Lunar profiles determined from annular solar eclipses of 1962 and 1963

mAavs 5, 334-359 (1966)

Lunar Profiles Determined from Annular Solar Eclipses of 1962 and 1963 ~ D. C A R S O N , ~ M . D A V I D S O N , * C. L. G O U 1 ) A S , *+ Z. KOPAL,*:~ .XXl) L. G. S T O D D A R D t

* Department'of Ast~vnomy, ~"niver~ity of Manche.ster, England Lunar-Planetary Branch, Basic Scie~mes Laborotory, Lockheed Ccdifor~d(t Company, Burbank, Califarnia, and ~Mathematics Research Labor,dory, Boeing Scict~lific Resc~lrch Laboratories, Seattle, Washinfflon Received October 26, 1965 The present paper contains a discussion of the techniques of measurements and the result of reductions of the photographs of the annular solar eclipses observed on July 31, 1962 in Senegal, West Africa, and January 25, 1963 in the Union of South Africa, by the Lockheed-Manchester Eclipse Expeditions, with the aim of improving our knowledge of the form of the Moon as seen in projection against the disc of the Sun. The lunar profile deduced from the 1962 eclipse plates was found to be insufficiently well defined due to the effects of irradiatien. The 1963 eclipse plates were, however, largely free from this defect, and their analysis led to satisfactory results. The outcome of the reductions revealed that the deviations of the lunar profile from a circle were of the form +0'.'54 sin 2/~ -- 0'.'16 cos 23 + 0':27 sin 33 -- 0':05 cos 32 -- 0':16 sin 43 -b 0':11 cos 4/~, where 3 denotes the lunar position angle measured from the eastern part of the equator in the counterclockwise direction. These deviations were compared with the preceding lunar limb determination, and found to be in satisfactory agreement; in particular, they agree within the limits of observational errors with the profiles obtained by Potter and Watts. The elliptical component of the profile is found to have its major axis inclined by approximately 37 ° to the hmar axis of rotation--a value which is almost identical with that given 1)y Watts. of t h e E a r t h , b y visual or p h o t o g r a p h i c m e t h o d s , a n d is r e q u i r e d to m e e t a n u m b e r I t is s c a r c e l y n e c e s s a r y to u n d e r l i n e t h e of p u r e l y t e r r e s t r i a l needs. need for i m p r o v i n g o u r k n o w l e d g e of t h e T h u s , in a s t r o n o m e t r i e o b s e r v a t i o n s of t h e g l o b a l s h a p e of t h e M o o n a t t h e p r e s e n t t i m e , M o o n , t h e p o s i t i o n of its c e n t e r m u s t be on t h e eve of t h e efforts t o l a n d m e n on t h e d e t e r n f i n e d f r o m o b s e r v e d t r a n s i t s (or altisurface of o u r satellite, w h e n spacecraft t u d e s ) of its l i m b ; a n d a t r a n s f e r f r o m t h e l a u n c h e d f r o m t h e E a r t h are to o p e r a t e soon l i m b to t h e c e n t e r ( i n v o l v e d in a n y c o m p a r i in close p r o x i m i t y of t h e l u n a r surface. U n t i l son of t h e o b s e r v e d p o s i t i o n s of t h e M o o n these f e a t s will h a v e b e e n a c c o m p l i s h e d , with the ephemeris) presupposes a knowlhowever, all i n f o r m a t i o n we possess on t h e edge of t h e l i m b ' s g e o m e t r y . Secondly, w h e n s h a p e of t h e M o o n m u s t be o b t a i n e d b y c o n s t r u c t i n g t h e m a p s of t h e m a r g i n a l zones a s t r o n o m i c a l o b s e r v a t i o n s from t h e d i s t a n c e of t h e M o o n ( H a y n , 1907; W e i m e r , 1952; N e f e d j e v , 1957; W a t t s , 1963) we h a v e to 1 Investigation supported by Contract AF 61 (052)524 between the Terrestrial Sciences Laboratory, refer t h e h e i g h t s of t h e i n d i v i d u a l p o i n t s to Air Force Cambridge Research Laboratories and the a m e a n l i m b defined for all sections of t h e Department of Astronomy, University of Man- l i m b a n d all angles of l i b r a t i o n . T h i r d , inchester; and Contract AF 19(628)-4162 between the v e s t i g a t i o n s of t h e axial r o t a t i o n of t h e M o o n same agency and the Lockheed Califi),'nia Company. a n d of its p h y s i c a l lit)ration are b a s e d on t h e 334 I. INTRODUCTION

L U N A R PROFILES FROM SOLAR ECLIPSES OF 1 9 6 2 AND 1 9 6 3

heliometric measurements of apparent motion of lunar points relative to the limb, and require again a knowledge of its general form. Last but not least, a knowledge of the exact figure of the Moon is very important for theories of the origin and evolution of our satellite. A complete survey of the methods by which such a knowledge can be obtained is wholly beyond the scope of the present communication. Our aim will, nmre modestly, be to present certain new contributions to the problem of the exact form of the Moon's limb, as obtained from the observations of the annular solar eclipses of 1962 and 1963, and to compare them with the outcome of previous work. A photographic determination of the form of the Moon's limb at any particular libration is, in principle, possible from plates of sufficiently high quality at any phase by measuring with sufficient accuracy the shape of its sunlit limb. However, on account of the phase effect, only one (i.e., the illuminated) half of the lunar circumference can be measured at any time--including the time of full-moon, at which phase effect becomes mininmm, but is still sufficiently large to invalidate the measurements of more than one-half of the entire circumference. Due to the geometry of the problem, no sunlit full moon is ever completely full, and their actual defect may fluctuate from month to month. Almost complete full moons of a phase approaching zero can occur only when the Moon enters the shadow cone of the Earth; and zero phase is attained at the midpoint of a central lunar eclipse, when the low residual brightness of the lunar face (due to its illumination through the aureola of the terrestrial atmosphere) would necessitate relatively long exposures for photographic registration. It is evident that the only times when the apparent lunar disk is free from a phase effect and the circunfference of its limb can, therefore, be measured over arcs more than 180 ° in length occur when the centers of the Sun, Earth, and the Moon lie on the same line. Central eclipses of the Moon represent one class of such opportunities; the other are, of course, central eclipses of the Sun. If such eclipses are total, the dark limb of tl:e

335

Moon can be measured only on the background of the solar corona; the requisite exposure times are again relatively long, and (similarly as in the case of total eclipses of the Moon) determinations of the form of the lm~b would require precise angular measurements over distances up to half a degree. If, however, the eclipse of the Sun becomes annular, a much more favorable situation obtains, for not only does the limb of the Moon continue to be free fi'om any phase effect, but the shape of the entire circumference of the Moon can be measured differentially with respect to the adjacent limb of the Sun, which should not deviate significantly from a circle. As a rotating mass of gas the Sun must, to be sure, be slightly flattened at the poles; but the actual difference between the equatorial and polar sere;axes due to axial rotation should be equal to only

~(¢o2/27rapm)A2 = 2.10 )< 10-5 ial terms of the apparent solar radius (for G = 6.668 )< 10- s em3/gm see s, p,, --- 1.41 gm/cm3; ~ = 2.85 × 10-6 see -1, and As = 1.02 for the latest models of the interval structure of the Sun), which for the average value of ro = 900" corresponds to an angular difference of less than 0'.'02. If quantities of this order can be regarded as ignorable (they are within the diffraction limits of a 300-inch aperture for visible light, and have not so far been measured), the limb of the Sun can be regarded as a conveniently situated reference circle, and all measurements carried out differentially with respect to it. This should constitute an inestimable advantage, for not only are the exposure times for photographs of the Moon on the bright background of the Sun thousand times shorter than those that would be required to record the image of the totally eclipsed Moon on the dark background of the sky, but all troublesome atmospheric as well as inslrumental effects (such as air turbulence, off-axis field aberrations, emulsion shifts, and differential refraction, etc.) which might seriously impair the precision of astrometric measures extending over half a degree, enter only differentially (inasmuch as they affect the adjacent porlicl:s ~f the limbs of the Sun and the Moon

336

n. CARSON ET AL.

by different amounts) and affect the results the less, the smaller the difference between the angular size of the Sun and the Moon at the time of the eclipse. With these considerations which emphasize an overwhelming advantage of annular eclipses as a tool for the studies of lunar figure, one of us (Z. K.), then serving as consultant to the Lunar-Planetary Branch of the Basic Sciences Laboratory, Lockheed California Corp., suggested to Dr. L. G. Stoddard, then head of the Lunar-Planetary Branch (and now head of the Astronomical Sciences Laboratory) that suitably equipped expeditions be sent out to Africa to observe the annular eclipses of the Sun on July 31, 1962 and January 25, 1963. This project was approved by Dr. L. Larmore, then Chief Scientist of the Lockheed California Corp., and was granted logistic support by two branches of the U. S. Air Force---the Terrestrial Sciences Laboratory of the Air Force Office of Aerospace Research (Mr. Mahlon S. Hunt, Monitor) and the Aeronautical Chart and Information Center at St. Louis (Mr. R. W. Carder). Both expeditions, organized and led by Dr. Stoddard, ably assisted by Mr. D. Carson, met with a large measure of success. The results of the observations were then measured and reduced in the Department of Astronomy, University of Manchester. Preliminary results were obtained by Messrs. S. A. Brooks and M. Davidson in their unpublished M.Sc. Theses written under the supervision of Professor Kopal and submitted to the Faculty of Sciences at Manchester in 1963, while the final results as presented in this paper were obtained later by a collaborative effort between Dr. Goudas, Professor Kopal, and Mr. Davidson at the University of Manchester and the Boeing Scientific Research Laboratories in Seattle, in 1965. The second section of this paper, describing the circumstances of both eclipses and the methods of their observation, has been contributed by Dr. Stoddard; while all subsequent parts dealing with the measurements of the photographic material, their reduction and extraction of the final results have been contributed by Mr. Davidson, Dr. Goudas, and Professor Kopal. The re-

sults obtained in the course of this study, their comparison with previous data and conclusions arrived at, have already been submitted in the abstract; in what follows they will be substantiated in detail. II. T H E CIRCUMSTANCES OF THE ECLIPSES AND THEIR OBSERVATIONS

The Annular Eclipse of 1962 July 31 This eclipse was observed from the Woloff village of Kheur Soce, in the flat savanna of Senegal about 10 miles south of Kaolack, and about 125 miles south east of Dakar. The expedition assembled at Wiesbaden, Germany and was flown by a U. S. Air Force C-130 airplane to Dakar. The return trip to Germany was also by C-130. Also provided by the Air Force and flown in on the airplane were provisions, as well as a truck and trailer for ground transportation to the site. Members of the expedition team were: Dr. L. G. Stoddard (Expedition Leader) and Mr. D. G. Carson, from the Basic Sciences Laboratory of Lockheed California Co., together with Mr. James Hallows from the Department of Astronomy, University of Manchester. The coordinates of the observing site were Longitude: 16° 04' Latitude: 13° 59'

12" 04"

W N.

The longitude was obtained from Air Force maps and the latitude from star observations with a theodolite. The annular phase occurred about an hour before noon local time. It had been raining in the early morning, but it stopped about 2 hr before the annular phase with the sky remaining overcast. However, breaks in the clouds did occur and about 20 satisfactory frames were obtained from the approximately 100 exposures during the annular phase. Since the magazine of the K-22 aerial camera used for photography had a capacity of about 200 frames, about 50 frames were exposed before second contact and a similar number after third contact. Few of these were satisfactory and none were used in the study. The temperature was about 90°F with high humidity and low wind velocity. The ground was grass-covered, and ground

LUNAR PROFILES FROM SOLAR ECLIPSES OF 1 9 6 2 AND

radiation was too low to seriously affect the quality of the images. The pertinent data for this eclipse are as follows: Place of observation: Kheur Soce, Senegal UT of middle of eclipse: 11h 46 m 17~.2 Duration of annular phase: 204 ~.6 Topocentric geometrical semidialueter of the Moon: 917'.'7 Topocentric geometrical semidiameter of the Sun: 945'.'4 Topocentric libration of the Moon: longitude ~-4758 l a t i t u d e - 0717 Position angle of the Moon's axis: q- 16732 Position angle of second contact: 27173 Position angle of third contact: 11079 Position angle of line of motion of the Moon's center: 101?1 Zenith distance of the Sun: 20?0 Position angle of zenith: 277?7

1963

337.

were obtained during the complete duration of the annular phase as well as before second and after third contact. Pertinent data for this eclipse are as follows: Place of observation: Kruisrivier, Republic of South Africa UT of middle of eclipse: 14h 48 m 59*.6 Duration of annular phase: 37~.7 Topocentric geometrical semidiameter of the Moon: 966~2 Topocentric geometrical semidiameter of the Sun: 974~7 Topocentric libration of the Moon: longitude -- 5?36 latitude q-0707 Position angle of the Moon's axis: - 15733 Position angle of second contact: 257?6 Position angle of third contact: 66?2 Position angle of line of motion of the Moon's center: 7179 Zenith distance of the Sun: 55?9 Position angle of zenith: 11873

The Annular Eclipse of 1963 January 25 This eclipse was observed from Kruisrivier about 15 miles west of the city of Oudtshoom in the Little Karoo of South Africa, about 300 miles east of Cape Town. Members of this expedition were: Dr. L. G. Stoddard (Expedition Leader), Messrs. D. G. Carson and G. A. Carroll from Lockheed, together with Professor Z. Kopal of the University of Manchester. Star observations were made to determine the coordinates of the site. However, Mr. Reginald Barry, an Oudtshoorn surveyor, kindly determined the position of the site with respect to the South African grid system. His determination yielded the following coordinates of the observing site: Longitude: 21 ° 58' Latitude: 33 ° 36'

10" 34"

E S.

The eclipse was observed in the middle of the afternoon with a beautiful clear and transparent sky. The temperature was about 90°F with the humidity low and a 10 mph wind blowing from the northwest. Ground cover was poor, consequently the ground radiation was greater than in Senegal and the image quality somewhat inferior. Frames

Optical Equipment The optical system was designed to produce a series of high-resolution photographs of the Sun during an annular eclipse. As both the Sun and the Moon subtend an angle of approximately half a degree, a small field-ofview system could be used. Furthermore, the specific object of photography was the bright ring of the Sun surrounding the Moon. Consequently, it was feasible to use a system of long focal length giving a large-scale image in short exposure times. For these conditions a long-focus achromatic mirror system has many advantages. To avoid moving this long optical system to track the Sun during the eclipse, the system was rigidly fixed and fed with a plane mirror driven by clockwork so that the reflected rays would continually fall on the imageforming concave mirror. This, in turn, would reflect and form the image on the film. Under annular eclipse conditions, scattered light is no problem since the Moon's presence across nmst of the Sun's disk reduces the total sunlight (and, hence, scattered light) by a factor between 100 and 1000. Thus, no light shielding was necessary.

338

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fAIL ~ON

ET

AL.

LARGE OPTICAL SYSTEM (PLAN VIEW) J J

K-22

CAMERA

SuN

5 7 El"

~ 2 C O N C A V £ MIRROR

N

w ~r--.- E

J

s

Fro. 1. The major optical system (Fig. l) employed a 17-inch diameter optically plane mirror loaned through the courtesy of the Mt. Wilson Observatory. This was mounted a s a clock-driven coelostat (Fig. 2) that reflected the Sun's rays horizontally to a 12-inch diameter concave-mirror approximately 60-ft away (Fig. 3). The concave mirror of 57-ft focal length formed an image

on a lensless K-22 aerial camera body mounted just in front of the coelostat (Fig. 4). This K-22 film transport system uses 9½-inch-wide film in 200-ft lengths and is capable of one exposure per 1.7 sec. The camera body had been modified by the addition of a microswitch so that the exposure times could be recorded. This system formed an image of the Sun (somewhat

Fro. 2.

L U N A R PROFILES FROM SOLAR ECLIPSES OF 1 9 6 2 AND 1 9 6 3

339

FIG, 3.

larger than 6 inches in dialneter) in the focal plane of the K-22. A smaller optical systein of the same type was also used primarily as a back-up system. It consisted of a 12-inch coelostat and an 8-inch secondary mirror of 26-ft focal length. Its 3½-inch solar image was recorded with a K-24 film transport system. This calnera body, also modified with a microswitch, used 5½-inch-wide film and could take three frames per second. Film systems were chosen in preference to plate systems. With modern Estar-based emulsions, dimensional stability is comparable to plates and the use of fihns offers

m a n y advantages. Reseau exposures were obtained at the South African site on the same roll as the eclipse films and were processed with them. Study of these reseau fihns indicate a negligible distortion o n the originals due to photographic effects. The prime reason for the choice of films, however, was to obtain the maximum number of exposures during the brief annular phase and thus minimize random atmospheric distortion effects. Since an intensively bright source was to be photographed, the choice w a s made to use a very slow, fine-grain emulsion in order

FIG. 4.

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D, CARSON ET AL.

~

%

V

q ~

.......

~ ; ¸¸L¸¸ ....

FIG. 5. to obtain maximum resolution and sharply defined limbs. This film also obviated the need to use filters and permitted the exposure times to remain within the capabilities of the camera shutters. The film supplied and processed by the Aeronautical Chart and Information Center (ACIC) of the U. S. Air Force was ideal for the purpose. It was the Eastman S0-105 emulsion, which has an ASA rating of 0.08; it is blue-sensitive, with a resolution of the order of 1500 lines/mm. This emulsion, with an exposure

of 1/350 see at both eclipses gave sharp outlines of the Sun and Moon with the larger system. The smaller system, being optically faster, was supplied with the even slower Eastman 649-0 emulsion. This emulsion proved somewhat too slow for the shutter capabilities of the K-24 and did not produce images as good as those of the larger system. Sky transparency can vary because of many factors. It is, therefore, necessary to determine exposure times immediately before an eclipse. For this purpose, ACIC

Vt~:. G.

LUNAR PROFILES FROM SOLAR ECLIPSES OF 1962 AND 1963

provided a special exposure-meter calibrated to the optical systems and film used. This meter, which could be inserted into any portion of the solar image forming rays just ahead of the film plane, could measure the intensity of light in any portion of the solar image. This versatility was required, because radiation from the solar disk is not uniform. Radiation is very intense at the center and diminishes to a small percentage at the edge which is the region to be photographed during the annular phase.

Miscellaneous Equipment The camera pulses were recorded on duplicate recorders which also recorded time from the two chronometers and the radios. At both eclipses radio reception was poor during the daytime, and gave nothing meaningful during either eclipse. Consequently, the two chronometers were used as an intermediary. Radio time signals from WWV were generally obtained in the morning, at night, and occasionally in the daytime. The chronometer rates were small and fairly constant with no definite isochronal error. Chronometer corrections were obtained after each eclipse and interpolated to each frame. It is felt that the chronometer error at each eclipse was accurate to within 0".02, and the times assigned to each frame accurate to 0:1. III. MEASUREMENTS OF THE DATA AND ERROR ANALYSIS

Measurements Typical examples of the photographic plates obtained from the 1962 and 1963 eclipses are shown in Figs. 5 and 6, respectively, and our task now is to make an accurate mathematical presentation of the inner boundary of the bright annulus, i.e., the lunar profile. For reasons stated in Sec. I it is sufficient to assume that the outer boundary is practically a circle. The inner boundary can be described by measurements of the coordinates of a series of points with respect to an arbitrary origin performed by means of a comparator. The same aim can essentially be accomplished by measuring the width or annulus of the bright belt versus

341

the position angle using initially an arbitrary origin which later will be transferred to the center of the limb. The definition of this center will be discussed in the analysis of the data at a later stage. The second method was found more practical as well as accurate in this case, and the width of the annulus between the solar and lunar limbs was measured by means of the Joyce-Loebl Mk IV twin-beam microdensitometer of the Astronomy Department of the University of Manchester. The origin was located at the center of the solar disc, because the width was always measured along a direction perpendicular to the solar limb. In more detail, the work involved in the measurement was carried out as follows: The original frames (about 180 from the 1962 eclipse and 25 from the 1963 one) were copied on two types of film: namely, the Kodak SO105 and S0278 which are high and low contrast, respectively. In the case of the 1962 eclipse plates it was found that only 15 of them, all on SO105 film, were suitable for measurement. Out of the 25 plates of the 1963 eclipse 20 were of good quality and were used for measurements. Each negative was placed on a sheet of graph paper, the lines of which were visible through the transparent annulus. With a little adjustment it was easy to find the lines on the graph paper which intersected each other within a millimeter from the center of the solar disc. These lines were marked on the actual emulsion using a fine needle and with the aid of a ruler. A protractor was then used to mark 30° intervals round the an nulus, measured from an arbitrary fiducial line coincident with one of the original markings, which was the same for all positives. The fiducial line was chosen to be parallel with the longest edge of the film. The microdensitometer table is transparent, runs on rails, and is connected by means of a lever arrangement to the pen recorder table so that a magnification of 50 is achieved. In addition, the table may be rotated, with respect to the rails, through an angle of -4-18° . The film was placed on the table and centered, as accurately as possible, by observing the magnified image (which is five times larger) of the central cross previously marked on the emulsion on a screen in the

D. CARSONET AL.

342

inicrodensitometer as the table was rotated. When the centering was satisfactory, i.e., when the shift of the central cross was unobservable for a 12° rotation of the table and the fiducial line was parallel to the rails, the film was clamped with the aid of a glass slab and several clips. The annulus at two points 180 ° apart was then traced on one piece of graph paper and then the procedure was repeated. Two typical traces for position angles 180 ° apart are given in Fig. 7. This last figure shows clearly t h a t the boundary of the annulus is very sharp a n d the width of the trace at the half-height defines accurately a quantity linearly related to the angular width of the annulus a t the given position angle. Because of the design of the apparatus, after five measurements had been made it

(-90"

270"

Fic.. 7. was necessary to recenter the film, this time with the datum line at an angle of 30 ° to the rails. This entire procedure was repeated until the whole annulus had been measured - - 6 0 measurements in all, per frame. The necessity of recentering the film every 30 ° made t h e Work time consuming but it also made centering errors less serious as these would n o t now extend round the entire annulus. T h e measurements were made by four different persons in order to diininish the probability of systematic errors arising from inaccurate centering. A f t e r the measurements of each frame were completed, the width of the annulus was plotted against the position angle from the fiducial 5ne, and a typical graph is shown in Fig. 8, Which corresponds to a 1962 plate. This graph immediately gives an idea of the

21

15

<

~

9 ......

i

i 31

120 °

240 °

360"

Position Angle, 0 Fro. 8.

extent to which the measurements can be affected by atmospheric turbulence and b y the relief of the lunar limb. As we shall presently show the curve on Fig. 8 should be a cosine wave if the profile of the Moon were circular. The marking of the center of the solar disc obviously cannot be exact; therefore, we have to determine the errors in the width of the annulus and in the position angle 0 resulting from it. The case is illustrated in Fig. 9 where S and M denote the centers of the Sun and Moon, respectively, and the distance S M - A. We adopt the point S as origin and the direction SM as the refery B

Fro. 9.

L U N A R PROFILES FROM SOLAI~ ECLIPSES OF 1 9 6 2 AND 1 9 6 3

ence line from which position angles 0 are measured. Let R and r be the radial distances between the centers S and M and the limb points B' and A ~, respectively. The latter points lie in the direction of the solar radius of position angle 0. The distance A ' B ' varies with 0 and, if both the solar and lunar profiles were circular, it would attain a minim u m value, (AB) = a, for 0 = 0 (although in reality it is not necessarily a minimum). If the line SB' is t a k e n as the x axis of a planar rectangular frame of reference then the coordinates of a point on the inner b o u n d a r y of the annulus, i.e., the lunar profile, satisfy the relation r2 =

(X - - A c o s 0) 2 -~-

(y -- A sin 0)2,

(1)

where r depends on 0. Also the x coordinate of the point A', if denoted b y xA,, is given from the expression,

XA, = r + A cos 0.

(2)

Therefore, the width a of the annulus is a = R -

(r + ~ cos 0) + ~z0.

--

A 2

sin s 0/2r2)~ ~

x * =Acos0+r(1--A 2sin 20/2r ~ + y "A sin O/r2)

annulus and its position angle. In particular, we find t h a t ~a = A -- [(A -- AB sin O/r) 2 + B2]m,

(6)

where A = R+~x0--C, B = ~y0{1 -- [(C + 6Xo)/R]}, and C = A cos 0 + r(1 -- A2 sin 2 0/2r2). On the other hand, the error ~0 in the position angle is given b y the expression ~0 = cot 0 -R(A cos 0 -- Sx0) -- Sy0(A sin 0 - ~y0). sin O[R2 + (6yo)211/2[(Acos 0 -- Sxo)" + (A sin 0 -- ~yo)2]I/2

(7)

The following numerical values have been employed for the calculation of the errors 8a and ~0: R=8cm,

r=8cm,

A =0.2cm, ~Xo = ~y0 = 0.02 cm.

(3)

The center M of the lunar profile will emerge as a by-product of our reduction, but the center S of the Sun m a y require the corrections ~A and ~0. We shall show that, in the present case, both corrections are smaller t h a n the resolution of the plates. Let it be assumed t h a t the coordinates of the true position S' of the center of the solar disc are ~x0, ~y0 instead of 0,0. The coordinates of the point B ~ (see Fig. 9) are not (R,0) but [ ( R 2 - - 8y02) 1/2 ~ - Sx0,0] and the correct a n n u l u s is the q u a n t i t y B ' A u and not B ' A ' . Finally the correct position angle is the angle B ' S ' M instead of B'SM. We shall first compute the error in the width of the annulus. T o do so we determine the coordinates (x',y~), (x',y ~) of the points A ~ and A ~, respectively. T h e y are x ~ ---- A cos 0 q- r(1 y' = 0

343

The numerical computations showed t h a t the errors in a never exceed 0.2 #, whereas the errors in 0 never become large enough to m a k e it necessary to a p p l y corrections to the measurements obtained. T h e size of the image of the film makes it possible to calculate its scale as well as the focal length of the telescope. Since the solar disc departs from a perfect circle b y ignorable amounts, the scale of the photographs can be determined from the solar diameter only after the factors distorting it are properly treated. Since the focal length f of the telescope can be defined as the ratio of the linear diameter of the solar disc in the focal plane to the a p p a r e n t (topocentrie) angular diameter of the Sun at the time of observation we obtain, f = 17.67 meters.

(4)

!

(8)

The scale s of the photographs is, on the other hand, the inverse of the focal length and the numerical values obtained are as follows:

y~ = ~yo(R -- r + ~ X o - - A c o s 0 ~i" (5) + A2 sin 20/2r) (R + $yo~ sin 0/4) ]

s = 11:74 per millimeter, for the 1962 eclipse

These coordinates suffice for the determination of both errors in the width of the

s = 11'.'55 per millimeter, for the 1963 eclipse.

(9)

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D. CARSON ET AL.

The ratio of the radii of the two discs for the same eclipse is found to be 0.971 and 0.991, respectively. All the figures given here will have to be slightly nmdified after the analysis of the errors involved in their estimate are analyzed and estimated in the following section.

Error Analysis [ The departures of the lunar profile from a circle are rarely larger than 2'.~0 geocentrically, and on the average are less than 1'.'0. It is, therefore, essential in the search for significant deformation of the lunar figure in its marginal zone that all sources of possible systematic errors capable of introducing effects larger than -4-0':2 be carefully considered. For this reason, a detailed study of errors was undertaken for the sets of photographs of both eclipses, concerning not only the systematic errors likely to be present, but also those which could possibly influence the final reductions by appreciable amounts. Our error analysis concentrated mainly on the following effects. (a) Emulsion Shifts. Such a shift after the exposure would be liable to distort appreciably the images of both the lunar and solar profile, and thus produce a measurable effect on the measurements. A uniform dilatation will, in general, change the scale of the photographs and leave the spectrum of the limb unaffected except for the zero harmonic, and for this reason need not be considered at all. On the other hand, nonuniform dilatation could be regarded as a single or multiple stretch and can be examined as follows: Since the two profiles recorded on the fihns, namely, that of the Sun and the Moon, are at a close proximity, any shift should deform both largely by the same amount. For this reason, measurements of the diameters of the solar disc at different directions for each frame were made. Analysis of these measurements showed that such an effect was not present in our case. In the chemical development of the photographs the emulsion m a y be subjected to irregular effects causing differential displacements on the film. These displacements, if random, should cancel out when measurements are averaged over the 15 (in the first case) and 20 (in the second case) sequential

photographs of each of the eclipses. If, on the other hand, substantial differential displacements are present, then their effect should appear on the solar profile and thus be detected. Such an effect has not been found either. (b) Refraction. The altitude of the Sun during the 1962 eclipse was 60 ° and during the 1963 eclipse 56 °. The maximum geocentric angular dimension of the annulus along the main direction of differential refraction was in both eclipses less than 15"; this effect introduces a decrease in the annulus of at most 0'.'2, which is again negligible and does not need to be taken into account.

However, the eclipse itself is expected to modify to an important extent the normal properties of a quiet atmosphere for the following reason: The sudden elimination of the energy flux inside the shadow cone will inevitably result in a fall of temperature and, hence, in local currents which are capable of causing anomalous refraction patterns on the recorded shape of the limbs. This effect was discussed by Schuster (1922) in connection with the relativistic deflection of stellar images in the neighborhood of the solar limb. He concluded that the effect was unimportant, but doubted if this was generally the case. (c) Turbulence. Atmospheric turbulence can result in differential displacements of limb details during short exposure times. In the case of long exposures the instantaneous profiles will be smeared out around the true position of the limb, and obliterate its details. For this reason, short exposures were taken sequentially every 1.5 sec. The effects of refraction anomalies due to atmospheric turbulence can easily be seen by a comparison of any two successive photographs. These, however (being of random nature), can be eliminated by averaging the measurements on the entire sequence of photographs. This has been found to be a powerful technique in lunar photogrammetry in other cases as well, and is expected to increase the accuracy of selenodetic work in the marginal zone as well as on the remainder of the lunar surface. Reduction of the amount of work necessary for the measurement of many sequential photographs can be achieved by

LUNAR PROFILES FROM SOLAR ECLIPSES OF 1 9 6 2 A N n 1 9 6 3

judicious choice of a small number of frames out of the entire series. The selection has to be based on the condition that the average of positions of features on the selected subject of photographs must be as close as possible to the average of the entire set. In order to do so, measurements of a certain number of identical points on all photographs are initially required. Their positions must then be harmonically analyzed and the important frequencies of the atmospheric turbulence during the exposures be determined. (d) Plate Tilt. If the optical axis of the telescope is not perpendicular to the photographic plate, the width of the annulus resulting from our measurements will not be correct. Let ~b (see Fig. 10) be the angle F

¢P

R'

--X

Fro. 10.

between the optical axis of the telescope and the normal to the plane of the film. On the assumption that their centers coincide, we find that

R -- r = (R' -- r')(1 -- ½~2),

(10)

where R ~ and r ~ are the radii of the solar and lunar images in the directions most affected by the plate tilt. If now the maximum error we can tolerate in R or r is about 0w.1 geocentric, then the angle 4~ must be less than 7 °. But such a plate tilt cannot have been present for the instrumental arrangements used to observe both eclipses. (e) Telescopic Aberrations. If we assume

345

that the mirror aberrations are linear in distance from the optical axis, these will simulate the stretching of the film, the effects of which we found before to be ignorable. Nonlinear (i.e., quadratic and higher) offaxis aberrations would, however, be reflected in the measured figure of the Moon. In our case since the images of the solar annulus were formed slightly off-axis (due to the Herschelian nature of our system)--off-axis effects are bound to be present and could become serious if measurements were to be performed across the entire diameter of the annulus. Since, however, we are using the outer limb of the annulus as our reference circle, the form of the lunar profiles emerging from our measurements should be affected by off-axis effect only in so far as these change across the width of the annulus; the latter amounts only to about one-hundredth of its diameter. Such differential effects have proved to be theoretically negligible, and no trace of them was found in the observed results. (f) Diffraction. Due to the finite resolving power of the telescope, there may be some distorting effect when the annulus is narrow. This is the ease with all the films available for both eclipses. Such distortion may, of course, be expected only when the width of the annulus is comparable in size with the diameter of the Airy disc imaging a light point; this may be true only shortly after the second contact and before the third, when the two discs have one common point at internal tangency. The diffraction effect may appear in parts of the photographs where the geocentric width of the annulus is less than, or equal to, 0'.'5 in blue light. The size of the error introduced by this effect can be computed in the following way: Let, in Fig. 11, the x axis be in the direction of the solar radius and the interval ( - a , Wa) be the width of the annulus along which the light intensity I0 is assumed to be distribution constant. The Fourier transform f(k) of this is 1 f(k) = V ~

~

.o

" / f(x)e-'k~dx

f-

01)

(e'ka -- e--lka).

346

D. CARSONET AL.

-(~

The aperture A ( x ) given in this form vanishes for qx = ~r and, as a result, if a is the Airy disc the integration in (14) can be terminated with k' = rca/a. The evaluation of this integral was performed numerically, but prior advantage was taken of the following consideration: Since the measurements of the photographs with the microdensitometer can be carried out at different light intensities, a nondimensional quantity z was introduced, defined so that

+o Fro. 11.

Since the aperture of the telescope is finite the image in the photographic emulsion will be given by this Fourier transform nmltiplied by the aperture function. If it is assumed that the optics of the telescope is perfect and that the highest spatial frequency received by the telescope is q, then the intensity f'(x) of the image in the enmlsion will be given by the expression

1 fq

f'(x) = ~

¢ika

__ (2--ika

il~

._~

~'k~dk,

02)

which, in turn, can be rewritten as

fr(x) 1 -

fq

sin lc(a -k x) -- sin/c(a -- x)

21r - - j - q

1,:

die; (13)

light intensity during measurements = zlo. (16) What we actually seek is the function x = x(a) for various values of the parameter z. In this way we shall have the relation between the width of the annulus recorded on the enmlsion, and its correct value a in the absence of diffraction. The results of the numerical integration performed with the Atlas Computer of Manchester University are graphically shown on Fig. 12 for four different values of the parameter z. The values of x and a are plotted in terms of the radius of the Airy disc. In the absence of diffraction, the slope of these curves should be equal to 45 °; but apparently this is not the case when the true size of the

and the substitution

It' =a]c,

!

2.0

q' = aq,

x' = ax,

i

transforms (13) further into

....

/ z=O

1.5

f

,(x') a

l foe' , , =~r .)(, sin ., ~ It'(1- + x')- + sin - k (1 -- x ) .a/c.

o

I

~

0.8

O 1.0

(14) We now introduce the aperture function A (k), defined so that A(/c) = 1 A(k) = 0

for for

Ikl < Iql, ]/~l > Iq]-

i

0.5

Therefore, 0

A(x) -

-x/~

1

f q eikxdk ~-q

q

(sin qx). \

qx /

(15)

0.5

1.0

1.5

Actual Width of the Annulus in Units of Airy's Disk

FIC,. 12.

2.0

LUNAR

PROFILES

FROM SOLAR ECLIPSES

annulus is less t h a n 0.7 Airy radii. When a is larger t h a n one Airy radius the measured annulus x is a linear function of a with 45 ° slope, indicating the diffraction effects to be negligible. The annulus on the photographs of both eclipses was never less t h a n 10 Airy radii--a fact which gave us confidence t h a t no such effects would appear in our reductions,

(g) Error in the position of the lunar center, The p a t h of the center of the lunar dis# can be assumed to a very good approximation to be a straight line, such as A ' A in Fig. 13. If now the adopted p a t h of this center were different (say, B'B) then a systematic error in the reduction of the measurement is in-

0

x

P

B'

Fro. 13. troduced which can be treated in the following way: Let P and Q be the adopted and true positions of the lunar center. The residuals of the points from the best-fit circle with P as its center would then contain an additional t e r m varying as cos 0, where 0 denotes the angle between the position vector of the limb points and the direction PQ. In particular, if (PQ) = p0 and the lunar limb were close to a circle, the radius p(O) from the point P to the limb point of position angle 0 could be written in the form p = r + p0 cos 0

½(po2/r) sin 2 0 -f- -~(po4/r3) sin 4 0 + . . . .

--

The presence of the t e r m p0 cos 0 would become immediately obvious in the ease where the constant p0 were sufficiently large. In such a case a new determination of the p a t h of the lunar center would be necessary. With the photographs of the 1963 eclipse, this did not prove to be the ease. However,

OF 1962

AND

~ ;o_2 ~ ~9 _~ 0 $ ~6 a) ~ 10-2 a ® ~ ~62~1o_2 o°

347

1963

....7

5

w >"

s4

96

120

108

132

144

Frame Number FIG. 14.

an examination of Figs. 14 and 15 which correspond to the 1962 eclipse showed t h a t a systematic error was present in the positions of the lunar centers, because the separate t r e a t m e n t of the frames of serial numbers from 93 to 99 and from 138 to 147 did not give comparable values for the slope of the straight line interpolated b y least squares. In addition, an inspection of the plot of the annulus versus position angle (see Fig. 8) shows clearly an " a t t r a c t i o n " of the lunar limb b y the solar limb. This appeared to be the case with all individual frames. The two groups of sequential photographs mentioned above resulted in two profiles which, together with their average, proved to be unreliable. These two profiles are barely compatible. The discrepancy between the slopes of the paths of the two centers as well as between the profiles in the case of the 1962 eclipse is due to the effect of irradiation, which we shall discuss next. (h) Irradiation in the emulsion. A careful s t u d y of all available photographs revealed t h a t the 1962 eclipse data cannot produce good profiles of the Moon because of un2x10-2 U

o

o yz × 2xlO 2

96

108

120

132

Frame Number

Fro. 15.

144

152

348

D. CARSON E T A L .

disclosed systematic effects. In more precise terms, first--against expectation--the centers of the Moon in the series of photographs were not sufficiently well aligned. Secondly, the mean radius of the Moon was found to vary from frame to frame; the value of r as borne out by measurements differed from the one listed in the Astronomical Ephemeris and Nautical Almanac by an amount greater than expected (although this could be explained in terms of the characteristic curve of the emulsion). Third, the profile of the Moon obtained from two sets of sequential photographs taken a few minutes apart in time are not in satisfactory agreement. In addition, none of the profiles determined from individual frames seemed to agree well enough with the others. All this can be explained by the effects of irradiation in the emulsion. Various authors (see, e.g., Selwyn, 1950) have studied the effect of irradiation on the size of photographic stellar images and derived a number of relationships between the diameter of stellar images and exposure times. The extent of the blurring is represented by the law I = I0 exp (--kx - (,x2),

(18)

can safely be adopted because for our photographs the exposure times (1/350 sec) do not allow the effects of atmospheric turbulence to be recorded on the emulsion. Consider a section of the annulus of width x0, along which the intensity I0 is uniform. The effect of irradiation is to blur the image in such a manner that the intensity I at a point whose distance from a small element of the annulus is x will be given by the expression I = 210{fo ~ exp [ - - k ( x 2 - k z~)X/2]dz}dx,

y

0

x

Fro.

x0

I* x

16.

side can be expressed in terms of Bessel functions, so that (to avoid complexity:of numerical calculation) the z term will be neglected. This should not, in principle, alter the predicted effects qualitatively, but will make the result only approximate. It is thus assumed that I = I0 exp ( - - k l x l ) d x .

(20)

Consider now Fig. 16 in which one of the limbs of the annulus sets in for x = 0; the other, for x = x0. The intensity at the point x contributed by the entire annulus is

(17)

where I0 and I are the intensities of the image at its geometrical center and at distance x from it, respectively; and k and a are constants, the latter representing the effects of atmospheric turbulence. Therefore, the law of irradiation of the form I = I0 exp (--lcx)

Io

(19)

where z is the width along the annulus which is assumed to be infinite. The right-hand

= I0 [2 -- e-kx -- e--k(:r°-x)].

(21)

If the point y were on the other side of the annulus, then I y = (Io/l¢)[e -ku - - e-k(u-x')];

(22)

whence, for y < 0, we obtain Iu = ( I o / k ) [ e -ku -

e-k(~°+Y)].

(23)

The total of the energy E incident on the emulsion (on the assumption that we work on the linear part of the reciprocity curve) is given by the expression

if: o[ 2 - - e - ~ . ~ - e -k(.... )]dx

E=n + ~

[e-ky -- e-k(x~-x)]dx

+/:

t,

which an evaluation of the integrals becomes E = 2nxoIo/k,

(24)

LUNAR PROFILES FROM SOLAR ECLIPSES OF 1 9 6 2 AND 1963

where the quantity n is a normalizing constant. If we now want to make the total energy equal to xolo, we assign to n the value k / 2. The effect of high contrast obtained in the copies selected for measurement had no effect on the results because the contrast of a copy merely provides a line of constant density in the original, thus making the measurements easier. It is necessary to assume here that not only the originals, but also the copies, had identical exposures, so that this quantity does not v a r y from positive to positive. Because of the relations (21) and (22), at the measured boundary of the annulus we have

/,

lo lk

elk

0.8

-~ 6/k

0.9

~ 4/k

/

2/k

/

o

2/k

4/k

l 6/k

I 8/k

I ~o/k

Actual Width

Fro. 17.

and, consequently, exp [ - - k ( x -t- ½x0)] = 2q exp (--½kxo)/[1 -- exp ( - k x ) ] . If the measured width of the annulus is denoted b y ~, then we can put (25)

and, as a consequence, we find t h a t k~ -- kxo = 2 log (1 -- e-k~) -- 2 log 2q.

(26) This formula relates the variables x and ~, the actual and measured widths, as long as the apparent width is greater than the actual one. If the opposite were true the corresponding relation could be obtained b y combining Eqs. (20) and (22) leading to the relation

be approximated, rendering it more convenient to standardize the functions so that, as xo approaches infinity, 8 approaches x0. The new functions are plotted in Fig. 18. The maximum cutoff corresponds to q = 0.5. These curves, as stated previously, cannot be expected to correspond exactly to the experimental curve, because of the initial assumptions, and also of the fact that solar limb-darkening and the characteristics of the emulsion will affect the experimental curves. A modified treatment which allows for solar limb-darkening m a y be performed 8/k

6/k

e-k=-- ( 1 - - q) -l- [ ( 1 - - q)2 -- e-k~] 1/2, (27)

which can, in turn, be combined with Eq. (25) to give the desired relation. The two functions coincide in the line 8 = Xo. The graphs a versus Xo, are shown in Fig. 17 for different values of q from 0.1 to 0.9. It will be seen that for x0 approaching infinity, /i exceeds x0 by the amounts 1/k log 2 ( 1 - - q) for q < 0 . 5 , - - 2 / k log 2q for q ~ 0.5.

:)10. 2

/

2q = e-k~ -- e-k(x°+~)

= Xo + 2x

349

(28) (29)

In any experimental curve that can be drawn these quaDtities m a y only roughly

:7

-(3

~ 4/k - -

:~

~

2/k

0

0

2/k

I

q=0.4

4/k

Actual Width

Fro. 18.

6/k

8/k

350

D. CARSON

ET AL.

along the following lines: The intensity Io, at a point on the solar disc where the heliocentric radius-vector makes an angle 8 with the line of sight, can be represented empirically by a relation of the form 1~ = I t ( l - - u - v + u c o s 0 +vcos 20+

. . .),

16

12

----

(30) .c_

where I~ is the intensity at the center of the solar disc, and u and v are parameters depending on the wavelength of the observation. This corresponds on our previous model

8

-no

4

tO

I ( x ) = A + B x ~/2 + Cx,

(31)

where the solar limb has been assigned the coordinate x and A, B, C are constants; and leads to integrations involving the term xll°-e-k~ which, again, has to be solved numerically. Further, if for a zone close to the limb we approximate this law by the (,xpression I(x) = A ' + C'x, (32) we can obtain a relationship between x0 and which is rather complicated. Such an investigation would, however, be of limited interest in the present case, since it is not possible to determine the necessary parameters by least squares. It is sufficient, at present, to investigate the effects of limb-darkening qualitatively. The energy which reaches the emulsion does not then depend linearly on the width of the annulus, but increases more sharply as the annulus width increases. It is to be expected, therefore, that the experimental curves will be steeper than the theoretical ones for values of x0 for which the distortion takes place. It m a y be noted that no frame among the SO105 or the S0278 positives shows any sign of limb-darkening. If the copies are assmned to have been exposed correctly (which is likely, since they were made under laboratory control) this implies that the originals were considerably overexposed--a fact which accounts for the importance of irradiation in the reductions. The experimental curves of ~ versus x0 do, in fact, appear steeper than the predicted curves. The curves for Senegal and South Africa are shown in Fig. 19. The

o

0

4

8

ActuaJ W i d t h

12

18

in Plate Units

FIG. 19.

methods of deriving the two curves are, however, different. In the case of the photographs of the South African eclipse there were a considerable number of high-contrast positives corresponding to partial phases. By measuring these, and extrapolating for the centers of the lunar images on the basis of the positions of the centers found for the central phases, we found it possible to plot the measured width against the width found by adopting the extrapolated centers. For the Senegal eclipse, there were no highcontrast positives of the partial phases. An attempt to use the low-contrast positives in a similar way failed because of a large dispersion on the tracings equal to that width found on the high-contrast positives of the same frame number. A histogram of these heights is shown on Fig. 20. This being the case, it was clear that measurements of the partial phases would not produce a very reliable calibration curve and an iterative process was instead employed. This will be discussed later. A further prediction, on the grounds of irradiation in the enmlsion, m a y be verified with reference to the low contrast positives. Equation (20) can be written in the form I(x) = I0x012 -- e-k~ - e-k(x°-x)],

(33)

which attains the maximum value I ..... = I0(1 -- e-~k~0)

(34)

LUNAR PROFILES FROM SOLAR ECLIPSES OF 1 9 6 2 AND 1 9 6 3

lated earlier, namely, from Eq. obtain

12

351

(28) we

A = ~ - - x 0 = [ k l o g w ( 1 - - q)]-~ = 0.007 cm,

10

where kxo >> 1. IV.

R E D U C T I O N OF MEASUREMENTS

The reduction of the measurements made with the aid of a microdensitometer can be performed in different ways. Let (see Fig. 22) the origin S of the Sxy rectangular frame of reference (whose axes are arbitrarily oriented) be identical with the center of the solar disc, and the point M(xo, yo) be the Y 0 ~

0

0.2

0.4

0.6

P (xp Yl)

0.8

Fractional H e i g h t FIG. 20.

for x = Xo/2. The q u a n t i t y x0 cannot be measured directly, but can be estimated b y plotting the m a x i m u m height of the trace against the measured width, bearing always in mind the effects of solar limb-darkening, etc. Such a graph is shown in Fig. 21. The continuous lines correspond to the theoretical curves [Eq. (28)] for k = 50 cm -1 and q = 0.2, 0.3, and 0.4. I t seems t h a t the parameters have their best values when k - 50 cm -~ and q = 0.3. However, it was concluded t h a t q was not independent of x0. With the aid of these parameters, it is possible to recalculate the quantities calcu-

L~

Xo,

Yo)

0 S

X FIG. 22.

center of the lunar disc. The radius of the Sun is denoted b y R and r0 is the mean radius of the lunar disc. Wo can, therefore, express the two limbs b y the equations x 2~_y2 = R 2, (x -- x0) 2 + (y - y0)2

=

(r0 q- ~r) 2,

(35) (36)

where ~r is the departure of the particular point of the limb from the mean circle. If now the point P(xl, yl) on the lunar limb corresponds to a position angle 0, the straight line passing through it and the origin is represented b y the equation y = x tan 0, and the coordinates of the point P are given b y the expressions

.~10°

xl = {x0 cos 0 + y0 sin O-t- [(r0 + ~r) 2 -- (Xo sin 0 -- Yo c o s 0)2]1/2} COS O, (37) FIG. 21.

yl = {x0 cos 0 + Y0 sin 0 + [(r0 + ~fr)2 - - (xo sin 0 -- Yo cos 0)2]1/2} sin 0. (38)

352

D.

CARSON

ET

AL.

( M e ) : -- (SM)~ + (SP) ~-

Therefore, a:~~ + yl 2 = X o C O S O + y o s i n O + [ ( r o + ~r) 2 - - (Xo sin 0 -- yo cos 0)~]~/~; and for =

R

--

2(MS)(SP) cos O,

which can be rewritten as (SP) = (MS) cos 0 + [(MP) 2 - (MS) 2 sin 01if2. (40)

( X l 2 -{- y 1 2 ) 1/2

we obtain ~r = [ ( R - - ~ ) : - - 2 ( R - - ~) X (x0 cos 0 + Y0 sin 0) + x02 +

-

y02] 1/2 - - r 0 .

Remembering that (SM) = A, we can derive the expression of the width of the annulus PQ in the form A2

I t also follows that

(PQ) = (R -- r) -- A cos 0 + ~ sin 20

tan 0 = y~ -- Y0 Xl ~

3:0'

+

3A4 ~

sin 40.

(41)

and, hence, tan 0 =

sin O[ro2 - (xo sin O - yo cos 0)21"~ cos O[ro2 -- (Xo sin 0 -- yo cos 0)2]in + x 0 s i n 0 c o s 0 - - y0cos 20 + y 0 s i n 0 c o s 0 - - x0cos 20" (39)

Assuming the Sun to appear as a circle, and the relative coordinates of the Sun and Moon as known, we can determine the deviations of the lunar profile from a circle. In theory, both the differences in celestial coordinates and the value of r0 can be determined from the Astronomical Ephemeris. In this case, each negative can be dealt with separately, and the results from each negative combined. The methods presently to be described do not make a n y assumption concerning the knowledge of the relative coordinates of the two bodies. Let, in Fig. 23, M be the position of the projected center of the Moon, and S, that of the Sun. If so, it evidently follows that Q

Thus, to a good approximation, as the Moon moves across the solar disc the width of the annulus changes according to a cosine law. As the distance between the centers changes the cosine will exhibit a phase change. We m a y generalize the formula to a=

(PQ) = ( R - r ) A c o s ( 0 - - 4)) + (A2/2r) sin 2 ( 0 - 4)) + • " • •

(42)

Thus if, for each negative, we plot the width of the annulus against the position angle, we obtain graphs resembling a cosine curve whose amplitude and phase varies from frame to frame. The angle 4) is, of course, equal to zero when the line joining the two centers is taken as the x axis. Equation (42) after elimination of the term containing A2 can be used as one condition for the determination of the mean radius and of the coordinates of the center of the lunar disc. If X j, Y~, and Z J"are defined by the expressions X~ = ~i cos ~J, YJ = AJ sin 4)5,

(43)

Z~ = -- (R~' -- r~), where the superscript j indicates the frame number, then the width a j of the annulus corresponding to position angle, 0j, will be related with these quantities b y the expression

aiJ -~- aiJXJ + b / y i + ZJ = 0, where FIo.

~.

aj

=

c o s O i j,

bi J = s i l l O i j.

(44)

LUNAR PROFILES FROM SOLAR ECLIPSES OF 1 9 6 2 AND 1963

By performing n measurements distributed along the entire annulus and applying the least-squares technique, we can determine the central and mean radius corresponding to the jth plate. The sequence of positions of the lunar center must practically define a straight line. In this way, we can determine corrections for the position of the lunar center for each frame and the new coordinates will be denoted by X d and Yd. The next step is to determine the radius r~J corresponding to position angle 0d. This can be done by means of the relation rd = a d - - Rd + ajXJ + bjY~. (45) By averaging over j we determine the radius r~ corresponding to the position angle 0,.. We must emphasize here that 0d" and 0d+k are not even approximately the same due to the variation of the line through the solar and lunar disc and, as a result, 0,. is by no means the average of the 0j's, but a position angle measured from an arbitrary direction which is common to all frames. An obvious alternative to the technique just described would consist of fitting a circle to the lunar limb for each frame, then referring all to the same origin and arbitrary directions with res pect to which the position angles are to be measured and, finally, obtaining the residuals for each position angle and their average over the entire sequence of frames. In order to ensure the good quality of the results obtained by the first technique, we independently made use of the second technique and the results were then compared. It was found that, for all position angles, the results from the two techniques were compatible within the marginal error 0'.'15 or 0.26 km in elevation of the limb points--an agreement which we consider satisfactory. A spectrum analysis of these residuals seems highly useful, because it may provide a comparison between the present reductions and results obtained independently on the basis of different or similar photographic material, or even different kinds of observation. Equation (42) shows that the values of the width of the annulus are already in a suitable form for Fourier analysis. Before presenting the results of such an analysis,

353

it is necessary to determine constants defining the particular sections of the marginal zone recorded on the photographs of the two eclipses as well as constants of the eclipses themselves. The Astronomical Ephemeris and Nautical Almanac gives the predicted values of all variables defining both eclipses. Quantities of importance are the geographic coordinates of the site, the exact time of the four contacts (especially the internal ones), the apparent path of the Moon across the solar disc, etc. The determinations of the latitude and longitude of the site, and the measurement of time, are not very critical since, unlike the cases when a determination of the difference between ephemeris time and universal time is intended, the main objective of our expeditions was to procure photographs of the central phase for the measurement of the lunar profiles. Precise prediction of the points of contact of the lunar and solar limbs, and a prediction of the apparent path of the Moon across the Sun are also not essential, but certainly desirable, in the reductions. These are required in the calculation of the orientation of the image on the plate, which may be calculated, as is to be seen later, by other means, but it is useful to have an independent confirmation. The geocentric right ascensions and declinations of the Moon and Sun are tabulated in the Ephemeris; it is necessary only to interpolate between the given values and determine them for the times of observations and then to compute the right ascensions and declinations as seen from the sites of observation in order to find the apparent path of the Moon across the Sun. In this calculation we shall have to allow for the oblateness of the Earth, which enters into the computation of the geographic latitude of the site of observation. Thus if ~ and ~' are the geographic and geocentric latitudes of the site, their relationship is expressed by the formula tan 4' = (I -- e2) tan ~b.

(46)

The determination of the topocentric coordinates of a point from its geocentric ones will be made as follows: Let ~ and ~ be the geocentric right ascension and declination

354

D.

CARSON

of the point in question, and a ' and ~' their topocentric values for an observing site whose geocentric distance is p. Let also O x y z be a geocentric frame of reference with the axis Oz coinciding with the axis of rotation, and the Ox axis pointing toward the point of vernal equinox. The rectangular coordinates x, y, z, of the point of geocentric distance A will then be X =

ACOS

~COSol~

y = A cos 5 sin a, z = A sin &

(47)

The coordinates x', y', z', of the same point with respect to a frame of axis parallel to those of the O x y z frame but with its origin at the observer's site, will be X t -~ A t COS ~l COSott~

y' = A' cos ~' sin a', z' = A' sin:O',

(48)

where h ' is the distance between the observer a n d the point. The transformation x' = x -- a, y' = y - - b, Zt =

(49)

ET

AL,

These equations make it possible to calculate the relative positions of the Sun and Moon, as seen in the sky, observed from a n y point on the surface of the Earth. The ephemeris gives values for the geocentric quantities used above in intervals of time of 1 hr or 1 day. I t is necessary to interpolate between these and in order to obtain accurate interpolations the problem was solved in the computer. The results of our reduction of the motion of the center of the lunar disc across t h a t of the Sun are plotted iD Figs. 24 and 25. The first corresponds to the 1962 eclipse, and the second to t h a t of 1963. T h e coordinates axes x and y are defined with respect to the photographic films and are common for all frames. The straight line fitted b y least squares to these points in either case is found to be identical within the permissible error of 011 with the predicted lines which were determined from Eqs. (51) and (52). The times of the second and third contact of each eclipse were computed as follows: The equation of the p a t h of the lunar center across the solar disc is expressed in terms of the time parameter, t, by the equations x = kxt q'- c1,

Z - - C~

y = k2t -q- c2,

can be established, where O-2cm

a = p cos 4)' cos l b = p cos 4)' sin t

(50)

ti

10

c = p sin 4)'

14 - ~

6

and t is the sidereal time at the observing station. F r o m Eqs. (47) to (50) sufficient relations can be established for the determination of the topocentric coordinates, the final expressions being

2 J

r

i I I

2

6

6 80 ..... •

Fro.

p cos 4)' sin (~ -- t) tan (a - a') = A cos ~ -- p~s4)' cos (a -- t)' -

-

~)

Neg

p sin ? sin (~ ?) A sin ? -- p sin 4)' cos (~ -- ? ) '

(52)

A' = A sin ($ -- ? ) / s i n (~t' -- ?),

(53)

-

-

where -

i

x

i

_

1()-2Cm

Theoretical Prediction Experimemtal Line Experimental Points for 3rd Iteration

24.

-3

31

:

-2

Y I 32

I --II

J0

1

2

3

4~,..,

I

6 Neg ]8

-

tan 4)' cos ½(a' ~) t a n ? = cos [½(a' + a) -- t]"

I I

cm xlO -2

(51) tall (~' =

i

I0...~.~-18

I -2

-

FIG. 25.

t

7

× crux10-2

355

LUNAR PROFILES FROM SOLAR ECLIPSES OF 1 9 6 2 AND 1 9 6 3

where x, y, are the coordinates ot the lunar center and the origin rests at the center of the Sun. If we denote b y t, the time at which a n y of the four contacts occurs, then the angle of contact of the limb of the Moon to t h a t of the Sun is equal to

N o w we can establish the following relations: --sin

5 = sin ¢ sin a q- cos q~cos a cos A,

(55)

sin z sin (~r -- q~)

sin A sin (½r q- 5) -

sin K sin (½7r -- a)"

(56)

tan-1 \xlt, qF o r a = 0 we have The U T of the second and third contact are found to be as follows: For the 1962 eclipse we had Time of second contact = llh Time Of third contact = llh

44m

3.2sec,

48m

1.1sec.

The angles of contact for the same eclipse were found to be 271 ° for the second contact and 111 ° for the third. In the case of the 1963 eclipse the equivalent times were Time of second contact = 14h Time of third contact -- 1 4 h

48m

41.2sec,

49m

19.4sec.

The angles of contact at the second and third contacts were 257 ° and 68 ° , respectively. We shall consider now the problem of determining the orientation of the lunar image on the photographic film. The case is illustrated in Fig. 26 where Z is the local zenith and z the angle between the normal to the film and the meridian plane. We shall make use of the following notations: P, the N o r t h pole; ¢, the latitude of the site of observation; 5, the declination of the Sun; H, the hour angle of the Sun; K, the hour angle of the telescope; A, the azimuth of the telescope; OT, the direction of the optical axis of the telescope; OM, the direction normal to the coelostat mirror; OS, the direction of the Sun; OT', the direction to which the telescope p o i n t s w h e n the altitude, a, of the telescope is equal to zero; i, t h e angle of incidence o f a r a y of light f r o m t h e S u n on the coelostat mirror,

cos A = --sin 5 sec ¢.

(57)

Therefore, Eq. (56) becomes sinz = --tanStanA = (cos2 q~ -- sin 2 5)1/2/cos 5.

(58)

The last relation enables us to determine z. But it is now necessary to know the spherical angle S M P which we shall denote Z

FIC. 26.

b y 7, and b y which the image of the Moon is rotated inside the plane of film due to the fact t h a t the normal to the eoelostat mirror is not in the meridian plane. The determination of ~ can be done b y means of the relations sin n cos 5

sin (K - H) sin i '

sin (½r -- 7) 1 sin 5 = sin i" Therefore, tan ~ = cot 5 sin ½(K -- H).

(59)

This formula was used f o r t h e determination of n in all cases . . . . . . . . For the determination of the particular section of the marginal zone recorded on the eclipse photographs we::!leed:.to know, in

356

D. CARSON ET AL.

addition, the geocentric zenith distance of the Moon ~, the parallactic angle Q (i.e., one between the vertical and the declination circle through the Moon's center, in other words, the angle ZSP of Fig. 26) a n d the topocentric parallax 7r', which ca n be calculated from the geocentric right ascension a, declination 6, and parallax r, of the Moon from the following formulas: cos ~ = sin ¢bsin 6 + cos ~ cos 6 cos H,

(60) sin Q = sin H cos 4~csc ~,

(61)

sin $ - cos z sin 6 sin z cos 6 '

(6 2)

7r' = ~(sin ~ + 0.0084 sin 2~).

(63)

cos Q =

If the geocentric librations of the Moon are l, b, C (longitude, latitude, and position angle) then the corrections to these, according to Atkinson (1951), are hl = - - r ' sin (Q - C) sec b,

(64)

Ab = ~' cos (Q - C),

(65)

AC = sin (b + 6b)Al -- ~' sin Q tan &

(66)

The numerical values of the topocentric libration angles are found to be, l' = 4?58 b' = --0717 C ' = 16732

(67)

for the 1962 eclipse, and l' = - - 5 7 3 6

b' = 0?07

(68)

C' = - - 1 5 7 3 3

for the 1963 eclipse. The topocentric libration angles are necessary mainly when one intends to compare the lunar profiles obtained here with those derived by other surveys of the marginal zone, such as the ones of H a y n (1907) and Watts (1963). V. ANALYSIS OF THE PROFILES AND COMPARISON WITH PRECEDING I~ESULTS

The numerical values of the elevations along the profile of the 1962 eclipse were analyzed in the same way as those of the 1963 onc the latter case will be discussed in

detail presently--but the answers obtained were not compatible with the corresponding data of H a y n (1907), Watts (1963), Yakovkin (1962), Dommanget (1962), Potter (1962), and others. This was anticipated by the extent of the distortion caused by unfavorable weather but mainly by the irradiation in the photographic emulsion. The magritude of the effect is much larger than the deviations of the profile from a circle which we originally set out to determine. The effect was made serious by overexposure. The same exposure times were used for the South African eclipse photographs but, in the latter case, the annnlus was much narrower than in Senegal, and hence the effect was negligible. The effect is difficult to correct because solar limb-darkening makes the construction of simple calibration curves impossible. Nonuniform irradiation impairs the quality of the statistics. The statistics are also deficient because clouds marred m a n y of the photographs of the central phase of the 1962 eclipse. The profile of the 1963 eclipse is given in Table I where the standard selenographic coordinates (~, ~, ~') of the 62 points are listed. The standard deviation of these points referred to a 1738.0 km mean radius is 2.1 km, and the mean error is 0.44 km or 0'.'23. A harmonic analysis of these data was performed with the aim of separating harmonics of orders from zero to four, since the accuracy of the data would not allow the accurate determination of higher terms. For comparison, we used the profiles determined from the surveys of H a y n (1907) and Watts (1963), which we list in Tables II and III. In order to check the results, the analysis was performed in two different ways. First, a separate Fourier analysis of the profiles presented by Hayn, Watts, and the 1963 eclipses was made and the results compared. Second, each profile was combined with the data of Schrutka-Rechtenstamm, the Army Map Service (AMS) (1964), and the Aeronautical Chart and Information Center (ACIC) (1965). An analysis into spherical harmonics was then applied in each case. The results were then used in order to determine the values of the low-order Fourier terms of the limb. In principle, the latter terms must be independent of the

LUNAR PROFILES FROM SOLAR ECLIPSES OF 1962 AND 1963 TABLE I POINTS OF Tr~. PROFILE OBTAINED FROM THE 1963 ANNULAR ECLIPSE

O. 03490 --0.13910

--

--0.24187

--0.34183 --0.43784

--0.52933 --0.61440 --0.69291 --O. 76417 --O. 82651 --0.87988 --0.92393 --0.95820 --0.98172 --0.99480 - - O . 99692 --0.98689 --0.96510 --0. 93398 --0.89561

34603 0. 78748 --0. 71806 -- 0. 64103 --0.55857 --0.47005 --0. 37522 --0. 27547 --0.17342 --0. 06973 0. 03491 0.13928 0. 24177 0. 34173 0. 43735 0. 52882 0. 61389 0. 69247 0. 76364 0. 82605 0. 87996 0. 92341 0. 95764 0. 98167 O. 99150 0. 99436 0. 98575 0. 96542 0. 93606 0. 89579 0. 34566 0. 78633 --O. --

O. 99936 O. 98973 0.97009 O. 93917 0.89770 O. 84710 0. 78639 0.71753 0.64121 0.55748 0.46734 0.37329 0.27476 0.17310 O. 06956 --0.03481 --0.13870 - O. 24063 --0. 33994 --0.43682 --0. 52866 -- 0.61525 --0. 69342 -- 0. 76395 --0.82812 --0.88403 -- 0. 92871 --0. 96068 --0. 98351 --0. 99724 --0. 99959 --0. 99105 --0. 96969 --0. 93888 --0. 89671 --0. 84629 --0. 78574 --0. 71707 --0. 64077 -- 0. 55718 --0. 46789 --0. 37308 --0. 27460 --0.17309 --0. 06933 0. 03472 0.13854 0. 24071 0. 34070 0. 43690 0. 52843 0. 61435

O. 00076 --0.00895 --0.01853 --0.02785 --0.03681 --0.64535 --0.05331 --0.06065 --0.06732 --0.07316 --0.07817 --0.08230 --0.08551 --0.08772 --0.08895 --0.08915 --0.08821 --0. 08617 --0. 08325 --0.07964 -- O. 07499 -- 0. 06949 --0. 06300 -- 0. 05580 --0.04809 --0.03981 -- 0. 03096 --0. 02167 --0. 01215 --0. 00249 0. 00727 0. 01700 0. 02655 0. 03588 0. 04480 0. 05334 0. 06129 0. 06865 0. 07532 0. 08117 0. 08623 0. 09031 0. 09352 0. 09579 O. 09670 0. 09697 0. 09617 0. 09425 0. 09150 0. 08771 0. 08301 0. 07745

TABLE I

0. 71865 0.64128 0. 55734 0.46912 0.37458 0. 27567 0.17366 0.06972

357

(Continued)

0. 69399 0. 76425 0. 82629 0.88228 0. 92712 0. 96137 0. 98488 0. 99711

0.07111 0. 06386 0. 05600 0. 04777 0. 03894 0. 02972 0. 02020 0. 01051

front side Control System with which the limb data are combined, but due to the finite number of points available for both areas, the values obtained for the same profile differed by amounts smaller than the mean error of the profile. For this reason, we shall present below the analytical form of the profile as obtained by means of the harmonic analysis of the entire figure. In such a case the radius r of the surface points whose selenographic coordinates will be denoted by ~,/3 will be given by the expression

i ffio $'=0

(J~i cosg~ -t- J'ij siny~)T~(~),

(69)

where r0 -- 1738.0 km and ~ --- sin/3. The e q u a t i o n of t h e profile for zero librations is then r = r0 -t- J00 + ¼J20 - ~J22 + ~ J 4 0 -- {--~J42 -F SlaJ44 -]- (Jlo -t- ~Jso -- -~J3~) sin ~ + (J'l, + ]J'al -- -~J'ss) cos/3 + (~J'21 + ]J'4~ (70) _ lO5 J'43) sin 213 + ( - I J 2 o - ]J~2 -- a~J4o -{- ~-~J42 -}- !~-~J44) cos 2/3 -P ( - ~ J 3 o -tx~Ys2) sin 3/3 + ( - - ~ J ' 8 1 -- aa~J'33) cos 3/3 "t- (--~-~J'41 _ i sosj,48) sin 48 -}- (~-J4o + 11°5J4~ -

-

_{_ 1 o 5j44) c o s

4f],

where ~ is no longer the selenographic latitude but a position angle varying counterclockwise between 0 ° and 360 ° from the positive direction of the O~ axis of the standard frame of reference. By substituting in Eq. (70) the numerical values of the coefficients determined from the profile of the 1963 eclipse and expressing the obtained coefficients in geometric seconds of arc, we

358

D, CARSON ET AL.

T A B L E II POINTS OF THE MARGINAL ZONE (ZERo LIBRATION~ ~" = 0) OBTAINED FROM HAYN'S MAPS

0.000000 --.087155 --.173666 --.258846

--.341983 --.422482 --.499892 --.573453

--.642718 --.707334 --.765798

--.818976 --.865468 --.906016 --.939491 --.965925

--.985124 --.996621 --1.000214 --.996515 --.984702 --.965407 --.938483 --.906016 --.866118 --.819591 --.766701 --.707182 --.642511 --.573268 --.500160 --.423071 --.342130

--.258819 --.173573 --.087071

.999571 .996194 .984913 .966029 .939591 .906016 .865839 .818976 .765962 .707334 .642580 ,573453 .499678 .422482 341946 .258819 .173704 .087193 0.000000 --.087183 --.173629 --.258680 --.341580 --.422482 --.500053 --.573883 --.643338 --.707182 --.765715 --.818712 --.866303 --.907279 --.939994 --.965925 --.984385 --.995233

0.000000 .087155 .173890 .259124 .342020 .422573 .500107 .573760 .642925 .706879 .765633 .818888 .865932 .906307 .939793 .966547 .985441 .995981 .999035 .995340 .984174 .965718 .939994 .906113 .866396 .819503 .766537 .707789 .642994 .573207 .499034 .422437 .342130 .258846 .173666 .087174

--1.000785 --.996194 --.986180 --.967065 --.939692 --.906210 --.866211 --.819415 --.766208 --.706879 --.642442 --.573391 --.499946 --.422618 --.342056 --.258985 --.173759 --.087137 0.000000 .087080 .173536 .258763 .342130 .422527 .500214 .573822 .643201 .707789 .766290 .818625 .864353 .905919 .939994 .966029 .984913 .996408

find t h e f o l l o w i n g e x p r e s s i o n f o r t h e l u n a r profile : r --- r0 -+ + --

0':21 0':54 0':27 0':16

sin sin sin sin

~ -- 0':11 cos fl 2f~ -- 0'.'16 cos 2/~ 3/3 -- 0':05 cos 3/~ 4~ + 0':11 cos 4/3.

(71)

T h e profile o b t a i n e d b y t h e a n a l y s i s of t h e d a t a o b t a i n e d f r o m H a y n ' s m a p s (1907), is expressed similarly by the equation r = ro -+ + --

0"21 0':38 0'.'32 0':05

sin sin Sin sin

~ + 0'.'05 cos/3 2/3'-- 0':38 cos 2/3 3/3 - - 0(t41 cos 3B 4~ + 0':16 cos 4¢.

(72)

TABLE III POINTS OF THE MARGINALZONE (ZERo LIBRATION, ~" = 0) TAKEN FROM THE MAPS BY WATTS

0.000000 --.087146 --.173611 --.258763 --.341910 --.422618 --.500000 --.573268

--.642925 --.707334 --.765798 --.818888 --.865653 --.906016 --.939894 --.965615 --.984913

--.997369 --1.000107 --.997476 --.984913

--.964579 --.939390 --.906307 --.866861

--.820820 --.766783 --.707485 --.642580

--.573453 --.500750 --.423071 --.342790 --.258930

--.173387 --.087081

.999785 .996087 .984596 .965718 .939390 .906307 .866025 .818712 .766208 .707334 .642580 .573391 .499785 .422482 .342093 .258735 .173666 .087258 0.000000 --.087267 --.173666 --.258458 --.341910 --.422618 --.500482 --.574744 --.643407 --.707485 --.765798 --.818976 --.867325 --.907279 --.941808 --.966340 --.983329 --.995340

0.000000 087118 173927 258791 342093 422165 499946 573760 .642649 706955 765389 818185 865746 906502 940297 966547 985863 994806 997104 994485 984068 964993 939491 906890 865839 819503 766783 708016 642443 573207 499142 422890 341946 .258930 .173648 .087127

--1.000857 --.995767 --.986391

--.965822 --.939894

--.905335 --.865932 --.819415

--.765880 --.706955 --.642236 --.572899 --.499839

--.422708 --.342240 --.258985

--.173834 --.087034

0.000000 .087006 .173517 .258569 .341946 .422890 .499892 .573822 .643407 .708016 .765633 .818625 .864539 .906890 .939491 .966340 .984807 .995874

A t h i r d e x p r e s s i o n of t h e s a m e profile is o b t a i n e d b y t h e a n a l y s i s of t h e d a t a o b t a i n e d f r o m t h e r e c e n t s u r v e y of t h e m a r g i n a l z o n e b y W a t t s (1963). I t s a c t u a l f o r m is r = r0 -+ + --

0':32 0':59 0':27 0':32

sin sin sin sin

¢~ -- 0':16 cos/3 2~ -- 0':21 cos 2~ 3~ -- 0':05 cos 3B 4~ + 0':21 cos 48.

(73)

Y a k o v k i n (1962) h a s s u g g e s t e d t h a t t h e l u n a r profile c a n be d e s c r i b e d b y a m o d e l w h o s e e x p a n s i o n in F o u r i e r series is ( G o u d a s , 1965)

LUNAR PROFILES FROM SOLAR ECLIPSES OF 1962 AND 1963 3a

r = r0 A- 16

a

4 cos 2(~ A- "y)

a + ~-~ cos 4(8 + ~,)

48a ~ j=l

sin (2j + 1)(8 + ~) (2j -- 3)(2j -- 1)(2j + 1)(2j + 3)(2j + 5)' (74) where ~/ denotes a phase constant. The numerical values of the constants involved h a v e been determined b y Yakovkin from a long series of observations of star occultations as well as measurements of photographs. I n particular, he concludes t h a t a = 0~.96 -~ 0.~08b

a = 0'.'96 + 0'.'02b for the western half. The phase constant is found to be 15 ° . Thus, an expression of the zero libration profile similar to these given by Eqs. (71), (72), and (73) is obtained

- -

(75)

T h e coefficients A'~ and B'I need not be discussed here. F r o m the figures quoted b y Potter (1962) we can present the limb in a Fourier form as follows: r =r0WA~lsinS+B"lcos8 -~ 0.~36 sin 28 -- 0."13 cos 2~.

The lower limit given by Yakovkin (1962) is based on an assumption about the shape of the M o o n and, of course, measurements. This assumption, which is undoubtedly unnecessary here, seems to have affected considerably the estimate of the above angle as well as the axes of the elliptical component of the limb. If we eliminate the estimates b y Y a k o v k i n and H a y n (23°), we find t h a t the recent measurements are in almost perfect agreement. In particular, we have estimates as follows: P o t t e r (1962) : 34 ° W a t t s (1963) : 35 ° Present determination (1965): 37 ° D o m m a n g e t (1962) also gave similar values, which correspond to libration angles different t h a n zero.

for the eastern half of the limb and

r = r0 + A'I sin 8 + B'I cos 8 -{- 0':17 sin 28 - 0'.'30 cos 28 + 0':44 sin 3~ -~ 0~.44 cos 38 0':05 sin 4f~ + 0."08 cos 4~.

359

(76)

This expansion corresponds to libration angles t - - 2 ? 5 and b = - 1 ? 4 and, therefore, to a profile different t h a n all others given above. Nevertheless, there is little difference between it and the profile corresponding to zero libration. The constants A ~, B x, need not be considered here, either. F r o m Eqs. (71) to (76) we find the major axis of the elliptical component of the limb at zero libration and the axis of rotation of Moon m a k e an angle between 15 ° and 37 ° measured clockwise from the N o r t h Pole.

REFERENCES Aa~Y MAP SERVICE (1964). Tech. Report 29, Part Two. Corps of Engineers, U. S. Army, Washington, D. C. AERONAUTICAL CHART AND INFORMATION CENTER (1965). Tech. Report 15, U. S. Air Force, St.

Louis, Missouri. ATKINSO~, R. D'E. (1951). Monthly Notices Roy. Astron. Soc. I I I , 448. DOMMANGET,J. (1962). Commun. Obs. Roy. Belg., No. 208. GOUDAS,C. L. (1965). Icarus 4, 218-220. ttAYN, F. (1907). "Selenographische Coordinaten, III," Abhandl. Saechs. Ges. Wiss. Math.-Phys. Kl. 30, 1-105. Nz~DmV, A. (1957). Bull. Engelhardt Obs. 30. PorrEa, H. I. (1962). In "The Moon" (Z. Kopal and Z. K. Mikhailov, eds.), pp. 63-66. Academic Press, London and New York. SCHVS~R, A. (1922). Proc. Phys. Soc. 32, 135. SBLwY.~, E. W. H. (1950). "Photography in Astronomy," p. 66. Eastman Kodak Co., Rochester, New York. WATTS, C. B. (1963). The marginal zone of the Moon. Astron. Papers 17 (Astronomical Ephemeris and Nautical Almanac, U. S. Govt. Printing Office, Washington, D. C.) WEIMER, TH. (1952). "Atlas de Profils Lunaires." Obs. de Paris. YAKOVKIN,A. A. (1962). In "The Moon, A Russian View" (A. V. Markov, ed.), pp. 3-53, Univ. of Chicago Press, Chicago, Illinois.