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An authoritative reference that covers essential concepts of orbital mechanics and explains how they relate to advanced space trajectory applications
Space Trajectories is the first book to offer a comprehensive exploration of orbital mechanics and trajectory optimization in a single volume. Beginning with a review of essential concepts, the book progresses to advanced space applications, highlighting methods used in today’s space missions.
The contents are organized into three parts. The first part delves into free orbital motion, covering topics such as Keplerian motion, perturbed motion, the three-body problem, orbit determination, and collision risks in orbit. The second part focuses on controlled orbital motion, discussing impulsive transfer, orbital rendezvous, thrust level optimization, low-thrust transfer, and space debris cleaning. The third part examines ascent and reentry, including launch into orbit, launcher staging, analytical solutions in flat Earth, interplanetary missions, and atmospheric reentry.
Each chapter is written in a modular way, featuring conclusion summaries, key points, and suggestions for further investigation. Examples are included with detailed solutions methods that readers can apply to solve their own trajectory problems.
Written by an expert of the topic who has performed guidance of Ariane launchers for 30 years, Space Trajectories includes information on:
Space Trajectories is an essential reference for students and researchers aiming to quickly understand the main issues in astrodynamics and the way to design trajectories, as well as space engineers seeking to consolidate their knowledge in the field of optimization and optimal control applied to aerospace and space missions.
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Seitenzahl: 733
Veröffentlichungsjahr: 2024
Max Cerf
ArianeGroup, Les Mureaux, France
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To Mallow, who witnessed the completion of this book in March 2024.
Max Cerf has been an engineer at ArianeGroup since 1990, specializing in optimization of space trajectories and spacecraft. He is also a university professor and author of a book on optimization techniques. He is graduate of Ecole Centrale de Paris (1989), holds a PhD (2012) and a “Habilitation à Diriger des Recherches” (2019), and was appointed “Chevalier de l’Ordre National du Mérite” (2015).
The conquest of space is a major adventure in the history of mankind.
In the 17th century, Newton discovered the laws governing the motion of celestial bodies, thus founding classical mechanics, notably the theory of universal gravitation. Although mathematically simple, these laws conceal an immense wealth of possible dynamics. The Newtonian gravitational field possesses a host of remarkable properties, discovered over the centuries through the development of increasingly sophisticated mathematics, which are of great interest to space agencies. For example, the existence of periodic orbits, points of equilibrium, invariant manifolds, and other sometimes unexpected properties make possible space missions that are awe-inspiring.
In this remarkable book, in his inimitably clear style, Max Cerf reveals all the secrets of space dynamics and mission design. Senior engineer at ArianeGroup, expert in mission analysis and optimization, Max Cerf is a master of mission design and space vehicle control. In addition to being an accomplished and renowned engineer and researcher, he is also an exceptional teacher, as the reader will appreciate from the pleasant and fascinating reading of this book. From rocket launch and orbiting, through problems of atmospheric reentry, orbital transfer, and space debris clean-up, to interplanetary missions, Max Cerf provides all the keys you need to quickly master all aspects and implementation.
This book will appeal to students, researchers, and engineers looking for a global view of trajectory design problems in astrodynamics, and to acquire the main techniques, particularly in the fields of optimization and optimal control. After reading this book, all you want to do is take part in this amazing adventure!
Emmanuel Trélat Sorbonne Université, Paris
I express my gratitude to the people who have supported me in one way or another during my career and contributed to the content of this book, and more especially:
Lauren Poplawski for accepting and managing the book’s publication by Wiley;
Nandhini Karuppiah, Govindanagaraj Deenadayalu, and Isabella Proietti for their assistance throughout the editorial process;
the many ArianeGroup colleagues with whom I worked over the years, with a special mention for Mathieu Verlet, Arnaud Ruiz, Stéphane Reynaud, and Hervé Gilibert;
Christophe Sauty and Ana Gomez of Paris Observatory, who enabled me to complete the master’s degree in astronomy in parallel with my daily work;
Emmanuel Trélat for his laudatory foreword and our deep collaboration on various space trajectory problems;
Thomas Haberkorn, Jean-Baptiste Caillau, and Hasnaa Zidani for our many fruitful exchanges
.
Space trajectories is a field at the crossroads of mechanics and mathematics. Its development can be divided into two periods.
The first period followed the discovery of Kepler’s laws and their demonstration by Newton who laid the foundations of classical mechanics. During this period spanning the 18th and the 19th centuries, research focused on celestial mechanics with the goal of predicting the motion of planets as precisely as possible.
The second period began with the invention of rocket propulsion at the beginning of the 20th century. The possibility of sending maneuvering vehicles in space stimulated the development of optimal control theory, which was originally mainly oriented toward space applications.
This book is aimed at students, researchers, and engineers interested in space applications and who wish to gain an overview of the field. It recalls the essential concepts of orbital mechanics and also presents advanced space applications involving trajectory optimization.
The content is divided into three parts, each of which composed of five chapters.
The first part deals with free orbital motion with the following topics: Keplerian motion, perturbed motion, three-body problem, orbit determination, and collision risks in orbit.
The second part deals with controlled orbital motion with the following topics: impulsive transfer, orbital rendezvous, thrust level optimization, low-thrust transfer, and space debris cleaning.
The third part deals with ascent and reentry with the following topics: launch into orbit, launcher staging, analytical solutions in flat Earth, interplanetary mission, and atmospheric reentry.
These topics have been selected for their practical relevance to space applications. Each chapter covers the necessary concepts, so that it can be read independently of the rest of the book.
The two-body problem is the fundamental model of orbital mechanics. It describes the motion of two bodies in pure gravitational interaction to the exclusion of all other forces. In the restricted two-body problem, the mass of one body is assumed to be negligible, while the other is considered as point-like. This modeling is appropriate to the motion of artificial satellites around a celestial body.
The two-body problem has an analytical solution called Keplerian motion. The trajectories are conics, satisfying properties of conservation of energy and angular momentum. The conic nature (circle, ellipse, parabola, or hyperbola) depends only on the initial conditions of position and velocity and the motion time law is determined by Kepler’s equation. The Keplerian orbit of a satellite around the attracting body is represented geometrically by its orbital parameters, which are analytically related to the position and velocity.
The first section presents the dynamic model and derives the main prime integrals of the Keplerian motion, namely those of angular momentum, eccentricity vector, and energy. The conic shape is then determined from the initial conditions of position and velocity.
The second section deals with the time law of the motion. Position and time are linked by Kepler’s equation, which is transcendental. Kepler’s equation takes different forms depending on the conic nature. It can also be expressed in a universal form valid for any kind of orbit. The solution of Kepler’s equation requires a numerical iterative method.
The third section defines the orbital parameters locating the orbit in space and establishes their relation to position and velocity. The classical orbital parameters have singularities for circular or equatorial orbits, which motivates the definition of the equinoctial parameters, devoid of such singularities. An overview of Earth’s orbits is finally presented, with a focus on Earth’s coordinate systems and on specific properties such as geo-synchronism or sun-synchronism.
The two-body problem, or Kepler’s problem, concerns the relative motion of two particles exerting between them the force of gravitational attraction. The resulting dynamics, called Keplerian motion, is presented in this section.
The two-body problem is formulated in a Galilean (or inertial) reference frame.
Both bodies are considered as material points of masses m1 and m2 and positions and , respectively. The gravitational force exerted by the second body on the first body is
The constant G is the universal gravitational constant and denotes the position of body 2 relative to body 1. Newton’s third law (action and reaction) implies that body 1 exerts an opposite force on body 2: .
The accurate knowledge of the value of G is of considerable importance, both for the computation of space trajectories and for the confirmation of theories such as general relativity.
The first reliable measurement of G was made in 1798 by Henry Cavendish, using a torsion pendulum. The possibility of on-board experiments carried out by satellites has greatly improved the accuracy. The reference value adopted in the international system is: G ≈ 6.674 08 ⋅ 10−11 m3 kg−1 s−2.
Newton’s second law (also called fundamental relation of dynamics) applied to each body in the inertial frame yields the two differential equations
After subtracting the first equality from the second, we obtain the differential equation for the motion of body 2 relative to body 1.
where μ = G(m1 + m2) is called the gravitational constant of the two-body system.
In the restricted two-body problem, the mass of body 2 is negligible compared with that of body 1 (main attracting body). The constant μ = Gm1 is then called the gravitational constant of the main attracting body. This model is well suited to practical applications studying the motion of artificial satellites.
The values of μ for Earth, Moon, and Sun are (in m3/s2):
The solution to the two-body problem is called Keplerian motion, after Johann Kepler, who first set out the laws of motion.
The laws of planetary motion around the Sun were set out empirically by Johann Kepler (1571–1630) in Astronomia Nova, published in 1609.
The three laws of Kepler are as follows:
the planets describe ellipses with the Sun as one of their foci;
the radius vector sweeps equal areas in equal times;
the square of the period of revolution is proportional to the cube of the semi-major axis.
Kepler had guessed these three laws from the collection of observations made by the Danish astronomer Tycho Brahé (1456–1601) on the motion of the planets.
Their demonstration came much later owing to Isaac Newton (1642–1727).
In Philosophiae Naturalis Principia Mathematica, published in 1687, Newton postulated the laws of motion (inertia, force, and reaction) and the law of gravitation, from which he rigorously established the solution to the two-body problem.
These fundamental results form the basis of orbital and celestial mechanics since the 18th century.
A prime integral of a dynamic system is a quantity that remains constant over time. Knowledge of prime integrals helps to delimit the system evolution, even when no analytical solution exists. The two-body problem admits several prime integrals, such as those of angular momentum, energy, and eccentricity vector.
The angular momentum per unit mass is , where is the velocity of body 2 relative to body 1. Calculating the derivative of with the assumption of a central acceleration given by (1.3) yields
The angular momentum vector is therefore constant throughout the motion.
If and are collinear, then and the motion takes place along a straight line. Such a trajectory is said to be degenerate. Unless otherwise specified, we assume in the sequel that .
As the position and the velocity are constantly perpendicular to , the motion takes place in a plane called the orbital plane.
Taking body 1 as origin O and a reference axis Ox in this plane (Figure 1.1):
the position of body 2 is defined by the polar angle θ and the radius vector r;
the velocity of body 2 has radial and orthoradial components and (where denotes the time derivative of x).
Figure 1.1 Orbital plane.
The modulus of the angular momentum is
In an infinitesimal time dt, the radius vector sweeps the area dS of the triangle with sides and . This area is
The areal velocity (area swept per unit time) is therefore constant, just as the angular momentum. This establishes Kepler’s second law or law of areas.
The mechanical energy per unit mass noted w is the sum of the kinetic energy and the potential energy , from which the gravitational force derives.
Taking the scalar product of each member of (1.3) with and using (Figure 1.1), we obtain
Instantaneous variations in kinetic and potential energy cancel each other out. This statement is in fact true for any force deriving from a potential.
The mechanical energy is therefore constant throughout the motion.
The accessible domain depends on the energy sign. To escape the main attracting body, the energy must be zero or positive (kinetic energy ≥ potential energy).
In this case, the escape velocity vesc associated with zero energy (w = 0) and the residual velocity at infinity v∞ (for r → ∞) are respectively given by
The eccentricity vector (also called Laplace vector) is defined as:
To calculate the derivative of , we observe that the radial unit vector , which lies in the orbital plane, rotates around the fixed axis with an angular velocity (see Figure 1.1).
Its derivative is therefore orthoradial and it can be expressed as:
Using (1.5) to substitute , then (1.3) to make appear, we obtain
By derivingequation (1.10), with from (1.4), and using (1.12), the terms cancel each other out and the derivative of is zero.
The eccentricity vector is therefore constant throughout the motion.
By squaring each member of equation (1.10), we obtain the useful relation
This relation can be rewritten by defining two constants p and a from the angular momentum and from the energy respectively, with the convention a = ∞ if w = 0.
Substituting in (1.13) yields a relation between the constants p, a, and e.
This section establishes the orbit equation in the orbital plane, successively in polar and Cartesian coordinates. The frame origin O is at the center of mass of the attracting body and the axis Ox is directed along the eccentricity vector . In the case , the axis Ox can be chosen arbitrarily. The axis Oy completes the frame in the orbital plane, which is orthogonal to the angular momentum (Figure 1.1).
The scalar product with of the two members of equation (1.10) yields
with the constant p defined in (1.14).
On the other hand, we have . Substituting in (1.16), we obtain
This equation in polar coordinates defines a conic of parameter p, eccentricity e and major axis parallel to . The angle θ from to is the true anomaly.
Kepler’s first law regarding the trajectory shape (conic) is thus established.
The eccentricity value determines the nature of the conic.
e < 1 yields an ellipse with w < 0 from
(1.13)
. With a negative energy, the central attraction cannot be escaped and the orbit is closed.
e = 1 yields a parabola with w = 0 from
(1.13)
. With a null energy, the central attraction is escaped with a null residual velocity at infinity
(1.9)
.
e > 1 yields a hyperbola with w > 0 from
(1.13)
. With a positive energy, the central attraction is escaped with a non-null residual velocity at infinity
(1.9)
.
The polar coordinates (r, θ) are transformed into Cartesian coordinates (x, y) by
Substituting in (1.16) yields: r = p − ex.
By squaring both members, we obtain the conic equation in Cartesian coordinates.
At this step, we need to distinguish the cases e < 1, e > 1 or e = 1, corresponding to an ellipse, a hyperbola and a parabola, respectively.
By substituting p = a(1 − e2) from (1.15), equation (1.19) becomes
and defining c = ae and , we get
This is the Cartesian equation of an ellipse of semi-major axis a along Ox and semi-minor axis b along Oy. The distance from the center to the focus is c.
Figure 1.2 shows the geometric elements of an elliptical orbit. The center C is located at (−c; 0), on the negative side of O. The focus O (attracting body) and the virtual focus O’ are located on either side of the center C.
The extreme points called apses are the periapsis or pericenter P (closest point to the attracting body O) and the apoapsis or apocenter A (farthest point from the attracting body O). Their names depend on the celestial body: apogee and perigee (Earth), aphelion and perihelion (Sun), apolune and perilune (Moon)…
The distances from the focus to the pericenter P and apocenter A are
Figure 1.2 Elliptical orbit.
The ellipse is deduced from the eccentric circle (of center C and radius a) by an affinity orthogonal to the major axis AP and of ratio b/a. The position is defined by the true anomaly θ or by the eccentric anomaly E, depicted in Figure 1.2.
Expressing the Cartesian coordinates with the true and eccentric anomalies
we deduce relations between the anomalies θ, E, and the radius vector r.
In the degenerate case (h = 0, e = 1), the ellipse flattens and reduces to the segment from A to P, yielding a rectilinear ellipse. The focus O coincides then with the pericenter (c = a). The position and velocity along the straight line are given by
In a similar way to the elliptical case, substituting p = a(1 − e2) in (1.19) yields
and defining c = − ae and (with a < 0, e > 1), we get
This is the Cartesian equation of a hyperbola of semi-major axis a along Ox and center-focus distance equal to c.
Figure 1.3 shows the geometric elements of a hyperbolic orbit. The center C is located at (c; 0), on the positive side of O. The focus O (attracting body) and the virtual focus O′ are located on either side of the center C. Only one hyperbola branch is drawn, corresponding to the actual trajectory around the attracting body.
The polar angle of the asymptote is found by taking the limit r → ∞ in (1.17).
The distance from the focus O to the pericenter P is (recalling that a < 0, e > 1)
The Cartesian coordinates can be expressed with the hyperbolic anomaly H.
Figure 1.3 Hyperbolic orbit.
Similarly to the elliptical case (1.23) and (1.24), angle E is replaced by angle H and the trigonometric functions are replaced by hyperbolic functions (with a < 0).
We deduce relations between the anomalies θ, H, and the radius vector r.
In the degenerate case (h = 0, e = 1), the hyperbola flattens and reduces to the half straight line ending at P, yielding a rectilinear hyperbola. The focus O coincides then with the pericenter P. The position and velocity along the straight line are given by
In that case, equation (1.19) yields
This is the Cartesian equation of a parabola of axis Ox with vertex P at .
A parametric form is obtained by introducing the parabolic anomaly B.
In the degenerate case (h = 0, e = 1), the parabola reduces to the half straight line ending at P, yielding a rectilinear parabola. The focus O coincides then with the pericenter P. The position and velocity along the straight line are obtained by integrating the energy equation .
Denoting tP the time at pericenter, we obtain
Before ending this section, we derive some useful formulas for the velocity vector and its modulus.
The vector product with of the two members of equation (1.10) yields
Expanding the double vector product yields the velocity vector as a function of , and the radial unit vector .
On the other hand, the velocity modulus can be expressed from the energy (1.14).
This formula is valid for any conic with the sign of a defined by the energy (1.14):
a > 0 for an ellipse (w < 0);
a = ∞ for a parabola (w = 0);
a < 0 for a hyperbola (w > 0).
From this formula, we deduce:
the
circular velocity
v
sat
at radius r to stay on a circular orbit (r = a = Cte);
the
escape velocity
v
esc
at radius r to escape to infinity (w = 0 and a = ∞).
Figure 1.4 Effect of the initial velocity.
At the Earth’s surface, the circular velocity is 7.905 km/s (taking for the Earth’s equatorial radius r = 6378.137 km). This is the strict minimum horizontal velocity required for a projectile to perfectly follow the surface of a hypothetical round Earth with no atmosphere.
Increasing the initial velocity allows the projectile to move away from the Earth’s surface before returning to the point of departure. On reaching the escape velocity, whose value is 11.180 km/s at the Earth’s surface, energy becomes zero, and the projectile goes to infinity.
Figure 1.4 illustrates the possible trajectories of a projectile launched horizontally from the Earth’s surface, as a function of initial velocity.
The orbit equation takes different forms, depending on the conic nature. In this section, we deal with the position on the orbit as a function of time. The elliptical, hyperbolic, and parabolic cases are first separately considered, then the universal formulation is presented.
We first calculate the period of revolution on an elliptical orbit by the law of areas. The constant areal velocity is equal to the total area S of the ellipse divided by the period T. Integrating equation (1.6) yields
We obtain Kepler’s third law linking the period and the semi-major axis.
The mean angular velocity (called mean motion) noted n is defined by
To derive the motion time law, we start from (1.5) giving the angular momentum.
The true anomaly differential dθ is then expressed as a function of the eccentric anomaly differential dE by deriving the equation for in (1.24).
Using the relations (1.24), we have
which is substituted in (1.44) to give
Substitutingdθ in (1.43) with , we obtain the differential equation linking time and eccentric anomaly, with n defined by (1.42).
Integrating from the time tP of passage at the pericenter (anomaly EP = 0) and defining the mean anomaly by M = n(t − tP), we obtain Kepler’s equation.
Kepler’s equation has no analytical solution. Because of its practical importance, it has been the subject of considerable research since the 18th century and many numerical methods have been investigated. Solutions by series expansions were useful until the advent of computers. Nowadays, iterative methods are preferred like Newton’s or Laguerre’s methods presented hereafter.
Newton’s method is applied to find a root of the function
The time t is given defining the mean anomaly M = n(t − tP). Newton’s iteration formula is
The starting point can be simply chosen as E0 = M. A more accurate initial guess is based on the observation that . The root is therefore in the interval [M, M + e] and it is estimated by linear interpolation as
Newton’s method may lack robustness, especially for eccentricities close to 1.
This motivates the alternative method of Laguerre.
Laguerre’s method aims at finding the root of polynomials. Its advantage is to be globally convergent (whereas Newton’s method is sensitive to the starting point) with a cubic convergence rate, whatever the polynomial degree. Although Kepler’s equation is transcendental, Laguerre’s method can be customized to it with some specific assumptions. It leads to the following iteration formula:
The denominator sign is chosen to minimize the difference |Ek + 1 − Ek|.
The starting point can be E0 = M, or it can be defined as above by (1.51).
Reference: An improved algorithm due to Laguerre for the solution of Kepler’s equation, B. Conway, Celestial Mechanics 39 (1986)
Let us consider a polynomial of degree n: g(x) = (x − x1)(x − x2)⋯(x − xn) and suppose we are interested in finding the specific root x1. By assuming (drastically) that all other roots are identical: x2 = x3 = ⋯ = xn, the function g(x) is replaced by G(x) = (x − x1)(x − x2)n − 1. Let us now define the functions G1 and G2 as
If x0 is an arbitrary estimate of the root x1 we are looking for, we have
Solving this system with two unknowns x1 and x2 yields for x1
Assume that the estimate x0 is already close to x1; the denominator sign is chosen to deviate as little as possible from x0. The iteration (1.52) is obtained by applying this formula to the function f (1.49), with G1 and G2 being the derivatives of ln f with the arbitrary choice n = 5, which works well in practice.
Extensive tests [R12] have shown that Laguerre’s method properties are retained when applied to Kepler’s equation. It is insensitive to the initial guess and the convergence requires on average less iterations than Newton’s method. But its computational cost per iteration is higher, so that Newton’s method remains competitive as long as eccentricity is not close to 1.
The time law equation is established in a similar way to the elliptical case, starting from (1.5). The true anomaly differential dθ is expressed as a function of the hyperbolic anomaly differential dH by deriving the equation for in (1.31).
Using the relations (1.31), we have
which is substituted in (1.53) to give (with )
Substitutingdθ in (1.5) with , we obtain the differential equation linking time and hyperbolic anomaly.
The mean motion n is defined in a similar way to the elliptical case, taking into account that a < 0 for a hyperbola.
Integrating from the time tP of passage at the pericenter (anomaly HP = 0) and defining the mean anomaly by M = n(t − tP), we obtain Kepler’s equation for a hyperbolic orbit.
Numerical solution methods (Newton, Laguerre) apply in a similar way to the elliptical case, with the function (1.49) replaced by
Here again, the time law equation is established starting from (1.5). The true anomaly differential dθ is expressed as a function of the parabolic anomaly differential dB by deriving the equation for in (1.34).
Using r expressed by (1.35), this can be written as:
Substituting dθ in equation (1.5), we obtain the differential equation linking time and parabolic anomaly.
The mean motion n is defined in a similar way to the elliptical case, using the parameter p instead of the semi-major axis a.
Integrating from the time tP of passage at the pericenter (anomaly BP = 0), and defining the mean anomaly by M = n(t − tP), we obtain Barker’s equation for a parabolic orbit.
Unlike Kepler’s equation, Barker’s equation has an analytic solution.
By the change of variable , equation (1.62) becomes
whose solutions are
Since the product of the roots is −1, both solutions in (1.64) yield the same value of B, so that the unique solution of (1.62) is expressed as:
The Keplerian trajectory is contained in the orbital plane defined by the initial conditions at a time t0. It is therefore possible to express the position and velocity at any time t as a linear combination of the initial conditions.
The functions F(t) and G(t) are called Lagrange coefficients and the functions Ft(t) and Gt(t) are their derivatives with respect to time.
To compute these functions, we use the perifocal frame defined from the eccentricity and angular momentum vectors (see Figure 1.5).
The unit vector is directed toward the pericenter .
The unit vector is along the angular momentum .
The unit vector complete the trihedron.
In the perifocal frame, the position vector is assessed by
and the velocity vector given in (1.37) is assessed by
where is the unit radial vector and is the unit orthoradial vector (orthogonal to in the orbital plane, directed along the motion).
Figure 1.5 Classical orbital parameters.
Solving (1.67) and (1.68) at the initial time for and , we obtain
Substituting these vectors and in (1.67) and (1.68) at time t yields
where Δθ = θ − θ0 denotes the true anomaly difference.
These formulas can be rearranged by introducing the variable .
Using (1.70), we have , so that .
From the conic equation (1.17), we have also .
After substituting in (1.70) the terms in e sin θ and e cos θ, with , and identifying with (1.66), we get the following expressions of Lagrange coefficients as functions of σ0, r0, r, and Δθ.
Another useful formula can be derived for Ft. For that purpose, we express the angular momentum from and from , using (1.66).
The coefficients F, G, Ft, and Gt are therefore linked by
This allows to calculate Ft by
Substituting F, G, and Gt given by (1.71), we obtain for Ft
The formulas (1.70) depending on the true anomaly are valid for any type of conic.
They can be specialized to the conic type by using either the eccentric anomaly E for ellipses, the hyperbolic anomaly H for hyperbolas or the parabolic anomaly B for parabolas.
For an ellipse (
e < 1, a > 0
), the position
(1.67)
and velocity
(1.68)
vectors are expressed with the eccentric anomaly using
formulas (1.24)
.
By calculations similar to (1.69)–(1.71), we obtain
For a hyperbola (
e > 1, a < 0
), the position
(1.67)
and velocity
(1.68)
vectors are expressed with the hyperbolic anomaly using
formulas (1.31)
.
By calculations similar to (1.69)–(1.71), we obtain
For a parabola (
e = 1
), the position and velocity vectors are given by
By calculations similar to (1.69)–(1.71), we obtain
The elliptical, hyperbolic, and parabolic cases can be merged using the universal variable formulation. This section follows the development presented in [R1].
Let us retrieve Kepler’s equations (1.47), (1.56), and (1.61) for each kind of conic, with their expressions for n and r.
The similarity of the three equations suggests defining a new variable χ by the differential equation
The change of variable from t to χ is called Sundman’s transformation, and χ is called the universal variable (or generalized anomaly) since it merges the three conic kinds. The goal is now to express Kepler’s equation in variable χ.
The first step consists in calculating the successive derivatives of the time t with respect to the variable χ, starting from the definition (1.83).
From , we have , so that
This expression was noted σ in Section 1.3.4, just after (1.70). By substitution in (1.84), we have
Equation (1.86) is derived in turn.
From , we have
so that . Substituting in (1.87), we obtain
The constant α denotes the inverse of the semi-major axis, which is defined from the energy by .
It is positive for an ellipse, negative for a hyperbola and null for a parabola.
Equation (1.88) is derived in turn, using (1.85).
We observe from (1.86) and (1.89) that the derivatives of t satisfy the relation
The solution of this linear fourth-order differential equation depends on four integration constants noted a0, a1, a2, a3. This solution can be expressed by introducing “universal functions” as follows.
Let us consider the function U0(χ, α) defined by the series
and the functions U1(χ, α), U2(χ, α)… defined by successive integrations
Their expressions by series are
These functions all satisfy equation (1.90). Their values in χ = 0 are
They are furthermore linearly independent.
Indeed, if a0U0 + a1U1 + a2U2 + ⋯ + anUn = 0, ∀ χ, then taking the value in χ = 0 yields a0 = 0. By deriving, we have a1U0 + a2U1 + ⋯ + anUn − 1 = 0, ∀ χ, and taking again the value in χ = 0, we get a1 = 0, and so on.
Since the functions Un(χ, α) are independent solutions of (1.90), the general solution of (1.90), which is of fourth order, can be searched under the form
To express a0, a1, a2, a3 in terms of initial conditions at t0, we associate the value χ = 0 to the initial time t0, which leads in (1.95) to a0 = 0. The other coefficients are obtained in three steps by successive derivations as follows.
Deriving (1.95) with from (1.84) yields
Applying this relation at the initial time (χ = 0) leads to a1 = r0.
Deriving (1.96) with from (1.85) yields
where we have also used , demonstrated from (1.93).
Applying this relation at the initial time (χ = 0) leads to a2 = σ0.
Deriving (1.97) with obtained from (1.86) and (1.84) yields
Applying this relation at the initial time (χ = 0) leads to a3 = 1.
We can now substitute the four coefficients a0, a1, a2, a3 in (1.95) to obtain the generalized Kepler’s equation.
This equation must be solved using the functions Un(χ, α) corresponding to the conic type. Their explicit expressions are deduced from the series formula (1.93).
For an ellipse
(α > 0)
For a hyperbola
(α < 0)
For a parabola
(α = 0)
A useful formula can be derived by summing Kepler’s equation (1.99) multiplied by α and (1.97), which gives σ = σ0U0(χ, α) + (1 − αr0)U1(χ, α).
From the series expression (1.93) of Un(χ, α), we observe that
which simplifies the terms in brackets in (1.103). We obtain an explicit relation between the variables t, χ, and σ.
Solving equation (1.99) with the functions Un(χ, α) corresponding to the conic nature (sign of α) yields the generalized anomaly χ.
In order to calculate the corresponding position and velocity by (1.66), we need expressions of Lagrange coefficients as functions of χ. For that purpose, we first calculate the derivatives of the position vector with respect to χ.
By using and , we obtain
so that
This linear third-order differential equation is similar to (1.90). Applying the same procedure as (1.91)–(1.94), the solution is searched under the form
with three constant vectors .
These vectors are determined by deriving (1.108) and identifying to (1.106) at the initial time (χ = 0).
This yields
Substituting these constants in (1.108), we obtain
By identification with (1.66), we find the Lagrange coefficients expressed with the generalized anomaly χ, and also their time derivatives using .
These coefficients define the position and velocity at any time t from initial conditions at a time t0, with the constants , and with the generalized anomaly χ solution of Kepler’s equation (1.99).
Comparing Lagrange coefficients in variable χ(1.112) with those obtained for each conic shape in (1.77), (1.79), and (1.81), we deduce the relations between χ and E, H, or B, with the convention χ = 0 at time t0.
The universal variable formulation will be further studied in Section 6.4.4 dealing with Lambert’s problem.
Until now, we have studied the Keplerian motion in its orbital plane independently of the attracting body frame. This section introduces the orbital parameters, which position the Keplerian conic in space. Their relation to position and velocity is established and some specific properties of Earth orbits are presented, such as geo-synchronism and sun-synchronism.
The classical orbital parameters are defined in an inertial frame linked to the attracting body. The origin O is the body center of mass, which is a focus of the conical orbit. The Z axis is directed toward the North pole and the X and Y axes are in the equatorial plane. These axes are fixed (not rotating with the attracting body) and they are defined at a given reference epoch. The X axis corresponds to a reference meridian (usually passing by the vernal point or by Greenwich in the case of the Earth).
The conical Keplerian orbit is positioned in the reference frame OXYZ through the classical orbital parameters, which are the right ascension of the ascending node Ω (RAAN), the inclination i, the argument of the pericenter ω, the semi-major axis a, and the eccentricity e. These five constants define the angular momentum and the eccentricity vector as follows (see Figure 1.5).
The frame (OXYZ) is transformed into the frame (OX′Y′Z′) by a first rotation of axis Z and angle Ω, and a second rotation of axis X′ and angle i. The X′ axis is the intersection of the equatorial and orbital planes.
The ascending node AN, opposite the descending node DN, is the orbit point where the equator is crossed from South to North.
The X′ axis is called the line of nodes. The Z′ axis defines the direction of the angular momentum , orthogonal to the orbital plane.
The frame (OX′Y′Z′) is transformed into the frame (OPQW) by a rotation of axis Z′ and angle ω. The axis OW is identical to OZ′ (orthogonal to the orbital plane). The axis OP is along the eccentricity vector , pointing toward the pericenter P.
The orbit shape is defined by the values of the semi-major axis a and of the eccentricity e, from which the angular momentum modulus is determined.
The five orbital parameters (a, e, i, Ω, ω) completely determine the Keplerian orbit in space. The current position on the orbit is defined by the true anomaly θ, which varies with time according to Kepler’s equation studied in Section 1.3.
In space applications, it is frequently needed to pass from orbital parameters to position and velocity, or conversely.
The transition from orbital parameters (a, e, i, Ω, ω, θ) to position and velocity in the frame (OXYZ) involves the orbit perifocal frame (OPQW), whose axes P, Q, and W are respectively oriented by the vectors , , and .
The orbit lies in the (OPQ) plane, with the P and Q axes corresponding to the x and y axes in Figures 1.2 and 1.3.
The position and velocity components in (OPQW) are given by (1.67)–(1.68).
The transition from (OXYZ) to (OPQW) is achieved by the three successive rotations Ω [Z], i [X′], ω [Z′] as explained earlier.
The change of coordinates from (OXYZ) to (OPQW) is obtained by multiplying the matrices of these rotations. By convention, the same letter (e.g. X) is used to designate the axis, the vector orienting the axis, or the coordinate along this axis.
The reverse change from (OPQW) to (OXYZ) is obtained by swapping the matrix product order and changing the angle signs.
Formulas (1.114) and (1.116) are combined to calculate the position and velocity components in the (OXYZ) reference frame from the six orbital parameters.
The transition from the position and velocity vectors expressed in (OXYZ) to the orbital parameters is achieved by successively calculating:
the energy and semi-major axis by (
equation (1.14)
);
the angular momentum and eccentricity by
h
2
= μp = μa(1 − e
2
)
;
the eccentricity vector (
equation 1.10
).
The inclination (comprised by convention between 0 and 180 deg) is obtained from . The line of nodes is oriented along the vector (OX′ axis in Figure 1.5), and the axis normal to the line of nodes in the orbital plane is oriented along .
The three vectors , , and are used to calculate:
the right ascension of the ascending node Ω by
the argument of the pericenter ω by
the true anomaly θ by
With the above relations, the angles Ω, ω, and θ are determined unambiguously from their cosines and sines, provided that the vectors and are not null.
The classical orbital parameters are indeterminate in two cases that occur very frequently in space applications:
for an equatorial orbit, the line of nodes is not defined because ;
for a circular orbit, the major axis is not defined because .
In order to avoid such singularities, the equinoctial orbital parameters are based on the components of the vectors and , instead of the angles i, Ω, and ω.
These parameters noted (n, P1, P2, Q1, Q2, L) are related to (a, e, i, Ω, ω, θ) by
The parameters (P1, P2) define the eccentricity vector, which can be null in the case of a circular orbit, whereas the parameters (Q1, Q2)