Applied Orbit Perturbation and Maintenance
Chia-Chun "George" Chao
Chapter 1: A Review of Two-Body Mechanics
1.1 Kepler's Laws
Johannes Kepler's three laws of orbital motion lay the foundation of the field of orbital mechanics. A review of two-body (Keplerian) mechanics requires familiarity with these laws:
- An orbit is an ellipse with a central body at one focus.
- An orbiting body's radius vector from a central body sweeps out equal areas in equal times.
- The square of an orbiting body's revolutionary period is proportional to the cube of the satellite's mean distance from the central body.
The content of this book is restricted to the motion of bodies in elliptical orbits with Earth as the central body. The reader may apply methods or theories discussed here to satellite orbits around other planets, with the understanding that changes to certain constants and assumptions will be necessary. It is important to add that Kepler's laws work in the inertial space with no perturbing forces. Once established, unperturbed elliptical orbits stay fixed in their inertial reference frames.
1.2 Equations of Motion in Relative Form
Through Newton's law of gravitation and his second law of motion (F = ma), one can derive the equations of motion of a space object moving under the influence of a central force field. As shown in several texts,1.1–1.5 the equations of motion of a Keplerian orbit can be given in relative form as:
| dr2/dt2 = -µr/r3, | (1.1) |
where r is the position vector of the space object with its origin at the center of mass of the primary body. The gravitational constant µ, sometimes called GM if the primary body is Earth, is defined by the following equation.
| µ = k2(m1 + m2), | (1.2) |
where k represents the Gaussian, or heliocentric, gravitational constant (0.01720209895 
); m1 is the mass of the primary body, or Earth; and m2 is the mass of the second body, or the satellite.
Figure 1.1 shows the position vector in an Earth-centered inertial (ECI) coordinate system. The x-axis is pointing to the vernal equinox, 
, and the y- and z-axes complete the right-handed system with the x-y axes in Earth's equatorial plane. In spherical coordinates, the corresponding equations of motion become:
| dr2/dt2 – r(dθ/dt)2 = -(µ/r2) | ||
| r d2θ/dt2 + 2(dr/dt) (dθ/dt) = 0, | (1.3) |
where θ is the angular variable measured from a reference axis that is usually the ascending node, or the intersection of the orbit with Earth's equatorial plane. Positive values for θ correspond to counterclockwise movement or movement in the direction of motion, and θ lies in the orbital plane. Therefore, two-body, or Keplerian, motion is two-dimensional if it is expressed in terms of spherical coordinates as described here. Figure 1.2 shows the geometry of the position vector in terms of these spherical coordinates.
Fig. 1.1. Geometry of ECI coordinates. |
Fig. 1.2. Geometry of spherical coordinates. |
1.3 Orbit Parameters
In accord with commonly used conventions, orbit parameters are denoted by the following symbols. The four angular variables are defined in Fig. 1.3.
Fig. 1.3. Orbit orientation and geometry in an inertial reference coordinate system. |
The six classic orbit elements that define an orbit in a three-dimensional inertial space are
- a, the semimajor axis
- e, eccentricity
- i, inclination (0 < i < 180 deg)
- W, right ascension of the ascending node
- ω, argument of perigee
- M, mean anomaly, or M0, mean anomaly at epoch t0
Orbit perturbations to be discussed in the later chapters of this book will be in terms of the deviations from those six classic elements.
Additional orbit parameters are used to compute the position and velocity of an orbiting object. Some of these parameters appear in the equations of motion with perturbations:
- E, eccentric anomaly
- v, true anomaly
- u, argument of latitude (= v + ω)
- p, semilatus rectum (= a[1 – e2])
- n, mean motion (= [µ/a3]½)
- P, period (= 2p/n = 2p[a3/µ]½)
- γ, flight-path angle
- Re, Earth equatorial radius
- hp, perigee altitude
- ha, apogee altitude
1.4 Conic Solutions
The solutions to the equations of motion (Eqs. [1.1] and [1.2]) are the conic solutions (i.e., ellipse, parabola, and hyperbola). This book's content is restricted to the ellipse. The mathematical derivations can be found in fundamental books on orbital mechanics or astrodynamics. Table 1.1 contains the commonly used relations, included here for quick reference (following Herrick1.2).
| Entity to be defined (or name of equation) | Equation |
|---|---|
| Vis viva energy integral | V2 = µ(2/r – 1/a) |
| Angular momentum | h = (µp)½ |
| Kepler's equation | M = E – e sin E |
| Radius equation | r = a(1 – e2)/(1 + e cos v) |
| Time rate of change of r | dr/dt = (µ/p)½ e sin v |
| Conversion of eccentric anomaly (E) to true anomaly (v) | cos v = (cos E – e)/(1 – e cos E) sin v = [(1 – e2)½ sin E]/(1 – e cos E) |
| Conversion of true anomaly (v) to eccentric anomaly (E) | cos E = (cos v + e)/(1 + e cos v) sin E = [(1 – e2)½ sin v]/(1 + e cos v) |
| Half-angle relation | tan (v/2) = [(1 + e)/(1 – e)]½ tan (E/2) |
| Flight-path angle | tan γ = e sin v/(1 + e cos v) = e sin E /(1 – e2)½ |
| Mean anomaly at t | M = M0 + n(t – t0) |
| Perigee altitude | hp = a(1 – e) – Re |
| Apogee altitude | ha = a(1 + e) – Re |
1.5 Conversion Between Earth-Centered Inertial Coordinates
An orbit analyst often needs to know how the perturbations in orbit elements translate into satellite position and velocity deviations. Although the computation can be done accurately by computer via calling subroutines, it is important to understand the fundamental relations between the two sets of orbit conditions. Some of the orbit perturbation and maintenance equations to be discussed in later chapters are derived from these relations. The conversion of classic orbit elements to ECI Cartesian coordinates may be accomplished through these equations (following Herrick1.2):
| x = r (cos W cos u – sin W sin u cos i) | ||
| y = r (sin W cos u + cos W sin u cos i) | ||
| z = r sin u sin i | (1.4) |
| dx/dt = V[(x/r) sin γ – cos γ (cos W sin u + sin W cos i cos u)] | ||
| dy/dt = V[(y/r) sin γ – cos γ (sin W sin u + cos W cos i cos u)] | ||
| dz/dt = V[(z/r) sin γ + cos γ cos u sin i] | (1.5) |
where V is the magnitude of the velocity (Table 1.1). A detailed derivation may be found in Chapter 4 of Orbital Mechanics.1.4 To convert ECI Cartesian coordinates to classical elements, one may use the following relations.
Solve for a using vis viva equation (Table 1.1).
Solve for e using:
| e cos E = rV2/µ – 1 and e sin E = r (dr/dt)/(µa)½ | (1.6) |
Solve for v using:
| cos v = a(cos E – e)/r and sin v = [a(1 – e2)½ sin E]/r | (1.7) |
Solve for i and W using the following relations:
| rxV = rVw | (1.8) |
| wx = sin i sin W | ||
| wy = –sin i cos W | ||
| wz = cos i | (1.9) |
Solve for u using Eq. (1.4).
Solve for ω through u = v + ω.
1.6 Types of Orbits
The following definitions of various orbit types are useful for discussing concepts related to the orbits of Earth satellites.
ACE (apogee at constant time-of-day equatorial) orbit: An elliptical orbit that lies in Earth's equatorial plane with a sun-pointing apogee. To satisfy the sun-pointing property, the secular rate of the apsidal rotation in the inertial reference frame must equal the rate of the right ascension of the sun.
frozen orbit: An Earth satellite orbit whose mean eccentricity and argument of perigee remain constant, such as NASA's Topex mission orbit.
GEO: Geostationary or geosynchronous orbit; one with an altitude of about 35,786 km. Its orbital mean motion equals the Earth's rotation rate. A geostationary satellite requires both longitude and latitude control, while a geosynchronous satellite requires only longitude stationkeeping. A geostationary satellite appears stationary to a ground observer. Most communication satellites, such as Intelsat and PanAmSat, are geostationary.
GTO: Geostationary transfer orbit; an elliptical orbit that completes a Hohmann and plane-change transfer from a low, circular parking orbit to a geosynchronous drift orbit. A geosynchronous or geostationary drift orbit is a circular orbit with a mean altitude either higher or lower than the stationary altitude required for a newly launched satellite to move to its desired longitude, usually at a rate of 3 deg/day, equivalent to an altitude of 234 km above or below GEO altitude.
HEO: Highly elliptical orbit; one with eccentricity larger than 0.5.
LEO: Low Earth orbit; one with altitude less than 1000 km, the level where atmospheric drag becomes significant.
Magic orbit: An orbit that has a period of about 3 hours, an inclination of 116.6 deg, and a nonzero eccentricity. Its semimajor axis and eccentricity values satisfy conditions for both sun-synchronous and frozen orbits.
MEO: Medium Earth orbit; one with an altitude between 1000 km and 35,286 km (500 km less than geostationary distance), such as the orbits of Galileo and GLONASS.
Molniya orbit: A highly elliptical orbit that has a 12-hour period and an inclination near the critical value (63.4 deg). It has an argument of perigee of 270 deg, and its ground traces repeat every other revolution.
sun-synchronous orbit: A satellite orbit whose nodal rate equals the angular rate of the mean sun, or one for which the local time of every ascending node crossing remains the same throughout the year, such as the weather satellite orbits.
supersynchronous orbit: A circular or nearly circular orbit with an altitude higher than that of the GEO orbit (about 35,786 km), such as the GEO disposal orbits.
Tundra orbit: An orbit with a 24-hour period, 30 to 70 deg inclination, and eccentricity from 0.13 to 0.5. Its primary purpose is to ensure good polar coverage in situations where regular GEO orbits cannot do so.
1.7 References
| 1.1 | D. Brouwer and G. M. Clemence, Methods of Celestial Mechanics (Academic Press, New York, 1961). |
| 1.2 | S. Herrick, Astrodynamics, Vol. 1 (Van Nostrand Reinhold Company, London, 1971). |
| 1.3 | R. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics (AIAA, Reston, VA, 1987). |
| 1.4 | V. A. Chobotov, ed., Orbital Mechanics, 3rd ed. (AIAA, Washington, 2002). |
| 1.5 | D. Vallado, Fundamentals of Astrodynamics and Applications, 2nd ed. (Space Technology Library, Microcosm, Inc., and Kluwer Academic Publishers, El Segundo, CA 2001). |
