Attitude Xyz


Fig. 3-17. Approximate Lifetimes for Earth Satellites. See text for explanation (adapted from Kendrick [1967]).


Fig. 3-17. Approximate Lifetimes for Earth Satellites. See text for explanation (adapted from Kendrick [1967]).

where P is the momentum flux from the Sun (4.4x 10~6 kg-m~1 • s-2, as discussed in Section 5.3), A is the cross-sectional area of the spacecraft perpendicular to the sunline, and K is a dimensionless constant in the range KK. 1 for translucent material, K=> 1 for perfectly absorbent material (black body), and K — 2 for materia] reflecting all light directly back toward the Sun. Echo I, a 60-kg, 30-m-diameter balloon satellite, was launched into a circular orbit at an altitude of 1600 km; however, solar radiation pressure reduced perigee to 1000 km at times [Glasstone, 1965].

Nonsphertcal Mass Distribution. For near-Earth satellites above several hundred kilometres, the major source of perturbations is the nonspherical shape of the Earth. This shape approximates an oblate spheroid which would be formed by a rotating fluid. Thus, the major correction for the nonspherical Earth is for the oblateness with much smaller corrections for the deviations from an oblate shape. (See Sections 4.3 and 5.2.)

The oblateness corrections for the Earth are grouped into three catagories. All of the instantaneous, or osculating, orbital elements undergo short period variations in which they fluctuate with true anomaly as the spacecraft moves in its orbit. For three of the six elements (a, e, and i), these short period variations average to zero over an orbit; the other three elements undergo cumulative secular variations in which the average value of the parameters changes monotonically. It is the secular variations that are of primary interest in following the gross motion of the satellite.

Because the gravitational force is conservative, the total energy and the mean values of the semimajor axis, the apogee and perigee heights, and, consequently, the eccentricity do not change due to oblateness. To understand the physical cause of the secular variations which occur in other elements, it is convenient to think of the Earth as a point mass and a ring of uniform density in the equatorial plane representing the Earth's equatorial bulge. The easiest perturbation to visualize is the rotation of the orbit plane. Figure 3-18 shows the direction of the orbital angular momentum for a satellite in a prograde orbit. When the satellite is south of the equator, the average torque, N = rXF, due to the pull, F, of the Earth's equatorial bulge produces a westward torque. The average perturbing force north of the equator also produces a westward torque, causing the angular momentum vector and the orbit plane to rotate westward without changing the inclination. This motion of the line of nodes opposite the direction of rotation of the satellite is called regression of the nodes.

The other major oblateness perturbation is a motion of the line of apsides. Consider first the case of a satellite with zero inclination. In the equatorial plane, the gravitational force from a ring of mass (i.e., the equatorial bulge) is larger than if all of the mass were concentrated at the center. As shown in Fig. 3-19(a), this added force causes the orbit to curve more strongly, that is, the angular velocity of the satellite about the Earth will be increased. Thus, as shown in Fig. 3-19(b), each successive apogee and perigee will occur farther around than formerly and the line of apsides rotates in the direction of the satellite's motion.

For a satellite over either of the Earth's poles, the distance to the Earth's equator is greater than the distance to the center of the Earth. Therefore, the gravitational force due to the equatorial bulge is less than it would be if the mass of the bulge were at the Earth's center. The smaller force causes the orbit to curve



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