An analysis of the observed contrast of solar surface granulation

CHINESE ASTRONOMY AND ASTROPHYSICS

PERGAMON Chinese Astronomy and Astrophysics 25 (2001) 439-445

An Analysis of the Observed Contrast of Solar Surface Granulation t *

RAO Chang-huil JIANG Wen-han ’ LING Ning ’

Jacques M. Beckers2 ‘Institute of Op tics and Electronics, Chinese Academy of Sciences, Chengdu 610209

2National Solar Observatory, U.S.A.

Abstract The contrast of the solar surface granulation detected in the focal

plane of the observing system as well as its relations with the aperture of the

observing system, the coherent length of atmospheric turbulence and the sensi-

tivity of the detecting system are analyzed. The results of numerical calculation

of the granulation contrast as functions of aperture, coherent length of atmo-

spheric turbulence and sensitivity of the detecting system are presented. Results

of a related observation are also given.

Key words: solar surface granulation-contrast-atmospheric coherent length-

optical transfer function

1. INTRODUCTION

In the study of solar magnetic activity it is necessary to very accurately measure the so-

lar magnetic vector by means of simultaneous observations of solar surface granulation in

differently polarized spectral lines. The latter is an extended object with observed low

contrast and evolving features. Due to the wave-front disturbance caused by atmospheric

turbulence and mechanical vibration of the telescope and to the influence of the aperture

and optical quality of the telescope, the contrast of granulation observed in the focal plane

of the observing system and the resolution of the imaging system are impaired. The tech-

nique of self-adaptive optics can eliminate the phase disturbance caused by the atmospheric

t Supported by National Natural Science Foundation Received 2000-07-03; revised version 2000-10-16

* A translation of Acta Astron. Sin. Vo1.42, No. 2, pp. 134-139, 2001

0275-1062/01/$-see front matter @ 2001 Elsevier Science B. V. All rights reserved. PII: SO275-1062(01)00097-2

440 RAO Chang-hui et al. / Chinese Astronomy and Astrophysics 25 (2001) 439-445

turbulencel’~21 and provides an effective route for the solution of this problem. In the explo-

ration of the imaging of solar surface granulation (generally by means of CCD), the contrast

is still restricted by the spatial optical transfer function. Besides, the accuracy of exploration

of the wave-front disturbance of atmospheric turbulence with self-adaptive optical system

is limited by the contrast of granulation. Hence the necessity of analyzing the observed

contrast of solar surface granulation.

The relation between the observed contrast of solar surface granulation and the coherent

length of atmospheric turbulence was analyzed in Refs. [3] and [4]. But the analysis was

made only for observing systems with comparatively large apertures (40 cm - 1 m) and

for comparatively small coherent lengths of atmospheric turbulence (smaller than 10cm).

Moreover, the influence of the spatial optical transfer function of the detector was not taken

into consideration. In this paper, not only are the relations among the observed contrast

of solar surface granulation and the observing aperture as well as the coherent length of

atmospheric turbulence analyzed, but also considered is the influence of the spatial optical

transfer function of the exploring system.

2. THEORETICAL BASIS

2.1 The Contrast of Solar Surface Granulation in the Focal Plane of the Ob-

serving System

Let the intensity distribution of the image of solar surface granulation under ideal condi-

tions be O(r), the point spread function (PSF) of the whole optical path including telescope

and atmosphere be S(r, t), and the intensity distribution of the observed granulation in the

focal plane of the observing system be I(r,t), then we have

I(r, t) = O(r) * S(r, t) ) (1) where r is the spatial two-dimensional coordinate, and * represents convolution.

According to Parseval’s theorem, the root-mean square of observed intensity of granu-

lation in the focal plane of the observing system (Al);,,,,, isl’l:

W):.,.,. = /(lf(f)12)df = / 1~(f)12(l~(f)12)~f, (2)

where X(f) represents the 2-D Fourier transform of X(r), f = If] is the spatial frequency

and ” 0” expresses the overall mean.

If the solar granulation is an isotropic phenomenon, then according to Ref. [l] the spatial

frequency spectrum of granules is:

I&]2 = We(f) = 0.019(1 - 0.68e-10fz)e-‘~ssf,

where f is in units of cycles/arcsec.

(3)

From statistical optics it is known that S(f) i.e. the optical transfer function (OTF)

of atmospheric turbulence and observing system, is given by

em”) = T2(f).

RAO Chang-hui et al. / Chinese Astronomy and Astrophysics 25 (2001) 439-445 441

Hence, from Eqs.(2), (3) and (4) the root-mean square value of the intensity contrast of

observed granulation in the focal plane of the observing system is

(A%.,.,. = [27r 1 .W~(f)r~(f)@]~‘~ . (5)

Here r(f) is the product of the optical transfer function of the observing telescope 70(f)

and the short exposure optical transfer function of atmospheric turbulence rs~(f):

T(f)‘= ~O(.f)%‘E(f), (6)

where

ro(f) = i{arccos 2 - g[l - (:)‘I’/‘},

vE(f) = ezp { -3.44($5/3($5~3[1 - ,@3]} .

(7)

In the above expressions, D is the aperture of the observing system, TO is the coherent length

of atmospheric turbulence, and the unit of f is cycles/rad.

It follows from the above deduction that the intensity contrast of the granulation in

the focal plane is a function of the aperture D and the atmospheric coherent length TO.

2.2 The Detected Contrast of Solar Surface Granulation

In the above an analysis of the intensity contrast of the granulation observed in the

focal plane of the observing system was made. In practice, when solar surface granulation is

imaged (which we suppose to be always made by a CCD), the measured contrast is limited

by the size of the pixels of the detector (the detector’s spatial optical transfer function). If

we suppose all the pixels have the same degree of responsiveness, then we need only consider

the influence of the size of the pixels. Then the root-mean-square value of the image contrast

of the solar surface granulation measured by the detector is

(A%.,.,. = [27r J fw0(f)?2(f)Pizel(f)df11’2, (9)

where Pixel(f) is the normalized value of the square of module of the Fourier transform of

the pixel window size function.

In the commonly used imaging CCDs, the pixels are generally square in shape. Let us

assume that the size of the pixel is d x d, then the window size function is

pizel(z, Y> = rect[~/(d/2)]rect[y/(d/2)] , (10) where rect(z) is the function of square aperture: 1 for 121 < 1, 0 otherwise.

For the sake of simplicity and convenience of analysis, we use the function of circular

aperture to approximate the function of square aperture. So we take

pixel(r) = circ[r/(d/2)],

where &c(r) = 1 for 0 < r < 1, and 0 otherwise; and T = dm.

(11)


For consistency with the units in Eq. (9), we need to change the unit of r in the window

size function from m to arcsec. We have

pixel(r) = circ[r/(S/2)] ,

where S = d/fi, and fi is the convergent focal distance.

Hence, we get,

(12)

Pi=l(.f) = [2J1(7rfS)/(7rfS)]2, (13)

where 51(x) is the Bessel function of the 1st kind.

Substituting Eqs.(3), (7), (8) and (13) into Eq.(9), we get the final expression for the

observed contrast of solar granulation. Obviously, besides the aperture of system D and the

coherent length of atmospheric turbulence TO, the contrast is dependent of the sensitivity of

the detecting system.

3. NUMERICAL COMPUTATION AND ANALYSIS

In order to estimate the influence of the aperture and sensitivity of the observing system

and of the coherent length of atmospheric turbulence on the observed contrast of solar

granulation, we made numerical calculations separately for the contrast in the focal plane

of the observing system and for the contrast measured by the detector.

Fig.1 presents the contrast in the focal plane (a) as a function of the system aperture D, for different values of the coherent length of atmospheric turbulence TO up to the diffraction

limit (D.L.) of no turbulence, and (b) as a function of r-0 for different values of D(system

apertures). The following may be inferred from the figure:

20 40 60 80 Diameter(cm)

100 0.

5 10 15 20 25

r,, (cm)

Fig. 1 The contrast of solar surface granulation in the focal plane of the observing system (b) as

a function of the aperture D for different coherent lengths of atmospheric turbulence TO , and (a) as a function of TO for different values of D

(1) For a given aperture, the granulation contrast increases with with increasing TO or

decreasing turbulence.

(2) For the same TO, the contrast increases with increasing aperture, markedly for small

apertures, and inconspicuously for large ones.


r ,) (cm) r. (cm)

Fig. 2 The contrast of solar surface granulation in the focal plane of the observing system as a

function of ~0 for different values of D

Fig. 3 A short exposure image of solar surface granulation taken with the Vacuum Tower

Telescope of U.S. National Solar Observatory on March 30, 2000

(3) When the aperture of the observing system is comparatively small (e.g., D =

80 mm), the contrast of granulation is mainly determined by the aperture.

Fig.2 illustrates the measured contrast of solar granulation as a function of TO for

different sensitivities of the detector for two apertures, D = 400mm (a) and D = 80 mm

(b). The figure shows the detected contrast to vary with the sensitivity of the detecting

system. The higher the sensitivity (the smaller the quantity S), the greater the contrast.

4. EXPERIMENTAL RESULTS

On March 30, 2000 we carried out observation of solar surface granulation with the Vacuum

Tower Telescope of the U.S. National Solar Observatory. This Observatory is sponsored by

the U.S. National Science Foundation and operated by the Association of Universities for

Research in Astronomy. The aperture of the telescope is 760 mm. Through the main optical


path of the telescope and the corresponding guide and imaging optical path, the solar image

was formed in the focal plane of the CCD detector. The pattern of solar granulation was

then stored by a real time image sampling system. Our adopted CCD exposure time is

about 8ms, the A/D figure has 16 digits, the readout time of data is approximately 1.2s

and the sensitivity of the detecting system is O.O9024arcsec/pixel.

Fig.3 presents an image of the observed solar surface granulation. Fig.4 displays the

contrast of solar surface granulation pattern yielded by 100 consecutive photographs taken

in 120 seconds. Fig.5 illustrates the time series of coherent length of atmospheric turbulence

corresponding to Fig.4.

7.5 12 3 h g Y E 7 Gl

z 2 6.5

2 f

6 g’0

g E ‘S 5.5 B

-=9 $& :: 5

.g

2 4.5 Average value:5.987% p. erage value:9,960cm

*

28 0” 4

0 20 40 60 80 100 120 s7 -t 0 20 40 Tint-$) 80 100 120

Time(s)

Fig. 4 A short exposure image of solar

surface granulation taken with the

Vacuum Tower Telescope of U.S. National

Solar Observatory on March 30, 2000

Fig. 5 The time series of atmospheric coherent

length corresponding to Fig.4

As may be seen from these figures, the average value of the contrast of granulation is

about 5.987% and the mean coherent length of atmospheric turbulence is 9.96cm.

Some new methods have been developed in recent years for the measurement of the

coherent length of atmospheric turbulence. From the error variance of the overall tilt of

wave front calculated with the algorithm of tracking correlation, the coherent length may

be computed, and this is a rather good method. Because the coherent length is a statistical

average, the value obtained with the statistical value of the time series of the overall tilt

of wave front is appropriate. In Ref.[5], the coherent length calculated from the variance

of error signal of the overall tilt is TO = 9.6crn. This agrees rather well with the value

TO = 9.96cm obtained from the contrast of solar surface granulation, as stated above.

5. CONCLUSIONS

The detected contrast of solar surface granulation by a detector is a function of the aper-

ture of the observing system D, the coherent length of atmospheric turbulence TO and the

sensitivity of the detector. For the same D, the contrast increases with increasing TO. For

the same TO, the contrast increases with D, very strongly if D is small, very weakly if D is

large. The detected contrast is also dependent on the sensitivity of the detecting system;

the higher the sensitivity system, the greater the contrast.


ACKNOWLEDGEMENTS In this work much assistance has been given by Colleagues

JIANG Ling-tao, XU Bing, JIANG Ping, REN Guo-qiang, ZHOU Wei-chao and TAO Hui-

ming of Institute of Optics and Electronics of Chinese Academy of Sciences, FANG Cheng,

DING Ming-de and GAO Xiu-fa of Astronomy Department of Nanjing University, as well

as Steven Hegwer and Tim Henry of U.S. National Solar Observatory. To all of them, our

sincere thanks.

References

1 Wenhan Jiang, Mingquan Li, Guomao Tang et al., Opt. Eng., 1995, 34, 15

2 Wenhan Jiang, Mingquan Li, Guomao Tang et al., SPIE Proc.,l996, 2828, 322

3 Ricort G., Aime C., Roddier C. et al., Solar Phys., 1981, 69, 223

4 Ricort G., Borgnino J., Aime C., Solar Phys., 1982, 75, 377

5 RAO Chang-hui, JIANG Wen-han, LING Ning et al., Acta Astron. Sin., 2001, 42, 329

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An analysis of the observed contrast of solar surface granulation