Boutonnet, A.

Colombo, C.

Masat, A.

English

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5th International Workshop on Key Topics in Orbit Propagation Applied to SSA, Logroño, June 22-24, 2022Flyby dynamical characterisation with Jacobian eigenvaluesAlessandro Masat*1, Camilla Colombo1, and Arnaud Boutonnet21Politecnico di Milano, Milano, Italy, 201562European Space Agency (ESA-ESOC), Darmstadt, Germany, 642931 IntroductionThe detection of flybys in the numerical propagationof interplanetary trajectories is a key aspect to en-able planetary protection analyses. Their much fasterdynamics, compared to the pure heliocentric motion,hinders the development of advanced orbital integra-tors: regularisation-based approaches and variation ofparameters implementations are extremely sensitiveon close approach events, and may fail in correctlypredicting their effect on the propagated trajectory.Switching the integration/regularisation centre to theflyby body in case of close approaches is an effec-tive workaround to retain all the benefits of advancedpropagation techniques [1]. Nonetheless, this philos-ophy requires to properly define and, consequently,detect possible flyby events.The commonly used concepts all rely on the defi-nition of a spherical region that surrounds any solarsystem planet. In general, the stronger the gravita-tional field of the minor body, the larger its associatedsphere. Sphere of influence (SOI) and Hill’s sphereare the two usually adopted definitions. The formerapproximates the distance from the minor body whereeither the planet or the Sun can be considered as aperturbation of the other dynamics. The latter ap-proximates the distance of the Lagrange points L1and L2 from the planet [3]. Differently from thesedefinitions, Debatin et al. [2] used the eigenvalues ofthe dynamics’ Jacobian to measure the relative mag-nitudes of the contribution of the different bodies inan N-body system, and used it as robust step controlmechanism. Similarly, Romano [5] used this criterionto detect flybys in planetary protection analyses.This work analyses flybys from a different perspec-tive. Rather than building on a concept based on equi-librium distances, the dynamical nature of the closeapproach is emphasised. The eigenvalues of the three-body problem dynamics’ Jacobian leads to an analyti-cal solution for a spheroidal locus of points that locally*Email: alessandro.masat@polimi.it. PhD Candidate, De-partment of Aerospace Science and Technology.Email: camilla.colombo@polimi.it. Associate Professor,Department of Aerospace Science and Technology.Email: arnaud.boutonnet@esa.int. Senior Mission Analyst,HSO-GFA, ESA-ESOC.highlights which body is contributing the most to thedynamics variation. The results are also comparedagainst the Circular Restricted Three-Body Problem(CR3BP) zero-velocity curves concept.2 Jacobian eigenvalues for flyby detectionThe dynamics’ Jacobian is traditionally linked to thestep size control for general numerical simulations.In particular, the maximum eigenvalue influences thestability of the numerical scheme [4]. Contrarily topredictor-corrector integrators (e.g. the Runge-Kuttafamily), the knowledge on the dynamics Jacobian isexploited to minimise the truncation error. In the or-bital dynamics case, the work of Debatin et al. [2]used an analytical approximation of the maximumJacobian eigenvalue of the N-body dynamics, build-ing a fast integration algorithm with step size control.They approximate the square of the maximum Jaco-bian eigenvalue λ2max as the sum of the squares of allthe separate two-body Jacobian eigenvalues:λ2max ≈N∑i=1λ2i =N∑i=12µi∣∣r− ri∣∣3 (1)with the subscript i denoting the i-th body, r theposition and µi the gravitational parameter. This ap-proximation becomes particularly reliable far from theboundaries of any sphere of influence/Hill’s sphere,since in these regions either the Sun or the planetflown by heavily dominates the dynamics.The later work of Romano [5] used a similar ap-proximation approach to implement a flyby detectioncriterion. If the ratio between the eigenvalues of agiven planet and the Sun grows above a user-specifiedtolerance, then a flyby event is detected. Romanoalso showed that this criterion encompasses the usu-ally defined sphere of influence/Hill’s sphere, in thecase of threshold set equal to 1.3 Dynamical meaning in the Three-body problemThe barycentric three-body Jacobian J is defined as:J =[0 IG 0](2)5th International Workshop on Key Topics in Orbit Propagation Applied to SSA, Logroño, June 22-24, 2022with 0 and I the 3 × 3 null and identity matrices,respectively, and G defined as:G =2∑i=1Gi =2∑i=1µi∣∣di|5 (|di∣∣2I− 3didTi ) (3)with di = r − ri. It can be proved that, in the two-body case, λj,J ≡ λ2j,G, with j = 1, 2, 3, and λ2max,Gis defined as in Equation (1) [5]. λ2 appears onlyon the right-hand side because the properties of thedeterminant of block-square matrices are introducedto compute the eigenvalues of J.Approximating the dynamics as if only the cur-rently dominant body 1 was present reflects on theapproximated Jacobian as G ≈ G1. The matrix Ja-cobian error becomes equal to G2. Since any Gi issymmetric (Equation (3)), its euclidean matrix normequals its spectral radius, i.e.∣∣Gi∣∣2 ≡ ρ(Gi) ≡λmax,Gi . In other words, the maximum eigenvalueof G2 directly measures the error of G ≈ G1.The color scale in Figure 1 shows the maximumJacobian error values in the space near Jupiter, i.e.max[λmax,Sun/λmax, λmax,Jup/λmax], on the planecontaining Jupiter’s orbit about the Sun. Rather thancrossing a fixed-shape spheroid, the proposed visuali-sation highlights the smooth transition nature of fly-bys. Even if fast, the whole continuous domain iscrossed, starting from the interplanetary space, pass-ing through a region where none of the two bodiesdominates the gravitational acceleration change, andfinally reaching a far greater planetary effect at low al-titudes. Two higher error regions, called ”Thickenedregions” in Figure 1, appear nearly perpendicularlyto the Sun-Jupiter direction. The possible reason oftheir appearence is explored in Section 5.Figure 1: Jupiter’s Jacobian percent error (colorscale), compared against Hill’s surfaces (dotted) andSOI (dashed).Equivalently, this criterion highlights the highestcurvature regions of the three-body gravitational po-tential V = −µ1/∣∣d1∣∣−µ2/∣∣d2∣∣, since it can be provedthat G ≡ Hessian(V ). In other words, the space re-gions determining a flyby may be identified by thelocal curvature of the gravitational potential.4 Analytical loci of pointsReplacing the G,J subscript notation with the bodynames, a direct comparison can be made with theparametrisation λmax,Jup/λmax,Sun = γ. As donein Figure 1, centering the reference frame on Jupiterleads to an analytical expression for the spheroidalloci with common γ, whose radius r̃ is:r̃(θ) = rSun−Jup−α cos θ +√α(1− α sin2 θ)1− α(4)with θ the angle between the direction of r̃ and thedirection identified by the line connecting the Sun andJupiter, and α = (γµJup/µSun)2/3. The 2/3 exponentarises because of the third power of the eigenvalueexpression (Equation (1)), and a squaring taken toremove the square root of the vector norm operator.Figure 2 extends Figure 1, comparing the analyticalloci of points obtained with Equation (4) against thecomputed values of the Jacobian error (on the colorscale), for γ equal to 0.1, 1, and 10. The cases γ =0.1, 10 (dashed red lines) well bound the regions wherethe Jacobian approximation error is higher than 10%.Thickened regions aside, the case γ = 1 (solid redline) perfectly predicts the ”critical” distances wherethe error is maximised.Figure 2: Jupiter’s Jacobian percent error (colorscale), compared against Hill’s surfaces (dotted), SOI(dashed), and analytical loci of points (red).The ”critical” spheroid (parametrised by γ = 1)can be used as robust flyby detection criterion, assess-ing if and when a propagation crosses that boundary,5th International Workshop on Key Topics in Orbit Propagation Applied to SSA, Logroño, June 22-24, 2022aiming to the minimisation of the truncation error in-troduced by the interrupted integration. In fact, forthe Sun-Earth case this distance falls within the min-imum error range identified by Amato et al. [1].5 Visualisation in the CR3BPThe CR3BP provides further insight on the natureof the thickened regions observed in Figures 1 and2. Figure 3 shows the perfect alignment of thezero-velocity surfaces, plotted as grey/shadowed ar-eas, with the Jacobian error, on the plane containingJupiter’s orbit about the Sun. The red lines repre-sent different values of γ, equal to 1 (solid), 0.1 or10 (dotted) and 0.01 (dashed). The selected Jacobiconstant to plot the zero-velocity surfaces was onlychosen to highlight their alignment with the thick-ened regions, without particular meaning. On thezero-velocity curves the kinetic energy content of thetest particle, the centrifugal reaction due to the non-inertial rotating frame, and the gravitational attrac-tion of both bodies all balance out. Simplifying thedynamical model along this curves may become inac-curate, particularly if close to the Hill sphere bound-aries, as highlighted by the thickened error regions.In other words, the mutual effect of the two bodies onhow the dynamics changes is more prominent alongthe zero-velocity surface direction, on a wider regionthan the sole ”critical spheroid”.Figure 3: Jupiter’s Jacobian percent error (colorscale), compared against Hill’s surfaces (dotted), SOI(dashed), analytical loci of points (red), and zero-velocity surfaces (grey/shadowed).6 ConclusionThis work proposes a flyby characterisation approachthat accounts for the dynamical nature of the en-counter, focussing on the changes on the dynamicscaused by the body flown by. The actual smoothtransition of the motion from interplanetary to plan-etary is also modelled, highlighting regions of spacewhere none of the two body is clearly dominating, andapproximating the dominance that each body has ineach point of the planet neighbourhood.Apart from the regions nearby the critical distanceand along the zero-velocity curves, the proposed para-metric analytical model accurately predicts the loci ofpoints of common Jacobian error. Setting the param-eter equal to 1 allows the use of the ”critical” spheroidas robust flyby detection criterion. Additionally, fu-ture works will analyse whether this criterion is also asuitable approach to improve the characterisation ofshallow encounters.Deeper details on the model will be given in theoral presentation, as well as further insight on thecomparison with the CR3BP regime and the relationwith zero-acceleration saddle points.AcknowledgmentsThe research leading to these results has receivedfunding from the European Research Council (ERC)under the European Union’s Horizon2020 researchand innovation programme as part of project COM-PASS (Grant agreement No 679086), www.compass.polimi.it, and the European Space Agency (ESA)through the Open Space Innovation Platform (OSIP)co-funded research project ”Robust trajectory designaccounting for generic evolving uncertainties”, Con-tract No. 4000135476/21/NL/GLC/my.References[1] Davide Amato, Giulio Baù, and Claudio Bom-bardelli. Accurate orbit propagation in the pres-ence of planetary close encounters. MonthlyNotices of the Royal Astronomical Society,470(2):2079–2099, Sep 2017.[2] Frank Debatin, A. Tilgner, and Friedhelm Hech-ler. Fast numerical integration of interplanetaryorbits. In ESA Special Publication, volume 255of ESA Special Publication, pages 329–333, Dec1986.[3] W.D. McClain and D.A. Vallado. Fundamentalsof Astrodynamics and Applications. Space Tech-nology Library. Springer Netherlands, 2001.[4] A Quarteroni, R Sacco, and F Saleri. NumericalMathematics Texts in Applied Mathematics. 2007.[5] Matteo Romano. Orbit propagation and uncer-tainty modelling for planetary protection compli-ance verification. PhD thesis, Politecnico di Mi-lano, Supervisors: Colombo, Camilla and SánchezPérez, José Manuel, Feb 2020.

FlyBy Dynamical Characterisation with Jacobian Eigenvalues

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FlyBy Dynamical Characterisation with Jacobian Eigenvalues

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