P O W E R S P E C T R A OF H~ D O P P L E R S H I F T S

I A N ELLIOTT*

Sacramento Peak Observatory, Sunspot, N.M., U.S.A.

(Received 21 June, 1968)

Abstract. The temporal characteristics of the chromospheric velocity field in a quiet region are studied by means of a carefully guided sequence of 215 Hc~ spectra of the disk centre lasting 54 min. The Doppler shifts of each frame at A2 -- 0.4 A_ are measured and the velocity history of each position on the sun is reconstructed. The velocity power spectrum is found for each of 256 points along the total slit length of 280000 km.

A steady downward velocity is associated with places where the amplitude of the fluctuating velocity is high. The average velocity power spectrum exhibits three main features: (1) A peak at 287 sec, (2) A group of high frequency peaks in the range 150-210 sec, and (3) A low frequency peak with a period of 900 sec.

The relationship of these features to the Ca K network is discussed.

1. Introduction

Recent w o r k on solar veloci ty fields has shown the impor tance of the large scale con-

vective flow in the pho tosphe re k n o w n as the supergranu la t ion (SIMON and LEIGHTON,

1964). The hor i zon ta l flow in supergranule cells can p lay a m a j o r role in the concen-

t r a t ion and dissolut ion o f magnet ic fields (L~IGHTON, 1964). I t has been shown tha t

the coarse br igh t ne twork o f the Ca K line and, to a lesser extent, the da rk ne twork

observed in the wings o f H a lie over the supergranule cell boundar ies .

In view of the above results, i t seemed useful to find out i f the ch romospher i c

veloci ty field is influenced by the supergranula t ion . The veloci ty field was observed

using EVANS and MICHARD'S (1962) technique of t ak ing a sequence o f spectra at one

pos i t ion o f a careful ly guided solar image. I f the spec t rograph slit is near the centre

o f the solar disk, the D o p p l e r shifts give a measure of the vert ical m o t i o n in the solar

a tmosphere . Since the ver t ical velocit ies o f the supergranu la t ion are small ( ~ O. 1 km/

sec), they are difficult to observe direct ly near the centre o f the disk, and the br ight K

ne twork was used as an ind ica t ion o f the supergranule boundar ies .

2. Observations

The sequence o f H a spect ra was ob ta ined at Sac ramen to Peak Obse rva to ry with the

30 cm hor i zon ta l te lescope fed by a 45 cm quar tz coelos ta t and second mir ror , and

the 13 m Li t t row spect rograph. One o f the best f rames o f the sequence is shown in

F igure 1. Re levan t obse rva t iona l da t a is given in Table I.

The or ien ta t ion o f the spec t rograph slit was roughly N o r t h - S o u t h at the centre

o f the disk. The solar image was guided in r ight ascension by means o f two pho to -

* Member of the High Altitude Observatory Solar Project at Sacramento Peak, on leave from Dunsink Observatory, Ireland.

Solar Physics 6 (1969) 28-40; �9 D. Reidel Publishing Company, Dordrecht-Holland

POWER SPECTRA OF H a DOPPLER SHIFTS 29

R e d

1.0 ,a,

S

Fig. I.

BLue

lOSk m N

Frame No. 176, one of the best of the sequence of Hc~ spectra. The thin vertical line parallel to the dispersion is a fiducial mark made by a hair crossing the spectrograph slit.

TABLE I Observational data for spectral sequence

Date: 1963 October 1 Time: 14.11-15.05 UT Grating: 300 line/ram Harrison, ninth order,

(dispersion = 7.67 mm//~) Slit width: 200/t Emulsion: Kodak IVE (70 ram), developed 8 min D19 Filter: RG-1 Exposure duration: 0.65-0.45 sec Exposure frequency: 15 sec Image diameter: 256 mm

electric cells which cont ro l led the second coelosta t mirror . Since the sun was at a low

al t i tude, a red filter was p laced over the guider cells to avoid d isp lacement due to

a tmospher ic dispersion. F ine guiding in decl inat ion was cont ro l led manual ly . The

decl ina t ion drif t was no t negligible since i t was near the au tumna l equinox when the

sun 's dec l ina t ion is decreasing rapidly. In order to cause min imum dis turbance, the

manua l ad jus tments were made at ten-minute intervals.

The effect o f solar ro ta t ion is impor t an t : after an hour ' s observa t ion with a 200 #

slit, the ini t ial slit pos i t ion will be displaced by abou t seven slit widths. The solar

image was moved relat ive to the slit a t minute intervals by sui table amounts to main- tain the ini t ial slit posi t ion.

Dur ing the per iod o f observat ion, the slit pos i t ion on the solar d isk was mon i to r ed

30 i. ELLIOTT

by photographing the reflecting slit jaws through a birefringent filter. The filter had a pass-band of 0.5 A and was centred at 0.5 A in the red wing of the He line. The filter- grams were recorded on IVE film in a 35 m m Acme camera. The Acme shutter was automatically tripped at the same time as the spectrograph shutter.

Information about the K network was obtained by taking spectroheliograms before and after the sequence of spectra. The Sacramento Peak spectroheliograph is of somewhat novel design, consisting essentially of a double monochromator of zero dispersion which gives very high spectral purity in pass-bands down to 0.05 A and avoids spectroscopic smearing of the image when using broad pass-bands. It is possible to obtain simultaneous spectroheliograms in three separate lines thus facilitating detailed comparison of features in each line.

Spectroheliograms were taken in the He and K lines and each scan took about two minutes. The He pass-band was placed in the red wing of the line to obtain maximum contrast and to permit comparison with the slit jaw filtergrams. A wide pass-band was used for the K line in order to include the K 2 emission peaks. The spectroheliograms are shown in Figure 2. Times and slit data are given in Table II.

TABLE II Observational data for spectroheliograms

UT First Slit Ha pass-band K pass-band

Scan 1 13.47-13.49 200/z (2800 kin) .06 A at 0.5 R .21/~ (W to E) line centre

Scan 2 15.14-15.16 100/t (1400 km) .06/~ at 0.5 R .21 ]k (E to W) line centre

3. Measurement and Reduction

Each spectrum was measured on the Recording Doppler Compara tor (RDC) of Sacramento Peak Observatory. This instrument scans a spectral line perpendicular to the dispersion with twin slits separated by 2A2 between which the profile is centred by means of a pair of photocells linked to a servomechanism. The displacement of the film parallel to the dispersion is a measure of the Doppler shift of the profile. In this case, the effective size of each slit was 545 km x 0.026 A and A2 was set at 0.4 A. The Doppler shifts were automatically digitized and recorded on punched cards with a sampling interval of 200 p (1090 km), giving 264 points for each scan. In the absence of an adequate theory of line formation for a moving atmosphere, the shifts were converted directly to velocities using the Doppler relation. I t should be noted that each R D C scan is made at constant line width and that both absolute wavelength and intensity may vary. Hence, the information f rom the R D C differs from that provided by a spectroheliogram or by an isophote trace of a spectrogram.

In general, the spectrum lines produced by a plane grating are curved on account of the effective change of groove Separation for oblique rays (MINKOWSKI, 1942).

H,:,

1348 UT

Ca K

Ha

1515 UT

Ca K

Fig. 2. Spectroheliograms taken before and after the sequence of spectra. The slit position was located by means of the slit jaw filtergrams.

32 I. ELLIOTT

The line curvature was found by fitting a second degree curve to an average R D C scan by the method of least squares. The zero velocity level was assumed to be the mean of the R D C scan. The net Doppler shifts were corrected for line curvature and punched as a new deck of cards.

Examination of the slit jaw photographs showed that, apart f rom the declination drift parallel to the slit, the guiding errors were mostly within 1". The original time sequence started at 14.00 UT and consisted of 259 frames ( t = 0 to 258) but a guiding break was detected near t = 42 and the analysis was restricted to 215 frames (t = 44 to 258, inclusive). A 16 m m cine film of the spectra was also used for evaluating guiding. In order to correct for image drift parallel to the slit, a computer programme was written to match successive R D C scans. The first part of the programme was designed to detect gross guiding errors, and it computed the cross-correlation coefficient between adjacent scans. Even with perfect guiding, the correlation coefficient would be less than unity on account of changes of the velocity field during the 15 sec interval between exposures; the coefficient generally lay between 0.6 and 0.8 and if it was less than 0.5, the matching was considered faulty. The second part of the programme improved matching at breaks by shifting adjacent scans relative to one another by unit values of x, the distance along the slit, in order to obtain maximum correlation. In general it was possible to find an improved match, but for six scans the correlation with both adjacent scans was low; inspection of the slit jaw photographs showed slight shifts perpendicular to the slit caused by bad seeing affecting the guider, and in these six cases the V(x) values were replaced by artificial values which were the means of the two adjacent scans. The matching was improved further by shifting adjacent scans by fractional values of x, using linear interpolation between points. The three highest correlation coefficients separated by unit values of x were used as a starting point; a parabola was fitted to the three points and the position of its maximum was used as origin for determining three more correlation coefficients which were separated by intervals of 1/4 unit. The procedure was repeated with an interval of 1/16 unit and the final maximum was taken as the best match. The relative shift of each pair of scans was found and hence, the cumulative displacement of each scan relative to the first.

The cumulative displacement of the last scan after matching with the computer amounted to 20000 km, but no drift in excess of 2000 km could be detected from comparison of the slit jaw photographs. No bias could be detected in the computer matching programme. One explanation is the possible existence of a systematic migra- tion of velocities along the slit - some evidence for this was seen on the cine film. The horizontal disturbances appeared to move mainly from South to Nor th along the slit and were most prominent between slit positions x = 80 and 150. The velocity of the disturbances along the slit was estimated as 100-200 km/sec from the cine film. For the final drift corrections, it was assumed that the overall drift between first and last scans was zero and all the scans were shifted relative to the first by the amount of the residual cumulative drift. As before, linear interpolation between scans was used for the fractional part of the drift correction.

The next step in the reduction of the data was the inversion of the array of V(x)-

P O W E R S P E C T R A OF Ha D O P P L E R S H I F T S 33

curves for each time t, into an array of V(t)-curves for each position x. This was accomplished with the limited capacity of an IBM 1620 computer by inverting the data in smaller blocks and rearranging the card output. The final output was a series of V(t)-curves for each of the 256 positions on the sun.

From the V(t)-curve for each position, we computed the steady velocity V~, the amplitude of the fluctuating component {v, the amplitude of the total velocity field {T, the autocorrelation function A(k ) and the power spectrum G(v). Thus,

i

~___ - - V~ 2 ~-

i = 1

(1)

where Vs is the average velocity for a given position:

and

n

1 Z v ~ = - v, n

i = 1

(2)

#T=[#~ + 2V~] }. (3)

The autocorrelation function for each position is defined as

A(k ) = Z V~V~+k V* 2 , i = I i

(4)

where n=number of scans (i.e., 215) and k is the lag number which runs from zero to M (i.e., 214). The power spectrum is the cosine Fourier transform ofA (k) multiplied by ~ . Using ORRALL'S (1966) approximation for the cosine transform when A(k ) consists of discrete values, the power spectrum becomes

M

k = O

(5)

where h=0, I, 2 . . .M is related to frequency v by h=2Tv (T=total observation time, i.e., 3210 sec) and e=0.5 if k = 0 or M, otherwise e= 1. The function (1-kZ/M2) 2 is an apodizing function introduced by CONNES (1961) to correct for effects due to the finite observing time.

In order to re/ate the chromospheric velocity field observed in Ha to the network of K emission, it was necessary to find the position of the spectrograph slit on the Ha and K spectroheliograms. The Ha filtergrams of the reflecting slit jaws provided the necessary link and the slit position is shown in Figure 2. The hairline that is used as a fiducial mark falls at position x = 151. Each K line spectroheliogram was scanned along the position of the slit using the Sacramento Peak microphotometer and after normalization the curves were combined to give the relative K intensity, I K.

4 . R e s u l t s a n d D i s c u s s i o n

i

G(~)

The average power spectrum of the Doppler shifts for all 256 positions is shown in Figure 3. The vertical bars indicate + 2 standard deviations of the averaged spectrum. The ratio of signal to noise lies in the range 10 to 20. The spectrum shows several interesting features and these will be discussed in turn.

0.1"

AVERAGE POWER SPECTRUM

period (sec) I looo 5~o 3~,o 2~o 1~o ,~o

(256 samples)

34 [. ELLIOTT

I "-,k_,_"

o ' ~ ' ' ~ . . . . lb ' '

v(units of 10-ZHz)

Fig. 3. The average power spectrum of Ha Doppler shifts for A), = 0.4/~. The vertical bars indicate 4- 2 standard deviations of the averaged spectrum. The intercept on the G(v) axis is 0.234.

The highest value of G(v) occurs at zero frequency where it reaches 0.234; this implies that there is considerable steady flow or that there are very slow oscillations with a period greater than the total observing time (3210 sec). We shall return to this point later in discussing the variation of Vs with position on the sun.

The most prominent peak of the spectrum has a period of 287 (-t-3) sec and is probably due to excitation by the underlying photospheric velocity field. The peak is asymmetric with a steeper slope on the low frequency side. EVANS et al. (1963) showed that the power spectrum of chromospheric lines changes progressively with increasing height with the resonance peak drifting towards smaller periods. The present value agrees well with values for lines formed near the same level in the atmosphere (NoyES, 1967).

Several peaks are superimposed on the high frequency tail; the most conspicuous have periods of 168 sec, 188 sec and 205 sec. Previously published power spectra of chromospheric lines have shown much broader peaks in this frequency band (NoYEs, 1967); it may be that the peaks are intrinsically narrow and only appear when the

POWER SPECTRA OF Hor DOPPLER SHIFTS 35

resolving power is sufficiently good. In the present observations, the half-width of the spectral window is 0.375 x 10 -3 Hz.

At low frequencies a peak occurs with a period of 900 sec. The peak is quite definite and cannot be due to side lobes of the spectral window of the zero frequency component. The spectral window is the Fourier transform of Connes' apodizing function, and it is plotted in Figure 4. It may be seen that the side lobe at 1.0 x 10 -3 Hz amounts to less than one per cent of the principal peak.

1.0

SPECTRAL WINDOW

0.5

Fig. 4.

0 0.5 1 .O

v(units of lO-3Hz)

The spectral window for an autocorrelation function extending to 54 min. The window is the Fourier transform of Connes' apodizing function.

In order to explore further the significance of the peaks in the average power spectrum, the individual power spectra were plotted as a contour diagram with power as a function of frequency and position on the sun (see Figure 5). The three parameters IK, V~ and IF are also plotted alongside, and we discuss them first. A glance at the V~ and IF curves suggests that downward Vs is often coincident with large values of IF. Comparison of Vs and IK indicates a slight tendency for downward Vs to occur near

36 I. ELLIOTT

50'

I 00 -

Z to

~) 150--

200--

250

Fig. 5.

FAINT

f

f

f

J

<-, /

/

% J

t

(

<

!

~HT DOWN

IK

?

- , . . . . .

km/sec 0 1.0 2.0

I I

>

<

0 5

~-) ~ ~ ' - ~ @ " CONTOUR LEVELS

O O

XX

O

I I 1 0

u(units of 10-3Hz)

Contour diagram of power spectral density as a function of frequency and position on the sun. The slit parameters are shown on the left side of the diagram.

TABLE III

Cross-correlation coefficients between slit parameters

1K(bright) Vs(down) ~F Vs (down) 0.168 1.000 0.326 ~F 0.060 0.326 1.000 ~T 0.116 0.494 0.932

POWER SPECTRA OF Ha DOPPLER SHIFTS 37

bright K emission. The cross-correlation coefficients in Table III confirm these im- pressions. Thus we conclude that downward steady velocity is strongly associated with large values of the fluctuating velocity and is weakly associated with regions of bright K emission.

In their study of supergranulation, SIMON and LEIGHTON (1964) found that the boundaries of the supergranules were associated with: (a) The bright K232 emission network, (b) A network of descending matter observed in Ha and Hfl, termed 'funnels', and (c) The longitudinal magnetic field pattern (1.5-15 gauss).

We may identify (a) with I K. In Ha and Hfl the descending matter was best seen at +0.7 A in Hc~ and +0.4 A in Hfl and it had a magnitude of -(1.2_+0.2)km/see relative to the quiescent background; we therefore identify (b) with downward Vs. The positive correlation between bright I X and downward Vs is consistent with the results of Simon and Leighton, so we regard the downward steady motion as the chromospheric extension of the downward motion in the photosphere at the super- granule boundaries.

We now turn to the contour diagram in Figure 5, and it appears that the low frequency part of the spectrum is associated with bright K emission and downward V s, i.e., with regions above supergranule boundaries. There also seems to be a tend- ency for the medium and high frequencies to be attenuated near these regions. In order to examine these notions more thoroughly, the 256 power spectra were ordered with respect to increasing values of the slit parameters Ix, Vs and IF and were grouped into eight groups, each of 32; the average of each group is shown in Figure 6.

The grouped averages are displayed in three columns: the upper rows of columns (1) and (2) refer to faint It~ and upward Vs and so may be considered characteristic of regions over supergranules; the lower rows refer to bright I x and downward V~ and may be associated with regions over the supergranule boundaries. The third column gives IF with low values at the top and high values at the bottom. We now look for the frequency components appearing in the average power spectrum. The 900 sec low frequency component is largest for row (h), so we associate it with the supergranule boundaries. From an analysis of K line spectra, ORRALL (1966) also found periods in excess of 500 sec where K2 is bright and in plages he found definite oscillations with periods as long as 900 sec. The 287 sec resonance component appears slightly attenuated where the K emission is bright so we assume that it avoids the supergranule boundaries. Although the behavior of the high frequency components is complicated, there seems to be a tendency for them to be least in the lower rows of columns (1) and (2), so we assume that they also favour regions over supergranules.

The relative power of each of the three frequency components is shown in Table IV as a function of I K, Vs and IF. The fractional power is tabulated for frequency bands 1.0 x 10-3 Hz wide centred at frequencies corresponding to periods of 900 sec, 287 sec and 186 sec.

The data given in Table IV are plotted in Figure 7 and support the following conclusions :

3 8 I . E L L I O T T

(1) The low frequency component (900 sec) shows a gradual increase of power with K brightness and also is largest for downward V~ and high values of IF.

(2) The resonance peak (287 sec) is slightly attenuated for bright I K and for large values of V~ (either up or down).

(3) The high frequency power is least where I K is bright and V~ is downwards.

(o)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(1)

o,2 l ' s~o 36o 26o 1~o IK faint16~ sec

~ ;=

o

o.I

o

o.i-

o I

o I

IK bright

(2) ' sbo 3~o 2bo ~o ]60 sec

1

I

i i

~ n - i i i i s lO

v (units of 10-3Hz)

(3)

lo

Fig. 6. Grouped power spectra. The 256 power spectra are ordered with respect to the slit para- meters IK, It's and ~F and averaged in eight groups of 32 each.

POWER SPECTRA OF Ha DOPPLER SHIFTS

TABLE IV

Fractional power in selected frequency bands

39

Low frequency (900 sec) Resonance (287 sec) High frequency (186 sec)

IK Vs ~F Ix Vs ~F IK Vs ~F

a .063 .074 .073 .178 .146 .160 .180 .123 .100 b .066 .062 .064 .210 .179 .185 .145 .156 .113 c .079 .058 .093 .187 .228 .160 .143 .166 .113 d .070 .094 .099 .209 .213 .191 .149 .134 .109 e .089 .057 .068 .185 .197 .200 .156 .188 .166 f .103 .108 .078 .175 .188 .192 .102 .157 .134 g .122 .132 .097 .144 .168 .188 .089 .115 .154 h .171 .158 .153 .143 .133 .151 .109 .070 .135

T o s u m m a r i z e , i t a p p e a r s t h a t s t e a d y d o w n w a r d m o t i o n a n d a 900 sec o s c i l l a t i o n

in t h e c h r o m o s p h e r e a re b o t h a s s o c i a t e d w i t h t h e p h o t o s p h e r i c s u p e r g r a n u l e b o u n d -

ar ies . O n t h e o t h e r h a n d , t h e p o w e r s o f t he 287 sec r e s o n a n c e o sc i l l a t i on a n d o f t h e

h i g h f r e q u e n c y c o m p o n e n t s in t h e r a n g e 150 - 210 sec a re a t t e n u a t e d in s u c h reg ions .

I t is p l a n n e d to e x t e n d t h e ana ly s i s to o t h e r p a r t s o f t h e H e prof i le .

IK v~ ~F

FAINT BRI6HT UP DOWN LOW H IGH

J 0 1 1 1 1 1 1

03

O/

03

0.'

O i i i i i n l a b c d e f g h

, I I I i I i i

I i i I i I

b c d e I g

I I i 1 I i i

I I I I I L I

b c d e l g h

Fig. 7. Variation of fractional power in three frequency bands as a function of 1K, Vs and ~F. Each frequency band has a width of 1.0 • 10 3 Hz and is centred at frequencies corresponding to periods

of 900 sec, 287 sec and 186 sec. The letters (a)-(h) refer to the eight groupings of Figure 6.

40 I. ELLIOTT

Acknowledgements

Most of this work was carried out whilst I was a visiting astronomer at Sacramento Peak, and I am most grateful to Dr J. W. Evans for introducing me to the study of wiggly lines and encouraging my faltering steps. I should like to thank Mr. H. Mauter for his skillful assistance in obtaining the observations. Part of the computing was carried out on the ICT 1909 computer of the Royal Greenwich Observatory and Mrs. D. Hobden 's help was much appreciated. I am also indebted to Dr G. W. Simon for the use of his contouring programme.

This research was supported by the Dublin Institute for Advanced Studies and also by the U.S. Air Force under contract F19628-67-C-0231.

References

CONNES, J.: 1961, Rev. Opt. 40, 45. EVANS, J. W. and MICHARD, R. : 1962, Astrophys. J. 136, 493. EVANS, J. W., MICHARD, R., and SERVAJEAN, R. : 1963, Ann. Astrophys. 26, 368. LEIGHTON, R. B.: 1964, Astrophys. d. 140, 1547. MINKOWSKI, R. : 1942, Astrophys. Y. 96, 306. NOYES, R. W. : 1967, LA. U. Symposium No. 23, p. 293. ORRAL, F. Q. : 1966, Astrophys. J. 143, 917. SIMON, G. W. and LEIGHTON, R. B.: 1964, Astrophys. J. 140, 1120.