Evaluation of the solar disk sextant concept

Hong-Yee Chiu

In this paper we evaluate the viability of the solar disk sextant concept and derive the optimum parametersrequired to carry out the solar variability studies which are the mission objectives. The experimental envi-ronment is first discussed, followed by the application of the finite Fourier transform definition (FFTD) tothe detector array data. The requirements on the optical system are studied next. A computer programwas carried out simulating solar edge data and FFTD. From this study we conclude that the required accu-racy of measurement may be reached using currently available detector array technology, a focal ratio of theoptical system in excess of 90, and an entrance aperture of 22 cm. The guidance error must be small enoughto require no more than a correction rate of 0.1 sec of arc/sec. All these conditions are well within currenttechnology.

1. Introduction

The sun is the source of energy to terrestrial life. Ona long-term basis, any variations in the sun's energyoutput (luminosity) will affect the climate structure ofthe earth. As a consequence of changes in solar lumi-nosity the solar diameter will change. From the stellarstructure point of view, the sun is in gravitationalequilibrium with respect to thermodynamic structuresof the interior (hydrostatic equilibrium). In view of theviolent events often observed on the surface of the sunand the lack of a complete understanding of the energyproducing mechanisms due to the failure of detectionof solar neutrinos, it is not totally unexpected that theluminosity of the sun might change resulting in a changeof the solar diameter on time scales shorter than thatfor stellar evolution. Indeed, from analyses of past solareclipses spanning several centuries, changes of the orderof 0.04% have been reported.'

In a separate article2 an experiment has been pro-posed to monitor the diameter of the sun by means ofa space based instrument. The motivation for movingthe experiment to space is to avoid the turbulent andrefractive atmosphere. Various aspects of the experi-ment are being discussed in a separate paper.3 In thispaper we will concentrate-chiefly on problems associ-ated with the data reduction. Indeed, once solar di-ameter data are available on a long-term basis, with

The author is with NASA Goddard Space Flight Center, Laboratoryfor Planetary Atmospheres, Solar Radiation Office, Greenbelt,Maryland 20771.

Received 26 September 1983.

measurements made in closely spaced time domains, thestudy of oscillations will become an important part ofthe data reduction process. However, before the os-cillation problems may be studied, we must have a firmidea as to how useful individual data points may beobtained, and this is the purpose of the presentpaper.

Previous work (where solar oblateness was the pri-mary concern) used electromechanical devices to obtainsolar data. For example, in a recent experiment 4 de-tector arrays were used in conjunction with mechanicalmovements. In our case, weight, complexity of in-strument, as well as versatility of data for subsequentanalysis lead us to the use of detector arrays without anymoving parts (except for the guidance motion of theoptics) as the prime detection device.

II. Experimental Environment

The target accuracy for the proposed experiment isARo/Ro = 10-5 (where Ro is the solar radius) or better.In terms of apparent angular diameter, the accuracy is-0.02 sec of arc. Since experimental verification gen-erally requires a 3a confidence level, the noise limit ofthe proposed experiment must be better than 0.006 secof arc.

The earth's orbital motion around the sun causes theapparent angular diameter of the sun to change yearlyby 3.34%, with the maximum occurring on 3 Jan. and theminimum on 5 July. The properties of this modulationare given in Table I. Additionally the instrument-sundistance will be modulated by approximately the earth'sdiameter once a day on a balloon-borne instrument, oronce per orbit on a low-altitude satellite. The modu-lation is larger (and also once per day) for a satellite ingeosynchronous orbit. The properties of these modu-lations are given in Table II.

1226 APPLIED OPTICS / Vol. 23, No. 8 / 15 April 1984

Table 1. Yearly Modulation of the Earth-Sun Distance due to the Earth's Orbital Motion

Time ui - d (sec') Oo dto

3 Jan. 0.983 (min) 0 1955 08 Apr. 1.00 -3.36 X 10- 9/sec (max) 1920 -6.4 X 10- 6/sec (max)6 July 1.017 (max) 0 1891 016 Oct. 0.997 3.3 X 10- 9/sec (max) 1928 6.4 X 10- 6/sec (max)

u is earth to sun distance measured in astronomical units. 00 is the apparent diameter of the sun (measured in seconds of arc):

do 1 dud _ dSo dt u dt

Table II. Earth-Sun Distance Modulation due to the Instrument Motionwith Respect to the Center of Mass of Earth

Low-earth GeosynchronousBalloon satellite satellite

Period 1 day -90 min 1 dayAmplitude 1-2 Ro t2 Ro 10 RoChange in 00 = 4.3 - 8.6 X 8.6 X i0-5 Oo 4.3 X 0-4 00

i0-5 00Ro = earth radius = 6.4 X 106 cm

function is implicitly contained within the formulationso that no actual curve matching takes place.

Let G(r) be the observed solar intensity profile de-scribing the fading of the solar edge into interplanetaryspace. G (r) contains, in addition to the actual fadingof the solar surface into space according to conditionsgoverning the surface structure (such as temperature,gravity, etc.), the instrument profile describing the pointspread function of the instrument. A Fourier coeffi-cient F(G;q,a) is formed:

I11. Physical Definition of the Solar Edge

The boundary of the sun is not sharp; the brightnessat the photosphere drops approximately exponentiallyinto interplanetary space, with an e-folding distance of-70 km (in constrast to the target accuracy of 4 km). Inaddition, the surface of the sun is filled with variousfeatures, such as sunspots, granulation, corona, faculae,and flares, producing localized luminous zones whichmay extend far above the solar limb. Some of thesefeatures are short-lived, whereas others have lifetimesranging from days to weeks. However, all phenomenaother than sunspots and granulation are associated withline emission. By limiting observations in the contin-uum region, far away from the line emission phenome-na, we need only concern ourselves with active regions(faculae and spots) and granulation; we disregard themeasurements of a diameter associated with an activeregion, and we assume that the granulation effect av-erages out. Indeed, this is the approach taken in pre-vious solar oblateness work.5

Because of the extension of the solar surface, a sharpedge definition is possible only if the edge of the sun canbe described by a time-invariant function. We assumethat such a time-invariant function exists in the con-tinuum as described above. The task of determiningthe position of the solar edge is the same as that ofmatching the position of the time-invariant function tothe observed solar edge profile; the parameter that de-termines the position of the time-invariant functiondefines the solar edge. Thus, it is not necessary thatthis parameter coincide exactly with the usual definitionof the solar edge. The solar edge defined in this waymay be referred to as the instrumentally defined solaredge. To facilitate the matching of the time-invariantfunction to the observed edge profile of the sun, Steb-bins6 and Hill et al. 7 developed the so-called finiteFourier transform definition (FFTD) to define the edgeof the sun. In their formulation, the time-invariant

51/2F(G;qa) = G(q + a sinlrs) cos(2lrs)ds, (1)

where a is a constant, q is a parametric variable, and sis the integration variable. The edge of the sun is de-fined to be the location q such that F(G;q,a) van-ishes:

F(G;q,a) = 0. (2)

Extensive discussions on the behavior of F and the va-lidity of this definition of the solar edge have been madeby Hill, Stebbins, 6 7 Brown,8 Hill et al.,9 etc. One of themany advantages of FFTD is that the function F can bedirectly determined by experiment using proper scan-ning technique. From another point of view, a greatadvantage of FFTD is that one does not need to havedetailed knowledge of the point spread function in theoptical system (as long as it is spatially invariant) or thelimb darkening function which describes the solar in-tensity profile near the edge. However, the solar edgedefined in this way is dependent on the optical systemchosen, and in order to measure the solar diameter overlong periods of time, the instrument must have long lifeexpectancy.

FFTD may be extended so that the functions sine andcosine are replaced by another periodic, orthonormalset of functions. The proper formulation is

F(G,q,a) = S G[q,a0(s)]0'(2s)w(s)ds, (3)

where is a member of an orthnormal set of functionsand w (s) is a weighing function.

IV. Optical System Requirements

As stated earlier, in our proposed experiment detectorarrays will be used to record solar edge data withoutmechanical scanning. The experimental environmentavailable in space makes the use of complicated me-chanical scanning systems undesirable, and self-scan-ning photodetector arrays will be used instead. Here,

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we are confronted with two chief problems, namely,calibration and the finite resolution of the detector el-ements.

While the problem of calibration must be resolvedexperimentally when the instrument becomes available,the problem of finite resolution of photodetector arraysmust be solved first, since it is intimately associatedwith system design. As described previously, theproblem of determining the position of the solar edgeis the same as that of matching a predetermined, tem-plate profile function to the observed profile function.FFTD only eliminates the need for an explicit expres-sion of this template profile function. In order that theedge determination be accurate, it is necessary to havean accurate observed edge profile function, includingthe instrument profile function. It turns out that thereis a minimum matching condition between optical pa-rameters and the photodetector arrays, as we will dis-cuss below.

The intensity of the surface of the sun falls approxi-mately as exp - z/H, where H is the scale height and zis the height above the surface. H is -70 km. In termsof angular dimensions, the intensity near the edge of thesun falls off as exp - O/O, where 0 is the angle awayfrom the surface and 0, is the angular scale height, -0.1sec of arc. Assuming a storage capacity of 106 photo-electrons per detector element, the maximum signal-to-noise ratio is -103. Over a range of 6 scale heightsthe intensity falls off by a factor of e- 6 = 2.5 X 10-3.

Thus, the maximum range of coverage of the solar edgeis at most 6 scale heights. The number of detector el-ements that fits over the observable range of the edgedetermines the accuracy of the observed profile andhence the accuracy of measurement. For example, ifthis observable range is covered by 6 elements, eachextending over an angular dimension of 0.2 sec of arc,in order to measure the angular diameter of the sun to0.001 sec of arc, the FFTD method must yield a positionaccuracy to within 1/200 of the element width. A cor-responding accuracy must be achieved in the registra-tion of the true edge profile of the sun. It is the purposeof this section to obtain an optimum match betweendetector spacing and optical resolution of the system.

Generally speaking, improvements of data resolutionmay be achieved by decreasing the spacing betweendetector elements. For a given optical configurationin 2-D arrays, the signal-to-noise ratio is proportionalto the detector dimension D (due to reduced number ofphotoelectrons), while the signal-to-noise ratio onlydecreases as D1 /2 in the 1-D case. The current indus-trial standard for detector spacing in 1-D arrays is 10Am.

Detector arrays are localized area integrators; eachindividual element integrates the radiation falling onits small albeit finite area. Let G(r) be the intensityprofile of the solar disk along its radius vector. Theneach element senses an integrated portion of G (r) suchthat, gk, the response of the kth detector, is given by

gh = G(r)h(r)dr. (4)

where h (r) includes calibration factor and instrumentprofile function.

The observed solar edge is a compounded effect of theactual solar edge and the instrument profile function.The observed solar edge is therefore a convolution in-tegral, G(0):

G(O) = f 1 .(O')S(O')P(0,0')dO'. (5)

While ,u(O) is the limb darkening function and S(O) isthe solar edge profile function, P(0,O') is the pointspread function of the instrument. G (0) is measuredby detector arrays in incremental steps, separated byan angular step 30. The limb darkening function Au(O')describes the brightness variation of the solar disk alonga radius vector and has been extensively discussedelsewhere.10 If the instrument profile were a 6 function(i.e., optical system with an infinite resolution),

P(O,O') = 6(0,0'). (6)

Then G(O) = u(0)S(0). In reality, in small instru-ments the effect of the point spread function usuallyoverwhelms that of S(O), so that S(O) can be takenliterally as a step function. The problem of determiningthe solar edge then reduces to determining G(0). Anaccurate determination of G (0) is possible if a largeenough sample of points is taken. We are next goingto determine the minimum number of sample pointsrequired to achieve our target accuracy.

This was done by means of a rather extensive recon-naissance computer simulation. A step function S(O)is used to simulate the solar edge, convoluted by aGaussian function P(AO') = exp - (0 - O') 2 /a2 , wherea is a parameter (a image size for a point source).The limb darkening function (a slowly varying function),(O) is taken to be unity. After the solar profile func-tion G(0) is generated,

G(O) = r P(0,0')u(0')S(O')dO', (7)

G (0) is integrated into discrete blocks simulating theresponses of the elements, resulting in a discrete set ofsimulated data points:

G(s) 40i+' G(0)dO,oi

Oi = , + i aO,

(8)

(9)

where 0 c is a constant. A functional fit on 0) wascarried out to reproduce an experimentally observedprofile G (b)(0). We found that, among the followingfitting functions: trigonometric, Tchebycheff, andLegendre polynomials, the Legendre polynomialsyielded the best fit. Finally, the two FFTD integralsF(G,q,a) [using Eq. (7) from G(O)] and F(b)(G,q,a)[using the experimentally determined G(b)(O)] arecompared. We find that the error AF/F defined as

AF F - F(b)F F

(10)

is AF/F <5 0.01 when the range of the fit is taken to beA3a, and the minimum number of detector elementsover the point spread range a is 6. Differences in zerocrossing points of the FFTD functions F and F(b) are

1228 APPLIED OPTICS / Vol. 23, No. 8 / 15 April 1984

<10-2 if a/b0 > 6 (i.e., there are more than 6 elementsover a range a). Let the focal length of the telescope beF, the detector spacing be 1, and the entrance aperturebe A, then

a = 1.22(X/A) rad, (11)

- = 60. (12)F

In the proposed experiment, the optical configurationmust fulfill the condition

a X F- = 1.22 - > 6. (13)60 Al1

Using X = 5.5 X 10-5 cm, I = 10-3 cm (10 Am), we findF/A > 90. F/A is the same as the focal ratio. In prin-ciple, an optical system of any aperture can be usedprovided the focal ratio is >90. However, with de-creasing focal length, the point spread function becomesextended, and eventually the solar edge information islost among the point spread function. Ideally, if thescale height angle is O3, the resolution of the telescopeshould also be 0, and the focal length of the telescopesystem should be such that FOs covers a/5O elements.This would require an entrance aperture of at least 102cm (40 in.). Next to this ideal situation we should haveFO, 1. Thus

1 10-3F =-= = 2 X 103 cm,

Os 5 X 10-7

(a/60) = O, = 1.22 A/A,yielding A 22 cm. The focal length f must be >2000cm. Such an optical system, if it is to be deployed inspace experiments, must be a compound optical system.In a separate paper3 we have evaluated the effects ofdistortions on the accuracy of measurement and con-clude that the system requirements are within thecurrent state of the art. The effect of space variationof the point spread function (which cannot be neglectedin a compound design) is going to be studied next.

V. Image Motion and Rectification

We have assumed that a perfect solar image is formedat the detector plane without image motion. In practicethis is not the case.

At the desired accuracy of measurement, it should beassumed that image motion blur always occurs. As-suming a guidance drift of 0.1 sec of arc/sec and an ex-posure time of 1/30 sec, the average image drift is 0.003sec of arc per frame, which is within the range of thedesired accuracy.

A guidance error that requires an average correctionin the amount 0.1 sec of arc/sec is possible if the systemis not subject to vibration motion. Otherwise, the datamay require substantial rectification before they can beused. Effects of vibration and image motion are beingstudied.

VI. Discussion

In this paper we have performed an analysis of theSDS concept. We found that the experiment can bechecked for its sensitivity using internal consistencyrequirements. We have found that a relatively smallaperture instrument (-22 cm) is adequate to yield thedesired accuracy. This makes it feasible to perform theexperiment in space with modest resources. Discretedetector arrays can be used to replace mechanicalscanning stages, thus simplifying design and other re-quirements on collection and preliminary handling ofdata.

In this work the limb darkening function M(0) istaken to be unity. In practice Mu(0) varies with time andthis variation may present the most severe problem.However, in this paper our stress is the definition of theinstrument, and a constant M(0), although a poor ap-proximation, adequately allows us to obtain a minimumrequirement for the instrument. The problem of spa-tial and temporal variation of /L(O) will be studied as theproject progresses.

References1. S. Sofia, D. W. Dunham, J. B. Dunham, and A. D. Fiala, Nature

(London) 304, 522 (1983).2. S. Sofia, H.-Y. Chiu, E. Maier, K. H. Schatten, P. Minott and A.

S. Endal, Appl. Opt. 23, 1235 (1984).3. H.-Y.Chiu, E. Maier, K. H. Schatten, and S. Sofia, Appl. Opt. 23,

1230 (1984).4. T. M. Brown, D. F. Elmore, L. Lacey, and H. Hull, Appl. Opt. 21,

3588 (1982).5. H. A. Hill and R. T. Stebbins, Astrophys. J. 200, 471 (1975).6. R. T. Stebbins, Ph.D. Thesis, "An Observational Investigation

of the Solar Oblateness," U. Colorado, available from NTIS(1975).

7. H. A. Hill, R. T. Stebbins, and J. R. Oleson, Astrophys. J. 200,484(1975).

8. T. M. Brown, Astron. Astrophys. 116, 260 (1982).9. H. A. Hill, P. T. Ballard, and S. D. Bryan, J. Opt. Soc. Am. 69,978

(1979).10. See, for example, G. A. Chapman and D. P. Klabunde, Astrophys.

J. 261, 387 (1982).

We would like to thank S. Sofia, E. Maier, and K.Schatten for helpful discussion and comments.

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