Impact Monitoring of Near-Earth Objects: old algorithms and new challenges

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UNIVERSITY OF PISA

MASTER OF SCIENCE IN MATHEMATICS

Master Thesis

Impact Monitoring of Near-Earth Objects: old

algorithms and new challenges

26 October 2018

Candidate: Alessia Bertolucci

Supervisor: Prof. Giacomo Tommei

Examiner:

Prof. Andrea Milani Comparetti

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Introduction

This thesis presents a review of the theory and of the techniques concerning the impact

monitoring of Near-Earth Objects (NEOs). The NEOs population includes asteroids with

a heliocentric orbit and comets both having perihelion distances q ≤ 1.3 au, and aphelion distances Q ≥ 0.983 au.

An asteroid that has just been discovered has a strongly undetermined orbit, being weakly constrained by the few available astrometric observations, and there is a set of possible orbits, all compatible with the observations, forming a confidence region in the 6-dimensional orbital elements space. This region can be sampled by a set of Virtual

Asteroids (VAs), that are orbits representative of the entire region; among them there is the

real orbit but we do not know which one. Thus, it might be the case that an impact on the Earth in the near future cannot be excluded.

The goal of impact monitoring is to investigate whether an asteroid may have an impact on the Earth in the future, that is to establish whether the confidence region contains some

Virtual Impactors (VIs), a subset of the initial conditions leading to a collision with the

Earth. A crucial issue is to be able to identify hazardous cases as soon as new objects are discovered or as new observations are added to prior discoveries; since a significant amount of new observations are submitted every day, this activity requires an automated system scanning continually the NEA (Near-Earth Asteroid) catalog. This has been achieved by CLOMON2 and Sentry, two independent impact monitoring systems that are operational at the University of Pisa (since 1999) and at NASA Jet Propulsion Laboratory (since 2002), respectively. During the time span over which observations are obtained, CLOMON2 and Sentry outcomes, eventually with the announcement that some asteroid has the possibility of impacting, are published on the web; in particular, CLOMON2 results are published on the on-line information system NEODyS1. These two systems, whose output is carefully compared, now guarantee that the potentially dangerous objects are identified very early and followed up.

Both the systems generate VAs by applying a 1-dimensional sampling method of the confidence region based upon the Line Of Variations (LOV), that is a differentiable curve representing a kind of “spine” of the confidence region. The LOV method is very useful when the confidence region is elongated and thin, but this is not the case when the observed

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iv CONTENTS arc is very short (≤ 1◦). When the set of observations of an object covers only a very short arc, the confidence region results to be wide in at least two directions; thus the LOV is not representative of the entire confidence region and its definition strongly depends upon the coordinates and units used. In this case, both CLOMON2 and Sentry do not perform very well. Therefore, it has been proposed a different technique, based upon the idea of changing the geometric object used in the sampling method and switching to a sampling by surfaces of the Admissible Region (AR), that is a 2-dimensional manifold containing orbits compatible with the observations.

By following this approach, systematic ranging methods have been developed, with the aim to optimally analyse objects having only short observed arcs available. Among their various applications, these methods are also used in detecting when a small asteroid just discovered may be an imminent impactor. In particular, softwares based upon such techniques have been successful in predicting the impact on Earth of small asteroid 2018 LA on June 2, 2018, that collided shortly after being discovered.

This thesis work has been carried out thanks to a short university internship made at spin-off company SpaceDyS2, giving the opportunity to study OrbFit software’s code (upon which is based the system CLOMON2) and to analyse its output files. SpaceDyS is currently working to the development of an impact monitoring software, called AstOD, in the context of a migration of the NEODyS activity at European Space Agency ESRIN; thus, it was also possible to make a theoretical comparison between the procedures implemented in OrbFit and AstOD. Moreover, we could also analyse the outcomes of the follow up activity of asteroid 2018 LA that has been obtained by using a software based upon a systematic ranging technique.

In the first part of this thesis we summarize the main theoretical tools developed to set up an impact monitoring procedure, as it has been implemented in CLOMON2 and Sentry. In Chapter 1 we present the Öpik’s theory of close encounters (Öpik [34]) and some specific tools and methods aiming to analyse a close approach of an asteroid to the Earth (Milani and Valsecchi [28]). Then we briefly illustrate the notions of resonant returns and

keyholes (Valsecchi et al. [41]) and their role in the framework of impact monitoring.

In Chapter 2 we show several definitions of the LOV with a uniform step-size sampling method and discuss the issue of selecting a metric for the LOV parameterization (Milani et al. [27], Milani and Gronchi, [23] chapter 10).

In Chapter 3 we illustrate an overview of the whole procedure of impact monitoring as implemented in CLOMON2 and Sentry (Tommei [39]), describing the mathematical theory applied in the development of these systems and focusing on certain methods recently introduced in CLOMON2 (Del Vigna et al. [9]). In the last part of the chapter we will provide a description of the implementation of such procedures in OrbFit’s code and of the structure of its output files, according to the analysis work done during the internship; we

2_{Space Dynamics Services s.r.l., via Mario Giuntini, Navacchio di Cascina, Pisa, Italy.}

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CONTENTS v also outline the main theoretical characteristics of the software AstOD.

In the second part of the thesis we illustrate recently developed techniques to deal with the issue of imminent impactors detection.

In Chapter 4 we discuss the problem of short observed arcs, namely Too Short Arcs (TSAs) and Very Short Arcs (VSAs), introducing the Admissible Region as a tool to obtain a set of orbits compatible with the observations. We shall present two techniques (proposed by Tommei [40], [39]) to sample the AR in VAs in the case of TSAs and VSAs.

In Chapter 5 we describe two different systematic ranging methods (Farnocchia et al [14], Spoto et al. [37]) to sample the AR; such techniques have been implemented in two services (Scout and NEOScan) that scan the Minor Planet Center NEO Confirmation Page3 (NEOCP) every two minutes, determining whether an observed object is a NEO, computing its impact probability, and also assigning a priority level to the follow up activity of the object. Lastly, we show some examples of application of NEOScan. These services have been proved to obtain good results and they constitute fundamental tools in detecting whether a small just discovered asteroid may be an imminent impactor.

In Chapter 6 we describe the case of the small asteroid 2018 LA, that we analysed during the internship. We show the results of the software implemented upon the systematic ranging method proposed by Spoto et al. ([37]) (which is the kernel of the service NEOScan) obtained during the follow up of the object. Furthermore, we present a semilinear method to predict the Impact Corridor (Del Vigna [8], Dimare et al. [11]), that is a stripe going from a chosen altitude in the atmosphere to the Earth’s surface containing possible impact locations; finally, we show the result, by applying this technique, of the prediction of the impact corridor of 2018 LA, discussing how the result is consistent with the actual impact location.

3_{It is a “catalog” of observed objects, available at http://www.minorplanetcenter.net/iau/NEO/}

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Chapter 1

Close encounters and Target

Planes

A close encounter of a Near-Earth Object (NEO) with the Earth is defined as a passage of the small object near our planet; more precisely, we mean a passage inside the sphere of influence of the Earth. The study of close approaches of celestial bodies has deep roots in the history of astronomy; since the end of the 18th century the study of a newly discovered small Solar System body, comet D/Lexell, made it clear that comets can pass very close to the Earth, and that the gravitational perturbations at close planetary encounters can modify their orbits significantly.

To achieve a satisfactory understanding of such phenomena, Öpik assumed a simpler mathematical setting, developing a theory of planetary encounters ([33], [34]) based on a piecewise 2-body approach; that is, the small body (asteroid, comet, meteoroid) is considered to be in a heliocentric ellipse until the time of the encounter with some planet, then its dynamics is switched to a planetocentric 2-body orbit, which is approximated as always hyperbolic. Then the standard formulas of 2-body scattering are applied to obtain the initial conditions of a new post-encounter heliocentric orbit.

Öpik’s theory was successfully used to study the statistical properties of the orbital changes resulting from close approaches, and to some extent it is still in use; the main limitation is that Öpik developed the theory only for the case the two orbits are actually touching, that is the Minimum Orbital Intersection Distance (MOID) is zero. The MOID (see [23] and [39] for a rigorous definition) is the minimum distance between the two osculating Keplerian orbits of a NEO and the Earth as curves in three-dimensional space without considering the positions the bodies occupy, that is, the absolute minimum of the Euclidean distance between a point on the first orbit and a point on the second one. The MOID can be used as an indicator for collisions; if a large value for the MOID between an asteroid and the Earth is found, then this asteroid will not collide with Earth in near future, otherwise, in the case of a small MOID value, the asteroid could have an impact on Earth.

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according to which the subsequent encounters of the small body with the same planet (or even with another one) are not independent of the occurrence of the previous ones; this idea was implicitly contained in the work of Lexell and Le Verrier on Lexell’s comet, and since the 70s was in fact used in spacecraft navigation, but was first applied to asteroid close approaches only at the end of the 90s. Valsecchi et al. ([41]) extended Öpik’s theory of close encounters to near misses, which can occur also for a finite value of the MOID, developing an analytical theory that they used to describe how each encounter changes the condition for the next encounter. They found that each subsequent return that could lead to an impact defines a keyhole on the target plane of the first encounter, such that an orbit through it would indeed collide with the planet; thus they gave an explicit, semi-analytic description of the keyholes for all possible resonant returns. In this thesis we will not deal with these issues, limiting ourselves to just a theoretical introduction to resonant (and non

resonant) returns and keyholes.

According to Öpik’s theory, the first thing to do when analysing a planetary encounter is to define an associated target plane. In this chapter we show two types of target planes: the one conceived in Öpik’s theory, which we refer as Target Plane (TP), and a modified version, called Modified Target Plane (MTP); we will also exhibit the most used conventional choices for the reference systems on such planes. One of the most important objectives of the target plane analysis of an encounter is to determine if a collision may occur and, if not, to examine the kind of encounter; this means to estimate NEO velocity with respect to the Earth, to acknowledge the geometry of such close approach (which is the matter of Section 3.4) and to forecast the post-encounter evolution in the dynamics of the asteroid.

In this chapter, we want to provide the fundamental tools in the analysis of an encounter, which may be used in the next chapters, in order to achieve a whole understanding of the impact monitoring theory; therefore, we start from the basis, by describing the main points of Öpik’s theory (we mainly follow the works by Valsecchi et al. [41] and Tommei [38]).

1.1 Öpik’s theory of close encounters

Close encounters of small objects with the planets in the Solar System are a key mechanism in the Solar System dynamics, since such encounters rule the evolution of comets and planet-crossing asteroids. However, analysing such phenomena could be quite complicated, given the presence of a great number of perturbative objects. Öpik in 1976 developed a simplified approach to this problem; his theory of close encounters ([34]) consists in modelling the motion of a small body approaching a planet as a planetocentric two-body scattering, while considering the heliocentric orbits of the body and the planet as constant and Keplerian before and after the encounter.

It is possible to use the initial heliocentric orbits of the planet and the asteroid to determine their relative velocity and the geometry of the close approach; if the minimum distance of the encounter is less than the sum of the radii of the two bodies, there would be

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a collision, otherwise, one can exploit these results, in the two-bodies approach, to compute the consequences of the encounter on the asteroid orbit. The relative velocity of the small body with respect to the planet defines the direction and speed of the incoming asymptote of the planetocentric hyperbolic orbit. The effect of the encounter is then computed as an instantaneous deflection of the velocity vector in the direction of the outgoing asymptote of the planetocentric hyperbolic orbit, ignoring the perturbation due to the Sun and the time it actually takes for the small body to travel along the curved path that “joins” the two asymptotes. Interestingly, the errors involved in such an approach are smaller for closer approaches, and Öpik’s theory is exact in the limit of the minimum approach distance going to zero.

1.1.1 The components of the geocentric velocity

Let us consider a small body encountering the Earth, and let us suppose that the latter moves on a circular orbit around the Sun. We use a particular system of units in order to simplify the formulas: the distance of the Earth from the Sun is 1 and the period of the planet is 2π; we also assume that both the mass of the Sun and the gravitational constant are equal to 1. We disregard the mass of the Earth in the heliocentric orbit of both the planet and the small body, thus the heliocentric velocity of the Earth is also 1.

We use a geocentric reference frame (X, Y, Z) such that the Y -axis coincides with the direction of motion of the Earth and the Sun is on the negative X-axis. Let a, e, i, ω, Ω be the orbital parameters of the asteroid. In this system, by considering that the two orbits are actually intersecting (i.e., the MOID is 0), the components of the unperturbed geocentric velocity vector ~U of the small body are (Carusi et al. [2])

    Ux Uy Uz     =     ±p 2 − 1/a − a(1 − e2₎ p a(1 − e2_{) cos i − 1} ±p a(1 − e2_{) sin i}    

and the geocentric velocity is

U = r 3 −1 a− 2 q a(1 − e2_{) cos i.}

This can be rewritten as U =√3 − T , where T is the Tisserand parameter with respect to the planet

T = 1 a + 2

q

a(1 − e2_{) cos i.}

The direction of the incoming asymptote is defined by two angles, θ and φ, such that

    Ux Uy Uz     =     U sin θ sin φ U cos θ U sin θ cos φ    

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and, conversely, " cos θ tan φ # = " Uy/U Ux/Uz # .

1.1.2 The Target Plane (TP)

We define the Target Plane (TP, also called b-plane) as the plane orthogonal to ~U

containing the center of the Earth. Let ~b be the vector that extends from the Earth to the

intersection of the incoming asymptote with the plane; b = |b| is called impact parameter. We use a geocentric coordinate system (ξ, η, ζ) such that (ξ, ζ) are coordinates on the TP and the η-axis is directed along ~U . The ζ-axis is in the direction opposite to the projection on

the TP of the heliocentric velocity of the Earth, while the ξ-axis completes the right-handed reference frame, being perpendicular to the heliocentric velocity of the planet.

In this way, the shortest segment joining the orbit of the Earth and that of the asteroid, corresponding to the MOID, turns out to be directed along the ξ-axis; this is because this axis is perpendicular, by definition, to both the Y -axis and ~U . The ζ-axis, then, can

be seen as a “time coordinate”, that is, a shift in the time of arrival of the small body at the target plane will mean a change only in its ζ coordinate, and not in ξ. In other words, this coordinate system decouples the two factors governing the possibility of a very close encounter, i.e. distance between orbits and encounter timing, mapping them into, respectively, the ξ and the ζ-axis.

Following Carusi et al. ([2]), we define the angle ψ by

" b sin ψ b cos ψ # = " ξ ζ # .

The transformation from the geocentric reference frame (X, Y, Z) to the TP frame (ξ, η, ζ) is accomplished by first rotating by an angle −φ about Y , then rotating by −θ about ξ (which is perpendicular to the old Y -axis and to ~U ). In matrix notation

    ξ η ζ     = ˆRξ(−θ) ˆRY(−φ)     X Y Z     . (1.1)

Similarly, the inverse transformation is achieved by rotating by θ about ξ and then by φ about Y . In matrix notation

    X Y Z     = ˆRY(φ) ˆRξ(θ)     ξ η ζ     . (1.2)

As a consequence of the encounter with the Earth, ~U is rotated into ~U0, aligned with the outgoing asymptote and having the same length: U = U0. The deflection angle γ between

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the two vectors is a function of U , the mass of the Earth m, and the impact parameter b according to tanγ 2 = m bU2 = c b, cos γ = 1 − tan 2 γ 2 1 + tan2 γ₂ = b2U4− m2 b2_U4_{+ m}2 = b2− c2 b2_{+ c}2, sin γ = 2 tan γ 2 1 + tan2 γ 2 = 2mbU 2 b2_U4_{+ m}2 = 2bc b2_{+ c}2,

where c = m/U2. This quantity plays the role of a characteristic length and it must be small in order to allow Öpik’s theory to apply; when b = c, the deflection angle γ is π/2. For U = 0.5, a typical value for many NEAs, c = 1.22 × 10−5au ' 0.29 Earth radii; this

means that, unless U is very low, large deflections at close Earth encounters cannot occur. The angles θ0 and φ0, defining the direction of the post-encounter velocity vector ~U0, can be obtained in terms of θ, φ, γ, ψ by (Carusi et al. [2])

cos θ0 = cos θ cos γ + sin θ sin γ cos ψ, tan(φ − φ0) = sin γ sin ψ

sin θ cos γ − cos θ sin γ cos ψ, tan φ0 = tan φ − tan(φ − φ

0₎

1 + tan φ tan(φ − φ0₎,

cos φ0 = _p 1 1 + tan2_φ0,

sin φ0 = cos φ0tan φ0.

U is an invariant of the problem and, once it is given, a is a function of cos θ only, and does

not depend on φ. In fact, in the geometric setup just described we have fixed the heliocentric distance of the asteroid, thus fixing its potential energy; to obtain its total energy (and this means to obtain a) we need to compute its kinetic energy, i.e. its heliocentric velocity. The latter is the vectorial sum of the heliocentric velocity of the Earth, whose components are (0, 1, 0), and of ~U ; for fixed U , the magnitude of the sum depends only on the angle between

the two vectors, i.e. on θ. Note that, as a consequence, also c is an invariant of the problem, due to the conservation of U.

Öpik’s theory of close encounters works as long as the region of space in which the encounter takes place is “small”, so that the interaction can be thought of as taking place in a point. This assumption breaks down as the Tisserand parameter approaches 3, that is, when the encounters take place at low geocentric velocity (and thus, for high values of

c), and therefore the assumption of a point-like interaction does not hold any more. The

theory is inapplicable for a Tisserand parameter exceeding 3.

Note that, in principle, this theory could be extended to the case of an elliptic orbit of the planet, preserving most of the formulae, provided the angles θ and φ are defined with

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respect to the velocity of the Earth at the time of the encounter, which would generally not be orthogonal to the Sun-planet direction.

1.2 The Modified Target Plane (MTP)

Let t∗ be the closest approach time, subsequent to the observations epoch t₀, of a nominal orbit with initial conditions X∗. We define, following Milani and Valsecchi ([28]), the Modified Target Plane (MTP) as the plane passing through the center of the Earth and perpendicular to the velocity vector of the encountering body at the closest approach time.

This definition is a modification of the TP used in Öpik’s theory of close encounters; the latter is the plane through the Earth and perpendicular to the unperturbed velocity vector, while the MTP is perpendicular to the perturbed velocity vector. The main difference between these two types of target planes is in the fact that the asteroid deflection caused by Earth (gravitational focusing) is directly pointed out by using the MTP, while it keeps somehow hidden by using the TP. Thus, the difference between the two planes is negligible for small asteroid deflections, that is for encounters occurring at either high velocities or great distances.

Remark 1. On the TP, we define the Earth impact cross section as a disk D⊕ of radius

b⊕= R⊕p1 + 2GM⊕/R⊕u2 (where u = |u| is the size of the escape velocity), larger than

the Earth radius R⊕ by a factor accounting for the gravitational focusing.

Some complications derive from the visibility of gravitational focusing effect on the MTP, due to the fact that nearby trajectories are deflected by different amounts. In fact, when the deflection is significant, as for deep encounters as well as for low velocities approaches, there is a substantial nonlinearity in mapping the asteroid pre-encounter state to its projection on the MTP. Furthermore, there might be great variations in the plane orientation for nearby trajectories, causing strong nonlinearities.

Of course, in very low velocity encounters, leading to a temporary capture of the asteroid, there are no asymptotes for the asteroid geocentric orbit, and thus Öpik’s theory cannot apply; in such cases we have to use necessarily the MTP. Therefore, we are able to distinguish the more suitable target plane depending upon the kind of asteroid deflection caused by the encounter. In the case of small deflections (i.e., high velocities or shallow encounters), these two planes are close to each other (almost indistinguishable), while in the case of moderate deflections we may prefer using one type of target plane rather than the other, under some circumstances. In the case of great deflections, the strong nonlinearity on the MTP suggests that the TP would be a better choice, but, as seen above, for very low velocities only the MTP is applicable.

Remark 2. To avoid geometric complications, we consider close only approaches with a

distance from the Earth center of mass not exceeding some value R_{T P}. Possible values for

RT P range between 0.05 au and 0.2 au, thus the target planes are in fact disks with a finite

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Once an MTP has been selected, we can select Cartesian orthogonal geocentric coord-inates (x, y, z) such that, on the nominal orbit at the encounter, the following equations hold y(t∗, X∗) = 0, d dtx(t ∗_{, X}∗_{) = 0,} d dtz(t ∗_{, X}∗_{) = 0.}

This is possible because the nominal orbit crosses the MTP orthogonally; we shall further assume that the crossing of the MTP is transversal (i.e., with non zero velocity):

d dty(t

∗_{, X}∗_{) 6= 0,}

that is, the y-axis is along the geocentric velocity vector. The choice of the orthogonal coordinates (x, z) in the MTP is arbitrary.

For orbits close to, but different from, the nominal one, the time at which the MTP is crossed can be different from t∗, and is a function of the initial conditions: t = t(X), implicitly defined by the equation

y(t(X), X) = 0, t(X∗) = t∗.

By applying the ordinary implicit function theorem, the gradient of the function t(X) can be computed in the following way

∂t ∂X = − 1 ∂y ∂t ∂y ∂X.

The map onto the MTP is expressed by the functions x(X), z(X), that can be shown to be differentiable as follows, provided the MTP crossing is transversal:

∂x ∂X y=0= ∂x ∂X(t ∗ , X∗) +∂x ∂t(t ∗ , X∗) ∂t ∂X(t ∗ , X∗),

and by using the above formula for the gradient of t(X), and the property that the nominal orbit crosses the MTP orthogonally,

∂x ∂X y=0 = ∂x ∂X(t ∗_{, X}∗_{) −} ∂x∂t(t ∗_{, X}∗₎ ∂y ∂t(t∗, X∗) ∂y ∂X(t ∗_{, X}∗_{) =} ∂x ∂X(t ∗_{, X}∗_).

In the same way

∂z ∂X y=0= ∂z ∂X(t ∗_{, X}∗_).

This means that, given the state transition matrix and the solution of the variational equation along the nominal orbit at time t = t∗, the partial derivatives of the map on the MTP are easily computed.

Note that it has been defined a continuous and differentiable map which allows us to transform from the MTP to the TP; in this way, we can perform a target plane analysis of

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an encounter on the TP starting with an analysis on the MTP. We refer to Tommei ([39]) for a detailed description of this map.

Remark 3. The transformation of coordinates rotating and rescaling the MTP into the TP is

not canonical, thus it is impossible to use the Hamiltonian formalism including coordinates on the TP (Tommei [39]).

1.3 Linear analysis on the target plane

In the following we shall briefly recall some of the notions used in the formulation of the orbit determination as a nonlinear least squares problem. We refer to Milani and Gronchi ([23], chapter 5) for a complete definition of the least squares problem.

Let x be the 6-vector of orbital parameters defining the initial condition for an asteroid orbit at some epoch t0; we can consider either Keplerian elements, or Equinoctial elements,

or Cartesian position and velocity (see Section 2.4). Let us assume that at some later time

t1 the asteroid has an encounter with the Earth.

Let Ψ be the orbital elements space, and W ⊂ Ψ an open subset, and let F : W → R2 be the function mapping an orbit x at epoch t0 to the point y on the target plane (TP or

MTP) of the close approach occurring at the epoch t1.

Let x∗ ∈ Ψ be the nominal solution, that is obtained, with its uncertainty, by applying the least squares method. We recall that the least squares method is a procedure minimizing a function fitting the model to the observations, namely the target function Q(x) =

1

mξ(x)TW ξ(x), where x ∈ Ψ is the vector of the fit parameters, m is the number of

observations, ξ is the vector of the “observed - computed” residuals, and W is the weight matrix. This can be done by applying an iterative method, the differential corrections procedure (or, alternatively, the Newton’s method), which solves the normal equation

C∆x = D, where C = BTW B, D = −BTW ξ, B = ∂ξ ∂x.

The uncertainty of x∗is expressed by the confidence region ZX(χ) = {x|m∆Q(x) ≤ χ2} ⊂ Ψ. The function F maps the orbital elements space onto the target plane, therefore it maps the confidence region ZX(χ) into a confidence region ZY(χ) on the target plane.

We now consider the linearised function DF; it maps the displacement from x∗ in the orbital elements space ∆x = x − x∗into linearised deviations from the projection y∗ = F(x∗)

∆y = y − y∗= DF(x∗)∆x,

and therefore maps the confidence ellipsoid around x∗ (that is the linear approximation of

ZX_(χ))

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onto a confidence ellipse around y∗ on the target plane

Z_linY (χ) = {y|∆yTCY(y∗)∆y ≤ χ2}

The matrix C_Y is the normal matrix for the projections y and the inverse Γ_Y = C_Y−1 is the corresponding covariance matrix. By a standard result from the theory of multivariate Gaussian distribution1 (Milani and Gronchi [23], section 3.3) the covariance matrix is transformed by

ΓY = DF ΓX(DF)T

We want to perform a close approach analysis on the target plane (TP or MTP) by assuming that the linear theory applies. As seen in the previous section, the function expressing the intersection of the asteroid orbit with the target plane is an explicitly computable differentiable function; thus it is possible to apply the linear approximation shown above, taking into account that y = [ξ, ζ] if we are using the TP, while y = [x, z] in the case of the MTP. The confidence ellipse Z_linY (χ) on the target plane is characterised by the nominal position y∗, which is the center of the ellipse, and by the covariance matrix ΓY, which defines dimension and orientation of the ellipse. The two eigenvalues of ΓY are

related to the length of the semimajor and semiminor axis of Z_linY (χ), and the corresponding eigenvectors indicate the orientation of these axis. We define the stretching and the width as the semimajor and semiminor axis of Z_linY (1), respectively, which in turn are the square roots of the eigenvalues of Γ_Y.

Other parameters involved in the target plane analysis are the angle α between y and the major axis of Z_linY (χ), and the asteroid distance d from the center of the Earth; note that the latter is given by d = kyk on the MTP and by d = b on the TP. Then the minimum distance between the semimajor axis of Z_linY (χ) and the origin of the reference system (that is the center of the Earth) is given by d sin α.

When an encounter occurs very far in the future, the orbital uncertainty is strongly ruled by mean anomaly uncertainty; in such cases, Z_linY (χ) is very elongated and thin, and thus, in the Earth vicinity, it can be assumed as a strip with constant width. This means that, moving along the major axis of the ellipse, only the asteroid anomaly changes; thus, the minimum distance from the center of the Earth along that axis (i.e., d sin α) represents the minimum possible distance of the encounter for all variations in the time of the encounter. This is exactly the definition of MOID, thus d sin α is a good approximation of the MOID in such situations; as a consequence, the width w represents the uncertainty of the MOID. Furthermore, still under these assumptions, the semimajor axis of Z_linY (χ) is approximately aligned with the projection on the target plane of the heliocentric velocity of the Earth, that is the z-axis on the MTP and the ζ-axis on the TP. Its length, given by the stretching S, expresses the quantity by which the nonlinear function F elongates the original confidence ellipsoid Z_linX (χ).

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These approximations may be inappropriate when, for instance, the encounter occurs near the observations epoch t0 (this means that the ellipse has low eccentricity), as well as

the function F is locally strongly nonlinear; in such cases there is no guarantee that the confidence ellipse is a good approximation of the projection of the real confidence region on the target plane.

1.4 Semilinear confidence boundary

As seen in the previous section, the projection of the confidence region on the target plane depends upon the function F; the easily computed elliptic disks Z_linY (χ) are good approximations whenever the nonlinearity of the function F is small. Unfortunately, this is not the case when the orbits have to be propagated for a long time, and especially when close approaches take place. Therefore, the image of the confidence ellipsoid may not be entirely contained in the confidence ellipse. Milani and Valsecchi ([28]) defined a semilinear

approximation, with the aim to introduce a good compromise between computational

efficiency and accurate representation of the nonlinear effects.

The boundary ellipse K_linY (χ) of the confidence disk Z_linY (χ) is the image, by the linear map DF, of an ellipse KX(χ) in the orbital elements space, which lies on the surface of the

ellipsoid Z_linX (χ). The semilinear confidence boundary K_N(χ) is by definition the nonlinear image F(K_X(χ)) in the target plane of the ellipse K_X(χ). According to Jordan’s curve theorem, the closed curve KN(χ) is the boundary of some subset ZN(χ) in the target

plane Y ; Z_N(χ) is a subset of the projection of the confidence ellipsoid Z_linX (χ) and it is a better approximation with respect to the elliptic disk Z_linY (χ). Thus, we use ZN(χ) as an

approximation of F (Z_linX (χ)), which is the set of all possible predictions on the target plane compatible with the observations.

To explicitly compute KN(χ) we can proceed as follows ([28]). Let us consider the

Jacobian matrix DF(x∗); it spans a 2-dimensional subspace in the orbital elements space

X ⊂ RN 2. We decompose the orbital elements space in a component E in the 2-dimensional space spanned by the rows of DF, and in a component L in the (N − 2)-dimensional orthogonal space: X = E ⊕ L. Let us make the following decomposition:

x − x∗ =

"

h − h∗ g − g∗

#

where x ∈ X, and g represents two coordinates in the space E and h represents N − 2 coordinates in the orthogonal space L. In this coordinate system the normal matrix C_X decomposes in this way:

CX =

"

Chh Chg

Cgh Cgg

#

2_{We consider either N = 6, if we select a set of six orbital elements, or N > 6 if some dynamical}

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The equation

h − h∗= −C_hh−1Chg(g − g∗)

defines a 2-dimensional subspace in X, called regression subspace of h given g ([23], section 5.4), containing the points of Z_linX (χ) with tangent space parallel to L.

The linear map DF can be described as the composition of the orthogonal projection

X → E and of an invertible linear map A : E → Y . Then KE = A−1(KlinY (χ)) is an ellipse

in the E space which represents the boundary of the orthogonal projection of the confidence ellipsoid on the space E. Thus, the ellipse K_X(χ) on the surface of the ellipsoid can be computed as the image of KE by the map

g − g∗→ " g − g∗ −C_hh−1Chg(g − g∗) # ,

which maps E to the regression subspace. Once K_X(χ) has been obtained, the semilinear confidence boundary can be computed by exploiting the images of the points on KX:

KN(χ) = F(KX(χ)). For further details, see the works of Milani and Valsecchi ([28]), Milani

([19]), and Tommei ([38]).

Note that, whatever the method of representation of the confidence region F (Z_linX (χ)), in the end we can only explore it by computing a finite number of orbits, since we would have to compute a complete orbit with a N-bodies model. To increase the level of resolution of this representation, however, the dimensionality of the space being sampled matters. The region ZX

lin(χ) is N-dimensional, and to increase the resolution by a factor 10 the number

of orbits grows by a factor 10N. On the other hand, the semilinear confidence boundary

KN(χ) is a 1-dimensional curve, and the resolving power increases linearly with the number

of orbits computed. In practice, even very complicated and strongly nonlinear examples can be dealt with only a few ten to a few hundred orbit propagations.

1.5 Returns and keyholes

Let us suppose that an asteroid have an encounter many years after the last observations; as a consequence, the confidence region on the target plane might be very elongated and the stretching might have high values. The asteroid possible orbits (that is, the points inside the confidence region) undergo different perturbation degrees, as a result of the close approach, going from an unperturbed orbit for a shallow encounter, to impulsive variations in the orbital elements for a deep encounter. After an encounter, the orbital periods of the possible orbits of the asteroid vary from a minimum P_min to a maximum P_max; every rational number within this range corresponds to a resonant return. For instance, if the period is such that P = h/k years, where k and h are integers, then after k periods of the asteroid h periods of the Earth have elapsed, and both the planet and the small body will be back again in the same position of the previous encounter. Also if the ratio of the period

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is not exactly h/k, but is close, a subsequent encounter can take place, but the planet will be earlier or later for the encounter than it was at the previous one.

In the case the asteroid have missed its first encounter, the second one may nonetheless be a close approach if the time quantity by which the body missed the first encounter is compensated; more precisely, if such time difference is ∆t, a resonant return will occur if the following condition holds

h + ∆t = kP,

where ∆t and P are measured in years.

Another important quantity to take into account in studying returns is the MOID; if its value keeps small, a resonant return will always be possible, since each real number P is arbitrarily well approximated with a rational number h/k.

It’s worth to observe that resonant returns are the main cause of collisional orbits; indeed a collision, usually, does not occur at first encounter, but is the result of a number of encounters.

Interestingly, there is also the possibility a non resonant return may occur; for instance, if the Earth has completed h + 1₂ revolutions, while the asteroid k + 1₂, both will be at the descending node to the same time. By considering the orbit eccentricities and the time spans requested by both the bodies for going from the ascending node to the descending one (we indicate t_E for the Earth and t_A for the asteroid), the condition to be fulfilled to have an encounter to the descending node is

h + tE+ ∆t = kP + tA.

Each close approach, resulting from a resonant return as well as a non resonant one, can be analysed by studying the previous encounters and the subsequent ones. We said that if the MOID keeps small for a long time, then returns can occur in this time span; in such cases, the returns sequence leads to a (almost) fractal structure in the confidence region. As said above, Valsecchi et al. ([41]) developed an analytic theory of resonant returns, by extending the Öpik’s theory to encounters with non-zero miss distances, providing relations between consecutive encounters. By using notations and results of Section 1.1, we show how to formulate a condition on the TP for a resonant return, according to [41].

Let us consider the TP reference frame defined in Section 1.1; it has the property that

ξ is the local MOID, while ζ is proportional to the delay of the asteroid in missing the

minimum distance to the Earth. A given resonance corresponds to specific values of a0, i.e. of θ0, that we denote with a0₀ and θ₀0. In fact, if we constrain the post encounter orbit in such a way that the ratio of the periods is k/h, then a0₀= p3

k2_/h2 _{and cos θ}0 0 =

1−U2_−1/a0 0

2U .

By using the formula of Section 1.1 for the post-encounter θ0, the relations for cos γ and sin γ (where γ is the deflection angle) and ζ = b cos ψ, we get

cos θ₀0 = cos θb

2_{− c}2

b2_{+ c}2 + sin θ

2cζ

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replacing b2 with ξ2+ ζ2 and rearranging terms, we can rewrite the above equation as the equation of a circle centered on the ζ-axis with radius R and D the value of the ζ-coordinate of its center:

ξ2+ ζ2− 2Dζ + D2 _{= R}2_,

where

D = c sin θ

cos θ₀0 − cos θ, and R =

c sin θ0₀ cos θ0₀− cos θ .

This is the locus of points on the pre-encounter TP leading to a given resonant return. These computations have been performed by disregarding the corrections due to the fact that the heliocentric distance of the small body at close encounter is not 1; if we take into account at the first order the non zero geocentric distance of the small body, the expression of the locus of points on the TP for a given resonant return would contain also third order terms in ξ, ζ, c. Since ξ, ζ, c are small numbers, in practice the effect of the third order terms is to distort the shape of the circles to some degree, without altering the overall geometry. Therefore, in the framework of a qualitative discussion, these terms can be neglected.

Chodas in 1999 ([4]) introduced the term keyhole to denote the small regions of the TP of a specific close encounter such that, if the asteroid passes through one of them, it will hit the planet at a subsequent return. The term keyhole may also be used to indicate a region leading not necessarily to collision, but to a very deep encounter, that is, we can generalize its definition by imposing that the successive encounter occur at a certain distance.

Note that a keyhole is simply one of the possible pre-images of the Earth’s cross section on the TP, thus, it is linked to a specific value of the post-encounter semimajor axis, which allows the occurrence of the next encounter at the given date.

As shown above, to compute the TP circles for a resonant return Valsecchi et al. used a Keplerian propagation between encounters, thus only concerning the timing of the next encounter (i.e., the ζ-coordinate) but leaving unchanged the MOID of the next encounter (i.e., the ξ-coordinate). This fact is not realistic, since the MOID is bound to vary between encounters because of secular perturbations, planetary perturbations and, for planets with massive satellites, the displacement of the planet with respect to the center of mass of the planet-satellite system. For the purpose of obtaining the size and shape of an impact keyhole we can, however, just model the secular variation of the MOID as a linear term affecting ξ00

ξ00= ξ0+dξ

dt(t

00 0− t00),

where ξ00 is the value of ξ at the subsequent encounter, while t0₀, t00₀ are the time of passage at the node for the orbit following the first encounter. The time derivative of ξ can be computed either from a suitable secular theory for crossing orbits (Gronchi and Milani [17]) or using a value deduced from a numerical integration.

A detailed description of the computation of the shape and the dimension of a keyhole can be found in the paper by Valsecchi et al. ([41]).

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Chapter 2

The Line Of Variations

When an asteroid is discovered, the first information obtained about its orbit is poor, since the orbit is weakly constrained by observations which span a short arc. Although a nominal orbital solution does exist, other orbits are acceptable as solutions, corresponding to RMS of the residuals not significantly above the minimum; thus, there is a set of possible orbits for the observed asteroid, all compatible with the observations. These “acceptable orbits” are, indeed, the points of the confidence region ZX(χ) in the orbital elements space.

In many applications we need to consider the set of the orbits with initial conditions in the confidence region as a whole; since the dynamic model for asteroid orbits (the N-body problem) is not integrable, there is no way to compute all the solutions for some time span in the future (or in the past), and we can only compute a finite number of orbits by numerical integration. Thus the confidence region is sampled by a finite number of the so called Virtual

Asteroids (VAs), each one with an initial condition in the confidence region.

Nevertheless, it must be defined how to select a finite number of VAs among the orbital elements spanning the confidence region; in practice, to describe all the points of the confidence region is not reasonable, since we should compute a number of solutions huge enough to sample uniformly the six-dimensional volume of the confidence region. A way is to select the VAs at random in the confidence region, following the so called Monte Carlo methods. However, an optimal method has been found by exploiting a basic property of the solutions of the N-body problem. During a sufficient long time span, the orbits of a swarm of VAs (in the case they all do not have the same semimajor axis) will spread mostly along track because of their different periods, causing the set of propagated VAs to form a sort of “necklace”, that is to spread along a wire which is similar to an arc of a heliocentric ellipse. Thus, the idea is to sample a one-dimensional subspace of the confidence region, a segment of a continuous curve in the initial conditions space, which could be representative of the entire six-dimensional confidence region; this is the Line Of Variations

(LOV), which is ideally defined as the image of a differentiable curve γ : [−σ_max, σmax] → X,

where [−σmax, σmax] ⊂ R and X is the orbital elements space. The main advantage of this

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to a differentiable curve along which interpolation is possible.

This idea of sampling the LOV by VAs, known as multiple solutions sampling, was introduced by Milani ([19]) and can be employed in different applications, such as asteroid identification, recovery of lost asteroids and detection of possible impacts.

In this chapter we shall expose the process leading to the definition, parameterization and sampling of the LOV; in particular, we shall also present the several possible different definitions of LOV so far formulated, and give details of a method to sample uniformly the LOV (i.e., to sample it by VAs at regular intervals). Another optimal method to sample the LOV, which is based upon probability, is presented in the next chapter, in the context of the performance of an impact monitoring system (Section 3.3).

2.1 Confidence Regions

Given any point x∗ in the orbital elements space, we can compute the 6 × 6 normal matrix

C(x∗); let us assume that the observation information is such that C(x∗) is positive definite. If the residuals and the confidence region are small, the latter can be well approximated by the confidence ellipsoid

ZL(χ) = {x|∆xTC(x∗)∆x ≤ χ2}.

Let the longest semiaxis of that ellipsoid be in the direction of the unit vector v₁, which is an eigenvector of C(x∗):

C(x∗)v1 = λ1v1.

The corresponding eigenvalue λ₁> 0 is the smallest among the eigenvalues of C(x∗): the longest semiaxis of the confidence ellipsoid Z_L(1) has length k₁= 1/√λ1. Note that v1 is

also an eigenvector of the covariance matrix Γ(x∗) = C−1(x∗) with the largest eigenvalue 1/λ₁ = k2₁, thus it defines the weak direction of the least squares fit.

Let H be the hyperplane spanned by the other eigenvectors vj, j = 2, . . . , 6; the tip of the

longest axis of the confidence ellipsoid x₁ = x∗+ k₁v₁ is the point of minimum of the target function restricted to the affine hyperplane x1+ H and is also a point of local minimum of

the target function restricted to the sphere |∆x| = k1. These properties, equivalent in the

linear regime, are not equivalent in general.

Remark 4. If the observational arc is short, but enough to compute an attributable (see

Chapter 4) and a preliminary orbit ([23], chapter 9), an approximate rank deficiency (see [23], chapter 6) can occur, with order 1 or at most 2; in fact, the observation information is above the minimum required to compute an attributable if the number of scalar observations is > 4, and in this case C(x) would have rank > 4. In such situations, a confidence region can be nonetheless defined, but it could present two directions of uncertainty, i.e. it could be wide in two directions.

Since the normal matrix C(x) is defined everywhere, for each point x of the space of initial conditions we can find the eigenvector v1(x), selecting it to be a unit vector,

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corresponding to the smallest eigenvalue λ₁(x) of C(x). Then F(x) = k₁(x)v₁(x), is a vector field defined for every x, where k₁(x) = 1/p

λ1(x). The unit eigenvector v1 is

not uniquely defined, as −v1 is also a unit eigenvector; thus k1(x)v1(x) is what is called an

axial vector, with well-defined length and direction but with an arbitrary sign. However,

given an axial vector field defined over a simply connected set, there is always a way to define a true vector field F(x) such that the function x 7→ F(x) is continuous, that is by expressing F(x) or −F(x). To this end, at an initial arbitrary point x we can select the sign according to some rule; for instance, we can choose it in such a way the directional derivative of the semimajor axis a is positive in the direction +v₁(x):

  

F(x) = +k1(x)v1(x) if ∇a(x) · v1(x) > 0

F(x) = −k₁(x)v₁(x) if ∇a(x) · v₁(x) < 0

Then the orientation is maintained by imposing that v₁(x) is continuous. In this way, F(x) is a well-defined and smooth vector field.

Remark 5. Some problems could arise, for some value of x, if the normal matrix C(x) had

its smallest eigenvalue of multiplicity 2. The exact equality of two eigenvalues does not occur generically, and even an approximate equality is rare in applications. Anyway, whenever the two smallest eigenvalues are of the same order of magnitude this method we are introducing has serious limitations, as we shall discuss in Section 2.4.

Given the vector field F(x), the differential equation

dx

dσ = F(x) (2.1)

has a unique solution for each initial condition, because the vector field is smooth. If a nominal solution x∗ has been found, let us select the initial condition as x(0) = x∗, that is

σ = 0 corresponds to the nominal solution; then there is a unique solution x(σ) with such

initial value. A first definition of the LOV could make use of the following approach: • In the linear approximation, the solution x(σ) is one tip of the major axis of the

confidence ellipsoid Z_L(σ); thus, the LOV is defined by direction v₁(x∗) and length 2pλ1(x∗).

• When the linear approximation does not apply, the LOV is the unique solution x(σ) which is indeed curved and can be computed only by numerical integration of the differential equation.

However, such a definition may not be a constructive one, because of two problems: first, the definition cannot be used unless the nominal solution x∗ is known, and second, there is a

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numerical instability in the algorithms solving equation (2.1) which may give solutions x(σ) that are not at all along the weak direction. This instability in the differential equation (2.1) is particularly remarkable in the cases where there is a largely dominant weak direction, that is the same cases in which the definition of the LOV is most useful.

In the next section we shall describe the constrained differential corrections procedure, which can provide stationary points x of the target function along the weak direction, and thus it may be used to give an alternative definition of LOV.

2.2 Constrained differential corrections and definition of LOV

Let us suppose that the vector field v1(x) is defined for all x; then, also the orthogonal

hyperplane H(x) is defined for all x:

H(x) = { y | (y − x) · v1(x) = 0 } .

Given an initial guess x, we want to perform a differential corrections procedure constrained to H(x); it is possible to compute one step of such procedure by defining the 5 × m matrix

Bh(x) with the partial derivatives of the residuals with respect to the coordinates of the vector h of H(x). Then the constrained normal equation is defined by the constrained normal matrix C_h = B_hTW Bh, which gives the restriction of the linear map associated with C(x) to the hyperplane H(x), and by the right-hand side Dh= BhTW ξ, which is the projection of the vector D(x) along the hyperplane:

Ch∆h = Dh, with solution ∆h = ΓhDh.

Note that the constrained covariance matrix Γ_h= C_h−1is not the restriction of the covariance matrix Γ to the hyperplane H(x) ([23], section 5.4). The computation of Ch and Dh can be performed by, first, computing a rotation to a basis where v1(x) is the first vector, and then

by removing the first row and the first column of C and the first coordinate from D. The constrained differential corrections process gives the corrected x0= x + ∆x, where ∆x coincides with ∆h along H(x) and has zero component along v₁(x). Then the weak direction v1(x0) and the hyperplane H(x0) are recomputed, and the next correction is

constrained to H(x0). This procedure is iterated until convergence1, that is ∆h = 0. Let ¯x be the convergence value; then D_h(¯x) = 0, that is the right-hand side D(¯x) of the unconstrained normal equation is parallel to the weak direction v1(¯x): D(¯x) k v1(¯x). This

equation is equivalent to the following property: the restriction of the target function to the hyperplane H(¯x) has a stationary point in ¯x.

As showed by Milani et al. ([27]), this constrained differential corrections procedure can be applied to define the LOV in a different way, as the set of points of convergence. Thus 1_{In a numerical procedure, convergence is defined as having the last iteration with a small enough}

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we can formulate this new definition for the Line Of Variations: the set

{ x | D(x) k v₁(x) } (2.2)

where the gradient of the target function is in the weak direction. If there is a nominal solution x∗, then D(x∗) = 0, and so it also belongs to the LOV. However, the LOV is defined independently from the existence of a local minimum of the target function. Thus, the definition (2.2) gives us the possibility to define the LOV even when the nominal solution either does not exist or anyway has not been found.

It’s worth to observe that the definition by equation (2.2) does not give the same curve as that resulting from the solutions of equation (2.1), unless the problem is linear. We shall make a more in-depth analysis.

2.2.1 Definitions of LOV

We enunciate different definitions of LOV and perform a comparison of their properties (Milani et al. [27]). To this end, we need to define the weak direction vector field for Newton’s

method.

Weak direction vector field for Newton’s method Let x be an arbitrary point of the space of initial conditions and consider the matrix CN(x) defined in the Newton’s method (see [23], section 5.2); let vN₁ (x), with |vN₁ | = 1, be such that

CN(x)vN₁ (x) = λN₁ (x)vN₁ (x)

where λN₁ is the lowest eigenvalue. Then we can define the weak direction vector field for Newton’s method as follows:

FN(x) = ±_q 1

λN₁ (x)

vN₁ (x).

Comparison of definitions Five different possible definitions have been so far developed. Definition 1. LOV1 is the solution of the differential equation

dx

dσ = F(x)

with initial conditions x(0) = x∗ (a nominal solution).

Definition 2. LOV2 is the solution of the differential equation

dx dσ = F

N_(x)

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Definition 3. LOV3 is the set of points x such that v₁(x) k D(x) Definition 4. LOV4 is the set of points x such that

vN₁ (x) k D(x)

Definition 5. LOV5 is a solution of the differential equation

dx

dσ = k(x)D(x),

where k(x) is a positive scalar function, and is called a curve of steepest descent. Such curves have as limit for σ → +∞ a nominal solution x∗ (almost always: exceptional curves can have a saddle as limit).

Generally, LOV1 and LOV2 are not the same curve; the two curves are close near x∗, provided the residuals are small, but they become very different for large residuals and especially near a saddle.

Not even LOV3 and LOV4 are the same curve. Moreover, they do not imply that the curve contains a nominal solution; indeed, they’re obtained using the constrained differential corrections method. A minimum may not exist, it could be beyond some singularity, such as e = 1 if the elements are Keplerian or Equinoctial; however, if these curves pass in the neighbourhood of a minimum, then they must pass through it.

Remark 6. In a linear case ξ = B(x − x∗) + ξ∗, with B constant, all the definitions LOV1, LOV2, LOV3, LOV4 are the same and they all are curves of steepest descent.

Theorem 1. If a curve satisfies LOV4 and either LOV2 or it is of steepest descent (LOV5), then it is a straight line.

Proof. If a curve is a LOV4, then there is a scalar function h(x) such that FN(x) = h(x)D; thus, LOV2 and LOV5 are equivalent.

We select the particular steepest descent curve

dx dσ = D

and we reparameterize it by arclength s, with

dx ds = 1 ⇐⇒ ds dσ = |D|; then dx ds = ˆD def = D |D|

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is the unit vector in the direction defined by D. Now we take into account the formulas ∂D ∂x = −C N_, dD ds = ∂D ∂x dx ds = −C N_D;_ˆ

thus, we can compute

d2x ds2 = d ds D |D| = 1 |D| dD ds − D,dD ds |D|3 D = = − 1 |D| h CND − h ˆˆ D, CNDi ˆˆ Di

and if we use LOV2 2, we get

CND = λ ˆˆ D =⇒ d 2_x ds2 = − 1 |D| h CND − h ˆˆ D, λ ˆDi ˆDi= = − 1 |D| h CND − λ ˆˆ Di= 0.

Thus we can conclude that the curve must be a straight line.

In conclusion, Milani et al. ([27]) have adopted LOV3 as definition of the LOV, because it is the one actually computable with standard tools, without computing the second derivatives of the residuals (as required for LOV4) and without incurring in the numerical instabilities found in computing LOV1 and LOV2.

Definitions LOV2 and LOV4 are not equivalent and they are indeed different curves apart from very special cases, where they are straight lines. So far it has not been shown that definitions LOV3 and LOV1 give different curves, but given the proven result LOV 2 6= LOV 4 we expect that also LOV 1 6= LOV 3, apart from some very special cases.

2.3 LOV sampling

In this section we show how to parameterize and sample the LOV starting from Defin-ition 3 by applying a uniform sampling strategy.

The equation v₁(x) k D(x) corresponds to five scalar equations in six unknowns, thus it has generically a smooth one-parameter set of solutions, that is a differentiable curve. However, we do know an analytical or anyway direct algorithm neither to compute the points of this curve nor to find some parameterization (e.g., by the arc length). From now on, we describe an algorithm ([23], section 10.1) to compute the LOV by continuation from one of its points x by selecting a uniform step.

The vector field F(x), deduced from the weak direction vector field v1(x), is orthogonal

to H(x). A step in the direction of F(x), such as an Euler step of the solution of the

2

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differential equation dx_dσ = F(x), that is x0 = x + δσF(x), does not provide another point on the LOV, unless the LOV itself is a straight line; this does not depend on the method employed to find a numerical solution of the differential equations and hence this would be true even if the step along the solutions of the differential equation is done with a higher order numerical integration method (a second-order implicit Runge-Kutta-Gauss is normally employed). Anyway, x0 will be close to another point x00 on the LOV, which can be obtained by applying the constrained differential corrections algorithm, starting from x0 and iterating until convergence.

If the LOV parameter of the starting point is σ₀, i.e. x is parameterized as x(σ₀), we can parameterize x00= x(σ0+ δσ), which is an approximation since the value σ0+ δσ actually

pertains to x0.

As an alternative, if we already know the nominal solution x∗ and the corresponding local minimum value of the target function Q(x∗), we can compute the χ parameter as a function of the value of the target function at x00:

χ =qm · [Q(x00_{) − Q(x}∗_)].

In the linear regime, the two definitions are related by σ = ±χ, but this is by no means the case in strongly nonlinear conditions. Thus we can adopt the definition σQ = ±χ, where the

sign is taken to be the same as that of σ, for an alternative parameterization of the LOV. Nevertheless, once we have defined the approximation of x00, we repeat the numerical integration step and constrained differential corrections to obtain another point on the LOV. If we assume that the probability density at the initial conditions x is an exponentially decreasing function of χ, as in the classical Gaussian distributions theory (see [23], chapter 3), then it is logical to terminate the sampling of the LOV at some value of χ. This means that we are considering the LOV as the intersection of the solution of the differential equation

dx

dσ = F(x) with the nonlinear confidence region Z

X_{(b), where b is the maximum χ value;}

when χ = |σ_Q| > b we stop sampling, even if |σ| < b, as the probability values associated with the LOV points would become too low and thus we would obtain no more significant sampling points.

This algorithm to compute the LOV can be used both (a) when a nominal solution x∗ is known, and (b) when it is unknown, even nonexistent.

(a) If the nominal solution x∗ is known, then we can set it as the origin of the param-eterization, x∗ = x(0), and proceed by using either σ or σQ as parameters for the

other points computed with the alternating sequence of numerical integration steps and constrained differential corrections.

(b) If a nominal solution is not available, we must first reach some point on the LOV by performing constrained differential corrections starting from some initial condition (a preliminary orbit, see [23]). Once on the LOV, we can begin moving along it as is done when starting from the nominal point. We set the LOV origin x(0) to whichever

Impact Monitoring of Near-Earth Objects: old algorithms and new challenges

UNIVERSITY OF PISA

MASTER OF SCIENCE IN MATHEMATICS

Master Thesis

Impact Monitoring of Near-Earth Objects: old

algorithms and new challenges

26 October 2018

Contents

Introduction

Chapter 1

Close encounters and Target

Planes

1.1

Öpik’s theory of close encounters

1.2

The Modified Target Plane (MTP)

1.3

Linear analysis on the target plane

1.4

Semilinear confidence boundary

1.5

Returns and keyholes

Chapter 2

The Line Of Variations

2.1

Confidence Regions

2.2

Constrained differential corrections and definition of LOV

2.3

LOV sampling