Basic Coordinate Systems

The (X, Y, Z) coordinate system used in the preceding sections has no preferred orientation for the simple description of the two-body motion. Nevertheless, a specific, well-defined orientation of the (X, Y, Z) axes is required in practice.

Consider the motion of the Earth about the Sun. Since the mass of the Sun is 328,000 times the mass of the Earth, it is quite natural to describe the motion of this system as the motion of the Earth relative to the Sun. But from the description of the two-body motion, there is no requirement that M1 be greater than M2 or vice versa. The two-body motion of the Earth-Sun system is illustrated in Fig. 2.2.5 from a geocentric point of view. The sphere representing the Earth includes a body-fix ed (Earth-fixed) coordinate system, with the (x, y) axes in the equatorial plane and with the x axis coincident with the intersection of the Greenwich

Figure 2.2.5: The Earth-Sun two-body motion. For convenience the motion can be described from the geocentric perspective to illustrate the relations between two-body parameters and astronomical conventions. The orbit plane defiled by the angular momentum of the two-body system is the ecliptic, the ascending node is the vernal equinox, and the inclination is the obliquity of the ecliptic. The bodies are not drawn to scale since the diameter of the Sun is 100 times the diameter of the Earth. The (X', Y') axes have no preferred orientation, but the X axis is chosen to coincide with the vernal equinox.

Figure 2.2.5: The Earth-Sun two-body motion. For convenience the motion can be described from the geocentric perspective to illustrate the relations between two-body parameters and astronomical conventions. The orbit plane defiled by the angular momentum of the two-body system is the ecliptic, the ascending node is the vernal equinox, and the inclination is the obliquity of the ecliptic. The bodies are not drawn to scale since the diameter of the Sun is 100 times the diameter of the Earth. The (X', Y') axes have no preferred orientation, but the X axis is chosen to coincide with the vernal equinox.

meridian and the equator. The z axis of the Earth-ix ed system coincides with the angular velocity vector of the Earth, we. The nonrotating (X', Y') axes lie in the Earth's equator, but there is no preferred direction for these axes, except the Z' axis coincides with the angular velocity of the Earth in this model. The Earth-Sun orbit plane intersects the equator at the line of nodes. The point where the Sun moves from the southern hemisphere into the northern hemisphere is the ascend ing node, denoted by AN in Fig. 2.2.5 from a geocentric viewpoint. Similarly, the descending node is identified by DN.

The description given in the preceding paragraph and illustrated in Fig. 2.2.5 contains the essential descriptions of a well-defned and consistent celestial coordinate system. Since the two celestial bodies are modeled as rigid spheres with constant density, the node locations are fix ed in space and the angular velocity vector, we, of the rotating Earth is constant in magnitude and direction.

Special terms are assigned to the orbit parameters illustrated in Fig. 2.2.5. The orbit plane of the Sun about the Earth (or the Earth about the Sun) is referred to as the ecliptic and the inclination is the obliquity of the ecliptic, usually denoted by e ( ~ 23.5°). The ascending node corresponds to the point known as the vernal equinox and the descending node is the autumnal equinox. The vernal equinox occurs about March 21 and the autumnal equinox occurs about September 21. The location of the Sun when it is 90° from the vernal equinox is the summer solstice and the location at 270° is the winter solstice, which occur about June 21 and December 21, respectively. It is somewhat prejudicial that these astronomical locations are identified with northern hemisphere seasons, since association with seasons is reversed in the southern hemisphere. Finally, the angle aG defines the orientation of the Earth-fix ed coordinate system (x, y, z) with respect to the vernal equinox (i.e., the x axis shown in Fig. 2.2.5). The angle aG is known as the Greenwich mean sidereal time (GMST), defined by aG = (t — 10 ) + aG0. The period of Earth rotation with respect to a fix ed direction in space is about 86,164 sec (23 hr 56 min 4 sec) (i.e., = 2n/86164 rad/sec).

With the X axis directed toward the vernal equinox and with both the z and Z axes coincident with the angular velocity vector, we, it follows that the relations between the nonrotating (X, Y, Z) axes and the Earth-fix ed, rotating (x, y, z) axes as illustrated in Fig. 2.2.5 are

For the two-body problem that has been described up to this point, the orientation of the ecliptic defined in a consistent manner with the two-body dynamics will be fix ed in space. Hence, the vernal equinox would be a fix ed direction in space and the obliquity of the ecliptic would also be fix ed. However, as described in Section 2.4, temporal variations exist in the location of the vernal equinox and in e. With such variations, it is necessary to designate a specific epoch to be assigned to the equinox and obliquity. A commonly used reference is the mean equator and equinox of 2000.0 (the vernal equinox on January 1, 2000, 12 hrs). The system defined for this date is referred to as the J2000 system. In some applications, the M50 system is used, which is defined by the mean equator and equinox of x

1950.0. The nonrotating J2000 or M50 systems are sometimes referred to as Earth-centered inertial (ECI) and the Earth-fk ed system (x, y, z) is referred to as Earth-centered, Earth-ix ed (ECF or ECEF or ECF (Earth-centered, ix ed)). We will use the notation J2000 to refer to the system and J2000.0 to refer to the date.

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