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Note that the heading is the angle by which the velocity vector deviates from being perpendicular to the radius vector. From Eq. (3-31), or the properties of an ellipse, a heading of 0 dsg implies that the spacecraft is at either apogee or perigee. The argument of perigee, u, may be obtained from the true anomaly by w=arc cos(Ó • R) - v (3-32)

where the first term on the right is in the range 0 deg to 180 deg if (ÓxR) Ñ>0 and 180 deg to 360 deg if (ÓxR)-Ñ<0. Finally, the mean anomaly may be obtained directly from Kepler's and Gauss' equations

where

It is clear from the foregoing principles that injection involving a substantial change in the semimajor axis will normally occur at apogee or perigee (/3=0 deg) and that changes in the inclination will normally occur at either the ascending or the descending node.

### 3.4 Orbit Perturbations

Real orbits never follow Kepler's Laws precisely, although at times they may come very close. The Keplerian elements of an orbit provide a convenient analytic approximation to the true orbit; in contrast, a definitive orbit is the best estimate that can be obtained with all of the available data on the actual path of a satellite. Because closed-form analytic solutions are almost never available for real orbit problems with multiple forces, definitive orbits are generated numerically based on both orbit theory and observations of the spacecraft Thus, definitive orbits are only generated for times that have past, although the information from a definitive orbit is frequently extrapolated into the future to produce a predicted orbit

A reference orbit is a relatively simple, precisely defined orbit (usually, though not necessarily, Keplerian) which is used as an initial approximation for the spacecraft motion. Orbit perturbations are the deviations of the true orbit from the reference orbit and may be classified according to specific causes, e.g., perturbations due to the Earth's oblateness, atmospheric drag, or the gravitational force of the Moon. In this section, we list the possible causes of orbit perturbations, describe qualitatively the effects of the various perturbations, and, where possible, provide formulae to determine the approximate effect of specific perturbations. Methods for the numerical treatment of perturbations may be found in the references at the end of the chapter.

Effects which modify simple Keplerian orbits may be divided into four classes: nongravitational forces, third-body interactions, nonspherical mass distributions, and relativistic mechanics. The first two effects may dominate the motion of a spacecraft, as in satellite reentiy into the atmosphere or motion about one of the stable Lagrange points. Although the effects of nonspherical mass distributions never dominate spacecraft motion, they provide the major perturbation, relative to Keplerian orbits, for most intermediate altitude satellites, i.e., those above where the atmosphere plays an important role and below where effects due to the Moon and the Sun become important. As indicated previously, relativistic mechanics may be completely neglected in most applications. The largest orbit perturbation in the solar system due to general relativity is the rotation of the perihelion of Mercuiy's orbit in the orbit plane by about 0.012 deg/centuiy or 3x 10~5 deg/orbit. Although a shift of this amount is measurable, it is well below the magnitude of the other effects which we will consider.

Although the relative importance of the three significant groups of perturbations will depend on the construction of the spacecraft, the details of its orbit, and even the level of solar activity*, the general effect of perturbing forces is clear. Atmospheric effects dominate the perturbing forces at altitudes below about 100 km and produce significant long-term perturbations on satellite orbits up to altitudes of about 1000 km. The major effect resulting from the nonspherical symmetiy of the Earth is due to the Earth's oblateness, which changes the gravitational potential by about 0.1% in the vicinity of the Earth. The ratio of the gravitational potential of the Moon to that of the Earth is 0.02% near the Earth's surface. As the satellite's altitude increases, the effect of oblateness decreases and the effect of the Moon increases; the magnitude of their effect on the gravitational potential is thé same at about 8000 km altitude. Lunar and solar perturbations are generally negligible at altitudes below about 700 km. (See Section 52.)

Nongravitational Forces. For near-Earth spacecraft, the principal nongravitational force is aerodynamic drag. Drag is a retarding force due to atmospheric friction and is in the direction opposite the spacecraft velocity vector. (If there is any component of the force perpendicular to the velocity vector, it is called lift.) In an elliptical .orbit, drag is most important at perigee because the density of the Earth's atmosphere, to which the drag is proportional, decreases exponentially with altitude. (See Section 4.4 for a discussion of atmosphere models.) Because drag forces are tangent to the orbit, opposite to the velocity, and applied near perigee, the qualitative effects of drag are similar to an impulsive in-plane transfer maneuver performed at perigee, as discussed in Section 33. As the drag slows the spacecraft at each perigee passage, the apogee height and, consequently, the semimajor axis and eccentricity are reduced. The perigee height and argument of perigee will remain approximately the same. In addition, neither the node nor the inclination will be affected, because the force is within the orbit plane (ignoring the small effect due to the rotation of the atmosphere).

'The level of solar activity significantly affects both the atmospheric density at spacecraft altitudes and the structure of the geomagnetic Field. See Sections 4.4 and S.l.

The general process of reducing the total energy and lowering apogee by atmospheric drag is called orbital decay. The orbital lifetime of a satellite is the time from launch until it penetrates deeply into the atmosphere such that the spacecraft either burns up or falls to the surface. Figure 3-17 gives the approximate lifetime of a satellite as a function of perigee height, hp, and eccentricity, e. The ordinate in the figure is the estimated number, N, of orbit revolutions in the satellite lifetime divided by the ballistic coefficient, m/(CDA), where m is the mass of the satellite and A is its cross-sectional area perpendicular to the velocity vector. The drag coefficient, CD, is a dimensionless number, usually between 1 and 2*. The ballistic coefficient is a measure of the ability of the spacecraft to overcome air resistance. For typical spacecraft, it ranges from 25 to 100 kg/m2. Given the value of N from Fig. 3-17, the satellite lifetime L, can be calculated directly from the period, P, or from the perigee height, hp, and eccentricity, e, by:

where hp is in kilometres and L is in days. For example, a satellite with a ballistic coefficient of 80 kg/m2 in a circular orbit at an altitude of 500 km (e=0, hp=500) has a value of N/80 from Fig. 3-17 of approximately 140; this gives an estimated lifetime of 11,000 revolutions (=80x140) or about 720 days. All formulas or graphs for spacecraft lifetimes are approximations, since the atmospheric density fluctuates considerably. For example, at 800 km, the density can fluctuate by a factor of 3 to 7 due to solar activity [Roy, 1965], Most lifetime estimates are in error by at least 10%. Simplified relations, such as that of Fig. 3-17, may be in error by 50%.

In addition to atmospheric drag, other nongravitational forces acting on a spacecraft are (1) drag due to induced eddy currents in the spacecraft interacting with the Earth's magnetic field, (2) drag due to the solar wind (Section 5.3) and micrometeoroids (interplanetary dust particles), and (3) solar radiation pressure. The first two effects are very small and are normally ignored. Solar radiation pressure can be important for some satellites, particularly those with large solar panels. For most satellites, the direction of the solar radiation force will be nearly radial away from the Sun*, although it is theoretically possible to build a solar "sail" which can tack in much the same fashion as a sailboat The magnitude of the force, VR, is given by:

* If no measured drag coefficient is available, CB=2 is a good estimate for satellites whose dimensions are large relative to the mean free path of atmospheric molecules.

tFor a spherical object, there will be a slight preferential scattering of light in the direction of the object's motion, which will produce an effective drag. While this Poynting-Roberlson effect is unimportant for spacecraft, it causes interplanetary meteoroids up to I cm in diameter to spiral into the Sun within about 20 million years from an initial distance of I AU.

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