Satellite Orbital Dvnamics and Observation Strategies in Support of Agricultural Applications

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JOHN C. PRICE Hydrology Laboratory USDA Agricultural Research Service Beltsville, MD 20705 Satellite Orbital Dvnamics and Observation strategies in Support of Agricultural Applications

Although operational satellite data are generally well suited to agricultural applications, changes in observing strategy should further improve the utility of the data.

INTRODUCTION vantages for monitoring and understanding certain LTHOUGH a large number of satellites have ac- aspects of the general status of global agriculture. A quired data relevant to surface phenomena Synchronous satellite data are well suited for es- and atmospheric variability (Allison et al., 1980), timation of solar-energy input (Gautier, et al., only in the last decade have observing capabilities 1980) and regional scale measurements of surface reached the technical quality appropriate for ag- temperature as it pertains to the seasonal pattern ricultural applications. Constraints of instrumen- of crop growth. High spatial-resolution, low repeat tation and orbital geometry tend to partition cycle data (e.g., from Landsat) is helpful for field- Earth-observing satellites into three classes: syn- by-field inference of crop type and land use chronous satellites, long repeat cycle (18 day) (MacDonald and Hall, 1980). The intermediate

ABSTRACT:The operational satellites (coes, NOAA) and the proto-operational satellite (Landsat) acquire data suitable for agricultural applications repre- senting a wide range of coverage frequenczes and spatial resolution. The geosynchronous (GOES) satellites provide highly repetitzve low spatial resolution data appropriate for monitoring atmospheric variability, the Landsats provide infrequent (80 my sputial resotz~tiondatu appropriate for detailed study of vegetation and surface characteristics, and the NOAA series (especially the instrument Advanced Very High Resolution Radiometer-AVHRR) represent an intermediate observing regime. A discussion of satellite orbital c1zaracteri.stics in the context of agriczcltural applications suggests acluantages to placing both the AVHRR and a high resolution imager (e.g., Multispectral Scanner of Landsat) on an early afternoon satellite having the capability for in-flight orbit maintenance. polar orbiters, and short repeat cycle (2 to 5 day) observing regime has potential for inference of the polar orbiters. Representative satellites from these surface temperature, surface moisture budget classes acquire (1) highly repetitive (Y2 hour) visi- (Price, 1980), and regional crop stress, and is espe- ble and thermal infrared data at relatively low cially well suited for routinely monitoring ag- spatial resolution; (2) low repetition rate (e.g., 18 ricultural status on a global basis. These capa- days from Landsat) visible and near infrared data bilities should prove useful by promoting a more of very high spatial resolution; and (3) two to five accurate assessment of global food and fiber day data in the visible and thermal infrared spec- production, and its resultant effect on commodities tral intervals at intermediate spatial resolution. prices and international trade. Each data-coverage type has characteristic ad- The general properties of data from geosyn- 1604 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING, 1982 chronous satellites are first covered. Then, orbital demonstrated the feasibility of using geosyn- characteristics of the Landsat series are discussed chronous satellite data for improving radiation as an example of the advantages of an appropriate budget estimates in agricultural areas. Regional orbit adjustment (maintenance) strategy. The scale estimates of maximum and minimum surface NOAA meteorological satellites lack this desirable temperature may also be inferred in order to pro- feature. However, it appears feasible to combine vide an indication of growth potential (degree the desirable characteristics of Landsat and the days), atmospheric moisture demand, etc. Surface NOAA satellites in a single appropriately designed temperature itself is especially significant in early observing system. spring for determining planting dates, and in late fall when frost may occur in cold-prone areas. In addition, work by Idso et a1 (1977) suggests the The GOES (Dismachek et al., 1980) satellites oc- possibility of using remotely sensed surface tem- cupy fixed positions in the sky over the eastern peratures as an indicator of yield in small grain and western United States, acquiring data at 30- crops. minute intervals over the entire United States as The basic orbital constraint for all Earth-fixed well as much of the western hemisphere. Figure 1 satellites is the balance between gravitational at- illustrates the viewing perspective from geosyn- traction and the centrift~galforce tending to pro- chronous orbit. The GOES satellites acquire visible duce straight line motion away from the Earth. A and thermal infrared data at spatial resolutions of circular, or nearly circular, orbit is assumed be- one kilometre and eight kilometres, respectively. cause a variable satellite-to-Earth distance would During operational processing, the visible data are cause an undesirable varying spatial scale in averaged to match the lower resolution of thermal image data products. The minor effects of orbital infrared data. eccentricity and the flattened shape of the Earth The high repetition rate and large area of cover- are discussed in the section on Orbit Selection: age from geosynchronous satellites is useful for Minimizing the Effect of Eccentricity. For a cir- inferring synoptic weather patterns, for monitor- cular orbit, the balance of forces is expressed as ing rapidly varying atmospheric features such as GMm = mo2 severe storms, and for tracking individual cloud a elements as indicators of atmospheric dynamics. These data are also well suited for estimation of where CM is the Earth's gravitational attraction, two agriculturally important properties: solar radi- 3.99 x 10%mVsec2,m is the satellite mass, a is the ation as affected by cloud cover, and regional scale distance from the Earth's center to the satellite, surface temperature. A cooperative experiment and o = 2?rl(periocl T) is defined in this section as between the Great Plains Regional Council on the angular velocity of the satellite about the Evapotranspiration and the National Oceanic and Earth's center. Dividing out the satellite mass Atmospheric Administration (Tarpley, 1979) has yields a simple relation between the orbital period and the Earth-to-satellite distance (Kepler's third law-White, 1948). Geosynchronous satellites are of interest because they maintain constant surveillance over an unvarying portion of the Earth. Setting the orbital period equal to that of the Earth's rotation and solving for a:

one finds a geosynchronous satellite distance of 42,200 km from the center of the Earth (42,200 - 6,400 = 35,800 km from the Earth's surface). In order to avoid an apparent daily north-south movement in the sky, the geosynchronous satellites are stationed above the Earth's equator. A negative, but unavoidable, consequence is the fact that areas at the limlx of observation (north and south poles, and far east and west of the subsatel- lite point along the equator) are observed ob- liquely. For such areas, this results in a lower ef- fective spatial resolution and an increased atmo- FIG.1. The western hemi~phereas viewed from geo- spheric influence as the atmospheric path length stationary orbit. increases. SATELLITE ORBITAL DYNAMICS 1605

Although highly repetitive measurements of track day after day, with no passes over the inter- multispectral reflectivity characteristics and high mediate area between successive ground tracks. resolution mappingljand-use classification are (Note that 24" divides exactly into 360"; 360724" = feasible from geosynchronous altitude, such 15.) This hypothetical satellite is a one-day re- possibilities are largely of academic interest be- peater, with the undesirable characteristic that cause surface properties such as land use, vegeta- points midway between adjacent orbits (separated tion cover, crop status, etc., do not change on an by 2,700 km at the equator) must be observed at hour-to-hour basis. In addition, for geosynchron- large nadir angles from the satellite. The result is ous observation, expensive and highly sophisti- poor spatial resolution at large nadir angles as well cated instrumentation would be required to pro- as the increasing atmospheric influence as the vide the high spatial resolution which is readily slant path through the atmosphere becomes obtainable from satellites closer to Earth. longer. More nearly vertical coverage in the region between adjacent satellite passes may be achieved by selecting a satellite altitude for which the cov- The Landsat satellites, which are planned to be- erage swath fills in over a number of days the gaps come operational in the next several years, repre- behveen successive orbits on a particular day. For sent low-observing frequency polar orbiters which example, a satellite with a 96-minute period com- acquire extremely high spatial resolution multi- pletes 24 hoursI(l.6 hours) = 15 orbits per day, at spectral data (U.S. Geological Survey, 1979). This intervals of 24O of longitude. If the period is type ofdata has a demonstrated utility for land-use changed to 2 daysl(29 orbits) = 99.3 minutes, then evaluation, crop discrimination, crop-area mensu- passes on alternate days fall midway between ration, and potentially, for crop-stress and crop- those of the day before, and the separation of yield estimation. Practical considerations, espe- ground tracks is 360°/29 = 12.4" of longitude. By cially design and cost of the observing instrument, varying the ratio (days per repeat cycle)/(number data transmission rates, and cost of data process- of orbits), any desired spacing between ground ing, all tend to restrict high spatial resolution sen- tracks may be achieved. In general, one obtains sors to a narrow swath of data acquisition. Thus, the orbital period for a pattern that repeats eveiy m the Landsat Multispectral Scanner with 80-metre days, after n orbits spaced at inten~alsof 3600/n, by resolution acquires data in a swath 185 kilometres equating n x (orbital period) with m x (24 hours). wide. As a consequence, a long revisit time re- For a given repeat cycle and approximate distance sults, i.e., 18 days in the case of Landsat. If shorter between coverage tracks, several choices of satel- repeat cycles, e.g., 6 days, were selected without a lite altitude are possible, depending on the inter- widened swath, then the two-thirds of the Earth leaving characteristics of the orbital tracks over a between coverage strips would not be observed period of days (King, 1976). Several of these op- from such an orbit. Such a coverage pattern ap- tions are discussed in the section on Short Revisit pears quite unacceptable, while a widened cover- Polar Orbiters. For Landsat 3, the repeat cycle is age pattern for complete coverage from a 6-day 18 clays and 251 revolutions, with the orbital track cycle would increase data acquisition by a factor of on a given day 36001251 = 1.43" to the west of the three. (The data rate from the Landsat D Thematic track for the orbit about 24 hours earlier. Mapper will be 8 x lo7 bitslsecond.) Over a long time interval, perturbing forces In order to clarify the options regarding cover- modify the satellite orhit, so that data acquisition age patterns for near-Earth polar orbiting satel- tracks tend to diverge from the regular repeating lites, Equation 2 is solved for the orbital peri- pattern. The repeating characteristic is not essen- od T: i.e., tial for meteorological obselvations where the in- tention is to monitor cloud cover and storm sys- tems, even at rather oblique viewing angles. Thus, the NOAA satellites, including Tiros-N and NOAA-6 and -7, are not maintained in a repeating orbit. As a The radius of the Earth is approximately 6,400 result, long-term accurate predictions of the NOAA km; thus, for low-Earth orbiters, a = 7,000 and the coverage patterns are not possible due to the slow, period is approximately 100 minutes. To a very but steady, decay of the satellite orbit. good appl.oximation, one may regard the satellite's In contrast, precise repetition of coverage is orbital plane as fixed in space while the Earth ro- necessary for many geologic and agl.icultural ap- tates under it. If the orbital period is 96 min~~tes,plications, as a ground observer must he able to corresponding to an altitude of 570 kilometres, schedule field activities days or weeks in advance then the Earth will rotate under the satellite (1.6 with certain knowleclge that a satellite overpass hours) x (360" of longitude124 hours) = 24" of lon- will occur. This requires a strategy of orbit main- gitude while the satellite conlpletes one orbit. tenance in order to maintain the desired coverage Such a satellite would trace out the same ground pattern. PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING, 1982

For most applications, analysis is greatly are subject to this gradual drift, a discussion of the simplified through data acquisition under rela- evolution of the ground track is appropriate. tively constant viewing conditions. In particular, From Equation 3, one finds the change in satel- the satellite orbit should pass overhead at a fixed lite period for a small change in mean orbital time rather than at varying times during the day radius, i.e., and night. Such orbits are termed sun synchro- 3 T ST= --&I (5) nous, as the satellite appears at a fixed time with 2 a respect to the sun. By definition, the orbital plane of such an orbit makes a fixed angle with the Thus, for a period change ST, the longitudinal sep- radius vector from the sun to the Earth. The re- aration AX between successive ground tracks is quirement that the orbital plane precess once per changed by 6(AA) (per orbit) = ST = (211124 year (to match the Earth's revolution around the hours) x [3Tl(2a)]b. A satellite at higher than sun) corresponds to a constraint on orbital inclina- nominal altitude (&I positive) will trace paths in- tion (i)with respect to the altitude of the satellite, creasingly to the west @(Ah)positive] of its nomi- where inclination is defined as the angle between nal track, and a satellite at lower than nominal al- the satellite motion as it crosses the equatorial titude will appear to be deflected to the east. plane from south to north and a vector parallel to Because of atmospheric drag, the orbit decays' the equator and pointed east (see Appendix, Fig- according to ure 6). Orbital precession results from the non- spherical nature of the Earth's gravitational field. The precession is determined by the equation (see Appendix and Escobal, 1965) where A is the appropriate cross section of the satellite, p is the atmospheric density at satellite 3 height, and m is the satellite mass. Although atmo- d = - jdGM/a3)U2 * cos i (4) 2 [ % ] spheric density is affected by solar variations (es- Where pecially sunspots), one may treat the right side of Equation 6 as constant over time intervals of fi = rate of motion of the orbital plane; = 36W weeks to months, so that by integration a(t) = a(to) (365.24 days) for sun synchronous observa- - [GMa(tO)]'l2Ap(t-to)lm where to is the initial tions; time. Substituting this expression into that for j2 = the quadrupole correction (Brower and 6(AA), and integrating with respect to time: Clemence, 1961) to the Earth's gravity field; = -0.0011; and S(AA) (degrees of longitude per day) Re = Earth radius = 6378 km. For orbiters in the altitude range 400 to 1000 km, the required inclination is between 95" and 100°, indicating that the satellites ascending motion crosses the equatorial place headed in a northwest where t is in seconds. direction, reaching maximum latitude excursions To maintain a desired coverage pattern over a at approximately 80" north and south latitudes. long period of time, it is necessary to initialize the Note that the Landsats acquire data during the de- orbit at a slightly higher than nominal altitude, scending pass (northeast to southwest). The con- leading to a slow drift to the west (first term on the straint fixed time of observation <=> orbital in- right). As time goes on, the small but steady effect clination causes Earth-observing satellites to go of drag forces brings the satellite down, leading to nearly over the poles, i.e., "polar orbiters." an increasing deflection to the east (second term). When the deviation to the east has reached the ORBIT MAINTENANCE-GROUND TRACK maximum permissable value, the onboard propul- In the absence of perturbing forces, a satellite at sion system is used to kick the satellite into a a properly selected altitude and orbital inclination higher orbit, leading again to the westward drift. will cross a given location at a fixed time of day at The deviation of ground track from nominal is thus regular intervals, e.g., every 18 days. In fact, ahno- a series of parabolas (Figure 2). If orbit adjust- spheric drag, the sun's influence, and the non- ments are not carried out, the coverage pattern symmetric nature of the earth's gravitational field continually shifts to the east as compared to a dl slowly modify the idealized orbital geometry, drag-free satellite. Unfortunately, atmospheric leading to an increasing divergence with time density, p, at satellite altitudes is not constant over between the desired observing conditions and the many months, so that Equation 7 may not be used actual. This slow divergence may be eliminated, to describe the satellite location accurately over a provided an orbit adjustment system is carried on long time interval: periodic orbital adjustments the satellite. Because the current NOM satellites are needed if this predictability is required. SATELLITE ORBITAL DYNAMICS 1607

rial plane and the plane of the Earth's motion about the sun (i, = 23.5"), and 4 is the angle between the ascending node of the satellite orbit and the Earth-sun line. An analysis similar to that for the ground track pattern shows that, as i increases linearly with time for Equation 8, the result is a quadratic falloff of the time of overpass from the desired sun synchronous condition (Equation 4). Figure 3 also illustrates the effect of the decision to adjust the Landsat-3 orbital inclination in De- cember of 1977, leading to a return toward the de- CALENDER TlME sired 9:30 crossing time. FIG.2. East-west variation of the Landsat 3 ground track. Cusps represent orbit adjustment maneuvers. OKBlT SELECTION: MINIMIZING THE EFFECT OF ECCENI'RICITY ORBIT MAINTENANCE-CROSSING TlME As Figure 3 illustrates, even a "sun synchro- Although a circular orbit may be regarded as iious" satellite will suffer a slow change from its optimum for Earth observations, such orbits do not predicted overpass time due to perturbing forces. persist in the Earth's asymmetric gravitational This occurs because the sun exerts a small but field. As orbital eccentricity builds up, the result is steady effect on the orbit, tending to change the a varying satellite-to-Earth distance, i.e., the sat- inclination angle. The force is essentially a tidal ellite altitude changes slightly from one overpass force, with the sun's attraction being greater on the to the next. This produces an apparent change of day half of the orbit than on the night half. The scale of image products unless resampling of the effect is described by (Wells, 1965) image data is carried out. Landsat-3, still operating at this time, has a typi- ; J z= - X cal eccentricity in the range 0.0011 to 0.0014, cor- 16 responding to an altitude variation of approxi- (a~lM,G)"2(Af,Clug) (COSis + sin i sin (2as) (8) mately 2 10 kilometres over the orbit. Because the location of closest approach, or perigee, gradually where M,and M, are the mass of Earth and sun, a, moves, with a period of approximately 300 days, and u, are the radius of satellite motion of the sat- the scale of imagery over a particular location ellite about the Earth and the Earth about the sun, changes by approximately *1 percent over this i, is the obliquity or the angle between the equato- time interval. Although the 21-km difference between the Earth's equatorial and polar radii also causes an apparent scale change in satellite irnag- ery, this variation is a function of location only, while a temporal variation of satellite altitude produces a time varying scale change at a given location. This temporal variability may be reduced to a negligible value if desired. The eccentricity, e, of the orbit changes according to

t = -b cos w

where b = 3I2(CM,)'l2 RZ I:, sin i (1-514 sin2i)luw'"

in which w is the angle between the satellite perigee and the equatorial plane, and13 is the third order correction to the Earth's gravitational field = 2.5 x corresponding to a small assy~netry 1975 1976 1977 1978 1979 1980 between the northern and southern hemispheres. (JAN) The eccentricity may be held at a fixed value (i = CALENDAR TlME 0) by setting cos w = 0, i.e., o = 90°, so that the FIG.3. Landsat 2 Time of descending node (equatorial apogee-perigee line coincides with the Earth's crossing), illustrating the effect of orbit adjustment in axis of rotation. In order for this condition to per- late 1977. sist, one must also have & = 0, where PHOTOCRAMMETRIC ENGINEERING & REMOTE SENSING, 1982

3J2R; GM, 34 5 ., (1 - -4 s~ni) duce higher leaf temperatures and thus a greater -(,) &= -(,) u2 evaporative tendency. Within the U.S. Department of Agriculture, the R J, sin i Foreign Agricultural Service is using AVHRR data [I+-] operationally to augment Landsat data in assessing the status of global agriculture. Efforts within the Setting the bracketed quantity equal to zero re- Agricultural Research Service are directed toward sults in a value for the eccentricity of approxi- increasing the utility of these data. Possible failure mately The resulting deviation from a circu- of Landsats 2 and 3 before launch of Landsat D lar orbit is not large. However, the resulting exact would leave only the data from NOAA satellites for reproducibility of the orbit is an advantage when evaluation of visible/near infrared reflectance satellite data are to be processed for maximum measurements of vegetation. (At press-Landsat 2 geometric fidelity. This benefit accrues both from has failed, Landsat 3 is marginally functional, the unvarying height of observation over each Landsat D has been launched successfully.) specific location on Earth, but also because satellite attitude (orientation with respect to the Earth's surface) tends to be coupled to satellite altitude through the horizon sensors used for attitude con- In many respects, the Landsat orbit is well trol on most satellites. suited to observations of global agriculture. The orbit maintenance strategy for assuring ground tracks and time of overpass is very desirable for coordination of satellite data with ground obser- Meteorological satellites such as NOAA 6 (Kid- vations. The suggested freezing of overpass al- well, 1979) provide a spatial scale and frequency titude at each location provides only a minor im- of observations falling nicely between those of the provement to data quality. However, it is "free," geostationary satellites and the Landsat's. The ob- i.e., no penalties result in other performance char- serving instrument, the Advanced Very High Res- acteristics. olution Radiometer (AVHRR),acquires multi- Although the observing characteristics of NOAA spectral data (five channels on NOAA-7)every few satellites are less favorable, this is consistent with days, depending on the swath width acceptable to the historical mission of the National Earth Satel- the user (horizon-to-horizon data are acquired). lite Service (formerly the National Environmental The datz are well suited for monitoring the gen- Satellite Service). The new responsibility coin- eral status of agriculture on a global basis because: cides with increasing quality and potential applications of data from the AVHRR. Obviously, data The visiblelnear infrared channels acquire data utilization would be facilitated through well de- which are indicative of crop canopy cover, and, if fined data coverage characteristics, such as those Landsat is a guide, to crop stress as it affects near infrared reflectance; and of Lanclsat. An additional improvement could be The thermal infrared channels yield surface tem- obtained by placing a high resolution instrument perature, which may pro\.ide an indication of (e.g., the Multispectral Scanner) and a wide swath surface moisture conditions, owing to the pro- instrument (AVI-IRR)on a single satellite. This nounced cooling effect of surface evaporation. would facilitate Although the spatial resolution of the AVHRR is Intercomparisons of radiometric calibration, in- somewhat coarse (1.1 km), the broad coverage asmuch as the instrumental spectral hands con- swath represents an advantage over Landsat by in- tain similar information, because Landsat bands creasing greatly the probability of acquiring 4 and 5 represent the spectral interval 0.5 to 0.7 cloud-free data. In addition, a three- to six-day micrometres, which is equivalent to AVHRR coverage frequency is well suited for monitoring band 1 (0.56 to 0.69), and Landsat hands 6 and 7 span the interval 0.7 to 1.1 micrometres, which surface variability induced by weather effects, in- is equivalent to AVHRR channel 2 (0.7 to 1.09). cluding need for irrigation, mapping of surface Such a comparison would add to the relatively temperatures as a key to spring planting time, and meager information available concerning calibra- in the fall as an indicator of susceptability to tion of satellite radiometers (Williamson, 1977). freezing due to microcli~natological variability. Extrapolation of inferences drawn from a narrow The afternoon passes of Tiros-N (2:30 P.M, at strip of high resolution data to the broader swath northern mid-latitudes) are reasonably well placed of lower spatial resolution data. In regions defi- for inference of surface moisture conditions, al- cient in surface observations, the high resolution data could be used for a better understanding of though a slightly earlier crossing time (1 to 2 P.M.) the spatial variability of AVHRR data. would provide temperature data less affected by the diurnal temperature wave, and crop stress If, in addition, the two instruments were placed could be detected more readily in the visiblelnear in an early afternoon orbit, this would render re- infrared data because higher radiation levels in- flectance measurements less susceptable to topo- SATELLITE ORBITAL DYNAMlCS

graphic effects, including shadowing. While this would provide less sensitivity to crop cover in the early season, as there would he less shadowing of the soil surface compared to midmorning observations. it would tend to ~rovidea better definition of crop growth stage at midseason when canopy cover reaches maximum values. In addition, observations in the noon to 230 P.M. time frame facilitate interpretation of thermal infrared data as they pertain to evapotranspiration and near- surface moisture. Finally, it should be noted that early afternoon observations will be affected somewhat adversely by mid-day growth of 5[,a a cumulus clouds. 2 It is not apparent that the possibility of frequent 1 satellite observations, as afforded by the broad 25.21 * West Longitude 0 coverage pattern of the AVHRR, is consistent with FIG.5. Ground track sequence for a hypothetical satel- the slowly stepping pattern of the narrow swath lite at 800 km. Path-to-path overlap occurs at seven-day Landsat coverage (see Figure 4). From the Land- intervals. sat orbit, a typical ground location could be observed at small nadir angles for 4 to 8 days, then only at highly oblique angles until the 18-day re- placement of a narrow- and wide-swath instru- peat cycle crossed the region again. ments on a single satellite would both tend to However, other coverage selections are possible facilitate agricultural utilization of remotely (King, 1976). For example, patterns exist which sensed data. combine the long period repeat cycle needed for narrow-swath observations with the large day-to- day longitudinal variation which is appropriate so I atn indebted to Richard Strafella for many that all areas are observed frequently at nearly helpful discussions. vertical (<30°)nadir angles. An example is the repeating orbit at approximately 800 km, which is essentially an 18-day and a 3 112-day repeater. Allison, L. J., A. Schnapf, B. C. Diesen. 111, P. S. Martin, Such a coverage pattern appears well suited to ag- A. Schwalb. and W. R. Bandeen, 1980. A4eteorologi- ricultural observations, both domestic and global cal Satellites, NASA Technical Memorandum (Figure 5). Clearly, a thorough understanding of 80704, Goddard Space Flight Center, Greenbelt, current satellite data is needed before tradeoffs in Md., 72 p. observing strategies may be evaluated with confi- Brower, D., and C. M. Clemence, 1961. iMethods oJCe- dence. However, it is also clear that 110th an orbit lestial Mechanics, Academic Press, New York. maintenance strategy for the NOAA satellites and Capellari, J. O., C. E. Velez, and A. J. Fuchs, 1976. Mathematicnl Theory oJ the Goddad Trrtjectory Determination. NASA X-582-76-77, Coddartl Space 1810.. Flight Center, Greenbelt, Md. a Dismachek, Dennis C., A. L. Booth, and J. A. Leese, a 1980. ArntionuI Environmental Sutellite Seroice 1s - a a Cntalog of Prodrrcts, Third Edition, NOAA Techni- a cal Memorandum NESS 109, pp. 11-21. L P) Escobal, Pedro R., 1965. Methods oJOrbit Determina- D -* /~~~~rhe~~~'~t40' letftude - tion, John Wiley & Sons, Inc., New York. $10 a Gautier, Catherine, G. Diak, and S. Masse, 1980. A Sim- C a ple Physical Model to Estimate Incident Solar Radi- % ation at the Surface from GOES Satellite Data, J. of m a 0 a Appl. Meteor., 19, pp. 1005-1012. 5 - a Idso, Sherwood B., R. D. Jackson, and R. J. Reginato, a 1977. Remote Sensing of Crop Yields, Science, 196, a pp. 19-25. 2 - a 1 a Kidwell, Katherine B, 1979. NOAA Polar Orbiter Duta - (Tiros-N and IllOAA-6) User's Guide. National Cli- 25.74 + West Longitude 0 mate Center, Washington, D.C. FIG.4. The figure illustrates the Landsat 1-3 westward King, Joseph C., 1976. Quantization and Symmetry in orbital motion of 25.74" in 18 days, from an arbitrary Periodic Coverage Patterns with Applications to starting point (longitude 0). At midlatitr~des,a 30 to 40 Earth Observation,]. jor tho Astronalrticol Sciences, percent coverage overlap occurs on successive days. ,WTV. 1610 PHOTOCRAMMETRIC ENGINEERING & REMOTE SENSING, 1982

MacDonald, R. G., and F. G. Hall, 1980. Global Crop fortunately, an initially circular orbit will not re- Forecasting, Science, 208, pp. 670-679. main so due to the third moment or egg-shaped Price, John C., 1980. The Potential of Remotely Sensed tendency of the Earth's gravitational field. Figure Thermal Infrared Data to Infer Surface Soil Mois- A-2 illustrates geometry within the orbit plane. ture and Evaporation, Water Resources Research, Expressions for the rate of change of a, e, o,i, 16, pp. 787-795. and Q due to a small deviation from the field of a Tarpley, J. D., 1979. Estimating Incident Solar Radiation joint mass are given in Escobal (1965), with ap- at the Surface from Geostationary Satellite Data, J. Appl. Meteor., 18, pp. 1172-1181. propriate formulas being utilized in this paper. One should note, in addition, that even with an U. S. Geological Survey, 1979. Landsat Data User's Handbook, Third Edition. orbit maintenance strategy, the satellite motion is not truly sun synchronous over the course of a Wells, William R., 1965. Annlytical Lifetime Studies of a Close Lunar Satellite. NASA TN-D 2805, Langley year. This effect occurs, not because of any im- Research Center, Langley, Va. perfection in satellite motion, but because of the White, Harvey E., 1948. Modern College Physics. Van apparent non-uniform motion of the sun with re- Nostrand Company, Inc., Princeton, N.J., p. 135. spect to the rotation of the Earth. The equation of Williamson, L. E., 1977. Calibration Technology for time, which is usually presented graphically, de- Meteorological Satellites, Atmospheric Sciences scribes the apparent deflection of the sun in the Laboratory, Monograph Series 3, White Sands Mis- sky from that expected by civil time. Thus, the sun sile Range, 139 p. may appear to cross the north-south line more than 15 minutes early in late November. (Received 30 October 1981; revised and accepted 18 The effect arises principally from two causes, March 1982) which may be closely approximated as (1) The non-uniform motion of the earth about APPENDIX the sun. From conservation of angular momentum The description of satellite motion represents a (i.e., Kepler's second law) straightforward application of classical mechanics, r2e = constant (A-1) the present consideration being selection of ap- proxin~ationsso that the relation between orbital where r is the Earth-sun distance = a,[l - e cos(8 parameters and ground observations is readily un- - O,)], as is the Earth-sun semimajor axis, e is the derstood. Although powerfill numerical solution eccentricity, and 6, is the phase with respect to methods are available (Cappellari et al., 1976), perihelion, which occurs on 2 January, so that Bo = they leave options and constraints for typical Earth (27r).(2 days1365 days). Because e is small (e - observation missions relatively obscure because of 1/60), a simple expansion and integration of their great generality. Equation A-1 is possible, leading to the time cor- Figure A-1 illustrates orbital geometry and the rection due to the non-uniform angular motion of definition of the angles, i, the north-south ten- the Earth about the sun; i.e., dency of the orbit with respect to the Earth's Ft, = -2e sin[2rr(~a~- 2)/365] (radians) equatorial plane, fl, the angle between the as- -8 sin[27r(Day - 2)/365] (minutes) cending node and a fixed direction in space (the vernal equinox), and w, the angle within the orbit (2) The second contributing factor is the solar plane between the ascending node and the point obliquity (a = 23.5'), which corresponds to the ap- of closest approach (perigee). This last is signifi- parent north-south motion of the sun as the sea- cant only for an eccentric (non-circ~~lar)orbit. Un- sons change. For this computation, the Earth's

T FIG. A-1. Conventional illustration of orbital angles FIG.A-2. Within-plane orbital geometry for an Earth (Escobal, 1965). orbiting satellite. SATELLITE ORBITAL DYNAMICS 1611

NORTH A

.6" Q EARTH'S EQUATORIAL PLANE FIG.A-3. The solar obliquity produces an apparent variation in the angular velocity of the Earth about the sun.

Jan 1 Apr 1 Jul 1 Oct 1

CALENDER TIME orbit about the sun may be considered to be cir- FIG.A-4. The difference between civil time and that cular. As illustrated in Figure A-3, uniform motion from a sundial-the equation of time. along the orbital path results in an apparent motion of the sun in the ecliptic plane of or a difference between A and 6 of approximately 0.087 cos 20. The phase difference (A - 0) is the time integral or 0.043 sin 20, where 0 is measured from the winter solstice, day 355, in the Earth's equatorial plane. The difference in minutes due to this effect is thus. 8 cosa - [lii (Dz5- 355) 1 + (cos2a- l)cos28 8t2 = 0.043 sin (radians)

By rearranging and using a double angle formula - 10 sin [47r (Day - 355)1365] (minutes) The sum of Ft, and Ft2 describes to within approximately one minute the variability in the n apparent sun with respect to mean solar time (1 + cos2a)l(2cosa) + (1 - cos2a) COS~~/(~COSCY)' (Figure A-4).

Errata A number of changes and additions to the July 1982 Yearbook Issue of Plzotogrammetric Engineering and Remote Sensing have been brought to our attention. On page 1148, in the "Roster of Honorary Members," the name JOHNT. SMITH,JR. 1981 should be added, and the dates after the names CLIFFORDJ. CRANDALLand ARTHURJ. MCNAIRshould be changed to 1982. On page 1163, the "1982 Roster of Members Emeritus" should include the name SPEKOKAPELAS. In the organization chart on page 1192 the name of the 2nd Deputy of the Geological Sciences Committee should be ADERBALC. CORKEA. On page 1033, in the list of "Officers of the Technical Commissions and Chairmen of Working Groups, 1980-1984" of the International Society for Photogrammetry and Remote Sensing, the follow- ing additions should be made:

Commission IV W.G. IV/2 Mapping Technology and Applications Jor Develop.ing Countries TOPOGRAPHICAND CARTOGRAPHICAPPLICATIONS General G. C. Agarawcll W.G. IVII Cost Models in the Mapping Surveyor General Process Post Box 37 Prof. Dr. H. C.]erie Delira Dun (U.P.), India ITC W.C. IVl3 Mapping Jrom Space Borne 350 Boulevard 1945 Imagery NL 7511 Enschede, Netherlands Mr. Raymond Batson