On the parameter-space of asteroid mitigation

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On the parameter-space of asteroid mitigation missions using ballistic kinetic impactors Alexandre Payez, Johannes Schoenmaekers European Space Agency, ESOC, Robert-Bosch-Str. 5, D-64293 Darmstadt, Germany

Abstract This second paper studies asteroid deflection by means of ballistic kinetic impactors, considering at once the entire parameter space of conceivable Earth-impacting orbits. The parametrisation devised in the first part of this work [1] is shown to efficiently facilitate the identification in parameter space of different broad classes of missions and to enable their connection to key physical drivers. This results from a combination of analytical and numerical investigations ignoring the phasing, highlighting the optimal deflection geometry that could be achieved with a given launcher. The assessment of the feasibility due to the illumination conditions at deflection and its decisive influence on what turns out to be the preferred mission-type in different regions of the parameter space is also discussed in depth.

Contents Contents 1

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Introduction 1.1 Mitigating the risk of collisions with our planet, using kinetic impactors 1.2 Designing optimal kinetic-impactor missions . . . . . . . . . . . . . . . . 1.3 Objective of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Approach, scope, and assumptions . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Ballistic trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Parametrisation of Earth-impacting orbits . . . . . . . . . . . . . . 1.4.3 Key quantities: ∆B and ϕSun . . . . . . . . . . . . . . . . . . . . . . 1.4.4 No phasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.5 Studying deflections in the vicinity of the nodes as initial guesses 1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Deflection merit 2.1 Linearised along-track displacement, ignoring non-secular effects 2.2 Corresponding B-plane displacement . . . . . . . . . . . . . . . . . 2.3 Merit function and B-plane yearly-drift . . . . . . . . . . . . . . . . 2.4 Asteroid mass and launcher performance used in the following .

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Page 1 of 68 Parameter-space of asteroid mitigation missions using ballistic kinetic impactors —Final Report II.

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European Space Agency Agence Spatiale Européenne

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Analytical deflection at the nodes without phasing 3.1 Strict 2π-transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Impact-point approximation . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Strict 2π-transfers, optimised analytically . . . . . . . . . . . . . . . . . 3.2 Strict π-transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Cheapest ballistic transfer to ⃗ropp . . . . . . . . . . . . . . . . . . . . . . 3.2.2 The spacecraft ballistic trajectory, seen as another Earth-crossing orbit 3.2.3 Strict π-transfers, optimised analytically . . . . . . . . . . . . . . . . . .

4 General numerical optimisation without phasing 4.1 Overview of some of the main results . . . . . . . . . . . . . . . . . . . . . . 4.2 Departures from the guesses: solar longitude at deflection . . . . . . . . . 4.3 Striking approximate symmetries in the optimal numerical results . . . . . 4.3.1 Relation between the optimal solar longitude at launch λD and fI 4.3.2 Solar aspect angle at deflection of the best deflection solutions . . 5

What have we learned?

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Appendices

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A Reminder: orbital energy & period variations due to a velocity change ∆⃗vA

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B Useful relations and PHA heliocentric velocities at the nodes

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C Linking difficulties in the daytime region to π-transfer feasibility in (P, e)

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D Optimised strict 2π-transfers: solutions assuming a cubic polynomial fit

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Ballistic spacecraft orbits seen as Earth-crossing: further relations

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Comparisons with some results including the phasing

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G Optimal π-transfers can subsist at higher i even for nighttime-impact PHAs

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References

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1

Introduction

1.1 Mitigating the risk of collisions with our planet, using kinetic impactors In the context of planetary defence, the kinetic-impactor concept is widely considered as being both the most technology-ready and the least controversial technique to mitigate medium-size threats [2]. The idea behind this impulsive technique is to ram a spacecraft with a large relative velocity into the target near-Earth object (NEO). The sought-after effect is a tiny instantaneous modification in the orbital motion of the minor body which effectively leads to a displacement that accumulates over time, such that with one or a handful of spacecrafts one might avoid a foreseen impact with the Earth, assuming a sufficiently early identification of the threat.



ESA UNCLASSIFIED – For Official Use While the method has not been demonstrated to date, that is precisely the objective of NASA’s DART mission [3], scheduled for 2022. A proper validation by means of a demonstration mission is crucially needed because even such a conceptually clear technique is not devoid of potential pitfalls. These include a dependence of the imparted momentum on the physical properties of the target asteroid1 , as well as a possibility of accidentally disrupting it, potentially making things even worse (see, e.g. Ref. [4, 5]).2 Having as much information as possible about the minor body and its composition is therefore particularly important, and a rendez-vous or at least a fly-by prior to deflection is strongly advised in case of an actual threat. As a matter of fact, the DART mission was originally part of the AIDA joint-mission concept between NASA and ESA [7], the second part being a proposed characterisation mission which would have benefited from the know-how of ESA’s Rosetta rendez-vous mission with a comet [8]. For the time being, what we already know is that such high-velocity collisions are technologically feasible. During NASA’s Deep Impact mission [9], a small impactor was indeed deployed and successfully crashed into the large comet Tempel 1 at over 10 km/s—though the aim of that mission was to study the comet interior, and not to quantify a transferred momentum. 1.2 Designing optimal kinetic-impactor missions Confronted with the difficult problem of designing optimal kinetic-impactor missions to deflect Earthcrossing asteroids, a truly wide breadth of approaches have been taken in the literature. Some in particular have not solely focused on working out what the optimal solution is on a case-by-case basis, but have also studied at least to some extent how that solution changes when varying the orbit of the targeted asteroid, hoping to derive some intuition about the physical problem at hand. This tends to come at the price of having to make a number of reasonable simplifications in order to proceed, especially when attempting to cast those results in analytical form. The aim is then indeed not to precisely calculate the deflection in a realistic setting, but rather to derive a broader understanding of the properties of mitigation missions. These are the kinds of works we shall mostly focus on here. There is absolutely no doubt that the most accurate computations possible will be needed to face an actual threat; however, when investigating the orbital dependence of mission properties in fictitious scenarios, such a machinery is arguably disproportionate, and would moreover tend to behave as a black box that might actually even hinder the study itself. Since the amount of displacement induced by kinetic impactors accumulates over time, early deflections obviously tend to be favourable. In the most extreme cases, such as last-minute deflections and interceptions, an optimal solution essentially means one that reaches the asteroid as soon as possible [10, 11]. With longer warning times however, while deflecting early does remain a major driver, the problem is actually more complicated, and the trajectories that optimise the asteroid deflection turn out to be completely different. It is then strongly beneficial to worry about where and how on the asteroid orbit the impulsive impact shall take place, even if that means deflecting the target later on. A trade-off appears: while an early deflection is still key (seeing that the effect accumulates over time), it may also be worth waiting for a later opportunity that might lead to a more efficient drift-rate in the asteroid motion. From this observation, a number of studies have therefore investigated in great detail where to most efficiently induce a change in the velocity of an Earth-impacting asteroid on its orbit, in order to avoid a collision. The series of seminal papers by Park and collaborators [12, 13, 14] are found among such 1

As in the first part of this work [1], any Earth-impacting NEO shall be referred to as “potentially hazardous asteroid” (PHA), or simply “asteroid”. 2 If the disruption risk is deemed too high, one might replace the original kinetic impactor by several downscaled spacecrafts; see e.g. Ref. [6].



ESA UNCLASSIFIED – For Official Use works in particular, as well as a number of successive refinements described e.g. in Refs. [15, 16, 17]. In a nutshell and as very well-known, if we could just arbitrarily decide where to induce a given ∆⃗vA to an asteroid, anywhere on its orbit, the ideal location in general would be to do so at its perihelion.3 The optimal solution tends to be an along-track deflection of the asteroid—that is, for sufficiently long warning times and in the absence of third-body effects due to pre-impact close-encounters which could dramatically alter this view [15, 16, 17]. If such conditions are satisfied and provided that the focus effect due to the gravity of the Earth is taken into account at the very end [13], using analytical formulae for the displacement as well as the 2-body description of Öpik’s theory [18] is actually very satisfactory [16, 17, 19]. However, the question of transfer feasibility, not addressed in those studies, should also be crucially taken into account. A deflection at the perihelion of the asteroid orbit with an actual kinetic-impactor mission would require getting the spacecraft there in the first place: a step which might actually be a real concern. The cost essentially comes as a reduced kinetic-impactor mass available at deflection; one can therefore expect that other locations, easier to reach from the Earth orbit, might lead to better results. Of course, a flyby sequence could in principle be designed to extend one’s reach if need be, but this would come at the cost of a reduced time between the deflection and the would-be impact with our planet—a limiting factor for any mitigation mission relying on accumulating a small effect over a long-enough period of time. Rather than starting from the determination of the optimal ∆⃗vA to construct their solutions, some authors have instead adopted more standard mission-analysis approaches to mission design, which directly take into account the launcher performance as part of the interplanetary trajectory optimisation. Among these in particular are found Refs. [19, 20, 21, 22], which also include some analytical derivations and formulae similar or complementary to what we shall use as we mathematically pose the problem. As argued in what precedes and as also discussed e.g. in Refs. [17, 20, 22], the optimal solutions obtained are then indeed not always characterised by a deflection around the perihelion in general, but can actually rather be found in the vicinity of the nodes for inclined asteroid orbits. Henceforth, we shall refer to the impact node ⃗rI and the opposite node ⃗ropp , as in the first part of this work [1].4 Finally, when taking into account the illumination conditions at deflection, dramatic differences can moreover be expected between daytime-impact asteroids and nighttime-impact asteroids, as discussed e.g. in the preparatory draft of a short study done at ESOC [23]. Deliberately adopting a number of similar assumptions to preserve some backward compatibility with the existing runs, the current work will otherwise adopt a completely different stance which will contribute to greatly extend the discussion. As a by-product, some of the preparatory results will moreover be corrected in the process, thereby allowing them to be rehabilitated. This will in turn enable the comparison and validation of our findings against the results of a full numerical optimisation with phasing. 1.3 Objective of this work The fact that we have not yet discovered all the existing asteroids and comets with orbits venturing in the vicinity of our planet gives an excellent motivation for undertaking a study of possible mitigation missions for all the conceivable Earth-impacting asteroid orbits, trying to map threat scenarios to mission types. 3

A direct parallel can actually be drawn with the Oberth effect, and this conclusion is obviously not limited to the kinetic-impactor mitigation technique. See also App. A. 4 An impact with our planet can obviously only take place in the ecliptic plane, meaning that the impact location necessarily corresponds to one of the nodes if the PHA orbit is inclined. Though the ⃗ropp location loses its appeal should the inclination be vanishing, one can obviously still call the impact point ⃗rI and keep the same set-up.



ESA UNCLASSIFIED – For Official Use With this work, our objective is to provide a preliminary kinetic-impactor mission design for any such PHA and to derive a better intuition of what constitutes the optimal deflection geometry over the entire parameter space, accounting for crucial feasibility constraints such as the launcher performance and the illumination conditions at deflection. This optimisation problem will be approached both analytically and numerically. A direct benefit of studying broad scenarios at once is that this greatly facilitates the identification of patterns in terms of the parameters, and can therefore help physically understand what are the main drivers leading to an optimal solution. As much as possible, we would like to be able to essentially guess where on any asteroid orbit the kinetic impactor should preferably be sent, given the necessarily limited launcher performance. We want to provide more refined and more reliable initial guesses than an overly simplistic targeting of the perihelion, though still ignoring the phasing. This could then enable the definition of general procedures for various classes of asteroid orbits, associated to different parts of parameter space. 1.4 1.4.1

Approach, scope, and assumptions Ballistic trajectories

From the launcher selection to the possible inclusion of an electric propulsion system on board, of planetary flybys, of deep-space manoeuvres, etc: the space of possible kinetic-impactor missions to even a single PHA is already huge if all options are left open, and it is even more so if the time to impact itself remains quite open as well, as is necessarily the case when studying the general problem posed by a fictitious threat. Here, as the objective is not to consider a few targets but rather all the possible Earth-impacting orbits, we choose to restrict ourselves to optimal ballistic trajectories, since they can be seen as useful preliminary solutions and can moreover be built upon and improved if needed —including as initial guesses for electric propulsion, as done e.g. in Ref. [22]. Such an approach will still leave us with a large breadth of cases to be considered. 1.4.2

Parametrisation of Earth-impacting orbits

A clear advantage of this work is that it actually uses the parametrisation specifically introduced in Ref. [1] for studying impact problems; we find that a lot can already be learned just from the new parametrisation and from geometry considerations. Its underlying assumptions are simply the following: the Earth-impacting asteroids are assumed to be moving on elliptical orbits around the Sun; the minimal orbit intersection distance (MOID), to be vanishingly small; and the Earth orbit, to be circular with a radius of one astronomical unit (AU). Whenever these conditions hold, any Earth-impacting orbit can be fully parametrised with as little as three physical dimensionless parameters: the inclination of the asteroid orbit i, its eccentricity e, and the asteroid true anomaly at impact fI . From (fI , e, i), the orbital elements can of course easily be recovered if needed. Using this parametrisation, the whole subset of conceivable Earth-impacting orbits actually becomes a very simple and bounded region in parameter space, which is moreover convex: any (fI , e, i) set corresponds to a valid impacting orbit, and all impacting orbits can take this form. Furthermore, it was also shown that using a polar representation of this parametrisation (with e as the radial variable, and fI as the angular variable) provides not only more insight into the physical problem, but also a one-to-one correspondence between each impacting orbit and each point on such a plot for a given inclination. The impacting orbits can moreover be divided in quadrants (see Fig. 1) that



ESA UNCLASSIFIED – For Official Use unambiguously separate them not only depending on whether they correspond to daytime or nighttime impacts, but also on whether their opposite node ⃗ropp lies inside or outside of the Earth orbit.

Figure 1: Unit-circle representation of fI : quadrants together with corresponding PHA-orbit properties [1].

The fact that fI (which indicates where the impact is to take place on the asteroid orbit) is both rooted in the impact problem and truly geometrically intrinsic to the PHA orbit brings many advantages, which furthermore greatly facilitate the derivation of analytical results; see also App. B. In comparison, using the argument of perihelion ω instead of fI would actually break a symmetry of the problem, artificially introducing the need for distinguishing between impacts at the ascending node and impacts at the descending node despite the fact that whether the asteroid hits the Earth from above or from below is completely irrelevant [1]. For more information and discussions, the reader is referred to that first paper, since the results presented there shall be heavily relied on, and used throughout. Note that as we explicitly assume the orbits to remain elliptical between the time of deflection and the time of impact, the results presented here must obviously not be used as they stand should there be any close-encounter with a planet prior to impact. In a real scenario, the presence of such close-encounters must absolutely be assessed for the individual asteroid. 1.4.3

Key quantities: ∆B and ϕSun

Yearly-drift in the B-plane—For each Earth-impacting orbit, what we actually want to identify in this work is the corresponding spacecraft transfer orbit which would lead to the largest B-plane displacementrate over time. In other words, we want to find out what would actually be the best conceivable deflection geometry that could be ballistically achieved with a given launcher, independently of the remaining time-to-impact. The results of the deflection will be given in terms of an optimised yearlydrift in the B-plane ∆B, defined in Sec. 2. Solar aspect angle—Another important quantity is the solar aspect angle at deflection ϕSun , since guaranteeing proper illumination conditions is obviously crucial for a successful mitigation mission of Page 6 of 68 Parameter-space of asteroid mitigation missions using ballistic kinetic impactors —Final Report II.


ESA UNCLASSIFIED – For Official Use this type. During the final approach, a key constraint is that the on-board cameras tracking the asteroid must not be pointing in a direction close to that of the Sun. At the time of deflection, if ⃗rSC,D and ⃗vSC,D are the heliocentric position and velocity of the spacecraft, ⃗vA,D , the heliocentric asteroid velocity, and ⃗v∞SC,D , the spacecraft velocity relative to the asteroid, one can for instance require that the solar aspect angle at deflection ( ) −⃗rSC,D · ⃗v∞SC,D ϕSun ≡ acos , with ⃗v∞SC,D = ⃗vSC,D − ⃗vA,D , (1) |⃗rSC,D | |⃗v∞SC,D | remains above a certain threshold, e.g. ϕSun ≥ 45° or even ϕSun ≥ 60°. The presence of such a threshold will strongly affect the definition and feasibility of what represents optimal missions. Given how the solar aspect angle and the phase angle relate to each other, the benefit of this is twofold. The more favourable the solar aspect angle is, the better the illumination of the target also turns out to be; both objectives are highly desirable and are reached simultaneously. 1.4.4

No phasing

Can we guess what makes an optimal transfer?—As mentioned earlier, as a rule of thumb, the perihelion of an Earth-impacting-asteroid orbit is widely regarded as the best deflection location. It is however important to realise that, strictly speaking, this would actually only truly hold when ignoring the launcher performance, ignoring the illumination conditions, and ignoring the phasing. Not only is an along-track momentum transfer easier to be said than done with an actual orbit, reaching at all the perihelion of any arbitrarily inclined Earth-impacting orbit can easily be a challenge or even downward impossible with a realistic launcher; in fact, even when feasible, it will often not be desirable at all since the cost is paid in terms of deflector mass. The insight that one can truly gain from this therefore remains necessarily rather limited. Can we replace this by something more realistic/accurate?—It would certainly be useful to refine this intuition of the problem, this time making sure that the considered trajectories can be achieved with an actual launcher, and that the deflections would moreover be feasible in terms of the solar aspect angle. The phasing will still not be taken into account though; it will therefore be an improvement but not the full answer since a feasibility in terms of warning time will not be enforced. In a way, such an approach essentially contributes to bridging the gap between the seminal studies searching for the optimal arbitrary ∆⃗vA to be applied to an asteroid, and more traditional mission-design strategies. Focusing strongly on the mission feasibility while otherwise ignoring the phasing actually brings a number of advantages. Essentially assuming long but otherwise unspecified warning times enables one to determine the feasible spacecraft trajectory which optimises the rate at which the asteroid motion changes over time, directly highlighting the best yearly-drift that one could ever hope to ballistically achieve with the considered launcher. Such a knowledge remains valuable even if some of the assumptions and simplifications are dropped, as long as they remain approximately correct (i.e. no dramatic change of strategy): it gives something to aim at. With this information at hand, the mission analyst might then indeed tweak the mission parameters in order to try and achieve something close to this optimal deflection geometry early on (e.g. adding small manoeuvres if needed), even if that means changing slightly the trajectory model.5 Knowing what would potentially be the best solution is more important than rigidly enforcing strict ballistic trajectories since, in reality, small manoeuvres will of course be considered. In comparison, an exploratory study that includes the phasing, being much more 5 Of course, if the model changes too much, one should expect that other deflection locations will eventually become preferable; requiring another study, more appropriate for such a model, to performed.



ESA UNCLASSIFIED – For Official Use rigid by design, will often settle for sub-optimal deflection geometries, the details of which will moreover strongly depend on the assumed warning time.6 Here, the presence of different classes of optimal missions will appear more clearly in different parts of the parameter space, and will therefore be more easily linked to geometrical and analytical expectations. From this, as missions types are summarised given their general properties, valuable lessons could then be learned, most notably how and where on its orbit it would be best to deflect each asteroid. Despite their limitations, results that ignore the phasing remain useful even when the warning time would not be sufficient to avoid an impact: as they tell what the best would have been, they provide an upper estimate for the outcome of any deflection with essentially ballistic trajectories. When the best solution is actually not good enough (even with multiple kinetic impactors), it immediately tells us that something more or less radically different must be done—e.g. involve flybys; consider electric propulsion; go retrograde; use another mitigation technique altogether; etc. Providing that time allows it, with more options, one could of course greatly improve the deflection results, though these typically require a more precise modelling, and thus possibly restricting the study to a handful of PHAs at a time. 1.4.5

Studying deflections in the vicinity of the nodes as initial guesses

Closely studying these initial guesses—In what follows, a central role is played by the study of deflections happening at either of the nodes of the asteroid orbit [1]: ⃗rI = 1 AU ⃗ex

or ⃗ropp = −

1 + e cos fI ⃗ex . 1 − e cos fI

(2)

When one wants to solve for any possible inclination, these locations indeed stand out, as a deflection in their immediate vicinity means that the cost required for ballistically venturing out of the ecliptic plane can be avoided. Here • strict 2π-transfers mean that ⃗rL = ⃗rD = ⃗rI ; • strict π-transfers mean that ⃗rL = ⃗rI , while ⃗rD = ⃗ropp ; where ⃗rL and ⃗rD respectively correspond to the launch location and to the deflection location. More generally, classes of solutions obtained from these initial guesses, or which can essentially be seen as deformations of these when relaxing the exact departure and arrival locations, will then be referred to as 2π- and π-transfers in the following.7 Figure 2 illustrates the simplest of such transfers, which are in fact simply the ballistic spacecraft orbits that maximise the kinetic impactor mass at deflection. They respectively coincide with the Earth orbit itself (simplest strict 2π-transfer) and with the first leg of a Hohmann transfer to ⃗ropp (simplest strict π-transfer). These simple orbits give a first rough idea of what to expect, of what the natural 6

When studying optimal solutions with phasing, one can observe sudden jumps between dramatically different types of solutions for points that are adjacent in parameter space, as e.g. in the preparatory draft [23]. While there is no question that the presence of such jumps is rigorously correct within the exact assumptions under which they have been obtained, their exact location and properties can be expected to be highly unstable under any slight change in the model or even uncertainty. They are basically an artefact of the method itself. The effects of the phasing would therefore be best taken into account by looking at a specific PHA at a time, in a refined model. 7 Later on, additional types of initial guesses shall also be introduced; while distinct from 2π- and π-transfers, they will also take the form of transfers targeting the nodes of the PHA orbit.




Figure 2: Ecliptic projection illustrating at once the simplest 2π- (blue) and π-transfers (red) a few days before the deflection takes place, respectively at ⃗rI (right panel) and at ⃗ropp (left panel). In this example, we furthermore show both the situation in the case of a daytime (brown) and in the case of a nighttime (grey) Earth-impacting orbit. The spacecrafts are shown with triangles, and the PHAs are indicated with circles (open for the daytimeimpact, and filled for the nighttime-impact); the perihelions (resp. aphelions) are shown with small open (filled) squares. Note that the situation for PHA orbits with |⃗ropp | > |⃗rI | would of course be extremely similar.

trend is; they actually are quite useful. Trying to depart significantly from the outcome of such transfers can be expected to come at a non-negligible cost. We mentioned the feasibility of reaching the perihelion as being a reason to worry. Since we want to use both nodes as initial guesses, we should worry about the reachability of the opposite node. This is discussed at length in the first paper [1] and will play a crucial role throughout this work as well when it comes to considering the illumination conditions at deflection; see also App. C for a complementary discussion. Strong motivation: the inclination can actually be sizeable—With Fig. 3, we want to strongly emphasise that PHA orbits should not be thought of as typically having vanishingly small inclinations: the median orbital inclination of the known objects is indeed close to 10°, and has in fact been growing over the last years with the growth of the database. In other words, as there are as many orbits with i ≳ 10° than there are orbits with i ≲ 10°, it would actually be a mistake to always conveniently assume that PHAs are on orbits with low inclinations. Dates on this plot correspond to the end of the corresponding year. To produce such a figure with a 2017 version of the MPC database, each PHA was filtered and included at a given date only if it was already known at the time (assumed discovery year given by the principal provisional designation).8 Surveys may actually have spent too much time focusing on the ecliptic plane [24]. Quoting Richard Wainscoat (Pan-STARRS) at the Planetary Defense Conference 2017: “Like everyone else in the busi8 Note that 6344 P-L (also known as 2007 RR9) has been included from 2007 onwards, and 6743 P-L (also known as 1983 TF2 or 5011 Ptah), from 1983.




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ness, we have focused perhaps too much on the ecliptic. [There are] more discoveries close to the ecliptic but the ecliptic is more often visited. Normalising the number of discoveries by the number of visits, [there is] much more than you would expect. It shows that there is merit in searching for NEOs well off the ecliptic, and that the amount of time that we have been spending on the ecliptic has been too much in the past (...) There’s more discovery space than people previously had thought.”.

0 2015

date Figure 3: Median inclination of the known PHA orbits and their total number as a function of time (see text). Data from the MPC pha_extended.json database (September 2017).

In what follows, we do not want to focus solely on what might be the most likely type of impactor orbit [25], be it Earth-like orbits or Atens with fI ∼ π (for which kinetic impactors might not be a well-suited response), but rather truly consider all the conceivable Earth-crossing orbits. 1.5 Outline With the context, the objectives, the scope, and the assumptions now presented, the remainder of this manuscript is organised as follows. First of all, Sec. 2 discusses the problem of optimising the asteroid deflection missions, and introduces in particular the yearly-drift ∆B, which shall then be the physical quantity optimised throughout. In Sec. 3, the problem is first studied by approaching it analytically without phasing, considering simple transfers to the nodes as initial guesses; these basic transfers are then also analytically optimised as a function of the launcher performance. The focus is strongly set on the physics behind the presence of different broad classes of missions in different parts of the parameter space. The discussions are not only about the properties of the yearly-drift over the entire parameter space of Earth-impacting asteroid orbits, but also on what the different illumination conditions for daytime and nighttime impacts entail for the selection of the optimal mission. Results of a full optimisation without phasing are then presented in Sec. 4. The properties of these optimal missions are then discussed, and the main analytical lessons from Sec. 3 are shown to provide a powerful physical basis for better understanding the numerical results. Departures from the initial guesses are also easily identified in different regions and are further discussed. A number of striking approximate symmetries in the numerical results are then also presented before we conclude. Page 10 of 68 Parameter-space of asteroid mitigation missions using ballistic kinetic impactors —Final Report II.



2

Deflection merit

2.1 Linearised along-track displacement, ignoring non-secular effects To avoid the impending impact of an asteroid with the Earth, we want to alter its course on its orbit. Typically, kinetic impactors cannot be expected to strongly alter the PHA orbital elements. In all the cases considered in this work and as usually found in the literature, the main outcome of the momentum transfer ∆⃗ pSC,D is actually assumed to be a small effect on the orbital period of the asteroid orbit: ∆P .9 When using this technique, the idea is indeed that, given enough time, the change in the orbital period will lead to an accumulated along-track displacement (or phase-shift) large enough to avoid the collision which had been foreseen. Quantifying tiny period changes is then unsurprisingly the objective pursued in different kinetic-impactor demonstration concepts such as DART [3] and NEOTωIST [26]. Throughout this work, the deflection shall always be considered to take place sufficiently in advance, letting the small effect on the orbital period enough time to accumulate. If this is the case, then the along-track displacement ( ) ⃗vundefl (t) ∆s(t) = ⃗rdefl (t) − ⃗rundefl (t) · , (3) |⃗vundefl (t)| which compares the projection along-track of the change in position for the PHA before (·undefl ) and after deflection (·defl ), can be estimated around the time of impact TI by ∆s(t = TI ) ≈ −|⃗vA,I | ∆P

TI − TD ; P

(4)

where TD is the time at deflection, and (TI − TD ) is thus the drifting time during which the effect accumulates. Similar formulae have been frequently obtained and discussed in the literature e.g. in Refs. [15, 19, 27, 28]. Such discussions also explain how to determine the induced change in orbital period ∆P for a PHA of mass mA . It relates to the momentum transferred to the asteroid, relative to its undeflected motion, by the kinetic impactor ∆⃗ pSC,D = mSC,D ⃗v∞SC,D ,

with ⃗v∞SC,D = ⃗vSC,D − ⃗vA,D

(5)

via

∆P 3a β ≈ (⃗vA,D · ∆⃗ pSC,D ), (6) P µSun mA where mSC,D is the spacecraft mass at deflection, and where one often introduces the so-called β-factor, equals to 1 in the case of the perfectly inelastic collisions always considered here.10 More precisely, Eq. (6) simply comes from the more general expression, valid for small arbitrary ∆⃗vA (see e.g. App. A), ∆P 3a ≈ (⃗vA,D · ∆⃗vA ) , P µSun

(7)

and the requirement of momentum conservation at deflection, which can be written (β = 1 case) [ ] mA ⃗vA,D + mSC,D ⃗vSC,D = (mA + mSC,D ) ⃗vA,D + ∆⃗vA ,

(8)

thereby giving ∆⃗vA =

mSC,D ⃗v∞SC,D (mA + mSC,D )

mA ≫mSC,D

∼

∆⃗ pSC,D . mA

(9)

9

Of course, this also translates into a tiny change in its semi-major axis a and its orbital energy. Values larger than 1 correspond to the case of elastic collisions, as the momentum transfer receives a boost provided by the presence of ejecta; to be more conservative one frequently finds β = 1 in the literature. 10



ESA UNCLASSIFIED – For Official Use Before moving on, let us actually go back to the expression of the linearised along-track displacement ∆s, in Eq. (4). Please note that accelerating an asteroid at the deflection point will increase its orbital period (∆P > 0), so that it will actually be late at its impact rendez-vous (∆s < 0), while decelerating it at the deflection point will reduce its orbital period (∆P < 0), meaning that the deflected PHA will pass the impact point earlier than if it had not been deflected (∆s > 0); this is also emphasised in Ref. [29]. This becomes particularly obvious if the deflection happens at the future impact location. Right after deflection, of course, an accelerated asteroid will move faster along track, and a decelerated one will move slower; but their orbits being slightly changed, the effect on the period will soon take over. We stress that this is different from what the preparatory draft [23] assumed, meaning that for the most part neither the quantitative nor the qualitative results presented there may actually be used as they stand, but must be corrected.

along-track displacement ∆s (km)

5000

PHA deflection, numerical propagation Idem, sampled at every PHA period Estimation of ∆s at ~rI

0 −5000 −10000 −15000 −20000 −25000 −30000 −7000

−6000

−5000

−4000

−3000

−2000

−1000

0

time from impact (mjd)

Figure 4: ∆s. Comparison of a numerical result sampled at every ⃗rI passage to the approximation used in this work. Very good agreement when the deflection happens an integer number of times the PHA period before impact (here, TD = TI − 15P ). For reference: i = 15°, e = 0.5, fI = 246.3° (P = 1.1 yr).

Figures 4–6 compare the analytical estimate for ∆s given in Eq. (4) to an actual numerical calculation in two specific examples in which an induced along-track ∆⃗vA of 1 cm/s was applied (∆P > 0). What we want to emphasise here is that, as seen in Fig. 6, there is an offset when the deflection is not taking place an integer number of PHA periods before impact (i.e. not at the impact location), and that this offset can actually have a different relative sign compared to the secular effect associated with ∆P . It means that the approximate ∆s is actually not conservative, since this might lead us to overestimate |∆B|. This is clearly something to be aware of. If the time of impact is close to the time of deflection, such a contribution should not be ignored; however, as the drifting time accumulates, it becomes more and more marginal, relatively speaking. Should a more precise description of the deflection outcome be needed, another approach altogether would be to obtain the new set of orbital elements that describe the asteroid motion after deflection; see e.g. Ref. [20]. Alternatively, one might want to follow Refs. [30, 31], for instance. Page 12 of 68 Parameter-space of asteroid mitigation missions using ballistic kinetic impactors —Final Report II.




along-track displacement ∆s (km)

0 −500 −1000 −1500 −2000 −2500 −3000 −3500 −4000 −6200

−6000

−5800

−5600

−5400

−5200

−5000


Figure 5: Zoom on Fig. 4. 500 along-track displacement ∆s (km)

0


−500 −1000 −1500 −2000 −2500 −3000 −3500 −4000 −1800 −1600 −1400 −1200 −1000 −800 −600 −400 −200

0


Figure 6: This time, the deflection does not happen an integer number of times the PHA period before impact (here, TD = TI − 4.25P ). An offset between the amount of along-track displacement at ⃗rI passage obtained numerically and the corresponding analytical estimation ignoring non-secular effects is clearly visible. Note that our aim here was to provide a case where the discrepancy would be as large as we could possibly find.

In this work however, since we are interested in studying the problem without considering the phasing and moreover always assume long drifting times (TI − TD ), we shall only consider the secular



ESA UNCLASSIFIED – For Official Use part described by Eq. (4). 2.2 Corresponding B-plane displacement Since our aim is to remain quite general, the discussion shall be done in terms of changes in the B-plane. The advantage is that this allows one to present results that can be used for any impacting asteroid (not only those with vanishing impact parameters), without having to specify what exactly its B-plane impact parameter B actually is. Instead, one can then discuss in terms of the worst possible case. Accounting for the sign of ∆s in Eq. (4), the linearised B-plane displacement ∆B = ∆s sin χ should in fact read: 3a β ∆B ≈ −|⃗vA,I | sin χ (⃗vA,D · ∆⃗ pSC,D )(TI − TD ), (10) µSun mA | {z } | {z } merit function M not/barely affected by the mission

where ∆s is projected onto the B-plane via ( ( )) ⃗vA,I · ⃗v∞A,I , sin χ ≡ sin acos |⃗vA,I | |⃗v∞A,I |

with ⃗v∞A,I = ⃗vA,I − ⃗vE,I .

(11)

Obviously, given the scaling of ∆B with mA in Eq. (10), one can easily rescale the results obtained assuming a given asteroid mass. As it just factors out of the merit function, one can immediately obtain the corresponding results in any another case (that is, provided that one can still assume that the asteroid does not break due to the high-velocity impact). A corollary of this is that, from deflection results derived assuming some given mA , one can deduce the asteroid mass up to which a desired |∆B| is satisfied, similarly to what is done in Ref. [32]. From the physical point of view, ∆B is a displacement in the B-plane, and is therefore necessarily independent of the influence of the Earth: it indeed corresponds to a situation at infinity. Here, we preserve this original physical meaning and provide at the same time a precise and meaningful measure of what is most relevant for the problem at hand—i.e. whether at close approach an impact can be avoided, given the gravitational pull of the Earth. There is actually no need to redefine distances in the B-plane as being normalised by the focus factor ρ, as is sometimes found in the literature; instead, to achieve the same end, we find conceptually much more satisfactory to simply express ∆B in units of b⊕ ≡ RE ρ(RE , |⃗v∞A,I |), with

√ ρ(rp , |⃗v∞A,I |) =

1+

2µE 2 , rp ⃗v∞A,I

(12)

(13)

where µE and RE are respectively the Earth gravitational parameter and radius; see also e.g. Ref. [15]. The advantage of b⊕ is that it is a physical length scale defined in the B-plane, which has a meaning that is directly relevant to the impact problem for each PHA. It is the impact parameter such that the perigee of the incoming hyperbola at close approach is given by the Earth radius RE ; see Fig. 7. In these units, the worst conceivable case then simply reads |∆B| > 2b⊕ ; this is when one would have to displace the perigee of an asteroid orbit all the way through the Earth, and beyond another rim [23]. Notice that units of b⊕ can obviously be used for the same benefit even when considering more precise calculations in which the impact parameter would be exactly determined for the perturbed/deflected (B∗ ) orbit. Avoiding an impact then simply requires that B∗ > b ⊕ , Page 14 of 68 Parameter-space of asteroid mitigation missions using ballistic kinetic impactors —Final Report II.

(14) European Space Agency Agence Spatiale Européenne


Figure 7: Sketch illustrating the definition of b⊕ : for a given |⃗v∞,A,I |, this specific impact parameter is the mapping in the B-plane of a perigee distance given by the Earth radius (Earth image credits: NASA).

where RE might moreover be chosen to further take into account the height of the atmosphere. Let us however note that, for any actual threat, the problem of keyholes [33] should of course crucially be assessed; in this study though, the discussion is only made in terms of ∆B. Before moving on, let us stress that the possibility of a inadvertent disruption is ignored when we use the merit function as it stands here above: the optimisation of this figure of merit indeed rewards larger momentum transfers.11 This is done because this work is only concerned about point masses and their trajectories, with no actual model for the asteroid itself; however, in any realistic case, the validity of this assumption must obviously first be assessed, especially since the specific physical properties of asteroids vary widely, from rubble-piles to strong rocks. For a given composition, it would become more critical the smaller the asteroid mass (smaller escape velocity). With this caveat, one can use this formula as long as the 2-body approximation makes sense, for sufficiently large drifting-times post-deflection. 2.3 Merit function and B-plane yearly-drift Maximising the B-plane displacement |∆B| with a kinetic impactor requires designing a deflection mission which optimises the merit function M: M = (⃗vA,D · ∆⃗ pSC,D ) (TI − TD ), | {z } | {z } ≡M

(15)

≡∆T

the other relevant quantities entering Eq. (10) being fixed by the PHA orbit and the associated relative geometry, or by the physical properties of the bodies. In particular, a kinetic impactor will not significantly alter the close-approach geometry. For a study that ignores the phasing, one can for instance choose the drifting time after deflection to be ∆T = TI − TD = 1 yr, or any other convenient time scale, and is then left with the optimisation of the quantity: M = ⃗vA,D · ⃗v∞SC,D mSC,D (|⃗v∞SC,L |), with ⃗v∞SC,D = ⃗vSC,D − ⃗vA,D ( ) = mSC,D (|⃗v∞SC,L |) ⃗vA,D · ⃗vSC,D − |⃗vA,D |2 ,

(16)

11

On a related note: large relative velocities at deflection are likely to present a challenge for the automatic navigation even with good tracking systems. As the navigation time becomes very short, putting several cameras would therefore help. Investigating this issue further is however beyond the scope of this work.



ESA UNCLASSIFIED – For Official Use where we now explicitly stress the dependence of the kinetic impactor mass at deflection on the hyperbolic excess velocity at launch ⃗v∞SC,L needed to achieve the corresponding ballistic trajectory, as given by the launcher performance. M can be seen as the geometric part of the merit function, in the sense that it only worries about optimising the projection of the momentum transfer onto the asteroid heliocentric velocity at deflection, irrespective of any sense of timing (including timing feasibility). When ignoring the phasing, it is furthermore useful to consider the B-plane displacement per interval of time following the deflection, which will be written ∆B. For convenience, its units will be chosen to be b⊕ /yr in what follows, so that our discussions shall be made in terms of the B-plane yearly-drift. It is simply obtained by replacing the merit function M by M in Eq. (10): ∆B ≈

3a β −|⃗vA,I | sin χ µ m | {z Sun A} not/barely affected by the mission

(⃗vA,D · ∆⃗ pSC,D ) . | {z }

(17)

M

Notice that the mission which optimises the yearly-drift ∆B is actually also the one that optimises the change in the PHA orbital period ∆P : these two quantities indeed essentially only depend on the mission parameters via M. A strong advantage of using the yearly-drift though is that it immediately tells something about the success of the deflection mission, and can be used at once as a meaningful measure for all the PHA orbits. Irrespective of whether the phasing is taken into account or not, let us stress that when the aim is to induce a tiny period change ∆P in the asteroid motion, one should not confuse the optimal orientation of ⃗vSC,D for that of ∆⃗vA . Again, since an orthogonal change in the asteroid velocity would lead to no secular effect, if one could arbitrarily apply a given ∆⃗vA , it would be preferable to make it along-track (acceleration) or opposite track (deceleration). However, this does not mean that when designing an actual trajectory the spacecraft heliocentric velocity at deflection ⃗vSC,D should preferentially be parallel or antiparallel to ⃗vA,D , and never orthogonal. Rather, recall that what enters the merit functions (15) and (16) is actually not ⃗vSC,D · ⃗vA,D but ( ) M ∝ ⃗v∞SC,D · ⃗vA,D = ⃗vSC,D · ⃗vA,D − |⃗vA,D |2 . (18) Focusing on what may be the optimal orientation at a given deflection location D, notice in particular that it may be difficult or even impossible (in particular with ballistic trajectories) to achieve ⃗vA,D · ⃗vSC,D < 0 when decelerating a PHA along-track (M < 0), in which case an advantageous deceleration geometry would actually correspond to ⃗vˆSC,D ⊥ ⃗vÂ,D , which would give then M ∝ −|⃗vA,D |2 ; see also Ref. [19] for an actual example with low-thrust. 2.4

Asteroid mass and launcher performance used in the following

Through optical observations alone, one only obtains the magnitude of an object, from which a sizeestimate can be inferred once an albedo is assumed. Threats are then commonly discussed in terms of the asteroid size, even though doing so hides an underlying assumption regarding the mass density (e.g. 2 t/m3 ). The appropriate measure to estimate an incoming threat is indeed the asteroid mass, not its size; it provides both the relevant scale for the momentum transfer required at deflection, and enters linearly in the kinetic energy at the time of impact with the Earth. From now on, when showing quantitative results, we shall use as benchmark the higher-end of the asteroid masses that one could hope to deflect with kinetic impactors. Since these would typically have correspondingly larger sizes and be brighter, the warning time would hopefully be somewhat larger Page 16 of 68 Parameter-space of asteroid mitigation missions using ballistic kinetic impactors —Final Report II.


ESA UNCLASSIFIED – For Official Use than with smaller-size, easier to deflect, asteroids. We choose to preserve some backward-compatibility for easier comparisons with the preparatory results with phasing: the asteroid mass will always be mA ≃ 7.8 × 107 t (diameter of about 400 m), and we shall explicitly consider the launcher performance of the SLS Block-1B with a cutoff at 10 km/s (see Fig. 8) [23].

35 30 mSC (t)

25 20 15 10 5 0

0

2

4 6 8 v∞SC,L (km/s)

10

12

Figure 8: Performance of the SLS Block-1B 8.4m Fairing EUS [34].

Again, the asteroid mass simply factors out when one does not consider disruption or plumes, so the results that follow can be easily rescaled as long as this holds true.




3 Analytical deflection at the nodes without phasing The objective here is to develop some general understanding of what the optimal ballistic mitigation missions might be expected to be in different parts of the parameter space. To this aim, we first approach the problem analytically while ignoring the phasing, meaning that the solutions are found for the entire PHA parameter space, without having to solve the Lambert problem. 3.1 Strict 2π-transfers In terms of launcher performance only, a 2π-transfer is always conceivable for any PHA since putting the deflection point at the impacted node obviously places it on the Earth orbit. From geometry it is clear that, in case of daytime impacts, unfavourable illumination conditions at deflection might however be sufficient to compromise such an option; see Figs. 9 and 10. For more discussions on this, the reader is also referred to the first paper [1].

Figure 9: Ecliptic projection illustrating at once the simplest 2π- (blue) and π-transfers (red) a few days before the deflection takes place, respectively at ⃗rI (right panel) and at ⃗ropp (left panel). In this example, we furthermore show both the situation in the case of a daytime (brown) and in the case of a nighttime (grey) Earth-impacting orbit. The spacecrafts are shown with triangles, and the PHAs are indicated with circles (open for the daytimeimpact, and filled for the nighttime-impact); the perihelions (resp. aphelions) are shown with small open (filled) squares. Note that the situation for PHA orbits with |⃗ropp | > |⃗rI | would of course be extremely similar.

3.1.1 Impact-point approximation To get a first rough idea of how much deflection could be achieved for different PHAs, one could imagine a strict 2π-transfer (⃗rL = ⃗rD = ⃗rI ) where the kinetic-impactor velocity at deflection is simply replaced by the Earth heliocentric velocity at that point: i.e. consider the spacecraft to be exactly on the Earth




Figure 10: Zoom on Fig. 9, with the direction of the projected relative spacecraft velocities with both PHAs now indicated by arrows. The left panel highlights the geometry slightly prior to a deflection with the simplest π-transfer, and the right panel shows the same, but in the case of the simplest 2π-transfer. The illumination conditions at deflection for nighttime-impact asteroids (grey) are naturally favourable in the vicinity of the impact location ⃗rI , as the spacecraft is crossing the asteroid orbit from the inside to the outside (guaranteed large solar aspect angle and small phase angle); for daytime-impact asteroids (brown), this is instead true around the opposite node ⃗ropp .

orbit. Then ⃗vSC,D = vE ⃗ey , so that

and

⃗v∞SC,D · ⃗vA,D = (vE ⃗ey − ⃗vA,I ) · ⃗vA,I

(19)

( ) 2 M = mSC,D vA,I,y vE − |⃗vA,I | .

(20)

This kind of zeroth-order estimate for the deflection was first discussed in Ref. [23], assuming a fixed kinetic-impactor mass of 30 tons to approximate the SLS Block-1B performance. The idea is that the actual optimal ballistic kinetic-impactor orbit will not be dramatically different from that of the Earth, the effect of the launcher being then seen as a perturbation. While certainly being a crude approximation, this was actually found to already provide, in different parts of the parameter space, a rather reliable estimate of the optimised deflection with phasing obtained numerically—at least for asteroid with nighttime impacts. The analytical results that we obtain within this approximation are shown in Figs. 11, 12 and 13, in terms of the secular B-plane displacement per year following the deflection.12 The regions of parameter space where PHAs are decelerated (⃗v∞SC,D · ⃗vA,D < 0, ∆B > 0) or accelerated (⃗v∞SC,D · ⃗vA,D > 0, ∆B < 0) along their path are clearly separated; the border between them in parameter space being heavily dependent on the inclination. Moreover, the yearly-drift ∆B tends to zero in specific regions: • One is the border separating acceleration and deceleration regions, where M = 0 because ⃗v∞SC,D is orthogonal to ⃗vA,D as a consequence of the deflection geometry, such that indeed no alongtrack displacement occurs; one could alleviate this issue, for instance by simply choosing another deflection location. 12 The expressions obtained for the asteroid heliocentric velocities at their orbital nodes [1] can be used directly; see also App. B.




≥ 0.5

0.3 0.6 e

0.2 0.4 0.1 0.2

∆B (b⊕ /yr)

0.4

0.8

0 i = 5.0°

0

−0.1 0

1

2

3

4

5

6

P (yr) Figure 11: Yearly-drift in the impact-point approximation, in a (P, e) plot; such a representation makes no distinction between daytime and nighttime impacts (though it is crucial for ϕSun ). Here mSC,D = 30 t. Note that the colour palette was chosen to improve the contrast at very small |∆B| values, in particular for the poorly-performing along-track acceleration region (light pink) to appear more clearly.

1

≥ 0.5

0.3 0.6 e

0.2 0.4 0.1 0.2

∆B (b⊕ /yr)

0.4

0.8

0 i = 5.0°

0

−0.1 0

π 4

π 2

3π 4

π

5π 4

3π 2

7π 4

2π

fI Figure 12: Same as Fig. 11 but in the new parametrisation (fI , e, i), clearly separating daytime- (fI < π) and nighttime-impact (fI > π) orbits.

• On the other hand, the unsatisfactory region corresponding to the fI → π Atiras limit for small PHA inclinations (and, to a lesser extent, to the fI → 0 or 2π Amor limit, for vanishingly small inclinations) is instead due to the geometry at close approach with the Earth, meaning that this should be considered as intrinsically problematic since a kinetic impactor is not designed to be able to significantly alter a PHA orbit. In other words, this one is not due to a zero of M but to Page 20 of 68 Parameter-space of asteroid mitigation missions using ballistic kinetic impactors —Final Report II.



1

≥ 0.5

≥ 0.5

i = 5.0°

0.4

0.4

0.3

0.3

0

0.2 0.1

−0.5

0.2 0.1

∆B (b⊕ /yr)

0

∆B (b⊕ /yr)

0.5 e sin fI

e sin fI

0.5

i = 10.0°

−0.5 0

−1 −1

−0.5

0

0.5

e cos fI

−1 −0.1 1 −1 −0.5

0 −0.1 0

0.5

1

e cos fI

Figure 13: Same as Fig. 12, but in the polar view [1] that shall be used in the remainder of this work, now for two values of the PHA-orbit inclination; see again Fig. 1 about the different quadrants. ∆B > 0 (blue) corresponds to decelerations along track, and ∆B < 0 (red), to accelerations along track (see text). The overwhelming majority of cases correspond to decelerations.

a zero of the prefactor in ∆B, therefore there is not much that can be done about it. On top of this, the focus effect due to the gravity of the Earth tends to aggravate the problem even further (i.e increase what b⊕ corresponds to for those asteroids). Similar remarks can be extended to their immediate surroundings in parameter space, including the small acceleration region. As well-known, for orbits with an impact taking place close to aphelion (fI ∼ π), as well as for Earth-like orbits (small PHA-orbit eccentricity e), the deflection can be challenging with kinetic impactors: |∆B| is indeed very small there. It might then be worth investigating other mitigation methods; see e.g. Ref. [35]. Beside assessing the induced deflection in the asteroid motion, we now moreover provide the analytical expression for another quantity playing an important role in the determination of the optimal mitigation mission: the corresponding solar aspect angle at deflection, given in Eq. (1). In the current case, see Fig. 14, it reduces to13 ( ) √ vA,I,x ⃗v∞SC,D = vA,I,x 2 + (vE − vA,I,y )2 + vA,I,z 2 . , ϕSun = acos with (21) ⃗v∞SC,D One indeed recovers what is expected from the relative geometry. For daytime-impact asteroids (fI ≤ π ⇒ vA,I,x ≥ 0), the solar aspect angle can be too small, especially for small PHA-orbit inclinations, to allow for a successful mitigation via a 2π-transfer with proper visibility at deflection. Another type of transfer would then be needed. On the other hand, ϕSun is always larger than 90° for nighttime impacts (fI ≥ π ⇒ vA,I,x ≤ 0), in which case a 2π-transfer will always be an option. From the geometric point of view, it is clear that this situation is general 13

Again, the analytical results in the new parametrisation make such calculations straightforward; see App. B.




1

180

180

i = 5.0°

150

150

120

120

−0.5

0

90

90

60

60

30

30

φSun (deg)

0

φSun (deg)

0.5 e sin fI

e sin fI

0.5

i = 10.0°

−0.5

−1 −1

−0.5

0

0.5

e cos fI

−1 0 1 −1

0 −0.5

0

0.5

1

e cos fI

Figure 14: Corresponding solar aspect angle at deflection in the impact-point approximation.

and will remain, not only for transfers targeting exactly the impact point, but even for more general (essentially) ballistic transfers with a deflection which would be (even very roughly) in the vicinity of ⃗rI . For completeness, we show how these results evolve with increased PHA-orbit inclinations in Fig. 15. In particular, we see that the PHA orbits tend to be more and more naturally decelerated by the deflection missions in this simple setting, and that the solar aspect angle at deflection becomes less and less extreme for both daytime and nighttime impacts (getting closer to 90°), so that illumination conditions around ⃗rI are notably slightly less unfavourable for some of the daytime-impact orbits.




1

1

≥ 0.5

≥ 0.5

i = 22.5°

0.4

0.4

0.3

0.3

0

0.2 0.1

−0.5

0.2 0.1

−0.5 0

−1 1 −1

−0.5

0 e cos fI

0

−1 −0.1 1 −1 180 −0.5

0.5

i = 22.5°

0 e cos fI

0.5

1

−0.1 180

i = 45.0°

150

150

120

120

−0.5

0

90

90

60

60

30

30

φSun (deg)

0

φSun (deg)

0.5 e sin fI

0.5 e sin fI

∆B (b⊕ /yr)

0

∆B (b⊕ /yr)

0.5 e sin fI

e sin fI

0.5

i = 45.0°

−0.5

−1 −1

−0.5

0

0.5

e cos fI

−1 0 1 −1

0 −0.5

0

0.5

1

e cos fI

Figure 15: Impact-point approximation: further cases.



ESA UNCLASSIFIED – For Official Use 3.1.2 Strict 2π-transfers, optimised analytically Let us now determine analytically, for any launcher performance, which hyperbolic excess velocity at launch ⃗v∞SC,L leads to an optimal ∆B if we focus solely on the deflection in the strict ⃗rL = ⃗rD = ⃗rI case. Note that the crucial question of feasibility in terms of ϕSun at the time of deflection is not included as an explicit constraint in this derivation, but is instead left for a later separate discussion. To achieve the current goal, one can exploit the fact that in this very specific setting where everything happens at the same location, the kinetic-impactor heliocentric velocities at launch and deflection are of course the same ⃗vSC,D = ⃗vSC,L , giving (22)

⃗vSC,D = vE ⃗ey + ⃗v∞SC,L . What one should then optimise is only a slight modification of Eq. (20); as expected: ( ) M = mSC,D (|⃗v∞SC,L |) vA,I,y vE − |⃗vA,I |2 + ⃗vA,I · ⃗v∞SC,L ,

(23)

from which it is clear that the optimal orientation of the kinetic-impactor excess velocity at launch should then necessarily be parallel or anti-parallel to the asteroid heliocentric velocity:14 ⃗v∞SC,L = ±|⃗v∞SC,L |

⃗vA,I ; |⃗vA,I |

(24)

the sign is explicitly introduced to clearly distinguish between two possible choices, that either tend to accelerate (+) or to decelerate (−) the PHA along its track. What remains to be done is then to derive the norm |⃗v∞SC,L | that optimises M in each case: Solve

∂M(|⃗v∞SC,L |) =0 ∂|⃗v∞SC,L |

⇔

∂mSC,D (|⃗v∞SC,L |) ∓mSC,D (|⃗v∞SC,L |) = , ∂|⃗v∞SC,L | ±|⃗v∞SC,L | + F (fI , e, i)

(25)

where all the dependencies on the PHA orbit have been grouped in a single term: F ≡ F (fI , e, i) =

vA,I,y vE − |⃗vA,I |. |⃗vA,I |

(26)

Using a polynomial fit of the launcher performance actually enables one to determine that optimum analytically. If we write x = |⃗v∞SC,L | for simplicity, and use a fit of order n: mSC,D (x; n) =

n ∑

ci xi ,

(27)

i=0

defined over a range [xmin , xmax ], an optimal solution x in that same range would respectively satisfy: • 1st order: x =

∓c0 −c1 F ±2c1 ;

• 2nd order: √ −(±c1 +c2 F )+ (±c1 +c2 F )2 −3c2 (c0 ±c1 F ) x1 = or x2 = ±3c2

−(±c1 +c2 F )−

√

(±c1 +c2 F )2 −3c2 (c0 ±c1 F ) ; ±3c2

• 3rd order: see App. D; this result is used in what follows. • (4th order: could also be done analytically, but was not considered here.) 14

A dependence of the launcher performance on the declination at departure is not considered here.



ESA UNCLASSIFIED – For Official Use For the SLS Block 1B (8.4m Fairing EUS) [34], we use xmin = 0 and xmax = 10 km/s, as indicated in Sec. 2.4. Now, whether the largest |∆B| is obtained with an acceleration or a deceleration along track (resp. + or − in Eq. (22)) can in fact be determined from the sign of the quantity F introduced in Eq. (26). This is what already distinguishes the regions in the impact point approximation: F actually takes an interesting meaning in that simple |⃗v∞SC,L | = 0 case, since it then corresponds to the projection of the kinetic impactor relative velocity along the asteroid-velocity direction at deflection—i.e. what governs Eq. (20) essentially. In that approximation, the deflection mission decelerates the asteroids for which F < 0, and accelerates the ones with F > 0, as can be understood from the relative geometry of their orbits with that of the Earth. For most of the PHA parameter space, an along-track deceleration actually turns out to be preferable when performing a strict 2π-transfer; the region that favours acceleration is indeed significantly smaller (concerning only a subset of Atens; no Apollo), and is furthermore monotonously shrinking with increasing i. The locus F = 0 is shown using lines that correspond to different values of the PHA-orbit inclination in Fig. 16.

1

e sin fI

0.5

0

−0.5

−1 −1

−0.5

0

0.5

1

e cos fI Figure 16: Impact-point approximation. Border between positive and negative ∆B, shown for increasing inclinations from 0° to 80° by steps of 10° (when going from the largest to the smallest enclosed region). The separation between Atens and Apollos is furthermore shown by a dashed curve.

The optimal B-plane yearly-drift that could be obtained with a strict 2π-transfer for a given PHA is then achieved when exploiting its predisposition for either an acceleration or a deceleration: decelerate it further along track if F < 0 (resp. accelerate if F > 0). Figure 17 illustrates this with three examples: • F < 0 (top) naturally favours along-track decelerations (i.e. ∆B > 0 when |⃗v∞SC,L | = 0); • F > 0 (middle) favours accelerations instead (∆B < 0 when |⃗v∞SC,L | = 0); Page 25 of 68 Parameter-space of asteroid mitigation missions using ballistic kinetic impactors —Final Report II.



0.080

acceleration deceleration best

∆B (b⊕ /yr)

0.060 0.040 0.020 0.000 −0.020 0

2

4

∆B (b⊕ /yr)

0.000

6

8

10

8

10

8

10

|~v∞SC,L |

−0.004 −0.008 −0.012 0

2

4

|~v∞SC,L |

0.010 ∆B (b⊕ /yr)

6

0.000

−0.010 0

2

4

6 |~v∞SC,L |

Figure 17: Sample results obtained considering either an acceleration or a deceleration along-track for 3 different PHA orbits: fI is equal to 0.3 (17°) (top), 2.8 (160°) (middle), or 2.467 (141°) (bottom), while i = 10°, and e = 0.5. These show the resulting ∆B, when we allow |⃗v∞SC,L | to take any possible value— here, using a thirdorder polynomial fit of the SLS Block 1B launcher performance mSC,D (|⃗v∞SC,L |). On each branch, the red square shows the corresponding optimum obtained via the analytical 2π-transfer optimisation presented in this section and in App. D. The one maximising |∆B| is considered the best ⃗v∞SC,L of all in this setting, and is found on the black line, easily determined from the sign of F (fI , e, i), given by Eq (26).




1

10 i = 10.0°

9 3

7

4

5

6

e sin fI

4

2

0

5

5 3 4

4

3

3

−0.5

|~v∞ SC,L | (km/s)

8

2

0.5

2 2

1

−1

0 −1

−0.5

0

0.5

1

e cos fI Figure 18: Norm of the hyperbolic excess velocity at launch analytically found to optimise ∆B, when only considering strict 2π-transfers. Here and in all the plots in this subsection, a third-order polynomial fit in the escape velocity was used to represent the SLS Block 1B launcher performance. Notice that within the assumptions made in this subsection, the optimal |⃗v∞SC,L | is actually never larger than about 6 km/s for that specific launcher (for which we took max(|⃗v∞SC,L |) = 10 km/s throughout).

• F = 0 (bottom) does not show any a priori preference (∆B = 0 when |⃗v∞SC,L | = 0). When the aim is to maximise |∆B|, fighting a strong natural tendency is obviously rather counterproductive. Should this become unavoidable, particularly poor deflection results are to be expected.15 On the other hand, for orbits which are such that ∆B ∼ 0 in the impact-point approximation, it is intuitively clear that both options may be considered equally (see again the bottom panel). For these orbits, one should however recognise that the resulting |∆B| can be expected to remain rather unsatisfactory with either choice—even worse than for the underperforming accelerating region. Notice moreover that the norm of the optimal hyperbolic excess velocity is sensibly larger when F is vanishingly small (almost 6 km/s), compared to the other best cases (about 3–4 km/s). This is easily understood. There must necessarily be a propensity to depart as much as possible from the relative deflection geometry of the impact-point approximation—even if that means a reduced kinetic-impactor mass. The deflection geometry in such cases was indeed identified as being among the worst possible ones for 2π-transfers in the previous subsection. Note on the other hand that a similar tendency for preferring large |⃗v∞SC,L | should not be expected in the case of PHA orbits for which we had ∆B ∼ 0 because of fI → π in the impact-point approximation: indeed, the rather poor results in those cases are instead due to the relative geometry at impact, essentially unaltered by the kinetic impactors; see again Eq. (10). All this is clearly seen in Figs. 18 and 19. The optimal transfers for each PHA are now fully identified in the case of strict 2π-transfers. For any conceivable PHA orbit, we indeed know which ⃗v∞SC,L shall maximise the yearly-drift. We show 15 As discussed later, this notably happens when including feasibility considerations from the launcher point of view while also taking into account the solar aspect angle at deflection.




1

36 i = 10.0°

32 28

32

0.5

28

32

20

24 28

0

24 24

16 28

mSC,D (t)

e sin fI

24

12 32

−0.5

8 4 0

−1 −1

0

−0.5

0.5

1

e cos fI Figure 19: Kinetic impactor mass at deflection corresponding to Fig 18.

1

≥ 0.5 i = 10.0°

0.4

0.25 0.1

0.5

0.3 e sin fI

−0.01

0

0.2

−0.01

0.01

0.1 −0.5

∆B (b⊕ /yr)

0.5

0.1

0

0.25 0.5

−1

−0.1 −1

−0.5

0

0.5

1

e cos fI Figure 20: Corresponding optimal B-plane yearly-drift ∆B that could be achieved ballistically with a strict 2π-transfer if we ignore the phasing, assuming the SLS performance. Again, the maximisation of |∆B| was the only focus here; the feasibility of such transfers regarding the solar aspect angle at deflection must be addressed separately.



ESA UNCLASSIFIED – For Official Use the corresponding optimal ∆B in Fig. 20, which, as expected from what precedes, simply strengthens the tendencies already appearing in Fig. 13. Once more, to enable the assessment of the feasibility of such transfers in terms of illumination at deflection, we can provide the corresponding analytical expression for the solar aspect angle at deflection. We now obtain ( ) vA,I,x (1 + ξ) ϕSun = acos , (28) ⃗v∞SC,D and

√ ⃗v∞SC,D = (vA,I,x (1 + ξ))2 + (vE − vA,I,y (1 + ξ))2 + (vA,I,z (1 + ξ))2 ,

where we write ξ=∓

|⃗v∞SC,L | , |⃗vAI |

(29) (30)

which has a sign that depends on whether the PHA is accelerated or decelerated along-track (choice made in Eq. (24)). We immediately see that Eq. (28) is a simple generalisation of the result that we obtained in Eq. (21) for the impact-point approximation, which is of course recovered in the limit ξ → 0. This generalised result is shown in Fig. 21, using for each PHA orbit the hyperbolic excess velocity at launch ⃗v∞SC,L found to optimise ∆B. This should be compared to Fig. 14. Overall, despite their differences, the general trends are already quite well captured in the simple impact-point approximation. One of them is that the most extreme values are now shifted towards the second and third quadrants; another one is that, in the region favouring an along-track acceleration, the solar aspect angle is then closer to 90°.

1

180 i = 10.0°

150 0.5 e sin fI

0

90

90°

120°

150°

φSun (deg)

120

30°

60°

60

−0.5 30 0

−1 −1

−0.5

0

0.5

1

e cos fI Figure 21: Optimised strict 2π-transfers: corresponding solar aspect angle at deflection.

Finally, all the results presented here being analytical, one can get the corresponding answer by means of function calls for any inclination and any launcher; which makes such an assessment particPage 29 of 68 Parameter-space of asteroid mitigation missions using ballistic kinetic impactors —Final Report II.


ESA UNCLASSIFIED – For Official Use ularly fast. These optimal analytical results are moreover exact: they match what is obtained with a numerical optimisation procedure when ⃗rL = ⃗rD = ⃗rI . Furthermore, we argue that whenever the deflection is taking place in the vicinity of ⃗rI , these findings can be expected to remain qualitatively unaltered for the most part, even when considering more general ballistic trajectories.16 They should provide rather good initial guesses, at the very least. Our hope is that with these analytical results, we could obtain a better understanding and develop an overall feel of what the optimal deflection geometry should be in different parts of the parameter space. This is especially true for |∆B|, which tends to be much more difficult to change significantly than other key quantities such as ϕSun for instance (which can nonetheless still be guessed qualitatively). 3.2 Strict π-transfers From the illumination point of view alone, there is no doubt that the general class of ballistic transfers with a deflection close to ⃗ropp is particularly favourable for mitigating daytime-impact asteroids; see again Figs. 9 and 10. The situation is however not exactly reversed with respect to deflections in the vicinity of ⃗rI . This is purely because of the limited launcher performance. These π-transfers are indeed not as straightforwardly available as the 2π-transfers are, since already trying to reach ⃗ropp might be quite difficult. As they are then simply not an option in some parts of the PHA-orbit parameter space, potential problems can be foreseen for some PHAs with daytime impacts; see again Ref. [1]. 3.2.1 Cheapest ballistic transfer to ⃗ropp Since both nodes were argued to constitute good initial guesses in general, and since a detailed analytical study of simple transfers to ⃗rI was already shown to be quite useful to build an intuition of the problem, there is a quite strong incentive to proceed with an analogous study, now targeting ⃗ropp . As with the impact-point approximation for 2π-transfers, let us start with the simplest kind of strict π-transfers in the ecliptic plane (i.e. most easily achieved with a given launcher if we ignore the phasing). That would be to launch at ⃗rI and deflect at ⃗ropp , such that these are resp. the aphelion and the perihelion of the transfer ellipse, or vice versa depending on which is furthest away from the Sun (which we know analytically thanks to Eq.(2)), as in the first leg of a Hohmann transfer, maximising the kinetic-impactor mass at the opposite node. What follows holds for any ⃗ropp , and therefore for any PHA: |⃗ropp | + |⃗rI | |⃗rI | (SC) (SC) a = and e = 1 − (SC) , (31) 2 a while the heliocentric velocities of the spacecraft at departure and at the opposing node are √ √ µSun |⃗ropp | |⃗vSC,I | = ; ⃗vSC,I = |⃗vSC,I |⃗ey |⃗rI | a(SC) and

√ |⃗vSC,opp | =

√ µSun a(SC)

|⃗rI | ; |⃗ropp |

⃗vSC,opp = −|⃗vSC,opp |⃗ey .

(32)

(33)

16 Technically, stronger deviations will appear where mitigation results are unsatisfactory (e.g. tiny deceleration instead of tiny acceleration); in those cases, the |∆B| results will tend to remain quite unsatisfactory nonetheless.



ESA UNCLASSIFIED – For Official Use The required hyperbolic excess velocity at launch to reach any ⃗ropp , using this specific transfer, then simply follows:   √ √ √ |⃗ropp | µSun  2 − 1 ⃗ey , (34) ⃗v∞SC,L = ⃗vSC,I − ⃗vE,I = |⃗ ropp | AU |⃗rI | +1 |⃗ rI |

so that, for a given launcher, the impactor mass mSC,D (|⃗v∞SC,L )| that can be brought for a deflection at ⃗ropp is also readily obtained. Let us explicitly write these results in terms of the (fI , e) parameters of the target PHA orbit. The spacecraft velocities at ⃗rI and ⃗ropp read √ µSun √ |⃗vSC,I | = 1 + e cos fI ; ⃗vSC,I = |⃗vSC,I |⃗ey (35) AU and

√ |⃗vSC,opp | =

µSun 1 − e cos fI √ ; AU 1 + e cos fI

⃗vSC,opp = −|⃗vSC,opp |⃗ey ,

(36)

while the required hyperbolic excess velocity at launch to reach the corresponding ⃗ropp is very simply √ ) µSun (√ ⃗v∞SC,L = ⃗vSC,I − ⃗vE,I = 1 + e cos fI − 1 ⃗ey . (37) AU Comparing the spacecraft heliocentric velocities at both nodes with the PHA ones, it is clear that for very small inclinations, this specific kinetic-impactor orbit would essentially be the worst possible choice for dealing with PHA orbits for which the impact location corresponds to either the perihelion or aphelion.17 Both orbits would then indeed coincide, and the relative velocity at deflection, therefore tend to zero. The norm of Eq. (37) is shown in Fig. 22, and the corresponding kinetic-impactor mass is shown in Fig. 23. Moreover, as given in Eq. (37), the projection ⃗v∞SC,L · ⃗ey must necessarily be negative if the opposite node lies inside the Earth orbit (when e cos fI ≤ 0) since we need to decrease the orbital energy to reach this point, and is positive otherwise (when e cos fI ≥ 0). Note that the vertical black and white lines in those figures correspond respectively to the largest aphelion and smallest perihelion that could be achieved ballistically with |⃗v∞SC,L | ≤ 10 km/s; see Ref. [1] for a derivation. They will be shown in black in what follows. Now, for these unoptimised ballistic transfers in the ecliptic plane, one finds ( ) M = mSC,D (|⃗v∞SC,L |) vA,opp,y vSC,opp,y − |⃗vA,opp |2 ( ) = mSC,D (|⃗v∞SC,L |) vA,opp,f 2 cos(i) − |⃗vA,opp |2 ,

(38)

which is actually always negative. From this observation and the analogue discussion that was made concerning the 2π-transfers, we can already deduce that more general variants of π-transfers will then naturally favour an along-track deceleration if they are optimised to maximise |∆B|.

17 Obviously, with a deflection technique requiring a rendez-vous, these would instead be interesting, especially for PHAs impacting Earth at their aphelion, for which a mitigation with kinetic impactors is notoriously difficult.




1

≥ 10 9 8

e sin fI

0.5

7 6

0

5 4 3

−0.5

2 1

−1

min |~v∞ SC,L | to reach ~ropp (km/s)

∀i

0 −1

−0.5

0

0.5

1

e cos fI Figure 22: Minimum required hyperbolic excess velocity at launch to reach ⃗ropp ballistically.

1

36 ∀i

28

0.5 e sin fI

24 20

0

16 12

−0.5

8

max mSC,D at ~ropp (t)

32

4 0

−1 −1

−0.5

0

0.5

1

e cos fI Figure 23: Maximum kinetic impactor mass mSC,D in tons that could be brought to the corresponding opposite node of any PHA, when considering the performance of the SLS Block-1B 8.4m Fairing EUS.




1

≥ 0.5

≥ 0.5

i = 10.0°

i = 22.5°

0.25 0.5

0.25 0.5

0.4

0.1

0.3 e sin fI

e sin fI

0.01

0

0

0.2

0.01

0.1 −0.5

0.3 0.01

0.2 0.01

0.1

−0.5

∆B (b⊕ /yr)

0.5 ∆B (b⊕ /yr)

0.5

0.4

0.1

0.1

0.1

0 0.5

−1 −1

−0.5

0

0.25

0.25 0.5

0

0.5

e cos fI

−1 −0.1 1 −1 −0.5

−0.1 0

0.5

1

e cos fI Figure 24: Simplest π-transfers: ∆B.

Figure 24 shows the results for ∆B obtained with the simplest strict π-transfers. What must be stressed here is that the region in parameter space where the deflection is found to be the most efficient with these transfers is for PHAs with |⃗ropp | < |⃗rI |. This appears driven by the fact that, as the opposite node is the one closest to the perihelion, a deflection there becomes potentially more appealing simply because |⃗vA,opp | > |⃗vA,I |; see also Fig. 25.18 In fact, despite the necessary price to pay to get to ⃗ropp , for a number of such orbits the resulting deflection is then actually even more efficient than with a 2πtransfer. Understandably though, when ⃗ropp becomes too difficult to reach ballistically (which happens rather quickly, even with the SLS-Block-1B performance), the outcome of the deflection eventually suffers and its quality degrades; the preferred solution then becomes a 2π-transfer again. Later, when discussing the absolute best solutions, we shall see that this explains the presence of a stripe of πtransfers in parameter space for both daytime- and nighttime-impact asteroids when no constraints on the solar aspect angle are enforced. As was also discussed earlier, it is also clear in Fig. 24 that, for PHAs impacting the Earth at their perihelion or aphelion (fI close to 0 or π), a deflection at the opposite node with such a simple transfer does not perform well if i(PHA) is small. This slightly improves for larger inclinations, and deteriorates for smaller ones. These results actually give the general trend to be expected from ballistic transfers to the opposite node. Again, what we learn here is interesting because, fundamentally, even numerically optimised transfers without constraints on ⃗rL and ⃗rD (⃗rL being much more free than ⃗rD ) will tend not to differ too significantly from transfers to either node, due to the limited transfer performance. The geometry at deflection is what matters most, and is difficult to strongly alter it while preserving satisfactory mitigation results. On the other hand, the solar aspect angle is found to be ( ) √ −vA,opp,x ϕSun = acos , with |⃗v∞SC,D | = vA,opp,x 2 + 2 vA,opp,f 2 (1 − cos(i)). (39) |⃗v∞SC,D | 18

A large heliocentric velocity is reason why the perihelion itself would be appealing in the first place; see App. A.




≥ 10

1

1.4 ∀i

∀i

1.2 0.5

0.5

0.4

−0.5

0

1

|~vA,opp |/vE

0.6

e sin fI

e sin fI

0.8 0

|~vA,I |/vE

1

−0.5

0.2 −1 −0.5

0

0.5

≤ 0.1

−1

0 −1

−1

1

−0.5

e cos fI

0

0.5

1

e cos fI

Figure 25: Asteroid heliocentric velocities at the nodes [1].

Compared to the impact point approximation (21), the situation is qualitatively reversed.19 More favourable illumination conditions at deflection are now found for daytime-impact asteroids; this is shown in Fig. 26. 1

1

180

180

i = 10.0°

i = 22.5°

150 0.5

150

0.5 150°

0 60°

0

90

30°

120°

90 60°

60 −0.5

60

−0.5

30°

30 −1 −1

−0.5

0

0.5

φSun (deg)

e sin fI

e sin fI

120°

120 φSun (deg)

120

150°

−1 0 1 −1

e cos fI

30 0 −0.5

0

0.5

1

e cos fI

Figure 26: Simplest π-transfers: corresponding solar aspect angle at deflection.

3.2.2 The spacecraft ballistic trajectory, seen as another Earth-crossing orbit To analytically determine the spacecraft hyperbolic-escape-velocity vector at launch that leads to the optimal ∆B for a π-transfer, there is an added difficulty. It is not as straightforward to relate it to 19

Remember that vA,I,x = vA,opp,x holds for any impacting PHA.



ESA UNCLASSIFIED – For Official Use what happens at deflection since, unlike what happens with 2π-transfers, launch and deflection points now correspond to two different locations on the spacecraft orbit. First of all, it is worth realising that, given that a spacecraft on a ballistic trajectory is also strictly speaking on an impacting orbit with the Earth, the formalism introduced for PHAs might be used for the spacecraft as well. In particular, it means that the velocity of the spacecraft at the nodes of its own orbit are also given by App. B, but where e is then the spacecraft-orbit eccentricity and where fI for the spacecraft is understood as its true anomaly where it intersects the Earth orbit, i.e. at launch. This result is general. For clarity one should write fL instead of fI for the spacecraft if one ever wants to describe the spacecraft orbit with this formalism in cases where the launch would not take place exactly at the node where the asteroid impact has been foreseen ⃗rI .20 The discussion about strict π-transfers that follows in this section would not apply to such generalised cases however, since it precisely relies on exploiting properties that require that the Earth orbit is also crossed by the spacecraft exactly at ⃗rI . Secondly, we exploit the fact that, with any strict π-transfer, the nodes of the spacecraft orbit coincide with those of the PHA orbit.21 The spacecraft and the asteroid orbits therefore necessarily verify: (PHA) (SC) e(PHA) cos fI = e(SC) cos fI , (40) because sharing the same ⃗rI and ⃗ropp fixes e cos fI . That implies that they furthermore have the same ⃗vIf and the same ⃗voppf since these only depend on e cos fI , as seen earlier. What this then shows is that, for any kind of strict π-transfer, there are in fact only two degrees of freedom left: the heliocentric radial velocity at ⃗rI , namely vSC,I,x = vSC,opp,x , and the inclination of the spacecraft orbit. Applications— • In particular, it is then straightforward to retrieve Eqs. (35) and (36) for the transfer discussed in Sec. 3.2.1, simply using App. B where the asteroid is replaced by a spacecraft with vanishing spacecraft-orbit inclination and heliocentric radial velocity. • Equation (37) gives e

(PHA)

(PHA) cos fI

(PHA)

for asteroids with e(PHA) cos fI e (PHA)

if e(PHA) cos fI

(PHA)

( ) |⃗v∞SC,L | 2 = 1+ − 1, |v⃗E |

(41)

≥ 0 (i.e. with |⃗ropp | ≥ 1 AU), while

(PHA) cos fI

( ) |⃗v∞SC,L | 2 = 1− − 1, |v⃗E |

(42)

≤ 0 (i.e. with |⃗ropp | ≤ 1 AU).

Since a π-transfer requires that the equality (40) holds, replacing Eqs. (41) and (42) in the expression of |⃗ropp | in Eq. (72) for a chosen launcher max(|⃗v∞SC,L |) provides respectively the largest aphelion and the smallest perihelion that could be reached. The idea can also be pushed further; see App. E. 20

To extend such a formalism to launch locations ⃗rL that would differ from ⃗rI , one would simply rotate by the solar longitude at launch. 21 It is easy to convince oneself that the following results remain valid even in the i → 0 limit.



ESA UNCLASSIFIED – For Official Use 3.2.3 Strict π-transfers, optimised analytically One therefore does not need to solve a single Lambert problem to get the complete set of conceivable strict π-transfer orbits to any PHA—i.e. any ballistic transfer such that ⃗rL = ⃗rI and ⃗rD = ⃗ropp . These are indeed simply given by22 ( ) (SC) (SC) (PHA) any fI , e(SC) , i(SC) with e(SC) cos fI = e(PHA) cos fI , (43) and for all of these, ⃗vSC,opp and ⃗vSC,I are given analytically in App. B, via simple relations. So now, for strict π-transfers, in the most general case, what one wants to optimise is ( ) M = mSC,D (|⃗v∞SC,L |) v∞SC,L,x vA,opp,x + vA,opp,f 2 cos(i(PHA) − i(SC) ) − |⃗vA,opp |2 , with |⃗v∞SC,L | =

√

v∞SC,L,x 2 + vA,I,f 2 + vE 2 − 2 vE |vA,I,f | cos(i(SC) ).

When, correspondingly, the solar aspect angle at deflection can be shown to obey: ) ( v∞SC,L,x − vA,opp,x , ϕSun = acos |⃗v∞SC,D | with |⃗v∞SC,D | =

√

(v∞SC,L,x − vA,opp,x )2 + 2 vA,opp,f 2 (1 − cos(i(PHA) − i(SC) )).

(44)

(45)

(46)

(47)

As we just discussed, for any such transfer, there are two degrees of freedom: v∞SC,L,x and i(SC) ; all the rest is fixed for a given PHA. Notice that the last two terms in Eq. (44), which generalises what was found in the opposite-node approximation (38), are again always negative for any i(SC) . As discussed earlier, this therefore a priori favours values of v∞SC,L,x that would further enhance along-track decelerations. Approximate analytical solution to the strict-π-transfer optimisation problem—As was done in the case of strict 2π-transfers, we would like to optimise strict π-transfers analytically. This time however, let us make an approximation: we take the limit of vanishing transfer-orbit inclination i(SC) = 0: ⃗v∞SC,L = v∞SC,L,x⃗ex + (vSC,I,f − vE )⃗ey , leaving us with only one degree of freedom, since √ (PHA) vSC,I,f = vE 1 + e(PHA) cos fI = vA,I,f ,

(48)

(49)

is required for a π-transfer. While doing so will not lead to the absolute optimum, we argue that most of the time the true optimum will not be far—at least when such transfers are going to be an appealing option.23 In particular, this assumption conceptually fails when vA,opp,x → 0, namely if the impact location is very close to either the perihelion or the aphelion of the PHA orbit; i.e. orbits with small e sin fI . Under such circumstances, Eq. (44) clearly shows that changing v∞SC,L,x would then bear no benefit, and that in those cases, one should rather try and find the optimal i(SC) . However, the 22

Which, in other words, corresponds exactly to the set of PHA orbits sharing the same opposite node. For large asteroid-orbit inclinations i(PHA) in particular, such a simple approximation can be expected to be rather satisfying. 23



ESA UNCLASSIFIED – For Official Use inclination of the transfer orbit cannot be changed easily: it is indeed very expensive in terms of kineticimpactor mass to significantly change the inclination with respect to the ecliptic plane ballistically (this already motivated the choice of 2π- and π-transfers as initial guesses). The B-plane displacement is then actually bound to remain close to the results of the simple opposite-node approximation (i(SC) = 0 and v∞SC,L,x = 0). In other words, for such asteroids, while the true optimal |⃗v∞SC,L | for a strict π-transfer can be expected to be sensibly larger that what Eq. (37) gives, the outcome of the deflection would nonetheless barely improve with respect to the cheapest π-transfers. That situation is even worse for orbits with i(PHA) close to zero. Now, similarly to what was done before, one can explicitly write, assuming vA,opp,x ̸= 0: v∞SC,L,x = ±α giving

vA,opp,x , |vA,opp,x |

α ≡ |v∞SC,L,x |,

(50)

( ) M = mSC,D (α) ±α |vA,opp,x | + vA,opp,f 2 cos(i(PHA) ) − |⃗vA,opp |2 .

(51)

where one can choose to slightly accelerate (+) the asteroid along its x-axis direction or further decelerate it (−); the sum of the last two terms in Eq. (51) is indeed always negative, as discussed earlier. The optimisation problem can for instance be solved analytically by simply assuming a polynomial fit (up to second order) of the launcher performance in terms of C3 = |⃗v∞SC,L |2 : mSC,D (C3) = c0 + c1 C3 + c2 C32

(52)

which, using Eqs. (48) to (50), can be written: 2

mSC,D (α) = c0 + c1 (α2 + (vA,I,f − vE )2 ) + c2 (α2 + (vA,I,f − vE )2 ) ,

(53)

so that one should solve ∂M(α) =0 ∂α where H ≡ H(fI , e, i) =

⇔

∂mSC,D (α) ∓mSC,D (α) = , ∂α ±α + H(fI , e, i)

vA,opp,f 2 cos(i(PHA) ) − |⃗vA,opp |2 , |vA,opp,x |

(54)

which is ≤ 0 for all PHAs.

(55)

To provide convenient and readily usable analytical formulae, let us show results assuming a simple first-order fit in C3. Results with a second-order fit of the performance would lead to a fourth order polynomial which, though analytically solvable, will give solutions that are arguably already too tedious to be genuinely useful—especially since the result is an estimate which has its limitations. For a firstorder fit (c2 = 0; c1 , c0 ̸= 0), we find: √ −H ± H 2 − 3( cc10 + (vA,I,f − vE )2 ) vA,opp,x with α1,2 = (56) v∞SC,L,x = +α |vA,opp,x | 3 and v∞SC,L,x

vA,opp,x = −α |vA,opp,x |

with α1,2 =


−H ±

√

H 2 − 3( cc01 + (vA,I,f − vE )2 ) −3

,

(57)


ESA UNCLASSIFIED – For Official Use where, once again, the quantity (vA,I,f − vE ) is the minimum |⃗v∞SC,L | required for any strict π-transfer. Note that the sign of v∞SC,L,x in these equations really is with respect to ⃗ex . As expected, a deceleration for daytime impactors, should be v∞SC,L,x < 0, and v∞SC,L,x > 0 for nighttime ones; and the opposite for an acceleration. Finally, one of course has to check whether the solutions (56) and (57) for α are positive (to be physically meaningful), and also make sure that the corresponding total ⃗v∞SC,L is feasible for the considered launcher. In practice, decelerations (57) are the solutions to be used when optimising π-transfers. Indeed, with the SLS Block 1B performance limited to |⃗v∞SC,L | = 10 km/s, accelerations (56) are not only found unappealing (again, going against a natural tendency), but are moreover only possible in a small part of the parameter space. As before, the different results are shown in Figs. 27–28. For validation of the approximation made here, we moreover compared them to numerical optimisation results of strict π-transfers (⃗rL = ⃗rI and ⃗rD = ⃗ropp fixed). As emphasised earlier, restricting to transfers with i(SC) = 0 cannot be expected to hold for PHAs straddling the region of small e sin fI , but since the ∆B results with strict π-transfers tend to be unsatisfactory for such orbits, other types of transfer might then be preferable anyway.24 For the rest of the PHA orbits, while it is clear that there is no reason why the actual optimal solution should exactly have a vanishing inclination if we allow for this additional degree of freedom, the associated gains are actually tiny: indeed, ∆B, our main concern, is well described at larger e sin fI , if numerical results are compared to our analytical approximation. Compared to unoptimised π-transfers, the tendency of having a larger |∆B| for orbits with |⃗ropp | < |⃗rI | (i.e. opposite node closer to perihelion) is actually further strengthened. We however immediately notice that the improvements brought by the optimisation of strict π-transfers are clearly inferior compared to what we had in the case of strict 2π-transfers. Finally, while the solar aspect angle is more subject to change than ∆B, π-transfers still always provide favourable illumination conditions for daytime-impact asteroid and would usually be particularly difficult for nighttime-impact asteroids, at least at low PHA-orbit inclinations. More importantly than the exact numerical value for the solar aspect angle, which will anyway be different for an actual mission (no longer restricted at the nodes), we indeed argue that this general feasibility assessment is what mostly matters about ϕSun : for identifying successful mitigation missions among the various options found to optimise ∆B. While remaining fairly simple analytically, this i(SC) = 0 approximation captures the essence of what one can expect from a strict π-transfer in general. Even though the description might be improved, it is important to realise that real difficulties actually arise for ∆B in limits in which this approximation is not suitable: i.e. when vA,opp,x (or equivalently e sin fI ) is small. As the inclination that can be ballistically achieved with a launcher is bound to be quite modest, the deflection with π-transfers for such regions in parameter space will usually be rather poor. Still targeting the opposite node, it would then be more satisfactory to change the relative geometry by changing ⃗rL , which is much more free in reality than ⃗rD —even if that increases the required minimum |⃗v∞SC,L | needed to reach ⃗ropp . This is done in the next section, where numerical optimisation results have been obtained without restrictions on ⃗rL and ⃗rD , but still starting from simple transfers such as the ones discussed here, as initial guesses. 24

The hyperbolic excess velocity at launch is larger, for orbits with small e sin fI (i.e. small vA,opp,x ), when i(SC) is not constrained — even when ⃗ropp is easily reached. This is because, exactly as we had in Fig. 18 for optimised 2π-transfers, there is a propensity to depart as much as possible from an unoptimised strict π-transfer — essentially the worst possible transfer for these orbits. Here there is an attempt to leave the ecliptic plane, since, as argued, v∞SC,L,x ̸= 0 would not help in these cases. Again, the resulting |∆B| is nonetheless bound to remain quite poor.




1

≥ 0.5 i = 10.0° 0.25 0.5

0.4

0.1

0.3 e sin fI

0.01

0

0.2 0.01

0.1

∆B (b⊕ /yr)

0.5

−0.5 0.1

0 0.25 0.5

−1 1 −1

−0.5

0

0.5

e cos fI

1

−0.1 180

i = 10.0°

150 0.5 e sin fI

120°

0

90

60° 30°

φSun (deg)

120 150°

60 −0.5 30 0

−1 −1

−0.5

0

0.5

1

e cos fI Figure 27: Approximate analytical optimisation results for strict π-transfers (restricted case i(SC) = 0): yearlydrift ∆B and solar aspect angle at deflection ϕSun . Again: this is an simplification, known not to describe the optimal solution for PHAs with smaller values of e sin fI , for which ∆B would be slightly higher in reality. In all these figures, a 1st-order fit in C3 for the launcher performance of the SLS Block 1B provided an estimate of the optimal v∞SC,L,x , thereby providing |⃗v∞SC,L |; from there, the performance was then estimated with exact same fit used for 2π-transfers.




1

10 i = 10.0°

9 8

5 8

4

7

6

3

7

8

|~v∞ SC,L | (km/s)

0.5 e sin fI

9

6 3 2 1 12 34

0

5 5 6

4

7 8 9

3

7

9

−0.5

8

6 5

4

2

3

1 −0.5

0 28

0.5 i = 10.0°

e cos fI 32

1

32 28

0.5

24

28

20 16

e sin fI

12

8

24 20

32

0

0 36

32

28

16

24 20

16

12

mSC,D (t)

−1 1 −1

12 8 1216

−0.5

20 24 28

8

32

4 0

−1 −1

−0.5

0

0.5

1

e cos fI Figure 28: Same as Fig. 27 but for the norm of the hyperbolic excess velocity at launch that optimises ∆B and the corresponding kinetic-impactor mass at deflection. The trough in |⃗v∞SC,L | at small values of e sin fI and the corresponding elevation in mass are artefacts of the approximation used here; see text.




4

General numerical optimisation without phasing

It is clear that the optimal solutions for the spacecraft orbits are not given by transfers strictly anchored at the nodes; the actual answers are obviously much more complex than this. For actual ballistic transfers, the departure location ⃗rL , in particular, is understandably much less constrained than the deflection one ⃗rD , which can still be expected to remain in the vicinity of either of the nodes when the inclination is sizeable. Let us now try to understand the actual best transfers found numerically, given the properties of what we already know from our study of our initial guesses; to derive an overall feel of the type of optimal solutions that can be associated to different parts of the full PHA parameter space. 1

1

≥ 0.5

≥ 0.5 i = 10.0°

i = 10.0° 0.5

0.5

0.4 −0.01

e sin fI

−0.01 −0.01 0.01

0

0.1

0.1

−0.5

0 e cos fI

0.1

0.1

0.25

0

0.5

−1 −0.1 1 −1 180 −0.5

0.5

i = 10.0°

0 e cos fI

0.5

1

−0.1 180

i = 10.0°

150

150

120

120

0

−0.5

0

90

90

60

60

30

30

φSun (deg)

0.5 e sin fI

0.5 e sin fI

0.2

−0.01 0.01

0

0.5

−0.5

−0.01

−0.5

0.25

−1 1 −1

0.2

0.3

φSun (deg)

e sin fI

0.3 0

0.4

0.1

0.5

0.1

∆B (b⊕ /yr)

0.5

0.25

∆B (b⊕ /yr)

0.25

−0.5

−1 −1

−0.5

0

0.5

e cos fI

−1 0 1 −1

0 −0.5

0

0.5

1

e cos fI

Figure 29: Optimal solutions that maximise |∆B| for all the conceivable Earth-impacting orbits with i = 10°, as found with a numerical optimisation procedure that ignores the phasing, and the corresponding solar aspect angle: (left) without constraints on ϕSun ; (right) when explicitly constraining ϕSun ≥ 60°. Since there can be abrupt changes in the combined optimal solutions, these figures show directly the different mesh points for which we have a calculation, to avoid any erroneous interpolation. As they are given a finite size, these points are visible as moderate artefacts, especially at the edges between regions.



ESA UNCLASSIFIED – For Official Use For the numerical optimisation, we have found additional initial guesses which we will use alongside 2π- and π-transfers. They shall prove useful since they can provide distinct types of solutions that may actually lead to the global optimum, which can under certain circumstances escape a numerical optimisation procedure that would rely only on 2π- and π-transfers as initial guesses—especially at higher inclinations. These new kinds of guesses are actually most easily introduced when considering double-crossing PHA orbits (i.e. with fI = π2 or 3π rI but also ⃗ropp exactly lie on 2 ), for which not only ⃗ the Earth orbit; they are: • strict opposite 2π-transfers, which correspond to ⃗rL = ⃗rD = ⃗ropp ; • strict opposite π-transfers, which correspond to ⃗rL = ⃗ropp , while ⃗rD = ⃗rI . Of the two, the first type is the most interesting: they are essentially strict 2π-transfers that are taking place around ⃗ropp instead of ⃗rI , and can then understandably bring more flexibility when targeting ⃗ropp (seeing that ⃗v∞SC,L would then be less constrained than with π-transfers). Only a handful of PHAs do have their opposite node ⃗ropp on the Earth orbit of course, meaning that a strict opposite 2π-transfer is obviously not possible most of the time; one has to relax the constraint that the launch happens exactly at ⃗ropp . This kind of transfers then splits into two separate initial guesses with displaced solar longitudes at launch λL : one smaller and one larger than 180°.25 While this is important for the numerical optimisation procedure, broadly speaking though, in our discussions, the solutions can be grouped according to the targeted node (either ⃗rI or ⃗ropp ), as this is what matters the most in terms of |∆B| (which is very difficult to significantly alter quantitatively) and ϕSun (which is determined by the targeted ⃗rD qualitatively speaking). Since no confusion is possible, we will tend to use ‘2π-transfers’ and ‘π-transfers’ more generally as shorthands for optimal transfers respectively found numerically when initially targeting either ⃗rI or ⃗ropp , though it is a slight misnomer. 4.1

Overview of some of the main results

Figure 29 shows the results of a numerical optimisation without phasing obtained for all the Earthimpacting orbits having a 10° inclination.26 These actually already feel familiar given all the results that we presented when studying in detail the deflection at both nodes analytically, and we can immediately make the following observations. Let us start with the left panels: • When there is no explicit constraint on the solar aspect angle, the optimal results for most PHAs are obtained by means of along-track decelerations with 2π-transfers and orbits similarly targeting the vicinity of ⃗rI , broadly speaking. Additionally, a stripe of transfers to the opposite node, again broadly speaking, also very clearly appears in the polar plot (cf. ϕSun ): these also correspond to decelerations, and appear as soon as |⃗ropp | becomes smaller than |⃗rI |. Notice also the symmetries between daytime- and nighttime-impact parts of parameter space, both in ∆B (mirror) and ϕSun (they add up to 180°). • The presence of an along-track-acceleration region persists for a subset of Atens, but quite clearly shrunk in size. Let us note however that when comparing with numerical optimisation results that include the phasing, the size of the acceleration region actually appears closer to what we find with the analytical results targeting ⃗rI (for which we gave the locus); see Fig. 42 in App. F. This 25

For double-crossing orbits, launching at ⃗ropp corresponds to λL = 180° in our reference frame. As before, the choice i = 10° for illustrating our results is motivated by the fact that it represents the median inclination in the MPC database; see again Fig. 3. Our discussion is however based on our study of the entire parameter space, even very high i, and reflects upon those findings as well. 26



ESA UNCLASSIFIED – For Official Use indicates that for the Aten asteroids which would now appear to settle with decelerations alongtrack, achieving the best deflection conditions found without phasing may actually be difficult.27 For these objects, the amount of displacement tends to remain rather unsatisfactory in any case; they do not benefit from a strong preference for either accelerations or decelerations with the trajectories that we consider. Again, another strategy, e.g. not limited to ballistic trajectories, or another technique altogether might be preferable for mitigating the risk that such objects represent. Now, turning to the right panels: • That being said, one of the most interesting features that we can see is that, when the solar aspect angle is required to be above a certain value (here ϕSun ≥ 60°), an acceleration region suddenly appears for daytime-impact PHAs that would otherwise naturally strongly favour a deceleration (both with 2π-transfers). For any such cases where one would have to fight the natural tendency associated to a given type of transfers, we already stressed that the results in terms of ∆B can be expected to be extremely poor. • The polar representation shows very clearly that this region ends with a vertical border that matches our π-transfer feasibility assessment (throughout, max(|⃗v∞SC,L |) = 10 km/s). For daytime-impact asteroids with e cos fI below this limit (i.e. ⃗ropp too close to the Sun), π-transfers are no longer an option (due to the launcher performance), and neither are decelerating 2πtransfers (due to the unfavourable ϕSun ); the only remaining trajectories then being underperforming solutions obtained by numerically optimising accelerating 2π-transfers. • For many daytime-impact asteroids, 2π-transfers are no longer accessible when one cuts on ϕSun , and must then essentially be replaced by transfers to the opposite node. This type of solutions therefore covers a much more extended part of parameter space, entering even in the |⃗ropp | > |⃗rI | domain (i.e. e cos fI > 0).28 In that domain, these solutions are only slightly sub-optimal compared to 2π-transfers, while being doable ϕSun -wise. • Similarly, as expected for nighttime-impact asteroids, due to poor illumination conditions at deflection alone, 2π-transfers are then replacing π-transfers in the |⃗ropp | < |⃗rI | domain (i.e. e cos fI < 0). On this specific plot, we see that no π-transfer survives for nighttime-impact objects, but we stress that this is specific to this inclination (and understandably also at lower inclinations as well). At higher inclination however, as shown in App. G, π-transfers remain the absolute best solution for some orbits with nighttime impacts, even with the stringent constraint ϕSun ≥ 60°. We stress that such findings are quite general and stable. They similarly hold when assuming the launcher performance of Ariane 5; the π-transfer region being then of course accordingly reduced. Earth-impacting orbits for which we found no solution In the difficult part of parameter space in which daytime-impact asteroids strongly suffer from the absence of possible π-transfers, one may actually even find a number of asteroids for which no ballistic solution was obtained at all, even before including any notion of phasing into the picture. For these, 27

In other words, in that region, the optimal deflection geometry found when optimising ∆B appears difficult to realise in reality due to timing; this is especially true given that Ref. [23] assumed a warning time as large as 20 yr. 28 One can also notice a small abrupt change in ϕSun inside the first quadrant due to a transition from opposite-2πtransfers back to π-transfers.



ESA UNCLASSIFIED – For Official Use the problem stems from the fact that none of our initial guesses correspond to a possible transfer, both due to the solar aspect angle around ⃗rI and to the impossibility to reach ⃗ropp . Striking examples of this can be seen in Fig. 29: they are found at fI roughly spanning ∼125–140° and large eccentricities. Moreover, at other orbital inclinations (or should we settle for another requirement on ϕSun ), there might easily be no solution once the phasing is included, even when no such gap is found in this part of parameter space when ignoring the phasing. Note that actual PHAs are populating such orbits. As mentioned in the first part [1], as the present study was being carried out, asteroid 2014 JO25 [36] (fMOID = 129°, and e = 0.89) actually made a very close approach of the Earth: a close shave by less than 5 lunar distances. With a size of about 850 m, it was on a significantly inclined orbit (i = 25.27°). Taking as a proxy a similar case for which we already had results, there appears to be no feasible ballistic trajectory for such an orbit when including the phasing, even with ϕSun allowed to be as low as 45° and assuming a very long warning time of 20 years (note that, in reality, that asteroid was only discovered 3 years before); we show that in App. F, see Fig. 41 (right). It is important to recognise that special care should be taken where our initial guesses do not correspond to any valid—even sub-optimal—solutions. It might be interesting to dedicate a study to these specific cases, to see if one could find a more reliable type of initial guess, better suited for that region. Since these orbits are confined in a small part of the parameter space, they are quite similar. Alternatively, for such orbits, it might make more sense to consider dropping the ballistic-transfer restriction, rather than trying to provide what might turn out to be a rather artificial initial guess for the ballistic transfers. 4.2

Departures from the guesses: solar longitude at deflection

As already discussed earlier, ⃗rL is much more free than ⃗rD , and what truly matters is mostly roughly where the deflection happens. Making too big of a change there indeed necessarily leads to very different results, already at the qualitative level—much more so than when changing the launch location, which plays a relatively minor role in comparison. Initial guesses for the impactor mission can then be grouped according to what their initial target deflection point is, since the results tend to bear similar properties when the deflection takes place in the same vicinity. To better understand the optimal solutions found numerically, we now focus on the exact deflection location, to identify potential trends in the systematic departures from the initial guesses. More precisely, to determine how close the actual optimal deflection is to the initial-guess assumption of a deflection in the vicinity of either node, let us consider the solar longitude at deflection λD . The closer λD is to 0° (resp. 180°), the closer the deflection indeed stays close to ⃗rI (resp. ⃗ropp ). Some remarks about the results shown in Fig. 30. While there are jumps at boundaries, the separate curves are continuous and smoothly evolving for neighbouring points. For daytime PHAs: • when the optimal deflection is in the vicinity of ⃗rI (e.g. 2π-like transfers), the solar longitude at deflection is always negative: the location that corresponds to an optimal deflection is systematically at a true anomaly f ≲ fI (corresponding to a point closer to perihelion than ⃗rI itself is). • in the stripe where the optimal deflection is in the vicinity of ⃗ropp , as we go from e cos fI = 0 to negative values, the solar longitude at deflection is first negative (λD ≳ −180°), then corresponds to ⃗ropp , and continues on, then being positive (λD ≲ 180°). In other words, as ⃗ropp moves closer and closer to the Sun (decreasing e cos fI ), we have the following situation: 1) when |⃗ropp | ≲ |⃗rI |, Page 44 of 68 Parameter-space of asteroid mitigation missions using ballistic kinetic impactors —Final Report II.



180 150 120 90 60 30 0 −30 −60 −90 −120 −150 −180

solar longitude at deflection (deg)

i = 10°

e sin fI

0.5

0

−0.5

−1 −1

−0.5

0

0.5

1

e cos fI Figure 30: The solar longitude at deflection λD that corresponds to each of the optimal solutions shown in Fig. 29, in the unconstrained solar-aspect-angle case. In a nutshell, extremely light colours show where the optimal deflection location is very close to ⃗rI (λD = 0°), and extremely dark, to ⃗ropp (λD = 180°).

the optimal deflection point is first closer to perihelion than ⃗ropp itself is. 2) As ⃗ropp is moved further inside the Solar System, the optimal deflection point eventually coincides with ⃗ropp . 3) As |⃗ropp | continues to decrease, the optimal deflection point is then found further away from the Sun that the node itself is. 4) Eventually, the trade-off required for reaching such a point while preserving an advantageous relative geometry is no longer worth it (or even not possible at all), and 2π-transfers become more appealing, purely because of the limited performance, so that this breaking point therefore differs with different launchers. The sign of λD is reversed for nighttime PHAs, so that the physical situation is equivalent. Obviously as the PHA-orbit inclination increases, the initial assumption of a deflection at the nodes is getting better and better; see Fig. 31. The ballistically reachable arcs on the PHA orbit then indeed necessarily shrink and concentrate more and more around both nodes, so that the optimal results found numerically tend to match the initial guesses simply because they become the only feasible solutions in the most extreme case.29 The analytical optimisation results presented before then become more and more useful. In a nutshell, to sum things up there is a clear imbalance between the two nodes, favouring ⃗rI . Not considering the solar aspect angle yet, for 2π-transfers, there are two cases: one either goes closer to perihelion than ⃗rI itself is, or deflects at ⃗rI . In no case is it further from the perihelion, since ⃗rI can always be reached (being on the Earth orbit). For orbits such that these transfers are an option when adding the solar aspect angle constraint into the picture, these remarks will remain. Only that constraint could favour a deflection point further away from the Sun than ⃗rI (phasing also could, of course; but again, here we are looking for the reachable location close to which a deflection is found to give the optimal yearly-drift). On the other hand, when considering a deflection in the vicinity of 29 This would of course happen even faster (i.e. starting at even lower i) should the launcher performance be inferior to that of the SLS Block-1B, assumed here.




0.5 e sin fI

e sin fI

0.5

0

−0.5

0

−0.5

−1 −1

−0.5

0

0.5

e cos fI

−1 1 −1

180 150 120 90 60 30 0 −30 −60 −90 −120 −150 −180

180 150 120 90 60 30 0 −30 −60 −90 −120 −150 −180

i = 45°

−0.5


1 i = 22.5°


1

0

0.5

1

e cos fI

Figure 31: Same as Fig. 30 (with a rougher mesh), only at higher inclinations.

⃗ropp , from the start there is an additional cost to bear because the spacecraft has to reach that location in the first place; something which might not be possible with a ballistic trajectory—even with nearfuture generations of launchers. Obviously, it might be possible to reach the opposite node with a flyby sequence at the cost of an added amount of time (which might be a sufficient reason to discard any similar transfer once the phasing is considered), but it does not change the fact that a deflection near ⃗rI will most probably remain easier to pull off and more efficient. 4.3

Striking approximate symmetries in the optimal numerical results

Here we report some approximate symmetries that actually appear in the optimal transfers found numerically, when there is no artificial constraint on λL and λD , which are then free to take any value. Within our model, these are considered to be the absolute best ballistic transfers when no explicit threshold on the solar aspect angle is applied beforehand. Any accurate analytical description of these optimal solutions would therefore mean having a direct access to what makes the actual best conceivable ballistic transfers in general when ignoring ϕSun . While there is no guarantee that a full analytical description of this output of numerical optimisation could be attained, any step in this direction is clearly desirable. We already know for a fact that using the nodes as initial guesses is very satisfactory for PHA orbits with large inclinations, up to the point where these become the only possible deflection locations; this, in turn, would then be another limit: the approximate symmetries that we find indeed appear more accurate the smaller the inclination, where they are especially striking. 4.3.1

Relation between the optimal solar longitude at launch λD and fI

Especially at low inclinations (see Fig. 32), it appears that there exists a tight relation between fI and the solar longitude at launch λL for the optimal transfer found numerically when ignoring the phasing and without imposing explicit constraints on ϕSun .




1

180 150 120 0.5 90 60 30 0 0 −30 −60 −0.5 −90 −120 −150 −1 −180 0.5 1 −1 180 −0.5 i = 10° 150 120 0.5 90 60 30 0 0 −30 −60 −0.5 −90 −120 −150 −1 −180 0.5 1 −1 −0.5

−0.5

−1 1 −1

−0.5

0 e cos fI

e sin fI

e sin fI

0.5

0

−0.5

−1 −1

−0.5

0 e cos fI

−λL (deg)

180 150 120 90 60 30 0 −30 −60 −90 −120 −150 −180 180 150 120 90 60 30 0 −30 −60 −90 −120 −150 −180

0

0.5

1 i = 10°

e cos fI

−λL (deg)

0

−λL (deg)

e sin fI

e sin fI

0.5

i = 5°

−λL (deg)

i = 5°

0

0.5

1

e cos fI

Figure 32: Highlighting a rather striking relation (when ϕSun is unconstrained) between the true anomaly at impact fI and the solar longitude at launch λL of the corresponding best solution found numerically. (Left) best transfer found numerically when only targeting the vicinity of ⃗rI , broadly speaking. (Right) best transfer found numerically, starting from all the initial guesses considered in this work (i.e. for i = 10° the corresponding ∆B and ϕSun are those shown in Fig. 29 (left).).

Over most of the parameter space, we indeed notice that the following approximate relation then holds, being both striking and extremely simple: λL ∼ −fI

mod 2π

for along-track decelerations;

(58)

in other words, the complex numerical optimisation procedure actually indicates that the optimal launch location on the Earth orbit is most of the time essentially found in front of the perihelion on the PHA orbit. This actually gets even better with increasing PHA-orbit eccentricities e. Indirectly, this relation (58) actually also highlights the fact that, even for PHA orbits with smaller inclinations, the optimal mitigation missions with ballistic trajectories thus only very rarely correspond to deflecting Page 47 of 68 Parameter-space of asteroid mitigation missions using ballistic kinetic impactors —Final Report II.


ESA UNCLASSIFIED – For Official Use strictly at the perihelion when preferentially decelerating the asteroids along-track (i.e. in the vast majority of cases).30 That the optimal ballistic solutions obtained in these cases tend to leave the Earth orbit at a location facing the perihelion of the target PHA orbit furthermore highlights that the corresponding angle between the spacecraft and the asteroid heliocentric velocities actually tends to be quite large when they cross.31 We moreover checked that such results also hold when one assumes the launcher performance of Ariane 5 instead of that of the SLS Block-1B—though the relation is then a bit more degraded (i = 10° is the case that we considered). 4.3.2

Solar aspect angle at deflection of the best deflection solutions

When studying the solutions that optimise ∆B over the entire parameter space, the resulting optimal deflection missions are qualitatively completely different, depending on whether the optimal solution is essentially targeting ⃗rI or ⃗ropp . Surprisingly, we nonetheless notice the presence of what appears to be a very striking (though not so trivial) underlying symmetry between these classes of optimal solutions when no threshold is imposed on the solar aspect angle at deflection. To illustrate this, we show in Fig. 33 (top) what we would find if we were to artificially swap, for optimal deflections in the vicinity of ⃗ropp only, the solar aspect angle at deflection for daytimeand nighttime-impact asteroids. The outcome is an extremely smooth curve32 for the resulting ϕSun , especially striking in the i = 5° case—it deteriorates at larger inclination, most notably at the |⃗ropp | < |⃗rI | boundary. Alternatively, a similar result is obtained by means of the transformation ϕπ-transfer → Sun π-transfer 180°−ϕSun (middle). What may be even more surprising is that the resulting ϕSun pattern moreover strongly resembles what is obtained when only considering numerically optimised 2π-transfers (bottom). We actually checked that the same kind of observation can be made from the results including the phasing, as we show in Fig. 34; we also made the same observation in the optimal results found with the Ariane 5 launcher performance instead of the SLS Block 1B. Again, let us stress that we are discussing the results of heavy numerical optimisation procedures here, with free ⃗rL and ⃗rD , and not a restricted analytical case. This is particularly striking since, while |∆B| is very difficult to alter significantly, there is much more latitude on the values that ϕSun can quantitatively take.33 Finding any such clear pattern in the solar aspect angle at deflection starting with very different initial guesses for different asteroid orbits was therefore unexpected. It would therefore seem that studying this further might lead to much more refined initial guesses of the actual best ballistic solution found numerically. This seems to be linked to a similar relative deflection geometry between what makes the optimal solution found when targeting ⃗rI and what makes the optimal solution when targeting ⃗ropp ; see Fig. 35. When a π-like-transfer is the global optimum in terms of resulting |∆B|, the corresponding relative geometry at deflection appears to be similar to that of the corresponding best (though slightly underperforming) 2π-transfer, and this observation seems even better at lower inclination.

30

For along-track accelerations on the other hand, ⃗rL remains close to the vicinity of ⃗rI (i.e. λL ≈ 0). Remember that what matters is ⃗v∞SC,D · ⃗vA,D , not ⃗vSC,D · ⃗vA,D ; see again Sec. 2.3. 32 The discontinuity in the resulting ϕSun surface can actually remain as small as ≤ 2°. 33 Compare for instance the exact values found when optimising exactly at the nodes and the exact values when ⃗rL and ⃗rD are free. 31




1

180

180

i = 5.0°

150

150

120

120

0

90 60

−0.5

90 60

−0.5 30

−1 1 −1

−0.5

0

−1 0 1 −1 180

0.5

i = 5.0°

e cos fI

30

−0.5

0 e cos fI

0.5

1

0 180

i = 10.0°

150

150

120

120

0

90 60

−0.5

90 60

“φSun ” (deg)

0

“φSun ” (deg)

0.5 e sin fI

−0.5 30

−1 1 −1

−0.5

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e cos fI

−0.5

0 e cos fI

0.5

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0 180

i = 10.0°

150

150

120

120

0

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90

φSun (deg)

0.5 e sin fI

0.5

30

90

60

60

30

30

φSun (deg)

e sin fI

0.5

e sin fI

“φSun ” (deg)

0

“φSun ” (deg)

0.5 e sin fI

e sin fI

0.5

i = 10.0°

−0.5

−1 −1

−0.5

0

0.5

e cos fI

−1 0 1 −1

0 −0.5

0

0.5

1

e cos fI

Figure 33: Hints of an underlying relation in the solar aspect angle at deflection between different types of solutions over the parameter space, as can be seen from its smooth evolution after transformation. This concerns the best transfers found numerically with unconstrained ϕSun (see Fig. 29, left). (Top) swapping daytime and nighttime for all π-like-transfer solutions; (middle) ϕπ-transfer → 180°−ϕπ-transfer . These can be further compared Sun Sun to the unaltered ϕSun found in the case of numerically optimised 2π-transfers (bottom). See text.




1

180 i = 5.0°

150

e sin fI

120 0

90 60

“φSun ” (deg)

0.5

−0.5 30 −1 1 −1

−0.5

0

0.5

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0 180

i = 5.0°

e cos fI

150

e sin fI

120 0

90 60

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0.5

−0.5 30 −1 1 −1

−0.5

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i = 5.0°

e cos fI

150 0.5

0

90

φSun (deg)

e sin fI

120

60 −0.5 30 0

−1 −1

−0.5

0

0.5

1

e cos fI

Figure 34: Same as Fig. 33; but where we now use the raw numerical results of the optimisation including phasing discussed in the preparatory draft [23]. See Fig. 42 for the actual ϕSun , before any swapping or transformation.




fI = 120°; e = 0.5; i = 5°

y (AU)

1

0

−1

−1

0

1

x (AU) Figure 35: Case of a daytime-impact asteroid orbit (brown) for which the optimal solution is a deflection in the vicinity of ⃗ropp ; the nighttime-impact equivalent is also shown (grey), which of course also favours ⃗ropp as not threshold on ϕSun is set in this example. We show the launch locations (on the Earth orbit; here, they almost exactly coincide) and deflection locations (on each asteroid orbit) corresponding to the best solution targeting ⃗rI (large circles) and ⃗ropp (small circles). Open circles correspond to the solutions found in the case of the daytimeimpact PHA; the closed ones, to the nighttime-impact one. Comparing the best transfer to ⃗rI in the nighttime case to the best transfer to ⃗ropp in the daytime case, we indeed see essentially the same geometry (though keep in mind that this is an ecliptic projection); except that in one case the target is ⃗rI and in the other, it is ⃗ropp , which is the closest to perihelion.




5 What have we learned? In this work, we considered at once the entire parameter space of Earth-impacting orbits and studied ballistic kinetic-impactor missions to deflect hazardous asteroids that might populate them. This allowed us to identify some of the main physical and geometrical drivers leading to the optimal deflection missions, while providing a preliminary-mission-design mapping for all the conceivable threat scenarios. Before we conclude, let us emphasise one last time that the optimal transfers presented here should not be assumed to hold if the time-to-impact is not long enough for non-secular effects on the PHA orbit or the phasing to be ignored. Indeed, under such conditions, rather than achieving the optimal deflection geometry, the need for a sufficient drifting time post-deflection will strongly drive what the optimal transfer would be, probably making this a very different problem overall. Similarly, special care should be taken if a close-encounter is known to take place before the foreseen impact, and the issue of keyholes should be carefully considered in any real scenario. Throughout, the general impact problem was thus studied by optimising what we called the B-plane yearly-drift ∆B, thereby highlighting the best feasible deflection geometry that one could actually hope to achieve with ballistic trajectories, given a specific launcher performance. Having a direct access to what the best drift rate could possibly be in units of b⊕ /yr is interesting. It can notably help quickly decide whether to proceed, with several spacecrafts if needed, or whether another strategy should rather be considered instead.34 Furthermore, and maybe most importantly, since it was already known and obvious that an early deflection is a strong driver, putting the emphasis on the geometry allowed us to access more readily some of the other drivers.35 Among the important questions to ask to determine what the optimal transfer might be are notably whether the PHA orbit corresponds to a daytime or a nighttime impact, whether its opposite node lies inside or outside of the Earth orbit, and whether it might be difficult to reach the opposite node for the considered launcher performance. Depending on the answers, the resulting optimal trajectories were indeed found to be qualitatively very different, especially when moreover ensuring a sufficiently large solar aspect angle at deflection. As expected, the new (fI , e, i) parametrisation for impacting orbits greatly helped when investigating this problem; the influence of these drivers appearing very distinctly in parameter space when using the polar representation. All this was shown by first analytically approaching the problem of deflecting the asteroids at either of their orbital nodes (including analytical optimisations), before moving on to a numerical optimisation problem relying on deflections at the nodes as initial guesses.36 For the analytical part, since they are the cheapest kinds of ballistic transfers to the nodes of any PHA orbit, we studied strict 2π-transfers and strict π-transfers in detail. In both cases, we first gave zeroth-order analytical assessments for ∆B as well as analytical formulae for the corresponding solar aspect angle at deflection ϕSun . With the cheapest strict 2π-transfers, we showed that the vast majority of PHAs would be naturally decelerated along-track, though there is also a potentially challenging subset of Atens which would impact our planet when they are close to their aphelion and which would instead slightly favour along-track accelerations. We also later gave the locus that defines the boundary between these two regions in parameter space as a function of the inclination. With the cheapest strict 34 Knowing what the optimal solution without phasing is may moreover give the mission analyst something to aim at when designing an actual trajectory, even under the constraint of a given warning time. 35 While also knowing that, as the available ∆T increases, the optimal missions with phasing will necessarily tend to the corresponding missions described here: lim∆T ↗ Optimal missions(M) = Optimal missions(M). 36 The nodes are quite reliable locations in general, especially since we did not only focus on what might be the most likely type of Earth-impacting orbits but all of them, including asteroid orbits with rather large inclinations.



ESA UNCLASSIFIED – For Official Use π-transfers, on the other hand, we saw that all PHAs would be naturally decelerated along-track. For these, there is also the question of transfer feasibility, given the added difficulty of reaching the opposite node. Working on these simple assessments, it also appeared that if it was only for the solar aspect angle at deflection, the best would quite clearly be for the spacecraft to hit as it is crossing the asteroid orbit from the inside to the outside. In other words, transfers to the vicinity of ⃗ropp (resp. ⃗rI ) would then be preferable to prevent daytime (resp. nighttime) impacts. Similarly, if it was only for ∆B, the best would rather obviously tend to be a deflection in the vicinity of the closest node to perihelion, except that in reality ⃗ropp will eventually get harder and harder to reach at some point, eventually losing its appeal in favour of ⃗rI . We then moved on and optimised both kinds of strict transfers. For strict 2π-transfers (⃗rL = ⃗rI ; ⃗rD = ⃗rI ), we solved this optimisation problem analytically, giving accurate formulae for ∆B, and updated the one for ϕSun accordingly. As could be seen there clearly, for a given asteroid orbit, going against the natural tendency associated to a type of transfers (either naturally accelerating or decelerating along-track) necessarily leads to much poorer results. For the analytical optimisation of strict π-transfers (⃗rL = ⃗rI ; ⃗rD = ⃗ropp ), we exploited the fact that a ballistic spacecraft orbit departing at ⃗rI is also technically an Earth-impacting orbit, and can therefore also be similarly parametrised by means of its true anomaly at launch, its eccentricity and its inclination. With this, we could then provide a similar discussion, although this time we only solved for ∆B under the reasonable, though approximate, assumption of ecliptic transfer orbits to ⃗ropp . While the analytical solutions provided in this work were obtained under assumptions that might be too restrictive (though they become more accurate at larger PHA-orbit inclination), these at least partially optimised trajectories are feasible in terms of launcher performance and can be called for any parameter or launcher. Such results are moreover extremely fast: these restricted solutions to a complex problem are obtained via simple function calls. They were however not optimised in a way which would allow for taking into account an additional inequality constraint on the solar aspect angle; if a threshold is set on ϕSun , one type of solution might become unfeasible. In such a case, the analytical solutions are then mostly useful qualitatively: providing a rule of thumb, and telling what may be the best location to target. For the general numerical optimisation procedure, in addition to the strict 2π- and π-transfers, we also introduced new distinct types of initial guesses, also targeting the nodes. The optimal solutions with unconstrained ⃗rL and ⃗rD obtained numerically from all these initial guesses were then discussed based on the targeted node, which is what matters the most, qualitatively speaking (e.g. when talking about mission types). We then discussed some of the main numerical results obtained, including how the optimal solution changes when constraining the solar aspect angle at deflection, greatly helped by the intuition gained during to our analytical investigations. Let us simply recall that, let alone the more pathological regions for which another approach might be best suited, one can say that along-track decelerations in the broad vicinity of ⃗rI are essentially the best for almost all asteroids when the illumination conditions at deflection are not yet taken into account, in particular for all asteroids with an opposite node outside of the Earth orbit. One should however not overlook that there is also a strip in parameter space for which the optimal solutions would then actually be along-track decelerations with π-like transfers, as can be implied from the values taken by ϕSun . Far from being second-class transfers which would only appear when the illumination or the timing would be unfavourable,37 they do lead to a |∆B| larger than 2π-transfers can provide in that part of parameter space. In the results of the numerical optimisation, 37 This is particularly clear when such solutions are found for nighttime PHA orbits when ϕSun is constrained: they are selected even though the illumination conditions tend to be more difficult for these transfers.



ESA UNCLASSIFIED – For Official Use this region starts as soon as ⃗ropp is closer to perihelion than ⃗rI is (i.e. the transition sharply happens where e cos fI becomes negative: in the second and third quadrants). Since these kinds of transfers get costlier and costlier as ⃗ropp is moved closer to the Sun, it is understandable that the more flexible 2π-transfers become a better option at some point, since one can then preserve a larger kinetic impactor mass at deflection. As we then saw, once taking ϕSun into account, the region covered by along-track decelerations with transfers targeting ⃗rI largely extends for nighttime impacts, while the opposite is true for transfers to ⃗ropp and daytime impacts. Finally, to understand even better the actual best numerical transfers, we took a closer look at how the numerically optimised deflection locations depart from the initial guesses, and further found a number of very interesting and striking approximate symmetries in the optimal numerical results for PHAs with low inclination (≲ 10°). Further investigation of these is left for future work. We would suggest starting with the general study of ballistic deflection missions, with i strictly vanishing.38 The aim should be to determine analytically in two dimensions what the best conceivable transfers are when no threshold is set on the solar aspect angle. Of course, overall, it will not be very different from the general results given here, but the point is rather to precisely determine the absolute best ⃗rL , ⃗rD and ⃗v∞SC,L in terms of launcher performance, and then study the properties of the orbit, including the solar aspect angle at deflection. The presence of these very simple symmetries indeed gives the hope that such an analytical derivation of the best ballistic transfers found numerically at low inclination might actually be achieved in very good approximation. For the time being, the results presented here could also be used as more refined initial guesses (that is, feasible in terms of launcher performance and solar aspect angle) for subsequent more accurate numerical modelling. In comparison, “targeting the perihelion” no matter the asteroid orbit seems far from satisfactory, as it is too simplistic and ignores completely any feasibility constraint coming from the necessarily limited launcher performance. That what is optimal would be to induce there a ∆⃗vA alongtrack or opposite-track is true only if we could select the optimal trajectory without acknowledging the cost associated to it. It simply cannot be assumed to be true in general, as feasibility can get in the way. As a matter of fact, it is interesting to notice that we essentially never find that the optimal location for a ballistic deflection mission which would lead to the maximal yearly-drift is at the perihelion. Of course, attempting to deflect close to perihelion still drives the optimal solution to some extent though: in particular regarding when determining in the broad vicinity of which node it would be preferable to deflect, when ignoring the illumination conditions at deflection.

Acknowledgements The work reported here was done in the context of a research fellowship in the Mission Analysis section at the European Space Operations Centre (ESOC). It is with great pleasure that we thank Michael Khan for a number of interesting discussions, comments, and inputs. Our thanks also go to Fabian Bach, who shared the preparatory raw numerical data that he generated at ESOC, and allowed for their correction, transformation, and use for comparison purposes with the analytical and numerical results obtained in this work. This research has made use of data and/or services provided by the International Astronomical Union’s Minor Planet Center.

38

Once again, note that with the (fI , e, i) parametrisation, one can take the limit i → 0, there is no singularity: the impact problem is independent of whether the impact happens at the ascending or descending node, only the location on the PHA orbit matters.




A

Reminder: orbital energy & period variations due to a velocity change ∆⃗vA

For completeness, recall that if at a given deflection point D on its orbit, Ekin = 21 |⃗vA,D |2 is the specific kinetic energy of the unperturbed asteroid, and ∆⃗vA is the tiny impulsive velocity change which will be induced there in the asteroid motion by means of a mitigation mission, ⃗vA,D → ⃗vA,D + ∆⃗vA ,

(59)

the corresponding variation in the asteroid specific mechanical energy, E = Ekin + Epot , 1 ∆E(⃗vA,D , ∆⃗vA ) = ⃗vA,D · ∆⃗vA + |∆⃗vA |2 2

(60)

trivially follows from 1 Ekin → Ekin + ∆Ekin = |⃗vA,D + ∆⃗vA |2 . (61) 2 As well-known, Eq. (60) shows clearly how the gain in specific kinetic energy ∆Ekin actually depends on the original asteroid velocity at the chosen deflection point ⃗vA,D , both in terms of direction and norm. Simply assuming that the velocities would be collinear, one would of course get 1 ∆E(vA,D , ∆vA ) = vA,D ∆vA + ∆vA 2 . 2

(62)

It is then clear why deflecting the asteroid at a location where its heliocentric velocity is large (the largest being at its perihelion) may actually be expected to translate into large gains. Notice however that the induced velocity change |∆⃗vA | is not independent but instead strongly depends of the deflection location, and also that any given orientation for this ∆⃗vA is not equally feasible everywhere on the PHA orbit with a given type of trajectories, such that a location with smaller |⃗vA,D | might actually be more favourable in reality. This change in orbital energy (60) can of course be translated into an induced variation in the asteroid orbital period P around the Sun, the leading term being ∆P ≈

3a P (⃗vA,D · ∆⃗vA ) , µSun

(63)

following for instance simply √ ∆P ≈ 3π

a µSun

∆a

and

∆a ≈

2a2 ∆E, µSun

(64)

as can be obtained from first-order Taylor expansions, using the textbook relations 2π P =√ a3/2 µSun


and

E =−

µSun . 2a

(65)



B

Useful relations and PHA heliocentric velocities at the nodes

For convenience, we provide some results derived in Ref. [1]. Setting ourselves in the same ecliptic frame,39 in all generality the position of any impacting PHA, anywhere on its orbit, can then in fact be written in a very compact way: [ ] ⃗r(f ) = r cos(f − fI ) ⃗ex + r sin(f − fI ) cos i ⃗ey ± sin i ⃗ez , (66) where r = r(a(fI , e), e, f ) is given by the conic equation, and where the positive (resp. negative) sign along z corresponds to an impact at the ascending (resp. descending) node of the PHA orbit —unimportant for the relative geometry and motions, as discussed earlier. From there, one can express ∂ ∂ ⃗r and ⃗ef (f ) = 1r ∂f ⃗r: the polar unit vectors ⃗er (f ) = ∂r [ ] ⃗er (f ) = cos(f − fI )⃗ex + sin(f − fI ) cos i ⃗ey ± sin i ⃗ez [ ] and ⃗ef (f ) = cos(f − fI ) cos i ⃗ey ± sin i ⃗ez − sin(f − fI )⃗ex .

(67)

Finally, from this Eq. (67) and the general expression for the velocity in an orbital frame (⃗er , ⃗ef ), √ √ µ µ e sin f ⃗er + (1 + e cos f ) ⃗ef (68) ⃗v (f ) = p p trivially follow what turn out to be the very compact expressions in the ecliptic frame (⃗ex , ⃗ey , ⃗ez ) for the PHA heliocentric velocity at the impact location ⃗vA,I = vA,I,x ⃗ex + vA,I,y ⃗ey + vA,I,z ⃗ez , and for the one at the opposite node ⃗vA,opp = vA,opp,x ⃗ex + vA,opp,y ⃗ey + vA,opp,z ⃗ez : √ e sin(fI ) µSun √ vA,I,x = = vA,opp,x (69) 1 AU 1 + e cos(fI ) √ √ µSun √ µSun 1 − e cos(fI ) √ vA,I,y = 1 + e cos(fI ) cos(i) and vA,opp,y = − cos(i) (70) 1 AU 1 AU 1 + e cos(fI ) √ √ µSun √ µSun 1 − e cos(fI ) √ vA,I,z = ± 1 + e cos(fI ) sin(i) and vA,opp,z = ∓ sin(i), (71) 1 AU 1 AU 1 + e cos(fI ) where µSun stands for the gravitational parameter of the Sun. This is always true. For studying missions with a deflection taking place at either node of any general PHA orbit analytically, such relations turn out to be particularly useful. The reader is referred to Ref. [1] for more relations, discussions and details.

39 That frame is such that the impact location ⃗rI is along the x-axis, the Earth velocity at the impact point is along the y-axis, and the angular momentum of the Earth is along the z-axis. See also Figs. 36 and 37.




1.5

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1

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Figure 36: Ecliptic projections of 2 orbits described by the same period P , eccentricity e, and inclination i: one corresponds to a daytime impact, the other to a nighttime impact. The circular Earth orbit is shown in blue. Empty (resp. full) squares indicate the perihelion (resp. aphelion).

0.4

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x (AU) 1.5

−1.5

Figure 37: Same as Fig. 36 in 3D, explicitly assuming for the sake of this example only that ⃗rI corresponds to the ascending node of these two PHA orbits.




C Linking difficulties in the daytime region to π-transfer feasibility in (P, e) From geometry alone, it is already clear that potential technical difficulties for reaching ⃗ropp can be expected to complicate the design of ballistic mitigation missions in some daytime-impact cases, and may actually even imply that none of our initial guesses lead to a feasible solution.40 Focusing on this issue, the appendix of Ref. [1] gave an analytical feasibility assessment of ballistic transfers to the opposite node of any Earth-impacting orbit, taking advantage of the new parametrisation. This was done by simply comparing the opposite-node location of each orbit: |⃗ropp | =

1 + e cos fI |⃗rI |, 1 − e cos fI

(72)

(SC)

to the smallest perihelion (|⃗ropp | < qmin ): (SC)

qmin =

1− AU, 1 + e(SC) e(SC)

with

e(SC) = 1 −

(√ )2 µSun 1 AU − max(|⃗ v |) ∞SC,L 1 AU µSun

(73)

(SC)

and the largest aphelion (|⃗ropp | > Qmax ): (SC)

Qmax =

e(SC)

1+ AU, 1 − e(SC)

with

e(SC) = −1 +

1 AU

(√

µSun 1 AU

)2 + max(|⃗v∞SC,L |) µSun

(74)

which would be reachable for a given maximal hyperbolic excess velocity at launch: max(|⃗v∞SC,L |). From this were then derived two relations, respectively ( ) (SC) 1 − q /AU 1 (SC) min (75) |⃗ropp | < qmin ⇒ e(PHA) > (SC) 1 + qmin /AU − cos fI (

and (SC)

|⃗ropp | > Qmax

⇒

e(PHA) >

(SC)

Qmax /AU − 1 (SC) Qmax /AU

+1

)

1 , cos fI

(76)

which identify extended regions of parameter space for which there exists no feasible ballistic transfer to the opposite node. Their usefulness also stems from the fact that we have derived the analytical expressions defining these regions, meaning that they can be used for any other max(|⃗v∞SC,L |) corresponding to the launcher that one would like to consider. Just for completeness, let us now determine to which parts of a (P, e) plot the regions of parameters for which a π-transfer cannot be performed due to the launcher performance correspond to. From what precedes, we can anticipate that this can help understand where complications (or even no solution) might be found. Expressed in terms of the period, Eq. (75)—which identifies the orbits for which a π-transfer is not possible because ⃗ropp is too far away in the inner Solar System—gives v (SC) u K −1 1 − qmin /AU u + 1, with K = , (77) e(PHA) > t 2 (SC) (PHA) 3 1 + q /AU min Pyr 40

For PHAs on orbits with sizeable inclination in particular this might be a real concern; cf. Ref. [1].



ESA UNCLASSIFIED – For Official Use while, for orbits with ⃗ropp being too far away in the outer Solar System, Eq. (76) leads to v (SC) u C +1 Qmax /AU − 1 u (PHA) e < t1 − with C = (SC) ; 2 , (PHA) 3 Qmax /AU + 1 Pyr

(78)

beware that the inequality sign changes from one case to the other.41 We can now put all these together in Fig. 38. For each of the two regions shown on this plot, the point of smallest eccentricity on the boundary is indicated by an arrow: the point marking the boundary on the left is given by e = e(SC) of 3 Eq. (73) and P = (1 + e)− 2 ; the one on the right is given by e = e(SC) of Eq. (74), with a period now 3 given by (1 − e)− 2 . Quite clearly, the new parametrisation is again better suited for this kind of assessments, as the boundaries are then very simply given by vertical lines in the polar-plot representation.

41

We exploited the fact that cos fI is always negative in the first case and always positive in the second case [1].




1 ropp | = 1 AU |~

0.8

e

0.6

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5

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9

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P (yr) Figure 38: Showing the π-transfer feasibility boundaries in (P, e), considering max(|⃗v∞SC,L |) = 10 km/s (top). We see in red the two regions for which no ballistic transfer to the opposite node is possible: the one appearing most prominent is when |⃗ropp | < 1 AU. Note that the other region that appears here (|⃗ropp | > 1 AU) is less problematic though, as shall be seen when we present our deflection results, since even for daytime impacts we find that the preferred solution for the corresponding PHAs is a 2π-transfer even when we constrain ϕSun ≥ 60°, at least when assuming such a modern launcher. (Bottom) other example, with max(|⃗v∞SC,L |) = 8 km/s.




D Optimised strict 2π-transfers: solutions assuming a cubic polynomial fit In the case of a cubic polynomial (i.e. c3 ̸= 0 and cj = 0 for j ≥ 4), from Eq. (25), we can obtain C0 + C1 x + C2 x2 + C3 x3 = 0,

writing Cl ≡ ±(1 + l)cl + (1 + l)cl+1 F,

(79)

so that, to determine the norm x = |⃗v∞SC,L | that optimises the yearly drift, one should solve x3 + a2 x2 + a1 x + a0 = 0,

(80)

±3c2 + 3c3 F C2 = C3 ±4c3 C1 ±2c1 + 2c2 F a1 = = C3 ±4c3 C0 ±c0 + c1 F a0 = = . C3 ±4c3

(81)

with a2 =

This can be solved analytically. For completeness, we quote the main quantities required to do so, and refer the reader to Ref. [37] for a derivation. Following this reference, one first defines p≡

3a1 − a22 3

and

and can then calculate Det(p, q) ≡

q≡

9a1 a2 − 27a0 − 2a32 27

( p )3

+

(82)

( q )2

. 3 2 If Det(p, q) ≤ 0, there are 3 real solutions (with some degeneracy if Det(p, q) = 0). Defining   q/2 , θ ≡ acos  √ 3 −(p/3) the solutions of Eq. (79), i.e. those that optimise M as given in Eq. (23), are then42 √ ( ) p θ a2 x1 = 2 − cos − 3 3 3 √ ( ) p θ + 2π a2 x2 = 2 − cos − 3 3 3 √ ( ) p θ + 4π a2 x3 = 2 − cos − . 3 3 3

(83)

(84)

(85)

We take the smallest meaningful one. Assuming a fit for the SLS launcher for instance, there tends to be one negative solution, one larger than xmax ∼ 10 km/s, and an acceptable one. Else, as again shown in Ref. [37], there is only one real solution, first define ( )3 q 3 2 C(p, q) ≡ . 2 |p| 42

(86)

There seems to be a typo in Ref. [37], in their Eq. (84): one should read zi , given their notation, not xi .



ESA UNCLASSIFIED – For Official Use If Det(p, q) > 0 and C ≤ −1, then √ x=2

|p| 3

( ( )) 1 a2 −1 − cosh cosh (|C|) − . 3 3

(87)

Lastly, if Det(p, q) > 0 and C ≥ 1, √ x=2

E

|p| cosh 3

(

) 1 a2 −1 cosh (C) − . 3 3

(88)

Ballistic spacecraft orbits seen as Earth-crossing: further relations

Here, we parametrise the ballistic spacecraft orbits by means of the parametrisation introduced in the context of this work [1], pursuing further the idea discussed in Sec. 3.2.2. We still explicitly consider only ⃗rL = ⃗rI (which we for instance have with strict 2π- and π-transfers), because we want to make clear that more information is then easily gained when compared to PHA orbits. Assuming a launch at ⃗rI , one can for instance show that, for a( given max(|⃗v∞SC,L ) |) which corre(SC)

sponds to some launcher, all the feasible ballistic spacecraft orbits fI

(SC)

, eI

, i(SC) must verify

[( )2 (√ )2 ] ( ( )2 ) max(|⃗ v |) ∞SC,L (SC) (SC) (SC) e(SC) sin fI ≤ − 1 + e(SC) cos fI −1 1 + e(SC) cos fI ; (89) vE (SC)

note that the whole corresponding range of e(SC) sin fI values given in this equation is however actually only allowed if the transfer is moreover assumed to take place in the ecliptic plane: i(SC) = 0. The corresponding range is otherwise restricted even further with growing i(SC) . Figure 39 uses that result: it shows all the spacecraft orbits in the ecliptic plane which a launcher like the SLS Block 1B could populate ballistically (assuming that max(|⃗v∞SC,L |) = 10 km/s). Note that any allowed ballistic spacecraft orbit with ⃗rL = ⃗rI necessarily has • 2 intersections (both at ⃗rI and at ⃗ropp ) with any PHA orbit for which (PHA)

e(PHA) cos fI

(SC)

= e(SC) cos fI

;

(40)

• 1 intersection (at ⃗rI ) with any other PHA orbit. Having this result also provide information about the range of vSC,I,x that are actually feasible with (SC) the same launcher, for any given e(SC) cos fI —which is fixed for a strict π-transfer. For instance, (SC) assuming i = 0°, the full range of allowed values is then given by √( √ )2 ) (√ max(|⃗v∞SC,L |) 2 µSun (SC) (SC) |v∞SC,L,x | ≤ 1+e cos fI −1 . (90) − AU vE This result is used to get Fig. 40. Again, note that the region in this figure would of course shrink, should an out-of-plane velocity component be allowed (i(SC) ̸= 0°).




1.00

0.67

e(SC) sin fI

(SC)

0.33

0.00

−0.33

−0.67

−1.00 −1.00 −0.67 −0.33 e

(SC)

0.00

0.33

0.67

1.00

(SC) cos fI

Figure 39: Spacecraft orbits with i(SC) = 0 that can be populated ballistically if max(|⃗v∞SC,L |) = 10 km/s. Understandably, if part of ⃗v∞SC,L was used to give an out-of-plane component to the spacecraft heliocentric velocity (i(SC) ̸= 0), the allowed region in this figure would then shrink.

10 8 6 vSC,I,x (km/s)

4 2 0 −2 −4 −6 −8 −10 −1 −0.8 −0.6 −0.4 −0.2 e

(SC)

0

0.2

0.4

0.6

0.8

1

(SC) cos fI

Figure 40: Values for the component of the spacecraft heliocentric velocity v∞SC,L,x at launch ⃗rL = ⃗rI which are (SC) feasible with ballistic transfers in the ecliptic plane. These are shown for all the possible values of e(SC) cos fI (i.e. which correspond to all the locations ⃗ropp that are reachable with the same launcher).




F

Comparisons with some results including the phasing

After correcting the sign and making sure that the focus effect due to the gravity of the Earth is taken into account, the raw numerical results with phasing obtained separately in Ref. [23] for daytime- and nighttime-impact asteroids can then be transformed into our new parametrisation and shown together on a single plot. In the new parametrisation, one does not have to worry about whether the impact is taking place at the ascending or descending node of the asteroid orbit. As the PHA period was restricted to be below 6 years, they only give a partial coverage but which corresponds to an important part of the parameter space. As for our own results, for the purpose of comparison, we take: ∆B = ∆B × ∆T,

with ∆T = 20 yr,

(91)

as we already did in Ref. [38] when showing the cases seen in Fig. 41, which compares results with and without phasing; the results of a numerical optimisation without phasing with ⃗rL and ⃗rD restricted at the nodes are also shown in that figure. As expected, the general results without phasing give the best |∆B| that could be ballistically achieved, while when strictly restricted at the nodes, the results are more conservative. As a side note, notice that in the example shown in right panel no solution taking into account the phasing could actually be found in the vicinity of fI ∼ 140°, as shown in red (cf. Sec. 4.1); the same is true for the numerical optimisation restricted at the nodes (i.e. none of the initial guesses correspond to a feasible solution). Additionally, we show Fig. 42, where are compared the results with and without phasing (unconstrained ϕSun case). In addition, we show the results of the analytically optimised strict 2π-transfers derived in Sec 3.1.2 which, despite their simplicity are quite informative, even at such a low PHA-orbit inclination (i = 5°). 2.5

28 no phasing (general) no phasing (nodes) phasing

no phasing (general) no phasing (nodes) phasing

24

2

1.5

|∆B| (b⊕ )

|∆B| (b⊕ )

20

1

16 12 8

0.5

Here: i = 5° e = 0.5

Here: i = 22.5° e = 0.9

4

0

0 0

45

90

135

180

225

270

315

360

fI (deg)

0

45

90

135

180

225

270

315

360

fI (deg)

Figure 41: Two example comparisons (with the explicit constraint ϕSun ≥ 45°).



≥ 10

1

i = 5.0°

8

−0.5

0

−1 −2 1 −1 180

0.5

−0.5

0

0.5

i = 5.0°

e cos fI

e sin fI

φSun (deg)

e sin fI

e sin fI

90

30


−1

−0.5

0 e cos fI

0.5

1 i = 5.0°

150

150

120

120

0

90

90

60

60

30

30

−0.5

−0.5

−1

0.5

−2 180

0.5

60 −0.5

0 e cos fI

120 0

2 0

−0.1 −1 1 −1 180 −0.5

0.5

0

4

0

150 0.5

0.2 0.1

i = 5.0°

e cos fI

6

−0.5 0

−1 1 −1

0.3

∆B (b⊕ )

0

2 −0.5

−0.5

8

∆B (b⊕ /yr)

4

e sin fI

0

∆B (b⊕ )

e sin fI

e sin fI

6 0

0.4

0.5

0.5

φSun (deg)

0.5

≥ 10

≥ 0.5

i = 5.0°

φSun (deg)

1 i = 5.0°

−1 0 1 −1

−0.5

0 e cos fI

0.5

−1 0 1 −1

0 −0.5

0

0.5

1

e cos fI

Figure 42: Comparisons for i = 5° (unconstrained ϕSun ). (Left) preparatory results with phasing [23], corrected; (center) optimal numerical solutions without phasing; (right) using the analytical optimisation formulae for strict 2π-transfers (quantitatively more accurate, the larger the PHA-orbit inclination).



1


G Optimal π-transfers can subsist at higher i even for nighttime-impact PHAs 1

1

≥ 0.5

≥ 0.5 i = 22.5°

i = 22.5° 0.25

0.25

0.5

0.4

0.4

0.5

0.1 0.5

0.1

0.01

0

0.2

0.01

0.1

0

0.25

0.5

0 e cos fI

−1 −0.1 1 −1 180 −0.5

0.5

i = 22.5°

0 e cos fI

0.5

1

−0.1 180

i = 22.5°

150

150

120

120

−0.5

0

90

90

60

60

30

30

φSun (deg)

0

φSun (deg)

0.5 e sin fI

0.5 e sin fI

0

0.25

0.5

−0.5

0.1

−0.5

0.1

−1 1 −1

0.2

−0.01

0.1 −0.5

∆B (b⊕ /yr)

e sin fI

e sin fI

−0.01

0

0.3

∆B (b⊕ /yr)

0.5

0.3

−0.5

−1 −1

−0.5

0

0.5

e cos fI

−1 0 1 −1

0 −0.5

0

0.5

1

e cos fI

Figure 43: Equivalent of Fig. 29, only at higher inclination. As clearly seen in the bottom right figure, even for nighttime impact asteroids there actually are π-transfers which remain the absolute best solution (notably better than 2π-transfers) despite a strongly constrained ϕSun —here required to be above 60°.




References [1] A. Payez, A simple and effective parametrisation for Earth-impacting orbits, Technical report, European Space Agency—Mission Analysis Section, ESOC (2017). [arXiv:1712.08436]. [2] National Research Council, Defending Planet Earth: Near-Earth Object Surveys and Hazard Mitigation Strategies (Washington, D.C. : The National Academies Press, 2010). [3] A. F. Cheng, P. Michel, M. Jutzi, A. S. Rivkin, et al., “Asteroid Impact & Deflection Assessment mission: kinetic impactor”, Planetary and Space Science 121 (2016) 27–35. [4] E. Asphaug, S. J. Ostro, R. S. Hudson, D. J. Scheeres, et al., “Disruption of kilometre-sized asteroids by energetic collisions”, Nature 393 (1998) 437–440. [5] M. Bruck Syal, J. M. Owen, and P. L. Miller, “Deflection by kinetic impact: sensitivity to asteroid properties”, Icarus 269 (2016) 50–61. [6] A. Carusi, “Early NEO deflections: a viable, lower-energy option”, Earth, Moon, and Planets 96 (2005) 81–94. [7] A. F. Cheng, J. Atchison, B. Kantsiper, A. S. Rivkin, et al., “Asteroid Impact & Deflection Assessment mission”, Acta Astronautica 226 (2015) 262–269. [8] K.-H. Glassmeier, H. Boehnhardt, D. Koschny, E. Kührt, et al., “The Rosetta mission: Flying towards the origin of the Solar System”, Space Science Reviews 128 (2007) 1–21. [9] M. F. A’Hearn, M. J. S. Belton, W. A. Delamere, J. Kissel, et al., “Deep Impact: excavating Comet Tempel 1”, Science 310 (2005) 258–264. [10] B. A. Conway, “Optimal low-thrust interception of Earth-crossing asteroids”, Journal of Guidance, Control, and Dynamics 20(5) (1997) 995–1002. [11] B. A. Conway, “Near-optimal deflection of Earth-approaching asteroids”, Journal of Guidance, Control, and Dynamics 24(5) (2001) 1035–1037. [12] S.-Y. Park and I. M. Ross, “Two-body optimization for deflecting Earth-crossing asteroids”, Journal of Guidance, Control, and Dynamics 22(3) (1999) 415–420. [13] I. M. Ross, S.-Y. Park, and S. D. Porter, “Gravitational effects of Earth in optimizing ∆V for deflecting Earth-crossing asteroids”, Journal of Spacecraft and Rockets 38(5) (2001) 759–764. [14] S.-Y. Park and D. D. Mazanek, Mission functionality for deflecting Earth-crossing asteroids/comets, Technical report 20050186572, NASA (2005). [15] A. Carusi, G. B. Valsecchi, G. D’Abramo, and A. Boattini, “Deflecting NEOs in route of collision with the Earth”, Icarus 159 (2002) 417–422. [16] R. Kahle, G. Hahn, and E. Kührt, “Optimal deflection of NEOs en route of collision with the Earth”, Icarus 182 (2006) 482–488. [17] A. Carusi, G. D’Abramo, and G. B. Valsecchi, “Orbital and mission planning constraints for the deflection of NEOs impacting on Earth”, Icarus 194 (2008) 450–462. See also references therein. [18] E. J. Öpik, “Interplanetary Encounters—Close-Range Gravitational Interactions”, in Z. Kopal and A. Cameron (editors), Developments in Solar System- and Space Science 2 (Amsterdam : Elsevier, 1976). [19] D. Izzo, “Optimization of interplanetary trajectories for impulsive and continuous asteroid deflection”, Journal of Guidance, Control, and Dynamics 30(2) (2007) 401–408. [20] M. Vasile and C. Colombo, “Optimal impact strategies for asteroid deflection”, Journal of Guidance, Control, and Dynamics 31(4) (2008) 858–872 [arXiv:1104.4670].



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