Smithsonian

Contributions to Astrophysics

VOLUME 3, NUMBER 5

THE DOPPLER WIDTHS OF SOLAR

ABSORPTION LINES

By BARBARA BELL AND ALAN MELTZER

SMITHSONIAN INSTITUTION

Washington, D. C.

1959

Publications of the Astrophysical Observatory

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The Doppler Widths of Solar Absorption LinesBy BARBARA BELL1 and ALAN MELTZER2

The precise profiles of solar and stellar ab-sorption lines depend upon many factors.Among them are the mechanism of radiativetransfer, the form of the line absorption coeffi-cient, atomic collision cross-sections, tempera-tures, densities, and their gradients in the solaratmosphere. For weak and medium-weaklines, the profiles are determined primarily bythe form of the absorption coefficient, whichdepends in turn upon the mechanisms acting tobroaden the lines. Spectral lines are broadenedintrinsically by three processes: the Dopplereffect due to line-of-sight velocities of the radi-ating atoms; natural width due to the finitewidth of the atomic energy levels; and externaleffects due to perturbation of the radiatingatom by neighboring atoms and ions. In thesolar atmosphere, Doppler broadening and col-lision broadening are far more important thannatural width.

This paper deals chiefly with Doppler broad-ening. From observations of the continuousspectrum, astronomers generally agree that thecontinuum is formed in regions of the solaratmosphere where the kinetic temperature can-not much exceed 5000° K. They also agree,however, that solar absorption lines show aDoppler width substantially in excess of thatgiven by 5000°. Since Doppler broadeningcan result equally well from the random line-of-sight velocities of individual atoms, or frommass motions of elements of gas, the conceptof turbulence, or mass motions of elements ofgas, has usually been invoked to explain theextra width of the lines. Any broadening pro-duced by turbulence must by definition be in-dependent of atomic weight, while, according

1 Harvard College Observatory, Cambridge, Mass.1 Formerly at Smithsonian Astrophyslcal Observatory; now at Rens-

selaer Polytechnic Institute, Troy, N. Y.

to kinetic theory, the mean square randomvelocities of individual atoms will be propor-tional to the kinetic temperature and inverselyproportional to the atomic weight. Thus itmight appear relatively simple to distinguishbetween these mechanisms by observing a fewwell chosen lines. Since the Doppler widthsusually have been derived from metallic atomsof nearly equal atomic weights, there has beenlittle evidence to test the correctness of theturbulence hypothesis.

In recent years several studies (Bell, 1951;Rogerson, 1957; Waddell, 1956, 1958) havebeen made of the profiles of weak and medium-weak solar absorption lines. For the first ofthese, the Utrecht Atlas (Minnaert et al.,1940) provided extensive observational ma-terial, while the other two used photoelectricprofiles traced by the respective authors. Belland Rogerson each used the Voigt profilemethod of analysis, which assumes in essencethat the shape of a weak line is the shape of itsabsorption coefficient, and involves no assump-tions about physical conditions in the solaratmosphere. Waddell, on the other hand, useda model atmosphere—with specific assumptionsabout temperatures and densities in the regionswhere his lines were formed—to computetheoretical profiles to compare with those ob-served. The success of this method presup-poses, of course, a reasonably correct idea ofphysical conditions in the solar atmosphere.These three studies give several contradictoryresults that are independent of the method ofanalysis.

Rogerson and Waddell each assumed kinetictemperatures of the order of 5000° K, as indi-cated by continuum observations, and derivedturbulent velocities of 1.4 and 1.8 km/secrespectively. Bell emphasized a comparison

39

40 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL. 8

of profiles from atoms of different atomicweights, from carbon to iron, and concludedthat her data indicated a kinetic temperatureof 10,000° K and a turbulence of less than 0.5km/sec for neutral lines of excitation potentialof 4 to 7 volts. Rogerson, however, foundthat the atomic weight dependence of his pro-files, from silicon to iron, was adequately ac-counted for by a kinetic temperature of 5000° K.

Furthermore, Bell found that the Dopplerwidths of iron lines varied with excitation po-tential (EP) and hence with depth. In a laterstudy van Regemorter (in press) obtained sim-ilar results, while Rogerson finds no evidence inhis data for such a variation. Bell found thedamping factor in iron lines to depend markedlyon the parity—and hence on the bindingenergy of the electron to the parent ion—andvery little on the excitation potential. Roger-son found the converse result, and has suggestedthat some of the discrepancies between hisresults and those of Bell may arise from the useof different tables of the Voigt functions (Elste,1953; van de Hulst and Reesinck, 1947), butwe have tried both tables on a few lines andfeel that this is not a probable explanation.Further observational study is needed to resolvethese discrepancies.

In the present work we have investigatedthe way in which the width of the line profilesdepends on the atomic weight of the atoms pro-ducing them.

The observational material was obtained atthe Sacramento Peak Observatory where theauthors were guest investigators—Meltzer dur-ing March and April 1957 and Bell fromSeptember to November 1957.

We made photoelectric tracings of the lineprofiles by means of a Lallamand cell and Brownrecorder attached to the large Littrow spectro-graph. A 1200-line grating was used in thesecond order, giving a dispersion of 4.45 mm/A.A Jena OG-2 filter prevented the overlappingof higher spectral orders. The entrance andexit slits were kept at a width of 30 micronsand at a length of one inch. The scale of thetracings was about 6 inches per A.

All tracings were made as near as practicableto the center of the sun's disc. Because thelevel of solar activity was high during 1957,

special care had to be taken to avoid plage andspot areas.

The instrumental profile was studied fromtracings of the mercury 5461 line of isotope 198.The half-width of the instrumental profile issmall, about 0.030A. The wings, unfortunately,do not fit a Voigt profile. For this reason aninstrumental correction to the Voigt dampingparameter cannot be determined, and we havenot attempted to study the damping from ourobserved line profiles. However, since the in-strumental profile is completely independent ofatomic weight, the unconnected profiles shouldsuffice to give the kinetic temperature, and theknown half-width of the instrumental functionpermits a reasonably good estimate of the tur-bulence.

We made at least six tracings of every line,reversing the direction of scan after each trace.For two reasons, we made additional tracingsfor some of the lines. Because of the possi-bility of getting into a weak unnoticed plagearea, Bell traced the vital 5380 line of carboneach day for comparison with other lines ob-served that day. Also, a random slippage inthe motion of the grating, and hence in the rateof scan, introduced errors into the individualtracings. Since we made two independent setsof observations which give essentially the sameresults, we feel justified in presenting our find-ings in spite of such errors in the individualtracings.

We analyzed our lines by the Voigt profilemethod. By definition, a Voigt function re-sults from the superposition of a damping ordispersion type function,

and a Gaussian probability function,

where eu c2, ft, and j82 are constants and x is thedistance from the line center. Thus Voigt func-tions are characterized by two parameters, ftand /32, and the strength of their wings dependson the ratio of /3,//32.

The line absorption coefficient similarly isgiven by the superposition of a Maxwellianvelocity distribution function and a dampingfunction. The parameters ft and & can thus

NO. 5 SOLAR ABSORPTION LINES 41

readily be equated to factors in the absorptioncoefficient, the first relating to the dampinghalf-width, the second to the Doppler half-width.

In an analysis by Voigt profiles, one assumesthat the profile of the observed line is the aver-age of the absorption-coefficient profile appro-priate to the regions of the solar atmosphereover which the line is formed. Thus themethod can properly be applied only to com-parison of lines that arise at the same averagedepth in the solar atmosphere.

It is difficult to find suitable lines sufficientlydifferent in atomic weight and reasonably sim-ilar in excitation level. For this study weselected lines of carbon, silicon, and iron havingexcitation potentials of 7.6, 4.9, and 4 to 5 volts,respectively, as shown in tables 1 and 2. Al-though the comparison of 7.6-volt carbon lineswith those of silicon and iron may be open tocriticism, we have included carbon because itprovides a long base line in atomic weight.However, the greater emphasis probably shouldbe given to results arising from comparison ofiron and silicon lines, since these are quitesimilar both in excitation and ionizationpotential.

A Voigt profile is completely specified bythree quantities: AX, the width at half intensity,generally called simply the half-width; e, thecentral depth; and p, a function of fr/fo, de-termined by the width of the profile in the

wings relative to that at the half-intensity pointand varying from 1.06 for a pure Gaussian to1.57 for a pure damping profile.

We measured the width of each trace at itshalf-intensity point and at four points in thewings, 0.1c, 0.15c, 0.2c, and 0.3c. The measuresfor the individual tracings were combined togive mean quantities for each line. Most ofthe mean observed profiles showed a good fitto a Voigt function. From Elste's (1953) tableswe obtained the Voigt parameters, p, bi, b\.From the measured half-width, the more con-venient quantity h=— 10* was obtained. We

A

then computed ft=bih and $=6|A,*. The ft and/Si of the line, after correction for the respectiveinstrumental ft and /3£, should yield a good firstapproximation to the damping constant andthe Doppler velocity, respectively. Our col-lection of lines is not suitable for testing theparity effect; furthermore, the instrumentalft is indeterminate, so we shall not further dis-cuss the ft's here, except to note that theysuggest that damping is not negligible even forweak lines.

The observational results are given in tables1 and 2, where the data obtained by Meltzerand by Bell are presented separately in orderto show clearly that, in spite of the gratingslippage, the independent sets of observationsgave results in close agreement.

Element

CSi

Fe

TABLE 1.

Wavelength

5380. 32•5665. 565772. 155793. 085386. 345441. 355752. 045793. 935852. 235856. 10

—Parameters of line profiles observed by Meltzer, April 1957

Excitationpotential

7.654.905.064. 914. 154.354.594.244.594 28

Observed quantities

c

0. 12.26.32.28.26.25.41.26.30.27

AX(A)

0. 142. 118. 130. 126.088.092. 106. 100. 108. 103

h*

700431505477270288340297342308

Voigt parameters

P

1. 181.251. 301.271.231.291.281.231.301.26

PI

18591.289. 693. 560. 752.062.866.559.562.0

Number oftracings

66

106658968

•Violet half only; red half blended.

42 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

TABLE 2.—Parameters of line profiles observed by Bell, September-November 1957

VOL. S

Element

C

Si

Fe

Wavelength

5052. 165380. 32*5665. 565690. 435701. 115772. 155793. 085293. 965295. 315386. 345395. 225401. 275441. 355464 295705. 485793. 935852. 235855. 095856. 105927. 806089. 58

Excitationpotential

7.657.654 904 914 915.064 914 124 354 154 354 354 354 124 244 244 594 594 284 595.00

Observed quantities

c

0.21. 13.26.34.26.32.27.25.24.26. 16.20.25.31. 29. 26.30. 17.26.31.29

AX(A)

0. 156. 156. 122. 121. 120. 132. 128.092.092.095.096.096. 101.095.099.099. 105. 100. 102. 104. 102

h2

950840466450443526488303306310319316344305302292324294303312287

Voigt parameters

P

1.201.231.251.241.221.301.271.271.221.231.251.261.291.241.241.261.311.241. 251.291.28

A

235190989610289965870706864626764585264635654

Number oftracings

122266711138818768661310111164

•Violet half only.

The quantity (% is proportional to the Dop-pler width of the line. From the definition ofVoigt functions it can easily be shown that

(1)

where T is the kinetic temperature, vT is theturbulent velocity, and /3f, is the instrumentalprofile correction. The quantity in parenthesesis independent of n, the atomic weight. If thisterm were zero, the /Sfs of the lines should be in-versely proportional to the atomic weight of theelement producing the line. When this quan-tity is not zero, the quantity [#}— {1%+(&)]should be inversely proportional to /*. Solvingequation (1) for the kinetic temperature, weobtain

T=5.41 [ ^ - ( 4 + / ^ , . ) ] M . (2)We computed a mean /3| for each element. Bytrial and error we determined that a correctionof (flr+0«)=26 would yield the best inverserelation between the quantity in square bracketsand the atomic weight, and the best agreeingvalues of T for the three elements. The results

appear in table 3. We also solved equation (2)for each pair of elements, with the results shownin table 4.

We obtain a kinetic temperature of the orderof 10,000 to 11,000° K, in substantial agreementwith BeD's (1951) earlier work on comparablelines.

If the entire quantity (vr+Pli) is due toturbulence (i. e. if the instrumental /3|, is negli-gible), we obtain vT=l.Q km/sec. This is, ofcourse, an upper limit to the turbulence. Themaximum instrumental correction will be ob-tained if we assume the instrumental profile is aGaussian, so that its entire half-width goes into/3|i- Such an assumption gives f%~\Q, and aturbulence of 1.2 km/sec. However inter-preted, our Doppler half-widths are somewhatlarger than those previously obtained. Thereason for this is unknown. If real, the effectmay be related to the remarkably high level ofgeneral solar activity observed throughout 1957.

In addition to the profiles observed at thecenter of the solar disc, in September we ob-served the profiles of 13 of these lines near the

NO. 6 SOLAR ABSORPTION LINES 43

north limb. The 1-inch long slit of the spectro-graph was centered one inch from the limb, on a10-inch image of the sun; thus, at r/R=0.8.

Table 5 gives the parameters of the lines,while the third section of table 4 gives the indi-cated kinetic temperatures. The half-widths

of the lines increase markedly to the limb, butthe weighted mean temperature is 11,400°, onlyslightly higher than that obtained at the centerof the disc. The turbulence increases to between1.9 and 2.1 km/sec, which is 0.6 to 0.7 km/secgreater than the center-of-disc values.

TABLE 3.—Determination of apparent kinetic temperature at center of disc

Observer

Meltzer

Bell

Element

CSiFe

CSiFe

Meanft

1859261

2069563

$-26

1596635

1806937

Ratio ofatomicweights(56/M)

4. 672.001. 00

4. 672.001. 00

Ratios of($-26)

4.601.901.00

4.871.861.00

Temperature

10, 300° K10,00010, 50010, 300 (mean)

11, 70010, 40011,20011,000 (mean)

Weight

62242

3443

122

Any thorough study of center-to-limb changesin the Doppler width of lines would of courserequire observations at several points along theradius. Until we obtain such data, we shallnot attempt to interpret the present north limbobservations. Our observations suggest, how-ever, that the well-known increase in line half-width to the limb (see Allen, 1949) resultslargely from increased turbulence rather thanfrom significantly higher kinetic temperatures.

Houtgast (1953) has criticized Bell's (1951)work on the grounds that some of the lines shecompared (specifically, carbon and oxygen withiron) differed in excitation potential and hencemust arise at different depths in the solar atmos-phere. This argument may apply to the car-bon lines, but it should not apply to the com-parison of silicon with high excitation lines ofiron which should arise from the same averagelevel in the atmosphere. (Bell's determinationof the kinetic temperature depended as muchon a comparison of silicon and iron as on thatof iron with carbon and oxygen; also, only 4-5volt lines of iron were used in these comparisonsprecisely because of the difficulty mentioned byHoutgast.)

A definitive analysis of a line profile cannotbe made without detailed consideration of the

processes of line formation and of the variationof the absorption coefficient with depth in anappropriate model atmosphere. The methodof this paper has been criticized as too simpleeven for a first approximation. Therefore, toexamine the relationship between results ob-tained by this method and one using a modelatmosphere, we made a Voigt profile analysisof a theoretical profile computed by Waddell(1956) for the titanium line, X6126, from astandard model atmosphere and Pecker's (1953)method of weighting functions (see Waddell,1958). We obtained a turbulence 0.2 km/secgreater than Waddell put into his line.

As a second test, we made a Voigt analysisof a pair of iron and silicon theoretical profiles,computed by Doherty and Hazen (unpub-lished) from the Claas (1951) model of thesolar atmosphere, with the assumption thatscattering plays no part in the formation of thelines. From comparison of their iron and sili-con profiles, we obtained 7r=6600°K, which isthe Claas temperature at the optical depth 1.2in the continuum; a turbulent velocity of 2.1km/sec, which is 0.35 above the assumed valueof 1.75 km/sec; and a damping of a=0.23,slightly below the original 0.25.

44 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL. 3

From this analysis we conclude that thesingle layer approximation, implicit in theVoigt profile method, is not unrealistic as afirst approximation for elements such as ironand silicon. Integration through an atmos-phere does make the width of the lines moder-ately greater than the width of their absorptioncoefficients, but when a Voigt analysis is usedthis width appears in a spuriously large turbu-lence rather than in a spuriously high kinetictemperature.

Our profiles have not been corrected forinstrumental scattered light. The Voigtmethod is insensitive to this correction, how-ever, so that its neglect results in only a minorerror at worst and cannot account for our hightemperatures. We tested this point by apply-ing an arbitrary scattered-light correction of 10percent to our mean observed silicon and ironprofiles. Voigt analysis of this pair of cor-rected profiles gave 7=12,000°, as comparedwith 10,600° from the same pair of profilesbefore correction. The correction decreasedthe apparent turbulence by about 0.2 km/sec.

Hazen (private communication) has pointedout that the Voigt profile method is verysensitive to small changes in the assumed levelof the continuum, because the shape of thewings is so important in determining the Voigtparameters. If the continuum were drawn 1or 2 percent higher, the two mean profiles wouldgive a temperature of 8,000 or 6,000°K respec-tively. In the spectral regions we studied, solarge a systematic error in drawing the con-tinuum appears extremely unlikely. How-ever, this sensitivity may account for much ofthe scatter among individual tracings (andindividual lines in the Utrecht Atlas), andindicates the importance of making severaltracings of each line. Moreover this source oferror would similarly distort the results frommore sophisticated methods of analysis, insofaras they attempted to fit the damping wings.

One other comment may be made on theVoigt profile method of analysis. Waddell, inhis model atmosphere analysis, assumed thatweak lines were undamped. In terms of Voigtparameters, this implies that /3i=0 and p = 1.06.Under the assumption of no damping, the entirehalf-width is considered to result from tem-perature and turbulence, and the wing meas-

ures may be disregarded. When we appliedthis assumption to our lines, Meltzer's observa-tions yielded temperatures of about 11,000°from carbon and iron and a substantially highertemperature for silicon, as one might expect,since its p's (Voigt parameters) are substantiallylarger than those for carbon and iron. FromBell's observations we can obtain good agree-ment among the three elements at around17,000°K. We thus feel that the dampingcannot properly be ignored, and indeed anyattempt to do so only aggravates the problemof high kinetic temperatures.

TABLE 4.—Apparent kinetic temperatures, determinedfrom comparison of Doppler half-widths of lines ofpairs of elements of different atomic weight

Observer

Meltzer(centerof disc)

Bell (centerof disc)

Bell (0.2Rfromnorthlimb)

Elements{paired)

C-SiC-FeSi-Fe

C-SiC-FeSi-Fe

C-SiC-FeSi-Fe

Temperature

10, 600° K10,3009,400

10, 000 (mean)

12,60011,9009,800

11, 100 (mean)

12,80011,80010, 30011, 400 (mean)

Weight

284864

77156165

436574

Waddell (1958) has pointed out a discrepancybetween observed and Voigt profiles in the farwings, within about one percent of the con-tinuum. But such a deviation is far smallerthan that between the observed profile and anytheoretical profile computed under the assump-tion of zero damping. While other interpreta-tions of the line wings are conceivable, dampingis the most simple and straightforward. Theparity effect (Bell, 1951; Carter, 1949) wouldappear to provide the best test of the correct-ness of this interpretation. According totheory, the damping is proportional to thebinding energy of the electron to its parent termof the ion. In iron these binding energies havevalues that lead one to expect the dampingarising from odd terms to be about double that

NO. 5 SOLAR ABSORPTION LINES 45

arising from even terms of comparable excita-tion potential, although parity in itself has nobearing on the problem (see Bell, 1951). Wehope to explore this question further with addi-tional observations.

If we accept the hypothesis that the 22 lines—or at least the 20 lines of silicon and iron—com-

pared in this paper are formed through thesame average layer of the solar atmosphere, weare faced with the conclusion that the atomspossess a kinetic temperature of 10,000 to11,000° K, although all continuum studiesindicate electron temperatures of the order of5000° K.

TABLE 5.—Parameters of line profiles observed at 0.2 of the solar radius from the north limb, September 1957

Element

C

Si

Fe

Wavelength

5052. 165380. 325665. 565772. 155793. 085386. 345395. 225441. 355464.295793. 935852. 235855. 095856. 10

Excitationpotential

7.657.654.905.064.914. 154.354.354. 124.244. 594.594.28

Observed quantities

c

0. 17. 10.25. 30. 26.26. 16.25.30. 25.28. 16.25

AX (A)

0. 154. 152. 134. 140. 134. 102. 101. 107. 105. I l l. 116. 109. 115

h*

930800555585542362348388370366393347385

Voigt parameters

P

1. 171. 171.231.281.221. 181.261.241.251. 181.251.211. 19

ft

2502151251111249470857795808398

Number oftracings

898

10875648666

Three possible explanations for this dis-crepancy present themselves: (1) equiparti-tion of energy between the atoms and electronsdoes not occur; (2) the electrons actually have akinetic temperature of 10,000°; or (3) the con-tinuous and the absorption-line spectra areformed in significantly different regions of thesolar atmosphere. Because of the great fre-quency of atom-electron collisions, the firsthypothesis cannot be regarded seriously (seeBhatnagar et al., 1955). With the secondhypothesis, the electrons would be much tooenergetic for capture in sufficient numbers byhydrogen to form H~, and it would seemvirtually impossible to account for the observedcontinuum. Only the third hypothesis appearsto offer any possibility of a reasonable explana-tion.

If we postulate that the lower chromospheremakes a substantial contribution to the absorp-tion lines, while the continuous absorptionarises from photospheric levels, there is hopethat the temperature differences can be re-conciled. Recent work by Zirker (1956, 1958)

on the metallic flash spectrum from the 1952eclipse and by Pecker (1957) on the CH andCN bands makes such a postulate appear moretenable than it did a few years ago. Thesimilarity between the intensity and velocityfluctuations in the strong magnesium X5167 lineand in medium-weak neighboring lines, ob-served with the vacuum spectrograph at theMcMath-Hulbert Observatory (McMath et al.,1956), also suggests that the chromosphere maymake significant contributions to the cores ofat least medium-weak solar absorption lines.

The mere existence of such velocity shiftsmakes it obvious that some form of macro-turbulence must be broadening the profiles oflines coming from an area of the sun largeenough to take in many granules. A completepicture of velocities can, therefore, come onlyfrom analysis of such detailed spectra.

As an alternative to chromospheric contri-butions, Pecker suggested (in conversation)that inhomogeneities in the photosphere suchas described by the 3-stream model of Bohm(1954) and Voigt (1956) might account for the