Applied Orbit Perturbation and
Maintenance
Chia-Chun “George” Chao
The Aerospace Press • El Segundo, California
American Institute of Aeronautics and Astronautics, Inc. • Reston, Virginia
Contents
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
1
A Review of Two-Body Mechanics . . . . . . . . . . . . . . . . . . . 1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Kepler’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Equations of Motion in Relative Form . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Orbit Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conic Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conversion Between Earth-Centered Inertial Coordinates. . . . . . . . . . . . .
Types of Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Equations of Motion with Perturbations . . . . . . . . . . . . . . 7
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
General Equations of Motion in Earth-Centered Inertial Coordinates for
Gravity Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Third-Body (Sun-Moon) Attractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Solar-Radiation Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Atmospheric Drag. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Other Small Forces for High-Precision Orbit Prediction . . . . . . . . . . . . . 10
Coordinates for Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Methods of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Numerical Integration Methods and Tools . . . . . . . . . . . . . . . . . . . . . . . . 15
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3
Averaged Equations of Motion in Classical Elements . . . . 19
3.1
3.2
3.3
3.4
Background of General Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Method of Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Kozai’s Method for Earth Zonal Harmonics . . . . . . . . . . . . . . . . . . . . . . 21
Extension of Kozai and Kaufman’s Approach to Sun-Moon Gravitational
Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Averaged Equations of Motion from Solar-Radiation Pressure . . . . . . . . 29
Simplified Averaged Equations for Earth Atmospheric Drag . . . . . . . . . 33
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5
3.6
3.7
vii
1
1
2
3
4
5
6
Contents
4
Resonant Tesseral Harmonics in Kaula’s Formulations . . . 37
4.1
4.2
4.3
4.4
4.5
4.6
4.7
G Function for Eccentricity Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
F Function for Inclination Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Generalized Equations of Variation in Terms of F and G Functions . . . .
Equilibrium Longitudes for 24-Hour Orbits . . . . . . . . . . . . . . . . . . . . . . .
Equilibrium Longitudes for 12-Hour Circular Orbits . . . . . . . . . . . . . . . .
Equilibrium Longitudes for 12-Hour Molniya Orbits. . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Application of Averaged Equations to Orbit Analysis . . . . 59
5.1
Long-Term Eccentricity and Inclination Variations in Geosynchronous
Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Long-Term Eccentricity and Inclination Variations in Medium Earth Orbits
(GPS Orbits) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Long-Term Eccentricity and Inclination Variations in Highly Elliptical
Orbits (Molniya and GTO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Resonance Effects in Gravity, Solar-Radiation Pressure, and Third-Body
Equations for Low Earth Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
J3 Effects and Frozen Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2
5.3
5.4
5.5
5.6
37
42
46
46
52
54
57
6
Orbit Maintenance of LEO, MEO, and HEO Satellites and
Constellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.1
6.2
6.3
6.4
6.5
Orbit Maintenance of LEO Satellites and Constellations . . . . . . . . . . . . .
Maintenance of GPS and Other MEO Constellations . . . . . . . . . . . . . . . .
Maintenance of Molniya Orbits and Other HEO Constellations . . . . . . .
Guidelines for Designing Orbit Analysis Tools . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
Stationkeeping of GEO Satellites . . . . . . . . . . . . . . . . . . 101
7.1
7.2
7.3
7.4
7.5
7.6
7.7
Longitude (East-West) Stationkeeping . . . . . . . . . . . . . . . . . . . . . . . . . .
Inclination (North-South) Stationkeeping . . . . . . . . . . . . . . . . . . . . . . . .
Solar-Radiation Pressure and the Sun-Pointing Strategy . . . . . . . . . . . .
Perturbations and Control of Tundra Orbits . . . . . . . . . . . . . . . . . . . . . .
Guidelines for Designing GEO Orbit Analysis Tools. . . . . . . . . . . . . . .
Stationkeeping Using Ion Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
79
85
91
95
98
101
107
108
112
116
123
124
Contents
8
Collocation of GEO Satellites . . . . . . . . . . . . . . . . . . . . . 125
8.1
8.2
8.3
8.4
8.5
ITU Policies and the Need for Collocation . . . . . . . . . . . . . . . . . . . . . . .
Strategies of GEO Collocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Operational Requirements for Collocation Maintenance . . . . . . . . . . . .
Collision Avoidance Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
Advanced Concepts of Orbit Control . . . . . . . . . . . . . . . 157
9.1
9.2
Autonomous Onboard Stationkeeping of GEO Satellites Using GPS . . . 157
Autonomous Formationkeeping of Cluster Satellites Through Relative
Ranging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
Ground Tracking of GEO Collocation Satellites via the Raven Telescope
System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
9.3
9.4
10
125
126
152
153
154
End-of-Life Disposal Orbits: Strategies and Long-Term
Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
10.1
10.2
10.3
10.4
Policies for End-of-Life Disposal of Satellites . . . . . . . . . . . . . . . . . . . .
Study 1: Stability of GEO Disposal Orbits . . . . . . . . . . . . . . . . . . . . . . .
Study 2: MEO Disposal Orbit Stability and Direct Reentry Strategy. . .
Study 3: Long-Term Evolution of Navigation Satellite Orbits: GPS/
GLONASS/Galileo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.5 Study 4: Reentry Disposal for LEO Spacecraft . . . . . . . . . . . . . . . . . . .
10.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
197
199
212
230
236
249
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
ix
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:
1. An orbit is an ellipse with a central body at one focus.
2. An orbiting body’s radius vector from a central body sweeps out equal areas
in equal times.
3. 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:
dr 2 ⁄ dt 2 = – µr ⁄ r 3 ,
(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 = k 2 ( m 1 + m 2 ),
(1.2)
where k represents the Gaussian, or heliocentric, gravitational constant
AU 3 ⁄ 2
- ); m1 is the mass of the primary body, or Earth; and m2 is
( 0.01720209895 ---------------------1⁄2
m sun day
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, ϒ, and the y- and zaxes complete the right-handed system with the x-y axes in Earth’s equatorial
plane. In spherical coordinates, the corresponding equations of motion become:
dr 2 ⁄ dt 2 – r ( dθ ⁄ dt ) 2 = – ( µ ⁄ r 2 )
rd 2 θ ⁄ dt 2 + 2 ( dr ⁄ dt ) ( dθ ⁄ dt ) = 0,
1
(1.3)
2
A Review of Two-Body Mechanics
z
r
y
ϒ x
Fig. 1.1. Geometry of ECI coordinates.
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.
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.
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)
• Ω, right ascension of the ascending node
r
θ
Ω or
Fig. 1.2. Geometry of spherical coordinates.
Conic Solutions
3
z
Perigee
axis
r
ν
y
ω
Ω
Vernal
equinox x
Orbital
plane
i
Ascending node
Fig. 1.3. Orbit orientation and geometry in an inertial reference coordinate system.
• 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]1/2)
• P, period (= 2π/n = 2π[a3/µ]1/2)
• 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).
4
A Review of Two-Body Mechanics
Table 1.1. Equations Commonly Used in Orbital Mechanics
Entity to be defined
(or name of equation)
Equation
Vis viva energy integral
V 2 = µ ( 2 ⁄ r – 1 ⁄ a ).
Angular momentum
h = ( µp ) 1 ⁄ 2 .
Kepler’s equation
M = E – e sin E.
Radius equation
r = a ( 1 – e 2 ) ⁄ ( 1 + e cos v ).
Time rate of change of r
dr ⁄ dt = ( µ ⁄ p ) 1 ⁄ 2 e sin v.
Conversion of eccentric anomaly
(E) to true anomaly (v)
cos v = ( cos E – e ) ⁄ ( 1 – e cos E ).
sin v = [ ( 1 – e 2 ) 1 ⁄ 2 sin E ] ⁄ ( 1 – e cos E ).
Conversion of true anomaly (v) to cos E = ( cos v + e ) ⁄ ( 1 + e cos v ) .
eccentric anomaly (E)
sin E = [ ( 1 – e 2 ) 1 ⁄ 2 sin v ] ⁄ ( 1 + e cos v ) .
Half-angle relation
tan ( v ⁄ 2 ) = [ ( 1 + e ) ⁄ ( 1 – e ) ] 1 ⁄ 2 tan ( E ⁄ 2 ).
Flight-path angle
tan γ = e sin v ⁄ ( 1 + e cos v ) = e sin E ⁄ ( 1 – e 2 ) 1 ⁄ 2 .
Mean anomaly at t
M = M 0 + n ( t – t 0 ).
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 Ω cos u – sin Ω sin u cos i ).
y = r ( sin Ω cos u + cos Ω sin u cos i ).
z = r sin u sin i.
(1.4)
Types of Orbits
dx ⁄ dt = V [ ( x ⁄ r ) sin γ – cos γ ( cos Ω sin u + sin Ω cos i cos u ) ].
dy ⁄ dt = V [ ( y ⁄ r ) sin γ – cos γ ( sin Ω sin u + cos Ω cos i cos u ) ].
dz ⁄ dt = V [ ( z ⁄ r ) sin γ + cos γ cos u sin i ].
5
(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 = rV 2 ⁄ µ – 1
and
e sin E = r ( dr ⁄ dt ) ⁄ ( µa ) 1 ⁄ 2 .
(1.6)
and
sin v = [ a ( 1 – e 2 ) 1 ⁄ 2 sin E ] ⁄ r.
(1.7)
Solve for v using:
cos v = a ( cos E – e ) ⁄ r
Solve for i and Ω using the following relations:
rxV = rVw.
(1.8)
w x = sin i sin Ω.
w y = – sin i cos Ω.
(1.9)
w z = cos i.
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 sunpointing 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.
6
A Review of Two-Body Mechanics
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).