![Introduction ix
Two Dynamical Systems in the Orbital Dynamics
In a dynamical system, the motion of a celestial body including spacecraft, whether
big or small, is usually controlled by more than two external forces. But in the Solar
System, where the Sun has been existing for more than 4.6 billion years and is the
dominator, the motion of a celestial body is mainly determined by no more than two
external forces. For the major planets, there is only one main source of force, i.e., the
Sun, the other sources of forces are regarded as perturbations, i.e., small disturbances.
For an asteroid, there is usually only one source, the Sun, or are two, the Sun and
one of the major planets. For a natural satellite, the main force is from a related
major planet. For most of the artificial Earth’s satellites, it is Earth. If the satellite
has a high Earth orbit (such as a lunar rover, its orbit needs to be changed during its
mission) there are two sources of force, Earth and the Moon. For a deep spacecraft
(to explore major planets or natural satellites) the external sources can be the Sun, or
the Sun and a major planet, or a major planet and one of its satellites. In all the cases
mentioned, besides the one or two main forces, other forces (including non-particle
gravitational forces and non-gravitational forces) can be treated as perturbing forces.
Therefore, from the perspective of orbital dynamics in the actual Solar System there
are only two rational dynamical models for studying the motion of various spacecraft.
One model is for the circular orbital problem (for artificial satellites including Earth’s
satellites and the Moon’s satellites, etc.) with one main force (i.e., the central celestial
body), which corresponds to “the perturbed two-body problem”. The other model
has one or two primary forces as in the cases of most deep-space spacecraft, which
corresponds to “the perturbed restricted three-body-problem” are regarded as
perturbations, i.e., small disturbances.
Mathematical Models for Satellite Motion: The Perturbed
Two-Body Problem [1–8]
As mentioned above, in the Solar System there is only one main force that controls
the motions of major planets, asteroids, satellites, and artificial spacecraft (artificial
Earth’s satellites, the Moon’s satellites, Mars’s satellites, and other orbiting space-
craft). For planets, the main force is the Sun, for natural satellites, it is the related
planet, and for spacecraft, it is the target celestial body. Compared to the main force,
other forces are small, therefore generally the N-body system (N ≥ 3) can be regarded
as a “perturbed two-body system”, which is mathematically called a perturbed two-
body problem. For distinguishing the two bodies, the main external source is called
the “central body”, denoted as P0, and its mass is denoted as m0; whereas the other
body, which is the object to study, is denoted as p, and its mass as m. Our research
object is the orbital motion of a celestial body, no matter it is a planet, a satellite, or
a circling spacecraft, controlled by the gravitational force of the central body and a
few other perturbing forces.](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-14-320.jpg)
![xiv Introduction
the perturbation function R (through M). With M as an independent element, the
perturbed equation can be simplified.
The Perturbed Restricted Three-Body Problem
in the Motion of Deep-Space Prober
The Restricted Three-Body Problem for Circular and Elliptical
Motions [9–12]
In a three-body problem with N = (2 + 1), there are two primary bodies and a small
body. Because the small body has no influence on the motions of the two primary
bodies, the motions of two primary bodies are defined by a simple two-body problem.
Each of the big bodies moves in a circle or an ellipse around their common barycenter,
but neither a parabola nor a hyperbola, realistically. This problem, therefore, is a
restricted circular three-body problem or a restricted elliptical three-body problem.
The motion of the third body (a small body) in this system is to be studied.
In the Sun-Earth-Moon three-body system the Moon’s mass (m) by comparison
is much smaller than the masses of the Sun and Earth, m1 and m2, respectively, that
m = 0.012 m2, approximately; and the eccentricity of Earth’s orbit around the Sun
is only 0.017, thus the motion of the Moon in this system can be treated in a circular
restricted three-body problem. Of cause, an elliptical restricted three-body problem is
closer to the real situation than a circular restricted three-body problem. This model
is also applied to the motion of an asteroid located in the asteroid belt (between
Mars’s orbit and Jupiter’s orbit, most asteroids are in this belt). The motion of an
asteroid is due to mainly the gravitational forces from the Sun and Jupiter. Because
the eccentricity of Jupiter’s orbit is relatively small, the motion of an asteroid can be
also treated as a circular restricted three-body problem.
The orbit of a deep-space spacecraft is more complicated. The whole process can
be divided into a few segments. For example, after launch a Moon’s prober has a
near Earth orbit like an Earth’s satellite; when it is near the Moon, it changes its
orbit and moves around the Moon, between the two orbits the motion of the prober is
in a typical restricted three-body system of Earth, the Moon, and the prober, which
can be a circular or an elliptical restricted three-body problem. Another example
is about a Mars’s prober. In the early stage after launch, it moves like an Earth’s
satellite. During the time it leaves the Earth-Moon system and before it reaches the
area of Mars’s gravitational field there is a long cruising period controlled by the
Sun’s attraction, and the motion is decided by a perturbed two-body problem with
the Sun as the central body. After it moves into Mars’s gravitational field its motion
then is provided by a typical restricted three-body problem of the Sun, Mars, and
the prober. There are many more examples like these in the exploration of the Solar
System.](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-19-320.jpg)
![Introduction xv
The equations of the above-mentioned motion models, including the simplest
circular restricted three-body problem, are unsolvable. There is only one solved
problem which is the restricted three-body problem with two “motionless” main
bodies. It is an approximate model when the motions of the two big bodies are much
slower than the small body. This is called the problem of two stationary main bodies
in the restricted three-body problem. The equation of this model is solved, but the
model is too simple to be used in solving any actual dynamic problem for a spacecraft
in the Solar System.
Models for the Restricted N-body Problem and the Perturbed
Restricted Three-Body Problem [13, 14]
In the restricted N-body (N ≥ 3) problem, there are n-big bodies and one small body.
One example of this problem is the motion of an asteroid in the asteroid belt. In order
to present the motion close to the real situation and to agree with the characteristics
of the distribution of asteroids (such as the Kirkwood gaps), the gravitational forces
should be included are not only the primary forces from the Sun and Jupiter but
also the forces from Saturn and Mars, thus a restricted problem of (4 + 1) bodies
is formed. Another example is about a Moon’s prober. The motion of the prober is
determined by gravitational forces from Earth, the Moon, and the Sun; therefore, it
is a kind of restricted problem of (3 + 1) bodies. In this kind of system, it does not
matter how many big bodies there are (i.e., different values of N), the motion of a
small body, which can be an asteroid or a Moon’s prober, is studied by assuming
that the motions of the big bodies are defined. In the first example, if the force from
the third body and the fourth body is not strong enough to cause obvious changes
to the results of the original restricted three-body problem, then their forces can be
treated as perturbations, and the N-body problem can be regarded as a perturbed
restricted three-body problem. Similarly in the second example, the force of the
third body can be treated as a perturbation. In reality, the motion of an asteroid
in the asteroid belt or a Moon’s prober is studied in this way. In other words, this
kind of problem is solved by the perturbation method using as much information
as possible from a restricted three-body problem. This method is applied to design
deep spacecraft orbits and some specific orbits for specific purposes (such as the
Halo orbit). From the above two examples, although using the five-body problem
model or the four-body problem model is seemly more precise and more attractive,
the corresponding four-body problem or three-body problem of the big bodies is
still unsolved to the present day. The fact is that in the Solar System to study the
motion of a natural celestial body or an artificial body the most commonly used
models are the perturbed restricted two-body problem model and the perturbed
restrictedthree-bodyproblem model,especiallythe perturbedcircularrestricted
three-body problem model.](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-20-320.jpg)
![xvi Introduction
The Restricted Problem of (n + k)-Bodies [15, 16]
The (n + k)-body problem is an N-body problem where N = n + k, and there are
n-big bodies and k-small bodies with n ≥ 2 and k ≥ 2. If k = 1, the problem is then
reduced to one of the above discussed two examples. This (n + k) system is actually
about motions of k-small bodies in an n-body system. The motions of these big bodies
are defined, and the small bodies are attracted by the big bodies. The masses of the
k small bodies are much smaller than those of the big bodies, thus the small bodies
do not affect the motions of the big bodies, but if the distances between the small
bodies are short, the gravitational forces between them should be considered. If the
gravitational forces of the small bodies can be ignored, then the motion of each small
body can be studied separately in an (n + 1)-body restricted problem.
The type of (n + k)-body problem exists in the Solar System. For example, to
studythemotionsoftwocloselylocatedasteroidsintheasteroidbelt,thegravitational
forces between them need to be considered, then the Sun, Jupiter, and the two small
asteroids make up a restricted problem of (2 + 2) bodies. Another example is about
launching two geosynchronous satellites at a fixed point high up above the equator. If
the weight of each satellite is a few tons, and the distance between them is about a few
hundred meters, then to obtain a high precision solution the influence between the
two small satellites needs to be included. In this case, Earth, an ellipsoid, is treated as
if there were two bodies, one is a sphere with evenly distributed mass, and the other
is a “body” formed by the non-spherical part around Earth’s equator, thus there is a
restricted problem of (2 + 2)-bodies. Similar situations also appear in the launching
of a few spacecraft at a specific point, the gravitational forces between them cannot
be ignored, if there are two related big bodies, then the spacecraft and the two big
bodies form a restricted problem of (2 + k)-bodies.
General Restricted Three-Body Problem
In a restricted three-body problem under Newton’s gravitational forces if a big body
has strong radiation, then its post-Newtonian effect (i.e., the post-Newtonian expan-
sion) should be considered, and the system might be called a generalized restricted
three-body problem. In this case, if the motion of the second big body is not affected
by the radiation (rigorously speaking, the effect is small enough to be omitted), then
the non-gravitational force on the small body should be included. The research of
the generalized restricted three-body problem is not discussed in the book.](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-21-320.jpg)
![Contents
1 Selections and Transformations of Coordinate Systems . . . . . . . . . . . 1
1.1 Time Systems and Julian Day [1, 2] . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Selection of Standard Time . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Time Reference Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Julian Day . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Space Coordinate Systems [2–6] . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Earth’s Coordinate Systems [2, 6–10] . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 The Realization of the Dynamical Reference
System and J2000.0 Mean Equatorial Reference
System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.2 The Intermediate Equator and Three Related
Datum Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.3 Three Geocentric Coordinate Systems . . . . . . . . . . . . . . . 12
1.3.4 Transformation of the Earth-Fixed Coordinate
System O-XYZ and the Geocentric Celestial
Coordinate System O-xyz . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.5 Relationship Between the IAU 1980 Model
and the IAU 2000 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.6 The Complicity in the Selection of Coordinate
System Due to the Wobble of Earth’s Equator . . . . . . . . . 24
1.3.7 Coordinate Systems Related to Satellite
Measurements, Attitudes, and Orbital Errors . . . . . . . . . . 24
1.4 The Moon’s Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.4.1 Definitions of the Three Selenocentric Coordinate
Systems [6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4.2 The Moon’s Physical Libration . . . . . . . . . . . . . . . . . . . . . 26
1.4.3 Transformations Between the Three Selenocentric
Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.5 Planets’ Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.5.1 Definitions of Three Mars-Centric Coordinate
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
xix](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-23-320.jpg)
![Contents xxi
2.5.2 Expressions of Satellite’s Position Measurements
from a Ground-Based Tracking Station . . . . . . . . . . . . . . 72
2.5.3 Equatorial Coordinates of the Sub-Satellite Point . . . . . . 73
2.5.4 Satellite’s Orbital Coordinate System . . . . . . . . . . . . . . . . 74
2.5.5 Expressions of Errors in Satellite Position . . . . . . . . . . . . 75
2.6 Parabolic Orbit and Hyperbolic Orbit . . . . . . . . . . . . . . . . . . . . . . . 75
2.6.1 The Parabolic Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.6.2 The Hyperbolic Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.6.3 Formulas for Calculating the Position Vector
and Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3 Analytical Methods of Constructing Solution of Perturbed
Satellite Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.1 The Method of the Variation of Arbitrary Constants
Applied to the Perturbed Two-Body Problem . . . . . . . . . . . . . . . . . 81
3.2 Common Forms of Perturbed Motion Equation . . . . . . . . . . . . . . . 84
3.2.1 Perturbed Motion Equations Formed
by Accelerations of the (S, T, W)-Version
and the (U, N, W)-Version . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.2.2 The Perturbation Motion Equations Formed
by ∂R/∂σ-Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.2.3 Canonical Equations of Perturbation Motion . . . . . . . . . . 88
3.2.4 Singularities in the Perturbation Equations . . . . . . . . . . . 88
3.3 Perturbation Method of Constructing Power Series
Solution with a Small Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.3.1 Perturbation Equations with a Small Parameter . . . . . . . . 94
3.3.2 Existence of Power Series Solution with a Small
Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.3.3 Construction of the Power Series Solution
with a Small Parameter: The Perturbation Method . . . . . 96
3.3.4 Secular Variations and Periodic Variations . . . . . . . . . . . . 99
3.4 An Improved Perturbation Method: The Method of Mean
Orbital Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.4.1 Introduction of the Method of Mean Orbital
Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.4.2 The Mean Values of Related Variables
in an Elliptic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.4.3 Construction of Formal Solution: The Method
of Mean Orbital Elements [3–8] . . . . . . . . . . . . . . . . . . . . . 106
3.4.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.4.5 Two Annotations About the Method of Mean
Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.5 The Method of Quasi-Mean Elements: The Structure
of the Formal Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-25-320.jpg)
![xxii Contents
3.5.1 Small Divisors in Expressions of Perturbation
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.5.2 Configuration of Formal Solution: The Method
of Quasi-Mean Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.6 Methods of Constructing Non-singularity Solutions
for a Perturbed Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.6.1 Configuration of the Non-singularity Perturbation
Solutions of the First Type . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.6.2 Configuration of the Non-singularity Perturbation
Solutions of the Second Type . . . . . . . . . . . . . . . . . . . . . . . 124
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4 Analytical Non-singularity Perturbation Solutions
for Extrapolation of Earth’s Satellite Orbital Motion . . . . . . . . . . . . . 129
4.1 The Complete Dynamic Model of Earth’s Satellite Motion . . . . . 129
4.1.1 Selection of Calculation Units in Satellite Orbit
Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.1.2 Analyses of Forces on Satellite’s Orbital Motion . . . . . . 132
4.1.3 Further Analyses of the Forces Acting on a Satellite . . . 136
4.2 The Perturbed Orbit Solution of the First-Order Due
to Earth’s Dynamical Form-Factor J2 Term . . . . . . . . . . . . . . . . . . 137
4.2.1 The Perturbed Orbit Solution of the First-Order
in Kepler Orbital Elements [1–5] . . . . . . . . . . . . . . . . . . . . 137
4.2.2 The Non-singularity Perturbation Solution
of the First Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.2.3 The Non-singularity Perturbation Solution
of the Second Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
4.3 The Perturbed Orbit Solution of the First-Order Due
to Earth’s Ellipticity J2,2 Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.3.1 The Perturbed Orbit Solution of the First-Order
in Kepler Orbital Elements . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.3.2 The Non-singularity Perturbation Solution
of the First Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4.3.3 The Non-singularity Perturbation Solution
of the Second Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
4.4 Additional Perturbation of the Coordinate System
for the First-Order Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
4.4.1 The Cause of the Additional Perturbation
of the Coordinate System [3, 8] . . . . . . . . . . . . . . . . . . . . . 168
4.4.2 The Additional Perturbation Solution in Kepler
Orbital Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
4.4.3 The Non-singularity Additional Perturbation
Solution of the First Type . . . . . . . . . . . . . . . . . . . . . . . . . . 173
4.4.4 The Non-singularity Additional Perturbation
Solution of the Second Type . . . . . . . . . . . . . . . . . . . . . . . . 176](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-26-320.jpg)
![xxiv Contents
4.8 The Perturbed Orbit Solution Due to Earth’s Deformation . . . . . . 230
4.8.1 Expression of the Additional Potential of Tidal
Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
4.8.2 Effect of the Main Term in the Additional Tidal
Deformation Potential (the Second-Order Term
of l = 2) on a Satellite Orbit . . . . . . . . . . . . . . . . . . . . . . . . 232
4.9 Post-Newtonian Effect on the Orbital Motion . . . . . . . . . . . . . . . . . 235
4.9.1 The Post-Newtonian Effect . . . . . . . . . . . . . . . . . . . . . . . . . 235
4.9.2 Perturbation Solution Due to the Post-Newtonian
Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
4.9.3 Other Post-Newtonian Effects on the Earth’s
Artificial Satellite Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 239
4.10 Perturbed Orbit Solution Due to the Solar Radiation
Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
4.10.1 Calculation of Radiation Pressure . . . . . . . . . . . . . . . . . . . 239
4.10.2 Two States of Radiation Pressure Perturbation . . . . . . . . 242
4.10.3 The Perturbation Solution Due to Radiation
Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
4.10.4 The Non-singularity Perturbation Solution
of the First Type Due to the Radiation Pressure . . . . . . . . 252
4.10.5 The Non-singularity Perturbation Solution
of the Second Type Due to the Radiation Pressure . . . . . 254
4.11 Perturbed Orbit Solution Due to Atmospheric Drag . . . . . . . . . . . 256
4.11.1 Damping Effect: Atmospheric Drag . . . . . . . . . . . . . . . . . 256
4.11.2 Atmosphere Density Model . . . . . . . . . . . . . . . . . . . . . . . . 258
4.11.3 Atmospheric Rotation and the Expression
of Atmospheric Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
4.11.4 Structure of the Perturbed Solution Due
to the Atmospheric Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
4.11.5 The Non-singularity Perturbation Solution
by the Atmospheric Drag . . . . . . . . . . . . . . . . . . . . . . . . . . 273
4.12 Orbital Variations Due to a Small Thruster . . . . . . . . . . . . . . . . . . . 274
4.12.1 The Perturbation Solution Due to an (S,T,W)-Type
Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
4.12.2 The Non-singularity Perturbation Solution Due
to an (S,T,W)-Type Thrust . . . . . . . . . . . . . . . . . . . . . . . . . 277
4.12.3 The Perturbation Solution by a U-type Thrust . . . . . . . . . 281
4.12.4 The Non-singularity Perturbation Solution Due
to a U-type Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
5 Satellite Orbit Design and Orbit Lifespan Estimation . . . . . . . . . . . . . 287
5.1 Sidereal Period and Nodal Period [1–3] . . . . . . . . . . . . . . . . . . . . . . 287
5.1.1 The Transformation Between the Sidereal Period
Ts and the Nodal Period Tϕ . . . . . . . . . . . . . . . . . . . . . . . . 288](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-28-320.jpg)
![Contents xxv
5.1.2 The Anomalistic Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
5.2 Orbital Characteristics of Polar Orbit Satellite [2, 3] . . . . . . . . . . . 293
5.2.1 Basic Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
5.2.2 Preservation of Polar Orbit . . . . . . . . . . . . . . . . . . . . . . . . . 295
5.3 Existence and Design of Sun-Synchronous Orbit [2–5] . . . . . . . . 296
5.3.1 Conditions of Forming a Sun-Synchronous Orbit . . . . . . 296
5.3.2 Sun-Synchronous Orbits for Different Celestial
Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
5.4 Existence and Design of Frozen Orbit [2–5] . . . . . . . . . . . . . . . . . . 300
5.4.1 Basic State of Frozen Orbit . . . . . . . . . . . . . . . . . . . . . . . . . 300
5.4.2 Basic Equations of a Possible Frozen Orbit . . . . . . . . . . . 301
5.4.3 A Particular Solution of Eq. (5.40): The Frozen
Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
5.4.4 Stability of Frozen Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
5.4.5 Frozen Orbit for Other Celestial Bodies . . . . . . . . . . . . . . 306
5.4.6 Characteristics and Applications of Satellite Orbit
with a Critical Inclination . . . . . . . . . . . . . . . . . . . . . . . . . . 306
5.5 Existence and Design of Central Body Synchronous Orbit . . . . . . 309
5.5.1 Basic State of Central Body Synchronous Satellite
Orbit [2, 3, 5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
5.5.2 Existence and Evolution of a Central Body
Synchronous Satellite (Earth, Mars) . . . . . . . . . . . . . . . . . 312
5.6 Estimation and Calculation of Satellite’s Lifespan Due
to the Mechanism of Gravitational Perturbation . . . . . . . . . . . . . . . 316
5.6.1 Definition and Mechanism of a Low Orbit Satellite
Lifespan Due to Gravitational Perturbations [6–10] . . . . 317
5.6.2 Overview of Low Orbit Satellite Lifespan
for Earth, the Moon, Mars, and Venus . . . . . . . . . . . . . . . . 319
5.6.3 Evolution Characteristics and Lifespans of Orbit
with a Large Eccentricity [2, 9] . . . . . . . . . . . . . . . . . . . . . 323
5.6.4 Evolution Characteristics and Lifespans of High
Earth Satellite Orbit [6, 10] . . . . . . . . . . . . . . . . . . . . . . . . . 330
5.6.5 Key Points About Estimating Satellite Orbit
Lifespan Due to Gravitational Perturbations . . . . . . . . . . 332
5.7 Estimation and Calculation of Satellite Orbit Lifespan
in the Perturbed Mechanism of Atmospheric Drag . . . . . . . . . . . . 332
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
6 Orbital Solutions of Satellites of the Moon, Mars, and Venus . . . . . . 335
6.1 Characteristics of Gravitational Fields of Earth, the Moon,
Mars, and Venus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
6.1.1 Basic Characteristics of Earth’s Gravity Potential . . . . . . 335
6.1.2 Basic Characteristics of the Moon’s Gravity
Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
6.1.3 Basic Characteristics of Mars’s Gravity Potential . . . . . . 338](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-29-320.jpg)
![xxvi Contents
6.1.4 Basic Characteristics of Venus’s Gravity Potential . . . . . 339
6.2 Perturbed Orbital Solution of the Moon’s Satellite . . . . . . . . . . . . . 340
6.2.1 Selection of Coordinate System . . . . . . . . . . . . . . . . . . . . . 341
6.2.2 Mathematical Model for the Perturbed Motion
of the Moon’s Satellite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
6.2.3 The Numerical Solution for the High Precise
Orbital Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
6.2.4 The Analytical Perturbation Solution
of the Moon’s Satellite Orbit . . . . . . . . . . . . . . . . . . . . . . . 346
6.2.5 Additional Perturbation of Coordinate System
[5, 6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
6.2.6 Applications of Analytical Orbital Solution
in Orbital Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
6.3 Perturbed Orbital Solution of Mars’s Satellite . . . . . . . . . . . . . . . . 371
6.3.1 Selection of Coordinate System . . . . . . . . . . . . . . . . . . . . . 372
6.3.2 The Mathematical Model of Perturbed Motion
for a Mars’s Satellite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
6.3.3 The Analytical Perturbation Solution of Mars’s
Satellite Orbit [7, 8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
6.4 Perturbed Orbital Solution of Venus’s Satellite . . . . . . . . . . . . . . . . 383
6.4.1 The Perturbation Function of Venus’s
Non-Spherical Gravity Potential . . . . . . . . . . . . . . . . . . . . 384
6.4.2 The Structure and Results of the Analytical
Perturbation Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
7 Orbital Motion and Calculation Method in the Restricted
Three-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
7.1 Selection of Coordinate System and Motion Equation
of a Small Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
7.1.1 The Motion Equation of a Small Body
in the Barycenter Inertial Coordinate System . . . . . . . . . . 391
7.1.2 The Motion Equation of a Small Body
in the Synodic Coordinate System . . . . . . . . . . . . . . . . . . . 393
7.2 Jacobi Integral and Solution Existence of the Circular
Restricted Three-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
7.2.1 Jacobi Integral in the Circular Restricted
Three-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
7.2.2 Existence of Solution of the Circular Restricted
Three-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-30-320.jpg)
![Contents xxvii
7.3 Calculation and Application of the Libration Point
Positions of the Circular Restricted Three-Body Problem . . . . . . . 397
7.3.1 Conditions of Existence for Libration Solutions . . . . . . . 398
7.3.2 The Positions of the Three Collinear Libration
Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
7.3.3 Two Triangle Libration Points . . . . . . . . . . . . . . . . . . . . . . 401
7.3.4 Dynamical Characteristics of the Five Libration
Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
7.3.5 Characteristics and Applications of the Stability
of the Five Libration Points . . . . . . . . . . . . . . . . . . . . . . . . 410
7.3.6 Calculations and Applications of Libration Points
in the Restricted Problem of (2+2) Bodies . . . . . . . . . . . . 419
7.4 Orbit Design for Formation Flying of Satellites
and Companion-Flying in the Exploration of Asteroids . . . . . . . . 422
7.4.1 The Principle of Satellite Formation Flying . . . . . . . . . . . 422
7.4.2 The Problem with the Eccentricity in Orbit Design
of Formation Flying of Satellites . . . . . . . . . . . . . . . . . . . . 426
7.4.3 Extension of the Principle and Related Orbit
Design Method of Satellite Formation Flying . . . . . . . . . 428
7.4.4 Orbital Problem of Companion Flying in Asteroid
Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
7.5 Geometric Characteristics of Libration Point Orbit
and Orbit Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
7.5.1 Geometric Characteristics of Libration Point
Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
7.5.2 Analysis of Forces on a Prober’s Motion
in a Libration Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
7.5.3 Orbit Determination and Forecast Method
of Libration Point Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
7.5.4 Orbit Determination of Libration Point Orbit
and Precision Examination of Short-Arc Forecast
[22] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
7.5.5 Orbital Transformation Between the Two
Coordinate Systems for a Libration Point Orbit
Prober . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
8 Numerical Method for Satellite Orbit Extrapolations . . . . . . . . . . . . . 441
8.1 Basic Knowledge of Numerical Method in Solving
the Motion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
8.1.1 Basic Principles of Numerical Method in Solving
Motion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
8.1.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-31-320.jpg)
![xxviii Contents
8.2 Conventional Singer-Step Method: The Runge–Kutta
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
8.2.1 The Fourth-Order RK Method (RK4) . . . . . . . . . . . . . . . . 446
8.2.2 The Runge–Kutta-Fehlberg (RKF) Method . . . . . . . . . . . 447
8.3 Linear Multistep Methods: Adams Method and Cowell
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
8.3.1 Adams Methods: Explicit Methods and Implicit
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
8.3.2 Cowell’s Method and Størmer’s Method . . . . . . . . . . . . . 453
8.3.3 Adams-Cowell Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
8.4 Key Issues in Applications of the Numerical Method
in Orbital Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
8.4.1 Selections of Variables and Corresponding Basic
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
8.4.2 Singularity Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
8.4.3 Homogenization of Step-Size . . . . . . . . . . . . . . . . . . . . . . . 463
8.4.4 Control of the Along-Track Errors . . . . . . . . . . . . . . . . . . . 464
8.5 Numerical Calculation of the Right-Side Function . . . . . . . . . . . . 465
8.5.1 The Perturbation Acceleration of the Zonal
Harmonic Term →
F1(Jl) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
8.5.2 The Perturbation Acceleration of the Tesseral
Harmonic Term →
F
(
Jl,m
)
. . . . . . . . . . . . . . . . . . . . . . . . . . . 468
8.5.3 The Recursive Formulas of Legendre Polynomials,
Pl(µ) and the Associated Legendre Polynomials
Pl,m(µ), and Their Derivatives [15, 16] . . . . . . . . . . . . . . . 469
8.5.4 The Perturbation Acceleration of the Tidal
Deformation →
F
(
k2, J2,m
)
. . . . . . . . . . . . . . . . . . . . . . . . . . . 470
8.6 The Role of the Hamiltonian Method in the Orbital
Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
9 Formulation and Calculation of Initial Orbit Determination . . . . . . . 473
9.1 Formulation of Orbit Determination . . . . . . . . . . . . . . . . . . . . . . . . . 473
9.2 A Review of Initial Orbit Calculation in the Sense
of the Two-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
9.2.1 Basic Conditions for Initial Orbit Determination . . . . . . . 475
9.2.2 Construction of the Basic Equation for an Initial
Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
9.3 Initial Orbit Determination for Perturbed Motion . . . . . . . . . . . . . . 478
9.3.1 Construction of the Basic Equation for Initial
Orbit Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
9.3.2 Initial Orbit Determination Using Angle Data
Over a Short-Arc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
9.3.3 Initial Orbit Determination Using (P, A, h) Data
or Navigation Information . . . . . . . . . . . . . . . . . . . . . . . . . 487](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-32-320.jpg)
![2 1 Selections and Transformations of Coordinate Systems
1.1 Time Systems and Julian Day [1, 2]
As mentioned above the two coordinate times are TCB and TCG, but for making
ephemerides and in motion equations the time variables are measured by the Barycen-
tric Dynamical Time (TDB) and the Terrestrial Time (TT) for BCRS and GCRS,
respectively. The terrestrial time was called Terrestrial Dynamical Time (TDT) but
changed to TT after 1991. The difference between the two time systems, TDB and
TT, is caused by the effect of relativity, and the transfer relationship can be defined
by the theory of gravitation. In practical applications, their relationship is given by
the International Astronomy Union (IAU) in 2000 as
TDB = TT + 0s
.001657 sin g + 0s
.000022 sin(L − LJ), (1.1)
where g is the mean anomaly of Earth’s orbit around the Sun, and (L − LJ) is the
difference of the Sun’s mean ecliptic longitude and Jupiter’s mean ecliptic longitude
that
{
g = 357◦
.53 + 0◦
.98560028t,
L − LJ = 246◦
.00 + 0◦
.90251792t,
(1.2)
t = JD(t) − 2451545.0. (1.3)
In (1.3), JD(t) is the related Julian Day Number of time t, the definition of Julian
Day is given in Sect. 1.1.3. Formula (1.1) is valid between 1980 and 2050, and the
error is less than 30 μs (10−6
s). Near Earth’s surface with errors in the order of ms
(10−3
s), there is approximately
TDB = TT. (1.4)
Modern space and time reference systems accept IAU 2009 Astronomy Constant
System (Appendix 1). In this system, the Astronomical Unit (AU) is provided by
IAU 2012 resolution and is directly related to the unit of length “meter (m)”. The
value of AU is now given by
1AU = 1.49597870700 × 1011
m. (1.5)
1.1.1 Selection of Standard Time
In the TT system, the time is realized by atomic time. The earliest atomic clock, which
used the period of atomic oscillation as the standard to measure time, was built in
1949. In 1967, International System of Units (SI) defined the base unit of time (a](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-37-320.jpg)
![1.1 Time Systems and Julian Day [1, 2] 3
second) as the duration of 9,192,631,770 cycles of the radiation corresponding to the
transition between two energy levels of the ground state of the cesium-133 atom. In
1997, the International Committee for Weights and Measures (EIPM) added that the
preceding definition refers to a cesium atom at rest at a temperature of absolute zero.
The atomic time (TAI, in French Temps Atomique International) uses the SI second
(s) as a unit and the universal time of 1 January 1958, 0:00:00 as the starting epoch.
Since 1971, TAI is provided by the International Bureau Weights and Measures
(BIPM, in French) as a weighted average of the time kept by over 400 atomic clocks
in over 50 national laboratories worldwide. The only difference between TT and TAI
is the starting points that
TT = TAI + 32s
.184. (1.6)
Tothepresentday,TAIisthemostaccurateanduniformstandardtime,itsaccuracy
is about 10−16
s, and its error would be less than 1 s over one billion years.
1.1.2 Time Reference Systems
To study the motions of celestial bodies including spacecraft, it is necessary to have a
time system with a uniform time scale. For an observatory on the surface of Earth, it
is also necessary to have a time system related to Earth’s rotation. Before the atomic
time became the standard time Earth’s rotation was the time basis for the two time
systems. But Earth’s rotation is non-uniform and the accuracy of measurements of
Earth’s rotation is continuously improving. Therefore, there is a need to build a time
system that uses the uniform unit scale TAI and is also related to Earth’s rotation in
a coordinative way.
Sidereal Time (ST). The definition of a “sidereal day” is the time interval for
the March equinox at the upper transit between two successive returns. Therefore,
ST is the angle measured along the celestial equator from the observer’s meridian
(at longitude λ) to the great circle that passes through the March equinox and both
poles, and is given by time. Its value (S) equals the right ascension (α) of a star at
the upper transit of the observatory, that
S = α, (1.7)
where S is the local sidereal time (LST), and the Greenwich sidereal time (SG, GST)
is given by
SG = S − λ. (1.8)
In fact, GST is the Greenwich apparent sidereal time and is different from the
Greenwich mean sidereal time (GMST). Because the sidereal time is defined by](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-38-320.jpg)
![1.1 Time Systems and Julian Day [1, 2] 5
Coordinated Universal Time (UTC). A uniform time system is appropriate for
high precession ephemerides, which require a uniform scale for time intervals, but
it cannot replace a time system related to the Earth’s non-uniform rotation. Thus,
the coordinated universal time (UTC) is introduced for solving this problem. To the
present day there are many suggestions, discussions, and arguments, but without a
definite conclusion. We still keep the sidereal time system and the universal time
system, as each of them has its merits.
The difference between TAI and UT2 (or UT1), given in (1.11), is 0 s
.0039 at
the beginning of 1 January 1958, it was near zero. Because Earth’s rotation has a
long-term slowness, the difference between TAI and UT2 increases then causes the
problem. In order to keep UT (UT1 or UT2) as close as possible to TAI and still,
use the uniform scale. In 1963, the international communities adopted a third-time
system, which is the coordinated Universal Time (UTC). UTC is still a time system
based on TAI but with leap seconds added at irregular intervals, such as 12 months
or 18 months, to make it as close as possible to UT. Since 1972 it has been required
that UTC must be kept within ±0.9 s of UT1. Actual adjustments of leap seconds are
given by the International Time Bureau based on observational information, which
can be found on the EOP web page. Until 1 January 2017, the adjustment is 37s
, that
TAI = UTC + 37s
.
The transformation process from UTC to UT1 is that first to download the newest
EOP (Earth Orientation Parameters) data (use the B data if the time is more than a
month earlier than present, and use the A data for other times), then to calculate the
adjustment ΔUT by interpolation, which gives UT1 as
UT1 = UTC + ΔUT. (1.12)
According to the international convention, if a measurement is given at time t, the
time system means UTC unless there is a special description.
1.1.3 Julian Day
Besides the time system, in solving the dynamical problem we often have to choose
an epoch and deal with the problem related to the length of different types of one
year. In Astronomy, there are several definitions of a year. One is the Besselian year,
which has a length of a tropical year, i.e., 365.2421988 mean solar days. The epoch
of a Besselian year is the moment when the Sun’s mean ecliptic longitude is 280°.
For example, the Besselian year 1950.0 does not mean 1 January 1950, 0:00:00 but
is 31 December 1949, 22:09:42 (UT), which corresponds to the Julian day number
(JDN) 2,433,282.4234. Another type of year is the Julian year, which has 365.25
mean solar days. The epoch of each Julian year is exactly the beginning of a year,
for example, 1950.0 means 1 January 1950, 00:00:00. Obviously, it is easier to use](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-40-320.jpg)
![6 1 Selections and Transformations of Coordinate Systems
Table 1.1 Besselian Epoch,
Julian Epoch, and Julian day
number
Besselian Epoch Julian Epoch Julian day number
1900.0 1900.000858 2,415,020.3135
1950.0 1949.999790 2,433,282.4234
2000.0 1999.998722 2,451,544.5333
1989.999142 1900.0 2,415,020.0
1950.000210 1950.0 2,433,282.5
2000.001278 2000.0 2,451,545.0
Julian years than Besselian years. Therefore, since 1984 Besselian year has been
replaced by the Julian year. Some correspondences of the Besselian epoch, Julian
epoch, and Julian day number are listed in Table 1.1.
For convenience, the Modified Julian Date (MJD) is introduced and defined as
MJD = JD − 2400000.5. (1.13)
As an example, JD(1950.0) corresponds to MJD = 33,282.0. The lengths of a
century of a Besselian year (tropical century) and a Julian year are 36,524.22 and
36,525 mean solar days, respectively.
1.2 Space Coordinate Systems [2–6]
A coordinate system is actually a mathematical representation of a theoretical
concept. A reference frame is the physical realization of a coordinate system, there-
fore, a reference system is an integrated system of a theoretical concept and a physical
frame. Although the concept of a reference system is different from that of a coor-
dinate system, in the practical application of most fields, as in this book, these two
systems are interchangeable without misunderstanding.
To study the motions of celestial bodies in the Solar System, there are commonly
accepted three types of the coordinate system, which are the horizontal coordinate
system, the equatorial coordinate system, and the ecliptic coordinate system. These
coordinates are applied to problems no matter from the point of view of Earth or
other celestial bodies (such as the major planets or the Moon). For each space coor-
dinate system, there are three key elements, the origin of the coordinate system, the
fundamental plane, i.e., the xy-plane, and the primary direction (the direction of the
x-axis). In this section, we introduce three coordinates with respect to Earth.
Horizontal system. A proper name for this system should be the topocentric hori-
zontal coordinate system. In this system the origin is at the center of an observatory
(or a sampling center), the fundamental plane is the local horizontal plane containing
the origin and is tangential to the ellipsoid of Earth (the horizon), and the primary
direction is towards the north (N) in the xy-plane. The direction of the z-axis is
towards the zenith (Z) (Fig. 1.1).](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-41-320.jpg)
![1.2 Space Coordinate Systems [2–6] 7
Fig. 1.1 The horizontal
system and the equatorial
system
Equatorial system. There are two equatorial systems, one is the topocentric equa-
torial system with the center at the location of an observatory, and the other is the
geocentric equatorial system with the center at the center of Earth. For both systems,
Earth’s equatorial plane is the reference plane, but for the topocentric system the
reference plane is parallel to the equatorial plane, and in the celestial sphere the two
planes converge into one, therefore the two systems are related only by a translation.
The primary direction of both systems is towards the March equinox ( ).
Ecliptic system. There are also two ecliptic systems, the geocentric ecliptic system
with its origin at the center of Earth and the heliocentric ecliptic system with its origin
at the center of the Sun. For both systems, the fundamental plane is the ecliptic plane
of Earth’s orbit around the Sun, and the primary direction is towards the March
equinox ( ).
The geometrical relationship of the horizontal system and the equatorial system
is illustrated in Fig. 1.1, and that of the equatorial system and the ecliptic system in
Fig. 1.2. The symbols in the figures are customarily used in Astronomy, therefore,
are not explained here.
The position of a celestial body in a space coordinate system can be presented
by its coordinate vector. In the horizontal coordinate system, the position vector of
a body is denoted by →
ρ with spherical coordinates (ρ, A, h, or E), where ρ is the
distance between the origin of the system and the body, A is the azimuth (do not be
confused with the equator AA’) measured from the north point eastward along the
horizontal circle (clockwise), and h is the altitude (i.e. the height angle E). In the
equatorial coordinate system, this vector is denoted by →
r with spherical coordinates
(r, α, δ), where r is the same as ρ, α is the right ascension measured from the March
equinox eastward along the equator (i.e., the arc D on the equator AA,
), and δ is the
declination angle. In the ecliptic coordinate system, the position vector is denoted
by →
R with spherical coordinates (R, λ, β), where R is the same as ρ, λ is the ecliptic
longitude measured from the March equinox eastward along the ecliptic, and β is
the ecliptic latitude. The relationships of the coordinates are given by](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-42-320.jpg)
![1.2 Space Coordinate Systems [2–6] 9
where S = α + t is the hour angle of the March equinox, which equals the sidereal
time at the observatory (i.e. the arc D + DA measured along the equator circle
AA’ in Fig. 1.1), ϕ is the astronomical latitude of the observatory, →
rA is the position
vector of the observatory from the Earth’s center, ε is the obliquity, and →
RE is the
position vector of Earth’s center in the heliocentric coordinate system.
The rotation matrices Rx, Ry, and Rz in (1.16) and (1.18) are given by
Rx (θ) =
⎛
⎝
1 0 0
0 cos θ sin θ
0 − sinθ cos θ
⎞
⎠, (1.20)
Ry(θ) =
⎛
⎝
cos θ 0 − sin θ
0 1 0
sin θ 0 cos θ
⎞
⎠, (1.21)
Rz(θ) =
⎛
⎝
cos θ sin θ 0
− sin θ cos θ 0
0 0 1
⎞
⎠, (1.22)
In the dynamics of the Solar System for studying the motions of the major planets
and asteroids, we use the heliocentric ecliptic coordinate system; whereas in the
dynamics of artificial satellites, we use the equatorial coordinate system centered
at the barycenter of the main body, such as the geocentric equatorial system, or the
Moon-centric equatorial system, or the Mars-centric equatorial system, etc.
Thecoordinatesystemsusedforartificialsatellitesaremainlythegeocentricceles-
tial coordinate system and the Earth-fixed equatorial coordinate system (see 1.3.3).
The origins of both systems are obviously at the center of Earth, but their funda-
mental planes and primary directions are affected by Earth’s precession, nutation,
and polar motion, which make these space coordinate systems rather complicated. As
we know that Earth is an ellipsoid with unevenly distributed mass. The gravitational
forces from the Sun, the Moon, and other major planes act on Earth’s non-spherical
part and produce two phenomena. One is an effect of a rigid body translation force,
resulting in an indirect perturbation of Earth’s oblateness. The other is an effect of
rotation torque of a rigid body due to Earth’s gyro-like motion, producing precession
and nutation. Because of precession and nutation, Earth’s equatorial plane vibrates.
Besides the two effects, Earth’s internal motion and motions on the surface produce
a slow shift of the rotational axis, i.e., the polar motion, which also influences the
selection of the coordinate system. The equators of Mars and the Moon have similar
variations. As a result, there are different types of equatorial coordinate systems.
The properties of equatorial coordinate systems with respect to Earth, the Moon, and
Mars, and the transformation between these systems are discussed in the following
sections.](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-44-320.jpg)
![10 1 Selections and Transformations of Coordinate Systems
1.3 Earth’s Coordinate Systems [2, 6–10]
1.3.1 The Realization of the Dynamical Reference System
and J2000.0 Mean Equatorial Reference System
Current observational measurements, such as Ephemerides of the planets in the Solar
System, are all provided in the International Celestial Reference System (ICRS). The
realization of this reference system is a reference frame called International Celestial
Reference Frame (ICRF). The origin of ICRS is at the barycenter of the Solar System,
the fundamental plane and the primary direction of X-axis are decided by precise
observations of a group of extragalactic radio sources to be as close as possible to
the J2000.0 mean equatorial plane and the mean March equinox, respectively. Here
J2000.0 means the Julia year 2000, 1 January 12:00:00. Because the radio sources
are so distant, they are stationary to our technology, the coordinate system and its
orientation of ICRF are relatively fixed in space, and free of the dynamics of the
Solar System and Earth’s precession and nutation, also unrelated to the traditional
concepts of the equator, March equinox, and ecliptic. As a result, this system is closer
to an inertia reference system than any other system.
Before the existence of ICRS and ICRF, the basic astronomy reference system
is the Fifth Fundamental Catalog dynamic system (FK5) (strictly speaking, this
is a reference system dynamically defined and includes the correction of sidereal
kinematics). FK5 is built up on observations of bright stars and the IAU 1976 Astro-
nomic constants. Its fundamental plane is the J2000.0 mean equatorial plane, and the
direction of X-axis points to J2000.0 mean March equinox. Obviously, this system is
relatedtotheepoch. Thepresent systemICRSis animprovement of FK5. Thedynam-
ical reference system is the J2000.0 mean equatorial reference system, usually
called J2000.0 mean equatorial coordinate system. Specifically, the fundamental
plane and the primary direction of the X-axis of ICRF are realized by observations of
the Very-Long-Baseline Interferometry (VLBI) from hundreds of extragalactic radio
sources. The deviation of its pole from the pole of the dynamical reference system
FK5 is only about 20 milliarcseconds. In order to keep the continuity of the reference
system, the fundamental plane and the primary direction of ICRF are kept as close
as possible to these of FK5, which are the J2000.0 mean equatorial plane and the
J2000.0 mean March equinox. The origin of ICRS (or the zero point, its definition is
given in Sect. 1.3.2) is chosen to be the average right ascension of 23 radio sources
thus being close to that of FK5. The relationship of the dynamical reference systems,
ICRS and FK5, depends on three parameters, which are the deviations of the celestial
pole, ξ0 and η0, and the zero right ascension deviation dα0. Their values are
⎧
⎨
⎩
ξ0 = −0,,
.016617 ± 0,,
.000010,
η0 = −0,,
.006819 ± 0,,
.000010,
dα0 = −0,,
.0146 ± 0,,
.0005.
(1.23)](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-45-320.jpg)
![1.3 Earth’s Coordinate Systems [2, 6–10] 11
The relationship of ICRS and the J2000.0 mean equatorial coordinate system can
be given by
{
→
rJ2000.0 = B→
rICRS,
B = Rx (−η0)Ry(ξ0)Rz(dα0),
(1.24)
where →
rJ2000.0 and →
rICRS are for the same vector but in different coordinate systems,
the constant matrix B is the deviation matrix of reference frame composed of the
three small rotation angles.
The J2000.0 mean equatorial coordinate system is the commonly accepted
geocentric celestial coordinate reference system (GCRS) in present Aerospace
Dynamics(especiallyforEarth’ssatellites).Ifunnecessary,theabove-givendeviation
matrix of the reference frame is not mentioned again.
1.3.2 The Intermediate Equator and Three Related Datum
Points
The intermediate equator is introduced to better describe the relationship between
the Celestial Reference System (CRS) and the Terrestrial Reference System. The
celestial axis is the extension of Earth’s rotation axis, the points of intersection
of the celestial axis and the celestial sphere are called celestial poles. Because of
Earth’s precession, the direction of Earth’s rotation axis changes over time in CRS,
which is instantaneous, therefore the celestial pole and the celestial equator are also
instantaneous. For clarity IAU 2003 named the instantaneous celestial pole and the
celestial equator as the Celestial Intermediate Pole (CIP) and the Intermediate
Equator, respectively.
In order to take measurements in the celestial reference system, it is necessary to
select a fixed point with respect to the celestial reference system on the intermediate
equator as the origin, which is called the Celestial Intermediate Origin (CIO).
Similarly, in the terrestrial system a point, which is fixed with respect to the system,
is needed and called the Terrestrial Intermediate Origin (TIO). CIO is decided
based on observations of a group of quasars and is close to the 0° right ascension,
i.e., the March equinox on the International Celestial Reference Frame; whereas
TIO is decided by a group of observatories on Earth, and is near the 0° longitude
(the prime meridian, i.e., the Greenwich meridian) on the International Terrestrial
Reference Frame. In Fig. 1.3 the intermediate equator is given by the circle, E is
Earth’s barycenter, and is the March equinox.
In the celestial reference system, the intermediate equator tied with CIO is called
the Celestial Intermediate Equator, TIO moves along the equator anti-clockwise,
its period is a sidereal day. In the terrestrial reference system, the intermediate equator
tied with TIO is called the Terrestrial Intermediate Equator, CIO moves along the
equatoroverthesameperiodofasiderealdaybutclockwise.Bothobservationsreflect](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-46-320.jpg)
![1.3 Earth’s Coordinate Systems [2, 6–10] 13
As mentioned above that the realization of ICRS requires ICRF. It is the same
that the Terrestrial Reference Frame (TRF) is needed for the realization of TRS. TRF
(used in navigation, survey, terrestrial physics, etc.) is defined by a group of fixed
points on Earth’s surface, whose positions are precisely determined in TRS. The first
TRF is given by the International Latitude Service. Based on the observations over
five years, 1900–1905, the International Latitude Service defined the Conventional
International Origin (CIO), which was the average direction of the third axis (z-
axis), i.e., the mean direction of Earth’s pole. It should be noticed that nowadays the
abbreviation CIO is given to the Celestial Intermediate Origin (see Sect. 1.3.2), so is
no longer for the Conventional International Origin.
In the Earth-fixed coordinate system, the origin of the frame is at the center of
Earth, the xy-plane is close to the 1900.0 mean equatorial plane, and the direction of
the x-axis points to the intersection of the Greenwich meridian and the equator, so can
be called the Greenwich meridian direction [2]. Several Earth’s gravitational models
and the related reference ellipsoid are defined in this system, thus these models are
self-consistent. If there is no specific explanation the Earth-fixed system in this book
agrees with the World Geodetic System 84 (also known as WGS 1984). For this
system there are
GE = 398600.4418
(
km3
/s2
)
,
ae = 6378.137(km),
1
f
= 298.257223563, (1.25)
where GE is the geocentric gravitational constant, ae and f are the equatorial radius
and the flattening factor of the reference ellipsoid, respectively.
In the Earth-fixed frame, the position vector of an observatory is given by
→
Re(H, λ, ϕ). For the position vector, the relationship between the rectangular
coordinates (Xe, Ye, Ze) and the spherical coordinates (H, λ, ϕ) is given by
⎧
⎨
⎩
Xe = (N + H) cos φ cos λ,
Ye = (N + H) cos φ sin λ,
Ze = [N
(
1 − f )2
+ H
]
sin φ,
(1.26)
with
N = ae
[
cos2
ϕ + (1 − f )2
sin2
ϕ
]− 1
2
= ae
[
1 − 2 f
(
1 −
f
2
)
sin2
ϕ
]− 1
2
, (1.27)
where ae and f are given in (1.25). The spherical coordinate H is the geodetic
height of the observatory, and λ and ϕ are the geodetic longitude and latitude of the
observatory, respectively. Their relationships with the rectangular coordinates are](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-48-320.jpg)
![14 1 Selections and Transformations of Coordinate Systems
given by
tan λ = Ye
Xe
, sin2
ϕ = Ze
[N(1− f )2
+H]
. (1.28)
(3) Geocentric ecliptic coordinate system O-x,
y,
z,
The origin of this system is also Earth’s barycenter, and there is only a translation
relationship between this system and the heliocentric ecliptic system. The x,
y,
-plane
is the epoch J2000.0 ecliptic plane, and the direction of the x,
-axis is the same as
in the celestial coordinate system O-xyz, which points to the epoch J2000.0 mean
March equinox.
1.3.4 Transformation of the Earth-Fixed Coordinate System
O-XYZ and the Geocentric Celestial Coordinate
System O-xyz
1.3.4.1 Transformation Relationship (I) by the IAU 1980 Model
Using →
r and →
R as the position vectors of a spacecraft in the geocentric celestial system
O-xyz and the Earth-fixed system O-XYZ, respectively, then the transformation
relationship is given by
→
R = (HG)→
r. (1.29)
The coordinate transformation matrix (HG) is given by four rotating matrices,
that
(HG) = (E P)(E R)(N R)(P R), (1.30)
where (PR) is the precession matrix, (NR) the nutation matrix, (ER) the Earth’s
rotation matrix, and (EP) the polar motion matrix, given by
(E P) = Ry
(
−xp
)
Rx
(
−yp
)
, (1.31)
(E R) = Rz(SG), (1.32)
(N R) = Rx (−Δε)Ry(Δθ)Rz(−Δμ)
= Rx (−(ε + Δε))Rz(−Δψ)Rx (ε), (1.33)
(P R) = Rz(−zA)Ry(θA)Rz(−ζA), (1.34)](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-49-320.jpg)
![1.3 Earth’s Coordinate Systems [2, 6–10] 15
In (1.31) xp and yp are the components of the polar shift vector. The Greenwich
sidereal time SG in (1.32) is given by
SG = SG + Δμ, (1.35)
where Δμ is the nutation of the right ascension; SG is the J2000.0 Greenwich mean
sidereal time given by
SG = 18h
.697374558 + 879000h
.051336907T + 0s
.093104T 2
, (1.36)
T =
1
36525.0
[JD(t) − JD(J2000.0)]. (1.37)
In these two formulas, t is the UT1 time, but for calculating other variables such
as precession and nutation, t is the TDT time. Time T is measured from J2000.0 but
uses a century as a unit.
The precession constants in (3.14) ζA, θA, and zA are given by
⎧
⎨
⎩
ζA = 2306
,,
.2181T + 0
,,
.30188T 2
,
θA = 2004
,,
.3109T − 0
,,
.42665T 2
,
zA = 2306
,,
.2181T + 1
,,
.09468T 2
,
(1.38)
where θA is the precession in declination, and μ (or mA) is the precession in right
ascension that
μ = ζA + zA
= 4612,,
.4362 T + 1,,
.39656 T 2
. (1.39)
In (1.33) ε is the mean obliquity. The nutation components in ecliptic longitude
Δψ and in obliquity Δε can be calculated using the sequences provided by the
IAU 1980 model, which has 106 terms with amplitudes greater than 0,,
.0001. For
the requirement of general orbital accuracy, only the terms with amplitudes greater
than 0,,
.005 need to be included, which are the first 20 terms. Because these terms
are periodic (the shortest period term is due to the Moon’s motion), so there is no
accumulative effect, and the error caused by the terms with amplitude less than 0,,
.005
is equivalent to the order of meter in ground-based positioning, and it is less than
0 s
.001 with respect to time. The first 20 terms are:
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
Δψ =
20
Σ
j=1
(
A0 j + A1 j t
)
sin
( 5
Σ
i=1
kji αi (t)
)
,
Δε =
20
Σ
j=1
(
B0 j + B1 j t
)
cos
( 5
Σ
i=1
kji αi (t)
)
,
(1.40)](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-50-320.jpg)
![16 1 Selections and Transformations of Coordinate Systems
where the components of nutation in right ascension and in inclination, Δμ and Δθ,
respectively, are given by
{
Δμ = Δψcosε,
Δθ = Δψsinε,
(1.41)
and the value of the obliquity ε is given by
ε = 23◦
26,
21,,
.448 − 46,,
.8150 t. (1.42)
In (1.40) there are five basic arguments αi (i = 1, · · · , 5) related to the positions
of the Sun and the Moon, which are given by:
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
α1 = 134◦
57,
46,,
.733 + (1325r
+ 198◦
52,
02,,
.633)t + 31,,
.310t2
,
α2 = 357◦
31,
39,,
.804 + (99r
+ 359◦
03,
01,,
.224)t − 0,,
.577t2
,
α3 = 93◦
16,
18,,
.877 + (1342r
+ 82◦
01,
03,,
.137)t − 13,,
.257t2
,
α4 = 297◦
51,
01,,
.307 + (1236r
+ 307◦
06,
41,,
.328)t − 6,,
.891t2
,
α5 = 125◦
02,
40,,
.280 − (5r
+ 134◦
08,
10,,
.539)t + 7,,
.455t2
,
(1.43)
where 1r
= 360°. The first 20 terms of the nutation sequences are listed in Table
1.2. To reach the above-mentioned accuracy of the order of meter, the terms on the
right side of (1.40) only A11 and B11 in Table 1.2 are needed, other terms of A1 j
and B1 j can be omitted. Specifically, the number of terms needed depends on not
only the required accuracy but also the capability of the software, such as the factor
of functional expansion. The time t in (1.40)–(1.43) is the same as T, the century
number given by (1.37), but is in TDT.
The formulas for calculating the rotational matrices Rx(θ), Ry(θ), and Rz(θ)
are given by (1.20)–(1.22). Note that they are orthogonal matrices, that RT
x (θ) =
R−1
x (θ) = Rx (−θ), · · · .
1.3.4.2 Transformation Relationship (II) by the IAU 2000 Model
By the IAU 2000 model, the transformation from the geocentric celestial reference
system (GCRS) to the International Terrestrial Reference System (ITRS) is given by
[ITRS] = W(t)R(t)M(t)[GCRS], (1.44)
where [GCRS] and [ITRS] correspond to the geocentric celestial coordinate system
and the Earth-fixed coordinate system by the IAU 1980 model, respectively. Using
→
r and →
R for the position vectors of a spacecraft in the two systems, respectively (the
symbols are used in Sect. 1.3.4.1, for consistency we use the same symbols here),
then the transformation relationship is given by](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-51-320.jpg)
![1.3 Earth’s Coordinate Systems [2, 6–10] 17
Table 1.2 The first 20 terms of the IAU 1980 Nutation sequence
j Period kj1 kj2 kj3 kj4 kj5 A0j A1j B0j B1j
(d) (0"0.0001) (0"0.0001)
1 6798.4 0 0 0 0 1 −171,996 −174.2 92,025 8.9
2 182.6 0 0 2 −2 2 −13,187 −1.6 5736 −3.1
3 13.7 0 0 2 0 2 −2274 −0.2 977 −0.5
4 3399.2 0 0 0 0 2 2062 0.2 − 895 0.5
5 365.2 0 1 0 0 0 1426 −3.4 54 −0.1
6 27.6 1 0 0 0 0 712 0.1 −7 0.0
7 121.7 0 1 2 −2 2 −517 1.2 224 −0.6
8 13.6 0 0 2 0 1 −386 −0.4 200 0.0
9 9.1 1 0 2 0 2 −301 0.0 129 −0.1
10 365.3 0 −1 2 −2 2 217 − 0.5 −95 0.3
11 31.8 1 0 0 −2 0 − 158 0.0 −1 0.0
12 177.8 0 0 2 −2 1 129 0.1 −70 0.0
13 27.1 −1 0 2 0 2 123 0.0 −53 0.0
14 27.7 1 0 0 0 1 63 0.1 −33 0.0
15 14.8 0 0 0 2 0 63 0.0 −2 0.0
16 9.6 −1 0 2 2 2 −59 0.0 26 0.0
17 27.4 −1 0 0 0 1 −58 − 0.1 32 0.0
18 9.1 1 0 2 0 1 −51 0.0 27 0.0
19 205.9 2 0 0 −2 0 48 0.0 1 0.0
20 1305.5 −2 0 2 0 1 46 0.0 −24 0.0
→
R = W(t)R(t)M(t)→
r, (1.45)
where M(t) is the precession and nutation matrix, R(t) is the Earth rotation matrix,
and W(t) is the polar shift matrix. Based on the transformation relationship of the
March equinox the matrix M(t) can be written as
M(t) = N(t)P(t)B, (1.46)
where N(t) is the nutation matrix, P(t) is the precession matrix, and B is the devi-
ation matrix of the reference frame defined in (1.24), which is a small constant
matrix. When the J2000.0 mean equatorial coordinate system is directly used as
the geocentric celestial coordinate system, the effect of B can be omitted then
M(t) = N(t)P(t). (1.47)
The calculation methods for these matrices are given as follows.](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-52-320.jpg)
![1.3 Earth’s Coordinate Systems [2, 6–10] 19
where nik are integers, and Fk are the five basic arguments related to the positions
of the Sun and the Moon given by
F1 ≡ l = 134◦
.96340251 + 1717915923,,
.2178t + 31,,
.8792t2
+0,,
.051635t3
− 0,,
.00024470t4
, (1.53)
F2 ≡ l,
= 357◦
.52910918 + 129596581,,
.0481t − 0,,
.5532t2
+0,,
.000136t3
− 0,,
.00001149t4 , (1.54)
F3 ≡ F = 93◦
.27209062 + 1739527262,,
.8478t − 12,,
.7512t2
−0,,
.001037t3
+ 0,,
.00000417t4 , (1.55)
F4 ≡ D = 297◦
.85019547 + 1602961601,,
.2090t − 6,,
.3706t2
+0,,
.006593t3
− 0,,
.00003169t4 , (1.56)
F5 ≡ Ω = 125◦
.04455501 − 6962890,,
.5431t + 7,,
.4722t2
+0,,
.007702t3
− 0,,
.00005939t4 , (1.57)
The five basic arguments are defined as: F1 the Moon’s mean anomaly, F2 the
Sun’s mean anomaly, F3 the angular distance of the Moon’s mean ascending node,
F4 the mean angular distance between the Sun and the Moon, and F5 the mean
ecliptic longitude of the Moon’s ascending node. Actually, Fk(k = 1, . . . , 5) are the
same five basic arguments αi (i = 1, . . . , 5) given by (1.43).
Table 1.3 gives the coefficients of the first 20 terms of the nutation sequence by
the IAU 2000B model for comparing with the IAU 1980 model in Table 1.2.
(2) Calculations of the precession matrix P(t)
➀ The classical formula for three rotations of P(t)
P(t) = Rz(ζA)Ry(−θA)Rz(zA), (1.58)
where the three rotation angles are given by (1.48).
➁ The formula for four rotations of P(t)
P(t) = Rx (−ε0)Rz(ψA)Rx (ωA)Rz(−χA), (1.59)
where the last three rotation angles are given as
ψA = 5038,,
.481507t − 1,,
.0790069t2
− 0,,
.00114045t3
+0,,
.000132851t4
− 0,,
.0000000951t5
ωA = ε0 − 0,,
.025754t + 0,,
.0512623t2
− 0,,
.007725036t3
−0,,
.000000467t4
+ 0,,
.0000003337t5
χA = 10,,
.556403t − 2,,
.3814292t2
− 0.00121197t3
+0,,
.000170663t4
− 0,,
.0000000560t5
(1.60)
and ε0 is given by (1.63), and t in (1.60) is given by (1.49).](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-54-320.jpg)
![NOTAS
[1] Borrow salió de Sevilla el 9 de Diciembre de 1836,
estuvo once días en Córdoba, de donde partió el 20,
llegando a Aranjuez el 25 y a Madrid el 26. (Knapp.)
[2] Número 16, piso 3.º (Knapp.)
[3] María Díaz murió en 1844. (Knapp.)
[4] El primer contrato para imprimir el Nuevo
Testamento lo hizo con Mr. Charles Wood, impresor del
gobierno español. El contrato con Borrego es de 17 de
Enero de 1837, para reproducir la edición de Londres
(1826) del N. T. de Scio. (Knapp.)
[5] Borrow pensó primeramente en dar por terminada
su misión en la Península con la impresión del Nuevo
Testamento, dejando a otros el cuidado de distribuir la
obra. Cambió de idea y se ofreció a desempeñar en
persona ese cometido; los directores de la Sociedad
Bíblica aceptaron su propuesta, recibiendo Borrow la
autorización oficial dos días después de terminarse la
tirada del libro. (Knapp.)
[6] Buena suerte, Antonio.
[7] He aquí la original copla bilingüe que damos
traducida en el texto:
The Romany chal to his horse did cry,
As he placed the bit in his horse’s jaw.
«Kosko gry! Romany gry!
Muk man kistur tute knaw!»
[8] Plural de chabó o chabé: mozo, joven, compañero.
[9] Soldados.](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-77-320.jpg)
![[10] Parugar: trocar, traficar. Graste: caballo.
[11] Feria.
[12] Caballero.
[13] Plural de Caloró: gitano.
[14] Bul; Bullati: el ano.
[15] Un hombre no gitano; un gentil.
[16] Granada.
[17] ¡Quita de ahí! ¡Déjame!
[18] Estos «cuadros de Murillo» son imaginarios,
observa el editor U. R. Burke.
[19] Posiblemente Cisneros o Calzada. (Nota del editor
Burke.)
[20] El nombre del arriero era Pedro Mato. La estatua es
de madera. (Nota del editor Burke.)
[21] Es un error: Lucus Augusti fué sólo capital de la
Galicia septentrional; Bracara Augusta (Braga), de la
meridional; el Miño las dividía. (Nota del editor Burke.)
[22] Vocablo del dialecto milanés, según Borrow y su
anotador Burke, equivalente a vagar sin rumbo.
[23] Alude a D. Pelayo Gómez de Sotomayor, primer
enviado de Enrique III cerca de Tamerlán.
[24] El abogado se llamaba D. Claudio González y
Zúñiga, autor de la Descripción Económica de la Provincia
de Pontevedra. Pontevedra, 1834. (Knapp).
[25] El alcalde de Corcubión no necesitaba saber inglés
para leer a Bentham, porque desde 1820 a 1837 gran
parte de sus escritos se habían traducido y publicado en
España. Las obras completas fueron publicadas en
español por Baltasar Anduaga Espinosa, Madrid, 1841-
1843, 14 vols. en 4.º El calificativo de «Solón inglés» que](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-78-320.jpg)
![Borrow pone en boca del alcalde está tomado de un
artículo del Monthly Magazine, que Borrow conocía bien.
Su indiferencia por Bentham nace de la secreta hostilidad
que Borrow profesaba al Dr. Bowring, uno de los agentes
principales de la introducción de las obras de Bentham en
la Península. (Knapp.)
[26] ¿Avilés?](https://image.slidesharecdn.com/25277788-250521114739-2eaefa28/85/Algorithms-For-Satellite-Orbital-Dynamics-Lin-Liu-79-320.jpg)
Algorithms For Satellite Orbital Dynamics Lin Liu
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- 5. Lin Liu Translated by Shengpan Zhang Springer Series in Astrophysics and Cosmology Algorithms for Satellite Orbital Dynamics
- 6. Springer Series in Astrophysics and Cosmology Series Editors Cosimo Bambi, Department of Physics, Fudan University, Shanghai, China Dipankar Bhattacharya, Inter-University Centre for Astronomy and Astrophysics, Pune, India Yifu Cai, Department of Astronomy, University of Science and Technology of China, Hefei, China Maurizio Falanga, (ISSI), International Space Science Institute, Bern, Bern, Switzerland Paolo Pani, Department of Physics, Sapienza University of Rome, Rome, Italy Renxin Xu, Department of Astronomy, Perkings University, Beijing, China Naoki Yoshida, University of Tokyo, Tokyo, Chiba, Japan Pengfei Chen, School of Astronomy and Space Science, Nanjing University, Nanjing, China
- 7. The series covers all areas of astrophysics and cosmology, including theory, observations, and instrumentation. It publishes monographs and edited volumes. All books are authored or edited by leading experts in the field and are primarily intended for researchers and graduate students.
- 8. Lin Liu Algorithms for Satellite Orbital Dynamics
- 9. Lin Liu Department of Astronomy Nanjing University Nanjing, Jiangsu, China Translated by Shengpan Zhang Department of Astronomy York University Toronto, ON, Canada ISSN 2731-734X ISSN 2731-7358 (electronic) Springer Series in Astrophysics and Cosmology ISBN 978-981-19-4838-1 ISBN 978-981-19-4839-8 (eBook) https://doi.org/10.1007/978-981-19-4839-8 Jointly published with Nanjing University Press The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: Nanjing University Press. ISBN of the Co-Publisher’s edition: 9787305222276 © Nanjing University Press 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore
- 10. Preface “Orbital Dynamics” is an essential and fundamental part of Aerospace Dynamics (Aerospace System Engineering). It includes the launching of spacecraft, the entire system of orbital design, orbital observation and control, and the effective application of all aspects. The Author teaches, researches, and works in this field for more than 50 years. Based on his abundant experience, this monograph summarizes the application part of Orbital Dynamics. The main content is about the orbital motion of spacecraft of different types in the Solar System (actually spacecraft are circling satellites with different center bodies). A systematic and effective calculation method is provided for orbital position telemetry, tracking, and orbital determination and prediction after a spacecraft is launched, and special orbital design, realization, and retainment, for all sorts of space projects. This book describes the essential theory and analytical results but omits some details of the calculation principle and the process of formula derivation. The main purpose is to provide calculation methods and formulas so technicians and engineers with basic knowledge of orbital dynamics in the Aerospace industry can directly apply them in their work. The book is also beneficial for related professionals to obtain the necessary understanding of orbital dynamics. References of this book include the author’s published academic research papers, teaching materials, and 11 books. The books published in Chinese are as follows. 1. Liu, L. and co-authors, Motion Theory of Earth’s Artificial Satellite, Science Press, Beijing, China, 1974. 2. Liu, L. and Zhao, D. Z., Orbital theory of Earth’s Artificial Satellite, teaching materials, Nanjing University, China, 1979. 3. Liu, L., Orbital Dynamics of Earth’s Artificial Satellite, Higher Education Press, Beijing, China, 1992. 4. Liu, L., Methods of Celestial Mechanics, Nanjing University Press, Nanjing, China, 1998. 5. Liu, L., Orbital Theory of Spacecraft, National Defense Industry Press, Beijing, China, 2000. v
- 11. vi Preface 6. Liu, L. and co-authors, Mathematical methods of Precision Orbital Determina- tion for Earth’s Artificial Satellite, PLA Press, Beijing, 2002. 7. Liu, L., Hu, S. J., and Wang, X., Introduction of Aerospace Dynamics, Nanjing University Press, Nanjing, China, 2006. 8. Liu, L. and Wang, X., Orbital Dynamics of a Moon Prober, National Defense Industry Press, Beijing, China, 2006. 9. Liu, L. and Tang, J. S., Satellite Orbital Theory and Application, Electronic Industry Press, Beijing, China, 2015. 10. Liu, L., Hu, S. J., and co-authors, Spacecraft Orbital Theory and Application, Electronic Industry Press, Beijing, China, 2015. 11. Liu, L. and Hou, X. Y., The Basics of Orbital Theory, Higher Education Press, Beijing, China, 2018. In the process of writing this monograph, two of the Author’s students, Yanrong Wang and Gongyou Wu, who both work in the Chinese Aerospace Industry and have first-hand experience, provided valuable suggestions; another student, Zhitao Yang, reviewed Chap. 4. This monograph is also supported by Astronomy and Space Science Institute, Nanjing University research project (NSFC J1210039), and Jiangsu Brand Professional Construction Project (TAPP). The author is grateful for their help. Nanjing, China Lin Liu
- 12. Introduction Orbital Dynamics in the Solar System From a general point of view, the motion of any celestial body, natural or artificial, includes two different types of states. One is about the motion of its barycenter, and the other is about the motion of any part of the body with respect to its barycenter. The first type is called the orbital motion, which is the subject of Orbital Dynamics, and the second is called the attitude motion, which is the subject of Attitude Dynamics. Ancient astronomers already observed attitude motions such as Earth’s precession and nutation, and the lunar physical libration. This book is about orbital dynamics in the Solar System. The primary content is about the characteristics of the barycenter motion of spacecraft for different purposes. Some parts of the book deal with the attitude motion, such as in the selection of a proper coordinate system when it is necessary to consider the precession and nutation due to the vibration of Earth’s equator; also, the orbital motion of a spacecraft may be related to its attitude when a surface force on the spacecraft acts as an external force. The Solar System is an extremely complicated dynamical system. In this system, besides the Sun which is the dominating body, there are eight major planets and a large number of asteroids, natural satellites, comets, and space debris. The primary subject of Celestial Mechanics is to study orbital motions of celestial bodies, big and small, in the Solar System and the evolution of their orbits. In the Space Era, the ever-increasing artificial bodies have been added to the Solar System. Although they can be regarded as small celestial bodies, the problem of their motions including the dynamical environment and the wide range of their usage is quite different from the natural celestial bodies. The existence of artificial bodies has expanded the research scope and content of the dynamics of the Solar System, and has made Celestial Mechanics closely linked to Aerospace Dynamics. As mentioned above, in the Solar System there are numerous celestial bodies with relatively small masses, including both natural and artificial bodies. From the point of view of dynamics, the question is what kind of celestial body can be regarded as a small body. The answer is as follows. If the mass of a body is too small to influence the vii
- 13. viii Introduction relateddynamicalsystemaswellasthemotionsofothercelestialbodiesinthesystem, then this body is regarded as a small body. Based on all artificial spacecraft launched from Earth, it is obvious that a spacecraft is a small body, its mass is relatively small therefore the motions of all other celestial bodies in the Solar System (including Earth) cannot be affected by a spacecraft. Inadynamical system, all bodies aresources of gravitational forces. If adynamical system is formed by a group of celestial objects, and each object is treated as a “particle” (for now we ignore details about non-particle gravitational forces and non-gravitational forces, which does not affect the content provided here), and all objects interact with each other gravitationally, therefore the problem of predicting the individual motions is called an N-body problem mathematically, and the system is called an N-body system, N is the total number of bodies in the system. If one of the objects is a small body whose mass can be omitted, and the motions of the other N-1 objects are defined, then the problem of predicting the motion of the small body is called a restricted N-body problem. The difference between an N-body problem and a restricted N-body problem is not merely in names but fundamental in research methods and concepts. When N = 3, this problem is the most famous “restricted three-body problem” in Celestial Mechanics. In the restricted three-body problem because the research goal is the motion of a small body whose mass can be ignored, the mathematical method and the motion property of the small body are significantly different from the general three-body problem. For example, in the general three- body problem, there are only 10 classical integrals, and there is no other dynamical information available. In the restricted three-body problem not only the motions of the two big bodies are defined but also the characteristics of the motion of the small body are also given. The available dynamical information of their motions is extremely important for studying the motions of natural small bodies, such as asteroids, and all sorts of spacecraft. The information is also closely related to the launch of deep-space spacecraft and to the formation of a specific orbit due to a specific purpose. Therefore, the Orbital Dynamics of deep-space exploration and the restricted N-body problem are inseparable. The above-described restricted N-body model is built up in a gravitational system of particles, which is a classical model. In reality, under certain circumstances the motion of a small body is also affected by the irregular shape and the uneven mass distribution of the big bodies, the non-gravitational forces (such as the radiation force), and the post-Newtonian effect, etc. Because these forces do not change the basic principles and methods applied in the classical model, it is unnecessary to restrict ourselves in the classical definition, and these forces can be treated as external forces and their effects can be expressed mathematically.
- 14. Introduction ix Two Dynamical Systems in the Orbital Dynamics In a dynamical system, the motion of a celestial body including spacecraft, whether big or small, is usually controlled by more than two external forces. But in the Solar System, where the Sun has been existing for more than 4.6 billion years and is the dominator, the motion of a celestial body is mainly determined by no more than two external forces. For the major planets, there is only one main source of force, i.e., the Sun, the other sources of forces are regarded as perturbations, i.e., small disturbances. For an asteroid, there is usually only one source, the Sun, or are two, the Sun and one of the major planets. For a natural satellite, the main force is from a related major planet. For most of the artificial Earth’s satellites, it is Earth. If the satellite has a high Earth orbit (such as a lunar rover, its orbit needs to be changed during its mission) there are two sources of force, Earth and the Moon. For a deep spacecraft (to explore major planets or natural satellites) the external sources can be the Sun, or the Sun and a major planet, or a major planet and one of its satellites. In all the cases mentioned, besides the one or two main forces, other forces (including non-particle gravitational forces and non-gravitational forces) can be treated as perturbing forces. Therefore, from the perspective of orbital dynamics in the actual Solar System there are only two rational dynamical models for studying the motion of various spacecraft. One model is for the circular orbital problem (for artificial satellites including Earth’s satellites and the Moon’s satellites, etc.) with one main force (i.e., the central celestial body), which corresponds to “the perturbed two-body problem”. The other model has one or two primary forces as in the cases of most deep-space spacecraft, which corresponds to “the perturbed restricted three-body-problem” are regarded as perturbations, i.e., small disturbances. Mathematical Models for Satellite Motion: The Perturbed Two-Body Problem [1–8] As mentioned above, in the Solar System there is only one main force that controls the motions of major planets, asteroids, satellites, and artificial spacecraft (artificial Earth’s satellites, the Moon’s satellites, Mars’s satellites, and other orbiting space- craft). For planets, the main force is the Sun, for natural satellites, it is the related planet, and for spacecraft, it is the target celestial body. Compared to the main force, other forces are small, therefore generally the N-body system (N ≥ 3) can be regarded as a “perturbed two-body system”, which is mathematically called a perturbed two- body problem. For distinguishing the two bodies, the main external source is called the “central body”, denoted as P0, and its mass is denoted as m0; whereas the other body, which is the object to study, is denoted as p, and its mass as m. Our research object is the orbital motion of a celestial body, no matter it is a planet, a satellite, or a circling spacecraft, controlled by the gravitational force of the central body and a few other perturbing forces.
- 15. x Introduction The orbital motion of the perturbed two-body problem can be presented by an ordinary differential equation, which is ¨ → r = − G(m0 + m) r3 → r + k Σ i=1 → Fi , (1) where G is the universal gravitational constant, → Fi is the i-th perturbing acceleration, k (>1) is the number of perturbing sources. The origin of the coordinate system is located at the barycenter of the central body P0, → r = → r(x, y, z) is the position vector of the moving body in the coordinate system. The initial values are given by → r(t0) = → r0, ˙ → r(t0) = ˙ → r0 . (2) By convention, we introduce a symbol μ defined by μ = G(m0 + m) . (3) Then Eq. (1) becomes ¨ → r = − μ r3 → r + k Σ i=1 → Fi . (4) For a small body p (representing any circling prober) with mass m = 0, we have μ = Gm0 which is the gravitational constant of the central body. In the motion problem of an artificial Earth satellite, the central body is Earth, therefore μ = Gm0 = GE, where E is Earth’s mass, and GE equals 3.98603 × 1014 (m3 /s2 ) is the Earth’s gravitational constant. For a low Earth orbit, if the altitude of a satellite is about 300 km, then Earth’s center gravitational acceleration (μ/r2 ) would be about 9 m/s2 . Denoting → Fi (i = 1, 2, · · ·) to the natural existing perturbing accelerations, then the largest among them is due to Earth’s non-spherical part, which is only 10−3 of the acceleration of the central force. We can say that Eq. (4) is for a typical perturbed two-body problem, and the corresponding motion orbit is a slowly changing ellipse. If the weight of the satellite is 1 ton, and it has a constant thrust of 100 N (like a mobile platform), then the mechanical acceleration of the thruster is about 0.1 m/s2 , which is 10−2 of the acceleration by the barycenter force, so this thrust can be also treated as a perturbation. Actually, when the Moon moves around Earth, Earth is the primary body, and the force from the Sun is a perturbation, which is about 2×10−2 of the barycenter force of Earth, greater than the mechanical thrusting acceleration on the satellite. Therefore, the perturbed two-body model can be applied to the motion of a satellite with a mobile platform.
- 16. Introduction xi The Two-Body Problem and Kepler Orbit The reference model of the perturbed two-body problem is a simple two-body problem, which is expressed by an ordinary differential equation as ¨ → r = − μ r3 → r (5) with the initial condition given by (2). This equation is completely solvable, and the solution of this equation is the well-known Kepler orbit. A Kepler orbit is a conic curve, i.e., an ellipse, a parabola, or a hyperbola, and can be presented as r = p 1+e cos f . (6) where f is the true anomaly, e is the eccentricity, and P is the semi-latus rectum given by p = a ( 1 − e2 ) , e < 1 ; (7) p = 2q, e = 1 ; (8) p = a ( e2 − 1 ) , e > 1. (9) The three curves are ellipse (e < 1), parabola (e = 1), and hyperbola (e > 1). In (7) and (9), a is denoted as the semi-major axis; in (8), q is denoted as the periapsis. Another key integral in the two-body problem is the anomaly, which is a function of time t and is directly related to the position of the orbiting body. The relationship of the anomaly and time t has three forms for ellipse, parabola, and hyperbola that E − e sin E = n(t − τ) = M , (10) 2tan f 2 + 3 2 tan3 f 2 = 2 √ μq− 3 2 (t − τ) , (11) e sinh E − E = n(t − τ) = M . (12) The three formulas are the three forms of the famous Kepler Equation, and the motion of the body is called Kepler orbit. In the three equations τ is the time when the moving body is at the periapsis; f, E, M and are the true anomaly, eccentric anomaly, and mean anomaly, respectively; and n is the mean angular speed given by
- 17. xii Introduction n = √ μa− 3 2 . (13) As described above, the two-body problem and the Kepler orbit are indiscrimi- native and both contain three types of orbits, i.e., ellipse, parabola, and hyperbola. The focus of attention is usually on the ellipse because this is the primary form of celestial motion in the Solar System. The Method of Solving the Perturbed Two-Body Problem For solving the perturbed two-body motion Eq. (1) so far, we do not have a very effi- cient method. Summarized here is the widely used perturbation method in advanced science and engineering. The accepted reference orbit is a Kepler orbit, the actual orbit is a slowly changing Kepler orbit. The related motion, at any given time, can be presented by an instantaneous Kepler orbit (such as an instantaneous ellipse orbit). Specifically based on the reference model, by the method of the variation of arbitrary constants we first transfer the original equation to a small parameter equation; then construct a required analytical solution according to the analytical theory of ordinary differential equation (Poincare Theorem) as power series of a small parameter of the first-order, the second-order, or the higher-order form. In the method of the variation of arbitrary constants, the basic parameters are usually the six constant integrals in the complete solution of the two-body problem. Note that for a perturbed two-body problem the six constant integrals are no long constants. These basic parameters have definite geometrical meanings and are called Kepler orbital elements denoted by a set σ that σ = (a, e, i, Ω, ω, M)T , (14) where the superscript T means the transposition of a matrix. In the barycenter celestial coordinate system, the definitions of the orbital elements are a the semi-major axis, e the eccentricity, i the inclination, Ω the longitude of the ascending node, ω the argument of the periapsis, and M the mean anomaly. The first three elements, a, e, and i, are angular momentums, and the other three elements Ω, ω, and M are angular variables (Ω and ω are slowly changing variables, and M a fast-changing variable). The perturbed two-body problem Eq. (1) can be converted into a system of equa- tions of small parameters using the method of the variation of arbitrary constants, written as σ̇ = f (σ, t, ε) , (15)
- 18. Introduction xiii where ε is a small parameter related to the perturbing acceleration → Fε. This system of equations has a few different forms and is discussed in depth in subsequent chapters. The initial value is denoted by σ(t0) = σ0 , (16) where σ0 is for the initial values of the six orbital elements. Equations (15) has other forms, as discussed in related chapters. The perturbed solution of orbital elements can be expressed as a small parameter power series by the classic perturbation method (or other improved perturbation methods) written as σ(t) = σ(0) + Δσ(1) + Δσ(2) + · · · + Δσ(k) , (17) where σ(0) is for the orbital elements of the reference orbit which is an unperturbed orbit. This classic method for the perturbation solution is still the best method in use, and is also applicable for both solutions of a varying ellipse and a varying hyperbola. In the development of Celestial Mechanics and the Satellite Orbital Dynamics, researchers have tried different methods. One of them is the “intermediary orbit” method. The so-called intermediary orbit is an orbit including some influences of perturbing forces therefore is closer to the actual orbit than the non-perturbed orbit. One of the successful examples is the Moon’s intermediary orbital solution (the Hill problem). The intermediary orbital method is also applied in forming the orbital solution for the artificial satellite when the effect of the non-spherical gravitational force of Earth is included in the intermediary reference orbit. The intermediary orbit, in fact, is a changing ellipse including some perturbing forces, thus it does not have any essential improvement, so is not necessary to be called a non-Kepler orbit. Actually, neither the Hill solution of the Moon orbiting around Earth nor the intermediary orbit for an artificial Earth satellite can be directly applied in practice. The practical method is still based on adding remaining perturbing forces to the changing ellipse orbit. Therefore, at the present time, the Kepler orbit is still the most desired reference orbit in solving the perturbed two-body problem. In dealing with an actual problem, the sixth orbital element of the perturbed orbit after using the method of the variation of arbitrary constants is neither τ (τ is the time when the moving satellite is at the periapsis) nor M0 = nτ, n = √ μa− 3 2 , but M, the mean anomaly, given by M = n(t − τ) . (18) TherearetworeasonsforusingM.Oneisthatτ and M0 havenopracticalmeanings in a perturbed motion, whereas M has a defined geometric meaning so is easy to use; the other is that M is a function of a and τ, therefore in the perturbed equation the operation of ∂ R/∂a no longer deals with the problem of inexplicitly including a in
- 19. xiv Introduction the perturbation function R (through M). With M as an independent element, the perturbed equation can be simplified. The Perturbed Restricted Three-Body Problem in the Motion of Deep-Space Prober The Restricted Three-Body Problem for Circular and Elliptical Motions [9–12] In a three-body problem with N = (2 + 1), there are two primary bodies and a small body. Because the small body has no influence on the motions of the two primary bodies, the motions of two primary bodies are defined by a simple two-body problem. Each of the big bodies moves in a circle or an ellipse around their common barycenter, but neither a parabola nor a hyperbola, realistically. This problem, therefore, is a restricted circular three-body problem or a restricted elliptical three-body problem. The motion of the third body (a small body) in this system is to be studied. In the Sun-Earth-Moon three-body system the Moon’s mass (m) by comparison is much smaller than the masses of the Sun and Earth, m1 and m2, respectively, that m = 0.012 m2, approximately; and the eccentricity of Earth’s orbit around the Sun is only 0.017, thus the motion of the Moon in this system can be treated in a circular restricted three-body problem. Of cause, an elliptical restricted three-body problem is closer to the real situation than a circular restricted three-body problem. This model is also applied to the motion of an asteroid located in the asteroid belt (between Mars’s orbit and Jupiter’s orbit, most asteroids are in this belt). The motion of an asteroid is due to mainly the gravitational forces from the Sun and Jupiter. Because the eccentricity of Jupiter’s orbit is relatively small, the motion of an asteroid can be also treated as a circular restricted three-body problem. The orbit of a deep-space spacecraft is more complicated. The whole process can be divided into a few segments. For example, after launch a Moon’s prober has a near Earth orbit like an Earth’s satellite; when it is near the Moon, it changes its orbit and moves around the Moon, between the two orbits the motion of the prober is in a typical restricted three-body system of Earth, the Moon, and the prober, which can be a circular or an elliptical restricted three-body problem. Another example is about a Mars’s prober. In the early stage after launch, it moves like an Earth’s satellite. During the time it leaves the Earth-Moon system and before it reaches the area of Mars’s gravitational field there is a long cruising period controlled by the Sun’s attraction, and the motion is decided by a perturbed two-body problem with the Sun as the central body. After it moves into Mars’s gravitational field its motion then is provided by a typical restricted three-body problem of the Sun, Mars, and the prober. There are many more examples like these in the exploration of the Solar System.
- 20. Introduction xv The equations of the above-mentioned motion models, including the simplest circular restricted three-body problem, are unsolvable. There is only one solved problem which is the restricted three-body problem with two “motionless” main bodies. It is an approximate model when the motions of the two big bodies are much slower than the small body. This is called the problem of two stationary main bodies in the restricted three-body problem. The equation of this model is solved, but the model is too simple to be used in solving any actual dynamic problem for a spacecraft in the Solar System. Models for the Restricted N-body Problem and the Perturbed Restricted Three-Body Problem [13, 14] In the restricted N-body (N ≥ 3) problem, there are n-big bodies and one small body. One example of this problem is the motion of an asteroid in the asteroid belt. In order to present the motion close to the real situation and to agree with the characteristics of the distribution of asteroids (such as the Kirkwood gaps), the gravitational forces should be included are not only the primary forces from the Sun and Jupiter but also the forces from Saturn and Mars, thus a restricted problem of (4 + 1) bodies is formed. Another example is about a Moon’s prober. The motion of the prober is determined by gravitational forces from Earth, the Moon, and the Sun; therefore, it is a kind of restricted problem of (3 + 1) bodies. In this kind of system, it does not matter how many big bodies there are (i.e., different values of N), the motion of a small body, which can be an asteroid or a Moon’s prober, is studied by assuming that the motions of the big bodies are defined. In the first example, if the force from the third body and the fourth body is not strong enough to cause obvious changes to the results of the original restricted three-body problem, then their forces can be treated as perturbations, and the N-body problem can be regarded as a perturbed restricted three-body problem. Similarly in the second example, the force of the third body can be treated as a perturbation. In reality, the motion of an asteroid in the asteroid belt or a Moon’s prober is studied in this way. In other words, this kind of problem is solved by the perturbation method using as much information as possible from a restricted three-body problem. This method is applied to design deep spacecraft orbits and some specific orbits for specific purposes (such as the Halo orbit). From the above two examples, although using the five-body problem model or the four-body problem model is seemly more precise and more attractive, the corresponding four-body problem or three-body problem of the big bodies is still unsolved to the present day. The fact is that in the Solar System to study the motion of a natural celestial body or an artificial body the most commonly used models are the perturbed restricted two-body problem model and the perturbed restrictedthree-bodyproblem model,especiallythe perturbedcircularrestricted three-body problem model.
- 21. xvi Introduction The Restricted Problem of (n + k)-Bodies [15, 16] The (n + k)-body problem is an N-body problem where N = n + k, and there are n-big bodies and k-small bodies with n ≥ 2 and k ≥ 2. If k = 1, the problem is then reduced to one of the above discussed two examples. This (n + k) system is actually about motions of k-small bodies in an n-body system. The motions of these big bodies are defined, and the small bodies are attracted by the big bodies. The masses of the k small bodies are much smaller than those of the big bodies, thus the small bodies do not affect the motions of the big bodies, but if the distances between the small bodies are short, the gravitational forces between them should be considered. If the gravitational forces of the small bodies can be ignored, then the motion of each small body can be studied separately in an (n + 1)-body restricted problem. The type of (n + k)-body problem exists in the Solar System. For example, to studythemotionsoftwocloselylocatedasteroidsintheasteroidbelt,thegravitational forces between them need to be considered, then the Sun, Jupiter, and the two small asteroids make up a restricted problem of (2 + 2) bodies. Another example is about launching two geosynchronous satellites at a fixed point high up above the equator. If the weight of each satellite is a few tons, and the distance between them is about a few hundred meters, then to obtain a high precision solution the influence between the two small satellites needs to be included. In this case, Earth, an ellipsoid, is treated as if there were two bodies, one is a sphere with evenly distributed mass, and the other is a “body” formed by the non-spherical part around Earth’s equator, thus there is a restricted problem of (2 + 2)-bodies. Similar situations also appear in the launching of a few spacecraft at a specific point, the gravitational forces between them cannot be ignored, if there are two related big bodies, then the spacecraft and the two big bodies form a restricted problem of (2 + k)-bodies. General Restricted Three-Body Problem In a restricted three-body problem under Newton’s gravitational forces if a big body has strong radiation, then its post-Newtonian effect (i.e., the post-Newtonian expan- sion) should be considered, and the system might be called a generalized restricted three-body problem. In this case, if the motion of the second big body is not affected by the radiation (rigorously speaking, the effect is small enough to be omitted), then the non-gravitational force on the small body should be included. The research of the generalized restricted three-body problem is not discussed in the book.
- 22. Introduction xvii References 1. Brouwer D, Clemence GM (1961) Methods of Celestial Mechanics. Academic Press, New York and London 2. Beutler G (2005) Methods of Celestial Mechanics. Springer-Verlag Berlin, Heidelberg 3. Vinti JP (1998) Orbital and Celestial Mechanics. AIAA Education Series, American Institute of Aeronautics and Astronautics, Inc., Reston, Virginia 4. Boccaletti D, Pucacco G (1999) Theory of Orbits. Vol.1–2, Springer-Verlag, Berlin, Heidelberg 5. Kozai Y (1959) The motion of a close earth satellite. Astron. J., 64 (9): 367–377 6. Liu L (1992) Orbital Dynamics of Earth’s Artificial Satellite. Higher Education Press, Beijing 7. Liu L (2000) Orbital Theory of Spacecraft. National Defense Industry Press, Beijing 8. Liu L, Hu SJ, Wang X (2006) Introduction of Aerospace Dynamics. Nanjing University Press, Nanjing 9. Szebehely V (1967) Theory of Orbit: The Restricted Problem of Three Bodies. Academic Press, New York, London 10. Brown EW (1896) An Introductory Treatise on Lunar Theory. Cambridge University Press 11. Murray CD, Dermott SF (1999) Solar System Dynamics. Cambridge University Press, 1999 12. Gómez G et al (2001) Dynamics and Mission Design near Libration Points, Vol. 1–4. World Scientific, Singapore, New Jersey, London, Hong Kong 13. Hou XY, Liu L (2008) Dynamical characteristics of collinear Lagrangian points and the application in the deep space exploration, Journal of Astronautics, 2008, 29(3): 461–466 14. Liu L, Hou XY (2012) Orbital Dynamics of Deep Spacecraft. Electronic Industry Press, Beijing 15. Whipple AL, Szebehely V (1984) The Restricted Problem of n + v Bodies. Celest Mech 32(2):137–144 16. Whipple AL (1984) Equilibrium Solutions of the Restricted Problem of 2+2 Bodies. Celest Mech 33(3):271–294
- 23. Contents 1 Selections and Transformations of Coordinate Systems . . . . . . . . . . . 1 1.1 Time Systems and Julian Day [1, 2] . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Selection of Standard Time . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Time Reference Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.3 Julian Day . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Space Coordinate Systems [2–6] . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Earth’s Coordinate Systems [2, 6–10] . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.1 The Realization of the Dynamical Reference System and J2000.0 Mean Equatorial Reference System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.2 The Intermediate Equator and Three Related Datum Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.3 Three Geocentric Coordinate Systems . . . . . . . . . . . . . . . 12 1.3.4 Transformation of the Earth-Fixed Coordinate System O-XYZ and the Geocentric Celestial Coordinate System O-xyz . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.5 Relationship Between the IAU 1980 Model and the IAU 2000 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3.6 The Complicity in the Selection of Coordinate System Due to the Wobble of Earth’s Equator . . . . . . . . . 24 1.3.7 Coordinate Systems Related to Satellite Measurements, Attitudes, and Orbital Errors . . . . . . . . . . 24 1.4 The Moon’s Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.4.1 Definitions of the Three Selenocentric Coordinate Systems [6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.4.2 The Moon’s Physical Libration . . . . . . . . . . . . . . . . . . . . . 26 1.4.3 Transformations Between the Three Selenocentric Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.5 Planets’ Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.5.1 Definitions of Three Mars-Centric Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 xix
- 24. xx Contents 1.5.2 Mars’s Precession Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.5.3 Transformation of the Mars-Centric Equatorial Coordinate System and the Mars-Fixed Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.5.4 Transformation of the Geocentric Coordinate System and the Mars-Centric Coordinate System . . . . . . 37 1.5.5 An Explanation of the Application of the IAU 2000 Orientation Models of Celestial Bodies . . . . . . . . . . 40 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2 The Complete Solution for the Two-Body Problem . . . . . . . . . . . . . . . 43 2.1 Six Integrals of the Two-Body Problem . . . . . . . . . . . . . . . . . . . . . . 43 2.1.1 The Angular Momentum Integral (the Areal Integral) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.1.2 The Orbital Integral in the Motion Plane and the Vis Viva Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.1.3 The Sixth Motion Integral: Kepler’s Equation . . . . . . . . . 49 2.2 Basic Formulas of the Elliptical Orbital Motion . . . . . . . . . . . . . . . 51 2.2.1 Geometric Relationships of the Orbital Elements in the Elliptical Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.2.2 Expressions of the Position Vector → r and Velocity ˙ → r . . . . 51 2.2.3 Partial Derivatives of Some Variables with Respect to Orbital Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.2.4 Derivatives of M, E, and F with Respect to Time t . . . . . 59 2.3 Expansions of Variables in the Elliptical Orbital Motion . . . . . . . 60 2.3.1 Expansions of Sin kE and Cos kE . . . . . . . . . . . . . . . . . . . 61 2.3.2 Expansions of E, r/a, and a/r . . . . . . . . . . . . . . . . . . . . . . . 62 2.3.3 Expansions of Sin F and Cos F . . . . . . . . . . . . . . . . . . . . . 62 2.3.4 The Expansion of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.3.5 Expansions of (r a )n cosm f and (r a )n sinm f . . . . . . . . . . . . 63 2.3.6 Expansions of (a r ) p, E, and (F − M) in the Trigonometric Function of F . . . . . . . . . . . . . . . . . . 66 2.4 Transformations from the Orbital Elements to the Position Vector and Velocity and Vice Versa . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.4.1 Calculations of the Position Vector → r(t) and Velocity ˙ → r(t) from Orbital Elements σ(t) . . . . . . . . . 67 2.4.2 Calculations of the Orbital Elements σ(t) from → r(t) and ˙ → r(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.4.3 Calculations of Orbital Elements σ(t0) from Two Position Vectors → r(t1) and → r(t2) . . . . . . . . . . . . . . . . . . . . . 69 2.4.4 Method to Solve Kepler’s Equation . . . . . . . . . . . . . . . . . . 70 2.5 Expressions and Calculations of Satellite Orbital Variables . . . . . 71 2.5.1 Two Expressions of the Longitude of Satellite’s Orbital Ascending Node . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
- 25. Contents xxi 2.5.2 Expressions of Satellite’s Position Measurements from a Ground-Based Tracking Station . . . . . . . . . . . . . . 72 2.5.3 Equatorial Coordinates of the Sub-Satellite Point . . . . . . 73 2.5.4 Satellite’s Orbital Coordinate System . . . . . . . . . . . . . . . . 74 2.5.5 Expressions of Errors in Satellite Position . . . . . . . . . . . . 75 2.6 Parabolic Orbit and Hyperbolic Orbit . . . . . . . . . . . . . . . . . . . . . . . 75 2.6.1 The Parabolic Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2.6.2 The Hyperbolic Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2.6.3 Formulas for Calculating the Position Vector and Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3 Analytical Methods of Constructing Solution of Perturbed Satellite Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.1 The Method of the Variation of Arbitrary Constants Applied to the Perturbed Two-Body Problem . . . . . . . . . . . . . . . . . 81 3.2 Common Forms of Perturbed Motion Equation . . . . . . . . . . . . . . . 84 3.2.1 Perturbed Motion Equations Formed by Accelerations of the (S, T, W)-Version and the (U, N, W)-Version . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.2.2 The Perturbation Motion Equations Formed by ∂R/∂σ-Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.2.3 Canonical Equations of Perturbation Motion . . . . . . . . . . 88 3.2.4 Singularities in the Perturbation Equations . . . . . . . . . . . 88 3.3 Perturbation Method of Constructing Power Series Solution with a Small Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.3.1 Perturbation Equations with a Small Parameter . . . . . . . . 94 3.3.2 Existence of Power Series Solution with a Small Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.3.3 Construction of the Power Series Solution with a Small Parameter: The Perturbation Method . . . . . 96 3.3.4 Secular Variations and Periodic Variations . . . . . . . . . . . . 99 3.4 An Improved Perturbation Method: The Method of Mean Orbital Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.4.1 Introduction of the Method of Mean Orbital Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.4.2 The Mean Values of Related Variables in an Elliptic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.4.3 Construction of Formal Solution: The Method of Mean Orbital Elements [3–8] . . . . . . . . . . . . . . . . . . . . . 106 3.4.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.4.5 Two Annotations About the Method of Mean Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.5 The Method of Quasi-Mean Elements: The Structure of the Formal Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
- 26. xxii Contents 3.5.1 Small Divisors in Expressions of Perturbation Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 3.5.2 Configuration of Formal Solution: The Method of Quasi-Mean Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 3.6 Methods of Constructing Non-singularity Solutions for a Perturbed Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.6.1 Configuration of the Non-singularity Perturbation Solutions of the First Type . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.6.2 Configuration of the Non-singularity Perturbation Solutions of the Second Type . . . . . . . . . . . . . . . . . . . . . . . 124 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4 Analytical Non-singularity Perturbation Solutions for Extrapolation of Earth’s Satellite Orbital Motion . . . . . . . . . . . . . 129 4.1 The Complete Dynamic Model of Earth’s Satellite Motion . . . . . 129 4.1.1 Selection of Calculation Units in Satellite Orbit Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.1.2 Analyses of Forces on Satellite’s Orbital Motion . . . . . . 132 4.1.3 Further Analyses of the Forces Acting on a Satellite . . . 136 4.2 The Perturbed Orbit Solution of the First-Order Due to Earth’s Dynamical Form-Factor J2 Term . . . . . . . . . . . . . . . . . . 137 4.2.1 The Perturbed Orbit Solution of the First-Order in Kepler Orbital Elements [1–5] . . . . . . . . . . . . . . . . . . . . 137 4.2.2 The Non-singularity Perturbation Solution of the First Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 4.2.3 The Non-singularity Perturbation Solution of the Second Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.3 The Perturbed Orbit Solution of the First-Order Due to Earth’s Ellipticity J2,2 Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 4.3.1 The Perturbed Orbit Solution of the First-Order in Kepler Orbital Elements . . . . . . . . . . . . . . . . . . . . . . . . . 160 4.3.2 The Non-singularity Perturbation Solution of the First Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.3.3 The Non-singularity Perturbation Solution of the Second Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 4.4 Additional Perturbation of the Coordinate System for the First-Order Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 4.4.1 The Cause of the Additional Perturbation of the Coordinate System [3, 8] . . . . . . . . . . . . . . . . . . . . . 168 4.4.2 The Additional Perturbation Solution in Kepler Orbital Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 4.4.3 The Non-singularity Additional Perturbation Solution of the First Type . . . . . . . . . . . . . . . . . . . . . . . . . . 173 4.4.4 The Non-singularity Additional Perturbation Solution of the Second Type . . . . . . . . . . . . . . . . . . . . . . . . 176
- 27. Contents xxiii 4.4.5 Selection of Coordinate System and Related Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 4.5 The Perturbation Orbit Solution Due to the Higher-Order Zonal Harmonic Terms Jl (l ≥ 3) of Earth’s Non-spherical Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.5.1 General Expression of the Perturbation Function of the Zonal Harmonic Terms Jl(l ≥ 3) . . . . . . . . . . . . . . 178 4.5.2 The Perturbation Solution of the Zonal Harmonic Jl(l ≥ 3) Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 4.5.3 The Non-singularity Perturbation Solution of the First Type by the Zonal Harmonic Terms Jl(l ≥ 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 4.5.4 The Non-singularity Perturbation Solution of the Second Type by Zonal Harmonic Terms Jl(l ≥ 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 4.5.5 The Perturbation Solution of the Main Zonal Harmonic Terms J3 and J4 in Kepler Elements . . . . . . . . 188 4.6 The Perturbation Solution Due to the High-Order Tesseral Harmonic Terms Jl,m (l ≥ 3, M = 1, 2, · · · , l) of Earth’s Non-spherical Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 4.6.1 The General Expression of the Perturbation Function of the Tesseral Harmonic Terms Jl,m (l ≥ 3, M = 1, 2, · · · , l) . . . . . . . . . . . . . . . . . . . . . . . . 196 4.6.2 The Perturbation Solution Due to the Tesseral Harmonic Terms Jl,m(l ≥ 3, m = 1 − l) . . . . . . . . . . . . . 197 4.6.3 The Non-singularity Perturbation Solution of the First Type Due to the Tesseral Harmonic Jl,m(l ≥ 3, m = 1 − l) Terms . . . . . . . . . . . . . . . . . . . . . . . 200 4.6.4 The Non-Singularity Perturbation Solution of the Second Type Due to the Tesseral Harmonic Jl,m(l ≥ 3, m = 1 − l) Terms . . . . . . . . . . . . . . . . . . . . . . . 200 4.6.5 The Perturbation Solution Due to the Tesseral Terms, J3,m (M = 1, 2, 3) and J4,m (M = 1, 2, 3, 4) in Kepler Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 4.6.6 The Non-singularity Perturbation Solution of the First Type Due to the Tesseral Harmonic Terms J3,m (M = 1, 2, 3) and J4,m (M = 1, 2, 3, 4) . . . . . 211 4.6.7 The Non-singularity Perturbation Solution of the Second Type Due to the Tesseral Harmonic Terms J3,m (m = 1, 2, 3) and J4,m (m = 1, 2, 3, 4) . . . . . . 211 4.7 The Perturbed Orbit Solution Due to the Gravitational Force of the Sun or the Moon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 4.7.1 The Perturbation Function and Its Decomposition . . . . . 212 4.7.2 The Perturbation Solution Due to the Gravity of the Sun or the Moon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
- 28. xxiv Contents 4.8 The Perturbed Orbit Solution Due to Earth’s Deformation . . . . . . 230 4.8.1 Expression of the Additional Potential of Tidal Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 4.8.2 Effect of the Main Term in the Additional Tidal Deformation Potential (the Second-Order Term of l = 2) on a Satellite Orbit . . . . . . . . . . . . . . . . . . . . . . . . 232 4.9 Post-Newtonian Effect on the Orbital Motion . . . . . . . . . . . . . . . . . 235 4.9.1 The Post-Newtonian Effect . . . . . . . . . . . . . . . . . . . . . . . . . 235 4.9.2 Perturbation Solution Due to the Post-Newtonian Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 4.9.3 Other Post-Newtonian Effects on the Earth’s Artificial Satellite Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 239 4.10 Perturbed Orbit Solution Due to the Solar Radiation Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 4.10.1 Calculation of Radiation Pressure . . . . . . . . . . . . . . . . . . . 239 4.10.2 Two States of Radiation Pressure Perturbation . . . . . . . . 242 4.10.3 The Perturbation Solution Due to Radiation Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 4.10.4 The Non-singularity Perturbation Solution of the First Type Due to the Radiation Pressure . . . . . . . . 252 4.10.5 The Non-singularity Perturbation Solution of the Second Type Due to the Radiation Pressure . . . . . 254 4.11 Perturbed Orbit Solution Due to Atmospheric Drag . . . . . . . . . . . 256 4.11.1 Damping Effect: Atmospheric Drag . . . . . . . . . . . . . . . . . 256 4.11.2 Atmosphere Density Model . . . . . . . . . . . . . . . . . . . . . . . . 258 4.11.3 Atmospheric Rotation and the Expression of Atmospheric Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 4.11.4 Structure of the Perturbed Solution Due to the Atmospheric Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 4.11.5 The Non-singularity Perturbation Solution by the Atmospheric Drag . . . . . . . . . . . . . . . . . . . . . . . . . . 273 4.12 Orbital Variations Due to a Small Thruster . . . . . . . . . . . . . . . . . . . 274 4.12.1 The Perturbation Solution Due to an (S,T,W)-Type Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 4.12.2 The Non-singularity Perturbation Solution Due to an (S,T,W)-Type Thrust . . . . . . . . . . . . . . . . . . . . . . . . . 277 4.12.3 The Perturbation Solution by a U-type Thrust . . . . . . . . . 281 4.12.4 The Non-singularity Perturbation Solution Due to a U-type Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 5 Satellite Orbit Design and Orbit Lifespan Estimation . . . . . . . . . . . . . 287 5.1 Sidereal Period and Nodal Period [1–3] . . . . . . . . . . . . . . . . . . . . . . 287 5.1.1 The Transformation Between the Sidereal Period Ts and the Nodal Period Tϕ . . . . . . . . . . . . . . . . . . . . . . . . 288
- 29. Contents xxv 5.1.2 The Anomalistic Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 5.2 Orbital Characteristics of Polar Orbit Satellite [2, 3] . . . . . . . . . . . 293 5.2.1 Basic Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 5.2.2 Preservation of Polar Orbit . . . . . . . . . . . . . . . . . . . . . . . . . 295 5.3 Existence and Design of Sun-Synchronous Orbit [2–5] . . . . . . . . 296 5.3.1 Conditions of Forming a Sun-Synchronous Orbit . . . . . . 296 5.3.2 Sun-Synchronous Orbits for Different Celestial Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 5.4 Existence and Design of Frozen Orbit [2–5] . . . . . . . . . . . . . . . . . . 300 5.4.1 Basic State of Frozen Orbit . . . . . . . . . . . . . . . . . . . . . . . . . 300 5.4.2 Basic Equations of a Possible Frozen Orbit . . . . . . . . . . . 301 5.4.3 A Particular Solution of Eq. (5.40): The Frozen Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 5.4.4 Stability of Frozen Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 5.4.5 Frozen Orbit for Other Celestial Bodies . . . . . . . . . . . . . . 306 5.4.6 Characteristics and Applications of Satellite Orbit with a Critical Inclination . . . . . . . . . . . . . . . . . . . . . . . . . . 306 5.5 Existence and Design of Central Body Synchronous Orbit . . . . . . 309 5.5.1 Basic State of Central Body Synchronous Satellite Orbit [2, 3, 5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 5.5.2 Existence and Evolution of a Central Body Synchronous Satellite (Earth, Mars) . . . . . . . . . . . . . . . . . 312 5.6 Estimation and Calculation of Satellite’s Lifespan Due to the Mechanism of Gravitational Perturbation . . . . . . . . . . . . . . . 316 5.6.1 Definition and Mechanism of a Low Orbit Satellite Lifespan Due to Gravitational Perturbations [6–10] . . . . 317 5.6.2 Overview of Low Orbit Satellite Lifespan for Earth, the Moon, Mars, and Venus . . . . . . . . . . . . . . . . 319 5.6.3 Evolution Characteristics and Lifespans of Orbit with a Large Eccentricity [2, 9] . . . . . . . . . . . . . . . . . . . . . 323 5.6.4 Evolution Characteristics and Lifespans of High Earth Satellite Orbit [6, 10] . . . . . . . . . . . . . . . . . . . . . . . . . 330 5.6.5 Key Points About Estimating Satellite Orbit Lifespan Due to Gravitational Perturbations . . . . . . . . . . 332 5.7 Estimation and Calculation of Satellite Orbit Lifespan in the Perturbed Mechanism of Atmospheric Drag . . . . . . . . . . . . 332 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 6 Orbital Solutions of Satellites of the Moon, Mars, and Venus . . . . . . 335 6.1 Characteristics of Gravitational Fields of Earth, the Moon, Mars, and Venus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 6.1.1 Basic Characteristics of Earth’s Gravity Potential . . . . . . 335 6.1.2 Basic Characteristics of the Moon’s Gravity Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 6.1.3 Basic Characteristics of Mars’s Gravity Potential . . . . . . 338
- 30. xxvi Contents 6.1.4 Basic Characteristics of Venus’s Gravity Potential . . . . . 339 6.2 Perturbed Orbital Solution of the Moon’s Satellite . . . . . . . . . . . . . 340 6.2.1 Selection of Coordinate System . . . . . . . . . . . . . . . . . . . . . 341 6.2.2 Mathematical Model for the Perturbed Motion of the Moon’s Satellite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 6.2.3 The Numerical Solution for the High Precise Orbital Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 6.2.4 The Analytical Perturbation Solution of the Moon’s Satellite Orbit . . . . . . . . . . . . . . . . . . . . . . . 346 6.2.5 Additional Perturbation of Coordinate System [5, 6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 6.2.6 Applications of Analytical Orbital Solution in Orbital Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 6.3 Perturbed Orbital Solution of Mars’s Satellite . . . . . . . . . . . . . . . . 371 6.3.1 Selection of Coordinate System . . . . . . . . . . . . . . . . . . . . . 372 6.3.2 The Mathematical Model of Perturbed Motion for a Mars’s Satellite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 6.3.3 The Analytical Perturbation Solution of Mars’s Satellite Orbit [7, 8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 6.4 Perturbed Orbital Solution of Venus’s Satellite . . . . . . . . . . . . . . . . 383 6.4.1 The Perturbation Function of Venus’s Non-Spherical Gravity Potential . . . . . . . . . . . . . . . . . . . . 384 6.4.2 The Structure and Results of the Analytical Perturbation Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 7 Orbital Motion and Calculation Method in the Restricted Three-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 7.1 Selection of Coordinate System and Motion Equation of a Small Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 7.1.1 The Motion Equation of a Small Body in the Barycenter Inertial Coordinate System . . . . . . . . . . 391 7.1.2 The Motion Equation of a Small Body in the Synodic Coordinate System . . . . . . . . . . . . . . . . . . . 393 7.2 Jacobi Integral and Solution Existence of the Circular Restricted Three-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 7.2.1 Jacobi Integral in the Circular Restricted Three-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 7.2.2 Existence of Solution of the Circular Restricted Three-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
- 31. Contents xxvii 7.3 Calculation and Application of the Libration Point Positions of the Circular Restricted Three-Body Problem . . . . . . . 397 7.3.1 Conditions of Existence for Libration Solutions . . . . . . . 398 7.3.2 The Positions of the Three Collinear Libration Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 7.3.3 Two Triangle Libration Points . . . . . . . . . . . . . . . . . . . . . . 401 7.3.4 Dynamical Characteristics of the Five Libration Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 7.3.5 Characteristics and Applications of the Stability of the Five Libration Points . . . . . . . . . . . . . . . . . . . . . . . . 410 7.3.6 Calculations and Applications of Libration Points in the Restricted Problem of (2+2) Bodies . . . . . . . . . . . . 419 7.4 Orbit Design for Formation Flying of Satellites and Companion-Flying in the Exploration of Asteroids . . . . . . . . 422 7.4.1 The Principle of Satellite Formation Flying . . . . . . . . . . . 422 7.4.2 The Problem with the Eccentricity in Orbit Design of Formation Flying of Satellites . . . . . . . . . . . . . . . . . . . . 426 7.4.3 Extension of the Principle and Related Orbit Design Method of Satellite Formation Flying . . . . . . . . . 428 7.4.4 Orbital Problem of Companion Flying in Asteroid Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 7.5 Geometric Characteristics of Libration Point Orbit and Orbit Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 7.5.1 Geometric Characteristics of Libration Point Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 7.5.2 Analysis of Forces on a Prober’s Motion in a Libration Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 7.5.3 Orbit Determination and Forecast Method of Libration Point Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 7.5.4 Orbit Determination of Libration Point Orbit and Precision Examination of Short-Arc Forecast [22] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 7.5.5 Orbital Transformation Between the Two Coordinate Systems for a Libration Point Orbit Prober . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 8 Numerical Method for Satellite Orbit Extrapolations . . . . . . . . . . . . . 441 8.1 Basic Knowledge of Numerical Method in Solving the Motion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 8.1.1 Basic Principles of Numerical Method in Solving Motion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 8.1.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
- 32. xxviii Contents 8.2 Conventional Singer-Step Method: The Runge–Kutta Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 8.2.1 The Fourth-Order RK Method (RK4) . . . . . . . . . . . . . . . . 446 8.2.2 The Runge–Kutta-Fehlberg (RKF) Method . . . . . . . . . . . 447 8.3 Linear Multistep Methods: Adams Method and Cowell Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 8.3.1 Adams Methods: Explicit Methods and Implicit Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 8.3.2 Cowell’s Method and Størmer’s Method . . . . . . . . . . . . . 453 8.3.3 Adams-Cowell Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 8.4 Key Issues in Applications of the Numerical Method in Orbital Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 8.4.1 Selections of Variables and Corresponding Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 8.4.2 Singularity Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 8.4.3 Homogenization of Step-Size . . . . . . . . . . . . . . . . . . . . . . . 463 8.4.4 Control of the Along-Track Errors . . . . . . . . . . . . . . . . . . . 464 8.5 Numerical Calculation of the Right-Side Function . . . . . . . . . . . . 465 8.5.1 The Perturbation Acceleration of the Zonal Harmonic Term → F1(Jl) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 8.5.2 The Perturbation Acceleration of the Tesseral Harmonic Term → F ( Jl,m ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 8.5.3 The Recursive Formulas of Legendre Polynomials, Pl(µ) and the Associated Legendre Polynomials Pl,m(µ), and Their Derivatives [15, 16] . . . . . . . . . . . . . . . 469 8.5.4 The Perturbation Acceleration of the Tidal Deformation → F ( k2, J2,m ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 8.6 The Role of the Hamiltonian Method in the Orbital Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 9 Formulation and Calculation of Initial Orbit Determination . . . . . . . 473 9.1 Formulation of Orbit Determination . . . . . . . . . . . . . . . . . . . . . . . . . 473 9.2 A Review of Initial Orbit Calculation in the Sense of the Two-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 9.2.1 Basic Conditions for Initial Orbit Determination . . . . . . . 475 9.2.2 Construction of the Basic Equation for an Initial Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 9.3 Initial Orbit Determination for Perturbed Motion . . . . . . . . . . . . . . 478 9.3.1 Construction of the Basic Equation for Initial Orbit Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 9.3.2 Initial Orbit Determination Using Angle Data Over a Short-Arc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 9.3.3 Initial Orbit Determination Using (P, A, h) Data or Navigation Information . . . . . . . . . . . . . . . . . . . . . . . . . 487
- 33. Contents xxix 9.3.4 Examination of Orbit Determination Method Using Actual Measurements . . . . . . . . . . . . . . . . . . . . . . . . 488 9.3.5 Initial Orbit Determination When a Deep-Space Prober is on a Transfer Orbit . . . . . . . . . . . . . . . . . . . . . . . 489 9.3.6 Initial Orbit Determination Using Space-Based Angle Measurements (α, δ) . . . . . . . . . . . . . . . . . . . . . . . . . 494 9.3.7 A Brief Summary of Initial Orbit Determination . . . . . . . 496 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 10 Precise Orbit Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 10.1 Precise Orbit Determination: Orbit Determination and Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 10.2 Theoretical Calculation of Measurement Variables . . . . . . . . . . . . 503 10.3 Calculation of Transformation Matrixes . . . . . . . . . . . . . . . . . . . . . 508 10.3.1 Matrix ∂Y ∂(→ r,˙ → r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 10.3.2 Matrix ( ∂(→ r,˙ → r) ∂σ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 10.3.3 State Transition Matrix Φ . . . . . . . . . . . . . . . . . . . . . . . . . . 514 10.4 Estimation of the State Variable: Calculation of Precise Orbit Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 10.4.1 Certainty of Solution in the Orbit Determination . . . . . . 521 10.4.2 Process of Calculating Solution in the Orbit Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 10.5 The Least Squares Estimator and Its Application in Precise Orbit Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 10.5.1 Estimation Theory and a Few Commonly Used Optimal Estimation Methods . . . . . . . . . . . . . . . . . . . . . . . 524 10.5.2 The Least Squares Estimator . . . . . . . . . . . . . . . . . . . . . . . 526 10.5.3 Two Processes of the Least Squares Estimator . . . . . . . . 530 10.5.4 Least Squares Estimator with a Priori State Value . . . . . . 531 10.6 Orbit Determination by Ground-Based and Space-Based Joint Network and Autonomous Orbit Determination by Star-To-Star Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 10.6.1 Outline of Space-Based Network of Orbit Tracking and Determination . . . . . . . . . . . . . . . . . . . . . . . . 534 10.6.2 Basic Principles of the Orbit Determination of Ground-Based and Space-Based Joint Network . . . . . 534 10.6.3 The Rank Deficiency in the Autonomous Orbit Determination by Start-To-Star Measurements . . . . . . . . 536 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
- 34. xxx Contents Appendix A: Astronomical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 Appendix B: Formulas of Mean Ephemeris of Major Celestial Bodies in the Solar System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 Appendix C: Orientation Models of Major Celestial bodies in the Solar System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553
- 35. About the Author Lin Liu is a Chinese Astronomer currently a Distinguished Professor at the Astronomy and Space Science Institute, Nanjing University, China. He is an expert in Celestial Dynamics and Spacecraft Orbital Determination both in theory and application. He is regarded as the main founder of this field in China and a highly respected teacher over 50 years. His work has profound influence especially on the Chinese Aerospace Industry. He has been involved in many important Chinese Space programs such as the Shenzhou Spacecraft, the Moon Exploration, etc. Inrecentdecades,ProfessorLiuhasworkedontheOrbitalDynamicsofdeepspace exploration. Many of his initial and ground-breaking research results are connected to the Chinese Aerospace Industry. He was in charge of several Aerospace research projects and research programs of the National Natural Science Foundation of China. He was a director of the Chinese Astronomy Society, a director of the Celestial Dynamics and Satellite Dynamics sections, and a director of the Chinese Aerospace Society. Currently, he is on the editorial board of the Chinese Astronomical Journal, a member of the Academic Committee of the Deep Space Exploration Joint Center, Ministry of Education, an external expert for the National Astronomical Observa- tory of Chinese Academy of Science, and a member of the Chinese Committee of COSPAR. ProfessorLiuhasmorethan250researchpublicationsinnationalandinternational journals and 11 monographs. There is a long list of awards Professor Liu has received including the National Science Congress Major Achievement Award in 1978, the Chinese Astronomical Society Zhang Yuzhe Award, and three times of the State Educational Commission awards, to name a few. In April of 2016, for recognising Professor Liu’s scientific achievements, the International Astronomy Union (IAU) named Asteroid 261936 Liu in his honor. xxxi
- 36. Chapter 1 Selections and Transformations of Coordinate Systems The main content of orbital dynamics is about solving a dynamical problem. The first step of solving a specific dynamical problem is to select a proper spatial reference frame and a time reference system. A small body, most likely an artificial satellite or a specific spacecraft, moves in an orbit. The orbit then can be presented in a reference system, such as the Earth reference system, the Moon reference system, a planet’s reference system, and the heliocentric reference system. This chapter intro- duces these reference systems and their relationships, and the formulas for mutual transformations. According to the general relativity theory, a reference system is a 4-dimensional space–time system. In the Solar System, there are two important inertial reference systems. One system is centered at the barycenter of the Solar System, and its orien- tation is decided by remote quasars. This system is the barycenter reference system of the Solar System called the Barycentric Celestial Reference System (BCRS) and is related to all motions of celestial bodies in the Solar System. The other system is centered at Earth’s barycenter and is called the Geocentric Celestial Reference System (GCRS). This system is related to all motions around and on Earth including the observers on the ground. The fourth demission of a reference system is the time variable called the Coordinate Time, whose variation is relative to the local gravita- tional field. Therefore, there are two Coordinate times for the two reference systems, the Barycentric Coordinate Time (TCB) for BCRS, and the Geocentric Coordinate Time (TCG) for GCRS. These are theoretical definitions, in the application, there can be some changes. In this chapter, the commonly used space coordinate systems and time systems are introduced and discussed. © Nanjing University Press 2023 L. Liu, Algorithms for Satellite Orbital Dynamics, Springer Series in Astrophysics and Cosmology, https://doi.org/10.1007/978-981-19-4839-8_1 1
- 37. 2 1 Selections and Transformations of Coordinate Systems 1.1 Time Systems and Julian Day [1, 2] As mentioned above the two coordinate times are TCB and TCG, but for making ephemerides and in motion equations the time variables are measured by the Barycen- tric Dynamical Time (TDB) and the Terrestrial Time (TT) for BCRS and GCRS, respectively. The terrestrial time was called Terrestrial Dynamical Time (TDT) but changed to TT after 1991. The difference between the two time systems, TDB and TT, is caused by the effect of relativity, and the transfer relationship can be defined by the theory of gravitation. In practical applications, their relationship is given by the International Astronomy Union (IAU) in 2000 as TDB = TT + 0s .001657 sin g + 0s .000022 sin(L − LJ), (1.1) where g is the mean anomaly of Earth’s orbit around the Sun, and (L − LJ) is the difference of the Sun’s mean ecliptic longitude and Jupiter’s mean ecliptic longitude that { g = 357◦ .53 + 0◦ .98560028t, L − LJ = 246◦ .00 + 0◦ .90251792t, (1.2) t = JD(t) − 2451545.0. (1.3) In (1.3), JD(t) is the related Julian Day Number of time t, the definition of Julian Day is given in Sect. 1.1.3. Formula (1.1) is valid between 1980 and 2050, and the error is less than 30 μs (10−6 s). Near Earth’s surface with errors in the order of ms (10−3 s), there is approximately TDB = TT. (1.4) Modern space and time reference systems accept IAU 2009 Astronomy Constant System (Appendix 1). In this system, the Astronomical Unit (AU) is provided by IAU 2012 resolution and is directly related to the unit of length “meter (m)”. The value of AU is now given by 1AU = 1.49597870700 × 1011 m. (1.5) 1.1.1 Selection of Standard Time In the TT system, the time is realized by atomic time. The earliest atomic clock, which used the period of atomic oscillation as the standard to measure time, was built in 1949. In 1967, International System of Units (SI) defined the base unit of time (a
- 38. 1.1 Time Systems and Julian Day [1, 2] 3 second) as the duration of 9,192,631,770 cycles of the radiation corresponding to the transition between two energy levels of the ground state of the cesium-133 atom. In 1997, the International Committee for Weights and Measures (EIPM) added that the preceding definition refers to a cesium atom at rest at a temperature of absolute zero. The atomic time (TAI, in French Temps Atomique International) uses the SI second (s) as a unit and the universal time of 1 January 1958, 0:00:00 as the starting epoch. Since 1971, TAI is provided by the International Bureau Weights and Measures (BIPM, in French) as a weighted average of the time kept by over 400 atomic clocks in over 50 national laboratories worldwide. The only difference between TT and TAI is the starting points that TT = TAI + 32s .184. (1.6) Tothepresentday,TAIisthemostaccurateanduniformstandardtime,itsaccuracy is about 10−16 s, and its error would be less than 1 s over one billion years. 1.1.2 Time Reference Systems To study the motions of celestial bodies including spacecraft, it is necessary to have a time system with a uniform time scale. For an observatory on the surface of Earth, it is also necessary to have a time system related to Earth’s rotation. Before the atomic time became the standard time Earth’s rotation was the time basis for the two time systems. But Earth’s rotation is non-uniform and the accuracy of measurements of Earth’s rotation is continuously improving. Therefore, there is a need to build a time system that uses the uniform unit scale TAI and is also related to Earth’s rotation in a coordinative way. Sidereal Time (ST). The definition of a “sidereal day” is the time interval for the March equinox at the upper transit between two successive returns. Therefore, ST is the angle measured along the celestial equator from the observer’s meridian (at longitude λ) to the great circle that passes through the March equinox and both poles, and is given by time. Its value (S) equals the right ascension (α) of a star at the upper transit of the observatory, that S = α, (1.7) where S is the local sidereal time (LST), and the Greenwich sidereal time (SG, GST) is given by SG = S − λ. (1.8) In fact, GST is the Greenwich apparent sidereal time and is different from the Greenwich mean sidereal time (GMST). Because the sidereal time is defined by
- 39. 4 1 Selections and Transformations of Coordinate Systems Earth’s rotation, then the non-uniformness of Earth’s rotation can be measured by the difference between the sidereal time and an unformed time. Universal Time (UT). It is, like ST, a time measured according to Earth’s rotation but chooses the mean solar day as its unit, therefore one second of UT is 1/86400 mean solar day. The astronomically measured universal time, UT0, is relative to the instantaneous polar meridian. UT0 is affected by the motion of the pole, to provide the universal time with respect to the mean pole, denoted by UT1, a correction is needed that UT1 = UT0 + Δλ, (1.9) where Δλ is a correction of the polar shift. UT1isnotauniformtimescalebecauseofthenon-uniformnessofEarth’srotation. There are three types of Earth’s rotation variation. The first is the slow long-term variation (the universal day increases 1.6 ms per 100 years); the second is the periodic variation (mainly the seasonal variation, about 0.001 s in a year, and some other smaller periodic variations); and the third is the irregular variations. These variations cannot be easily corrected, only the annual variation can be given by an empirical formula based on multi-year observations. If the annual variation is denoted by ΔTs, then the adjusted universal time, UT2, is UT2 = UT1 + ΔTs. (1.10) UT2 is a relatively uniform time scale, although it still includes the long-term variation of Earth’s rotation and the irregular variations. The physical cause of the irregularity is unknown therefore there is no way to adjust. For general requirements UT1 is commonly accepted as a united time system because its periodic variation ΔTs is rather small, also it is directly related to Earth’s instantaneous position. For high precision problems, even UT2 is not precise enough, and a more uniform time scale is required, thus it is necessary to introduce the atomic time TAI as the time basis. As mentioned above, TAI is defined at 1 January 1958, 0 h, which is very close to the UA2’s starting time, the difference is (TAI − UT2)1958.0 = −0s .0039. (1.11) TAI is defined in the geocentric coordinate system and is measured by the inter- national time unit. Since 1984, the ephemeris time system (ET) has been formally replaced by TAI which became the uniform scale required by researchers in the field of Dynamics. The Terrestrial Dynamical Time (TDT), therefore, was introduced (renamed as terrestrial time TT in 1991). The epoch 1 January 1977, 0:00:00 by TAI corresponds to January 1d .0003725, 1977 by TDT. This difference equals the difference between ET and TAI at that moment. With the definition of the beginning of TT, it is easy to use the TT system to replace the ET system.
- 40. 1.1 Time Systems and Julian Day [1, 2] 5 Coordinated Universal Time (UTC). A uniform time system is appropriate for high precession ephemerides, which require a uniform scale for time intervals, but it cannot replace a time system related to the Earth’s non-uniform rotation. Thus, the coordinated universal time (UTC) is introduced for solving this problem. To the present day there are many suggestions, discussions, and arguments, but without a definite conclusion. We still keep the sidereal time system and the universal time system, as each of them has its merits. The difference between TAI and UT2 (or UT1), given in (1.11), is 0 s .0039 at the beginning of 1 January 1958, it was near zero. Because Earth’s rotation has a long-term slowness, the difference between TAI and UT2 increases then causes the problem. In order to keep UT (UT1 or UT2) as close as possible to TAI and still, use the uniform scale. In 1963, the international communities adopted a third-time system, which is the coordinated Universal Time (UTC). UTC is still a time system based on TAI but with leap seconds added at irregular intervals, such as 12 months or 18 months, to make it as close as possible to UT. Since 1972 it has been required that UTC must be kept within ±0.9 s of UT1. Actual adjustments of leap seconds are given by the International Time Bureau based on observational information, which can be found on the EOP web page. Until 1 January 2017, the adjustment is 37s , that TAI = UTC + 37s . The transformation process from UTC to UT1 is that first to download the newest EOP (Earth Orientation Parameters) data (use the B data if the time is more than a month earlier than present, and use the A data for other times), then to calculate the adjustment ΔUT by interpolation, which gives UT1 as UT1 = UTC + ΔUT. (1.12) According to the international convention, if a measurement is given at time t, the time system means UTC unless there is a special description. 1.1.3 Julian Day Besides the time system, in solving the dynamical problem we often have to choose an epoch and deal with the problem related to the length of different types of one year. In Astronomy, there are several definitions of a year. One is the Besselian year, which has a length of a tropical year, i.e., 365.2421988 mean solar days. The epoch of a Besselian year is the moment when the Sun’s mean ecliptic longitude is 280°. For example, the Besselian year 1950.0 does not mean 1 January 1950, 0:00:00 but is 31 December 1949, 22:09:42 (UT), which corresponds to the Julian day number (JDN) 2,433,282.4234. Another type of year is the Julian year, which has 365.25 mean solar days. The epoch of each Julian year is exactly the beginning of a year, for example, 1950.0 means 1 January 1950, 00:00:00. Obviously, it is easier to use
- 41. 6 1 Selections and Transformations of Coordinate Systems Table 1.1 Besselian Epoch, Julian Epoch, and Julian day number Besselian Epoch Julian Epoch Julian day number 1900.0 1900.000858 2,415,020.3135 1950.0 1949.999790 2,433,282.4234 2000.0 1999.998722 2,451,544.5333 1989.999142 1900.0 2,415,020.0 1950.000210 1950.0 2,433,282.5 2000.001278 2000.0 2,451,545.0 Julian years than Besselian years. Therefore, since 1984 Besselian year has been replaced by the Julian year. Some correspondences of the Besselian epoch, Julian epoch, and Julian day number are listed in Table 1.1. For convenience, the Modified Julian Date (MJD) is introduced and defined as MJD = JD − 2400000.5. (1.13) As an example, JD(1950.0) corresponds to MJD = 33,282.0. The lengths of a century of a Besselian year (tropical century) and a Julian year are 36,524.22 and 36,525 mean solar days, respectively. 1.2 Space Coordinate Systems [2–6] A coordinate system is actually a mathematical representation of a theoretical concept. A reference frame is the physical realization of a coordinate system, there- fore, a reference system is an integrated system of a theoretical concept and a physical frame. Although the concept of a reference system is different from that of a coor- dinate system, in the practical application of most fields, as in this book, these two systems are interchangeable without misunderstanding. To study the motions of celestial bodies in the Solar System, there are commonly accepted three types of the coordinate system, which are the horizontal coordinate system, the equatorial coordinate system, and the ecliptic coordinate system. These coordinates are applied to problems no matter from the point of view of Earth or other celestial bodies (such as the major planets or the Moon). For each space coor- dinate system, there are three key elements, the origin of the coordinate system, the fundamental plane, i.e., the xy-plane, and the primary direction (the direction of the x-axis). In this section, we introduce three coordinates with respect to Earth. Horizontal system. A proper name for this system should be the topocentric hori- zontal coordinate system. In this system the origin is at the center of an observatory (or a sampling center), the fundamental plane is the local horizontal plane containing the origin and is tangential to the ellipsoid of Earth (the horizon), and the primary direction is towards the north (N) in the xy-plane. The direction of the z-axis is towards the zenith (Z) (Fig. 1.1).
- 42. 1.2 Space Coordinate Systems [2–6] 7 Fig. 1.1 The horizontal system and the equatorial system Equatorial system. There are two equatorial systems, one is the topocentric equa- torial system with the center at the location of an observatory, and the other is the geocentric equatorial system with the center at the center of Earth. For both systems, Earth’s equatorial plane is the reference plane, but for the topocentric system the reference plane is parallel to the equatorial plane, and in the celestial sphere the two planes converge into one, therefore the two systems are related only by a translation. The primary direction of both systems is towards the March equinox ( ). Ecliptic system. There are also two ecliptic systems, the geocentric ecliptic system with its origin at the center of Earth and the heliocentric ecliptic system with its origin at the center of the Sun. For both systems, the fundamental plane is the ecliptic plane of Earth’s orbit around the Sun, and the primary direction is towards the March equinox ( ). The geometrical relationship of the horizontal system and the equatorial system is illustrated in Fig. 1.1, and that of the equatorial system and the ecliptic system in Fig. 1.2. The symbols in the figures are customarily used in Astronomy, therefore, are not explained here. The position of a celestial body in a space coordinate system can be presented by its coordinate vector. In the horizontal coordinate system, the position vector of a body is denoted by → ρ with spherical coordinates (ρ, A, h, or E), where ρ is the distance between the origin of the system and the body, A is the azimuth (do not be confused with the equator AA’) measured from the north point eastward along the horizontal circle (clockwise), and h is the altitude (i.e. the height angle E). In the equatorial coordinate system, this vector is denoted by → r with spherical coordinates (r, α, δ), where r is the same as ρ, α is the right ascension measured from the March equinox eastward along the equator (i.e., the arc D on the equator AA, ), and δ is the declination angle. In the ecliptic coordinate system, the position vector is denoted by → R with spherical coordinates (R, λ, β), where R is the same as ρ, λ is the ecliptic longitude measured from the March equinox eastward along the ecliptic, and β is the ecliptic latitude. The relationships of the coordinates are given by
- 43. 8 1 Selections and Transformations of Coordinate Systems Fig. 1.2 The equatorial system and the ecliptic system → ρ = ρ ⎛ ⎝ cos h cos A − cos h sin A sin h ⎞ ⎠, → r = r ⎛ ⎝ cos δ cos α cos δ sin α sin δ ⎞ ⎠, → R = R ⎛ ⎝ cos β cos λ cos β sin λ sin β ⎞ ⎠. (1.14) The azimuth A sometimes is measured from the south point (S) eastward along the horizontal circle (anti-clockwise), then → ρ is given by → ρ = ρ ⎛ ⎝ cos h cos A cos h sin A sin h ⎞ ⎠. (1.15) In the topocentric equatorial coordinate system and the geocentric ecliptic coordi- nate system, the position vectors of a body can be presented by → r, and → R,, respectively, and the corresponding relationship is similar to that for → r and → R, but r and R should be replaced by r, and R, , respectively, then α and δ are for the topocentric equatorial system, and λ and β for the geocentric ecliptic system. The transformation relationships between these coordinate systems are simple and only involve translations and rotations, that → r, = Rz(π − S)Ry (π 2 − ϕ ) → ρ, (1.16) → r = → r, + → rA, (1.17) → R, = Rz(ε)→ r, (1.18) → R = → R, + → RE, (1.19)
- 44. 1.2 Space Coordinate Systems [2–6] 9 where S = α + t is the hour angle of the March equinox, which equals the sidereal time at the observatory (i.e. the arc D + DA measured along the equator circle AA’ in Fig. 1.1), ϕ is the astronomical latitude of the observatory, → rA is the position vector of the observatory from the Earth’s center, ε is the obliquity, and → RE is the position vector of Earth’s center in the heliocentric coordinate system. The rotation matrices Rx, Ry, and Rz in (1.16) and (1.18) are given by Rx (θ) = ⎛ ⎝ 1 0 0 0 cos θ sin θ 0 − sinθ cos θ ⎞ ⎠, (1.20) Ry(θ) = ⎛ ⎝ cos θ 0 − sin θ 0 1 0 sin θ 0 cos θ ⎞ ⎠, (1.21) Rz(θ) = ⎛ ⎝ cos θ sin θ 0 − sin θ cos θ 0 0 0 1 ⎞ ⎠, (1.22) In the dynamics of the Solar System for studying the motions of the major planets and asteroids, we use the heliocentric ecliptic coordinate system; whereas in the dynamics of artificial satellites, we use the equatorial coordinate system centered at the barycenter of the main body, such as the geocentric equatorial system, or the Moon-centric equatorial system, or the Mars-centric equatorial system, etc. Thecoordinatesystemsusedforartificialsatellitesaremainlythegeocentricceles- tial coordinate system and the Earth-fixed equatorial coordinate system (see 1.3.3). The origins of both systems are obviously at the center of Earth, but their funda- mental planes and primary directions are affected by Earth’s precession, nutation, and polar motion, which make these space coordinate systems rather complicated. As we know that Earth is an ellipsoid with unevenly distributed mass. The gravitational forces from the Sun, the Moon, and other major planes act on Earth’s non-spherical part and produce two phenomena. One is an effect of a rigid body translation force, resulting in an indirect perturbation of Earth’s oblateness. The other is an effect of rotation torque of a rigid body due to Earth’s gyro-like motion, producing precession and nutation. Because of precession and nutation, Earth’s equatorial plane vibrates. Besides the two effects, Earth’s internal motion and motions on the surface produce a slow shift of the rotational axis, i.e., the polar motion, which also influences the selection of the coordinate system. The equators of Mars and the Moon have similar variations. As a result, there are different types of equatorial coordinate systems. The properties of equatorial coordinate systems with respect to Earth, the Moon, and Mars, and the transformation between these systems are discussed in the following sections.
- 45. 10 1 Selections and Transformations of Coordinate Systems 1.3 Earth’s Coordinate Systems [2, 6–10] 1.3.1 The Realization of the Dynamical Reference System and J2000.0 Mean Equatorial Reference System Current observational measurements, such as Ephemerides of the planets in the Solar System, are all provided in the International Celestial Reference System (ICRS). The realization of this reference system is a reference frame called International Celestial Reference Frame (ICRF). The origin of ICRS is at the barycenter of the Solar System, the fundamental plane and the primary direction of X-axis are decided by precise observations of a group of extragalactic radio sources to be as close as possible to the J2000.0 mean equatorial plane and the mean March equinox, respectively. Here J2000.0 means the Julia year 2000, 1 January 12:00:00. Because the radio sources are so distant, they are stationary to our technology, the coordinate system and its orientation of ICRF are relatively fixed in space, and free of the dynamics of the Solar System and Earth’s precession and nutation, also unrelated to the traditional concepts of the equator, March equinox, and ecliptic. As a result, this system is closer to an inertia reference system than any other system. Before the existence of ICRS and ICRF, the basic astronomy reference system is the Fifth Fundamental Catalog dynamic system (FK5) (strictly speaking, this is a reference system dynamically defined and includes the correction of sidereal kinematics). FK5 is built up on observations of bright stars and the IAU 1976 Astro- nomic constants. Its fundamental plane is the J2000.0 mean equatorial plane, and the direction of X-axis points to J2000.0 mean March equinox. Obviously, this system is relatedtotheepoch. Thepresent systemICRSis animprovement of FK5. Thedynam- ical reference system is the J2000.0 mean equatorial reference system, usually called J2000.0 mean equatorial coordinate system. Specifically, the fundamental plane and the primary direction of the X-axis of ICRF are realized by observations of the Very-Long-Baseline Interferometry (VLBI) from hundreds of extragalactic radio sources. The deviation of its pole from the pole of the dynamical reference system FK5 is only about 20 milliarcseconds. In order to keep the continuity of the reference system, the fundamental plane and the primary direction of ICRF are kept as close as possible to these of FK5, which are the J2000.0 mean equatorial plane and the J2000.0 mean March equinox. The origin of ICRS (or the zero point, its definition is given in Sect. 1.3.2) is chosen to be the average right ascension of 23 radio sources thus being close to that of FK5. The relationship of the dynamical reference systems, ICRS and FK5, depends on three parameters, which are the deviations of the celestial pole, ξ0 and η0, and the zero right ascension deviation dα0. Their values are ⎧ ⎨ ⎩ ξ0 = −0,, .016617 ± 0,, .000010, η0 = −0,, .006819 ± 0,, .000010, dα0 = −0,, .0146 ± 0,, .0005. (1.23)
- 46. 1.3 Earth’s Coordinate Systems [2, 6–10] 11 The relationship of ICRS and the J2000.0 mean equatorial coordinate system can be given by { → rJ2000.0 = B→ rICRS, B = Rx (−η0)Ry(ξ0)Rz(dα0), (1.24) where → rJ2000.0 and → rICRS are for the same vector but in different coordinate systems, the constant matrix B is the deviation matrix of reference frame composed of the three small rotation angles. The J2000.0 mean equatorial coordinate system is the commonly accepted geocentric celestial coordinate reference system (GCRS) in present Aerospace Dynamics(especiallyforEarth’ssatellites).Ifunnecessary,theabove-givendeviation matrix of the reference frame is not mentioned again. 1.3.2 The Intermediate Equator and Three Related Datum Points The intermediate equator is introduced to better describe the relationship between the Celestial Reference System (CRS) and the Terrestrial Reference System. The celestial axis is the extension of Earth’s rotation axis, the points of intersection of the celestial axis and the celestial sphere are called celestial poles. Because of Earth’s precession, the direction of Earth’s rotation axis changes over time in CRS, which is instantaneous, therefore the celestial pole and the celestial equator are also instantaneous. For clarity IAU 2003 named the instantaneous celestial pole and the celestial equator as the Celestial Intermediate Pole (CIP) and the Intermediate Equator, respectively. In order to take measurements in the celestial reference system, it is necessary to select a fixed point with respect to the celestial reference system on the intermediate equator as the origin, which is called the Celestial Intermediate Origin (CIO). Similarly, in the terrestrial system a point, which is fixed with respect to the system, is needed and called the Terrestrial Intermediate Origin (TIO). CIO is decided based on observations of a group of quasars and is close to the 0° right ascension, i.e., the March equinox on the International Celestial Reference Frame; whereas TIO is decided by a group of observatories on Earth, and is near the 0° longitude (the prime meridian, i.e., the Greenwich meridian) on the International Terrestrial Reference Frame. In Fig. 1.3 the intermediate equator is given by the circle, E is Earth’s barycenter, and is the March equinox. In the celestial reference system, the intermediate equator tied with CIO is called the Celestial Intermediate Equator, TIO moves along the equator anti-clockwise, its period is a sidereal day. In the terrestrial reference system, the intermediate equator tied with TIO is called the Terrestrial Intermediate Equator, CIO moves along the equatoroverthesameperiodofasiderealdaybutclockwise.Bothobservationsreflect
- 47. 12 1 Selections and Transformations of Coordinate Systems Fig. 1.3 Illustration of the intermediate equator Earth’s rotation, and the angle between CIO and TIO is called Earth Rotation Angle (ERA). 1.3.3 Three Geocentric Coordinate Systems (1) The geocentric celestial coordinate system O-xyz This system is actually the above-mentioned epoch J2000.0 mean equatorial refer- ence system, also called the geocentric celestial coordinate system. Its origin is Earth’s barycenter, the xy-plane is the epoch J2000.0 mean equatorial plane, the direction of x-axis points to the epoch J2000.0 mean March equinox , which is the intersection of the epoch J2000.0 mean equator and the epoch J2000.0 instantaneous ecliptic. This system, in a certain sense, is a “fixed system” (because it eliminates the rotation of the frame caused by the vibration of Earth’s equator), thus the motion orbits of a celestial body (such as a satellite) at different times can be displayed in the same frame and the actual variation of the orbit can be compared. The geocen- tric celestial system is the adopted space coordinate system worldwide. It should be noticed that in this system the gravitation potential due to Earth’s non-spherical part is variable. (2) The Earth-fixed geocentric coordinate system O-XYZ This system is the Terrestrial Reference System (TRS), which is a space reference system rotating with Earth, commonly called the Earth-fixed coordinate system. In this system, the position of an observatory is fixed on the surface of Earth, except for some minor variations due to the tidal force or Earth’s physical deformation force.
- 48. 1.3 Earth’s Coordinate Systems [2, 6–10] 13 As mentioned above that the realization of ICRS requires ICRF. It is the same that the Terrestrial Reference Frame (TRF) is needed for the realization of TRS. TRF (used in navigation, survey, terrestrial physics, etc.) is defined by a group of fixed points on Earth’s surface, whose positions are precisely determined in TRS. The first TRF is given by the International Latitude Service. Based on the observations over five years, 1900–1905, the International Latitude Service defined the Conventional International Origin (CIO), which was the average direction of the third axis (z- axis), i.e., the mean direction of Earth’s pole. It should be noticed that nowadays the abbreviation CIO is given to the Celestial Intermediate Origin (see Sect. 1.3.2), so is no longer for the Conventional International Origin. In the Earth-fixed coordinate system, the origin of the frame is at the center of Earth, the xy-plane is close to the 1900.0 mean equatorial plane, and the direction of the x-axis points to the intersection of the Greenwich meridian and the equator, so can be called the Greenwich meridian direction [2]. Several Earth’s gravitational models and the related reference ellipsoid are defined in this system, thus these models are self-consistent. If there is no specific explanation the Earth-fixed system in this book agrees with the World Geodetic System 84 (also known as WGS 1984). For this system there are GE = 398600.4418 ( km3 /s2 ) , ae = 6378.137(km), 1 f = 298.257223563, (1.25) where GE is the geocentric gravitational constant, ae and f are the equatorial radius and the flattening factor of the reference ellipsoid, respectively. In the Earth-fixed frame, the position vector of an observatory is given by → Re(H, λ, ϕ). For the position vector, the relationship between the rectangular coordinates (Xe, Ye, Ze) and the spherical coordinates (H, λ, ϕ) is given by ⎧ ⎨ ⎩ Xe = (N + H) cos φ cos λ, Ye = (N + H) cos φ sin λ, Ze = [N ( 1 − f )2 + H ] sin φ, (1.26) with N = ae [ cos2 ϕ + (1 − f )2 sin2 ϕ ]− 1 2 = ae [ 1 − 2 f ( 1 − f 2 ) sin2 ϕ ]− 1 2 , (1.27) where ae and f are given in (1.25). The spherical coordinate H is the geodetic height of the observatory, and λ and ϕ are the geodetic longitude and latitude of the observatory, respectively. Their relationships with the rectangular coordinates are
- 49. 14 1 Selections and Transformations of Coordinate Systems given by tan λ = Ye Xe , sin2 ϕ = Ze [N(1− f )2 +H] . (1.28) (3) Geocentric ecliptic coordinate system O-x, y, z, The origin of this system is also Earth’s barycenter, and there is only a translation relationship between this system and the heliocentric ecliptic system. The x, y, -plane is the epoch J2000.0 ecliptic plane, and the direction of the x, -axis is the same as in the celestial coordinate system O-xyz, which points to the epoch J2000.0 mean March equinox. 1.3.4 Transformation of the Earth-Fixed Coordinate System O-XYZ and the Geocentric Celestial Coordinate System O-xyz 1.3.4.1 Transformation Relationship (I) by the IAU 1980 Model Using → r and → R as the position vectors of a spacecraft in the geocentric celestial system O-xyz and the Earth-fixed system O-XYZ, respectively, then the transformation relationship is given by → R = (HG)→ r. (1.29) The coordinate transformation matrix (HG) is given by four rotating matrices, that (HG) = (E P)(E R)(N R)(P R), (1.30) where (PR) is the precession matrix, (NR) the nutation matrix, (ER) the Earth’s rotation matrix, and (EP) the polar motion matrix, given by (E P) = Ry ( −xp ) Rx ( −yp ) , (1.31) (E R) = Rz(SG), (1.32) (N R) = Rx (−Δε)Ry(Δθ)Rz(−Δμ) = Rx (−(ε + Δε))Rz(−Δψ)Rx (ε), (1.33) (P R) = Rz(−zA)Ry(θA)Rz(−ζA), (1.34)
- 50. 1.3 Earth’s Coordinate Systems [2, 6–10] 15 In (1.31) xp and yp are the components of the polar shift vector. The Greenwich sidereal time SG in (1.32) is given by SG = SG + Δμ, (1.35) where Δμ is the nutation of the right ascension; SG is the J2000.0 Greenwich mean sidereal time given by SG = 18h .697374558 + 879000h .051336907T + 0s .093104T 2 , (1.36) T = 1 36525.0 [JD(t) − JD(J2000.0)]. (1.37) In these two formulas, t is the UT1 time, but for calculating other variables such as precession and nutation, t is the TDT time. Time T is measured from J2000.0 but uses a century as a unit. The precession constants in (3.14) ζA, θA, and zA are given by ⎧ ⎨ ⎩ ζA = 2306 ,, .2181T + 0 ,, .30188T 2 , θA = 2004 ,, .3109T − 0 ,, .42665T 2 , zA = 2306 ,, .2181T + 1 ,, .09468T 2 , (1.38) where θA is the precession in declination, and μ (or mA) is the precession in right ascension that μ = ζA + zA = 4612,, .4362 T + 1,, .39656 T 2 . (1.39) In (1.33) ε is the mean obliquity. The nutation components in ecliptic longitude Δψ and in obliquity Δε can be calculated using the sequences provided by the IAU 1980 model, which has 106 terms with amplitudes greater than 0,, .0001. For the requirement of general orbital accuracy, only the terms with amplitudes greater than 0,, .005 need to be included, which are the first 20 terms. Because these terms are periodic (the shortest period term is due to the Moon’s motion), so there is no accumulative effect, and the error caused by the terms with amplitude less than 0,, .005 is equivalent to the order of meter in ground-based positioning, and it is less than 0 s .001 with respect to time. The first 20 terms are: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Δψ = 20 Σ j=1 ( A0 j + A1 j t ) sin ( 5 Σ i=1 kji αi (t) ) , Δε = 20 Σ j=1 ( B0 j + B1 j t ) cos ( 5 Σ i=1 kji αi (t) ) , (1.40)
- 51. 16 1 Selections and Transformations of Coordinate Systems where the components of nutation in right ascension and in inclination, Δμ and Δθ, respectively, are given by { Δμ = Δψcosε, Δθ = Δψsinε, (1.41) and the value of the obliquity ε is given by ε = 23◦ 26, 21,, .448 − 46,, .8150 t. (1.42) In (1.40) there are five basic arguments αi (i = 1, · · · , 5) related to the positions of the Sun and the Moon, which are given by: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ α1 = 134◦ 57, 46,, .733 + (1325r + 198◦ 52, 02,, .633)t + 31,, .310t2 , α2 = 357◦ 31, 39,, .804 + (99r + 359◦ 03, 01,, .224)t − 0,, .577t2 , α3 = 93◦ 16, 18,, .877 + (1342r + 82◦ 01, 03,, .137)t − 13,, .257t2 , α4 = 297◦ 51, 01,, .307 + (1236r + 307◦ 06, 41,, .328)t − 6,, .891t2 , α5 = 125◦ 02, 40,, .280 − (5r + 134◦ 08, 10,, .539)t + 7,, .455t2 , (1.43) where 1r = 360°. The first 20 terms of the nutation sequences are listed in Table 1.2. To reach the above-mentioned accuracy of the order of meter, the terms on the right side of (1.40) only A11 and B11 in Table 1.2 are needed, other terms of A1 j and B1 j can be omitted. Specifically, the number of terms needed depends on not only the required accuracy but also the capability of the software, such as the factor of functional expansion. The time t in (1.40)–(1.43) is the same as T, the century number given by (1.37), but is in TDT. The formulas for calculating the rotational matrices Rx(θ), Ry(θ), and Rz(θ) are given by (1.20)–(1.22). Note that they are orthogonal matrices, that RT x (θ) = R−1 x (θ) = Rx (−θ), · · · . 1.3.4.2 Transformation Relationship (II) by the IAU 2000 Model By the IAU 2000 model, the transformation from the geocentric celestial reference system (GCRS) to the International Terrestrial Reference System (ITRS) is given by [ITRS] = W(t)R(t)M(t)[GCRS], (1.44) where [GCRS] and [ITRS] correspond to the geocentric celestial coordinate system and the Earth-fixed coordinate system by the IAU 1980 model, respectively. Using → r and → R for the position vectors of a spacecraft in the two systems, respectively (the symbols are used in Sect. 1.3.4.1, for consistency we use the same symbols here), then the transformation relationship is given by
- 52. 1.3 Earth’s Coordinate Systems [2, 6–10] 17 Table 1.2 The first 20 terms of the IAU 1980 Nutation sequence j Period kj1 kj2 kj3 kj4 kj5 A0j A1j B0j B1j (d) (0"0.0001) (0"0.0001) 1 6798.4 0 0 0 0 1 −171,996 −174.2 92,025 8.9 2 182.6 0 0 2 −2 2 −13,187 −1.6 5736 −3.1 3 13.7 0 0 2 0 2 −2274 −0.2 977 −0.5 4 3399.2 0 0 0 0 2 2062 0.2 − 895 0.5 5 365.2 0 1 0 0 0 1426 −3.4 54 −0.1 6 27.6 1 0 0 0 0 712 0.1 −7 0.0 7 121.7 0 1 2 −2 2 −517 1.2 224 −0.6 8 13.6 0 0 2 0 1 −386 −0.4 200 0.0 9 9.1 1 0 2 0 2 −301 0.0 129 −0.1 10 365.3 0 −1 2 −2 2 217 − 0.5 −95 0.3 11 31.8 1 0 0 −2 0 − 158 0.0 −1 0.0 12 177.8 0 0 2 −2 1 129 0.1 −70 0.0 13 27.1 −1 0 2 0 2 123 0.0 −53 0.0 14 27.7 1 0 0 0 1 63 0.1 −33 0.0 15 14.8 0 0 0 2 0 63 0.0 −2 0.0 16 9.6 −1 0 2 2 2 −59 0.0 26 0.0 17 27.4 −1 0 0 0 1 −58 − 0.1 32 0.0 18 9.1 1 0 2 0 1 −51 0.0 27 0.0 19 205.9 2 0 0 −2 0 48 0.0 1 0.0 20 1305.5 −2 0 2 0 1 46 0.0 −24 0.0 → R = W(t)R(t)M(t)→ r, (1.45) where M(t) is the precession and nutation matrix, R(t) is the Earth rotation matrix, and W(t) is the polar shift matrix. Based on the transformation relationship of the March equinox the matrix M(t) can be written as M(t) = N(t)P(t)B, (1.46) where N(t) is the nutation matrix, P(t) is the precession matrix, and B is the devi- ation matrix of the reference frame defined in (1.24), which is a small constant matrix. When the J2000.0 mean equatorial coordinate system is directly used as the geocentric celestial coordinate system, the effect of B can be omitted then M(t) = N(t)P(t). (1.47) The calculation methods for these matrices are given as follows.
- 53. 18 1 Selections and Transformations of Coordinate Systems (1) Calculations of precession and nutation The 24th IAU general assembly (August 2000, Manchester) decided that from 1 January 2003, the IAU 2000 Precession-Nutation model formally replaces the IAU 1976 Precession Model and the IAU 1980 Nutation Model. For different accuracy requirements, the IAU 2000 model includes two versions, IAU 2000A and IAU 2000B with accuracies of 0.2 mas (milliarcsecond) and 1 mas, respectively. In calculating the precession at a given epoch measured from J2000.0 the required three equatorial precession quantities ξA, zA, and θA for transforming mean equatorial coordinate systems are given by ζA = 2,, .650545 + 2306,, .083227t + 0,, .2988499t2 + 0,, .01801828t3 −0,, .000005971t4 − 0,, .0000003173t5 , θA = 2004,, .191903t − 0,, .4294934t2 − 0,, .04182264t3 −0,, .000007089t4 − 0,, .0000001274t5 , zA = 2,, .650545 + 2306,, .077181t + 1,, .0927348t2 + 0,, .01826837t3 −0,, .000028596t4 − 0,, .0000002904t5 , (1.48) where t is the Julian century number of the epoch time measured from J2000.0 (TT time) that t = (JD(TT) − 2452545.0)/36525. (1.49) The nutation components in ecliptic longitude ΔΨ and in obliquity Δε can be calculated by the IAU 2000 model as Δψ = Δψp + 77 Σ i=1 ( Ai + A, i t ) sin(αi ) + ( A,, i + A,,, i t ) cos(αi ), Δε = Δεp + 77 Σ i=1 ( Bi + B, i t ) cos(αi ) + ( B,, i + B,,, i t ) sin(αi ), (1.50) where t is the same as given by (1.49), ΔΨ p and Δεp are long period variations of nutation, that Δψp = −0,, .135 × 10−3 , Δεp = 0,, .388 × 10−3 . (1.51) The arguments αi in (1.50) is a linear combination of five basic arguments, that αi = 5 Σ k=1 nik Fk = ni1l + ni2l, + ni3 F + ni4 D + ni5Ω, (1.52)
- 54. 1.3 Earth’s Coordinate Systems [2, 6–10] 19 where nik are integers, and Fk are the five basic arguments related to the positions of the Sun and the Moon given by F1 ≡ l = 134◦ .96340251 + 1717915923,, .2178t + 31,, .8792t2 +0,, .051635t3 − 0,, .00024470t4 , (1.53) F2 ≡ l, = 357◦ .52910918 + 129596581,, .0481t − 0,, .5532t2 +0,, .000136t3 − 0,, .00001149t4 , (1.54) F3 ≡ F = 93◦ .27209062 + 1739527262,, .8478t − 12,, .7512t2 −0,, .001037t3 + 0,, .00000417t4 , (1.55) F4 ≡ D = 297◦ .85019547 + 1602961601,, .2090t − 6,, .3706t2 +0,, .006593t3 − 0,, .00003169t4 , (1.56) F5 ≡ Ω = 125◦ .04455501 − 6962890,, .5431t + 7,, .4722t2 +0,, .007702t3 − 0,, .00005939t4 , (1.57) The five basic arguments are defined as: F1 the Moon’s mean anomaly, F2 the Sun’s mean anomaly, F3 the angular distance of the Moon’s mean ascending node, F4 the mean angular distance between the Sun and the Moon, and F5 the mean ecliptic longitude of the Moon’s ascending node. Actually, Fk(k = 1, . . . , 5) are the same five basic arguments αi (i = 1, . . . , 5) given by (1.43). Table 1.3 gives the coefficients of the first 20 terms of the nutation sequence by the IAU 2000B model for comparing with the IAU 1980 model in Table 1.2. (2) Calculations of the precession matrix P(t) ➀ The classical formula for three rotations of P(t) P(t) = Rz(ζA)Ry(−θA)Rz(zA), (1.58) where the three rotation angles are given by (1.48). ➁ The formula for four rotations of P(t) P(t) = Rx (−ε0)Rz(ψA)Rx (ωA)Rz(−χA), (1.59) where the last three rotation angles are given as ψA = 5038,, .481507t − 1,, .0790069t2 − 0,, .00114045t3 +0,, .000132851t4 − 0,, .0000000951t5 ωA = ε0 − 0,, .025754t + 0,, .0512623t2 − 0,, .007725036t3 −0,, .000000467t4 + 0,, .0000003337t5 χA = 10,, .556403t − 2,, .3814292t2 − 0.00121197t3 +0,, .000170663t4 − 0,, .0000000560t5 (1.60) and ε0 is given by (1.63), and t in (1.60) is given by (1.49).
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- 56. peseta y sigue tu camino; pero cuidado con decir a nadie nada de nosotros, porque si no, ¡carracho!...» Descargó el trabuco por encima de mi cabeza, y tan cerca que durante un segundo me tuve por muerto. Luego, dando una gran voz, salieron al galope; sus caballos saltaban por los barrancos como si estuvieran poseídos de los demonios. Yo.—¿Qué le ocurrió a usted al llegar a La Coruña? Benedicto.—Al llegar a La Coruña pregunté por usted, lieber Herr, y me dijeron que precisamente el día anterior se había marchado usted a Oviedo; al oirlo se me heló el corazón, viéndome en el extremo más remoto de Galicia sin un amigo que me socorriera. Estuve un día o dos sin saber qué hacer; al fin resolví dirigirme a la frontera de Francia, pasando por Oviedo, donde esperaba verle a usted y pedirle consejo. Mendigué entre los alemanes establecidos en La Coruña un socorro para el camino, y saqué muy poco, sólo unos cuartos, menos de lo que los facciosos me dieron en el camino de Santiago; con eso salí para Asturias por el camino de Mondoñedo. Och, qué ciudad, ¡Mondoñedo!, llena de canónigos, de curas, de pfaffen, más carlistas todos que el propio don Carlos. »Un día fuí al palacio del obispo y hablé con él, diciéndole que volvía de una peregrinación a Santiago y le pedí un socorro. Díjome que no podía remediarme, y en cuanto a lo de ser peregrino de Santiago se holgó mucho de ello, esperando que fuese de gran provecho para mi alma. Salí de Mondoñedo y me metí por las montañas, pidiendo limosna a la puerta de cada choza que encontraba; decía a todos que era un peregrino procedente de Santiago, y mostraba mi pasaporte en prueba de que había estado allí. Lieber Herr, nadie me dió un cuarto, ni siquiera un pedazo de broa; gallegos y
- 57. asturianos se reían de Santiago y me dijeron que el nombre del santo no era ya un talismán en España. Me hubiera muerto de hambre a no ser porque de vez en cuando arrancaba una o dos mazorcas de algún maizal; también cogía tal cual racimo de las parras y moras de zarza; de este modo fuí tirando hasta llegar a las bellotas; allí encontré un cabrito perdido, lo maté y me comí un pedazo, crudo y todo, porque el hambre era mucha; me sentó muy mal, y estuve dos días postrado en un barranco, medio muerto, incapaz de valerme; fué una gran suerte que no me devorasen los lobos. Después, a campo traviesa, seguí a Oviedo; no sé cómo he llegado; parecía un espectro. La noche pasada dormí en una pocilga vacía, a unas dos leguas de aquí, y antes de abandonarla me hinqué de rodillas y pedí a Dios que me permitiese encontrarle a usted, lieber Herr, porque usted era mi última esperanza. Yo.—¿Y qué piensa usted hacer ahora? Benedicto.—¿Qué quiere usted que le diga, lieber Herr? No sé qué hacer. Me someto en todo a sus consejos. Yo.—Estaré en Oviedo unos pocos días más; durante ellos, puede usted alojarse en esta posada, y trate de recobrarse de las fatigas de tan desastrosos viajes; quizás antes de marcharme se me ocurra algún plan para sacarle a usted de esta situación tan apurada. Oviedo tiene unos quince mil habitantes. Está en una situación pintoresca, entre dos montañas: el Morcín y el Naranco; la primera es muy alta y escabrosa; durante la mayor parte del año se halla cubierta de nieve; las vertientes de la otra están cultivadas y plantadas de viñedo. El ornamento principal de la ciudad es la catedral; su torre, extremadamente alta, es
- 58. quizás uno de los más puros ejemplares de la arquitectura gótica que existen hoy en día. El interior de la catedral es decente y apropiado; pero muy sencillo y sin adornos. Sólo vi un cuadro: la Conversión de San Pablo. Una de las capillas es cementerio, donde descansan los huesos de once reyes godos. ¡Paz a sus almas! En La Coruña me habían dado una carta de recomendación para un comerciante de Oviedo, el cual me recibió con gran cortesía, y dedicó, por lo general, un rato todos los días a enseñarme las cosas notables de Oviedo. Una mañana me dijo: —Usted habrá oído, sin duda, hablar de Feijóo, el famoso filósofo benedictino, cuyos escritos han contribuido mucho a disipar las supersticiones y los errores populares, tanto tiempo acreditados en España; está enterrado en uno de los conventos de Oviedo, donde pasó gran parte de su vida. Venga usted conmigo y le enseñaré su retrato. Nuestro gran rey Carlos III envió desde Madrid a su pintor para que lo hiciera. Ahora pertenece a mi amigo el abogado don Ramón Valdés. Fuimos a casa de don Ramón Valdés, quien, muy cortésmente, me enseñó el retrato de Feijóo, de forma circular, como de un pie de diámetro, rodeado de un pequeño bastidor de cobre, algo así como el borde de una bacía de barbero. Tenía el semblante ancho y grueso, pero correcto; arqueadas las cejas, los ojos vivos y penetrantes, la nariz aguileña. Llevaba en la cabeza un gorro de seda; el cuello de la túnica apenas llegaba a verse. Era, sin duda, un cuadro bueno, y me llamó mucho la atención, como uno de los mejores ejemplares del moderno arte español que había visto hasta entonces.
- 59. Uno o dos días después dije a Benedicto Mol:— Mañana me voy a Santander. Es hora ya de que resuelva usted lo que ha de hacer: o volverse a Madrid o dirigirse rápidamente a Francia, y desde allí continuar hacia su país. —Lieber Herr—dijo Benedicto—, iré detrás de usted a Santander en jornadas cortas, porque en un país tan montañoso no puedo andar mucho; una vez allí, acaso encuentre medio de ir a Francia. En estos viajes tan horribles me sirve de mucho consuelo pensar que voy siguiendo las huellas de usted y la esperanza de alcanzarle de nuevo. Esta esperanza me salvó la vida en las bellotas, y sin eso no hubiera llegado jamás a Oviedo. Saldré de España lo antes posible y me iré a Lucerna, aunque es fuerte cosa dejar detrás de mí el Schatz en la tierra de los gallegos. Al separarnos le regalé unos pocos duros. —Benedicto es un hombre extraño—me dijo Antonio a la mañana siguiente, cuando, acompañados por un guía, salimos de Oviedo—. Es un hombre extraño, mon maître, el tal Benedicto. Ha llevado una vida extraña y le espera una muerte extraña también: lo lleva escrito en el rostro. No creo que se marche de España, y si se marcha será para volver, porque está embrujado con el tesoro. Anoche envió a buscar una sorcière, y delante de mí la consultó; le dijo que estaba destinado a encontrar el tesoro, pero que antes tenía que cruzar agua. Le puso en guardia contra un enemigo, que Benedicto supone que será el canónigo de Santiago. He oído hablar mucho del ansia de dinero de los suizos; este hombre es una prueba. Por todos los tesoros de España no sufriría yo lo que Benedicto ha sufrido en estos últimos viajes.
- 61. S CAPÍTULO XXXIV Salida de Oviedo.— Villaviciosa.— El joven de la posada. — La narración de Antonio.— El general y su familia. — Noticias deplorables.— Mañana moriremos.— San Vicente.— Santander.— Una arenga.— El irlandés Flinter. alimos, pues, de Oviedo e hicimos rumbo a Santander. El guía que llevábamos, y a quien había yo alquilado la jaca que montaba, nos lo recomendó mi amigo el comerciante de Oviedo. Resultó ser un individuo desidioso e indolente; iba, por lo general, doscientas o trescientas varas rezagado de nosotros, y en lugar de alegrarnos el camino con cantares y cuentos, como Martín de Ribadeo, apenas abrió los labios, salvo para decirnos que no fuésemos tan de prisa, o que le iba a reventar la jaca si le daba tantos espolazos. Además era ladrón, y aunque se ajustó para hacer el viaje a seco, o sea corriendo de su cuenta sus gastos personales y los del caballo, se las arregló de modo que, durante todo el viaje, unos y otros pesaron sobre mí. Cuando se viaja por España, el plan más barato es que en el ajuste entre la manutención del guía y de su caballo o mula, porque así el precio del alquiler disminuye lo menos un tercio, y las cuentas en el camino rara vez suben más por eso; mientras que, en
- 62. otro caso, el guía se embolsa la diferencia, y, no obstante, queda libre de su escote a expensas del viajero, gracias a la connivencia de los posaderos, unidos a los guías por una especie de espíritu de cuerpo. Entrada la tarde llegamos a Villaviciosa, ciudad pequeña y sucia, a ocho leguas de Oviedo, al borde de una ensenada que comunica con el golfo de Vizcaya. Suele llamarse a Villaviciosa la capital de las avellanas por la inmensa cantidad de ese fruto que se cosecha en su término; la mayor parte se exporta a Inglaterra. Al acercarnos al pueblo, dábamos alcance a numerosos carros de avellanas que llevaban la misma dirección que nosotros. Me dijeron que en la rada había anclados algunos barcos ingleses. Por extraño que parezca, y a pesar de hallarnos en la capital de las avellanas, nos fué muy difícil procurarnos un puñado de ellas para postre, y más de la mitad de las que nos dieron estaban hueras. Los de la posada nos dijeron que como las avellanas eran para la exportación, no se les ocurría siquiera comerlas ni ofrecérselas a los huéspedes. Al día siguiente llegamos muy temprano a Colunga, lindo pueblecito, situado en una elevación del terreno, entre frondosos castañares. El pueblo es famoso, al menos en Asturias, por ser cuna de Argüelles, padre de la Constitución española. Al desmontar a la puerta de la posada, donde pensábamos reparar las fuerzas, una persona, asomada a una ventana del piso alto, lanzó una exclamación y desapareció. Estábamos todavía en la puerta, cuando el mismo individuo llegó corriendo y se arrojó al cuello de Antonio. Era un joven bien parecido, de unos veinticinco años, vestido con elegancia y tocado con una gorra de montero. Antonio, después de mirarle un momento,
- 63. exclamó: Ah, monsieur, est ce bien vous?, y le dió un afectuoso apretón de manos. El desconocido le hizo señas de que le siguiera, y en el acto se fueron los dos al aposento de encima. Preguntándome lo que podría significar aquello, me senté a almorzar. Pasó una hora, y Antonio no volvía. Por entre las tablas que formaban el techo de la cocina, oía yo su voz y la de su amigo, y me parecía oír a veces sollozos entrecortados y gemidos. Hubo después un largo silencio. Ya empezaba a impacientarme e iba a llamar a Antonio, cuando el hombre se presentó; pero no le acompañaba el desconocido. —Sepamos, por todas las extravagancias de este mundo—pregunté—¿qué ha estado usted haciendo por ahí? ¿Quién es ese hombre? —Mon maître—dijo Antonio—, c’est un monsieur de ma connaissance. Con su permiso, voy a tomar un bocado, y por el camino le contaré a usted lo que sé de él. —Monsieur—dijo Antonio cuando cabalgábamos ya fuera de Colunga—, está usted impaciente por saber la historia de ese caballero a quien ha visto usted abrazarme en la posada. Sepa usted, mon maître, que estas guerras de carlistas y cristinos han causado muchas miserias y desventuras en este país; pero no creo que haya en toda España persona tan plenamente desdichada como ese pobre y joven caballero de la posada; todas sus desventuras provienen del espíritu de partido y de facción que en estos últimos tiempos prevalecía tanto. »Mon maître, como le he dicho a usted repetidas veces, he vivido en muchas casas y servido a muchos
- 64. amos; sucedió que hará unos diez años entré a servir al padre de ese caballero, muy niño entonces. La familia estaba en muy buena posición; el padre era general del ejército y bastante rico. Constituían la familia el padre, su señora y dos hijos; el más joven es el que usted ha visto; el otro le llevaba unos cuantos años. ¡Par Dieu! En aquella casa lo pasé muy bien; todos los individuos de la familia me trataban con bondad. De muchas casas me han despedido; pero de aquella, no; cosa notable. Las tres veces que me salí fué por mi libre voluntad. Me enfadaba con los otros criados, o con el perro o el gato. La última vez me fuí por culpa de una codorniz colgada en la ventana de madame, y que me despertaba todas las mañanas con su canto. Eh bien, mon maître, así corrieron las cosas durante los tres años que, con tales alternativas, estuve al servicio de la familia; al cabo de ese tiempo, decidieron que el señorito más joven se fuese a viajar, y se pensó que yo le acompañase como criado. Tenía yo muy buenas ganas de irme con él; mas, par malheur, me encontraba por aquellos días muy disgustado con madame, su madre, por causa de la codorniz, e insistí en que antes de acompañar al señorito matarían al pájaro y lo echarían al puchero. Madame se negó a esto de modo terminante; y hasta el pobre señorito, que siempre se había puesto de mi parte en tales ocasiones, dijo que eso era una extravagancia; me fuí de la casa muy amoscado, y no volví más. »Eh bien, mon maître, el señorito se fué a viajar y estuvo fuera varios años; desde su partida hasta que le he encontrado en Colunga, no había vuelto a verle ni oído hablar de él; pero sí tenía noticias de su familia: de monsieur, su padre; de madame, su madre, y de su hermano, oficial de caballería. Poco antes de la guerra civil, o sea antes de morir Fernando VII, monsieur,
- 65. padre de este joven, fué nombrado capitán general de La Coruña. Aunque muy buen amo, monsieur era bastante orgulloso, amigo de la disciplina, de la obediencia y de todas esas cosas. Además, no era amigo del populacho, de la canaille, y profesaba singular aversión a los nacionales. Por esto, al morir Fernando, se susurraba en La Coruña que el general no era liberal, y que era más amigo de Carlos que de Cristina. Eh bien: aconteció que un día se celebraba en la bahía una gran fête en la que tomaban parte los soldados y los nacionales; yo no sé cómo sucedió; el caso es que hubo una émeute, y los nacionales echaron mano a monsieur, el general, le ataron una cuerda al cuello, le zambulleron en el agua desde la falúa en que iba, y lo llevaron a remolque hasta que se ahogó. Entonces fueron a su casa, la saquearon, y maltrataron de tal modo a madame, que por entonces estaba enceinte, que a las pocas horas expiró. »Le digo a usted, mon maître, aunque le cueste trabajo creerlo, que al saber la desgracia de madame y del general, lloré por ellos, y sentí haberme despedido de la casa airadamente, por causa de la maldita codorniz. »Eh bien, mon maître, nous poursuivrons notre histoire. El hijo mayor, oficial de caballería, como le he dicho, y hombre enérgico, en cuanto supo la muerte de sus padres juró vengarse. ¡Pobre infeliz! No se le ocurrió más que desertar con dos o tres camaradas descontentos, y, metiéndose en Galicia, levantaron una pequeña facción y proclamaron a don Carlos. Por un poco de tiempo hicieron mucho daño a los liberales, quemando y arrasando sus propiedades, y dieron muerte a varios nacionales que cayeron en sus manos.
- 66. Pero esto duró poco; su facción fué dispersada y el jefe preso y ahorcado, y su cabeza clavada en un palo. »Nous sommes déjà presque au bout. Cuando llegamos a la posada, el joven me llevó a su cuarto, como usted vió, y durante un buen rato las lágrimas y los sollozos no le dejaron hablar. Su historia se cuenta en dos palabras: volvió de su viaje, y la primera noticia que le aguardaba a su regreso era que habían ahogado a su padre, asesinado a su madre y ahorcado a su hermano, y que, además, todos los bienes de la familia estaban confiscados. Y no era eso todo: donde quiera que iba le miraban como faccioso, y los nacionales le apaleaban. Acudió a sus parientes, y algunos, del bando carlista, le aconsejaron que se alistara en el ejército de don Carlos, y el mismo Pretendiente, que fué amigo de su padre, le ofreció un empleo en su ejército. Pero, mon maître, como le dije a usted antes, se trata de un joven pacífico, manso como un cordero, que aborrece el derramamiento de sangre. Además, no era de ideas carlistas, porque durante sus estudios había leído libros escritos en tiempos antiguos por algunos compatriotas míos, donde no se habla más que de repúblicas, de libertades y de derechos del hombre, de suerte que se inclinaba más al sistema liberal que al de don Carlos; declinó, por tanto, la oferta de don Carlos, y todos sus parientes le abandonaron, mientras los liberales le acosaban de pueblo en pueblo como a bestia salvaje. Al fin, vendió unas tierrecillas que le quedaban, y con el producto se retiró a Colunga, donde nadie le conoce; aquí lleva hace varios meses una vida muy triste; la lectura de dos o tres libros y correr de vez en cuando una liebre con su perro son todas sus distracciones. Me pidió consejo, pero no pude darle ninguno y no hice más que llorar con él. Al cabo, dijo: «Querido Antonio, para mí no hay remedio, ya lo veo. Dices que tu amo
- 67. está abajo; ruégale de mi parte que se espere hasta mañana; mandaremos llamar a las muchachas del pueblo, buscaremos un violín y una gaita, y bailaremos para olvidar nuestros cuidados un momento.» Entonces me dijo unas palabras en griego viejo; apenas las entendí, pero creo que significan algo así como: «Bebamos y comamos y alegrémonos, que mañana moriremos.» »Eh bien, mon maître: le dije que usted es un señor muy serio, que no se divierte nunca y que estaba de prisa. Lloró otra vez, y, abrazándome, nos dijimos adiós. Ya sabe usted, mon maître, la historia del joven de la posada.» Dormimos en Ribadesella, y al mediar el siguiente día llegamos a Llanes. El camino corría entre la costa y una inmensa cadena de montañas que alzaba su barrera formidable a una legua del mar. El terreno por donde íbamos era regularmente llano y parecía bien cultivado. Abundaban los viñedos y los árboles, y a cortos intervalos se alzaban los cortijos de los propietarios, edificios de piedra, de planta cuadrada, rodeados de un muro exterior. Llanes es una ciudad antigua, de gran importancia en otros tiempos. En sus cercanías está el convento de San Cilorio, uno de los edificios monásticos más grandes de España. Ahora está abandonado, y se alza solitario y desolado en una de las penínsulas de la costa cantábrica. Dejado Llanes, entramos a poco en una de las regiones más áridas y tristes que pueden imaginarse, donde todo era piedra y rocas, sin árboles ni hierba. La noche nos cogió en aquellos lugares. Continuamos la marcha, no obstante, hasta llegar a una aldea llamada Santo Colombo. Allí pasamos la noche en casa de un carabinero, hombre atlético, a quien encontramos a la puerta, armado de
- 68. fusil. Era castellano, con todo el ceremonioso formulismo y la grave urbanidad que en otro tiempo dieron tanta fama a sus compatriotas. Regañó a su mujer porque hablaba con la criada delante de nosotros de asuntos de la casa. «Bárbara—dijo—, esa conversación no puede interesarles a unos caballeros forasteros; cállate, o vete a otra parte con la muchacha.» No quiso aceptar remuneración alguna por su hospitalidad. «Soy un caballero como ustedes—dijo —. No acostumbro a albergar gente en mi casa para ganar dinero. A ustedes les admití porque se les había hecho de noche y la posada estaba lejos.» Madrugamos mucho y seguimos nuestra ruta por un terreno tan triste y pedregoso como el recorrido el día antes. En cuatro horas llegamos a San Vicente, pueblo grande y destrozado, habitado principalmente por miserables pescadores. Conserva, empero, notables reliquias de su pasada magnificencia; el puente, tendido sobre la profunda y ancha ría en cuya margen se alza la ciudad, no tiene menos de treinta y dos arcos, y es de granito gris. Su fábrica es muy antigua; se halla tan ruinoso en algunos sitios, que ofrece peligro. Dejando atrás San Vicente, caminamos unas cuantas leguas por la costa; a veces atravesábamos alguna angosta ría. El terreno comenzó a mejorar; en las cercanías de Santillana era ya fértil y ameno. Como una hora antes de llegar al país de Gil Blas, atravesamos un extenso bosque, con muchas rocas y precipicios. En un lugar como éste se hallaba la caverna de Rolando, según se cuenta en la novela. El bosque tenía mala fama; el guía nos dijo que en él se cometían robos; pero nada nos sucedió, y llegamos a Santillana a eso de las seis de la tarde.
- 69. No entramos en la ciudad; hicimos alto en una gran venta o posada, en las afueras, delante de la que se alzaba un fresno gigante. Apenas hospedados, estalló una espantosa tormenta de agua y viento, con muchos truenos y relámpagos, que se prolongó sin interrupción varias horas, y cuyos efectos observé durante el viaje del siguiente día: todos los ríos que encontramos iban muy crecidos; al borde del camino yacían descuajados algunos árboles. Santillana cuenta con cuatro mil habitantes, y dista de Santander, adonde llegamos al otro día temprano, seis leguas cortas. No hay cosa que contraste más con la región desolada y los pueblos medio en ruinas que acabábamos de atravesar, que el bullicio y la actividad de Santander, casi la única ciudad de España que no ha padecido con las guerras civiles, a pesar de hallarse en los confines de las Provincias Vascongadas, reducto del Pretendiente. Hasta las postrimerías del siglo pasado, Santander era poco más que una obscura ciudad de pescadores; pero en estos últimos años ha monopolizado casi por completo el comercio con las posesiones ultramarinas de España, especialmente con la Habana. La consecuencia de esto ha sido que, mientras Santander se enriquecía con rapidez, La Coruña y Cádiz han ido decayendo al mismo paso. Santander posee un muelle muy hermoso, sobre el que se alza una línea de soberbios edificios, mucho más suntuosos que los palacios de la aristocracia en Madrid; son de estilo francés, y en su mayoría los ocupan comerciantes. La población de Santander es de unos sesenta mil habitantes. El día de mi llegada comí en la table d’hôte de la fonda principal, regida por un genovés. La concurrencia era muy mezclada: franceses, alemanes y españoles
- 70. hablaban en sus idiomas respectivos, y en una punta de la mesa, sentados frente a frente, dos catalanes, uno de los cuales pesaría veinte arrobas, gruñían en su áspero dialecto. Mucho antes de terminar la comida, un individuo sentado junto al catalán corpulento monopolizó la atención y las conversaciones de todos. Era un hombre delgado, de mediana estatura, rubicundo y con una irregularidad en la mirada que, si no era estrabismo, se le parecía mucho. Llevaba uniforme militar, azul, y por el gusto de perorar se olvidaba de los manjares que tenía delante. Hablaba en correctísimo español, pero con un leve acento extranjero. Entretúvose un buen rato en discurrir acerca de la guerra y de sus particularidades, criticando con mucha libertad la conducta de los generales, tanto carlistas como cristinos, en la presente lucha, y, por último, exclamó: —Si el Gobierno me diese veinte mil hombres tan sólo, acababa yo la guerra en seis meses. —Dispense usted, señor—dijo un español sentado a la mesa—; la curiosidad me mueve a pedirle a usted el favor de decirnos su distinguido nombre. —Yo soy Flinter—contestó el militar—, nombre que las mujeres, los niños y los hombres de España traen de boca en boca. Soy Flinter el irlandés y acabo de escaparme de las garras de don Carlos en las Provincias Vascongadas. Al morir Fernando me declaré por Isabel, estimando que todo buen caballero irlandés al servicio de España debía hacer otro tanto. Todos ustedes han oído hablar de mis hazañas; permítanme ustedes decir que aún hubiese hecho mucho más si la envidia de mi gloria no hubiese trabajado para privarme de los medios de acción necesarios. Hace dos años me mandaron a Extremadura a organizar las milicias. Las partidas de
- 71. Gómez y de Cabrera entraron en la provincia, sembrando la devastación en torno; con todo, me encontraron en mi puesto, y si mis subalternos me hubieran secundado como era debido, los dos cabecillas no habrían vuelto ante su amo a jactarse de sus triunfos. Estando a la defensiva en mis atrincheramientos, se destacó de las filas carlistas un hombre y nos intimó la rendición. «¿Quién eres?»—le pregunté—. «Soy Cabrera»—respondió—. «Y yo soy Flinter—repliqué desenvainando el sable—; retírate a tus líneas o mueres inmediatamente.» Amedrentado, hizo lo que le mandé. Una hora después nos rendimos. Me llevaron prisionero a las Provincias Vascongadas, y los carlistas se regocijaron mucho con mi captura, porque el nombre de Flinter era muy sonado en sus filas. Me arrojaron en una mazmorra repugnante, donde estuve veinte meses. Hacía mucho frío, yo estaba desnudo, pero no me desanimé por eso: mi indomable espíritu no podía sentir tal flaqueza. Al cabo, mi carcelero se compadeció de mis desdichas. Díjome que «le apesadumbraba ver morir sin gloria a hombre tan valiente». Combinamos un plan de fuga, adquirimos unos disfraces y nos lanzamos juntos a la ventura. Pasamos inadvertidos hasta llegar a las líneas carlistas sobre Bilbao; allí nos dieron el alto. Pero mi presencia de ánimo no me abandonó. Iba yo disfrazado de carretero catalán, y la frialdad de mis respuestas engañó a mis interrogadores. Nos dejaron pasar y no tardamos en vernos en salvo dentro de los muros de Bilbao. Aquella noche hubo iluminación en la ciudad, porque el león había roto sus redes, Flinter se había escapado y volvía a reanimar una causa abatida. Acabo de llegar ahora de Santander, de paso para Madrid, donde voy a pedir al Gobierno el mando de veinte mil hombres.
- 72. ¡Pobre Flinter! Seguramente no se han visto juntos en el mismo cuerpo un corazón más intrépido ni una boca más fanfarrona. Se fué a Madrid y, por la influencia del embajador británico, amigo suyo, obtuvo el mando de una pequeña división, con la que se dió traza para sorprender y derrotar, en las cercanías de Toledo, un cuerpo de carlistas al mando de Orejita, tres veces superior en número a sus tropas. En pago de esa hazaña, el Gobierno, que era entonces moderado o juste milieu, le persiguió con incansable animosidad; el primer ministro, Ofalia, apoyó con toda su influencia numerosas y ridículas acusaciones de robos y saqueos aducidas contra el demasiado victorioso general por los canónigos carlistas de Toledo. Fué asimismo acusado de negligencia por haber consentido, después de la batalla de Valdepeñas, ganada también por él con gran intrepidez, que las fuerzas carlistas se posesionaran de las minas de Almadén; bien que el Gobierno, empeñado en perderle, hizo cuanto pudo para impedir que se aprovechara de la victoria, negándole todo género de recursos y refuerzos. Privado de los frutos de su victoria, cegáronse sus esperanzas, y una melancolía morbosa se apoderó del irlandés; resignó el mando, y menos de diez meses después de haberle visto en Santander, dió a sus cobardes y envidiosos enemigos un triunfo que los satisfizo, cortándose el cuello con una navaja de afeitar. ¡Almas ardorosas, nacidas en otros climas, que aspiráis a distinguiros al servicio de España y a ganar recompensas y honores, acordaos de la suerte de Colón y de otro no menos valiente y apasionado: Flinter!
- 73. T CAPÍTULO XXXV Salida de Santander.— Alarma nocturna.— La hoz tenebrosa. enía yo encargado que mandaran desde Madrid a Santander 200 Testamentos; con no pequeño disgusto hallé que no habían llegado, y supuse o que los carlistas se habían apoderado de ellos en el camino, o que mi carta se había extraviado. Pensé pedir a Inglaterra provisión de ellos; pero abandoné la idea por dos razones: en primer lugar, hubiera tenido que perder un mes aguardando, ocioso, su llegada, y la ciudad era muy cara, y en segundo lugar, me encontraba muy mal de salud y no podía procurarme buena asistencia médica en Santander. Desde que salí de La Coruña me afligía una disentería terrible, complicada últimamente con una oftalmía. Resolví, por tanto, marcharme a Madrid. Pero no era esto empresa fácil. Partidas del ejército de don Carlos, batidas en Castilla, merodeaban por la región que yo iba a cruzar, sobre todo por la parte llamada La Montaña, de modo que las comunicaciones de Santander con el Sur estaban cortadas. Sin embargo, determiné confiar, como siempre, en el Todopoderoso y afrontar el peligro. Compré un caballejo, y en compañía de Antonio me puse en camino.
- 74. Antes de marcharme hablé con los libreros para el caso de que me fuera posible enviarles un depósito de Testamentos desde Madrid; arregladas las cosas a gusto mío, me puse en manos de la Providencia. No me detendré en referir este viaje de 300 millas. Pasamos por en medio del fuego, aunque parezca raro, sin chamuscarnos un pelo de la cabeza. Delante, detrás y a cada lado de nosotros se cometían robos, muertes y todo género de atrocidades; pero ni siquiera nos ladró un perro, aunque en cierta ocasión se concertó un plan para cogernos. A unas cuatro leguas de Santander, mientras echábamos pienso a los caballos en la posada de un pueblo, vi salir corriendo a un hombre que había estado cuchicheando con el mozo que nos daba la cebada para las bestias. En el acto le pregunté lo que el hombre le había dicho; pero obtuve sólo respuestas evasivas. Luego resultó que hablaron de nosotros. Dos o tres leguas más lejos había otro pueblo y otra posada, donde tenía pensado detenerme, y de seguro lo dije así; pero al llegar a ella, como aún quedaba bastante sol, decidí continuar hasta otra posada que creía encontrar a una legua de distancia; me equivoqué en esto, porque no encontramos ninguna hasta Ontaneda, a nueve leguas y media de Santander, donde había un pequeño destacamento de soldados. A media noche nos despertó el grito de alarma; el faccioso estaba cerca; acababa de llegar un emisario del alcalde del pueblo inmediato, donde había tenido yo intención de pernoctar, diciendo que una partida carlista había sorprendido el lugar en busca de un espía inglés que suponían alojado en la posada. Al oír esto, el oficial que mandaba la tropa no se creyó seguro, y al instante reunió su gente y se retiró a un pueblo próximo fortificado, guarnecido por un destacamento más poderoso. Nosotros ensillamos los caballos y continuamos nuestro camino en la obscuridad. Si los carlistas llegan a cogerme me
- 75. hubieran fusilado en el acto, y arrojado mi cuerpo en las peñas para pasto de buitres y lobos. Pero «no estaba escrito», decía Antonio, que, como muchos de sus compatriotas, era fatalista. A la noche siguiente nos libramos también de buena: llegábamos cerca de la entrada de un paso horrible llamado El puerto de la puente de las tablas, que atraviesa una montaña pavorosa y negra, al otro lado de la cual está la ciudad de Oña, donde me proponía pasar la noche. Hacía un cuarto de hora que se había puesto el sol. De pronto un hombre, con el rostro lleno de sangre, salió precipitadamente de la hoz. —Vuélvase atrás, señor—dijo—, en nombre de Dios; en la hoz hay ladrones, y acaban de robarme la mula y todo lo que tengo; con trabajo he salido vivo de sus manos. No sé por qué no le hice caso, y sin responder seguí adelante; cierto que estaba yo tan cansado y enfermo que me importaba muy poco lo que pudiera sucederme. Entramos; a derecha e izquierda se alzaban las rocas a pico e interceptaban la escasa luz del crepúsculo, de suerte que en torno nuestro reinaban tinieblas sepulcrales o, más bien, las tinieblas del valle de la sombra de muerte, y no sabíamos por dónde íbamos; pero confiábamos en el instinto de los caballos, que avanzaban con las cabezas pegadas al suelo. No se oía más ruido que el fragor del agua al despeñarse por la hoz. A cada momento creía que iba a sentir un puñal en el cuello; pero «no estaba escrito». Atravesamos la hoz sin hallar ser humano, y a los tres cuartos de hora de haber entrado en ella nos encontrábamos en la posada de la ciudad de Oña, atestada de tropas y de paisanos armados en espera de un ataque del grueso del ejército carlista, que andaba muy cerca.
- 76. Bueno: llegamos a Burgos sin novedad; llegamos a Valladolid sin novedad; pasamos el Guadarrama sin novedad, y, por último, llegamos sin novedad a nuestra casa en Madrid. La gente ponderaba nuestra buena suerte; Antonio decía: «No estaba escrito»; pero yo digo: Loado sea el Señor por las mercedes que nos otorgó. FIN DEL TOMO SEGUNDO
- 77. NOTAS [1] Borrow salió de Sevilla el 9 de Diciembre de 1836, estuvo once días en Córdoba, de donde partió el 20, llegando a Aranjuez el 25 y a Madrid el 26. (Knapp.) [2] Número 16, piso 3.º (Knapp.) [3] María Díaz murió en 1844. (Knapp.) [4] El primer contrato para imprimir el Nuevo Testamento lo hizo con Mr. Charles Wood, impresor del gobierno español. El contrato con Borrego es de 17 de Enero de 1837, para reproducir la edición de Londres (1826) del N. T. de Scio. (Knapp.) [5] Borrow pensó primeramente en dar por terminada su misión en la Península con la impresión del Nuevo Testamento, dejando a otros el cuidado de distribuir la obra. Cambió de idea y se ofreció a desempeñar en persona ese cometido; los directores de la Sociedad Bíblica aceptaron su propuesta, recibiendo Borrow la autorización oficial dos días después de terminarse la tirada del libro. (Knapp.) [6] Buena suerte, Antonio. [7] He aquí la original copla bilingüe que damos traducida en el texto: The Romany chal to his horse did cry, As he placed the bit in his horse’s jaw. «Kosko gry! Romany gry! Muk man kistur tute knaw!» [8] Plural de chabó o chabé: mozo, joven, compañero. [9] Soldados.
- 78. [10] Parugar: trocar, traficar. Graste: caballo. [11] Feria. [12] Caballero. [13] Plural de Caloró: gitano. [14] Bul; Bullati: el ano. [15] Un hombre no gitano; un gentil. [16] Granada. [17] ¡Quita de ahí! ¡Déjame! [18] Estos «cuadros de Murillo» son imaginarios, observa el editor U. R. Burke. [19] Posiblemente Cisneros o Calzada. (Nota del editor Burke.) [20] El nombre del arriero era Pedro Mato. La estatua es de madera. (Nota del editor Burke.) [21] Es un error: Lucus Augusti fué sólo capital de la Galicia septentrional; Bracara Augusta (Braga), de la meridional; el Miño las dividía. (Nota del editor Burke.) [22] Vocablo del dialecto milanés, según Borrow y su anotador Burke, equivalente a vagar sin rumbo. [23] Alude a D. Pelayo Gómez de Sotomayor, primer enviado de Enrique III cerca de Tamerlán. [24] El abogado se llamaba D. Claudio González y Zúñiga, autor de la Descripción Económica de la Provincia de Pontevedra. Pontevedra, 1834. (Knapp). [25] El alcalde de Corcubión no necesitaba saber inglés para leer a Bentham, porque desde 1820 a 1837 gran parte de sus escritos se habían traducido y publicado en España. Las obras completas fueron publicadas en español por Baltasar Anduaga Espinosa, Madrid, 1841- 1843, 14 vols. en 4.º El calificativo de «Solón inglés» que
- 79. Borrow pone en boca del alcalde está tomado de un artículo del Monthly Magazine, que Borrow conocía bien. Su indiferencia por Bentham nace de la secreta hostilidad que Borrow profesaba al Dr. Bowring, uno de los agentes principales de la introducción de las obras de Bentham en la Península. (Knapp.) [26] ¿Avilés?
- 80. Nota de transcripción Se ha respetado la ortografía original, normalizándola a la grafía de mayor frecuencia. Los errores obvios de imprenta han sido corregidos sin avisar. Las páginas en blanco han sido eliminadas. Se ha reparado el emparejamiento de los puntos de admiración e interrogación, y de los paréntesis y comillas. El transcriptor ha creado la imagen de la cubierta y la sitúa en el dominio público.
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