasmas and conclude that some of spectra have densities in excess of the LDL. By including only spectra with R> 2.5, an acceptable fit to the distribution was obtained with R 0= 2.9±0.2. The number of spectra with R > 2.5 is 36 and changing to R cut-off value from 2.5 to 2.4 or to 2.6 does not significantly change the determined value of R0 or its uncertainty. The fitted Gaussian is plotted as a solid curve in Figure 1. All spectra are assumedto have a mean temperature Te= 5.5 MK. DISCUSSION The distribution of R values for 36 spectra with R
>
2.5 yields an empirical measure of R0=2.9±0.2.
This v~lii~i.s ~~nn c~t~nt~.,,th the. ,-,.lg’,.1,.t,.,I ~,al.rn r~f~ A (~.t T
—c
~ i~.,rw\F,~nm ~
ot ,,I
P7/
Th~
Coronal Density Measurements in Mg XI
(1)149
TABLE I. Results from Synthetic Spectral Fits for Spectra with Densitiesgreater than the LDL. Date
R Ratio1
6R1
3-Mar-86 7-Mar-86 11-Jul-86 13-Jul-86 13-Jul-86 19-Oct-86 19-Oct-86 16-Apr-87 16-Apr-87 17-Apr-87 18-Apr-87 18-Apr-87 21-May-87 21-May-87
2.37 1.81 2.04 2.35 2.48 2.49 2.17 1.91 1.40 2.31 2.28 2.09 2.20 2.12
0.19 0.25 0.28 0.22 0.29 0.31 0.30 0.30 0.47 0.29 0.38 0.43 0.46 0.56
Density (cm~)2 2.7x10’2 5.0x1012 3.7x1&2 2.5x10’2 1.5x1012 1.4x1012 3.0x1012 4.5x10’2 l.0x1013 2.2x1012 2.5x1012 3.OxlO’2 2.9x1012 2.6x1012
Emission Measure x 1050 0.02013 0.00068 0.00598 0.01320 0.01107 0.01069 0.00640 0.00457 0.00099 0.00812 0.00439 0.00537 0.00263 0.00828
Emission3 Volume 2.8x1023 2.7x1021 4.2x1022 2.1x1023 4.9x1023 5.4x1023 7.1x1022 2.3x1022 9.9x102° 1.6x1023
7.0x1022 6.0x1022 3.1x1022 1.2r1023
1, R Ratio, oR, and the Emission Measure were derived from the synthetic spectra fits. 2, Density was computed from eqn (1) (17/). 3, Emission Volume was derived from (Emission Measure)/ lle2~ Although the FCS data cannot be used to verify the relation between R and ,ie~it does verify Keenan et a!. ‘s /7/calculation in the low density limit. Furthermore, it implies that if spectra are observed to have measured values of the R ratio which are less than 2.5, then the emitting plasma density is greater than the LDL. From the analysis described above, 14 spectra were found to have R values which are excluded from the LDL distribution, and thus, have higher densities (see Table I). This is consistent with our earlier hypothesis that approximately 28 percent of the spectra analyzed had densities which were greater than the LDL. Keenan ci a!. ‘s relationship between R and ~e is used to derive electron densities for Table I. The occurrence of 14 spectra which had densities greater than the LDL (1012 cm3) is of interest since all of these spectra were obtained during the decay phase of flares. In particular, two long-duration M flares which were observed on 19 October 1986 and 16 April 1987 revealed high densities during the decay phase. Further analysis will be performed to determinethe nature of the loops within the field-ofview of the FCS. These observations may have been of hot, compact loops rather than the extended post-flare loop structures normally associated with long-duration events. In addition, there were several moderately sized long-duration events (GOES C) which also revealed high densities during the decay phase. The existence of such densities during the decay phase may provide some clue as to the nature of the magnetic reconnection which is presumably occurring during the decay phase. It may also provide some information about the existence of energy input well into the flare gradualphase. SUMMARY Of the 50 s~ctrastudied, 36 had densities consistent with the low density limit (LDL) which for Mg XI is flelO 2cm3. These 36 spectra have a distribution of R values whose mean is R 0=2.9±0.2.This value is consistent with recent calculations for Mg XI /7/ and empirically verifies that observations of R less than R~can be interpreted in terms of enhanced plasma densities. The remaining 14 (28%) of the
(1)150
0. A. Linford eta!. 15
I
H R Ratio
Fig 1. Frequency distribution of the R ratio for 50 Mg XI observations. Note that the density increases to the left on this plot, see eqn. (1). The fitted Gaussian for R0=2.9±0.2is plotted as a solid curve. The Gaussian is from the336and spectra with R >in2.5. The are tabulated Table I. remaining 14 spectra were found to have densities> 1012 cm Support for this paper was received from NASA grant NAS5-3043 1 and the Lockheed Independent Research Program. We thank M. D. Morrison for help with the software and, R. Martin and S. Peterson forprocessing the FCS data. REFERENCES 1. D.L. McKenzie, R.M. Brousard, P.B. Landecker, H.R. Rugge, and R.M. Young, Astrophys. J. (Letters), 238, L43 (1980).
2. G.A. Doschek, U. Feldman, P.B. Landecker, and D.L. McKenzie, Astrophys. J. 249, 372, (1981). 3. C.J. Wolfson, J.G. Doyle, J.W. Leibacher, and K.J.H. Phillips, Astrophys. J. 269, 319 (1983). 4. G.A. Linford and CJ. Wolfson, Astrophys. J. 331, 1036 (1988). 5. D.M. Zarro, G.L. Slater, and S.L. Freeland, Astrophys. J. (Letters) 333, L99 (1988).
6. A.H. Gabriel and C. Jordan, Mon, Nor. R. astro. Soc. 145, 241 (1969). 7. F.P. Keenan, S.S. Tayal, and A.E. Kingston, Mon. Not. R. astro. Soc. 207, 51(1984). 8. G.A. Linford, J.R. Lemen, and K.T. Strong, Adv. Space Res. 8, #11, 173 (1988)
0273—1177/91 $0.00 + .50 Copyright © 1991 COSPAR
A STATISTICAL STUDY OF HIGH CORONAL DENSITIES FROM X-RAY LINE-RATIOS OF Mg XI G. A. Linford, J. R. Lemen and K. T. Strong Lockheed Palo Alto Research Laboratory, Department 91-30, Building 256, 3251 Hanover Street, Palo Alto, CA 94304, U.S.A.
ABSTRACT We applied an X-ray line-ratio density diagnostic to 50 Mg XI spectra of flaring active regions on the Sun recorded by the Flat Crystal Spectrometer on the Solar Maximum Mission. The plasma density is derived from R, the flux ratio of the forbidden to intercombination lines of the He-like ion, Mg XI. The R ratio for Mg XI is only density sensitive when the electron density exceeds a critical value (about 1012 cm3), the low-density limit (LDL). This theoretical value of the low-density limit is uncertain as it depends on complex atomic theory. Reported coronal densities above 1012 cm3 are uncommon. In this study, we estimate the distribution of R ratio values about the LDL and derive the empirical values for the 1st and 2nd moments of this distribution from 50 Mg XI spectra. From these derived parameters we establish the percentage of our observations which indicated densities above this limit and discuss other aspects of these observations such as the types of events and the time duting which spectra were obtained (e.g. rise phase or decay). INTRODUCTION The importance of reliable electron density measurements of the solar flaring plasma cannot be overstated: the electron density is necessary to ascertain the flare energetics, thermodynamic properties of the loop, heating and cooling timescales of the plasma, and evolution of source volumes. Yet in most coronal studies, the electron density is rarely measured directly. Direct measurements of the coronal density have been made (e.g. /1,2,3,4,5/) from the X-ray line ratio, R /6/, which is the flux ratio of the forbidden (J) to intercombination (i) lines of the helium-like ions, e.g., 0 VII, Ne IX, and Mg XI. The most recent R-matrix calculations of f and i excitation rates for Mg XI find an LDL R ratio value of R 0=3.0, at
=
5.5 MK /7/. The relation between ~e and R, is given by (17/), 71e (cm3)
=
7.94 x
x (R 0IR
—
1);
(1)
3. The theoretical value of the LDL increases with for R0 = 3.0, the LDL corresponds ~e =1012as cm the atomic number of the ion and istouncertain it depends on complex atomic theory. It is very important to check this value empirically, firstly, to verify the atomic theory, and secondly, to establish that spectra whose values of R are significantly differently from R 0 can be positively identified as coming from high-density plasmas, and not due to an uncertainty in the atomic theory. This3.latter point applies particularly to studies of Mg XI spectra, since the LDL corresponds to n~= 1012 cm OBSERVATIONS
G. A. Linford eta!.
(1)148
studying the helium-like Mg XI (9.169 A) lines. For our analysis we have selected the 50 best spectra from 18 solar flares which varied in X-ray (GOES classification) magnitude from Cl to M5. These events varied from short impulsive C flares (lasting a few minutes) to long duration M flares (lasting over 10 hours) and several of these events may have been multiple flares. Many of the observations were not optimized for the collection of Mg XI spectra, so as a result the actual collection times of these spectra were primarily during the decay phase. Thus, for these data the electron distributions are expected to be Maxwellian. SYNTHETIC SPECTRAL FITS AND THE R RATIO DISTRIBUTION The synthesis of the soft X-ray spectrum from 9.1 to 9.35 A used to fit the observed spectra includes only Mg X-XI lines. We have extended the method described by Linford et al. /8/ to fit the widths of the lines independently of electron temperature. To compute the synthetic spectra for dielectronic satellites with n= 2, 3, and 4, we used the atomic data from Steenman-Clark et al. j9/ and the atomic data necessary to compute the contributions of inner-shell excitation of Mg X, the direct excitation of Mg XI, and the radiative recombination of Mg XII were taken from Faucher et al. /10/. The density dependence of the intercombination and forbidden line rates are estimated from the R and G ratio functions given by Keenan et a!. /7/. Voigt profiles are calculated for all the lines in the synthesized spectrum by convolving a Lorent.zian componentequal to the crystal rocking curve of the ADP crystal (FCS channel 3) with a Gaussian component equal to the thermal Doppler broadening. In order to fit an observed spectrum we computed many synthetic spectra covering a grid of electron temperatures (1 to 15 MK), R ratios which corresponded to densities (10~to 1013 cm ), and Doppler temperatures (1 to 20 MK). For each fit we minimized x2 by adjusting the emission measure and the background. The best fit parameters were determined by the fit which gave the smallest X2 statistic (X2min)~ Since for the purpose of this work we were primarily interested in a single quantity, namely, the R ratio, the uncertainty of the fitted ratio was derived from the confidence interval corresponding to x ~ + 1. These values are shown in Table I. In Figure 1 the fits for the electron density expressed in terms of the R ratio are shown as a frequency distribution for all 50 spectra. Note that density increases to the left on this plot (see eqn. [1]). Since many of the spectra were obtained from a late flare decay phase, most points cluster near R=3. The width of the bins (0.15) for the R ratio histogram was chosen so that several measurements are contained in each bin. From this distribution it is possible to obtain an empirical measurement of R 0. If we initially assume that all the measurements of R were from plasmas with low densities, and if we assume that there is a normal distribution of values about the true mean, then the first and second moments of the distribution can be estimated by fitting a Gaussian function. When all 13 bins are included in the fit, we obtain a Gaussian with a mean value of 2.94 and a l/e width of20.65. of 8.8.However, On the the acceptance of the thisoriginal fit is estimated to be 21 percent a measured x basis of this, welevel reject hypothesis thatonly all spectra werefrom emitted from low-density plasmas and conclude that some of spectra have densities in excess of the LDL. By including only spectra with R> 2.5, an acceptable fit to the distribution was obtained with R 0= 2.9±0.2. The number of spectra with R > 2.5 is 36 and changing to R cut-off value from 2.5 to 2.4 or to 2.6 does not significantly change the determined value of R0 or its uncertainty. The fitted Gaussian is plotted as a solid curve in Figure 1. All spectra are assumedto have a mean temperature Te= 5.5 MK. DISCUSSION The distribution of R values for 36 spectra with R
>
2.5 yields an empirical measure of R0=2.9±0.2.
This v~lii~i.s ~~nn c~t~nt~.,,th the. ,-,.lg’,.1,.t,.,I ~,al.rn r~f~ A (~.t T
—c
~ i~.,rw\F,~nm ~
ot ,,I
P7/
Th~
Coronal Density Measurements in Mg XI
(1)149
TABLE I. Results from Synthetic Spectral Fits for Spectra with Densitiesgreater than the LDL. Date
R Ratio1
6R1
3-Mar-86 7-Mar-86 11-Jul-86 13-Jul-86 13-Jul-86 19-Oct-86 19-Oct-86 16-Apr-87 16-Apr-87 17-Apr-87 18-Apr-87 18-Apr-87 21-May-87 21-May-87
2.37 1.81 2.04 2.35 2.48 2.49 2.17 1.91 1.40 2.31 2.28 2.09 2.20 2.12
0.19 0.25 0.28 0.22 0.29 0.31 0.30 0.30 0.47 0.29 0.38 0.43 0.46 0.56
Density (cm~)2 2.7x10’2 5.0x1012 3.7x1&2 2.5x10’2 1.5x1012 1.4x1012 3.0x1012 4.5x10’2 l.0x1013 2.2x1012 2.5x1012 3.OxlO’2 2.9x1012 2.6x1012
Emission Measure x 1050 0.02013 0.00068 0.00598 0.01320 0.01107 0.01069 0.00640 0.00457 0.00099 0.00812 0.00439 0.00537 0.00263 0.00828
Emission3 Volume 2.8x1023 2.7x1021 4.2x1022 2.1x1023 4.9x1023 5.4x1023 7.1x1022 2.3x1022 9.9x102° 1.6x1023
7.0x1022 6.0x1022 3.1x1022 1.2r1023
1, R Ratio, oR, and the Emission Measure were derived from the synthetic spectra fits. 2, Density was computed from eqn (1) (17/). 3, Emission Volume was derived from (Emission Measure)/ lle2~ Although the FCS data cannot be used to verify the relation between R and ,ie~it does verify Keenan et a!. ‘s /7/calculation in the low density limit. Furthermore, it implies that if spectra are observed to have measured values of the R ratio which are less than 2.5, then the emitting plasma density is greater than the LDL. From the analysis described above, 14 spectra were found to have R values which are excluded from the LDL distribution, and thus, have higher densities (see Table I). This is consistent with our earlier hypothesis that approximately 28 percent of the spectra analyzed had densities which were greater than the LDL. Keenan ci a!. ‘s relationship between R and ~e is used to derive electron densities for Table I. The occurrence of 14 spectra which had densities greater than the LDL (1012 cm3) is of interest since all of these spectra were obtained during the decay phase of flares. In particular, two long-duration M flares which were observed on 19 October 1986 and 16 April 1987 revealed high densities during the decay phase. Further analysis will be performed to determinethe nature of the loops within the field-ofview of the FCS. These observations may have been of hot, compact loops rather than the extended post-flare loop structures normally associated with long-duration events. In addition, there were several moderately sized long-duration events (GOES C) which also revealed high densities during the decay phase. The existence of such densities during the decay phase may provide some clue as to the nature of the magnetic reconnection which is presumably occurring during the decay phase. It may also provide some information about the existence of energy input well into the flare gradualphase. SUMMARY Of the 50 s~ctrastudied, 36 had densities consistent with the low density limit (LDL) which for Mg XI is flelO 2cm3. These 36 spectra have a distribution of R values whose mean is R 0=2.9±0.2.This value is consistent with recent calculations for Mg XI /7/ and empirically verifies that observations of R less than R~can be interpreted in terms of enhanced plasma densities. The remaining 14 (28%) of the
(1)150
0. A. Linford eta!. 15
I
H R Ratio
Fig 1. Frequency distribution of the R ratio for 50 Mg XI observations. Note that the density increases to the left on this plot, see eqn. (1). The fitted Gaussian for R0=2.9±0.2is plotted as a solid curve. The Gaussian is from the336and spectra with R >in2.5. The are tabulated Table I. remaining 14 spectra were found to have densities> 1012 cm Support for this paper was received from NASA grant NAS5-3043 1 and the Lockheed Independent Research Program. We thank M. D. Morrison for help with the software and, R. Martin and S. Peterson forprocessing the FCS data. REFERENCES 1. D.L. McKenzie, R.M. Brousard, P.B. Landecker, H.R. Rugge, and R.M. Young, Astrophys. J. (Letters), 238, L43 (1980).
2. G.A. Doschek, U. Feldman, P.B. Landecker, and D.L. McKenzie, Astrophys. J. 249, 372, (1981). 3. C.J. Wolfson, J.G. Doyle, J.W. Leibacher, and K.J.H. Phillips, Astrophys. J. 269, 319 (1983). 4. G.A. Linford and CJ. Wolfson, Astrophys. J. 331, 1036 (1988). 5. D.M. Zarro, G.L. Slater, and S.L. Freeland, Astrophys. J. (Letters) 333, L99 (1988).
6. A.H. Gabriel and C. Jordan, Mon, Nor. R. astro. Soc. 145, 241 (1969). 7. F.P. Keenan, S.S. Tayal, and A.E. Kingston, Mon. Not. R. astro. Soc. 207, 51(1984). 8. G.A. Linford, J.R. Lemen, and K.T. Strong, Adv. Space Res. 8, #11, 173 (1988)