coronal information from euv disk spectral line intensities

C O R O N A L I N F O R M A T I O N F R O M E U V D I S K

S P E C T R A L L I N E I N T E N S I T I E S

D. E. BILLINGS and MANUEL ALVAREZ* Department of Astro-Geophysics, University of Colorado, Botdder, Colo. 80302, U.S.A.

(Received 14 August, 1974)

Abstract. Center of disk EUV line intensities from quiet and active regions are used for determining an analytical expression for the variation of temperature with height in the lower corona, including the corona-chromosphere transition region. This approach imposes two coronal temperature regimes in both quiet and active regions. In each case the lower temperature regime is a continuation of the transition region, reaching a maximum of about 1.4 million deg in the quiet and 1.7 million deg in the active region. In the quiet region the high temperature regime, assumed isothermal, has a temperature of about 2.4 million deg, and in the active region, about 4.2 million deg.

1. Introduction

The technique initiated by Pottasch (1963, 1964) and successively modified by Athay

(1966), Dupree (1972), and Kopp (1972) for analysis of EUV line intensities has proved

extremely useful for the study of the chromosphere-corona transition layer. It not only

has provided a great deal of information about the thermal structure of that remark-

able region, but has resolved a number of questions concerning the chemical abun-

dance of the solar atmosphere. The amount of information the technique has yielded

about the corona, however, has been much more limited because of two assumptions

inherent in the method: (1) The coronal temperature is considered to increase mono-

tonically outward throughout the region of application of the method. (2) In the

Athay modification of the Pottasch method, the temperature range in the solar atmo-

sphere is assumed to include all temperatures at which ions emitting lines used in the analysis have a significant population.

Because of these assumptions, the method gives coronal temperatures which in-

crease with height but never go through a maximum. Furthermore, ions which exist

near the temperature maximum do not fall into the pattern defined by lower-temper-

ature ions, and are therefore not of use in the method. This is an unfortunate weakness

of the method, since these are just the ions of most use in studying the corona.

Two additional ideas which are approximately true in the transition layer but not

at all applicable to the corona have been frequently incorporated into the method: (1) that the thermal flux is a constant, and (2) that the pressure is a constant.

Within these limitations, various investigators have resorted to a variety of tech- niques for deducing information about the corona from EUV observations. Athay

(1971) and Jordan and Wilson (1971) set the pressure in the transition region as inter-

mediate between that typical for the upper chromosphere and the lower corona.

* Now at: Observatorio Astronomico Nacional, APTO, Postal 877, Ensenada, B. C., Mexico

Solar Physics 40 (1975) 23-38. All Rights Reserved Copyright .~ 1975 by D. Reidel Publishing Company, Dordrecht-Holland

24 D.E. BILLINGS AND MANUEL ALVAREZ

Dupree (1972) determined the density in the transition region by the ratio of intensities of two lines from beryllium-like atoms. Withbroe (1970a) used EUV line intensity ratios at the limb to determine the temperature of an isothermal corona which he appended to a Dupree-Goldberg transition region. Noyes (1971) describes more detailed work by Withbroe in which the intensities of lines on the limb relative to their disk intensities are coupled with line intensity ratios on the limb to determine both coronal temperatures and transition-zone thermal flux.

The purpose of the present study is to introduce and illustrate a further modification of the Pottasch technique - one specifically designed to carry smoothly from the transition region into the corona and derive additional information about the corona from published EUV line intensities as measured on the solar disk.

2. Analytical Formulation of Temperature vs Height Relationship

We will follow Kopp's (1972) notation in which the variables describing the solar atmosphere are q= T 7/2 and f = (ap2/(d~7/dh)). Here T is the electron temperature, p = NT, where N is the electron density, and a is the fraction of the observed surface emitting in the line. (ap2/(dq/dh)), averaged over the temperature range for which a particular spectral line is emitted, is determined experimentally for each emission line observed by

I f = C ~ T1/2G(T) d T ' (1)

where I is the line intensity, and C incorporates various atomic parameters. Tile value of ~/ associated by Kopp with a line is the 7/2 power of the temperature at which the quantity G (T) = T - 1/2 (N/Na) e- (a Elk r) maximizes. Here Ni/N ~, the fraction of atoms

in the state of ionization from which the line is emitted, is the function of temperature which dominates G (T).

In his Figure l, Kopp (1972) shows that from a combination of two whole-disk observations of the EUV spectrum, the plot l o g f vs log~ I can be fit nicely by a quadratic curve. Such a fit is a violation of the Athay (1966) concept of constant pressure and thermal flux through the transition zone, since dtT/dh is proportional to the thermal flux.

In our Figure I we show a plot of logfvs log ~/for data given by Dupree et al. (1973) from the observations of a quiet region at the center of the Sun, taken in August 1969 by OSO-6. If we wish to fit these points with a smooth curve we have three choices : (1) to fit all of the points with a quadratic curve, as Kopp did, with considerable scatter at the high-temperature end of the curve. (2) fit a quadratic curve to all but the high temperature points, and explain the progressive drop-off of the higher temperature points from the resulting curve, as Dupree did, by the temperature range for emission of these lines being only partially covered by the corona; or (3) fit the entire set of points by some cubic or higher order curve which drops off at the higher temperature range.

C O R O N A L I N F O R M A T I O N F R O M E U V D I S K S P E C T R A L LINE INTENSITIES 2 5

~/~ 2 5 . 0

'ID

o_ 2 0 . 0 O

V li

4.-

0

Fig. 1.

1 5 0 14 .0

QUIET REGION

X X-: ~ /

J I i I i I r l 1 6 . 0 18 .0 2 0 . 0 2 2 . 0 24 .C

l o g -r/; " r / = T 7 / 2

Quiet region logfvs logll, log f = ,a +fl log~l + m/2 (logr/)~; a = 97.0625; f l~ 8.1683; m = 0.4160.

ACTIVE REGION A

c- 2 5 . 0 i i / i I I I I I I

g--

0,1

X 20o f x• v \ ~ tl -- X

4- x ~ _ ~ Y J ~

0 - - 1 5 0 i I i I i I i I

4.o 16.0 18.0 20.0 22.o log 7/; -q=T 7/2

I

2 4 . 0

Fig. 2. Active region logfvs log0. a = 100.6557; fl = -- 8.5750; m = 0.4438.

We choose the second alternative. The third can be rejected easily by noting that

the function f , as defined, must increase as one approaches the temperature maximum,

since dq/dh approaches zero, whereas p is clearly non-zero at the temperature max-

imum.

A possible objection to the use of Dupree 's data in this context is that she is dealing with specific intensity rather than emergent flux, whereas it is the latter that is propor-

tional to the number o f transitions along the line of sight. The relation between the two can differ by a factor o f two for optically thick vs optically thin lines, but this dif- ference would affect the curve we draw only if there is a strong, systematic relationship

between the optical depths o f the emitting regions for the various lines and the temper-


atures at which the lines are produced. "['he optical depths given by Withbroe (1970b) show no strong relationship of this kind. Thus, if they constitute a fair sample of EUV lines, there can be no objections to our use of Dupree's data.

The points in Figure 2 are deduced from emission line data in a very active region,

as reported by Dupree et al. (1973) and corrected for / l =0.65. The curves drawn in Figures 1 and 2 are quadratic leasts quares fits to the plotted points with the encircled points deleted. The deleted points correspond to lines of Fe xvl, Si xIr and Al xI, the

three highest-temperature ions in the two studies, for reasons already given. The most striking thing one notices is that even though the high activity curve is shifted upward

in f by about two orders of magnitude, and downward slightly in q, the two curves

are almost identical in shape. Thus the slope of such a curve, fl (q), appears to be a fundamental property of the upper solar atmosphere.

Since

d In f ap 2 - and f -

fl d i n q dq/dh '

. , , , - , [ ; o . ,, da fl('ll="kdl;) dh-\-dh) dlflJ+a(,1-- )dq" (2)

But if hydrostatic equilibrium holds,

1 dp mg - 2 . ~ - - col ( 3 )

p dh k T

Hence, Equation (2) can be rewritten as follows:

d2,l F f l ( q ) ldal(d',~2 a d,l dh 2 + _ - O. (4) k q a dqJ \ d h ] + v I"12/7 dh

There is a unique analytical solution of Equation (4) which we give in the Appendix for the case

fl(q) = rio + m logq. (5)

The slope of a l o g f vs log q curve, combined with the assumption of hydrostatic equilibrium, defines a family of q vs h curves (the equivalent of temperature vs height curves). The members of the family are conveniently identified by (dtT/dh)o at a speci-

fied r/o, or equivalently by Po at qo due to our definition of f by Equation (2). Each of these curves comes to a very nearly constant r/a - corresponding to a tem-

perature which we shall designate as T a - at a very short distance above the transition region, in contrast to the curves obtained when one considers a constant flux throughout the transition region, the change in gradient being more abrupt than that in the constant-flux case. The steeper the temperature gradient at the reference temperature - or the greater the assumed reference pressure - the greater the temperature Ta, as was pointed out by Jordan (1971).

CORONAL INFORMATION FROM EUV DISK SPECTRAL LINE INTENSITIES 27

7.0

o o o 0 o o LO ( M - -

i i I

I I I

QUIET REGION HEIGHT (kin)

0 O 0

0 0 0 0 o 0 o ~

0 -- O0 I"-- i"O

I I I I I

I--

0

6.0

6 28

6 26

6 24

6 22

v

" 6 .20

- - 6 1 8

6.18

6 1 4

612

5 . 0 - -

4 . 0 13.0

I 12.0

Fig. 3.

I S O T H E R M A L R E G I O N ( H )

I S O T H E R M A L R E G I O N ( L )

T ION R E G I O N ( r )

V" I ] I I I I 1 .0 I 0 . 0 9 . 0 8 . 0 7 . 0 6 .0

l o g N ( c m - 3 )

I 5.0 4 . 0

Quiet region temperature-density model.

I I I

-- QUIET REGION

log T L : a + b Po+cPo 2

' a = 5.84-72

b = 0.080.'.'5 c : - 0 . 0 0 4 - 2

] I I I I I 1 I ~ I I ] 1

ACTIVE REGION

log T L = o + b P0+cP02 G = 5 . 9 8 4 7

b = 0 . 0 0 9 6 c =-7.4-1 xlO -5 J

I [ I I I I I I J i J_ i I 1 4 0 5.0 6 0 30.0 35.0 40 .0

P o : N o TO ( x l O I 4 c m - 3 K )

, J

Fig. 4. Relationship between the two boundary conditions: temperature of region L and pressure at 10SK level.


Figure 3 is the temperature distribution that we obtain from the analysis of the quiet region described here. The important point here is that of the three quantities Po, (dq/dh)o and TL, there is only one independent parameter for a given region on the Sun with an observed EUV spectrum when described by the l o g f vs logq curve of the type we have drawn in Figures 1 and 2. Figure 4 is a plot of log TL vs Po for the quiet and active regions which we are analyzing here. It would appear from this analysis that given f (q) and the coronal temperature T z for a set of EUV data, the physical characteristics of the transition region and lower corona would be completely described. However, as one might infer from the work of Withbroe and Gurman (1973) as well as from several other papers by the HCO group, such a model alone would not give the observed line intensities. It is necessary to add another free parameter. Our procedure differs from that of others in the manner by which this is accomplished.

3. Determination of T L and Evidence for a High Temperature Regime

The temperature distribution as shown in Figure 3, comes to a constant maximum temperature. However, this conclusion must be considered somewhat fictitious, since it was derived from a curve which could not possibly describe the situation in the vicinity of the maximum coronal temperature. As h increases dq/dh ~ 0 and hence

f -'-> O'~.

For a more realistic analysis, we assume (1) that a curve computed according to section 2, with proper boundary condition, is accurate over a region which we have designated by r in Figure 3, extending up to a point fairly near the maximum coronal temperature (TL). We also assume (2) that the coronal temperature goes through a fairly broad maximum which we designate as region L and which we approximate by an isothermal at temperature T L.

In principle, the boundary between regions r and L should be independent of the ions or emission lines used in the analysis. The data of this study were not sufficiently precise to show where this boundary should be placed, but the results are quite in- sensitive to its location, provided it is in the vicinity of maximum change of temperature gradient.

Finally we introduce a third assumption which we will later modify: (3) the points in Figure 1 representing lines from high-energy ions drop progressively away from the curve because the temperature ranges over which these ions exist are only partially covered by the corona. In our first approximation we consider that Mg x, AI xL Si Xll and Fe xvI are the ions to which this applies.

As an example of how we make use of these assumptions, we will consider the 2550 line of A1 xl. Suppose we wish to test 1.5 • 106K as a possible value for TL. In Figure 5 we show a plot of TI/2G (T), vs T, derived by interpolation into an isoelec- tronic sequence from the computations of Allen and Dupree (1969). Now, referring to Equation (1), f was computed by dividing the observed line intensity by the product of C, a constant appropriate for this line, and the entire area under the curve in Figure 5. However, if the maximum temperature was really only 1.5 million deg,


Fig. 5.

0 . 2 0 I I T I I I I I I E I [

0 1 8 -

0 1 6 -

0 .14 -

~" 0 . 1 2 -

(.9 e4 0 .10 -

I-- 0 . 0 8 -

0 . 0 6 -

0 . 0 4 -

0 . 0 2

A I "g"l- X550

~f',l { I I [ I I I I 0 I 2 5 4 5 6 7 8 9 I0 II 12 15 14

T E M P E R A T U R E ( x l O 6 K)

G(T) curve for 2550 of AI xI, showing region of integration for TL = 1.5 • 106K.

f should have been determined by dividing by the product of C and the shaded area

only. Thus f should now be corrected by multiplying by ry, the ratio of the total area

to the shaded area; in this case rf = 13.25. Hence fc, the 'corrected ' value of f is given

as follows: T2

S T1/2G(T) dT Tt f c = f X r z = f x r . (6) I TI/2G(T) dT

Tl

Furthermore, the abscissa for the point in Figure 1 should not be that corresponding to the maximum G (T), or maximum Tt/ZG(T), but that corresponding to the tem-

perature Tc at the centroid o f the shaded area. When these two corrections are intro-

duced, the point representing the emission line in Figure 1 is moved to well above the

curve to the position marked G in the figure with coordinates log fc = 17.84 and log q~ = =21.28.

We now divide the observed intensity o f this l ine , / , into two parts I, and IL, such

that Ir would yield a corrected logf~, using Equat ion (6), which would fall precisely

on the curve. In our example, this value in log f , = 17.43. We attribute the emission I, to region r as defined above. The remainder, we attribute to region L. Hence, calling

r= f/(fc--fr)~Ir/IL we can write IL=I/(r+ 1). Since the region L is considered


Fig. 6.

29.0

28.0 ..J

q.)

2 7 . 0

26.0 I.I

QUIET REGION

1 [ I 1 I I I 1

~ ( I .535, 2 6 . 6 8 )

Mg X 625.3) I I I ~ [

1.2 3 1.4- t.5 1.6 1.7 1.8 1.9 T L (x106 K)

2.0

First iteration for determination of TL for quiet region, logeL vs TL for Mg x, A[ Xl and Si Xli. The curves for the second iteration are not shown.

i sothermal , we can write ez=IL/7CG L where GL=G(TL) and eL=SL NZe dh is the

emission measure to be a t t r ibu ted to region L.

We now compute eL for each assumed TL for lines of Mg x, AI xt, SI xn and

Fe xvI.

A plot o f the logar i thm of such emission measures against assumed temperatures ,

for Mg x, A1 xI, and Si xII is given in Figure 6 for the quiet region, and Figure 7 for

the active region. I f our assumpt ions made at the beginning of this sect ion were correct ,

curves for all four ions would intersect at the same point , since the emission measure

in region L, and also T L, should be independent of the ion used in de te rmining them.

We find, however, that the Si x[I line is d isplaced to too high a t empera ture to pass

th rough the intersection of the Mg x and AI x[ curves, and that the Fe xvI could no t

exist at all at the t empera ture of this intersection. We conclude, as a first app rox ima-

t ion, that the intersect ion of the Mg x and A1 x[ curves define TL and eL of a low tern-

pera ture regime, and that a f rac t ion q of the emission in the Si xn line comes f rom this

low- tempera ture regime, (LTR). The remainder , and all of the Fe xvI emission, we

a t t r ibute to a high tempera ture regime, (HTR) , which we assume in this s tudy to be

i so thermal with a tempera ture T u. To determine q, Tt,, and eft, the emission measure

to be a t t r ibu ted to H T R , we use the fol lowing procedure :

C O R O N A L I N F O R M A T I O N FROM EUV DISK SPECTRAL LINE INTENSITIES 3 |

_1

0

30.0

290

28.0

27,C

ACTIVE REGION

"4 , , , I I I I I I

/ /

(I.65, 28.33)"

I I I I I I I . 2 1 .3 1 . 4 1 .5 1 .6 1 .7

T L (x106 K)

Fig. 7. Same as Figure 6 for active region.

I I .8 1.9 2.0

First we choose q so that the L T R port ion of the Si x~[ emission would give curves

in Figures 6 and 7 passing through the intersection o f the Mg x and A1 xl curves.

Next we choose Tn such that we get the same emission measure en for the H T R , whether f rom the H T R part of Si xn or f rom all Fe XVL This latter condit ion is sum-

marized by

(1 - q) eiz0t2 = e16~016, (7)

where e,2 and g,6 are the published emission measures for the Si xIr and Fe xvI lines,

respectively, and 012 and O t 6 are factors for converting emission measures which have been computed f rom an assumed range of temperatures including that for which

G(T) maximizes to a single temperature, i.e.

0.7 G (Tmax) Fexv~

OJ6 ----- G(Tn) F e x v I (8)

We have applied this analysis both to the quiet and the active regions studied by Dupree et al. (1973). The results are summarized in Table I. We should note that

T L and eL depend critically on the relative abundance of Mg x and A1 XL We have taken Dupree 's (1972) abundances.

The quantities in Table IA are computed on the assumption that all of the emission

in lines o f Mg x and A1 xt arises f rom the low-temperature regimes r and L. However,

32 D. E. BILLINGS AND MANUEL ALVAREZ

TABLE I

Summary of the physical conditions of the lower corona

Region T (• 106K)

loge % emission

Mg x (625) AI xl (550) Si xn (499) Fe xvl (362)

A. First iteration

Quiet 1.535 26.6850 (i) 100.0 (i) 100.0 (ii) 6.5 (i) 0.0 LTR Quiet 2.435 27.1668 (i) 0.0 (i) 0.0 (ii) 93.5 (i) 100.0 HTR Active 1.65 28.3350 (i) 100.0 (i) 100.0 (ii) 24.3 (i) 0.0 LTR Active 4.70 29.0860 (i) 0.0 (i) 0.0 (ii) 75.7 (i) 100.0 HTR

B. Second iteration

Quiet 1.365 26.4301 (iii) 69.7 (iii) 44.8 (ii) 1.8 (iii) 0.0 LTR Quiet 2.415 27.1911 (iii) 30.3 (iii) 55.2 (ii) 98.2 (iii) I00.0 HTR Active 1.70 27.99 (iii) 76.8 (iii) 73.4 (ii) 16.3 (iii) 0.0 LTR Active 4.23 29.0246 (iii) 23.2 (iii) 26.6 (ii) 83.7 (iii) 100.0 HTR

(i) Assumed ~/o emission from the low temperature regime (LTR) and high temperature regime (HTR) for the first iteration. (ii) Computed /o~ consistent with (LTR) derived from Mg x 2625 and AI xi~ 2550. (iii) Computed ~ emission from LTR and HTR assumed for the second iteration.

when we actually compute the line intensities for the model atmosphere obtained

under this first approximation, we find that there is an impor tant contr ibut ion to the

line emission f rom the HTR. As a second iteration we compute the line emission f rom

this H T R and substract it f rom the total line intensity as computed under the first

iteration. We consider this corrected value of the intensity and recompute the curves

in Figures 6 and 7. The results of this second iteration are in Table IB and Figure 3.

We consider the low and high temperature coronal regimes give in Table IB, coupled with the transition region models given in Table II, to be as detailed a description o f

the quiet and active region studied as the data justify. With more precise observations, abundances, and atomic parameters, the method of analysis is capable of disclosing

additional temperature regimes, or a distribution o f temperatures, in the corona. In Table I I [ we compare observed line intensities with those computed f rom our

models. We would not expect the logarithms of our computed intensities to agree with the logarithms of observed intensities more closely than the curves of Figures 1 and 2 agree with the plotted values of log f in those figures, except for the coronal ions. A plot o f A l o g / v s A log f , Figure 8, shows that descrepancies between our computed


TABLE II

Temperature, density and conductive flux distribution of the quiet and active regions described in this work

Quiet region Active region

Height log T log N Height log T log N (km) (K) (cm -~) (km) (K) (cm -3)

i

-- 537.6 4.00 10.99 77.0 4.00 11.59 -- 333.3 4.10 10.74 --35.1 4.10 11.46 --202.0 4.20 10.56 --17.7 4.20 11.35 -- 114.5 4.30 10.41 --10.0 4.30 11.24 -- 70.7 4.40 10.30 --6.1 4.40 11.14

- - 45.1 4.50 10.18 --4.1 4.50 11.04 -- 30.5 4.60 10.09 --2.8 4.60 10.95 - - 19.6 4.70 9.98 --1.8 4.70 10.84 -- 10.9 4.80 9.88 --1.0 4.80 10.74

- - 3.6 4.90 9.78 --0.4 4.90 10.64 0.0 4.955 9.73 0.0 4.955 10.59 3.7 5.01 9.67 0.4 5.00 10.54 7.4 5.06 9.62 0.8 5.06 10.49

11.5 5.11 9.57 1.2 5.10 10.44 21.2 5.20 9.47 2.3 5.20 10.34 34.9 5.30 9.37 4.0 5.30 10.24 56.6 5.40 9.27 6.9 5.40 10.14 95.0 5.50 9.17 12.6 5.50 10.04

172.4 5.60 9.07 25.2 5.60 9.94 347.7 5.70 8.97 57.0 5.70 9.83 801.4 5.80 8.86 147.9 5.80 9.73

2164.9 5.90 8.74 445.5 5.90 9.63 6962.0 6.00 8.60 1572.9 6.00 9.52

~- 13919.6 6.05 8.50 6858.0 6.10 9.38 o "~0 34799.0 6.10 8.31 13919.6 6.15 9.30 ~9 ~, 202000.0 6.135 7.46 34799.0 6.19 9.14

454000.0 6.23 7.85

and the observed intensities are to be explained almost entirely by scatter in l o g f rel-

ative to the curves in Figures 1 and 2. Had we limited our data to the sequence of ions

used by Withbroe and G u r m a n (1973), curves like those in Figures 1 and 2 would have

fit the t ransi t ion region ions much more precisely than was the case with the larger

group of ions we used, and our computed intensities would have agreed very closely

with the observed.

We can relate our conclusions to optical line observations. We note that the ioniza-

t ion curves for Si xH and Fe x lv are quite similar. Consequently, in the quiet region

we would expect the corona l green line profiles to correspond to temperatures of about

2.4 • 10 6 deg, but extend upward to 3.0 • 10 6 or more in active regions. This conclu-

sion is consistent with Billings (1959) observations. Also the temperature in the high

temperature regime of the active region is adequate for the product ion of Ca xv, in

agreement with the observat ion o f Waldmeier (1956) that the coronal yellow line is

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visible in almost every region of high activity when there are suitable observing con-

ditions. In the low temperature regime of the quiet region, Fe x ions would be very

abundant. In the active region LTR, the Fe x abundance would still be quite high. This leads us to the interesting conclusion that the coronal structures see in 26374 of Fe x connect thermally to the part of the transition zone from which most of the

Q U I E T R E G I O N

1.0

0 . 6

-I .0 O . 2

�9 E I I I ' t .

- I . 0

A logI

/x, log f , I ' I ~ "

I 0.2 0 .6 I.O

Li ISO-ELECTRONIC SEQUENCE

X B ISO-ELECTRONIC SEQUENCE

o OTHERS

Fig. 8. zJ logfvs A log/for the quiet region, where 3 log f - - logf (curve in Figure 2) -- logf (observations), A logI= log/(computed) log/(observed).

EUV radiation originates. Also 26374 should extend to very close to the Sun - a conclusion reached by Athay and Roberts (1955) many years ago. I f there is any portion of the transition zone through which the high temperature regime connects thermally to the chromosphere, the temperature gradient in it would be so steep that its contri- bution to transition zone EUV line intensities would be negligible.

36 D. ]E. BILLINGS AND MANUEL ALVAREZ

4. Summary and Conclusions

The basic features of the models which we have presented here can be brought out most clearly by a comparison with those of Withbroe and Gurman (1973). In their paper, they use what Withbroe (1970a) had earlier called the modified Dupree- Goldberg method, characterized by three parameters: the pressure at a defined temperature level in the transition zone, the transition zone thermal conductive(flux- assumed constant, and the temperature of an isothermal corona. These parameters are adjusted independently to fit the EUV line intensities.

The point of departure in our approach arises from our not accepting the hypothesis of a constant thermal flux. We regard this hypothesis as a linear approximation of the data only. In contrast, we use a quadratic description of the data, in which the quantity (pressure)Z/thermal flux increases as the maximum coronal temperature is

approached. This quadratic description introduced in our work has the effect of making the

temperature become almost constant with height abruptly above the transition zone, allowing us to go smoothly from the transition region up to the corona in the LTR. In addition, the relationship that exists between the pressure and the thermal conductive flux reduces the independent parameters describing the "main component ' of the atmosphere to a single one, either the temperature of the coronal LTR or the pressure 'or conductive flux) at some temperature level within the transition region. To this (main component' of the atmosphere, we add a high temperature regime (HTR). We note that for both the quiet and active Sun models, the Withbroe-Gurman single coronal temperature model lies between our LTR and HTR temperatures. The hypothesis of a multi-temperature corona is amply justified, both by optical (Dollfus, 1971; Zirker, 1971) and soft X-ray data (Evans and Pounds, 1968; Widing and Sandlin, 1968; Batstone et al., 1970; Walker, 1972; Walker et al., 1974; Wolff, 1974).

Since our study is based on two regions only, one quiet and one rather active, we are not able to demonstrate the ability of our model to reproduce line intensities for regions of intermediate activity, but a study of this kind is now under way.

Appendix

Solution to Equation (4): Equation (4),

(i) dh 2 + + - O, a d'7 \dh/ q2/7 dh

by the change of variable y = dq/dh, becomes

(ii) dy + Y - A /72./7 , dq


where

1 da A 07) -

a ( '7 ) dt7

is assumed to be a wel l -behaved funct ion of q.

I f the funct ion l o g f vs l o g , / i s adequa te ly represented by a quadra t ic funct ion,

/7('1) =/70 + m logq

and if we subst i tute x = log q, Equa t ion (ii) becomes

d y (iii) + y In 10x [(/7 0 + mx) - 10A (x) ] = - 2c~ In 10 x 10 (5/7)x .

dx

Equa t ion (iii) has the so lu t ion:

(iv) y = yo exp I X 2 + X 2 - (F (Xo) - F (X)) ] - Y

- C exp I X 2 - F (X) ] f dY' exp [112 - F ( y l ) ] ,

go

where

x = 4 t , , ,,,x + ,

2,n10 [ C = 2c~ exp

m 7 2m

+ )

and F is an a rb i t r a ry funct ion which can be evaluated, p rov ided A (x) is known.

Acknowledgements

The au thors wish to express their thanks to Drs Roger K o p p and J. A. Eddy of the

High Al t i tude Observa to ry for their interest and advice, to the Na t iona l Center for

A tmosphe r i c Research for compute r facilities, and to A. K. D u p r e e for supplying us

with unpubl i shed ion iza t ion tables. The research was sponso red by the Na t iona l Sci-

ence F o u n d a t i o n , G r a n t GA-31477.

References

Allen, J. W. and Dupree, A. K. : 1969, Astrophys. J. 155, 27. Athay, R. G. : 1966, Astrophys. J. 145, 784. Athay, R. G. : 1971, in C. J. Macris (ed.), Physics of the Solar Corona, D. Reidel Publ. Co., Dordrecht,

Holland, p. 36. Athay, R. G., and Roberts, W. O.: 1955, Astrophys. J. 121, 231. Batstone, R. M., Evans, K., Parkinson, J. H., and Pounds, K. A.: 1970, Solar Phys. 13, 389. Billings, D. E.: 1959, Astrophys. J. 130, 961. Dollfus, A. : 1971, in C. J. Macris (ed.), Physics of the Solar Corona, D. Reidel Publ. Co., Dordrecht,

Holland, p. 97.

38 D.E.BILLINGS AND MANUEL ALVAREZ

Dupree, A. K.: 1972, Astrophys. J. 178, 527. Dupree, A. K. and Goldberg, L.: 1967, SolarPhys. 1, 229. Dupree, A. K., Huber, M. C. E., Noyes, R. W., Parkinson, W. H., Reeves, E. M., and Withbroe,

G. L.: 1973, Astrophys. J. 182, 321. Evans, K. and Pounds, K. A.: 1968, Astrophys. J. 152, 319. Jordan, C. and Wilson, R.: 1971, in C. J. Macris (ed.), Physics of the Solar Corona, D. Reidel Publ.

Co., Dordrecht, Holland, p. 219. Kopp, R. A. : 1972, Solar Phys. 27, 373. Noyes, R. W. : 1971, in C. J. Macris (ed.), Physics of the Solar Corona, D. Reidel Publ. Co., Dordrecht,

Holland, p. 192. Pottasch, S. R.: 1963, Astrophys. J. 137, 945. Pottasch, S. R. : 1964, Space Sci. Rev. 3, 816. Waldmeier, M.: 1956, Z. Astrophys. 39, 219. Walker, A. B. C., Jr.: 1972, Space Sei. Rev. 13, 672. Walker, A. B. C., Jr., Rugge, H. R., and Weiss, K.: 1974, Astrophys. J. 188, 423. Widing, K. G. and Sandlin, G. D. : 1968, Astrophys. J. 152, 545. Withbroe, G. L.: 1970a, SolarPhys. 11, 42. Withbroe, G. L. : 1970b, Solar Phys. 11,208. Withbroe, G. L. and Gurman, J. B. : 1973, Astrophys. J. 183, 279. Wolff, R. S.: 1974, Solar Phys. 34, 163. Zirker, J. B. : 1971, in C. J. Macris (ed.), Physics of the Solar Corona, D. Reidel Publ. Co., Dordrecht,

Holland, p. 140.

coronal information from euv disk spectral line intensities

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