The Manifold Of Variations: impact location of short-term impactors
TTHE MANIFOLD OF VARIATIONS:IMPACT LOCATION OF SHORT-TERM IMPACTORS
ALESSIO DEL VIGNA, LINDA DIMARE, AND DAVIDE BRACALI CIOCI
Abstract.
The interest in the problem of small asteroids observed shortly before a deep close approach oran impact with the Earth has grown a lot in recent years. Since the observational dataset of such objectsis very limited, they deserve dedicated orbit determination and hazard assessment methods. The currentlyavailable systems are based on the systematic ranging, a technique providing a 2-dimensional manifold oforbits compatible with the observations, the so-called Manifold Of Variations. In this paper we first reviewthe Manifold Of Variations method, to then show how this set of virtual asteroids can be used to predict theimpact location of short-term impactors, and compare the results with those of already existent methods. Introduction
Astronomers observe the sky every night to search for new asteroids or for already known objects. TheMinor Planet Center (MPC) collects the observations coming from all over the world and then tries tocompute orbits and to determine the nature of the observed objects. New discoveries which could beNear-Earth Asteroids (NEAs) are posted in the NEO Confirmation Page (NEOCP ), which thus containsobservational data of unconfirmed objects. They could be real asteroids as well as non-real objects, andcannot be officially designated until additional observations are enough to compute a reliable orbit andconfirm the discovery.Some asteroids with an Earth-crossing orbit may impact our planet and the goal of impact monitoring isto identify potentially hazardous cases and solicit follow-up observations. Two automated and independentsystems continually scan the catalogue of known NEAs with this purpose, namely clomon -2 and Sentry ,which are respectively operational at the University of Pisa/SpaceDyS and at the NASA JPL. What thesetwo systems do not do is to scan the objects waiting for confirmation in the NEOCP, but some of themcould be on a collision trajectory with the Earth, with the impact occurring just after a few hours fromthe discovery. This is exactly what happened for asteroids 2008 TC , 2014 AA, 2018 LA, and 2019 MO,all discovered less than one day prior to striking the Earth and prior to being officially designated by theMPC. Thus being able to perform a reliable hazard assessment also in these cases is a fundamental issue,but needs dedicated techniques due to the very different nature of the problem. Indeed when an asteroid isfirst observed, the available astrometric observations are few and cover a short time interval. This amountof information is usually not enough to allow the determination of a well-constrained orbit and in fact theorbit determination process shows some kind of degeneracies. The few observations constrain the positionand motion of the object on the celestial sphere, but leave practically unknown the topocentric range andrange-rate. As a consequence the set of orbits compatible with the observations forms a region in the orbitalelements space which has a two-dimensional structure, so that every one-dimensional representation of theregion such as the Line Of Variations (LOV, Milani et al. (2005b)) would be unreliable.Two systems are now publicly operational and dedicated to the orbit determination and hazard assessmentof unconfirmed objects: Scout at the NASA JPL (Farnocchia et al., 2015) and NEOScan at SpaceDyS (Spotoet al., 2018). They are both based on the systematic ranging, an orbit determination method which exploresa subset of admissible values for the range and range-rate. NEOScan makes use of the Admissible Region https://minorplanetcenter.net/iau/NEO/toconfirm_tabular.html https://newton.spacedys.com/neodys/index.php?pc=0 http://cneos.jpl.nasa.gov/sentry/ https://newton.spacedys.com/neodys2/NEOScan/ a r X i v : . [ a s t r o - ph . E P ] F e b ALESSIO DEL VIGNA, LINDA DIMARE, AND DAVIDE BRACALI CIOCI (AR, Milani et al. (2004)) as a starting point to explore the range and range-rate space. Then the shortarc orbit determination process ends with the computation of the Manifold Of Variations (MOV, Tommei(2006)), a 2-dimensional compact manifold of orbits parameterized over the AR. A finite sampling of theMOV is thus a suitable representation of the confidence region, because it accounts for its two-dimensionalstructure, and thus the resulting set of virtual asteroids can be used for the short-term hazard assessment ofsuch objects (Spoto et al., 2018; Del Vigna, 2018, 2020). A part of this activity is the prediction of the impactlocation of a potential impactor, especially when the associated impact probability is high. A method topredict the impact corridor of an asteroid has been developed in Dimare et al. (2020), by which the impactregion is given by semilinear boundaries on the impact surface at a given altitude above the Earth andcorresponding to different confidence levels. The algorithm is conceived to be a continuation of the impactmonitoring algorithm at the basis of the clomon -2 system, since the semilinear method requires a nominalorbit obtained by full differential corrections and an impacting orbit , as provided by the LOV method(Milani et al., 2005a). When the observational arc is very short and the object is waiting for confirmation inthe NEOCP, the semilinear method could become inapplicable because very often a reliable nominal orbitsimply does not exist. In this paper we propose a method which uses the MOV to predict the impact locationof an imminent impactor, and we then test this technique using the data available for the four impactedasteroids so far, namely 2008TC , 2014AA, 2018LA, and 2019MO.In Section 2 we give a brief recap about the AR and the MOV method. In Section 3 we introduce theimpact surface at a given height over the Earth and the impact map, to then describe how to exploit theMOV sampling for impact location predictions. Section 4 contains the results of our method applied tothe impacted asteroids 2008 TC , 2014 AA, 2018 LA, and 2019 MO. When possible, we also compare theimpact regions with those computed with the semilinear method (Dimare et al., 2020) and with a MonteCarlo simulation. Lastly, Section 5 is dedicated to the conclusions.2. The Manifold Of Variations method
Suppose we have a short arc of observations, typically 3 to 5 observations over a time span shorter thanone hour. In most cases the arc is too short to allow the determination of a full orbit and it is called aToo Short Arc (TSA, Milani et al. (2007)). When in presence of a TSA, either preliminary orbit methodsfail or the differential corrections do not converge to a nominal orbit. Nevertheless, as anticipated in theintroduction, the little information contained in the arc can be summarised in the attributable , the vector A := ( α, δ, ˙ α, ˙ δ ) ∈ S × (cid:0) − π , π (cid:1) × R formed by the angular position and velocity of the object at the mean observational time. Note that if thereis at least one measurement of the apparent magnitude, the attributable could also contain an average valuefor this quantity. The topocentric range ρ and range-rate ˙ ρ are thus not known, otherwise we would have hada full orbit. The AR comes into play here, to provide a set of possible values of ( ρ, ˙ ρ ) by imposing generalconditions on the object’s orbit. It can be shown that the AR is a compact subset of R ≥ × R , which canhave at most two connected components. For the precise definition of the AR and the proof of its propertiesthe reader can refer to Milani et al. (2004), here we limit ourself to the general idea. We essentially imposethat the object is a Solar System body and that it is sufficiently large to be source of meteorites. Indeedshort-term impactors are usually very small asteroids, with a few meters in diameter, thus the main interestin such objects is not for planetary defence, but rather to observe them during the last part of their impacttrajectory and possibly to recover meteorites on ground, as it happened for asteroid 2008 TC .The AR is sampled by a finite number of points and with different techniques. In case of a TSA weexplore the whole AR with rectangular grids: first we cover the entire region and compute the correspondingsample of orbits, then we identify the subregion corresponding to the best-fitting orbits with respect tothe observations, and lastly we cover this subregion with a second grid. The two-step procedure is shownin Figure 1. Even if this is not the most common case, it may happen that the short arc of observationsallows the computation of a preliminary orbit and even of a full orbit. In this case we consider the nominal More precisely a representative of the virtual impactor, a connected subset of the asteroid confidence region made up ofimpacting orbits.
HE MANIFOLD OF VARIATIONS: IMPACT LOCATION OF SHORT-TERM IMPACTORS 3 solution as reliable only if the arc curvature is significant (Milani et al., 2007). Given its importance fororbit determination purposes, we give more weight to the geodesic curvature with respect to the along-trackacceleration, imposing that the signal-to-noise ratio of the geodesic curvature is greater than 3. In thiscase we switch to a different sampling method to exploit the additional strong information coming with thereliable nominal orbit. Indeed we consider the marginal covariance of the orbit in the range and range-ratespace, whose probability density function has level curves which are concentric ellipses around the nominalrange and range-rate. We select a maximum confidence threshold and sample the AR by following theseellipses up to this confidence level. This cobweb technique is shown in Figure 2. Full details about thesampling of the AR in the various cases can be found in Spoto et al. (2018) and Del Vigna (2018).
Figure 1.
Admissible Region sampling with the rectangular grids.
Left . First step, withthe rectangular grid covering the entire AR.
Right . Second grid, covering the subregion ofgood orbits identified in the first step. In both plots the sample points are marked in bluewhen χ ≤ < χ <
5. The orange cross marks the orbit with theminimum χ value. Figure 2.
Sampling of the AR with a cobweb around the nominal solution . The samplepoints are marked in blue when χ ≤ < χ <
5. The orange crossmarks the orbit with the minimum χ value.Now we describe how to obtain a sampling of orbits, namely the MOV, once a sampling of the AR isgiven. We start with some notation. The target function is Q ( x ) := 1 m ξ ( x ) (cid:62) ξ ( x ) , where x are the orbital elements, m is the number of observations used for the fit, and ξ is the vector of thenormalised observed-computed debiased astrometric residuals. Let A be the attributable computed from ALESSIO DEL VIGNA, LINDA DIMARE, AND DAVIDE BRACALI CIOCI the arc of observations. The AR sampling is a set K of admissible values of the range and range-rate. Foreach ρ = ( ρ , ˙ ρ ) ∈ K we consider the full orbit ( A , ρ , ˙ ρ ) and fit only the first four components, that isthe attributable part, with a constrained differential corrections procedure. Definition 2.1.
Let K be a subset of the AR. The Manifold Of Variations is the set M of points ( A ∗ ( ρ ) , ρ )such that ρ ∈ K and A ∗ ( ρ ) is the local minimum of the target function Q ( A , ρ ) | ρ = ρ , when it exists.We call K (cid:48) ⊆ K the set of values of ( ρ, ˙ ρ ) such that the constrained differential corrections converge,giving an orbit on M . To estimate the goodness of the fit to the observations, for each orbit x ∈ M wecompute χ ( x ) := (cid:112) m ( Q ( x ) − Q ∗ ) , where Q ∗ is the minimum value of the target function over K (cid:48) . We always consider MOV orbits having χ <
5, thus corresponding at most to the 5 σ confidence level. For hazard assessment, this guarantees to findimpact possibilities with a probability > − , the so-called completeness level of the impact monitoringsystem (Del Vigna et al., 2019).Hereafter we summarise the main steps of the Manifold Of Variations method, as it is implemented in thesoftware system NEOScan.(i) The MPC NEO Confirmation Page is scanned every two minutes to look for new objects or for newobservations to add to previous detections.(ii) Computation of the attributable and sampling of the Admissible Region with rectangular grids or withthe cobweb techinque, depending on the existence of a reliable nominal solution.(iii) Computation of the Manifold Of Variations by constrained differential corrections, obtaining a set ofvirtual asteroids with a two-dimensional structure.(iv) Propagation of the virtual asteroids for 30 days and projection of the propagated MOV sampling onthe Modified Target Plane .(v) Searching for impacting solutions and, if any, computation of the impact probability. The computationof this quantity is done with a propagation of the probability density function from the normalisedresiduals space to the sampling space and by integrating the resulting density over the set of impactingsample points (Del Vigna, 2020).3. Impact location prediction
Before describing how to exploit the MOV sampling for impact predictions, we emphasise a key differencein the treatment of short-term impactors with respect to the long-term hazard monitoring of known asteroids.The MOV method does not use interpolation of the sample to find impacting orbits as it is done for the LOVmethod (Milani et al., 2005a), but just consider the impacting orbits of the sampling. For instance, if noneof the MOV sample points impacts the Earth, we assign a null impact probability. Indeed we adopt a densesampling, thus if a set of impacting orbits is not detected by the sampling it has a too small probability tobe interesting. This is a crucial difference with respect to the LOV method at the basis of long-term impactmonitoring. Many of the NEAs in the catalogue are big objects and thus also impact events with probabilityof few parts per million are worth detecting. Since nearby LOV orbits are separated by the chaotic dynamicsintroduced by close approaches, the prediction of impacts with such low probabilities, especially if far intime with respect to the time of the observations, would require an extremely high number of sample pointsalong the LOV. Hence if we limited the analysis to the sample points only, the computations would be tooheavy. As a consequence, interpolation of the LOV sampling is essential , but with the MOV method we areallowed to consider just the impacting orbits of the sampling. For an asteroid having a hyperbolic close approach with a planet, the Modified Target Plane is the plane passing throughthe centre of the planet and orthogonal to the velocity of the body at the time of closest approach. Actually in some extremely non-linear cases this is not enough, and LOV sampling densification techniques have beendeveloped to overcome the problem and to guarantee that the preset completeness level is reached (Del Vigna et al., 2020).
HE MANIFOLD OF VARIATIONS: IMPACT LOCATION OF SHORT-TERM IMPACTORS 5
On the basis of the above comment, the short-term hazard analysis performed with the MOV methodends with the identification of a subset V of impacting orbits among the MOV sampling (see step (v) at theend of the previous section). The idea of our location prediction method is actually very simple: once theset V is given, we just propagate the orbits of V until the impact and compute the geodetic coordinates ofthe impacting points.For such predictions we consider the WGS 84 model (NIMA - National Imagery and Mapping Agency,2000), for which the Earth surface is approximated by a geocentric oblate ellipsoid with semimajor axisequal to 6378 .
137 km and flattening parameter f defined by 1 /f = 298 . e of theellipsoid can be computed as e = f (2 − f ). Definition 3.1.
Let h ≥
0. The impact surface S h at altitude h above ground is the set of points at altitude h above the WGS 84 Earth ellipsoid.The impact region will be a subset of the impact surface S h for a given value of the altitude h and pointson S h are given by means of the geodetic coordinates. To compute this region we use the impact mapintroduced in Dimare et al. (2020), that is F h : V ⊆ M → S h . This map takes an impacting orbit x ∈ V , computes the time t ( x ) at which the orbit reaches the surface S h ,applies the integral flow of the system to propagate the orbit to the time t ( x ), and lastly converts the statevector into the geodetic coordinates on S h . Thus our MOV impact region is the set F h ( V ) ⊆ S h , that is theset of impacting MOV orbits propagated to the surface S h .3.1. Semilinear boundaries for the impact corridor prediction.
We briefly recap the semilinearmethod applied to the problem of the impact location prediction, described in Dimare et al. (2020).Let X be the orbital elements space, let Y ⊆ R be the target space, and let F : W → Y be a differentiablefunction defined on an open subset W ⊆ X , to be thought of as the prediction function. The dimension N of X is 6 if we consider the six orbital elements, or is > x ∗ ∈ X provided with its N × N covariance matrix Γ X . The uncertainty of x ∗ can be described through the confidence region Z X ( σ ) := (cid:26) x ∈ X : Q ( x ) − Q ( x ∗ ) ≤ σ m (cid:27) , where σ > F maps the orbital elements space ontothe target space, and thus maps the confidence region Z X ( σ ) to Z Y ( σ ) := F ( Z X ( σ )). The semilinearmethod provides an approximation of the boundary ∂Z Y ( σ ) of the non-linear prediction region (Milani andValsecchi, 1999).To explain the construction of the semilinear boundaries we start with the notion of linear prediction.The inverse of Γ X , that is C X = Γ − X , is the normal matrix and defines the linear confidence ellipsoid Z Xlin ( σ ) := (cid:8) x ∈ X : ( x − x ∗ ) · C X ( x − x ∗ ) ≤ σ (cid:9) . The differential DF x ∗ , assumed to be full rank, maps Z Xlin ( σ ) to the ellipse Z Ylin ( σ ) := (cid:8) y ∈ Y : ( y − y ∗ ) · C Y ( y − y ∗ ) ≤ σ (cid:9) , where y ∗ = F ( x ∗ ) is the nominal prediciton, C Y = Γ − Y , and Γ Y = DF x ∗ Γ X DF (cid:62) x ∗ according to the covariancepropagation law. When F is strongly non-linear, this linear prediction region is a poor approximation of Z Y ( σ ) and here the semilinear method comes into play. The boundary ellipse ∂Z Ylin ( σ ) is the image by thelinear map DF x ∗ of an ellipse E X ( σ ) ⊆ X which lies on the surface ∂Z Xlin ( σ ). The semilinear boundary K ( σ )is the non-linear image in the target space of the ellipse E X ( σ ), that is K ( σ ) := F ( E X ( σ )) . If the curve K ( σ ) is simple (no self-intersections) and closed, then by the Jordan curve theorem it is theboundary of a connected subset Z ( σ ) ⊆ Y , which we use as an approximation of Z Y ( σ ). ALESSIO DEL VIGNA, LINDA DIMARE, AND DAVIDE BRACALI CIOCI
Remark . The differential DF x is clearly not injective. More precisely, each point of Y has a ( N − X , and thus in principle the selection of the ellipse E X ( σ ) could be done in infinitelymany ways. The choice foreseen in the semilinear method consists in selecting as E X ( σ ) the ellipse resultingfrom the intersection between a suitable regression subspace in X and the confidence ellipsoid Z Xlin ( σ ). SeeMilani and Valsecchi (1999) for all the details.The application of this method to the impact corridor problem is described in Dimare et al. (2020) andworks as follows. We have a nominal orbit x of an asteroid with a non-negligible chance of impacting theEarth, and thus a virtual impactor representative x imp as provided by the LOV method. In this problemthe prediction map is the impact map F h : W → S h defined as above, where W ⊆ X is an open neighbourhood of x imp . The semilinear method can now beapplied, with the following adaptation. For the linear prediction on the impact surface and the selection ofthe ellipse E X ( σ ) we use the differential ( DF h ) x imp of the impact map at the virtual impactor representativeorbit x imp . The result of this method are curves on the surface S h corresponding to different values of σ .Note that these curves need not be closed in this context, because in general not all the orbits on E X ( σ )impact. 4. Numerical tests
We test the MOV method for the impact location prediction of four asteroids, namely 2008 TC , 2014 AA,2018 LA and 2019 MO. When we have a reliable nominal solution we also compare these results with thoseof the semilinear method and, additionally, to an observational Monte Carlo simulation. The latter amountsto adding noise to each observation to then compute a new orbit based upon the revised observations and usethis orbit as a virtual asteroid. When the observed arc is short the uncertainty in the orbit determinationis typically so large that the true uncertainty is not represented by the confidence ellipsoid. In this casethe observational Monte Carlo is the best approach to follow, definitely better than an orbital Monte Carlo,which works well when the orbital uncertainty is fairly good.4.1. Asteroid 2008 TC . The small asteroid 2008 TC was discovered by Richard A. Kowalski at theCatalina Sky Survey on October 6, 2008. The preliminary orbit computations done at the MPC immediatelyrevealed that the object was going to impact the Earth within 21 hours. Thus the astronomical communitymade a great effort to observe it and the currently available dataset contains nearly 900 observations, thoughnot all of them are of high quality, and need to be properly treated for a precise estimate of the trajectoryof 2008 TC (Farnocchia et al., 2017).When asteroid 2008 TC was recognised to be an impactor it only had few observations and for ouranalysis we consider its first seven observations. They are enough to compute a reliable nominal solution, sothat our algorithm samples the AR with the cobweb technique. The application of the MOV method resultsin an impact probability of 99.7%, which means that the vast majority of the MOV orbits impacts the Earth.We then propagate the MOV sampling to the Earth surface, i.e. we set h = 0, and obtain the result shownin Figure 3. Since the impact region is very extended, we limit our analysis to the MOV orbits with χ < χ value. In the right figure we show the results of a Monte Carlo simulation with 10,000sample points, to compare with the MOV impact region. Moreover, thanks to the existence of a nominalorbit we can also apply the semilinear method, thus in both plots of Figure 3 we also draw the semilinearboundaries on ground corresponding to the confidence levels 1, 2, and 3. The result is a strong agreementof the three methods.We notice that the semilinear boundaries enclose a region which is larger than that obtained with theMOV method. This happens also in most of the other examples which we present in the next sections.We can explain this behaviour as follows. Recall that the basic idea of the semilinear method is to selecta curve in the orbital elements space to propagate non-linearly to approximate the boundary of the non-linear prediction region. The choice of this curve is made by considering the boundary of the marginaluncertainty on a suitable space, which is equivalent to considering the corresponding regression subspace HE MANIFOLD OF VARIATIONS: IMPACT LOCATION OF SHORT-TERM IMPACTORS 7 (see Remark 3.2). The marginal covariance is the largest projected uncertainty on a given space thus, so tosay, is the most conservative choice. -150 -100 -50 0 50 100 150
East Longitude [deg] La t i t ude [ deg ] <11 < <22 < <3 IC1 sigmaIC2 sigmaIC3 sigma -150 -100 -50 0 50 100 150 East Longitude [deg] La t i t ude [ deg ] Monte Carlo pointsIC1 sigmaIC2 sigmaIC3 sigma
Figure 3.
Impact location prediction on ground for asteroid 2008 TC . Left.
Comparisonbetween the impact region computed with the MOV orbits having χ <
Right.
Comparison betweena Monte Carlo run and the same semilinear boundaries.4.2.
Asteroid 2014 AA.
Asteroid 2014 AA was discovered by R. Kowalski at the Catalina Sky Survey onJanuary 1, 2014 at 06:18 UTC. Similarly to 2008 TC , also 2014 AA impacted the Earth just 21 hours afterits first detection. But very differently from 2008 TC , due to the particular night in which 2014 AA wasspotted, it was not recognised to be an impactor. As a consequence the astrometric dataset is very limited,containing just 7 observations.Also in this case we can fit a nominal orbit and thus the MOV method starts with the cobweb sampling ofthe AR. All the MOV orbits are impacting, resulting in an impact probability of 100%. The impact regionson ground of 2014 AA computed with the MOV or with the Monte Carlo method are shown in Figure 4.Again we also show the semilinear boundaries for comparison, this time corresponding to the confidencelevels σ = 1, 3, and 5.The plot deserves some comment to avoid misunderstadings, due to the particular shape of the impact region.It is known that semilinear boundaries are not necessarily simple curves. As a consequence, if this is thecase, the curve cannot be the boundary of the non-linear prediction and indeed the impact regions extendoutside the drawn boundaries. This behaviour is confirmed by the fact that, in the vicinity of the torsion,the boundaries with lower σ extends outside the ones with higher σ . With a heuristic approach we canstate that the impact region extends outside the region delimited by the boundary, on the side of the lowerconfidence level boundaries. As a consequence this result is somewhat unsatisfactory, because we can justguess the actual impact area. Of course this issue disappears as soon as we start with a sampling of the wholeconfidence region and not with just the sampling of a curve. As we can see, the impact region computedwith the MOV gives a clear representation of the impact area and also shows why the semilinear boundariesare twisted. Indeed the confidence region folds on itself before being projected on the impact surface S h ,and this is a consequence of the non-linear effects due to the ongoing close approach. This behaviour is alsoconfiermed by the Monte Carlo simulation shown in the right plot of Figure 4.4.3. Asteroid 2018 LA.
Asteroid 2018 LA was discovered by the Mt. Lemmon Observatory of the CatalinaSky Survey just 8 hours before its impact in Botswana. For this asteroid we also have the fireball reportshown in Table 1, which took place at 28.7 km of altitude.
ALESSIO DEL VIGNA, LINDA DIMARE, AND DAVIDE BRACALI CIOCI -150 -100 -50 0 50 100 150
East Longitude [deg] -10-505101520 La t i t ude [ deg ] <11 < <33 < <5IC1 sigmaIC3 sigmaIC5 sigma ✲✶✵✵ ✵ ✶✵✵✲✶✵✲✺✵✺✶✵✶✺✷✵ ❊❛s(cid:0) ▲♦♥❣✐(cid:0)✉❞❡ ❬❞❡❣❪✁✂t✄t☎✆✝✞✆✝✟✠ ✡☛☞✹❆❆▼✌✍✎✏ ❈✑r❧✌ ♣✌✒✍✎✓■❈✔ ✓✒✕♠✑■❈✸ ✓✒✕♠✑■❈✖ ✓✒✕♠✑ Figure 4.
Impact region on ground of asteroid 2014 AA computed with the MOV methodand by the semilinear method.
Left.
Comparison between the impact region computed withthe MOV orbits having χ <
Right.
Comparison between a Monte Carlo run and the same semilinearboundaries. -40 -20 0 20 40-30-25-20-15-10-5 East Longitude [deg] La t i t ude [ deg ] χ <11 < χ <33 < χ <5Fireball location Figure 5.
Impact region at h = 28 . h = 28 . χ < HE MANIFOLD OF VARIATIONS: IMPACT LOCATION OF SHORT-TERM IMPACTORS 9 -10 0 10 20 30 40
East Longitude [deg] -22-21-20-19-18-17-16 La t i t ude [ deg ] <11 < <33 < <5IC1 sigmaIC3 sigmaIC5 sigmaFireball location -10 0 10 20 30 40 East Longitude [deg] -22-21-20-19-18-17-16 La t i t ude [ deg ] Monte Carlo pointsIC1 sigmaIC3 sigmaIC5 sigmaFireball location
Figure 6.
Impact location prediction at h = 28 . Left.
Comparison between the impact region computed with the MOV orbitshaving χ <
Right.
Comparison between a Monte Carlo run and the same semilinear boundaries. Thestar marks the location of the fireball event. ✶(cid:0) ✲✁ ✶(cid:0) ✲✂ ✶(cid:0) ✲✄ ✶(cid:0) ✲☎✥✆✝✞✟(cid:0)✠(cid:0)✡✟(cid:0)✠(cid:0)✶☛✟(cid:0)✠(cid:0)✶✟(cid:0)✠(cid:0)(cid:0)☛(cid:0)(cid:0)✠(cid:0)(cid:0)☛(cid:0)✠(cid:0)✶(cid:0)✠(cid:0)✶☛(cid:0)✠(cid:0)✡(cid:0)✠(cid:0)✡☛☞✌✍✎✏✌✑✒ ✷✓✔✕✖✗ ✗✘✙✚✛✛✚✜✢✣ ✤✣✦✚✧★ ✩✩✩✪✩✫✩✬✸✭✸✩✸✪✮✯
Figure 7.
Admissible Region sampling of 2018 LA with 12 observations, second grid. Thesample points are marked in blue when χ ≤ < χ <
5. The red circlesmark the impacting orbits.4.4.
Asteroid 2019 MO.
Asteroid 2019 MO was discovered from the ATLAS Mauna Loa observatory onJune 22, 2019, less than 12 hours before impacting the Earth between Jamaica and the south Americancoast. Also in this case we have a fireball event reported in Table 1, which took place at 25 km of altitude.Figure 9 shows the MOV impact region at h = 25 km extending above the south American coast, togetherwith the fireball location. Figure 10 shows the comparison between the MOV method, the semilinearboundaries and a Monte Carlo run with 10,000 sample points, which are fully compatible. -50 0 50 100 150-40-20020 East Longitude [deg] La t i t ude [ deg ] χ <11 < χ <33 < χ <5Fireball location Figure 8.
Impact region at altitude h = 28 . -110 -100 -90 -80 -70 -60 -50 -40 East Longitude [deg] La t i t ude [ deg ] <11 < <33 < <5Fireball location Figure 9.
Impact region at altitude h = 25 km of asteroid 2019 MO. The black star marksthe location of the fireball event. Table 1.
Fireball reports corresponding to 2018 LA and 2019 MO impacts, extracted fromthe JPL web page https://cneos.jpl.nasa.gov/fireballs/ . Peak Brightness Date/Time Latitude Longitude Altitude (UTC) (deg) (deg) (km)2019-06-22 21:25:48 14.9N 66.2W 25.02018-06-02 16:44:12 21.2S 23.3E 28.7 Conclusions
Very small asteroids can be only observed during a deep close approach with the Earth and it may bethe case that an impact occurs a few days after the discovery. In this paper we considered the problem ofpredicting the impact location of such objects by exploiting the MOV, a set of virtual asteroids representingthe orbital uncertainty. Once a MOV sampling is available it suffices to propagate each orbit for a givenamount of time and, for the impacting orbits, to compute the geodetic coordinates of the impacting points.The impact region is thus given by a set of points on the impact surface at a certain height over the Earth.
HE MANIFOLD OF VARIATIONS: IMPACT LOCATION OF SHORT-TERM IMPACTORS 11 -110 -100 -90 -80 -70 -60 -50 -40
East Longitude [deg] La t i t ude [ deg ] <11 < <33 < <5IC1 sigmaIC3 sigmaIC5 sigmaFireball location -110 -100 -90 -80 -70 -60 -50 -40 East Longitude [deg] La t i t ude [ deg ] Monte Carlo pointsIC1 sigmaIC3 sigmaIC5 sigmaFireball location
Figure 10.
Impact location prediction at altitude h = 25 km for asteroid 2019 MO. Left.
Comparison between the impact region computed with the MOV orbits having χ <
Right.
Comparisonbetween a Monte Carlo run and the same semilinear boundaries. The star marks the locationof the fireball event.The advantage of the MOV method with respect to already existing techniques, like for example thesemilinear method of Dimare et al. (2020), is that it can be used even when it is impossible to fit a nominalorbit to the few available astrometric observations (see the example presented in Section 4.3 and shown inFigure 8).We tested our method using the data available for the four impacted asteroids so far, namely 2008TC ,2014AA, 2018LA, and 2019MO. Since these data are also enough to constrain a full orbit, we compared theresults of the MOV method with the semilinear boundaries and with the outcome of an observational MonteCarlo simultation, getting a very strong consistency among the three methods. References
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Space Dynamics Services s.r.l., via Mario Giuntini, Navacchio di Cascina, Pisa, ItalyDipartimento di Matematica, Universit`a di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
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