Preliminary Design of Interplanetary Trajectories with Deep Space Manoeuvres

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Universit`

a di Pisa

FACOLT `A DI INGEGNERIA

Corso di Laurea Specialistica in Ingegneria Aerospaziale

Dipartimento di Ingegneria Aerospaziale

Preliminary Design of Interplanetary Trajectories

with Deep Space Manoeuvres

Relatori:

Prof. Giovanni Mengali

Dr. Alessandro A. Quarta

Supervisore all’estero:

Prof. Ron Noomen

Candidato:

Elisabetta Iorfida

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Abstract

The aim of the present work has been to design high-thrust interplanetary trajectories with Deep Space Manoeuvres (DSMs). The work has been developed at the department of Astrodynamics & Space Missions, Delft University of Technology, The Netherlands, within the context of Tudat, TU Delft Astrodynamics Toolbox. It is a C++ library intended to provide the functionality to perform simulations for various astrodynamics applications, e.g., interplanetary trajectory optimization, asteroid-rendezvous missions, re-entry phase, etc. During this work, about 9000 lines of code have been written, able to simulate a complete interplanetary trajectory. To model the DSMs, it has been used an indirect optimization method, the primer vector variational approach. This method allows a strong decrease in the search space, when a global optimization is applied. Many verifications about the written codes have been made, in particular to test the entire developed package, a simulation about the current Phobos-Grunt mission has been carried out.

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List of Figures

1 The Messenger MGADSM trajectory. [8]. . . 2

1.1 Conic sections and their principal characteristics [22]. . . 9

1.2 Orbital Elements [4]. . . 11

1.3 Cartesian Coordinates System (Earth-centered ) [4]. . . 12

1.4 Position and Velocity Vectors in a Cartesian reference frame and other Orbital Elements [4]. . . 13

1.5 Longitude of periapsis, $ for equatorial orbits [22]. . . 16

1.6 Relative positions of three bodies [22]. . . 17

2.1 Scheme of J-2 rocket engine, a major component of the Saturn V rocket used in the Apollo program to send men to the Moon. [27]. . 19

2.2 HETs firing in Alta IV-4 facility [10]. . . 20

2.3 Scheme of a rocket [18]. . . 20

2.4 Geometry of 2D interplanetary trajectory [21]. . . 23

2.5 Configuration of Lambert’s problem, adapted from [14]. . . 27

2.6 Plot of T vs x for selected value of q and m [14]. . . 30

2.7 Geometry of an hyperbolic encounter trajectory [21]. . . 33

2.8 Vector diagram for a gravity assist behind the target planet [21]. . . 33

2.9 Configuration of a powered gravity assist, with v∞_i < v∞_f [21]. . . . 35

2.10 Geometry of the velocity change in a powered gravity assist. Case of α > αM AX [21]. . . 37

3.1 DSM localization within a certain trajectory [26]. . . 39

3.2 Solution space for a) two-impulse transfer and b) three-impulse transfer (DSM) [28]. . . 40

3.3 Contour plot of total ∆V based on a grid sampled search space for kV∞_Lk and η [23]. . . 42

3.4 Two-dimensional view of the contour plot of Fig. 3.3, where the values of the total ∆V are indicated by the color bar [23]. . . 42

4.1 Thrust-vector orientation [16]. . . 45

4.2 Three-burn CSI switching function and thrust profile [18]. . . 48

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List of Figures vi

4.3 Optimal three-impulse trajectory and primer magnitude [18]. . . 49

4.4 A fixed-time impulsive rendezvous trajectory [18]. . . 54

4.5 Reference and perturbed trajectories (fixed conditions), with fixed transfer time T = tf − t0[16]. . . 56

4.6 Primer magnitude (non-optimal) characteristics [16]. . . 59

4.7 Optimal three-impulse primer’s magnitude [26]. . . 60

5.1 Basic block-diagram of Tudat. . . 62

5.2 Example of modularity of Tudat. . . 63

5.3 Block-diagram of Astrodynamics group. . . 67

5.4 Block-diagram of Mission Segments and States classes (sub-groups). 68 6.1 Examples of unit tests about coordinate conversions, values taken from [22]. . . 70

6.2 Benchmark of Lambert’s problem code. . . 71

6.3 Benchmark of Lambert’s problem code compared to Hohmann results. 71 6.4 Earth to Jupiter Hohmann transfer. . . 72

6.5 Percentage deviation of semi-major axis compared with aHOM vs time-step size. . . 73

6.6 CPU time spent vs time-step size. . . 73

6.7 Position and velocity percentage deviation between a STM propaga-tor and an Euler integrapropaga-tor. . . 74

6.8 Flow chart of the algorithm used to implement DSM. . . 75

6.9 Propagation of p for an Earth to Jupiter Hohmann transfer. . . 76

6.10 Phobos-Grunt mission trajectory. . . 78

6.11 Phobos-Grunt mission: first propagation of p. . . 78

6.12 Phobos-Grunt mission: second propagation of p. . . 79

6.13 Phobos-Grunt mission: third propagation of p. . . 80

6.14 Phobos-Grunt mission: final propagation of p. . . 80

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List of Tables

1.1 Conics classification based on the value assumed by e [22]. . . 8

1.2 Conics classification based on the value assumed by a and E [22]. . 10

1.3 Radii of the spheres of influence of the Planets in the Solar System

[22]. . . 17

2.1 Solution cases of Lambert’s problem with relation to x (according

to the definition of Gooding, [14]). . . 28

2.2 Periapsis distance limits [21]. . . 34

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List of Symbols

α Slack Variable [kg2 _s-2_] Γ Thrust Acceleration [m s-2_] B LRL Vector [m3 _s-2_] g Gravitational Acceleration [m s-2] h Angular Momentum [kg m2 s-1] p Primer Vector [-]

E Specific Mechanical Energy [m2 _s-2_]

T Orbital Period [s]

µ Gravitational parameter [m3 _s-2_]

Ω Longitude of the Ascending Node [rad]

ω Argument of Periapsis [rad]

θ True Anomaly [rad]

$ Longitude of Periapsis [rad]

a Semi-Major Axis [m]

C Effective Exhaust Velocity [m s-1_]

e Eccentricity [-]

G Universal Gravitational constant [m3 s-2 kg-1]

i Inclination [rad]

Isp Specific Impulse [s]

l True Longitude [rad]

m0 Initial Propellant Mass [kg]

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List of Tables ix

mb Burn-out Propellant Mass [kg]

p Semi-Latus Rectum [m]

T Thrust [kg m s-2]

tb Burn-out Time [s]

u Argument of Latitude [rad]

v∞ Hyperbolic excess velocity [m s-1_]

vc Circular Velocity [m s-1]

vesc Escape Velocity [m s-1]

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Introduction

The human being has always been fascinated by the surrounding space and the last century can be even considered as the begin of the space age. The exploration, firstly of our planet and its satellite and afterwards of the entire Solar System (and beyond...), has been carried out by many space missions. The analysis and optimization of interplanetary space trajectories is then one of the most challenging topic that astrodynamics scientists deal with.

An interplanetary trajectory is usually designed in the frame of the patched-conics technique. In this method, different conic arcs, solutions of a number of Lambert’s problems, are linked together to define the whole transfer trajectory suitable for the considered mission. The patched-conics method is based on a two-body representation and on impulsive velocity changes provided by chemical high thrust engines; in addition it allows the definition of multiple gravity assist (MGA) transfers: complex trajectories made up by a sequence of planet-to-planet transfers in which the spacecraft exploits each planet encounter to achieve a velocity change. This method is commonly used in astrodynamics, and several past missions flew along MGA trajectories to reach both inner and outer planets. In the last decades, mission designer scientists have exploited the benefits of approaching complex MGA problems from a global optimization point of view. Nowadays, the aim of the trajectory design is not only to find a solution, but also to find the “best” solution in terms of propellant consumption: the problem consists in looking for the optimal solution in the regions of the search space that satisfy the problem constraints. Unfortunately, the MGA problems are characterized by an objective function with a large number of clustered minima which are prevalently associated to the complex relative motion of the planets involved in the transfer, and to the non-linearities governing the simple Kepler’s problem. Furthermore the class of MGA transfers does not cover all the possible trajectories for a chemical propelled spacecraft, an important feature to take into account is the possibility to perform deep space manoeuvres (DSMs) that are usually carried out to improve the performances of a particular trajectory. When DSMs are included into a MGA transfer, the resulting trajectory is usually referred to as MGADSM.

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Introduction 2

The importance of MGADSM is illustrated with the example of Fig. 4.2, it shows the nominal transfer trajectory for the NASA’s Messenger spacecraft that has inserted into an orbit around Mercury in March 2011 after flying-by the Earth (one time), Venus (two times), and Mercury itself (three times). One of the most challenging tasks in astrodynamics is finding a feasible solution to such a problem. Probably, the solution would not have been possible if several DSMs had not been included. As clearly shown in Fig. 4.2, the nominal solution involves five DSMs to be performed during the whole transfer. The aim of each DSM is the correction of the trajectory to get an optimal approach with the body to be encountered. It is

Figure 1: The Messenger MGADSM trajectory. [8].

clear that the introduction of DSMs, for chemical propelled spacecraft, gives some degrees of freedom that can be exploited not only to perform better gravity assists, but also to have encounters that would not be possible otherwise. A problem of this type would cover all the possible trajectories for a chemical propelled spacecraft. Furthermore, a MGADSM trajectory can be applied as first guess solution for the design of low-thrust gravity assist transfers, where the thrust arcs can spread the impulsive velocity changes given by the DSM [8].

The search space of an optimum solution when DSMs are included is not a trivial task. First of all, its dimension increases because the generic trajectory leg is described by four or even five variables if manoeuvres out-of-plane are included.

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Introduction 3

Consequently, difficulties are introduced in the manoeuvre modelling as well as in the definition of suitable constraints for the introduced variables. Another characteristic associated to the MGADSM problem is the proliferation of local minima, that makes difficult the detection of the best solution. Because of its complexity, there are many studies about models for this kind of problems, such as [16], [19], [26], [28], [29].

Outline of the work

The idea of this thesis came up with the necessity to be able to analyse and model this kind of trajectory, within the contest of the Tudat library [3]. Tudat stands for “TU Delft Astrodynamics Toolbox” and it is developed by students and staff of the Astrodynamics & Space missions section of the faculty of Aerospace Engineering, Delft University of Technology, The Netherlands and it is written in a C++ programming language.

A previous study of MGADSM transfers has been made at TU Delft [23], but the results obtained were not completely exhaustive. The model developed in [23] brought to a complex cost function with many clustered minima, and even a genetic algorithm (GA) optimization method was not able to find the best condition. Due to this considerations, the initial aim of this work was to check and continue the study of [23], trying to find a smoother cost function and carrying out an optimization process.

At the beginning of the study, it was not taken into account the additional comple-xity given by the development of all the codes within the “Tudat context”. The procedure linked to the coding inside this C++ library (coding protocols [17]), can delay the research: the code is firstly written according to the commandments of Tudat, afterwards it is checked by a (or more) different person involved in the project and lastly it is loaded inside the repository of the website by the use of Bazaar and Tortoise SVN.

When this thesis started (October 2010), the project of Tudat was at its very beginning and in the repository there was a big lack about the basic codes necessary to design an interplanetary trajectory. Therefore, the requirement to develop those fundamental codes became the priority. This is the reason why at the end this thesis was not focused on an optimization study but only on the modelling of a control method, able to reduce the search space of the cost function when DSMs are included. The approach examined and developed has been the one proposed by Lawden [19], that is the primer vector method.

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Introduction 4

Chapter 1 and Chapter 2 summarize the theory needed to model interplane-tary trajectory, that is the basic codes developed in Tudat, Chapter 3 introduces the DSMs, while Chapter 4 shows the primer vector method and the approaches followed to develop it. Chapter 5 defines the layout of Tudat and the procedure that must be followed to write a code inside it and Chapter 6 illustrates some of the benchmarks developed for the codes, with a final test-case made up as overall simulation of the model. Lastly, in Chapter 7 are presented the conclusions reached at the end of this work, and are given also some advises and recommendations for future development.

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Chapter 1 Principles of Astrodynamics

In order to analyse an interplanetary trajectory, it is necessary to use the basic tools given by the theory of astrodynamics. This chapter gives an overview of these fundamental principles, underlining the necessary ones for this thesis.

1.1 The n -body problem

In a system of n bodies, it is assumed that the i -th body has mass mi and

coordinates (xi, yi, zi) in an inertial reference frame T (O; x, y, z). All the bodies are approximated as be material points. Let’s consider the motion of the point

of mass mi under the gravitational attraction of the other n − 1 points (of mass

m1, m2, . . . , mn) and also under the effects of other non-gravitational forces (such as atmospheric drag, propulsive thrust, solar radiation pressure, etc.). The total force exerted on the i -th body, F is

F = F∗+ n X j=1 j6=i Fgj (1.1)

where F∗is the resulting non-gravitational force and Fgjrepresents the gravitational

force of the j -th body. The Newton’s law of universal gravitation states that the j -th body exerts on the i -th body an attractive force, that can be expressed by

Fgj = −G

mimj rij3

rji (1.2)

where rij is the distance vector from body i to body j and G is the universal

gravitational constant (in the SI, G = 6.6726 × 10−11 m3 _s-2 _kg-1_{). If we consider}

the case of a body in space, subjected only to mutual attraction forces with other

bodies, F∗ can be neglected. Defining rj as the position of the generic i -th body in

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1.2. The two-body problem 6

the inertial frame, for the i -th body it holds

mid 2_ri dt2 = n X j=1 j6=i Gmimj rij3 rij (1.3)

Therefore there are 3n second-order coupled differential equations of motion (3 for each body) and this system in general cannot be solved analytically. The only way to do that is approaching numerical solutions (although in some restricted three-body problems and all two-body problems, analytical solutions exist).

1.2 The two-body problem

If we consider only two bodies, neglecting all the other bodies mutual attractions, Eq. 1.3 reduces to:

¨

r = −GM + m

r3 r (1.4)

where r represents the position vector of the secondary body of mass m respect to the primary body of mass M . Due to the fact that usually the primary body has a mass much bigger then the secondary one (M m), we can assume that M + m ≈ M . We can then define the gravitational parameter as

µ , G(M + m) (1.5)

Using µ, in addition to the approximation on the masses, Eq. 1.4 becomes ¨r = −µ

r3r (1.6)

In the case of the Earth (whose astronomical symbol is ⊕), its gravitational

parameter is µ⊕ = 3.986 × 1014 m3 s-2. Eq. 1.6 is the fundamental equation for the

two-body problem.

1.2.1 Constants of the motion

Applying some mathematical operations on Eq. 1.6, yields expressions for the two mechanical constants of the motion:

E , v 2 2 − µ r = const h , r × v = const (1.7)

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where the first equation represents the conservation of the specific mechanical energy, E (being the first term on the right-hand side the kinetic energy and the second term the potential energy), and the second one shows the conservation of the angular momentum, h, where r (and its magnitude r ) represents the position vector, as already said at the beginning of the section, and v the velocity vector.

1.2.2 Trajectory equation

A combination of Eq. 1.6 and Eq. 1.7 in polar coordinates, results in the equation of the trajectory

r = h

2_/µ

1 + B

µ cos (φ − ω)

(1.8)

where B represents the magnitude of the constant Laplace - Runge - Lenz vector

(LRL), B1_{, r and φ the polar coordinates and ω a constant of integration (note}

that φ − ω is the angle between B and r). The shape of Eq. 1.8 resembles the formula of a conic section in polar coordinates

r = p

1 + e cos θ (1.9)

being p the semi-latus rectum, e the eccentricity and θ the true anomaly. This means that the most generic trajectory, solution of the two-body problem, is a conic trajectory. The eccentricity can be seen as the constant ratio of the distance from any point on the conic section to the focus and the distance from that point

to the directrix: e = r/d = p/d0. Comparing Eq. 1.8 and Eq. 1.9 it follows that:

p , h

2

µ ; e ,

B

µ ; θ = φ − ω (1.10)

The parameter p determines the size of the conic section and e establishes univocally the shape of the conic, see Tab. 1.1.

For φ = ω, the value of r is minimum and it is called periapsis. In this point the value assumed by θ is zero, and then it is measured from here. The angle ω indicates the position of the periapsis and, if the conic section is closed (circumference and

1_{For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion,}

meaning that it is the same no matter where it is calculated on the orbit; equivalently, IT is said to be conserved. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them. It is used to integrate the equation

d dt( ˙r × h) = µ d dt( r r) into ⇒ ˙r × h = µ r r + B It can be connected to the eccentricity vector by B/µ = e [22].

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Table 1.1: Conics classification based on the value assumed by e [22].

Eccentricity Conic Section

e = 0 circumference

0 < e < 1 ellipse

e = 1 parabola

e > 1 hyperbola

ellipse), exactly opposite to the periapsis can be found the apoapsis, where r is maximum. The line that joins the periapsis and the apoapsis is called line of apsides. These conic sections representing orbits in space are often called Keplerian orbits, since J. Kepler was the first to understand the movement of the planets. He described them by empirical laws, including the one which states that the orbit of a planet around the Sun can be approximated by an ellipse.

1.2.3 Keplerian orbits

Eq. 1.9 implies that the origin of the reference systems is one of the foci of the conic and, in the two-body problem case, it means that the primary body lies in this specific point. The position of the primary body is then called primary focus,

F , while the other focus is called secondary focus, F0. As it can be seen in Fig. 6.1,

the geometric definition of the p is clear. For each conic can be also defined the

semi-major axis |a|, and the distance between the two foci, F F0 _{= |2c|. For ellipse}

and circumference, both a and c are positive, on the other hand they are negative in the hyperbolic case. For all the conics (except parabola), the eccentricity can be

defined as e = c/a and the semi-latus rectum is p = a(1 − e2). In the parabolic

case e = 1 (see Tab. 1.1), a → ∞, c → ∞ and p is the one shown in Fig. 6.1. For elliptical orbits, it can be useful to express the radius at periapsis and one at apoapsis, combining together the expression of p and Eq. 1.9 as

rp = p 1 + e cos 0 = p 1 + e = a(1 − e) rp = p 1 + e cos π = p 1 − e = a(1 + e) (1.11)

From Eq. 1.11 we can also derive that e = c/a = (ra− rp)/(ra+ rp).

It can be shown that in the case of an ellipse, the total time needed to complete one revolution (empirically found for the first time by Kepler and later on confirmed

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Figure 1.1: Conic sections and their principal characteristics [22].

by Newton’s laws of motion) is given by

T = 2π s

a3

µ (1.12)

This specific time is called orbital period.

1.2.4 Characteristic velocities

In an astrodynamics problem, there are some common peculiar properties that can be useful to solve it, such as particular values of the magnitude of the velocities. The necessary velocity to keep a satellite on a circular orbit, is called circular velocity and it is

vc = r µ

r (1.13)

The velocity needed to escape an attracting body is called escape velocity and it can be defined as the necessary velocity to put a satellite on an open orbit. Considering that the parabola is the first open curve when increasing e monotonically, it will be the velocity at which e = 1, p is finite and a → ∞. It

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1.2. The two-body problem 10 yields vesc = r 2µ r = √ 2vc (1.14)

If the satellite is at a distance r0 respect to the central body, and it holds a velocity

v0 > vesc, that means that it is on an hyperbolic trajectory. In this case the velocity

takes a maximum value at the periapsis of the hyperbola, vp and a minimum value

at r → ∞, v∞ vp = r −µ a e + 1 e − 1 ; v∞ = r −µ a (1.15)

This means that at an infinite distance there is still a relative velocity between the two bodies, that is called hyperbolic excess velocity. Combining together Eq. 1.10, 1.11 and the expression of p for elliptical and hyperbolic orbits we obtain

E = −µ

2a (1.16)

this expression is valid for all orbit types and it is summarized in Tab. 1.2.

Table 1.2: Conics classification based on the value assumed by a and E [22].

Conic Section Semi-Major Axis Energy

ellipse (and circumference) a > 0 E < 0

parabola a → ∞ E = 0

hyperbola a < 0 E > 0

Eq. 1.7 together with Eq. 1.16 yields to the important Vis-Viva equation (also referred to as “orbital energy conservation equation”)

v2 = µ2 r − 1 a (1.17) Combining together Eq. 1.14, 1.15 and 1.17 one obtains

v2− vesc2 = v∞2 , C3 (1.18)

it implies that at any point in the hyperbolic trajectory, the local velocity can be expressed in terms of local escape velocity and hyperbolic excess velocity. The

parameter C3 (the departure energy [15]) has the dimension of a square velocity

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1.3. Orbital Elements 11

1.3 Orbital Elements

To be able to unambiguously define the motion of a satellite (secondary body), it is necessary to have six scalar quantities, such as the three components of the velocity vector and the position vector at a certain time in a specific Cartesian reference frame. As an alternative, it is possible to use an other set of elements that univocally defines the shape and orientation of the orbit and the position of the satellite on it [22]. We can define the set I of orbital classical elements as

I = {a, e, i, Ω, ω, θ}

In Fig. 6.3 some of them are shown in the case of a closed orbit (where the position of the satellite is indicated by S ).

Figure 1.2: Orbital Elements [4].

The elements are defined as a : semi-major axis e : eccentricity

i : inclination

Ω : longitude of the ascending node ω : argument of periapsis

θ : true anomaly

The elements a, e and θ are already known from the previous paragraphs.

The inclination, i ∈ [0, π] is the angle between the reference plane and the orbital plane. For i 6= (0, π), the intersection between the orbital plane and the reference

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plane is called line of nodes. This line intersects the orbit in only one point if we are in the case of open orbits and in two points in the case of closed orbits. These points of intersection are called orbital nodes and if a reference direction from one side of the plane of reference to the other is defined, the two nodes can be distinguished. For geocentric (around the Earth) and heliocentric (around the Sun)

orbits, the ascending node _{(or North node), is where the orbiting body moves}

North through the plane of reference, and the descending node _{(or South node)}

is where it moves South through the plane. In the case of bodies outside the Solar System, the ascending node is the node where the secondary orbit goes beyond the observer and the descending node is the node where it moves towards the observer. The longitude of the ascending node, Ω ∈ [0, 2π] is the angle from a reference direction, called origin of longitude, to the direction of the ascending node, measured in the reference plane. Note that for a geocentric orbit , the reference plane is the

Earth’s equatorial plane and the origin of longitude is the First Point of Aries _.

In this case, Ω is also called the right ascension of the ascending node, or RAAN. The angle is measured eastwards (or, as seen from the north, counter-clockwise)

from_{to the node.}

The argument of periapsis, ω ∈ [0, 2π] is defined as the angle between the line of nodes and the direction of the eccentricity vector e, that coincides with the line of apsides in the direction of the periapsis of the orbit. The angle is measured in the orbital plane and in the direction of motion.

1.3.1 Orbital Element Conversions

Given the position and velocity at any time, we can determine in principle the orbital elements and, from that, the position and velocity at all future times [30]. In practice the limitation are both on the orbit determination process and on the accuracy with which the forces on the spacecraft are known. The biggest problem with determining the elements is in the fact that the errors grow with

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time, but in general for many orbits and for most aspects of orbit analysis, simple computations can provide good approximations of the final results. Let’s begin considering a Cartesian Coordinates reference system, where the X and the Y axes

lie in the equatorial plane, the X axis points at the direction of_{, the Z axis is at}

90◦ degrees respect to the equatorial plane and extends through the North Pole,

see Fig. 6.4. This system is called Geocentric-Equatorial coordinates system. In the case of interplanetary missions, the reference system has its origin in the

center of the Sun, the plane (Xε, Yε) coincides with the ecliptic plane2, the Xε is

defined by the intersection between the ecliptic plane and the Earth’s equatorial

plane and the Zε is pointing the North. This frame is called Heliocentric-Ecliptic

coordinates system [22].

Orbital Elements from Position and Velocity Vectors

The basic geometry of determining the orbital elements from the position, r and velocity, v is shown in Fig. 6.5.

Figure 1.4: Position and Velocity Vectors in a Cartesian reference frame and other Orbital Elements [4].

2_{The ecliptic plane is the geometric plane containing the mean orbit of the Earth around its}

real motion around the Sun. It is inclined respect to the equatorial plane of an angle ε ∼= 23.5◦ [22].

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Firstly it is necessary to find the magnitude of the position and velocity vectors: krk = r

kvk = v (1.19)

After that the angular momentum is determined by Eq. 1.7, and the unit vector to

the ascending node _bn is found by

n , bZ × bh = bZ × h

khk ⇒ n =b

n

knk (1.20)

The first orbital element to be determined is the eccentricity, e as magnitude of the eccentricity vector, e which lies in the direction of periapsis and determines the fundamental directions in space:

e , v × r

µ −

r

r ⇒ e = kek (1.21)

Combining together Eq. 1.7 and Eq. 1.16 we can find the semi-major axis, a

a = − µ 2E = 2 r − v2 µ −1 (1.22) All the other orbital elements can be determined combining together the previous quantities as (the subscript stands for the component of that specific axis):

i = arccos hz khk Ω = atan2 (ny, nx) ω = sign_ω· a cos (_be ·_bn) θ = sign_θ· a cos (_br ·_be) (1.23) where sign_ω = ( +1 if n(n × e) · h > 0_b −1 if n(n × e) · h < 0_b sign_θ = ( +1 if (e × r) · h > 0 −1 if (e × r) · h > 0 (1.24)

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Position and Velocity Vectors from Orbital Elements

The reverse problem of the previous section consists in determining the position and velocity vectors, given the orbital elements at a specific time. In the absence of perturbation the only orbital element that changes in time is the true anomaly, θ. Given a and e it is firstly necessary to compute the semi-latus rectum by its definition

p = a(1 − e2_{). Defining an auxiliary reference frame, called Perifocal coordinate}

system (PQW) as a system centered at the focus of the orbit (principal body),

whose XPQW axis is directed towards the periapsis, the ZPQW axis is perpendicular

to the orbital plane and has the direction of h and YPQW lies in the orbital plane.

The component of the position in the PQW system are given by

rPQW= p 1 + e cos θ " cos θ sin θ # (1.25)

and of the velocity by

vPQW= r µ p " − sin θ e + cos θ # (1.26)

The ZPQW components is obviously zero for both the position and velocity vectors.

It is necessary now to transform the value just found into Cartesian coordinates, and it is possible by the coordinate transformation matrix, TPQW-CAR

r = TPQW-CARrPQW

v = TPQW-CARvPQW

(1.27)

where the 3x3 transformation matrix is given by

TPQW-CAR =               cos Ω cos ω

− sin Ω sin ω cos i

− cos Ω sin ω

− sin Ω cos ω cos i sin Ω sin i

sin Ω cos ω

+ cos Ω sin ω cos i

− sin Ω sin ω + cos Ω cos ω cos i

− cos Ω sin i

sin ω sin i cos ω sin i cos i

              (1.28)

Auxiliary orbital elements

The transformations defined in the previous sections are not valid for all kind of orbits. The first evident difference has to be done for parabolic orbit, for which a → ∞ and its value cannot be used in the equations. In this case instead of a is used the semi-latus rectum p, that is derived from Eq. 1.10 and can be directly used in Eq. 1.25 and Eq. 1.26. An other particular case occurs if the orbit is

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1.4. Sphere of Influence 16

Figure 1.5: Longitude of periapsis, $ for equatorial orbits [22].

equatorial (i = 0 or i = π). In this case both ω and Ω are not defined, it is then necessary to introduce an auxiliary element, that is the longitude of periapsis, $

see Fig. 6.6. It is defined by cos $ , bX ·_be, but for small value of i, $ ∼= ω + Ω.

For elliptical equatorial orbit, Ω can be set to be 0, and ω ≡ $. If the orbit is not only equatorial but even circular, also the periapsis is not defined (and with it the direction of e). In this case the auxiliary element is the true longitude, l.

It is defined by cos l , bX ·_br, but for small value of i, l ∼= Ω + ω + θ. Due to this

fact, Ω and ω can be set as zero, and l ≡ θ. The last special case are the inclined circular orbit, for which ω is not defined. It can be then introduced the argument

of latitude, u, defined as cosu ,n ·b br. For elliptical orbit in which the angle ω is

defined, the relation u = ω + θ is valid. This relation can be used in approximation for circular orbit, too, setting ω = 0 and u ≡ θ [22].

1.4 Sphere of Influence

The Keplerian orbits described in the previous sections are only valid for a two-body system. In reality there are multiple bodies to take into account, but during the different stages of an interplanetary trajectory it is possible to make an approximation of a two-body problem if the central body is identified by its gravitational effects. In those cases the Keplerian orbits are able to give an accurate description of the actual orbits and the other bodies cause only minor effects. In order to understand how to make this approximation, it is necessary to introduce the concept of sphere of influence. Inside this region, the spacecraft is in orbit around a specific planet and its motion can be considered effected only by the

gravitational effect of this celestial body (see Fig. 1.8). The body P2 represents

the secondary body (the spacecraft) and its motion is described with respect to

the central body P1 and the body P3 is a disturbing body. Denoting the masses

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1.4. Sphere of Influence 17

Table 1.3: Radii of the spheres of influence of the Planets in the Solar System [22].

Planet rSOI [km] Mercury 112 500 Venus 616 400 Earth 924 600 Mars 577 400 Jupiter 48 233 000 Saturn 54 432 000 Uranus 51 792 000 Neptune 86 668 000

expression for the radius of the sphere of influence has been derived in literature rSOI = r13

m₁ m3

2/5

(1.29)

where r13 is the distance between P1 and P3. As definition, on the boundary of this

sphere, the ratio of disturbance accelerations of P3 with respect to the accelerations

of the central body P1 are the same. The sphere of influence then describes a

virtual sphere about body P1 with respect to body P3 and inside this volume the

gravitational acceleration of P1 is dominant over the acceleration of P3. Tab. 1.3

shows the radii of the spheres of influence of all the Planets in the Solar System.

Referring to the previous notation, the planets represent the body P1 and the Sun

the body P3.

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Chapter 2 High Thrust Interplanetary

Trajectories

The propulsion subsystem is a fundamental one in the design of a spacecraft, indeed it is crucial to define which kind of motion (and consequently which thrust) characterized the spacecraft itself. The first distinction has to be made between high thrust and low thrust trajectories. In this chapter it will first explained the difference between those two big classes and afterwards it will be shown how the principles of orbital mechanics, discussed in the previous chapter, can be applied to high-thrust trajectories and how they can be computed and analysed. There will be explained also the principles of the patched conics method with all the relative astrodynamics consequences.

2.1 Chemical Propulsion vs Electric Propulsion

At the beginning of the XX century, K. Tsiolkovsky and R. H. Goddard were the first to conceive the principles of rocketry and their contribution made the chemical propulsion to be the primary method for space transportation. It is based on Newton’s third law, that is mutual forces of action and reaction are equal, opposite and collinear. The propulsive force exerted on the vehicle results from changes in system momentum through a high exhaust velocity. In the case of chemical propulsion the combustion of chemical propellants, solid or liquid, at high pressure, liberates large quantities of energy in a small time interval. The subsequent expansion of these high temperature products of combustion through a nozzle, converts thermal energy into kinetic energy [21].

In Fig. 2.1 it is shown a chemical thruster, specifically one of the most important liquid rocket engine. It was the first production engine to use liquid hydrogen and liquid oxygen. Note the lower part of the nozzle is not shown here.

Chemical propulsion is said to be “energy limited” because the chemical reactants 18

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2.1. Chemical Propulsion vs Electric Propulsion 19

Figure 2.1: Scheme of J-2 rocket engine, a major component of the Saturn V rocket used in the Apollo program to send men to the Moon. [27].

have a fixed amount of energy per unit mass, which limits the achievable exhaust velocity or specific impulse. However, because the propellants are their own energy source, the rate at which energy can be supplied to the propellant (which is ultimately limited by reaction kinetics) is independent of the mass of propellant, so very high powers and thrust levels can be achieved [2].

K. Tsiolkovsky, R. H. Goddard, H. Oberth and many others, were not interested only in chemical propulsion, but they were also pioneers in electric propulsion field. The 1950s are the beginning of the so-called “golden age” for propulsion, and the first ion engine was built in 1959 by Dr. Harold Kaufman, an engineer at NASA Glenn [6]. In a very general sense, all types of electric thruster can be described as plasma thrusters. The most important implication of this is to assume that the working fluid employed in the thruster is an electrically conducting medium which remains quasi-neutral throughout the acceleration process [7]. In Fig. 2.2 are

shown two Hall Effect thrusters, in operation at Alta’s1 laboratory. Oppositely to

chemical propulsion, electric propulsion systems are not energy limited. Neglecting component lifetime considerations, an arbitrarily large amount of energy can be delivered (from a solar or nuclear power system) to a given mass of propellant so that the exhaust velocity (or specific impulse) can be much larger than the one available from a chemical propulsion system. Electric propulsion systems are termed “power limited” because the rate at which energy from the external source is supplied to the propellant is limited by the mass available for the power system.

1_{Alta S.p.A. is the leading European small company in the aerospace propulsion sector.}

They operate in research and development on space electric propulsion, chemical propulsion and aerothermodynamics. The company provides testing services and space systems design and computational simulation tools. With a successful record of technology transfer cases, the company is also active in the energy and industrial plasma sectors [1].

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Figure 2.2: HETs firing in Alta IV-4 facility [10].

This has the result of limiting the thrust of the electric propulsion system for a given spacecraft mass. Because of this, electric propulsion vehicles tend to be low thrust to mass ratio (and therefore low acceleration) vehicles [2].

2.1.1 Orbital Manoeuvres

Let us consider a generic spacecraft moving at a velocity v(t), subjected to a certain number of external forces (gravity, atmospheric drag, etc.) whose resultant is F. One of the most important characteristic useful in the description of its motion, is the thrust, T. It can be defined as the amount of force applied to the rocket based on the expulsion of gases, and its magnitude can be calculated as:

T = ˙mve+ Ae(pe− pa) (2.1)

where Ae is the nozzle exit area, ve is the propellant exhaust velocity, pe the gas

pressure at the nozzle exit, pa the ambient pressure and ˙m is the propellant mass

flow rate, see Fig. 2.3.

Figure 2.3: Scheme of a rocket [18].

This expression may be simplified by the definition of an effective exhaust velocity, C as the combination of the actual exhaust velocity and a term representing the

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pressure force acting on the spacecraft

C , ve+ Ae

˙

m(pe− pa) (2.2)

Eq. 2.1 reduces then to

T = ˙mC (2.3)

For the study of the motion, it can be introduced an other important quantity, the

specific impulse, Isp which represents the total impulse per unit propellant weight

consumed: Isp , Z t 0 T (t) dt g0 Z t 0 ˙ m(t)dt (2.4)

where g0 is the acceleration due to the gravity at the central body surface. In case

of constant thrust and exhaust mass flow rate, Isp ≡ T /(g0m) = C/g˙ 0. Integrating

the momentum equation of the spacecraft in terms of C = const, over the duration of spacecraft motor firing, it yields

∆V = C lnm0 mb + Z tb 0 F mdt (2.5)

where m0 and mb are, respectively, the initial and burn-out propellant masses and

tb represents the burn-out time. In the case of a short burn time (the external

forces are not impulsive, that is F dt → 0) or if there are no such external forces, we can obtain the fundamental Tsiolkovsky rocket equation:

∆V = C ln m₀ mb ≡ Ispg0ln m₀ mb (2.6)

The quantity m0/mb takes the name of mass ratio. It is quite evident from Eq. 2.6,

that a high C is desirable. ∆V is usually prescribed by the mission, whilst C is limited by the available propulsion system and propellants [18].

An orbital manoeuvre is the definition of the strategy acted to modify some (or all) orbital parameter, through the application of one or more propulsive action as well as the determination of the total velocity ∆V , necessary to be able to accomplish the manoeuvre. From Eq. 2.6 is evident that an increasing ∆V yields to a higher propellant mass, due to that effect ∆V is a precise indication of the cost of the manoeuvre itself.

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2.2. Patched Conics 22

2.1.2 Thrust Level

It is useful to be able to determine the conditions under which the approximation of impulsive manoeuvre is possible, and, if not, when it is necessary to have a more refined model, and probably more complex, too. According to [12], a useful relation to quantify the degree of thrust provided by the propulsive subsystem is the value of the Thrust-to-Weight ratio. It shows the ratio between the magnitude of the

available thrust, T, and the local weight of the spacecraft, Ws evaluated as

Ws, m

µ

r2 (2.7)

where m is the instantaneous mass of the vehicle and r is its distance from the centre of mass of the primary body (whose gravitational parameter is µ).

The distinction can be made as:

High Thrust: (T /Ws) ≥ 0.5

In this case, when the engine is on, the thrust is the dominant effect and the behaviour of the vehicle can be studied, in first approximation, as an impulsive manoeuvre. This is mostly the case of chemical propulsion subsystems.

Low Thrust: (T /Ws) ≤ 10−5

If this is the case, to be able to obtain a significant modification of the orbital parameters and to make the manoeuvre, the thrust has to be continuous on the spacecraft and has to work for a long time. In this situation the thrust is so low that it can be even treated as an orbital perturbation, in the way to obtain semi-analytical models. This is mostly the case characterizing electrical propulsion subsystems.

In all the intermediate cases, it is not possible to analyse the motion of the space-craft through simple analytical models, and, due to that, it is necessary to use numerical simulations.

After the distinction between chemical and electrical propulsion subsystems, it is quite evident that the approximation of impulsive manoeuvre can be made only

in the former case, where the engines have an average Isp = 250 s. In the case of

electric thrusters, the maximum thrust magnitude is around 20 N, but the Isp can

even reach value around 10 000 s [22].

2.2 Patched Conics

The hypothesis of a two-body dynamic (Kepler orbits) in the case of interplane-tary trajectories is not always achievable, because the spacecraft can be subjected

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to the influence of different bodies of the Solar System depending on its operating position. As already mentioned in Sec. 1.4, an interplanetary mission can be subdivided in a convenient number of stages; during each stage the spacecraft can be considered to be subjected to the gravitational attraction of only one celestial body and, thanks to that, a keplerian analysis can be carried out. This is possible because the sphere of influences are quite small with respect to the interplanetary distances, for example in the case of the Earth it is less than 1% (924 600 km vs 150 000 000 km). Therefore the spacecraft will fly along different conic arcs, that are consequently patched to not have discontinuities along the overall path. The choice on where and how those “patching points” have to be, is the most tricky phase of the analysis and it can introduce some errors.

This way to proceed is called patched conics method, and it has the advantage to be enough accurate and decent during a preliminary analysis, being also quite simple. The concept of sphere of influence, already explained in Sec. 1.4, gives the criterion on which is the primary body for each phase. In this section it is proposed the ordinary method to evaluate the ∆V s necessary to conduct an hypothetical interplanetary mission using impulsive manoeuvres.

Furthermore it is assumed that the planets move around the Sun with circular and coplanar orbits on the ecliptic plane (already defined in the previous chapter). The most simple case of interplanetary trajectory can be then approximate by three consecutive conic sections (see Fig. 2.4):

Figure 2.4: Geometry of 2D interplanetary trajectory [21].

1. planetocentric escape phase around the departure planet; 2. heliocentric transfer phase;

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3. planetocentric capture phase around the target planet.

2.2.1 Launch ∆V

Let’s first consider the ∆V required to get launched from a parking orbit

around the initial planet into the interplanetary trajectory, ∆VI. For this, both

the velocities prior to and after the application of ∆VI have to be known.

Before applying the ∆VI at issue, the velocity equals the velocity at periapsis of

the initial parking orbit

vIp = vIc √ 1 + eI = r µI rI √ 1 + eI (2.8)

where vIc is the local circular velocity at periapsis altitude, µI the gravitational

parameter of the departure planet, eI the eccentricity of the parking orbit and rI

the distance at periapsis. To obtain the velocity after ∆VI is applied, we have to

consider the heliocentric phase of the spacecraft, when it describes an hyperbolic orbit with respect to the departure body. The velocity in such an hyperbolic leg is obtained with the use of Eq. 1.18:

vH2 = vesc2+ v∞2 (2.9)

with vesc the local escape velocity and v∞ the hyperbolic excess velocity. At the

periapsis altitude of the initial parking orbit, the local escape velocity can be found

using vIc combined to Eq. 1.14, while the hyperbolic excess velocity is given by

vectorially subtracting the heliocentric velocity of the departure planet, vd, from

the heliocentric velocity in the hyperbolic leg

v∞= kvI− vdk (2.10)

Substituting in Eq. 2.9, the Eq. 2.10 and C3 as defined in Eq. 1.18 (a subscript is

added to this notation to indicate that it considers the C3 value for the insertion

into interplanetary orbit, C3I), the hyperbolic velocity at periapsis altitude of the

initial orbit is given

vHp =

r 2µI

rI + C3I (2.11)

Now that both the velocity before and after the application of ∆VI are known, it

can be computed as ∆VI = vHp− vIp = r 2µI rI + C3I − r µI rI √ 1 + eI (2.12)

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2.2.2 Capture ∆V

The same approach as given above for the launch ∆V is needed to calculate the necessary ∆V to get captured into an orbit around the target planet at the end of the interplanetary transfer trajectory. The only difference is that in this case, the velocity before the application of the ∆V equals the hyperbolic velocity at periapsis altitude of the final orbit and the velocity after the application of the ∆V equals the velocity at the periapsis of this final parking orbit.

To obtain ∆VF, Eq. 2.8 and Eq. 2.11 can be used in case that rI and eI are

substituted respectively by rF and eF and v∞ is calculated using the heliocentric

velocity at the end of the interplanetary trajectory, vF and the heliocentric velocity

of the target planet, vt. By squaring this v∞, C3F is obtained and ∆VF can be

then calculated as ∆VF = r 2µF rF + C3F − r µF rF p1 + ef (2.13)

2.2.3 Hohmann transfer orbit

Looking at Fig. 2.4, the geometry of the heliocentric section has some variable involved. It can be shown that the most energy-efficient trajectory is the one where

γI = γF = 0 [22]. This results in half of an ellipse and is called Hohmann trajectory

whose semi-major axis follows to be

aHOM =

rd+ rt

2 (2.14)

where rd and rt are respectively the distance of the departure planet and of the

target planet from the Sun. The necessary condition for this trajectory to occur is that the hyperbolic excess velocity has to be aligned with the orbital velocity of the departure planet around the Sun. Therefore

v∞d = vI− vt= r µ ad r 2 − ad aHOM − 1 (2.15)

where adis the semi-major axis of the departure planet (that in the case of circular

orbit, as we are considered ad≡ rd and µ is the gravitational parameter of the

Sun. In a similar way, the hyperbolic excess velocity with respect to the target planet can be found:

v∞t = vt− vF =r µ at 1 − r 2 − at aHOM (2.16)

with at the semi-major axis of the orbit of the target planet. The required ∆V for

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2.3. Lambert’s problem 26

∆VF founded in Eq. 2.12 and Eq. 2.13, where C3I and C3F are replaced respectively

by (v∞d2_{) and (v∞t}2_).

The Hohmann orbit has been introduced as the minimum energy two-impulse transfer between two planets. As said at the beginning of the subsection, the resulting trajectory is simply an ellipse with one planet at the perihelion and the other one at aphelion, but this construction does not take into account whether the target planet is going to be in that position after a specific time of flight. For example in a case where the Earth is considered as departure planet and Neptune as the target one, assuming a parking orbit around the Earth at an altitude of 185 km and the capture orbit around Neptune at 1.1 times r[ (where r[ is the radius of Neptune, ∼ 25000km) the ∆V s can be easily found using Eq. 2.12 and 2.13

∆VI = 8.25km/s ; ∆VF = 6.98km/s ⇒ ∆VT OT = 15.23km/s (2.17)

This value is appreciably large and hard (if not impossible) to be obtained using current launchers. The flight time follows by Eq. 1.12, and it is found to be 30.6 years. This is quite a long time, which hints at the direction of faster trajectories, requiring even larger ∆V s or smarter solutions.

2.2.4 Three-Dimensions interplanetary trajectories

The assumption made at the beginning of this section (that the planetary orbits are circular and lie in the same plane) can introduce significant errors, although the eccentricities and inclinations of the planets are quite small. Because of that, it is necessary to make a three-dimensional analysis that takes into account the

orbits of the planets and therefore an ephemerides2 _{model is used.}

2.3 Lambert’s problem

As shown in the previous section, although the Hohmann trajectory is the one with minimum energy, it is not always the most convenient, because for example the necessary time of flight can be too long for a real mission. Therefore it is necessary to have a method that allows to fly along a (preferably keplerian) trajectory in a limited amount of time. This can be done by means of the Lambert’s orbital boundary-value problem, which is described by Gooding [14] as:

“[...] an unperturbed orbit, about a given inverse-square-law centre of force, F say, is

to be found connecting two given points, P1 and P2, with a flight time ∆t(= t2− t1)

that has been specified. [...] ”

2_{The ephemerides are tables of values that give the positions of astronomical objects in the}

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The general configuration of the problem is shown in Fig. 2.5. In the position F,

focus of the orbit, lies the central body, while P1 and P2 in the practical case are

the position of two planets at times t1 and t2 respectively, from which the time of

flight easily follows. The position vectors are indicated by r1 and r2.

Figure 2.5: Configuration of Lambert’s problem, adapted from [14].

Lambert stated a theorem regarding this problem, that is the transfer time ∆t

from P1 to P2 is function of only 3 parameters: the semi-major axis of the arc, a,

the chord c (see Fig. 2.5) and the sum of the distances r1 and r2. This theorem

can be written in the form of

t2− t1 = ∆t = f (a, c, r1+ r2) (2.18)

In particular the time of flight is independent from the eccentricity of the transfer orbit.

It should be enough clear that at least two solutions exist: one is shown in Fig.

2.5 as a continuous line, and goes in the anti-clockwise direction from P1 to P2

subtending an angle θ, the other is represented by a black dotted line, goes in the clockwise direction and subtends an angle of 2π − θ. This duality can be avoided, specifying in advance the direction of the motion. Since all the planets orbit around the Sun in an anti-clockwise direction, a wise choice is that the direction of the spacecraft is also anti-clockwise. In this way only the “θ” case is considered. This fact does not mean that 0 < θ < 2π, but multi-revolutions arcs are also possible solutions to the problems. It can be shown that for a specific number of full revolutions (1 or higher) two unique solutions exist for a predefined direction of motion, provided to the fact that the time of flight ∆t is long enough. In this thesis the case of multi-revolution Lambert arcs is not considered.

The algorithm required for solving the Lambert’s problem can take many forms since a direct solution does not exist, and all the different methods have their pros and con. For this thesis, the one proposed by Gooding in 1990 will be

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applied, which is summarized in the rest of the section. For a full derivation of the algorithm, it is suggested to look at the paper and at the references mentioned in it [14].

The analysis of Lambert’s problem starts with a formula that relates the time of flight and the orbital element

∆t = s

a3

µ[E2− E1− e(sin E2− sin E1)] (2.19)

where µ is, as usually, the gravitational parameter of the primary body and E the

eccentric anomaly3_{. In Eq. 2.19 ∆t and µ are known while a, e, E need to be}

found. Accordingly to Lambert’s theorem the fundamental parameter is represented by the semi-major axis a, that in [14] is non-dimensionalized by the following transformation

x2 = 1 − s

2a (2.20)

where s is the semi-perimeter of the triangle P1F P2 and is defined as

s = r1+ r2+ c

2 (2.21)

Depending on the values assumed by x, different conic arcs become solution of the problem (see Tab. 2.1).

Table 2.1: Solution cases of Lambert’s problem with relation to x (according to the definition of Gooding, [14]).

Parameter x Conic Section

|x| < 1 ellipse

x = 1 parabola

x > 1 hyperbola

The other variable that is non-dimensionalized is the time of flight ∆t (see also Eq. 1.12) as

T∗ =

r 8µ

s3∆t (2.22)

After an exhaustive derivation, Eq. 2.19 can now be expressed by means of the non-dimensional parameters of Eq. 2.20 and Eq. 2.22 as

T = 2(x − qz − d/y)

ELam

(2.23)

3_{The eccentric anomaly is the angle between the direction of periapsis and the current}

position of an object on its orbit, projected onto the ellipse’s circumscribing circle perpendicularly to the major axis, measured at the center of the ellipse itself.

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Eq. 2.23 in turn introduces several new parameters defined as follow

q = √ r1r2 s cos(θ/2) (2.24) ELam = x2− 1 (2.25) y =p|ELam| (2.26) z =p1 − q2_{+ q}2_x2 _(2.27) d = ( arctan(f /g) if ELam< 0 ln(f + g) if ELam> 0 (2.28)

In Eq. 2.28 are introduced two new variables, f and g. They are given by

f = y(z − qx) (2.29)

g = xz − qELam (2.30)

It is worth noting that the case of ELam < 0 corresponds to an elliptical orbit

and ELam> 0 to an hyperbolic one. In order to come up with a solution to the

problem, it is needed an iteration process solved by a root-finding algorithm such as the Newton-Rapson method that can be stated as

xn+1= xn−

f (xn)

f0_(xn) (2.31)

In this procedure, f (xn) follows from Eq. 2.23 and is given by

f (xn) = T −

2(xn− qnzn− dn/yn) ELamn

(2.32)

while f0(xn) represents the first derivative of f respect to x. To ensure even faster

convergence, appropriate starters have been developed in [14]. In Fig. 2.6 it is shown a plot of T versus x, for different values of q and m (which represents the number of revolutions), this plot can even be used to determine the starters and it is clear that the sign of x has a great influence on T .

In the case of single-revolution starters (m = 0), the procedure to follow can be stated as 1. computation of T∗ by Eq. 2.22 2. computation of T (x) at x = 0 by Eq. 2.23 ⇒ T0 3. • if T0 > T∗ ⇒ x0 = T0(T0− T∗) 4

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Figure 2.6: Plot of T vs x for selected value of q and m [14].

• if T0 < T∗ x01 = −(T∗− T0)/(T∗− T0+ 4) x02 = −p(T∗_{− T0)/(T}∗_{+ 0.5 T0)} ϕ = 2 · atan2(1 − q2, 2q) W = x01+ 1.7p2 − ϕ/π if W ≥ 0 ⇒ x03= x01 if W < 0 ⇒ x03 = x01+ (−W )1/16· (x02− x01) λ = 1 + 0.5x03(1 + x01) − 0.03x032 √ 1 + x01 ⇒ x0 = λ · x03

The found x0 has to be used as starter value in the Newton-Rapson method (Eq.

2.31). After the convergence of the procedure, a value of x, and thus for the semi-major axis of the transfer arc a (Eq. 2.20), is obtained. In [14] is shown also

a method to find the velocity components in radial and transverse direction at P1

and P2 vR1 = γ (qz − x) − ρ(qz + x) r1 vT 1 = γσ(z + qx) r1 vR2 = −γ (qz − x) − ρ(qz + x) r2 vT 2 = γσ(z + qx) r2 (2.33)

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where the new parameters γ, ρ and σ are defined as

γ =r µs 2 ; ρ = r1− r2 c ; σ = 2 r r₁r2 c2 sin(θ/2) (2.34)

The velocity components vR1, vT 1, vR2 and vT 2 are defined in the planed

spanned by the triangle P1F P2, but the state of the spacecraft (and its trajectory),

is determined in an inertial, heliocentric reference frame. Therefore the value of the velocities found in Eq. 2.33 have to be converted into the frame of interests.

The procedure of the conversion starts with the position vectors of P1 and P2

r1 =    x1 y1 z1    (2.35) r2 =    x2 y2 z2    (2.36)

The longitude, λ, of P1 and P2 are given by

λ1 = atan2(y1, x1) (2.37)

λ2 = atan2(y2, x2) (2.38)

The latitude, δ, of P1 and P2 are given by

δ1 = atan2(z1, p x12+ y12) (2.39) δ2 = atan2(z2, p x22+ y22) (2.40)

The components of the radial and tangential velocities in the inertial reference frame can be converted from Eq. 2.33 as

vR1I = vR1·    cos δ1cos λ1 cos δ1sin λ1 sin δ1    (2.41) vR2I = vR2·    cos δ2cos λ2 cos δ2sin λ2 sin δ2    (2.42) vT 1I = vT 1· _r 1× r2 kr1× r2k × r1 kr1k (2.43)

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2.4. Gravity Assist 32 vT 2I = vT 2· _r 1× r2 kr1× r2k × r2 kr2k (2.44)

Finally, the inertial, heliocentric velocity vectors of P1 and P2 can be computed

using

vI1 = vT 1I + vR1I

vI2 = vT 2I + vR2I

(2.45)

2.4 Gravity Assist

When a spacecraft enters in the sphere of influence of a planet, it can exploit the gravitational action to increase or decrease its heliocentric velocity. This effect is known as gravity assist (or hyperbolic encounter, or fly-by) and it is based on an exchange of momentum between the spacecraft itself and the planet. The reference equation can be written as:

msvsi + mtvti = msvsf + mtvtf (2.46)

where ms is the mass of the satellite, mt is the mass of the target planet, vs is

the satellite velocity and vt the target planet velocity. Both velocities are taken

respect to an inertial heliocentric reference frame, and the subscripts “i” and “f ” represent initial and final conditions. Eq. 2.46 can be rewritten as

vtf − vti =

ms mt

(vsi− vsf) (2.47)

Since the mass of the satellite can be neglected respect to the mass of the planet

(usually ms is in the order of 103 kg, while mt∼ 1024− 1027 kg), it can be assumed

that ms/mt ≈ 0, hence vti = vtf. Using the velocity-addition formula, we can

write that

vs= vsp+ vt (2.48)

where vsp is the relative velocity of the satellite respect to the considered planet.

Therefore two situations can happen:

1. Transit in front of the planet: vsi > vsf.

This means that the satellite is losing kinetic energy during its motion. 2. Transit behind the planet: vsi < vsf.

In this case the satellite is acquiring kinetic energy during its motion (see Fig. 2.8).

Fig. 2.7 shows a plot of the generic configuration of the problem, which is presented

in the plane spanned by the incoming and outgoing hyperbolic excess velocity (v∞_i

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2.4. Gravity Assist 33

Figure 2.7: Geometry of an hyperbolic encounter trajectory [21].

The velocity of the satellite with respect to the planet, vsp, equals the hyperbolic

excess velocity v∞ if we consider that the distance from the center of mass of the

planet acquired by the planet on the sphere of influence, is large enough to be considered infinite. Geometrically talking, it means that at the entrance of the

sphere of influence, the direction of vsp is parallel to the asymptote of the incoming

hyperbola. Applying energy conservation in the planet reference frame, it results

that, although the directions are different, the magnitude of v∞i and v∞f are the

same. In Fig. 2.7 it is shown B, that represents the impact parameter. It is the

Figure 2.8: Vector diagram for a gravity assist behind the target planet [21].

perpendicular distance from the target planet to the incoming hyperbolic leg. From the conservation of momentum we can write that

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Using the vis-viva equation (Eq. 1.17), the distance at periapsis can be written as

rp = − µt v∞2 + s µt2 v∞4 + B 2 _(2.50)

Applying the definition of the hyperbolic excess velocity (Eq. 1.15) and other geometrical relations for the hyperbola, we obtain that

sinα 2 = 1 e = 1/ 1 + rpv∞ 2 µt = 1/1 + B 2_v ∞4 µt (2.51)

where α is the bending angle. It can be seen that if the flyby is closer (smaller B),

α is larger. For heavier planets, α is also larger and v∞ is lower. If rp is equal or

less than the planet’s radius Rt, the spacecraft will impact on the planet’s surface.

From Eq. 2.50, the condition to prevent impact can be written as

B ≥ Rt

s

1 + vescs2

v∞2 (2.52)

where vescs = p2µt/Rt is the escape velocity at the planet’s surface. When

designing a gravity assist, it must be kept in mind the restriction about the limiting distance and the requirement becomes even more stringent when taking into account the environment of a planet.

Table 2.2: Periapsis distance limits [21].

Planet Rt [km] r∗ Mission Mercury 2440 1.082 Messenger Venus 6052 1.047 Cassini Earth 6378 1.048 Galileo Mars 3397 1.076 Rosetta Jupiter 71492 1.60 Pioneer 11 Saturn 60268 1.342 Pioneer 11 Uranus 25559 4.190 Voyager 2 Neptune 24764 1.181 Voyager 2

As a reference, the smallest fly-by distances of previous and current missions

are taken into account to define the smallest allowable periapsis distance, rp,limit

which is given by the product of the radius of the planet, Rt, with an appropriate

coefficient, r∗ (see Tab. 2.2). In some cases, a mission could get closer than the

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2.4.1 Powered Gravity Assist

The theory described in the first part of this section can be applied only if two necessary conditions are satisfied:

1. v∞_i = v∞_f 2. α < αM AX where αM AX = αi 2 + αf 2

The first condition states the matching between the arrival and departure velocities at infinity, while the second indicates that the required bending angle has to be smaller than the maximum achievable. Using Eq. 2.51, the second condition becomes αM AX = arcsin 1 1 + rp,limitv∞i 2 µt + arcsin 1 1 + rp,limitv∞f 2 µt (2.53)

with rp,limit substituted for rp.

The geometry of a swing-by in which the first condition is not valid, can be seen in Fig. 2.9. As explained in [20] (where this problem is referred to as vis viva

matching) this match does not happen if the v∞_i, given as output of the Lambert’s

problem of the first leg is different from the v∞_f, given as requirement of the mission

and that can be directly related to the energy (see Eq. 1.16, 1.17 and 1.18). If the required bending angle, α, associated with this two hyperbolic excess velocity

(composed by two components, one related to v∞i and the other to v∞f) is larger

than the maximum one, rp is simply equal to rp,limit while if it is smaller than

αM AX, the value for rp that results in this angle (see Eq. 2.51) has to be found.

In this way it is achieved the required bending angle, function of the distance at pericenter.

Figure 2.9: Configuration of a powered gravity assist, with v∞i < v∞f [21].

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needed rp. The equation that needs to be solved is

α = arcsin 1 1 + rpv∞i 2 µt + arcsin 1 1 + rpv∞f 2 µt (2.54)

To be able to come out with a solution, it is necessary to use another variable, that

is the eccentricity e. Since v∞_i 6= v∞_f, the incoming hyperbolic leg has a different

eccentricity than the one of the outgoing leg, but they are related by (see Eq. 1.11)

rp = ai(1 − ei) = af(1 − ef) (2.55)

From the definition of the hyperbolic excess velocity (Eq. 1.15) the semi-major axes of the two different legs, can be easily computed as

ai = − µt v∞i2 ; af = − µt v∞_f2 (2.56)

With the use of Eq. 2.51 and Eq. 2.55, Eq. 2.54 can be rewritten as

α = arcsin 1 ei + arcsin 1 ef = arcsin 1 ei + arcsin 1 1 − ai af(1 − ei) (2.57)

The iterative method can now be applied: in [21] a Newton-Rapson root-finder has been chosen (see Eq. 2.31), that means

ein+1 = ein−

f (ein)

f0_(e in)

(2.58)

where in this specific case f (ein) = arcsin 1 ein + arcsin 1 1 − ai af(1 − ein) − α (2.59)

and f0(ein) is simply the derivative of f (ein) with respect to ein. Considering

that we are dealing with hyperbolic orbit (e > 1), the starting value ei0 can be

arbitrarily set to 1.01. Using the described iteration scheme, the eccentricities of both incoming and outgoing hyperbolic legs are computed. The next step is to find out the velocities at periapsis with Eq. 1.15

vpi 2 _{= v∞} i 2ei+ 1 ei− 1 vpf 2 _{= v∞} f 2ef + 1 ef − 1 (2.60)

Preliminary Design of Interplanetary Trajectories with Deep Space Manoeuvres

Universit`

a di Pisa

Preliminary Design of Interplanetary Trajectories

with Deep Space Manoeuvres

Relatori:

Supervisore all’estero:

Candidato:

Abstract

Contents

List of Figures

List of Tables

List of Symbols

Introduction

Outline of the work

Chapter 1

Principles of Astrodynamics

1.1

The n -body problem

1.2

The two-body problem

1.2.1

Constants of the motion

1.2.2

Trajectory equation

1.2.3

Keplerian orbits

1.2.4

Characteristic velocities

1.3

Orbital Elements

1.3.1

Orbital Element Conversions

1.4

Sphere of Influence

Chapter 2

High Thrust Interplanetary

Trajectories

2.1

Chemical Propulsion vs Electric Propulsion

2.1.1

Orbital Manoeuvres

2.1.2

Thrust Level

2.2

Patched Conics

2.2.1

Launch ∆V

2.2.2

Capture ∆V

2.2.3

Hohmann transfer orbit

2.2.4

Three-Dimensions interplanetary trajectories

2.3

Lambert’s problem

2.4

Gravity Assist

2.4.1

Powered Gravity Assist