WATTS CHARTS The Marginal Zone of the Moon [Notes provided by D. Herald, of Canberra Australia - May 2006] 1. INTRODUCTION In his epochal work taking some 17 years, Chester Watts (of the US Naval Observatory) published his charts of 'The Marginal Zone of the Moon' - in Astronomical Papers prepared for the use of the American Ephemeris and Nautical Almanac, Volume XVII, 1963. These charts are generically referred to as the 'Watts Charts'. The charts as published consist of 1800 charts corresponding to each 0.2 degrees in position angle around the limb of the moon, measured from the north pole. Each chart has a grid covering the maximum longitude and latitude librations (+/- 9 degrees, and 8 degrees, respectively). The charts represent the elevation of the lunar limb by way of contour lines - with a contour interval of 0.2". In a project completed in 1970, the HM Nautical Almanac Office of Royal Greenwich Observatory digitized the Watts charts. Details of this project are described in 'A digitised version of C.B. Watts' Charts of the Marginal Zone of the Moon', The Moon 2 (1971) 463-467. This digitisation recorded the (x,y) coordinates along the length of each contour line - effectively representing the contours as a series of straight line segments. In some regions they drew and digitized intermediate contour lines. The extraction of a height for a particular libration value required a complex interpolation routine. Subsequently, staff at the United States Naval Observatory (using a 2- dimensional interpolation program written by Ron Abileah) converted the digitized data into point heights at each 0.2 degree interval of the longitude and latitude libration covered by each chart - for each of the 1800 charts. In this process any intermediate contour lines inserted by HMNAO were ignored. It is this version of the Watts charts (referred to below as the USNO file) that is provided in this file. The USNO file was distributed in binary format. For each data point the height was held as a 10-bit binary integer, and two accuracy codes were each held in 3 bits - making a total of 16 bits. That is, each data point was stored in two bytes. For archiving, the data has been converted to ASCII. NOTE: When constructing the charts, Watts intended that the angle used in his charts would correspond to the position angle with respect to the lunar north pole. However in fact there is a small difference. In the following, the angle used in the Watts Charts is referred to as the Watts Angle (WA), whereas the position angle from the axis of the moon is referred to as the Axis Angle (AA).] 2. FORMAT OF WATTS.TXT Each point on the lunar limb is represented by a 6-byte field, with the following definitions: -------------------------------------------------------------------------- Bytes Format Units Label Explanations -------------------------------------------------------------------------- 1 A1 --- sgn sign of the height [+ or -] 2- 4 I3 --- Ht height of the limb, in units of 0.01" 5 I1 --- a Accuracy code a [0-7]. See below 6 I1 --- b Accuracy code b [0-7]. See below -------------------------------------------------------------------------- The data is organized as follows: 1. There are 91 blocks of data for each longitude libration from +9.0 to -9.0, at 0.2 degree intervals; 2. For each longitude block of data, there are 81 blocks of data for each latitude libration from +8.0 to -8.0, at 0.2 degree intervals; 3. For each latitude block of data, there are 1800 data points - corresponding to each chart around the limb of the moon (from 0.0 to 359.8) For any longitude libration L, latitude libration B and Watts Angle value WA - expressed in multiples of 0.2 degree intervals (and not fractions thereof) within the ranges specified above, the record number in the file corresponding to that libration value is found as: Record = 729000 * (9.0 - L) + 9000 (8.0 - B) + 5 * WA + 1 The total number of valid records is 13,267,800, for a file size of 79,606,800 bytes. [The file includes an extra block of 1800 data points at the end (for unknown reasons) - each with a height of 0, Accuracy Code a = 0 and Accuracy Code b = 1.] 3. ACCURACY CODE The Accuracy code has two parts, a and b - each with possible values between 0 and 7. The following explanation was provided with the file. *************** The contours on each chart have a preferential direction, inclined at an angle to the horizontal of the chart equal to the Watts angle. If the given point requires extrapolation from the last contour toward the chart edge in a direction perpendicular to the preferential direction, b = 4 or 5; for interpolation in that direction, b = 1 or 2. If extrapolation is required in the preferential direction itself, b = 2 or 5; for interpolation, b = 1 or 4. If the interval between contours is so wide that a parabola cannot be fit to the heights perpendicular to the preferential direction (or if such a parabola would imply missing contours), then b = 0 or 3. It has the higher value (3) when the extra-wide interval between contours occurs in an extrapolated part of the chart. If the number of contours is so few in some region of the charts that one cannot tell from a single cut through the contours perpendicular to the preferential direction whether the slope is increasing or decreasing, then b = 6. Complete absence of contours intersected in such a cut or other type of error detected is indicated by b = 7. For b = 0,1, or 7, then a/2 is the maximum density of contours in the neighborhood of the given point, measured in number of contours per degree. For b = 2,3,4,5, or 6, a/2 is the maximum extrapolation distance in degrees. ************** 3.1 Accuracy Code for chart 36.0 For the Chart 36.0 the USNO data was incorrect - see (5) below. The accuracy codes for the 36.0 chart have unique values and definitions. * Code b is 7 * Code a has the following possible values and meanings: 5 = accurate; 6 = reasonable extrapolation; 7 = poor extrapolation. NOTE If code b is 7 and code a <5, the chart is NOT for 36.0, and the usual code definitions apply. Code a >4 with code b =7 can ONLY occur with the 36.0 chart. 3.2 A simplified accuracy code A simplified accuracy code can be derived from the a and b codes as follows: Code Meaning a and b conditions 1 Reliable b = 0 or 1; or 1 2; or b=7 and a=7; or height >3.4 or <-3.4 4. COMMENTS ON THE TRANSFORMATION FROM CONTOURS TO GRID POINTS The following comments were provided with the USNO file. ***************** *INTERPOLATION OF CONTOURS. In transforming from contour charts (as printed) to grid profiles (as stored on disk), an elaborate interpolation algorithm was used, which took advantage of the constraint that in the direction perpendicular to the preferential direction, the heights must lie on a series of parabolas of known shape. Consistency checking of heights of consecutive contours was also performed. Contours with odd heights, added by HMNAO, have been ignored. Hence, in general, heights between contours will be read as accurately as heights on contours, and no allowance need be made in the accuracy code for the distance from the nearest contour (unless b=0 or 3). This is true even near maxima and minima of the heights. For an ordinary use with no extrapolations (b=1), the reading error will equal the tracing error, about 0".02 or so on the average. *EXTRAPOLATION OF CONTOURS. Here, the situation is not as satisfactory as with interpolation. However, since Watts had already extrapolated the observational data as far as he thought advisable in preparing the printed charts, it probably matters little how the data are extrapolated. The problem faced by the computer was the complex one of pattern recognition: there was no simple means available to extrapolate "qualitatively", as a human would do intuitively, utilizing knowledge of the behavior of the entire contour, as well as its relationship to, and the behavior of, adjacent contours. This being the case, a variety of simple extrapolation procedures were tried. However, only one procedure simple enough to implement could be found which satisfied the very useful constraint that contours must not intersect, namely, RADIAL extrapolation from the chart center through the contour end points to the chart edge. It will be objected that this produces "kinks" in the contours, and is often quite unrealistic. However, considering the alternatives in which contours can intersect, at least the radial method will usually be closest over large extrapolation distances to the best guess "qualitative" extrapolation. ***************** 5. USNO FILE IN ERROR AT WA = 36.0 Circa 1991, D Herald noted that the data in the USNO file for Watts Angle 36.0 deg was erroneous. While the cause of the error is not known, the USNO data appears to be similar to that for the 0.0 chart. To address this issue, Herald created a file containing heights manually read from the paper chart, together with a code for the accuracy of the height. The data in the USNO file was replaced with data derived (using 4-point interpolation in both L and B) from this manually entered data. This means that while the data for the 36.0 chart is now a good representation of the 36.0 chart, it has been generated differently to the remainder of the dataset. The main consequence is that the accuracy codes have a unique set of values specific to this chart. See 3.1 above. 6. USING THE CHARTS The following considerations should be taken into account when using the data. a. Scale The Watts chart limb corrections are for the Moon's mean distance. They must be scaled by the ratio of mean to true distance to obtain the apparent limb height. The relevant mean distance that was (presumably) used for the construction of the charts is 384,400km, corresponding to a parallax of 3422.70". b. Orientation of the zero point of the Watts Angle Early studies of occultation data (especially grazing occultations) indicated that the zero point in position angle of the Watts Charts did not correspond with the lunar north pole. See, for example: "Some Notes on the Use of the Watts Limb-Correction Charts" by T. C. Van Flandern, A.J. 75, 744-746, 1970). "On the orientation of C.B. Watts' charts of the marginal zone of the Moon" L.V. Morrison MNRAS (1970) 149, 81-90 The primary source of this error was explained by David K Scott in Astron J 95, 1567-1568 (1988), where the error is shown to be 0.214 deg. Consequently it is necessary to add 0.21 deg. to a computed Axis Angle to obtain the corresponding Watts Angle. Additionally, Watts constructed the charts using 1.564 deg as the inclination of the lunar equator (I). [Watts, C.B. 1955 Astron J 60, 443- 445] The inclination adopted in the current Astronomical Almanac is 1.542 deg - a difference from the Watts value of 0.022 deg. The charts should be entered with a value of the Watts Angle that is consistent with the value of I used to create the charts. This can be done by either (i) computing the librations using the value of 1.564, or (ii) when converting from Axis Angle to Watts Angle, adding the additional amount of -.022 cos(L - W) where L-W is the elongation of the moon from the ascending node of its orbit on the ecliptic. c. Accuracy in the Cassini regions The Watts charts generally have no contours covering the Cassini regions of the Moon. However grazing occultation observations indicate that: At the northern limb with positive latitude librations, the profile is generally higher than that indicated by the Watts charts - and the extent of the Cassini region is incorrectly identified, with the negative height contours given in the charts in the range of 345 to 355 deg. for positive latitude librations generally being lower than the actual limb of the moon. At the southern limb with negative latitude librations, the profile is generally lower than that indicated by the Watts charts. d. Systematic corrections Several studies have suggested the existence of systematic corrections to the limb profiles. Illustrative of these are: "Analysis of Lunar Occultations - III. Systematic corrections to Watts' limb-profiles for the Moon" L.V. Morrison & G.M. Appleby, MNRAS (1981) 196, 1013-1020. "Analysis of Lunar Occultations - V. Grazing Occultations" 1964 - 1977 G.M. Appleby & L.V. Morrison, MNRAS (1983) 205, 57-65. "An analysis of lunar occultations in the years 1955-1980 using the new lunar ephemeris ELP2000", M. Soma, Celestial Mech,35(1985) 45-88. "Corrections to Watts' datum from photoelectric occultations", G. Rossello & C. Jordi, The Moon and the Planets 27(1982) 131-134. "Corrections to Watts' Charts varying with Libration", G. Rossello, C. Jordi, A. Salazar, Astrophysics & Space Science 177:331-338, 1991. "Examination into Error of Watts' Datum and Recomputation of Moon's Position", Yoshio Kubo, Report of Hydrographic Researches No.30, pp.37-47 (1994). 7. WATTSPHOTOS.dat The printed Watts charts includes a table of the photographs used in the survey, including the libration value and arc of the usable limb. This data has been OCR'd and sorted by longitude libration. It is provided in the file WATTSPHOTOS.dat. The format of this file is: -------------------------------------------------------------------------- Bytes Format Units Label Explanations -------------------------------------------------------------------------- 1- 3 I3 --- Ordinal sequence in the printed table 4- 9 I6 --- Identification No. in the printed table 10- 14 I5 --- Yr Year of the photograph 15- 18 A4 --- Mth Month of the photograph 19- 26 F8.4 --- Day Day of the photograph 27 A1 --- Flag: * used as a 'standard' # used to test circularity of 2nd datum $ used for both the above 28- 33 F6.2 deg l Longitude libration 34- 39 F6.2 deg b Latitude libration 40- 43 I4 deg Arc1 Start of measured arc 44- 47 I4 deg Arc2 End of measured arc 48- 49 --- CR/LF sequence -------------------------------------------------------------------------- The Table as printed is sorted by date for each of the instruments used in the survey. The ordinal sequence range for each instrument is as follows: 1-336 Washington 337-476 Johannesburg 477-503 Flagstaff To confirm the accuracy of the OCR, the topocentric librations were independently computed for the given date and telescope and compared to the librations given in the table. After correcting all OCR errors, there were a number of instances where the difference in either libration exceeded 0.1 deg: Table Computed Seq Ref Date L B L B 71 93 1949 Feb 12.1376 -4.24 -6.19 -4.22 -6.30 89 127 1949 Aug 8.2279* 4.13 6.33 4.59 6.99 350 35942 1934 Aug 24.9210* 2.32 -2.75 2.47 -2.89 416 59265 1946 Sep 10.9100# -3.84 6.29 -4.83 6.30 442 63827 1949 May 12.9299 3.63 3.56 3.78 3.53 466 65751 1951 Jun 19.9651 1.03 6.11 0.80 6.12 It may be remarked that the 1946 Sep 10 instance may be a publication error - the difference in L being essentially 1.0 deg. The other instances appear to be computational errors. While the difference for most is less than about 0.2 deg, the difference for the 1949 August instance is much larger. Furthermore this instance was used as a 'standard' in the preparation of the charts. The effect of this apparent error is not known at this time - although it may be noted that there are 5 other photos within the range of +4 to +5 in L, +6 to +7 in B, that cover a significant portion of the WA range of the 1949 August instance. 8. PREFACE AND INTRODUCTION TO THE WATTS CHARTS. For completion of the historical basis for this data set, the following are the preface and introduction from the printed Watts charts. ***************** PREFACE A proposal that the Naval Observatory undertake a survey of the marginal zone of the Moon was made by Dr. Watts in 1942, but circumstances caused the project to be postponed until 1946. The work has since been carried on without interruption, although various unforeseen technical and theoretical problems have delayed its completion. The USAF Aeronautical Chart and Information Center expressed an early interest in the survey of the marginal zone. This agency recognized that the results would be useful in connection with survey projects involving the 'Moon in combination with stars or the Sun, and in furnishing basic topographic information in the limb areas. Since 1954, therefore, ACIC has provided extensive technical services in direct support of the survey. This assistance has brought about the completion of the task at an earlier date than would have otherwise been possible. Following his retirement in 1959, WATTS was appointed a Research Associate by Yale University, which desired to further the completion of the survey. The interest of the university in the project was also demonstrated by the loan of lunar photographs made at the Yale-Columbia Southern Station. It is hoped that the results of the survey will advance materially many researches related to the Moon. T. S. BASKETT, Captain, U.S. Navy, Superintendent, Naval Observatory Washington, March, 1963. THE MARGINAL ZONE OF THE MOON INTRODUCTION The marginal zone of the Moon contains those portions of the physical surface that contribute, from time to time, to the outline of the limb as seen from any part of the Earth. It also contains numerous areas that are prevented from doing so by neighboring features of greater elevation, as well as some that can contribute only to the outline of the dark limb. The desire to evaluate the errors introduced in several classes of observations by limb irregularities has resulted in a number of investigations of the topography of the zone. The problem presented is somewhat difficult, inasmuch as this area, when viewed from the Earth, is greatly foreshortened. The widest portions, which are midway between the poles and the equator, include only about 0.08 of the radius of the disk when completely exposed to view, the farther half of the zone taking up about one-fourth of this amount. Nevertheless, the zone contains about 0.30 of the portion of the Moon's surface that is visible from the earth. A preliminary chart of the zone, constructed by F. HAYN (1907) at Leipzig, was based on heliometer measures joining the craterlet Mosting A to points on the limb. This was followed by one derived by E. PRZYBYLLOK (1908) from occultation residuals. A much improved chart was next prepared by HAYN (1914) from measures of photographs obtained on 49 nights during the period 1908-1912, to which he added measures of the limb in photographs of the solar eclipse of 1912. April 17. Recently a chart of the same type was derived by NEFEDIEV (1957) from heliometer measures made at the Engelhardt Observatory. All of these charts are referred to a system of spherical coordinates related to the Moon's equator system. A different approach to the problem, by WEIMER (1952), consists of a series of profiles each of which represents the outline derived from one of the Paris photographs. There are 139 profiles, arranged according to the librations associated with them. making possible the interpolation of a limb correction for a particular position angle and combination of the librations. The Washington survey was undertaken at the Naval Observatory in 1946 with two principal objects in view. These were the delineation of the Moon's limb area in greater detail than had previously been found practicable, and the establishment of a well-defined datum to which the indicated elevations could be referred. The results are based on measures of photographs obtained on 503 nights during the period 1927- 1956. In this work the limb corrections are presented in a form differing from any previously used. The 1800 small charts have as their horizontal and vertical arguments the topocentric librations in longitude and latitude respectively. As each chart corresponds to a single position angle, the user can interpolate limb corrections derived from adjacent charts. Acknowledgments are due to many for assistance in the survey. The photographs from the Yale-Columbia Southern Station at Johannesburg were kindly made available by Professor BROUWER. They were exposed with the 26-inch photographic refractor by H. L. ALDEN during the period 1927 to 1945 and by CYRIL JACKSON thereafter. For the use of the Lowell Observatory plates, which were exposed by ARTHUR L. BENNETT, I am indebted to the late C. O. LAMPLAND. Support was furnished by the Office of Naval Research and by the Aeronautical Chart and Information Center during a considerable portion of the program. Several suggestions made by R. D'E. ATKINSON during the early stages of the work were most helpful. The assistance of my colleague A. N. ADAMS in connection with the design of the electronic apparatus used in the measurement of the photographs contributed greatly to the accuracy of the results. Miss CAROLYN S. MCGREGOR, who assisted with the project throughout its course and became familiar With its many details, rendered invaluable service. The contributions of the following persons are also gratefully acknowledged. J. W. KITCHENS, U. S. LYONS, DAVID K. SCOTT, and WARREN W. TOLAND devoted a considerable portion of their time to the work. S. M. BESTUL, MISS JOSEPHINE P. BROWN, MISS WILHELMINE I. BURGAT, MAHLON S. HUNT, CLIFFORD W. KING, Miss REBECCA LICHTENSTEIN, Miss MARIE PRYTULAK, Miss IDA E. RAY, and ROGER D. STEWART were associated with the survey during shorter periods. The measuring and plotting apparatus was constructed in the Naval Observatory instrument shop under the direction of GEORGE E. STEINACKER. **************