Line of Variations: Definitions and Applications

Line of Variations: deﬁnitions and applications

Giacomo Tommei

e-mail: [email protected] web: people.unipi.it/tommei 1 Introduction

2 The Line Of Variations (LOV)

3 Selection of a metric

4 Applications of the constrained solutions Orbit determination Orbit identiﬁcation Qualitative analysis Impact monitoring

5 Manifold of Variations (MOV) Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) The orbit determination problem

Orbits y = y(t, y0, µ) solution of dy = f(y, t, µ) y(t ) = y dt 0 0 p y ∈ R : state vector p0 µ ∈ R : dynamical parameters (e.g. geopotential coeﬃcients) t ∈ R: time p y0 ∈ R : initial conditions at time t0 Observations (observation function)

R(y, t, ν)

p00 ν ∈ R : kinematical parameters Prediction function (R ◦ y)

r(t) = R(y(t), t, ν)

Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) The orbit determination problem

Residuals ξi = ri − R(y(ti), ti, ν) , i = 1, . . . , m . 0 00 ξ = (ξi)i=1,...,m depends on the p + p + p variables (y0, µ, ν). Target function (Gauss, 1809)

m 1 T 1 X 2 Q(ξ) = ξ ξ = ξ = Q(y , µ, ν) m m i 0 i=1 Parameters to be ﬁt to the data N p+p0+p00 x ∈ R : subvector of (y0, µ, ν) ∈ R Q(x) = Q(ξ(x))

p+p0+p00−N k ∈ R : consider parameters ﬁxed at assumed value ∗ N The minimum principle selects as nominal solution the point x ∈ R where the target function Q(x) has its minimum value Q∗. Optimization interpretation → Conﬁdence region

Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) Classiﬁcation of the problem

By central body Heliocentric orbits → Near Earth Objects (NEOs) Earth satellite orbits → Space Debris Satellite orbits of other planets → BepiColombo and Juno mission Orbits around another star Cases without a dominant central body By observational data Population orbit determination → NEOs, Space Debris Collaborative orbit determination → BepiColombo and Juno mission

Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) Population orbit determination

Note that: usually the fit parameters x are just the initial conditions for an orbit, thus x ∈ R6; there are at least enough observations to compute an attributable; if the arc is short, an approximate rank deficiency can occur, with order 1 or at most 2. Contents of this lecture Special techniques used to handle this kind of weak orbit determination, how to sample large confidence regions, the origin of such weakness, typically in too short observations time span, and the impact on the quality of the orbit solution.

Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) Conﬁdence region

When an asteroid has just been discovered, its orbit is weakly constrained by observations spanning only a short arc. Although in many cases a nominal orbital solution (corresponding to the least squares principle) exists, other orbits are acceptable as solutions, in that they correspond to RMS of the residuals not signiﬁcantly above the minimum. Conﬁdence region Z(χ) in the orbital elements space such that initial orbital elements belong to Z(χ) if the penalty does not exceed some threshold depending upon the parameter χ (probabilistic or optimization interpretation)

Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) Conﬁdence region

Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) Virtual asteroids

Problem: we need to consider the set of the orbits with initial conditions in the confidence region as a whole (observations prediction, close approach or impact monitoring,..) The N-body problem is not integrable: there is no way to compute all the solutions for some time span in the future or past; we can only compute a finite number of orbits by numerical integration. Concept of Virtual Asteroid (VA): the confidence region is sampled by a finite number of VAs, each one with an initial condition in the confidence region. Problem: how to select the Vas (not unique answer, it depends upon the nature and the time of the prediction to be computed)

Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) Virtual asteroids

We will select a string, that is a one-dimensional segment of a (curved) line in the initial conditions space, the Line Of Variations (LOV). There is a number of different ways to define, and to practically compute, the LOV: LOV as solution of a differential equation LOV from a constrained differential correction algorithm (LOV exists also in the case in which the nominal solution is not available) but the general idea is that a segment of this line is a kind of spine of the confidence region. Once the LOV is defined, it is quite natural to sample it by VAs at regular intervals in the variable which is used to parameterize the curve.

Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) The power of LOV

Using the LOV tool in

(1) orbit determination

(2) orbit identiﬁcation

(3) qualitative analysis

(4) impact monitoring

Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) Newton’s method and diﬀerential corrections

Least squares method: minimization the weighted RMS of the m observation residuals Ξ = (ξi), i = 1, . . . , m introducing the cost function

1 Q = ΞT W Ξ , m

W square, symmetric (but not necessarily diagonal), positive-deﬁnite m × m matrix; it should reﬂect the a priori RMS and correlations of the observation errors.

Design matrix (m × 6 matrix)

∂Ξ B = (X) , ∂X

Gradient of the cost function: ∂Q 2 = ΞT WB. ∂X m The stationary points of the cost function Q are solutions of the system of nonlinear equations ∂Q/∂X = 0

ITERATIVE PROCEDURE (NEWTON METHOD)

∂2Q 2 ∂B 2 = BT WB + ΞT W = CN , ∂X2 m ∂X m

CN : 6 × 6 matrix, positive deﬁnite in the neighborhood of a local minimum.

h i−1 m ∂Q T X −→ X + ∆X, ∆X = CN D,D = −BT W Ξ = − 2 ∂X

PSEUDO-NEWTON METHOD (DIFFERENTIAL CORRECTIONS)

∆X = −(BT WB)−1BT W Ξ . Diﬀerence between Newton’s method and the diﬀerential corrections method: usage of the normal matrix C = BT WB instead of the matrix N −1 N −1 C , and the covariance matrix Γ = C instead of ΓN = [C ] .

Note that the RHS D of the normal equations is the same.

Motivation: the computation of the three-index arrays of second derivatives ∂B/∂X = ∂2Ξ/∂X2 requires to solve 216 scalar diﬀerential equations (on top of the usual 6 + 36 for the equations of motion and the variational equations for the ﬁrst derivatives).

Convergence ⇒ the limit X∗ is a stationary point of the cost function:D(X∗) = 0. X∗ is a local minimum of Q(X): a best-ﬁtting or nominal solution.

Problems 1) Nominal solutions may not be unique (multiple minima). 2) Stationary points of Q(X) are saddles: CN has some negative eigenvalue. The term Ξ W ∂B/∂X provides a signiﬁcant contribution to CN , to the point of changing the sign of at least one eigenvalue. This can happen more easily when the residuals Ξ are large (the saddle corresponds to a value of Q well above the minimum). If the matrix C is badly conditioned, a very small eigenvalue of C can be perturbed into a negative eigenvalue of CN even with moderate residuals Ξ (we will see an example).

Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) The conﬁdence region

Expansion of the cost function at a point X = X∗ + ∆X in a neighborhood of X∗: 1 Q(X) = Q(X∗) + ∆XT CN ∆X + ... = m 1 = Q(X∗) + ∆XT C ∆X + ... = Q(X∗) + ∆Q(X) , m Conﬁdence region:

Z(χ) = X | ∆Q(X) ≤ χ2/m .

Conﬁdence region small + residuals small ⇒ conﬁdence ellipsoid

n T ∗ 2o ZL(χ) = X | ∆X C(X ) ∆X ≤ χ .

Unit vector V1: direction of the longest semiaxis of the conﬁdence ellipsoid and eigenvector of the normal matrix C(X∗), computed at the nominal solution ∗ C(X ) V1 = λ1V1 λ1 < λj , j = 2,..., 6

Length of the longest semiaxis of the conﬁdence ellipsoid ZL(1) √ k1 = 1/ λ1

∗ 2 (Note that V1 is also an eigenvector of Γ(X )) with eigenvalue 1/λ1 = k1

H: the hyperplane spanned by the other eigenvectors Vj , j = 2,..., 6.

Properties in linear regime ∗ The tip of the longest axis of the conﬁdence ellipsoid X1 = X + k1 V1 has the property of being the point of minimum of the cost function restricted to the aﬃne space X1 + H. It is also the point of minimum of the cost function restricted to the sphere |∆X| = k1. These properties, equivalent in the linear regime, are not equivalent in general.

Let us consider the vector k1(X) V1(X)

It can be defined at every point of the space of initial conditions X: the normal matrix C(X) is defined everywhere, thus we can find at each point X the smallest eigenvalue of C(X): 1 C(X) V1(X) = λ1(X) V1(X) = 2 V1(X) k1(X) and the product k1(X) V1(X) is a vector field defined for every X.

Problem 1 F (X) = k1(X) V1(X) is an axial vector: well defined length and direction but undefined sign. Proposition: 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. Rules to select the sign: the directional derivative of the semimajor axis a is positive in the direction +V1(X); the heliocentric distance is increasing.

Our two impact monitoring systems, CLOMON2 and Sentry use diﬀerent conventions for the orientation of the LOV. Thus, in comparing the outputs of CLOMON2 and Sentry, it is necessary to check whether the orientation of the LOV is the same.

Problem 2

Eigenvalues of the normal matrix C(X): for some value of the initial condition X, the smallest eigenvalue has alegbraic multiplicity equal to 2 (rare case);

for some value of the initial condition X, the two smallest eigenvalues are of the same order of magnitude.

The LOV method has serious limitations in these cases (switch to Manifold of Variations, called also Surface of Variations).

The diﬀerential equation dX = F (X) (1) dσ has a unique solution for each initial condition (vector ﬁeld smooth).

X(0) = X∗ ⇒ X = X(σ)

Linear approximation: the solution X(σ) is the tip of the major axis of the conﬁdence ellipse ZL(σ).

Non linear regime: X(σ) is indeed curved and can be computed only by numerical integration of the diﬀerential equation.

Problem: numerical instability in the computation.

For weakly determined orbits the graph of the cost function is like a very steep valley with an almost flat river bed at the bottom. The river valley is steeper than any canyon you can find on Earth; so steep that the smallest deviation from the stream line sends you up the valley slopes by a great deal. This problem cannot be efficiently solved by brute force, that is by increasing the order or decreasing the stepsize in the numerical integration of the differential equation. The only way is to slide down the steepest slopes until the river bed is reached again, which is the intuitive analog of the new definition.

Orthogonal hyperplane H(X) to V1(X):

H(X) = {Y |(Y − X) · V1(X) = 0} .

Diﬀerential correction constrained to H(X): 5 × m matrix BH (X) (pd of the residuals wrt the coord. of the vector H on H(X)). Constrained normal equations:

T T CH = BH WBH ,DH = −BH W Ξ ,CH ∆H = DH

−1 ∆H = ΓH DH , ΓH = CH

(the constrained covariance matrix ΓH is not the restriction of the covariance matrix Γ to the hyperplane)

Computation of CH , DH : rotation to a new basis in which V1(X) is the first vector, then CH is obtained by removing the first row and the first column of C, DH by removing the first coordinate from D.

Constrained diﬀerential correction algorithm We compute the corrected X0 = X + ∆X where ∆X coincides with ∆H along H(X) and has zero component along V1(X). The weak direction and the hyperplane are recomputed: 0 0 0 V1(X ),H(X ) and the next correction is constrained to H(X ). This procedure is iterated until convergence.

If X is the convergence value, then DH (X) = 0, that is the right hand side of the unconstrained normal equation is parallel to the weak direction

D(X) || V1(X) . (2)

Eq. (2) is equivalent to the following property: the restriction of the cost function to the hyperplane H(X) has a stationary point in X: the constrained corrections correspond to the intuitive idea of “falling down to the river”. Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) The new deﬁnition of LOV

Deﬁnition The Line Of Variations (LOV) is the set

{X | D(X) = s V1(X) for some s ∈ R} , (3)

where the gradient of the cost function is in the weak direction.

Note that: If there is a nominal solution X∗, then D(X∗) = 0, thus it belongs to the LOV. The LOV is deﬁned independently from the existence of a local minimum of the cost function.

Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) Parameterizing and sampling the LOV

D(X) = s V1(X): five scalar equations in six unknowns, thus it has generically a smooth one parameter set of solutions (a differentiable curve). Problem: we do not know an analytic or anyway direct algorithm neither to compute the points of this curve nor to find some natural parameterization. Algorithm to compute the LOV by continuation from one of its points F (X) ⊥ H(X): a step in the direction of F (X) (Euler step of the solution of the differential equation (1)), that is X0 = X + δσ F (X), is not providing another point on the LOV (unless the LOV itself is a straight line) and this would be true even if the step along the solutions of the differential equation is done with a higher order numerical integration method, such as a Runge-Kutta (we use a second order implicit Runge-Kutta-Gauss). 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. Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) Parameterizing and sampling the LOV

Multiple solutions, 2 steps 0.7 Top: starting from X∗ (circle), the LOV 0.6 solutions are obtained by propagation of 0.5 eq. (1) followed by constrained diﬀerential corrections (each iteration is a cross); they 0.4

eccentricity converge to the “river” (continuous line), 0.3 whose points have been computed by the same procedure with a much smaller step. 0.2 2.5 3 3.5 4 semimajor axis, AU Convergence of multiple solutions 1000 Bottom: the RMS of the residuals is large 800 at the starting point of each constrained

600 diﬀerential corrections procedure, and rapidly converges towards the much smaller 400 values obtained along the “river” line

RMS of residuals 200 (circles).

0 2.5 3 3.5 4 semimajor axis, AU

If X was parameterized as X(σ), we can parameterize X00 = X(σ + δσ), which is an approximation since the value σ + δσ actually pertains to X0. If we already know the nominal solution X∗ and the corresponding local minimum value of the cost function Q(X∗), we can compute the χ parameter as a function of the value of the cost function at X00:

χ = pm · [Q(X00) − Q(X∗)] .

Linear regime: the two deﬁnitions are related by σ = ±χ

Non linear regime: we can adopt the deﬁnition σQ = ±χ, where the sign is taken to be the same as that of σ, for an alternate parameterization of the LOV. Assuming a probability density at the initial conditions, it is logical to terminate the sampling of the LOV at some value of χ.

The algorithm described above can actually be used in two cases.

1) When a nominal solution is known: if he nominal solution X∗ is known, then we set it as the origin of the parameterization, ∗ 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.

2) When a nominal solution is unknown, even nonexistent: we must ﬁrst reach some point on the LOV by making constrained diﬀerential corrections starting from some potentially arbitrary initial condition. Once on the LOV we can begin navigating along it in the same manner as is done when starting from the nominal point.

Once on the LOV we can begin navigating along it in the same manner as is done when starting from the nominal point:

we set the LOV origin X(0) to whichever point X of the LOV we have ﬁrst found with constrained diﬀerential corrections, when starting from the initial guess;

we then compute the other points as above and use the parameterization σ with arbitrary origin (unfortunately, the parameterization σQ cannot be computed; however, it can be derived a posteriori).

Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) Comparison of LOV deﬁnitions

Deﬁnition LOV1 The solution of the diﬀerential equation dX = F (X) (4) dσ with initial conditions X(0) = X∗ (a nominal solution).

Deﬁnition LOV2 The solution of the diﬀerential equation dX = F N (X) (5) dσ

N N N N N 1 N C (X) V1 (X) = λ1 (X) V1 (X) F (X) = ± V1 (X) p N λ1 (X) with initial conditions X(0) = X∗.

Deﬁnition LOV3 The set of points X such that

V1(X)||D (6)

Deﬁnition LOV4 The set of points X such that

N V1 (X)||D (7)

Deﬁnition A solution of the diﬀerential equation dX = k(X)D(X) , (8) dσ with k(X) a (positive) scalar function, is called a curve of steepest descent. Such curves have as limit for s → +∞ a nominal solution X∗ (almost always; exceptional curves can have a saddle as limit).

LOV1 and LOV2 are not the same curve The two curves are close near X∗ (provided the residuals are small), they become very diﬀerent for large residuals and especially near a saddle. LOV3 and LOV4 are not the same curve LOV3 and LOV4 do not imply that the curve contains a nominal solution; indeed a minimum may not exist (it may be beyond some singularity, such as e = 1 if the elements are Keplerian/Equinoctial). However, if these curves pass in the neighborhood of a minimum, then they must pass through it. 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 If a curve satisﬁes LOV4 and either LOV2 or it is of steepest descent, then it is a straight line.

Proof If LOV4, then there is a scalar function h(X) such that F N (X) = h(X) D, thus LOV2 and being of steepest descent are equivalent. Let us select the particular steepest descent curve dX = D dσ and let us reparameterize the curve by arclength s, with

dX ds = 1 ⇐⇒ = |D| ds dσ then dX = Dˆ ds the unit vector in the direction deﬁned by D. Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) Comparison of LOV deﬁnitions

Taking into account that ∂D dD ∂D dX = −CN , = = −CN D,ˆ ∂X ds ∂X ds let us compute

d2X d D 1 dD < D, dD > = = − ds D = ds2 ds |D| |D| ds |D|3 −1 h i = CN Dˆ− < D,Cˆ N Dˆ > Dˆ |D| and if we use LOV2 (or even a weaker condition that D is parallel to some eigenvector of CN )

d2X −1 h i CN Dˆ = λDˆ ⇒ = CN Dˆ− < D,ˆ λDˆ > Dˆ = 0 ds2 |D| thus the curve must be a straight line. QED

We 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 and without incurring in the numerical instabilities found in computing LOV1-LOV2. Definitions LOV2 and LOV4 are not equivalent, and they are indeed different curves apart from very special cases, where they are straight lines. We have not been able to prove that definitions LOV3 and LOV1 give different curves, but given the proven result LOV2 6= LOV4 we expect that also LOV1 6= LOV3 apart from some very special cases.

The eigenvalues λj of the normal matrix C are not invariant under a coordinate change. The weak direction and the deﬁnition of LOV depend upon the coordinates used for the elements X Linear coordinate change: Y = SX, the normal and covariance matrices are transformed by

T −1T −1 ΓY = S ΓX S ,CY = S CX S

The eigenvalues, solutions of det [CY − λ I] = 0 are the same if S−1 = ST , (isometric transformations). Nonlinear coordinate change: the eigenvalues in the Y space are not the same, and the eigenvectors are not the image by S of the eigenvectors in the X space. The weak direction and the LOV in the Y space do not correspond by S−1 to the weak direction and the LOV in the X space.

Scaling: transformation changing the units along each axis, represented by a diagonal matrix S. The choice of units should be based on natural units appropriate for each coordinate. Coordinates used Cartesian heliocentric coordinates (position, velocity)

Cometary elements (q, e, I, Ω, ω, tp, with tp the time of passage to perihelion) Keplerian elements (a, e, I, Ω, ω, `, with ` the mean anomaly) Equinoctial elements (a, h = e sin($), k = e cos($), p = tan(I/2) sin(Ω), q = tan(I/2) cos(Ω), λ = ` + $, with $ = Ω + ω) Attributable elements (α, δ, α,˙ δ,˙ r, r˙, with α the right ascension, δ the declination, r the range, the dots indicate time derivatives)

Cartesian x y z vx vy vz Units AU AU AU AU/d AU/d AU/d Scaling r r r v v v Cometary e q tp Ω ω i Units - AU d rad rad rad Scaling 1 q Z 2π 2π π Keplerian e a M Ω ω i Units - AU rad rad rad rad Scaling 1 a 2π 2π 2π π Equinoctial a h k p q λ Units AU - - - - rad Scaling a 1 1 1 1 2π Attributable α δ α˙ δ˙ r r˙ Units rad rad rad/d rad/d AU AU/d Scaling 2π π n⊕ n⊕ 1 n⊕

Note: Here r and v are the heliocentric distance and velocity, respectively, d is one day. The angular rate n⊕ = 0.01720209895 rad/day, is approximately the mean motion of the Earth and is numerically equivalent to the Gaussian gravitational constant, 3 2 −1/2 3/2 −1 −1/2 k = 0.01720209895 (AU /d ) . The quantity Z = 2πq n⊕ (1 − e) is a characteristic time for a large eccentricity orbit.

Nonlinear coordinate change

∂Φ Y = Φ(X) ,S(X) = (X) , Γ = S(X)Γ S(X)T ∂X Y X ⇒ ∆Y can be computed accordingly.

Once the constrained diﬀerential correction ∆Y has been computed, we need to pull it back to X. If ∆Y is small, as is typically the case when taking modest steps along the LOV, then this can be done linearly X0 = X + S−1∆Y.

When the constrained diﬀerential corrections are large, as is likely to be the case when the initial point is not near the LOV then the correction ∆Y must be pulled back to X nonlinearly, that is X0 = Φ−1(Y + ∆Y ).

Problem (complex): to select the most effective coordinates in a specific case of orbit determination and for a specific use. (1) Arc drawn on the celestial sphere by the apparent asteroid position is small (≤ 1◦): coordinate systems which represent instantaneous initial conditions Cartesian coordinates, Attributable elements (the angular variables α, δ, α,˙ δ˙ well determined, r, r˙ poorly determined). (2) observed arc comparatively wide (tens of degrees): orbital elements solving exactly the two body problem Cometary elements (suitable for high eccentricity orbits), Equinoctial elements (avoid singularity for e = 0 and I = 0), Keplerian elements (poor metric properties for both e ' 0 and e ' 1).

Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) Diﬀerent LOVs ◦ 2004FU4 not scaled LOVs (ﬁrst 17 observations, time span of ' 3 days, arc of ' 1 )

Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) Diﬀerent LOVs ◦ 2004FU4 scaled LOVs (ﬁrst 17 observations, time span of ' 3 days, arc of ' 1 )

Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) Diﬀerent LOVs ◦ 2002NT7 not scaled LOVs (ﬁrst 113 observations, 15 days, arc of 9 )

Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) Diﬀerent LOVs ◦ 2002NT7 scaled LOVs (ﬁrst 113 observations, 15 days, arc of 9 )

Giacomo Tommei Weak solutions and LOV Introduction Orbit determination The Line Of Variations (LOV) Orbit identiﬁcation Selection of a metric Qualitative analysis Applications of the constrained solutions Impact monitoring Manifold of Variations (MOV) LOV solutions from preliminary orbits

The constrained diﬀerential corrections procedure can start from: an arbitrary initial guess X, which can be provided by some preliminary orbit;

from a known LOV solution (be it the nominal or not) as part of the continuation method to obtain alternate LOV solutions.

In both cases it can provide a richer orbit determination procedure.

1) If some orbit X is already available, it is used as preliminary. 2) If no orbit is available, a preliminary orbit X is computed: classical methods or other methods for short arcs. 3) If the preliminary orbit algorithm fails, the orbit determination procedure is considered failed (unless some other preliminary orbit algorithm is available). 4) If the preliminary orbit X is available, constrained differential corrections are computed starting from X as first guess. 5) If constrained differential corrections converge to a LOV solution XLOV (with RMS of the residuals ≤ Σ) then a full differential correction is attempted by using XLOV as first guess.

6) If the full diﬀerential corrections converge to a nominal solution X∗ (with RMS of the residuals ≤ Σ) then this is adopted as orbit (with uncertainty described by its covariance).

7) If the full differential corrections fail, then XLOV is adopted (if available, and with RMS ≤ Σ) as orbit (the covariance provides information on the uncertainty but it is not possible to define formally a confidence region because the minimum value of the cost function is not known and may not exist). 8) If the constrained differential corrections fail to converge we start differential corrections with the preliminary orbit X as first guess. 9) If these differential corrections converge to X∗ (with RMS of the residuals ≤ Σ), it is then adopted as orbit. 10) If this last attempt fails the orbit determination procedure is considered to be failed.

Some orbit available Yes No Preliminary orbit

Computed No Failure X

Yes

Constrained Diff. Corr.

Diff. Corr. YesConverging No Diff. Corr. from X from X LOV to X LOV

Yes Yes Converging Adopt X * Converging to X * to X *

No No

Adopt X Failure LOV

Data: all the astrometric data on unnumbered asteroids as made public by the Minor Planet Center on the 9th of November, 2003 (7.5 million observations). As many least squares orbits as possible among the 233, 411 designations corresponding to independent asteroid discoveries. Equinoctial coordinates (unscaled), control value for the normalized RMS of the residuals set to Σ = 3. The modern (after 1950) observations have been weighed at 1 arcsec, thus normalized RMS = 3 essentially means unnormalized RMS = 3 arcsec. If more than one preliminary orbit is available, from Gauss’ and from other preliminary orbit methods, in case of failure we repeat the procedure starting from each preliminary orbit. Final outcome: full orbit, constrained orbit, complete failure.

Quality Code No. Nights Arc Length (d) No. Obs.

1 ≥ 5 ≥ 180 ≥ 10 2 ≥ 4 ≥ 20 ≥ 8 3 ≥ 3 ≥ 6.5 ≥ 6 4 ≥ 3 ≥ 1.5 ≥ 6 5 ≥ 2 ≥ 0.5 ≥ 4 6 ≥ 2 ≥ 3 7 = 1 ≥ 3

Classiﬁcation of the designated asteroids according to the amount and timing of the available astrometry

QC Objects w/N. w/o N. D. C. M.S. 1 78, 672 78, 672 0 78, 672 N.A.

2 52, 927 52, 906 14 52, 888 20.0

3 18, 760 18, 642 66 18, 543 19.8

4 7, 818 7, 684 120 7, 518 18.9

5 7, 452 6, 199 972 5, 189 14.7

6 56, 098 29, 097 19, 068 22, 240 13.4

7 2, 200 62 57 9 7.2

Tot 223, 927 193, 262 20, 297 185, 059 17.3

QC: the quality code; w/N, number of nominal solutions obtained with diﬀerential corrections using the LOV solutions as intermediary; w/o N, number of cases in which a LOV solution could be computed but a nominal solutions could not; D.C., number of nominal solutions obtained by the classical procedure, full diﬀerential corrections without intermediaries; M.S., average number of multiple solutions computed per object with some least squares orbit, taking into account that the target was to compute 21 solutions per object.

Designations: correspond to independent asteroid discoveries, but they do not necessarily correspond to physically different objects. Identification: process by which two (or more) designations are found to contain observations of the same object, by computing a least squares fit to all the observations, with acceptable residuals. Problem: how to decide among all the couples which ones should be tested as candidates for identification. We need some filtering procedure, selecting couples of designations on the basis of some metric describing similarity between the two.

Procedures to propose identiﬁcations belong to diﬀerent types, depending upon the quality of observational data available for each designation.

Orbit Identification, when the observations of both arcs are sufficient to separately solve for two least squares orbits Attribution, when an amount of data insufficient to compute an unique orbit for one arc is compared to an orbit already computed for the other arc Recovery of a lost asteroid, by finding it either in the sky or in the image archives Linkage, when two arcs of observations, both too short to perform OD, are to bejoined into an arc good enough to compute an orbit

Background: predictions with large uncertainty and extremely nonlinear (this happens when the conﬁdence region is very large, either at the initial epoch or at some subsequent time, after the propagation has stretched the conﬁdence region preferentially in the along track direction).

Typical use of multiple solutions: to compute observation predictions, that is ephemerides.

For each observation epoch t, we can compute the 2p + 1 points

yk = F (xk(t))

on the celestial sphere, and plot the line joining these points (see next ﬁgure).

This method has been found useful to search for precoveries in plate archives (Boattini et al. 2001).

Asteroid 1992BU 10

-10

-20

-30

-40 Differences in declination (DEG) from 27.9900

-50 -100 -80 -60 -40 -20 0 20 40 60 80 100 120 Differences in right ascension (DEG) from 43.9589

Multiple solutions for the asteroid 1992BU give multiple ephemerides 31 years before, when 4161P LS was discovered. The actual observation of 4161 PLS is marked with a cross.

Identiﬁcations could be achieved by comparing multiple solutions for two asteroids, observed during short and widely separated arcs.

Example: case of 4161 PLS = 1992 BU The two curves, plotted in the (a, e) plane (the inclination and node are typically better determined), are the multiple solutions computed for both single opposition orbits (the gaps correspond to the nominal). The two lines cross in only one point: we select, among the multiple solutions computed, the two which are closest to this intersection point. From them by the linear identiﬁcation formula

−1 X0 = C0 (C1 X1 + C2 X2) C0 = C1 + C2 we compute the first guess for the least squares fit to all the observations of both arcs: the differential corrections converge to an orbit, shown with a cross.

Multiple solutions (up to σ = 3) for the asteroids 4161 PLS and 1992 BU: the projections of the LOV on the (a, e) plane have a single intersection point which is close to the least square fit (with the observations of both arcs). They were identified with a different procedure.

Better method To compute orbit identiﬁcation penalties from

−1 −1 K = ∆X · C ∆XC = C1 − C1 C0 C1 = C2 − C2 C0 C2 for each couple of multiple solutions. To find the minimum of the (2p + 1)2 penalties, for a given couple of objects, and proposes the couple as identification if this minimum is below some control value. In Milani et al. 2005 we discuss the systematic application of this class of methods to a large data set of asteroid orbits, with considerable success (' 1 500 confirmed identifications found in a single run).

Linkage problem: after computing the preliminary orbits (VAs method or Prime Integrals method), the next step is to compute, starting from the preliminary orbits, least squares solutions. Problem: in most cases the observational data available are very limited even after the identification, e.g., just enough to compute two attributables. Thus constrained differential corrections are necessary as first step, and in most cases the LOV solutions are the only ones achievable. Centaur (31824) Elatus: discovered in October 1999 (designation 1999 UG5), followed up until a good orbit could be computed, allowing to attribute to it prediscovery observations from October 1998. Test: linking the data from the discovery night with the precovery data one year earlier, with the method of the admissible region.

1999 UG5

Linkage of the 4 discovery observations with one night of data one year earlier.

The continuous lines join the nodes with identiﬁcation penalty Ki < 52: none of them belongs to the ﬁrst connected component of the admissible region, the one closer to the Earth.

The nominal least squares solution, marked “best”, is a hyperbolic orbit: it is not close to the true solution, which is marked with a crossed square near the node number 10.

The LOV has been approximated by a straight line best ﬁtting all the LOV points obtained in this way.

The sampling along the LOV is also useful to understand the situation whenever the orbit determination is extremely nonlinear.

Problem of multiple local minima

Example: asteroid 1998 XB (discovered on December 1, 1998, while it was at an elongation of ' 93◦ from the Sun) The ﬁrst orbit published by the MPC, based on observations over a time span of 10 days, had a semimajor axis a = 1.021 AU In the following days the orbit was repeatedly revised by the MPC, with semimajor axis gradually decreasing until 0.989 when the observation time span was 13 days. With the addition of observations extending the time span to 16 days, the semimajor axis jumped to 0.906.

The RMS of the residuals (in arcsec), as a 1.4 function of the LOV parameter σ, for diﬀerent amounts of observational data.

1.2 The lines are: red plusses (9 days); 1 green crosses (10 days); blue stars (11 days); RMS of residuals 0.8 purple boxes (13 days);

0.6 light blue full boxes (14 days); yellow circles (16 days).

0.4 -10 -5 0 5 10 15 20 25 30 LOV sigma

From ﬁgure: the RMS of the residuals along the LOV has a double minimum: the secondary minimum moves, as the data increase, in direction of lower a, but not as far as the location of the other minimum. The secondary disappears only with 16 days of data, and then the diﬀerential corrections lead to the other solution.

Note: Gauss’ method for determining an orbit from three observations can have two distinct solutions when the elongation is below 120◦. When applied with three observations selected in the shorter time spans, it can provide preliminary orbits close to both the primary and the secondary minimum.

This example confirms that the region with elongation around 90◦ is especially difficult for orbit determination, but also shows that the LOV can provide a very efficient tool to understand complex nonlinearities.

The sampling along the LOV is an essential tool whenever the predictions are extremely nonlinear: confidence region very large, at least in one direction, at the initial epoch (very limited observational data) confidence region very large, at some subsequent time (chotic propagation, close encounters). The use of the LOV in impact monitoring was introduced in 1999 and today, both CLOMON2 and SENTRY explore the geometry of the confidence region with this tool. NEXT LECTURES (JANUARY 2014)

Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) Manifold Of Variations

Problem: if the conﬁdence region is not elongated in one direction much more than in the others, whatever LOV we select may not be representative of the entire conﬁdence region.

Short arc of observations, attributable elements (A, ρ, ρ˙), where A is the attributable. Manifold (surface) of variations: the set S of the points where the target function has a local minimum with respect to changes of A, for each ﬁxed (ρ, ρ˙), with minimum RMS of the residuals below some control Σ. S is, generically, a 2-dimensional manifold. When there is little information beyond A, S is parameterized by (A(ρ, ρ˙), ρ, ρ˙), deﬁned on a subset B of the (ρ, ρ˙) plane: B is an open set, not necessarily connected.

Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) Manifold Of Variations

The surface S can be computed point by point: for each (ρ0, ρ˙0) we can correct only A, i.e., perform doubly constrained diﬀerential corrections, with normal equation

T T CA ∆A = DA ,CA = BA BA ,DA = −BA ξ ,BA = ∂ξ/∂A .

If these corrections converge to a point of minimum A(ρ0, ρ˙0), and if the RMS of the residuals at this minimum is < Σ, the point (A(ρ0, ρ˙0), ρ0, ρ˙0) belongs to the manifold of variations, and (ρ0, ρ˙0) belongs to B.

Giacomo Tommei Weak solutions and LOV Introduction The Line Of Variations (LOV) Selection of a metric Applications of the constrained solutions Manifold of Variations (MOV) Manifold Of Variations

To compute the surface of variations it is not required to compute the admissible region.

We can start from a set of points sampling the (ρ, ρ˙) plane in any convenient way: rectangular grid (used by Tholen in the free software KNOBS); triangulation or cobweb (our group); random selection.

Giacomo Tommei Weak solutions and LOV