2 Celestial Relationships2.1 Coordinate Systems Several coordinate systems are used in the study of the motions of the Earthand other celestial bodies. The heliocentric-ecliptic coordinate system, illustratedin Fig. 2.1, has its origin at the center of the s u n . 1 The X~Y~fundamental planecoincides with the ecliptic plane, which is the plane of the Earths revolution aroundthe sun. The line of intersection of the ecliptic plane and the Earths equatorialplane defines the direction of the X~ axis. On the first day of spring, a line joiningthe center of the Earth and the center of the sun points in the direction of thepositive X~ axis. This is called the vernal equinox direction and is denoted by thesymbol of a rams head by astronomers because it points in the general directionof the constellation Aries. All Earth locations experience identical durations ofdaylight and darkness. The Earths spin axis wobbles slightly and shifts in directionslowly over centuries. This effect is known as precession, and it causes the line ofintersection of the Earths equatorial plane and the ecliptic plane to shift slowly.As a result, the heliocentric-ecliptic system is not an inertial reference frame.Where extreme precision is required, it is necessary to specify the coordinates ofan object based on the vernal equinox direction of a particular year or epoch.ECI System The geocentric-equatorial coordinate system, on the other hand, has its originat the Earths center. The fundamental plane is the equator, and the positive Xaxis points in the vernal equinox direction. The Z axis points in the direction ofthe North Pole. (At equinox all Earth locations experience identical durations ofdaylight and darkness.) This is shown in Fig. 2.2. It is important to keep in mind,when looking at Fig. 2.2, that the X, Y, Z system is not fixed to the Earth andturning with it; rather, the geocentric-equatorial frame is nonrotating with respectto the stars (except for precession of the equinoxes), and the Earth turns relativeto it. The two angles needed to define the location of an object along some directionfrom the origin of the celestial sphere are defined as follows: ot (right ascension) = the angle measured eastward in the plane of the equatorfrom a fixed inertial axis in space (vernal equinox) to a plane normal to the equator(meridian), which contains the object; 0 deg < ot < 360 deg. (declination) = the angle between the object and equatorial plane measured(positive above the equator) in the meridional plane, which contains the object;- 9 0 deg < 3 < 90 deg. r (radial distance) = the distance between the origin of the coordinate systemand the location of a point (object) within the coordinate system; r > 0. To an observer fixed in an inertial frame (i.e., one that is at rest or moves withconstant velocity relative to distant galaxies), the Earth revolves about the sun 11
12 ORBITAL MECHANICS FIRST DAY | T ze FIRST DAY O F ~ S U M M E R VERNAL ON DIRECTI NOX EQUI (Seasons forNorthern sphere) ere Hemi Fig. 2.1 Heliocentric ecliptic coordinate system (from Ref. 1).in a nearly circular orbit in the ecliptic plane. The Earth rotates at an essentiallyconstant rate about its polar axis. However, that axis of rotation is tilted away fromthe normal to the ecliptic. This tilt causes the equatorial and ecliptic planes to forma dihedral angle that is conventionally known as the obliquity of the ecliptic andis denoted by e(~23.5 deg). On account of this angle between the two planes, the sun, as viewed by an ob-server on Earth, appears to shuttle northward and southward across the equatorialplane (called the celestial equator). This gives rise to the changing seasons duringthe course of the Earths yearly revolution about the sun. Of particular interest is the time each year when the sun moves northwardthrough the equatorial plane; this event, called the vernal equinox, marks thebeginning of spring. It is customary to associate a direction in inertial space with Z CELESTIALNORTHPOLE CELESTIALS H R ~ P E E ~MERIDIONAL PLANE W. L Fig. 2.2 Earth-centered inertial system (from Ref. 2).
CELESTIAL RELATIONSHIPS 13 Z Cele s tial I North Pole :cliptlc Plane / Y Celestial / Eq,,- tot s e~ / VernAl X Equinox Fig. 2.3 Motion of the sun in the ECI reference frame (from Ref. 3).this event by noting the position of the sun as seen from Earth against the fieldof infinitely distant background galaxies. Figure 2.3 illustrates the motion of thesun in the ECI frame frequently used in defining celestial relationships. Note thatthe X axis is directed toward the vernal equinox and that the Z axis lies along theEarths polar axis of rotation; the Y axis completes the right-handed triad. Thethree arcs drawn in the three references planes bound an octant of the celestialsphere that is centered at the Earth and of arbitrary radius. The path of the sunsmotion is shown by the dashed arc on the celestial sphere. About three months after the vernal equinox, the sun reaches its northernmostposition (and, at local noon, is directly overhead to observers on the Tropic ofCancer at about 23.5°N); this event is called the summer solstice, since the sunappears to stand still momentarily as it reverses its direction of motion fromnorthward to southward. Then, about three months after the summer solstice, thesun crosses the celestial equator from north to south at the autumnal equinox.The sun reaches its southernmost position about three months later at the wintersolstice, which marks the start of winter. At that time, the sun is directly overheadto observers, at local noon, on the Tropic of Capricorn at about 23.5°S. Referring again to Fig. 2.3, it is customary to indicate the suns position bya pair of angles similar to the familiar longitude and latitude system used forterrestrial measurements. The right ascension of the sun is the angle measured
14 ORBITAL MECHANICS 5OO 20 ~ Declln~tion 40O 2~ Right 300 Ascension Autumnal Vernal ~oo ~oo Winter -20- Solstice J I 80 1~0 1~ 0 2b0 210 2~0 310 3~0 315 . 7~ Day Numbe~ Marl Apt [ May I Jun I Jul [ Aug 1Sep I Oct INov I Dec I Jan I Feb 1 Mar [ Month Date Fig. 2.4 Solar declination and right ascension vs date (from Ref. 3)positively eastward along the celestial equator to the meridional plane (throughthe celestial polar axis), which passes through the sun; this angle, which is thecounterpart of terrestrial longitude, is denoted as Ors. The second angle used toindicate the suns position is the declination measured positively northward fromthe celestial equator; this angle, which is the counterpart of geocentric latitude, isdenoted by ~,. Figure 2.4 shows the variation with time of solar declination andright ascension.Geographic CoordinateSystem It is customary to locate an object relative to the Earth by two angular co-ordinates (latitude-longitude) and altitude above (or perhaps below) the adoptedreference ellipsoid. The present discussion considers the Earth to be spherical.The origin of the latitude-longitude coordinate system is the geocenter (Fig. 2.5).The fundamental plane is the equator, and the principal axis in the fundamentalplane points toward the Greenwich meridian. The two angles required to define the location of a point along some ray fromthe geocenter are defined as follows: ~b (geocentric latitude) = the acute angle measured perpendicular to the equa-torial plane between the equator and a ray connecting the geocenter with a pointon the Earths surface; - 9 0 deg < ~ < 90 deg. )~E (east longitude) -----the angle measured eastward from the prime meridian inthe equatorial plane to the meridian containing the surface point; 0 deg < )~E <360 deg. The observers coordinate frame is related to the previously discussed inertialframe through the angle ®, the sidereal time; ® is measured eastward in theequatorial plane from the vernal equinox to the observers meridian and rangesfrom Oh to 24 h (or, equivalently, from 0 to 360 deg). The distance from thegeocenter to the object is called the range and is denoted by r.
C L S IR L T N H S EE T L A O S P AE I I 15 Z6 GREENWICH # NORTH POLE MERIDIA~ G E O C E N T ~ ~YG XX ~ ~ a G ~ E UT R QAO VERNAL EQUINOX Fig. 2.5 Latitude-longitude coordinate system (from Ref. 2).Azimuth-Elevation Coordinate System An observer standing at a particular point on the surface of a rotating planet seesobjects in a rotating coordinate system. In this system, the observer is at the originof the system, and the fundamental plane is the local horizon (Fig. 2.6). Such acoordinate system is referred to as a topocentric system. Generally, the principalaxis or direction is taken as pointing due south. Relative to an observer, the objectis in a meridional plane that contains the object and passes through the zenith ofthe observer. ZZT E IH N x I I ^./JI NORTH Y lEast) / ~ - - PLANE THE OF ~ LOCALHORIZON NADIR Fig. 2.6 Azimuth-elevation topocentric coordinate system (from Ref. 2).
16 ORBITAL MECHANICS Table 2.1 Constants and conversion factors Earth Gravitational Constant: a /z = 5.53042885 x 10 -3 ER 3 (equatorial Earth Radius)3/min 2 = 3.986005 x 105 km3/s 2 = 1.40764438 x 1016 ft3/s 2 = 6.27501680 × 104 (n.mi.)3/s 2 Mean Equatorial Earth Radius: ER = 6.37813700 × 103 km = 3.44391847 x 103 n.mi. Rotational Rate of the Earth: wE = 7.29211515 × 10 -5 rad/s Mean Obliquity of the Ecliptic (1 January 1976): 8 = 23 ° 26 32.66" = 23.442405 ° Time Conversions: 1 mean solar day = 86,400 ephemeris s 1 mean sidereal day = 86164.09054 ephemeris s 1 tropical yr = 365.2421988 mean solar days 1 calendar yr = 365 mean solar days 1 mean solar s = 1.1574074 × 10 -5 mean solar day = 1.1605763 × 10 -5 sidereal day = 1.002737 sidereal s Julian day = continuous count of the number of days since noon, 1 January 4713 B.C. alAU (1976) Astrodynamics Standards. The two angles needed to define the location of an object along s o m e ray f r o mthe origin are defined as follows: Az (azimuth) = the angle, eastward f r o m north to the o b j e c t s meridian, asm e a s u r e d in the local horizontal plane, which is tangent to the sphere at theobservers position; 0 deg < Az < 360 deg. El (elevation) = angular elevation (measured positively upward in the mefid-ional plane) o f an object above the local horizontal plane, which is tangent to thesphere at the o b s e r v e r s position; - 9 0 deg < El < 90 deg. The distance f r o m the o b s e r v e r to the object is called the range and is usuallydenoted by p. p > 0.
CELESTIAL RELATIONSHIPS 17Coordinate Transformations The satellite state vector at a given time is obtained by integrating the equationsof motion that equate the acceleration of the vehicle to the sum of the variousaccelerations acting on the vehicle. The integration must be performed in an inertial(nonrotating) reference frame. However, the principal acceleration due to gravityand aerodynamic drag is expressed mostly in the rotating (body-fixed) systems.Transformations from body-fixed coordinates to inertial and back are thereforerequired. These transformations involve the motion of the equinox, which is dueto the combined motions of the Earths equatorial plane and the ecliptic plane,the equinox being defined as the intersection of these planes. The motion of theequatorial plane is due to the gravitational attraction of the sun and moon onthe Earths equatorial bulge. It consists of the lunisolar precession and nutation.The former is the smooth, long-period westward motion of the equators meanpole around the ecliptic pole, with an amplitude of about 23.5 deg and a periodof about 26,000 yr. The latter (nutation) is a relatively short-period motion thatcarries the actual (or true) pole around the mean pole in a somewhat irregularcurve, with an amplitude of approximately 9 s of arc and a period of about 18.6 yr.The motion of the ecliptic (i.e., the mean plane of the Earths orbit) is due to theplanets gravitational attraction on the Earth and consists of a slow rotation of theecliptic. This motion is known as planetary precession and consists of an eastwardmovement of the equinox of about 12 s of arc a century and a decrease of theobliquity of the ecliptic, the angle between the ecliptic and the Earths equator, ofapproximately 47 s of arc a century. The "true" equator and equinox are obtainedby correcting the mean equator and equinox for nutation.2.2 Time Systems Astronomers also specify the location of stars by their right ascensions and de-clinations, which are essentially invariant. (Very gradual changes in star locationsoccur as a result of the precession of the equinoxes, which is caused by coningof the Earths spin axis in inertial space.) The angular coordinates of the sun,however, vary considerably in the course of each year, as shown in Fig. 2.4 or bythe tabulations of sun position given in a solar ephemeris. As indicated earlier,the sun moves at an irregular rate (as a result of noncircularity of the Earths orbitabout the sun) along the dashed arc in Fig. 2.3. The time between two successivepassages through the vernal equinox is called a tropical year (about 365.25 days).Because of the precession of the equinoxes, the tropical year is about 20 minshorter than the orbital period of the Earth relative to the fixed-star or siderealyear.4 On account of the irregular motion of the true sun along the ecliptic, astronomershave introduced a fictitious mean sun, which moves along the celestial equatorat a uniform rate and with exactly the same period as the true sun. It is relativeto this fictitious mean sun that time is reckoned. This can best be explained bythe geometry illustrated in Fig. 2.7, in which the celestial sphere is viewed by anobserver at the celestial North Pole looking in the direction of the negative Z axisof the Earth-centered inertial (ECI) frame. The rotational orientation of the Earth is conventionally determined by spec-ification of the Greenwich hour angle of the vernal equinox (GHAVE), whichis measured positively westward along the celestial equator from the Greenwich
18 ORBITAL MECHANICS Greenwich Merid~n Y V i e w of the C e l e e t i l l Sphere from the C e l e s ~ l North Pule (+Z a x i s )Fig. 2.7 Orientation of the m e a n and true sun and the Greenwich meridian in theECI frame (from Ref. 3).meridian to the vernal equinox. (Greenwich hour angle is measured either indegrees or in hours, the conversion factor being exactly 15 deg/h). Similarly, theinstantaneous locations of the mean and the true suns are determined by Greenwichhour angle of the mean sun (GHAMS) and Greenwich hour angle of the true sun(GHATS), respectively. Incidentally, the difference ( G H A T S - GHAMS), alsoknown as the equation of time, can be appreciated as the discrepancy betweentime reckoned by a sundial (based on the true sun position) and conventionallydetermined time (based on the position of the mean sun). Greenwich mean time(GMT) is linearly related to the GHAMS as follows3: GMT(h) = [12 + GHAMS(deg)/15]mod 24 (2.1)Noon GMT occurs when the mean sun is at upper transit of the Greenwich meridian(that is, the mean sun is at maximum elevation to an observer located anywhereon the Greenwich meridian); when the mean sun is 90 deg west of Greenwich,the GMT is 18 h or 6 p.m., and so on. The notation "mod 24" in the precedingequation simply means that integer multiples of 24 are to be added or subtractedas necessary to give GMT a value in the range 0-24 h. A mean solar day is defined as the time during which the Earth makes a completerotation relative to the mean sun; the mean sun, of course, moves eastward alongthe celestial equator at a rate of 360 deg in about 365.25 mean solar days or slightlyless than 1 deg/day. The length of a mean solar day is 24 mean solar hours. Asidereal day is the time during which the Earth makes a complete rotation relativeto a fixed direction, e.g., the vernal equinox; clearly, it is shorter than a mean solar
CELESTIAL RELATIONSHIPS 19day by about 4 min, since the sidereal reference meridian is fixed in inertial space.The length of a sidereal day is 24 sidereal hours or about 23 h, 56 min, of meansolar time. Another useful term carried over from astronomy is epoch, an instant of time ora date selected as a point of reference. Quantities defining an orbit or some othercelestial relationship are often specified at a particular instant. For example, theGHAVE is tabulated in ephemerides for zero hours on every day of a given year.Since GHAVE increases linearly with mean solar time, it is possible to establishan important relationship between time, longitude, and fight ascension of somecelestial event. Refer to Fig. 2.8, and suppose that an observer at longitude k°E observes a starat upper transit of his meridian at t h GMT on a particular date; let this eventbe regarded as epoch and the day on which it occurs as epoch date. The rightascension of the star is just the sum of the Greenwich hour angle at midnight ofthe epoch date plus the Earth rotation occurring in time t plus the east longitudeof the observer. Symbolically, this becomes ~* = GHAVE0 + WEt + k (2.2)where the Earths rotational rate ((BE) is about 15.04 deg/mean solar hour. Knowl-edge of any three of the quantities (or, GHAVE, t, and k) enables the remainingquantity to be determined. If the star is recognized based on its relation to a knownconstellation, one can consult a star catalog to find its right ascension. Similarly,on a given epoch date, there is a unique value of GHAVE at midnight at the start ofthat day. Finally, if the time t of upper transit is known, the observer can infer hislongitude. This is, in fact, the way mariners who are far from known landmarksuse celestial observations to determine their longitude. G r e e n w i c h Meridian &t t h r | G~,~Ton the Epoch D a t e / a,jr ~ ~ - ~ ~ ~ Greenwich Met icLian / / I ~ ~ St Midnight of O b s e r v e r Viewing S t a r &t Upper TrLnsic at t brs GMT I ",d.,-: ~ .... o I I ~_ Ve,~l ~ Eq~tor View of the E&r~h from A b o v e r.he Nor.h Pole Fig. 2.8 Relation among time, longitude, and celestial position (from Ref. 3).
20 ORBITAL MECHANICS Computer programs use the atomic time system A.1 in the integration of theequations of motion. However, the input-output data sets are often referenced toephemeris time (ET) for solar/lunar ephemerides, universal time (UT) for com-puting Greenwich sidereal time, and universal time coordinated (UTC) for input-output epochs and tracking data. The unit of ET time is defined as the ephemerissecond, which is a fraction 1/31,556, 925.9747 of the tropical year for 12h ET ofJan 0 d, 1900. The A. 1 time is one of several types of atomic time. In 1958, the U.S.Naval Observatory established the A. 1 system based on an assumed frequency of9, 192, 631,770 oscillations of the isotope 133 of cesium atom/A.1 s. The UTCtime scale is kept in synchronism with the rotation of the Earth to within 4-0.9 sby step-time adjustments of exactly 1 s, when needed. It is often referred to as theGMT, or ZULU time. It is defined by the atomic second and leap second (-4-1) toapproximate UT 1, the fundamental unit of which is the mean solar day correctedfor seasonal variation of Earths rotation. An error in UT1 translates directly intoan error in Earth orientation relative to vernal equinox. Thus, a 1-s error in UT1may mean a 400- to 500-mile error in satellite position. This is not likely to occur,however, since the time adjustments to UT1 are known in advance. 4 In the selection of reference coordinate frames for orbital mechanics calcu-lations, a compromise is often made between extreme accuracy and ease ofcomputation. A frame completely free of known errors and approximations iscomputationally time-consuming and is unnecessary for many applications. Thisis the reason why there are often numerous options available to the user, dependingon his needs. However, it is essential to understand these options if the resultantpositional errors are to be resolved in application to specific missions. The coordinates used in all orbit determination programs consist of the funda-mental astronomical reference systems. For example, the input ephemerides of theplanets are heliocentric and refer to the mean equator and equinox of 1950.0. Thenotation .0 after the year refers to the beginning of the Besselian solar year at theinstant at which the right ascension of the fictitious mean sun is 18h 40 m. As of 1 January 1984, the new unit of time has been designated as the JulianEphemeris Century, which equals 36,525 days or 3.15576 x 109 S of InternationalAtomic Time (TAI). The origin of time for precessions and nutation is noon,1 January A.D. 2000 or 2000.0 exactly. The input observational data are usually in a topocentric coordinate system. Theintegration is done in either geocentric, heliocentric, or other planetary coordinatesreferred to the mean equator and equinox of 2000.0 or a specified epoch. The forcemodel includes terms referred to a coordinate system that is fixed in the rotatingEarth and terms that are referred to the moon and planets. References 1Bate, R. R., Mueller, D., and White, J. E., Fundamentals of Astrodynamics, Dover, NewYork, 1971. ZCantafio, L. (ed.), Space-Based Radar Handbook, Artech House, Norwood, MA, 1989. 3Ginsberg, L. J., and Luders, R. D., Orbit Planners Handbook, The Aerospace Corpo-ration Technical Memorandum, 1976. 4Wertz, J. R. (ed.), SpacecrafiAttitude Determination and Control, D. Reidel Publishing,Dordrecht, Holland, 1980.