NOAA TR NMFS SSRF-669 A UNITED STATES DEPARTMENT OF COMMERCE PUBLICATION V ' 4;.; r W NOAA Technical Report NMFS SSRF-669 LiB/vARY U.S. DEPARTMENT OF COMMERCE National Oceanic and Atmospheric Administration ij MOV 1 9 Itsyo National Marine Fisheries Service ' '^ Woods Hole, Mass. ^ Subpoint Prediction for Direct Readout Meteorological Satellites L. E. EBER SEATTLE, WA August 1973 NOAA TECHNICAL REPORTS National Marine Fisheries Service, Special Scientific Report-Fisheries Series The major responsibilities of the National Marine Fisheries Service (NMFS) are to monitor and assess the abundance and geographic distribution of fishery resomxes, to understand and predict fluctuations in the quantity and distribution of these resources, and to establish levels for optimum use of the resources. NMFS is also charged with the development and implementation of policies for managing national fishing grounds, develop- ment and enforcement of domestic fisheries regulations, surveillance of foreign fishing off United States coastal waters, and the development and enforcement of international fishery agreements and policies. NMFS also as- sists the fishing industry through marketing service and economic analysis programs, and mortgage insurance and vessel construction subsidies. It collects, analyzes, and publishes statistics on various phases of the industry. The Special Scientific Report — Fisheries series was established in 1949. The series carries reports on scien- tific investigations that docuinent long-term continuing programs of NMFS, or intensive .scientific reports on studies of restricted scope. The reports may deal with applied fishery problems. The series is also used as a medium for the publication of bibliographies of a specialized scientific nature. NOAA Technical Reports NMFS SSRF are available free in limited numbers to governmental agencies, both Federal and State. They are also available in exchange for other scientific and technical publications in the marine sciences. Individual copies may be obtained (unless otherwise noted) from NOAA Publications Section, Rockville, Md. 20852. Recent SSRF's are: 619 Macrozooplankton and small nekton in the coastal waters off Vancouver Island (Canada) and Washington, spring and fall of 1963. By Donald S. Day, January 1971, iii + 94 pp., 19 figs., 13 tables. 620 The Trade Wind Zone Oceanography Pilot Study. Part IX: The sea-level wind field and wind stress values, July 1963 to June 1965. By Gunter R. Seckel. June 1970, iii -p 66 pp., 5 figs. 621 Predation by sculpins on fall chinook salmon, Oncorhynchus tshawytseha, fry of hatchery oi-- igin. By Benjamin G. Patten. February 1971, iii + 14 pp., 6 figs., 9 tables. 622 Number and lengths, by season, of fishes caught with an otter trawl near Woods Hole, Massa- chusetts, September 1961 to December 1962. By F. E. Lux and F. E. Nichy. February 1971, iii + 15 pp., 3 figs., 19 tables. 623 Apparent abundance, distribution, and migra- tions of albacore, Tluiimus akdiinga, on the North Pacific longline grounds. By Brian J. Rothschild and Marian Y. Y. Yong. September 1970, v -(- 37 pp., 19 figs., 5 tables. 624 Influence of mechanical processing on the quality and yield of bay scallop meats. Bv N. B. Webb and F. B. Thomas. April 1971, iii + 11 pp., 9 figs., 3 tables. 625 Distribution of salmon and related oceanographic features in the North Pacific Ocean, spring 1968. By Robert R. French, Richard G. Bakkala, Ma- sanao Osako, and Jun Ito. March 1971, iii -f- 22 pp., 19 figs., 3 tables. 626 Commercial fishery and biology of the fresh- water shrimp, Mdcrohrachium, in the Lower St. Paul River, Liberia, 1952-53. Bv George C. Miller. February 1971, iii + 13 pp., 8 figs., 7 tables. 627 Calico scallops of the Southeastern United States, 1959-69. Bv Robert Cummins, Jr. June 1971, iii + 22 pp., 23 figs., 3 tables. 628 Fur Seal Investigations, 1969. By NMFS, Ma- rine Mammal Biological Laboi'atory. August 1971, 82 pp., 20 figs., 44 tables, 23 appendix A tables, 10 appendix B tables. G29 Analysis of the operations of seven Hawaiian skipjack tuna fishing vessels, June-August 1967. By Richard N. Uchida and Ray F. Sumida. March 1971, v + 25 pp., 14 figs., 21 tables. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 - 35 cents. 630 Blue crab meat. I. Preservation bv freezing. July 1971, iii + 13 pp., 5 figs., 2 tables." II. Effect of chemical treatments on acceptability. By Jurgen H. Strasser, Jean S. Lennon, and Fred- erick J. King. July 1971, iii -|- 12 pp., 1 fig., 9 tables. 631 Occurrence of thiaminase in some common aquat- ic animals of the United States and Canada. By R. A. Greig and R. H. Gnaedinger. July 1971, iii -f 7 pp., 2 tables. 632 An annotated bibliography of attempts to rear the larvae of marine fishes in the laboratory. By Robert C. May. August 1971, iii + 24 pp., 1 ap- pendix I table, 1 appendix II table. For sale by the Superintendent of Documents, U.S. Govern- ment Printing Office, Washington, D.C. 20402 - 35 cents. 633 Blueing of processed crab meat. II. Identification of some factors involved in the blue discoloration of canned crab meat Callinectes sapidus. By Melvin E. Waters. May 1971, iii + 7 pp., 1 fig., 3 tables. 634 Age composition, weight, length, and sex of her- ring, Chq^ea pallnsii, used for reduction in Alas- ka, 1929-66. By Gerald M. Reid. July 1971, iii + 25 pp., 4 figs., 18 tables. 635 A bibliography of the blackfin tuna, TImnnus atlanticus (Lesson). By Grant L. Beardsley and David C. Simmons. August 1971, 10 pp. For sale by the Superintendent of Documents, U.S. Government Printing Oflice, Washington, D.C. 20402 - 25 cents. Continued on inside back cover. _^0MM0SP^ U.S. DEPARTMENT OF COMMERCE Frederick B. Dent, Secretary NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION Robert M. White, Administrator NATIONAL MARINE FISHERIES SERVICE NOAA Technical Report NMFS SSRF-669 Subpoint Prediction for Direct Readout Meteorological Satellites L E. EBER SEATTLE, WA August 1973 For sale by the Superintendent of Documents, U.S. Government Printing Office Washington. U.C. 20402 The National Marine Fisheries Service (NMFS) does not approve, rec- ommend or endorse any proprietary product or proprietary material mentioned in this publication. No reference shall be made to NMFS, or to this publication furnished by NMFS, in any advertising or sales pro- motion which would indicate or imply that NMFS approves, recommends or endorses any proprietary product or proprietary material mentioned herein, or which has as its purpose an intent to cause directly or indirectly the advertised product to be used or purchased because of this NMFS publication. CONTENTS Page Introduction 1 Undisturbed orbital motion 1 Disturbed orliitai motion 2 Subpoint location 3 Orbit prediction 4 Application 5 Literature cited 7 111 Subpoint Prediction for Direct Readout Meteorological Satellites L. E. EBER' ABSTRACT The National Environmental Satellite Service (NESS) provides orbital information on me- teoroloKical satellites with direct transmission systems, through APT (Automatic Picture Trans- mission) Predict messages sent over standard weather communications networks. With peri- odic access to this information, operators of independent APT ground receiving stations can extrapolate, by means of nodal period and nodal increment, to determine future orbits within receiving range of their station. A technique for the prediction of subpoint location along an orbit as a function of time after ascending node was developed from consideration of Kepler's laws and derived expressions for the force due to the earth's gravitational potential. Subpoint latitudes and longitudes computed by this technique are within 0.1 degree of those given in NESS predictions. INTRODUCTION Users of APT (Automatic Picture Transmis- sion) and Direct Readout Scanning Radiometer data from meteoroio,) R + 2R^ (3) where M is the mass of the earth, K^ is the constant of gravitation, I is the moment of inertia about the polar axis, J is the moment of inertia about any dia- meter in the earth's equatorial plane, and 4) is the angle between the equatorial plane and a line from the earth's cen- ter to the satellite (i.e., cf) = latitude). The expression (I —J) can be related to the earth's mass M, the equatorial radius a, the earth's flattening/, and the ratio q of the earth's centripetal acceleration at the equator to gravity, as follows: I -J =(2/3) Ma^ if -q/2). (4) Substituting Equation (4) in Equation (3), we get U = K^ M R + IP Ma^ {f -q/2) (1^3 sin^ (/>) 3R' (5) A more precise form of expression for the earth's gravitational potential is given in terms of spherical harmonics by U = IP M IP M R R QO 2 J n -^ I Pn (sin ) = (3/2) sin^ - 1/2. If terms in higher n are omitted, the expression for gravitational potential becomes U IP M IP M a2 J, (1 -3sin2 )\ R' J and the angular motion is d_a_B_ / K^M dt ^ R \ R^A K^M aM / - q/2) (1-3 sin^ (/>) > SUBPOINT LOCATION Satellite orbit predictions are given in terms of subpoint locations at fixed intervals of time following the northward equator crossing, re- ferred to as the ascending node. Numerical inte- gration of Equation (9), carried out with suitably small time increments, can be made to yield suc- cessive values of a requisite for subpoint compu- tation. If a is taken to be zero when the satellite is at perigee, then the plus sign preceding e cos a in Equation (2) is applicable and R is given by R A (1 -e') 1 +e cos a (10) R ( I a dt IP M B^ R^ A (8) The right side of Equation (8) can be considered as the product of {B^/RA) and the quantity (K^M/R-). The latter is equivalent, except in sign, to the first term on the right side of Equa- tion (6) and represents the gravitational accel- eration for undisturbed motion. The correction for the ellipticity of an orbit is contained in the factor B^/RA. Its value approaches 1 when an orbit is nearly circular. By applying this factor to the right side of Equation (6) we can relate the R can be computed from Equation (10) for each iteration in the numerical integration of Equa- tion (9) using the value of a from the preceding iteration increased by one-half the average angular displacement [360 ( A t)/2P\ during the selected time increment, A^- In order to start the numerical integration of Equation (9) at the ascending node, an initial value of a must be de- termined from the orientation of the orbital el- lipse. The latter is expressed as the argument of perigee, which is the geocentric angle of the perigee measured in the orbital plane from the ascending node in the direction of motion. The Figure 1. — Right spherical triangle formed by the equator, the orbital subpoint track and the meridian through the sub- point ((/), A). initial value of a is obtained by subtracting the argument of perigee from 360 . The gravity effect of the earth's equatorial bulge causes the perigee to shift slightly during each orbit. Consequently, it is necessary to know the rate at which the orbital ellipse is shifting in order to determine the correct initial values of a for prediction of future orbits. The locus of orbital subpoints for an earth satellite traces a great circle on a sphere concen- tric with the earth. Each subpoint can be repre- sented by a pair of coordinates (4>, A), where 4> is the geocentric latitude and A is the longitudinal displacement of the subpoint measured from the longitude of the ascending node. Subpoint co- ordinates corresponding to any point of the orbit can be determined from its angular displacement, measured along the orbital track from the as- cending node. If a is the angular coordinate (measured from perigee, as defined in the pre- ceding section) and a,, is the value of a at the as- cending node, the angular displacement is ( «—«„). Figure 1 shows these quantities as com- ponents of a right spherical triangle whose ver- tices are defined by the subpoint ((/>, A), the co- ordinates of the ascending node (0, 0) and the point (0, A). The two sides forming the right angle are (/) and A respectively. The opposite side is ( a —a„) and is a segment of the subpoint track. The angle opposite , denoted by /, is the inclina- tion of the orbital plane to the equatorial plane. Application of the trigonometric formulas de- rived from Napier's rules yields the following re- lationships: sin = tan(i )sin A. (13) Equations (11) and (12) give explicit formulas for (}> and A as a function of (/) and ( a — a„). Equa- tion (13) represents the great circle traced by the orbital subpoints. ORBIT PREDICTION The concepts developed in the preceding sec- tions deal with the computation of satellite sub- points relative to the time and longitude of the ascending node. Application of these concepts to advance preparation of orbit schedules re- quires, therefore, prediction of the times and longitudes of ascending nodes for future orbits. The APT Predict messages give orbital period to the nearest second and longitudinal displace- ment per orbit to the nearest hundredth of a degree. This precision is adequate to extrapolate a few orbits ahead. Greater precision is needed for satisfactory extrapolation beyond a hundred orbits and can be obtained from NESS or can be gained by empirical adjustments. The orbital elements necessary for computa- tion of subpoint locations for extrapolated orbits, also obtainable from NESS, include the length of the semimajor axis, eccentricity, orbital in- clination, argument of perigee at a known time or reference orbit, and the rate of change of perigee. The latter is equal to the rate of change of a,„ the value of the angular displacement a at the ascending node, which can be computed from the following expression (Runcorn, 1967): (/ a„/dt = 4.98(3M)3.^ (1 ^e2) (degrees per day). ^ (5 cos^ i—l) where e is the eccentricity of the orbital ellipse, i is the inclination of the orbital plane to the earth's equatorial plane, a is the earth's equatorial radius, and A is the semimajor axis of the ellipse, as previously defined. For the satellite ESSA-8, the value of / is 78.3° , A = 7815.37 km, and e = 0.00323. With these values and taking 6378 km for a, we compute an ^T— = — 1.943 deg per day. Given the nodal period of ESSA-8 as 115.703 min we find, alternately, that d «( dt 0.15476 deg per orbit. The minus sign indicates that the change is op- posite to the angular motion of the satellite. Having determined a„ for a selected orbit and starting at the ascending node where a =a,„ the radial distance R can be computed from Equa- tion (10) for use in the numerical integration of Equation (9), to yield the angular displacement ( a — a„) for successive intervals of desired length after the time of ascending node. The values of (^ and A can then be obtained with Equa- tions (11) and (12). These are the spherical co- ordinates of the point of the orbit relative to a nonrotating frame of reference with origin at the earth's center. In order to plot the subpoint track on a chart, the spherical coordinates of points on the orbit need to be transformed to the geographic (geo- detic) latitudes and longitudes of the correspond- ing subpoints. The latter are the locations on the earth's surface where the local vertical passes through the satellite. The angle which local verti- cal makes with the equatorial plane defines the geodetic latitude of the subpoint (Bowditch, 1958). Owing to the earth's ellipsoidal shape, a line from the satellite to the center of the earth intersects the surface at a point north of the satellite subpoint. The geodetic latitude at this point of intersection is a close approximation to that of the subpoint and can be computed from the following expression: geodetic latitude = arc ctn [(1 ~ q) ctn 4)] where q is the ratio of centripetal acceleration at the equator to gravity, as defined earlier. This expression is derived from the vector difference between the true force of gravitation on a unit mass at a point on the earth and the apparent force of gravity at that point, as a result of the earth's rotation. Taking q = 3.468 X 10 "' the maximum correction for geodetic latitude computed with the above expression is about 0.1 degree at latitude 45 degrees. The longitude of the satellite subpoint can be obtained by adding the longitude of the ascend- ing node to the value of A computed with Equa- tion (12). Adjustment for the earth's rotation is made by adding to this sum the product of the rate of rotation (0.25 degree of longitude per min- ute) and the elapsed time from the ascending node to the point of the orbit for which the sub- point location is desired. A correction must be made for the precession of the orbital plane which, in a sun-synchronous orbit, amounts to 360 degrees in 365 days and is subtractive. APPLICATION The foregoing procedures have been imple- mented in a computer program for preparing or- bit schedules at the Southwest Fisheries Center, La Jolla Laboratory, National Marine Fisheries Service. The program includes a section for each active satellite in which appropriate values are assigned for semimajor axis, eccentricity, inclina- tion, anomalistic period, argument of perigee at a known reference orbit and the rate of change of the argument of perigee. All of these quanti- ties are taken directly from information sheets furnished by NESS. In running the program, a data card is submit- ted for each schedule to be printed. On it are punched the name of the satellite and the orbit number, day, hour, minute, second, and longi- tude of the ascending node for a particular orbit as read from a recent or current APT Predict message. The card is also punched with the stand- ard (or daylight) time zone at the station for con- version from universal to local time, and a speci- fication of the number of minutes after ascending node to which computed orbit reference times will apply. The latter tells the APT ground station operator when to expect to begin receiving trans- mission. 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