orbital mechanics 2. celestial relationships

2 Celestial Relationships

2.1 Coordinate Systems Several coordinate systems are used in the study of the motions of the Earth

and other celestial bodies. The heliocentric-ecliptic coordinate system, illustrated in Fig. 2.1, has its origin at the center of the s u n . 1 The X~Y~fundamental plane coincides with the ecliptic plane, which is the plane of the Earth's revolution around the sun. The line of intersection of the ecliptic plane and the Earth's equatorial plane defines the direction of the X~ axis. On the first day of spring, a line joining the center of the Earth and the center of the sun points in the direction of the positive X~ axis. This is called the vernal equinox direction and is denoted by the symbol of a ram's head by astronomers because it points in the general direction of the constellation Aries. All Earth locations experience identical durations of daylight and darkness. The Earth's spin axis wobbles slightly and shifts in direction slowly over centuries. This effect is known as precession, and it causes the line of intersection of the Earth's 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 of an 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 origin

at the Earth's center. The fundamental plane is the equator, and the positive X axis points in the vernal equinox direction. The Z axis points in the direction of the North Pole. (At equinox all Earth locations experience identical durations of daylight 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 and turning with it; rather, the geocentric-equatorial frame is nonrotating with respect to the stars (except for precession of the equinoxes), and the Earth turns relative to it.

The two angles needed to define the location of an object along some direction from the origin of the celestial sphere are defined as follows:

ot (right ascension) = the angle measured eastward in the plane of the equator from 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 system and 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 with constant 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 EQUINOX (Seasons ere for Northern Hemisphere) DIRECTION Fig. 2.1 Heliocentric ecliptic coordinate system (from Ref. 1).

in a nearly circular orbit in the ecliptic plane. The Earth rotates at an essentially constant rate about its polar axis. However, that axis of rotation is tilted away from the normal to the ecliptic. This tilt causes the equatorial and ecliptic planes to form a dihedral angle that is conventionally known as the obliquity of the ecliptic and is denoted by e(~23.5 deg).

On account of this angle between the two planes, the sun, as viewed by an observer on Earth, appears to shuttle northward and southward across the equatorial plane (called the celestial equator). This gives rise to the changing seasons during the course of the Earth's yearly revolution about the sun.

Of particular interest is the time each year when the sun moves northward through the equatorial plane; this event, called the vernal equinox, marks the beginning of spring. It is customary to associate a direction in inertial space with

Z CELESTIAL NORTH POLE

CELESTIAL SPHERE ~ ~ M E R I D I O N A L

W. L

Fig. 2.2 Earth-centered inertial system (from Ref. 2).

PLANE

CELESTIAL RELATIONSHIPS 13

Z Cele s t ial I North

Pole

/

:cliptlc Plane

Y

/ \ Celes t ia l '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 field of infinitely distant background galaxies. Figure 2.3 illustrates the motion of the sun in the ECI frame frequently used in defining celestial relationships. Note that the X axis is directed toward the vernal equinox and that the Z axis lies along the Earth's polar axis of rotation; the Y axis completes the right-handed triad. The three arcs drawn in the three references planes bound an octant of the celestial sphere that is centered at the Earth and of arbitrary radius. The path of the sun's motion is shown by the dashed arc on the celestial sphere.

About three months after the vernal equinox, the sun reaches its northernmost position (and, at local noon, is directly overhead to observers on the Tropic of Cancer at about 23.5°N); this event is called the summer solstice, since the sun appears to stand still momentarily as it reverses its direction of motion from northward to southward. Then, about three months after the summer solstice, the sun crosses the celestial equator from north to south at the autumnal equinox. The sun reaches its southernmost position about three months later at the winter solstice, which marks the start of winter. At that time, the sun is directly overhead to 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 sun's position by a pair of angles similar to the familiar longitude and latitude system used for terrestrial measurements. The right ascension of the sun is the angle measured


5OO

20 ~ Declln~tion 40O 2

~ Right 300 Ascension

Autumnal Vernal

~oo

~oo Winter

-20- Solstice

J I 8 0 ' ' 1 ~ 0 ' ' '1~ 0 ' ' ' 2 b 0 ' ' ' 2 1 0 ' ' ' 2 ~ 0 ' ' ' 3 1 0 ' ' ' 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 (through the celestial polar axis), which passes through the sun; this angle, which is the counterpart of terrestrial longitude, is denoted as Ors. The second angle used to indicate the sun's position is the declination measured positively northward from the celestial equator; this angle, which is the counterpart of geocentric latitude, is denoted by ~,. Figure 2.4 shows the variation with time of solar declination and right ascension.

Geographic Coordinate System 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 adopted reference 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 fundamental plane points toward the Greenwich meridian.

The two angles required to define the location of a point along some ray from the geocenter are defined as follows:

~b (geocentric latitude) = the acute angle measured perpendicular to the equatorial plane between the equator and a ray connecting the geocenter with a point on the Earth's surface; - 9 0 deg < ~ < 90 deg.

)~E (east longitude) ----- the angle measured eastward from the prime meridian in the equatorial plane to the meridian containing the surface point; 0 deg < )~E < 360 deg.

The observer's coordinate frame is related to the previously discussed inertial frame through the angle ®, the sidereal time; ® is measured eastward in the equatorial plane from the vernal equinox to the observer's meridian and ranges from O h to 24 h (or, equivalently, from 0 to 360 deg). The distance from the geocenter to the object is called the range and is denoted by r.

CELESTIAL RELATIONSHIPS Z6

GREENWICH # NORTH POLE M E R I D I A ~

G E O C E N T ~

XG X ~ ~ a ~

VERNAL EQUINOX Fig. 2.5

~ YG EQUATOR

Latitude-longitude coordinate system (from Ref. 2).

15

Azimuth-Elevation Coordinate System An observer standing at a particular point on the surface of a rotating planet sees

objects in a rotating coordinate system. In this system, the observer is at the origin of the system, and the fundamental plane is the local horizon (Fig. 2.6). Such a coordinate system is referred to as a topocentric system. Generally, the principal axis or direction is taken as pointing due south. Relative to an observer, the object is in a meridional plane that contains the object and passes through the zenith of the observer.

ZENITH Z

x I I ^ . / J I NORTH

Y lEast) / ~ - - PLANE OF THE ~ LOCAL HORIZON

NADIR Fig. 2.6 Azimuth-elevation topocentric coordinate system (from Ref. 2).


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 locat ion of an object a long some ray f rom the origin are def ined as follows:

Az (azimuth) = the angle, eastward f rom north to the objec t ' s meridian, as measured in the local horizontal plane, which is tangent to the sphere at the observer ' s posi t ion; 0 deg < Az < 360 deg.

El (elevation) = angular e levat ion (measured posi t ively upward in the mefid- ional plane) o f an object above the local horizontal plane, which is tangent to the sphere at the observer ' s posit ion; - 9 0 deg < El < 90 deg.

The distance f rom the observer to the object is cal led the range and is usually denoted by p. p > 0.


Coordinate Transformations

The satellite state vector at a given time is obtained by integrating the equations of motion that equate the acceleration of the vehicle to the sum of the various accelerations acting on the vehicle. The integration must be performed in an inertial (nonrotating) reference frame. However, the principal acceleration due to gravity and aerodynamic drag is expressed mostly in the rotating (body-fixed) systems. Transformations from body-fixed coordinates to inertial and back are therefore required. These transformations involve the motion of the equinox, which is due to the combined motions of the Earth's equatorial plane and the ecliptic plane, the equinox being defined as the intersection of these planes. The motion of the equatorial plane is due to the gravitational attraction of the sun and moon on the Earth's equatorial bulge. It consists of the lunisolar precession and nutation. The former is the smooth, long-period westward motion of the equator's mean pole around the ecliptic pole, with an amplitude of about 23.5 deg and a period of about 26,000 yr. The latter (nutation) is a relatively short-period motion that carries the actual (or true) pole around the mean pole in a somewhat irregular curve, 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 Earth's orbit) is due to the planet's gravitational attraction on the Earth and consists of a slow rotation of the ecliptic. This motion is known as planetary precession and consists of an eastward movement of the equinox of about 12 s of arc a century and a decrease of the obliquity of the ecliptic, the angle between the ecliptic and the Earth's equator, of approximately 47 s of arc a century. The "true" equator and equinox are obtained by 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 locations occur as a result of the precession of the equinoxes, which is caused by coning of the Earth's 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 by the 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 Earth's orbit about the sun) along the dashed arc in Fig. 2.3. The time between two successive passages 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 min shorter than the orbital period of the Earth relative to the fixed-star or sidereal year. 4

On account of the irregular motion of the true sun along the ecliptic, astronomers have introduced a fictitious mean sun, which moves along the celestial equator at a uniform rate and with exactly the same period as the true sun. It is relative to this fictitious mean sun that time is reckoned. This can best be explained by the geometry illustrated in Fig. 2.7, in which the celestial sphere is viewed by an observer at the celestial North Pole looking in the direction of the negative Z axis of 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), which is measured positively westward along the celestial equator from the Greenwich


G r e e n w i c h M e r i d ~ n

Y

View of the C e l e e t i l l Sphere from the C e l e s ~ l North Pule (+Z ax i s )

Fig. 2.7 Orientation of the mean and true sun and the Greenwich meridian in the ECI frame (from Ref. 3).

meridian to the vernal equinox. (Greenwich hour angle is measured either in degrees or in hours, the conversion factor being exactly 15 deg/h). Similarly, the instantaneous locations of the mean and the true suns are determined by Greenwich hour 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), also known as the equation of time, can be appreciated as the discrepancy between time reckoned by a sundial (based on the true sun position) and conventionally determined 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 anywhere on 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 preceding equation simply means that integer multiples of 24 are to be added or subtracted as 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 complete rotation relative to the mean sun; the mean sun, of course, moves eastward along the celestial equator at a rate of 360 deg in about 365.25 mean solar days or slightly less than 1 deg/day. The length of a mean solar day is 24 mean solar hours. A sidereal day is the time during which the Earth makes a complete rotation relative to a fixed direction, e.g., the vernal equinox; clearly, it is shorter than a mean solar


day 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 mean solar time.

Another useful term carried over from astronomy is epoch, an instant of time or a date selected as a point of reference. Quantities defining an orbit or some other celestial relationship are often specified at a particular instant. For example, the GHAVE 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 establish an important relationship between time, longitude, and fight ascension of some celestial event.

Refer to Fig. 2.8, and suppose that an observer at longitude k°E observes a star at upper transit of his meridian at t h GMT on a particular date; let this event be regarded as epoch and the day on which it occurs as epoch date. The right ascension of the star is just the sum of the Greenwich hour angle at midnight of the epoch date plus the Earth rotation occurring in time t plus the east longitude of the observer. Symbolically, this becomes

~* = GHAVE0 + WEt + k (2.2)

where the Earth's 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 remaining quantity to be determined. If the star is recognized based on its relation to a known constellation, 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 of that day. Finally, if the time t of upper transit is known, the observer can infer his longitude. This is, in fact, the way mariners who are far from known landmarks use celestial observations to determine their longitude.

G r e e n w i c h Meridian &t t hr | G~,~T on 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 ar &t Upper

TrLnsic at t brs GMT

I " , d . , - ' : ~ . . . . o I I ~'_ V e , ~ 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).


Computer programs use the atomic time system A.1 in the integration of the equations of motion. However, the input-output data sets are often referenced to ephemeris 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 ephemeris second, which is a fraction 1/31,556, 925.9747 of the tropical year for 12 h ET of Jan 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 of 9, 192, 631,770 oscillations of the isotope 133 of cesium atom/A.1 s. The UTC time scale is kept in synchronism with the rotation of the Earth to within 4-0.9 s by step-time adjustments of exactly 1 s, when needed. It is often referred to as the GMT, or ZULU time. It is defined by the atomic second and leap second (-4-1) to approximate UT 1, the fundamental unit of which is the mean solar day corrected for seasonal variation of Earth's rotation. An error in UT1 translates directly into an error in Earth orientation relative to vernal equinox. Thus, a 1-s error in UT1 may 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 of computation. A frame completely free of known errors and approximations is computationally time-consuming and is unnecessary for many applications. This is the reason why there are often numerous options available to the user, depending on his needs. However, it is essential to understand these options if the resultant positional errors are to be resolved in application to specific missions.

The coordinates used in all orbit determination programs consist of the fundamental astronomical reference systems. For example, the input ephemerides of the planets are heliocentric and refer to the mean equator and equinox of 1950.0. The notation .0 after the year refers to the beginning of the Besselian solar year at the instant at which the right ascension of the fictitious mean sun is 18 h 40 m.

As of 1 January 1984, the new unit of time has been designated as the Julian Ephemeris Century, which equals 36,525 days or 3.15576 x 109 S of International Atomic 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. The integration is done in either geocentric, heliocentric, or other planetary coordinates referred to the mean equator and equinox of 2000.0 or a specified epoch. The force model includes terms referred to a coordinate system that is fixed in the rotating Earth and terms that are referred to the moon and planets.

References

1Bate, R. R., Mueller, D., and White, J. E., Fundamentals of Astrodynamics, Dover, New York, 1971.

ZCantafio, L. (ed.), Space-Based Radar Handbook, Artech House, Norwood, MA, 1989. 3Ginsberg, L. J., and Luders, R. D., Orbit Planner's Handbook, The Aerospace Corpo-

ration Technical Memorandum, 1976. 4Wertz, J. R. (ed.), SpacecrafiAttitude Determination and Control, D. Reidel Publishing,

Dordrecht, Holland, 1980.

orbital mechanics 2. celestial relationships

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