true anomaly

v => M+2esinM + 125ezsin(2M)


As an example, we find the time it takes a satellite to go from perigee to an angle 90 deg from perigee, for an orbit with a semimajor axis of7,000 km and an eccentricity of 0.1. For this example v0 = = Mq = 0.0 rad t0 =0.0s v = 1.5708 rad E = 1.4706 rad

Finding the position in an orbit after a specified period is more complex. For this problem, we calculate the mean anomaly, M, using time of flight and the mean motion using Eq. (6-12). Next we determine the true anomaly, v, using the series expansion shown in Table 6-3, a good approximation for small eccentricity (the error is of the order e3). If we need greater accuracy, we must solve the equation in Table 6-3 relating mean anomaly to eccentric anomaly. Because this is a transcendental function, we must use an iterative solution to find the eccentric anomaly, after which we can calculate the true anomaly directly.

6.1.6 Orbit Determination

Up to this point we have assumed that we know both the position and velocity of the satellite in inertial space or the classical orbital elements. But we often cannot directly observe the satellite's inertial position and velocity. Instead, we commonly receive data from radar, telemetry, optics, or GPS. Radar and telemetry data consists of range, azimuth, elevation, and possibly the rates of change of one or more of these quantities, relative to a site attached to the rotating Earth. GPS receivers give GO latitude, longitude, and altitude. Optical data consists of right ascension and declination relative to the celestial sphere. In any case, we must combine and convert this data to inertial position and velocity before determining the orbital elements. Bate, Mueller, and White [ 1971 ] and Escobal [ 1965] cover methods for combining data, so I will not cover them here.

The type of data we use for orbit determination depends on the orbit selected, accuracy requirements, and weight restrictions on the spacecraft Because radar and optical systems collect data passively, they require no additional spacecraft weight but they are also the least accurate methods of orbit determination. Conversely, GPS

data is more accurate but also requires a receiver and processor, which add weight We can also use GPS for semiautonomous orbit determination because it requires no ground support An alternative method for fully autonomous navigation is given by Tai and Noerdlinger [1989] (See Sec. 11.2.)

6.2 Orbit Perturbations

The Keplerian orbit discussed above provides an excellent reference, but other forces act on the satellite to perturb it away from the nominal orbit We can classify these perturbations, or variations in the orbital elements, based on how they affect the Keplerian elements.

Figure 6-6 illustrates a typical variation in one of the orbital elements because of a perturbing force. Secular variationsrepresent a linear variation in the element Short-period variations are periodic in the element with a period less than or equal to the orbital period. Long-period variations have a period greater than the orbital period. Because secular variations have long-term effects on orbit prediction (the orbital elements affected continue to increase or decrease), I will discuss them in detail. If the satellite mission demands that we precisely determine the orbit we must include the periodic variations as well. Battin [1999], Danby [1962], and Escobal [1965] describe methods of determining and predicting orbits for perturbed Keplerian motion.

Fig. 6-6. Secular and Periodic Variations of an Orbital Element Secular variations represent linear variations In the element, short-period variations have a period less than the orbital period, and long-period variations have a period longer than the orbital period.

When we consider perturbing forces, the classical orbital elements vary with time.

To predict the orbit we must determine this time variation using techniques of either special or general perturbations. Special perturbations employ direct numerical integration of the equations of motion. Most common is Cowell's method, in which the accelerations are integrated directly to obtain velocity and again to obtain position.

General perturbations analytically solve some aspects of the motion of a satellite subjected to pertuibing forces. For example, the polar equation of a conic applies to the two-body equations of motion. Unfortunately, most pertuibing forces don't yield to a direct analytic solution but to series expansions and approximations. Because the orbital elements are nearly constant, general perturbation techniques usually solve directly for the orbital elements rather than the inertial position and velocity. The oibital elements are more difficult to describe mathematically and approximate, but they allow us to better understand how pertuibations affect a large class of orbits. We can also obtain solutions much faster than with special perturbations.

The primary forces which perturb a satellite orbit arise from third bodies such as the Sun and the Moon, the nonspherical mass distribution of the Earth, atmospheric drag, and solar radiation pressure. We describe each of these below.

6.2.1 Third-Body Perturbations

The gravitational forces of the Sun and the Moon cause periodic variations in all of the oibital elements, but only the right ascension of the ascending node, argument of perigee, and mean anomaly experience secular variations. These secular variations arise from a gyroscopic precession of the orbit about the ecliptic pole. The secular variation in mean anomaly is much smaller than the mean motion and has little effect on the orbit; however, the secular variations in right ascension of the ascending node and argument of perigee are important, especially for high-altitude oibits.

For nearly circular orbits, e1 is almost zero and the resulting error is of the order e2. In this case, the equations for the secular rates of change resulting from the Sun and Moon are:

• Right ascension of the ascending node:

• Argument of perigee:

where i is the oibital inclination, n is the number of orbit revolutions per day, and Q and (o are in deg/day. These equations are only approximate; they neglect the variation caused.by the changing orientation of the oibital plane with respect to both the Moon's oibital plane and the ecliptic plane.

6.2.2 Perturbations Because of a Nonspherical Earth

When developing the two-body equations of motion, we assumed the Earth has a spherically symmetric mass distribution. In fact, the Earth has a bulge at the equator.

a slight pear shape, and flattening at the poles. We can find a satellite's acceleration by taking the gradient of the gravitational potential function, i>. One widely used form of the geopotential function is:


where p is Earth's gravitational constant, RE is Earth's equatorial radius, P„ are Legendre polynomials, L is geocentric latitude, and 4 are dimensionless geopotential coefficients of which the first three are:

J2= 0.001 082 63 J3 = -0.000 002 54 J4 = -0.0000)1 61

This form of the geopotential function depends on latitude, and we call the geopotential coefficients, jj,, zonal coefficients. Other, more general expressions for the geopotential include sectoral and tesseral terms in the expansion. The sectoral terms divide the Earth into slices and depend only on longitude. The tesseral terms in the expansion represent sections that depend on longitude and latitude. They divide the Earth into a checkerboard pattern of regions that alternately add to and subtract from the two-body potential. A geopotential model consists of a matrix of coefficients in the spherical harmonic expansion. The widely used Goddard Earth Model 10B, or GEM 10B, is called a "21 x21 model" because it consists of a 21 x21 matrix of coefficients. In order to achieve high accuracy mapping of the ocean surface and wave properties, the TOPEX mission required creating a 100 x 100 geopotential model.

The potential generated by the nonspherical Earth causes periodic variations in all of the orbital elements. The dominant effects, however, are secular variations in right ascension of the ascending node and argument of perigee because of the Earth's oblateness, represented by the \ term in the geopotential expansion. The rates of change of £2 and a due to \ are

= 1.032 37 x 1014a"7/2(4 - 5 sin2 i) (1 - e2)"2

where r is mean motion in deg/day, RE is Earth's equatorial radius, a is semimajor axis in km, e is eccentricity, i is inclination, and Q and o) are in deg/day. Table 6-4 compares the rates of change of right ascension of the ascending node and argument of perigee resulting from the Earth's oblateness, the Sun, and the Moon. For satellites in GEO and below, the J2 perturbations dominate; for satellites above GEO the Sun and Moon perturbations dominate.

Molniya orbits are highly eccentric (e = 0.75) with approximately 12 hour periods (2 revolutions/day). Orbit designers choose the orbital inclination so the rate of change of perigee, Eq. (6-20X is zero. This condition occurs at inclinations of 63.4 deg and 116.6 deg. For these orbits we typically place perigee in the Southern Hemisphere, so

TABLE 6-4. Secular Variations In Right Ascension of the Ascending Node and Argument of Perigee. Note that these secular variations form the basis for Sun-synchronous and Molniya orbits. For Sun-synchronous orbits the nodal precession rate is set to 0.986 deg/day to match the general motion of the Sun.

TABLE 6-4. Secular Variations In Right Ascension of the Ascending Node and Argument of Perigee. Note that these secular variations form the basis for Sun-synchronous and Molniya orbits. For Sun-synchronous orbits the nodal precession rate is set to 0.986 deg/day to match the general motion of the Sun.


Effect of Jj (Eqs. 6-19,6-20) (deg/day)

Effect of the Moon (Eqs. 6-14,6-16) (deg/day)

Effect of the Sun (Eqs. 6-15,6-17) (deg/day)


a = 6,700 km, e=0.0, /=

28 deg

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