The position of any point on the sphere is given in terms of two components equivalent to latitude and longitude on the surface of the Earth. The arc length distance above or below the equator is called the latitude or elevation component. The angular distance around the equator between the meridian passing through a particular point and an arbitrary reference meridian, or prime meridian, is known as the longitude or azimuth component. For example, the reference meridian for longitude on the surface of the Earth is the one passing through the center of the former Royal Greenwich Observatory in London. Thus, we may define the positions of the points P, and P2 in terms of azimuth, <f>, and elevation, X, as:

Note that in most spherical coordinate systems, the azimuth coordinate is measured from 0 deg to 360 deg and the elevation component is measured from + 90 deg to —90 deg. The intersection of the reference meridian and the equator in any system is called the reference point and has coordinates (0°,0°).

Several properties of spherical coordinate systems are shown in Fig. 2-2. For example, a degree of elevation is a degree of arc length in that the angular separation between two points on the same meridian is just the difference between the elevation of the two points. Thus, Py at (75°,60°) is 25° from Pv However, 1-deg separation in azimuth will be less than 1 deg of arc, except along the equator. Point P4 at (50°, 35°) is less than 25 deg in arc length from P,. Specific equations fbr the angular distance along a parallel or between two arbitrary points are given in Appendix A (Eqs. (A-l), (A-4), and (A-5)). In using these equations, a parallel at elevation X is a small circle of angular radius 90°-|X|. The distortion in the azimuth component becomes particularly strong near the pole of any coordinate system. At either pole, the azimuth is undefined.

An alternative procedure for specifying the position of a point on the celestial sphere involves three components of a vector of unit length from the center of the sphere to the point on the surface of the sphere. Ordinarily, the x,y, and z axes of such a rectangular coordinate system are defined such that the z axis is toward the + 90-deg pole of the spherical coordinate system, the x axis is toward the reference point, and the y axis is chosen perpendicular to x and z such that the coordinate system is right handed (i.e., for unit vectors along the x,y, and z axes, z = x x y). A summary of vector notation used in this book is given in the Preface and the coordinate transformations between spherical and rectangular coordinate systems are given in Appendix E. Of course, only two of the three components of the unit vector are independent, since the length (but not the sign) of the third component is determined by requiring that the magnitude of the vector be 1. This constraint on the magnitude is a convenient check that any unit vector has been correctly calculated.

In principle, either spherical or rectangular coordinates can be used for any application. In practice, however, each system has advantages in specific circumstances. In general, computer calculations in long programs should be done in rectangular coordinates because there are fewer trigonometric functions to evaluate and these are relatively time consuming for the computer. Carrying the third component around is conveniently done in computer arrays. However, most input and output and most data intended for people to read are in spherical coordinates, since the geometrical relationships are usually clearer when visualized in terms of a coordinate picture similar to Fig. 2-2. For calculations external to the computer or in short computer runs, spherical coordinates give less likelihood of error because the quantities involved are more easily visualized. Many geometrical theorems, such as those in Chapter 11, are more easily done in spherical geometry than in vector geometry, although either system may be better for any specific problem.

Any spherical coordinate system (or its rectangular equivalent) is fully specified by indicating the positive pole and the choice of either reference meridian or reference point at the intersection of the reference meridian and the equator. On the surface of a sphere, the choice of poles and prime meridian is arbitrary, and any point on the sphere may be used as the pole for a spherical coordinate system. For the Earth, a system defined by the Earth's rotation axis is the most convenient. However, for the sky as viewed by the spacecraft, a variety of alternative coordinate systems are convenient for various uses, as described below.

The three basic types of coordinate systems centered on the spacecraft are those fixed relative to the body of the spacecraft, those fixed in inertial space, and those defined relative to the orbit and not fixed relative to either the spacecraft or inertial space.

Spacecraft-Fixed Coordinates. Coordinate systems fixed in the spacecraft are used to define the orientation of attitude determination and control hardware and are the systems in which attitude measurements are made. Throughout this book, spacecraft-fixed spherical coordinates will use <j> for the azimuth component and X for the elevation. Alternatively, 9 will be used for the: coelevation; that is, 9 =90° -X. For spinning spacecraft, the positive pole of the coordinate system will be the positive spin vector, unless otherwise specified. The reference meridian is taken as passing through an arbitrary reference point on the spin plane which is the equator of the coordinate system. The three components of a rectangular spacecraft fixed coordinate system will be represented by x, y, and z, with the relation between spherical and rectangular coordinates as defined in Section 2.2.1. For three-axis stabilized (nonspinning) spacecraft, no standard orientation is defined. For attitude-sensing hardware, it is the orientation of the field of view of the hardware in the spacecraft system that is important, not the location of the hardware within the spacecraft.

Inertial Coordinatès. The most common inertial coordinate system is the system of celestial coordinates defined relative to the rotation axis of the Earth, as shown in Fig. 2-3. Recall that the spacecraft is at the center of the sphere in Fig. 2-3. The axis of the spacecraft-centered celestial coordinate system joining the north and south celestial poles is defined as parallel to the rotation axis of the

Earth. Thus, the north celestial pole is approximately 1 deg from the bright star Polaris, the Pole Star. To fully define the coordinate system, we must also define the reference meridian or reference point. The point on the celestial equator chosen as the reference is the point where the ecliptic, or plane of the Earth's orbit about the Sun (see Chapter 3), crosses the equator going from south to north, known as the vernal equinox. This is the direction parallel to the line from the center of the Earth to the Sun on the first day of spring.

Unfortunately, the celestial coordinate system is not truly inertial in that it is not fixed relative to the mean positions of the stars in the vicinity of the Sun. The gravitational force of the Moon and the Sun on the Earth's equatorial bulge causes a torque which results in the slow rotation of the Earth's spin axis about the ecliptic pole, taking 26,000 years for one complete period for the motion of the axis. This phenomenon is known as the precession of the equinoxes, since it results in the vernal equinox sliding along the ecliptic relative to the fixed stars at the rate of approximately 50 sec of arc per year. When the zodiacal constellations were given their present names several thousand years ago, the vernal equinox was in the constellation of Aries, the Ram. Thus, the zodiacal symbol for the Ram, T, is used astronomically for the vernal equinox, which is also called the First Point of Aries. Since that time, the vernal equinox has moved through the constellation of Pisces and is now entering Aquarius, bringing the dawn of the zodiacal "Age of Aquarius." The other intersection of the ecliptic and the equator is called the autumnal equinox and is represented by the zodiacal symbol for Libra. (See Fig. 3-10.)

The importance of the precession of the equinoxes is that it causes a slow change in the celestial coordinates of "fixed" objects, such as stars, which must be taken into account for accurate determination of orientation. Thus, celestial coordinates require that a date be attached in order to accurately define the position of the vernal equinox. The most commonly used systems are 1950 coordinates, 2000 coordinates, and true of date, or TOD. The latter coordinates are

defined at the epoch time of the orbit and are commonly used in spacecraft work, where the small corrections required to maintain TOD coordinates are conveniently done with standard computer subroutines.* (See subroutine EQU1N in Section 20.3.)

The elevation or latitude component of celestial coordinates is universally known as the declination. 8. Similarly, the azimuth component is known as right ascension, a. Although right' ascension is measured in degrees in all spacecraft work, in most astronomical tables it is measured in hours, minutes, and seconds where 1 hour=l5 deg, 1 min=l/60th hour =1/4 deg, and I sec = I/60th min =

0.0041666...deg. Each of these measurements corresponds ¡o the amount of rotation of the Earth in that period of time. Note that minutes and seconds of right ascension are not equivalent to minutes and seconds of arc, even along the equator.

Although celestial coordinates are the most widely used, several other inertia! coordinate systems are used for special purposes. These systems are summarized in Table 2-1.

Orbit-Defined Coordinates. The l,b,n system of coordinates is a system for which the plane of the spacecraft orbit is the equatorial plane of the coordinate system. The / axis is parallel to the line from the center of the Earth to the ascending node* of the spacecraft orbit, the n axis is parallel to the orbit normal,

1.e., perpendicular to the orbit plane, and the b axis is such that for unit vectors along the axes, b=nxl. The l,b,n system would be inertial if the spacecraft orbit were fixed in inertial space. In fact, perturbations on the orbit due to nonsphericity of the central body, gravitational attractions of other bodies, etc., cause the orbit to rotate slowly as described in Section 3.4, so the l,b,n system is not absolutely inertial.

Lastly, we define a system of coordinates that maintain their orientation relative to the Earth as the spacecraft moves in its orbit. These coordinates are

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