Identifying Near-term Missions and Impact Keyholes for Asteroid99942 Apophis

J. Davis, P. Singla and J. JunkinsAerospace Engineering, Texas A&M University, USA

Abstract

We consider the challenges posed by dangerous asteroids that make a sequence ofclose encounters, such as asteroid 99942 Apophis which passes very close to Earthin 2029 and is considered a possible impacter at many subsequent encounters. Inparticular, we evaluate the energy costs for possible pre-2029 missions, and we alsostudy refined methods to compute probability of collision. We find two time windowscentered on 2011 and 2019 in which low energy missions are feasible. We also givepreliminary ideas on a new approach to approximate the collision probability for suchasteroids.

Introduction

Asteroid 99942 Apophis, hereafter referred to as Apophis, made headlines in Decem-ber 2004 when it was first discovered and a probability of collision on April 13, 2029was estimated as high as 2.7% [1]. Subsequent observations ruled out risk of impactin 2029 and confirmed that it will instead pass within≈ 5.9±0.35 Earth radii [1].The close encounter perturbs the orbit such that many resonant returns are possible inthe following years, the most probable of which occurs in 2036. In order to impact theEarth in 2036, Apophis must pass through a region≈ 0.6 km wide on the 2029 targetplane, known as a “keyhole” [1, 2]. Deflection from impact in 2036 is considerablyeasier if undertaken prior to or during the 2029 encounter.

Mitigation of the threat due to asteroid impact must begin with an accurate assess-ment of the probability of impact. Current methods use computationally expensivenonlinear particle methods by populating the entire regionof uncertainty with virtualasteroids (VAs) and propagating them forward through time as discussed in [3, 4].Instead, we propose that the keyholes be defined in size and shape at an earlier en-counter and the probability density function be integratedover this region to improvethe accuracy of the probability estimate.

Identifying keyholes is currently approached through a two-body approximation sim-ilar to Opik’s method, but this only provides a 2-D projection of thekeyhole ratherthan the 6-D definition we seek. We have developed a method that utilizes patched-conic approximations to define the size and shape of the keyhole in 6-D space, andthe boundaries of this region can be refined through use of an n-body propagator.

We first address opportunities for missions to Apophis priorto 2029 and expand onthe work presented in [5]. We then discuss current methods ofimpact probability

analysis and then our efforts to identify impact keyholes inlocation, size, and shapeby utilizing a patched-conic solution to inform a restricted search with a higher fi-delity model.

Feasible missions

A mission that places a radio transponder on Apophis prior to2029 would have twomajor results. First, such a mission would reduce the orbit uncertainty by more thana factor of 10 [2]. Second, should Apophis indeed be on a high probability of impacttrajectory, it could provide accurate information regarding the asteroid’s size, struc-ture, and composition that is required for mitigation. A third motivation for such amission is that it would provide valuable preparation for dealing with future Earth-impacting asteroids. The European Space Agency is currently studying monitoringand deflection missions to non-threatening asteroids to test such capabilities [6].

Preliminary mission analysis has been completed by considering a two-point bound-ary value problem with the known position of the Earth at a given launch date andthe position of Apophis after a prescribed∆t as shown in Figure 1. The Lambert al-

⊕

Asteroid

Orbit

Earth

Orbit

Transfer

Orbit

tlaunch

tarrive = tlaunch + tflight

V /A = VA -Varrive

V / = Vdepart -V

rA(tarrive)

r (tlaunch)

⊕

Asteroid

Orbit

Earth

Orbit

Transfer

Orbit

tlaunch

tarrive = tlaunch + tflight

V /A = VA -Varrive

V / = Vdepart -V

rA(tarrive)

r (tlaunch)

8

8

Figure 1: Lambert method

gorithm [7] was used to find the heliocentric transfer trajectory for a specified launchdate and time of flight by solving this boundary value problem. The launch datesfrom 2008 to 2025 were swept daily and the transfer orbit solved for times of flightbetween 120 and 1095 days (≈ 3 yrs). For longer times of flight, multi-revolutionsolutions were found up to 2+ revolutions of the spacecraft about the Sun.

Referring again to Figure 1, the magnitude of the vector difference between the ini-tial heliocentric transfer orbit velocity and the Earth’s heliocentric velocity (V∞/⊕)provides a measure of how much energy would be required to place a spacecraft onthe given transfer orbit, whereas the magnitude of the vector difference between the

arrival transfer orbit velocity and Apophis’ velocity (V∞/A) provides a measure ofenergy required to rendezvous with the asteroid.

“Porkchop” surfaces were generated as seen in Figures 2 in which contours show therequired departureC3 = V2

∞/⊕ and arrivalV∞/A for a spacecraft launching throughout2011 and over the entire range of time of flight. These plots are similar to those foundin [5], but have been expanded to include multi-revolution solutions.

C3 = 0.04km2/s2

Launch date (2011)

Tim

e of

flig

ht (

days

)

C3 > 60 km2/s2 (Unfeasible)

Jan 1 Apr 1 July 1 Oct 1 Dec 31

200

400

600

800

1000

(a) DepartureC3. Minimum point shown.

V∞/A > 10 km/s

Launch date (2011)

Tim

e of

flig

ht (

days

)

Jan 1 Apr 1 July 1 Oct 1 Dec 31

200

400

600

800

1000

(b) Arrival V∞/A

Figure 2: Porkchop plots for 2011. Lighter regions correspond to smaller values.

1/1/2008 1/1/2011 1/1/2014 1/1/2017 1/1/2020 1/1/20230

2

4

6

8

10

12

14

Launch date

C3 (

km2 /s

2 )

Partial rev1+ rev2+ rev

Figure 3: MinimumC3 for each year for different number of revolutions

For each year, a minimumC3 orbit was found, as seen in Figure 2(a), and these pointsare plotted in Figure 3, with each line denoting a partial (◦), 1+ (×), and 2+ (4) revo-lutions of the spacecraft about the Sun prior to rendezvous,respectively. It is evidentthat two sets of extremely low energy launch opportunities occur in the years 2010-2012 and 2018-2020. Further, multi-rev solutions show significant improvement overthe partial revolution case and expand the size of the feasible launch windows sub-stantially.

The first opportunity (∼ 2011) is attractive for a small spacecraft to land a transponderand conduct geological characterization experiments. Fora deflection mission withconsiderably higher required mass, the second set of launchopportunities (∼ 2019)would be very attractive. Between mid-November 2017 and theend of September2020, there is an almost continuous launch window withC3 < 1 km2/s2 with only acombined 5 weeks time out of those three years where a rendezvous would be moreexpensive. Additionally, during those 3 years the arrival dates fall into 4 discretearrival windows approximately two weeks long in late May 2020, late April 2021,late March 2022, and early February 2023. Cooperating with mother nature suggestswe focus on∼ 2011 and∼ 2019 as exceedingly important opportunities.

Calculating impact probability

Current efforts to calculate impact probability utilize Monte Carlo and other numer-ical methods discussed in [3, 4, 8]. The uncertainty region for a given asteroid ispopulated with a number of VAs, distributed either throughout the 6-D space or alongthe 1-D Line of Variations (LOV). Where linear error theory applies, the LOV cor-responds to the major axis of the uncertainty ellipsoid, which is generally orders ofmagnitude larger than the other two. The VAs’ trajectories are propagated until aclose encounter is found, and when distributed along the LOV, their trajectories canbe interpolated to find a virtual impactor (VI) if one exists.A probability densityfunction (PDF) - typically Gaussian - is then assumed and integrated over the portionof the uncertainty region that intersects the Earth when projected onto the target planeat the time of encounter. With such a method, the detectable probabilities are on theorder of the inverse of the number of VAs. Therefore, it typically requires tens ofthousands of VAs and multiple hours of CPU time [8], and thereare always residualconcerns regarding convergence (due to finite sample size statistics).

As mentioned in [8], most impacts with Earth result from resonant returns after anasteroid passes through a keyhole in an earlier near impact encounter. The new ap-proach to calculate probability of impact proposed here is to deterministically locatethe keyholes at a first close encounter, corresponding to theset of states (that exceeda conservative probability threshold) that result in impact in a subsequent encounter,and integrate the PDF over these keyholes to define the probability of impact. In ad-dition to providing an improved probability estimate, it quickly identifies the locationand size of such keyholes which has important implications in deflection scenariosand for ruling out impacts when subsequent observations aremade.

Identifying keyholes

Current efforts to identify keyholes begin with a modifiedOpik’s method [9], whichassumes the original encounter instantaneously alters therelative velocity vector fromthe incoming hyperbolic asymptote to the outgoing one. The keyhole is defined on a

target plane centered on the Earth, perpendicular to the incoming velocity asymptote.However, we aim to integrate over the entire 6-D uncertaintyregion to which thiskeyhole belongs, rather than the 2-D projection offered byOpik’s method to providea more accurate assessment of the probability of impact.

Method overview

To define the 6-D size and shape of a keyhole, the covariance - which can be ap-proximated as a 6-D ellipsoid - was propagated to the edge of the Earth’s sphere ofinfluence (SOI)1 following Apophis’ 2029 encounter as seen in Figure 4. The keyholewill be defined as a subset of this 6-D ellipsoid at SOI following the 2029 encounter.

Edge of Sphere

of Influence

Nominal

trajectory

Uncertainty

region at

exit of SOI

Edge of Sphere

of Influence

Nominal

trajectory

Uncertainty

region at

exit of SOI

Figure 4: Propagating covariance to the edge of SOI in 2029

The impact time was iterated to find an initial guess for the most probable impacttime, and the position sub-space of the uncertainty region was searched utilizingLambert’s method to determine the required velocity leading to direct (“dead-center”)Earth impact at the specified time. The metric used to locate the center of the key-hole was the∆V required for a direct Earth impact, which was minimized. Futureefforts will instead maximize the probability of impact. The PDF is approximated asGaussian, which for the full 6-D state space has the form,

p =1

6√

2π|P|1/2e−

12 [xxx−xxx]TP−1[xxx−xxx] (1)

wherex is the state of Apophis in 2029 that leads to impact,x is the estimated state ofApophis as given, for example, by JPL’s ephemeris generator[10], andP is the errorcovariance matrix.

After locating a preliminary point for the center of the keyhole, a modified Lambertalgorithm was employed to find the deterministic velocity leading to direct impactwith the center of the Earth using a patched-conic approximation at arrival. Next,the region of space surrounding the point at which the asteroid enters the Earth’s SOIin 2036 was investigated to determine the initial states in 2029 that lead to indirectimpact with Earth in 2036. This defines a set of initial velocities in 2029 for eachpoint in the position sub-space that could result in impact with the Earth in 2036.

1The sphere of influence of a body is roughly defined as the region in space around that body where itsgravitational force dominates the dynamics of a smaller satellite. See [7] for details on calculatingrSOI.

Finding impact time

Before a keyhole could be identified using the Lambert method, the flight time be-tween 2029 and the impact in 2036 had to be assumed. To providean estimate forthe time that Apophis is most likely to impact, the∆V required to impact the Earthdirectly - assuming only a heliocentric orbit - was calculated along the major axis ofthe position uncertainty sub-pace (roughly analogous to the LOV) every two hours onApril 13, 2036 as seen in Figure 5(a). Horizontal lines represent each time, and thecircles show the minimum∆V for that time.

−3 −2 −1 0 1 2 30

0.2

0.4

0.6

0.810:00

12:00

14:00

16:00

18:00

20:00

σ−space

∆V

(a) Minimum∆V along major axis of uncertaintyellipsoid every 2 hours

σ−space

Arr

ival

tim

e on

Apr

il 13

, 203

6 (G

MT

)

−3 −2 −1 0 1 2 314:00

15:00

16:00

17:00

18:00

(b) ∆V contours for time of flight versus departurepoint along major axis of uncertainty

Figure 5: Finding approximate time of flight

Ephemeris data available from JPL’s HORIZONS system [10] provided the state ofthe Earth at each time investigated. This sweep narrowed theimpact time window tobetween 14:00 and 18:00 GMT and suggested that there was onlya single keyholethat leads to a 2036 impact. A subsequent comprehensive search between those twotimes as represented in Figure 5(b) led to a time of impact calculated to the nearestminute. This final time was held constant through the next several steps.

Locating the center of the keyhole

A preliminary search of the position sub-space was completed to find the approximatelocation of the keyhole assuming only a heliocentric trajectory leading to Earth im-pact. Figure 6 shows the location inσ-space and associated∆V of the keyhole basedon a 1-D search along the major axis of uncertainty and also the refined location ofthe keyhole upon completion of a 3-D search using Powell’s method.

The location inσ-space of the refined point can be loosely compared to the loca-tion of the 2036 keyhole along the LOV shown in Figure 2 of [1] which serves as apositive sanity check at this stage. This refined point served as the initial point forthe patched-conic Lambert method and the remaining analysis. It is anticipated thatinsights derived from the patched-conic solutions, while not exact, will be useful toguide the analysis with the final n-body integrations.

−1 −0.8 −0.6 −0.4 −0.2 00.074

0.075

0.076

0.077

0.078

0.079

0.0753 km/s

0.0750 km/s

σ−space

∆V (

km/s

)

1−D minimum

3−D minimum

Figure 6: Preliminary location of the center of the 2036 keyhole inσ-space along themajor axis of uncertainty and its associated∆V using 2-body approximations.

Patched-conic Lambert’s method

The velocities found above that lead to Earth-impact were found assuming the Sunwas the only body acting on the asteroid. While this is a reasonable approximationthroughout much of Apophis’ solar orbit between 2029 and 2036, the Earth dominatesApophis’ motion after it enters Earth’s SOI in 2036. Therefore, a heliocentric orbitis assumed for the first portion of the orbit, whereas a geocentric orbit is assumedduring the time within Earth’s SOI; this is known as a patched-conic approach and isdiagrammed qualitatively in Figure 7.

Sun 20292036

∆t1

∆t2

Target point

on SOI

Elliptic heliocentric

orbitRectilinear

geocentric

orbit

Sun 20292036

∆t1

∆t2

Target point

on SOI

Elliptic heliocentric

orbitRectilinear

geocentric

orbit

Figure 7: Patched-conic two-point boundary value problem

For Apophis to directly impact the center of the Earth under the patched-conic as-sumption, it must enter the Earth’s SOI on an approximately rectilinear orbit relativeto the Earth. Therefore, the relative velocity and positionvectors must be collinear.Locating the point on the SOI where a trajectory from an initial point in 2029 satisfiesthis condition defines the patched-conic Lambert problem. Additionally, the sum ofthe time of flight from initial point to SOI (∆t1) and from SOI to the center of theEarth (∆t2) must equal the original∆t.

The patched-conic Lambert method developed starts with theheliocentric orbit thatleads to collision with the center of the Earth in∆t. The time that the asteroid reachesEarth’s SOI on this trajectory is then approximated and usedas the initial guess for∆t1. The point on SOI where this trajectory’s relative velocitywould be collinearwith the relative position is specified as the new target point for the classic Lambertalgorithm using∆t1 as the time. This new trajectory provides a velocity vector relativeto the Earth that can be used to calculate∆t2 assuming a rectilinear orbit, whichthen updates∆t1. The process is iterated until the time and the position converge.Convergence has been shown to be well-behaved for arbitraryinitial conditions anda solution was always found within< 20 iterations.

Gravitational focusing

Once the location on the sphere of influence was found that leads to impact with thecenter of Earth - the target point - the remaining Earth-impacting trajectories had tobe found. Defining a plane perpendicular to the relative velocity vector centered at thetarget point, it is intuitive that targeting a point on that plane over an area larger thanthe area of the Earth would still lead to impact due to gravitational focusing. Indeed,if the relative velocity were to be held constant, a “gravitational cross-section” ofradius

b = r⊕

√

2µ⊕

v2∞

(

1r⊕

− 1rSOI

)

+1 (2)

would define the set of points such that all trajectories (having the sameV∞) passinginside the circle of radiusb impact Earth [11], as shown in Figure 8.

Grazing

trajectory

Rectilinear

trajectory

STP

Target

point

Boundary of

impact

trajectories

Edge

of SOI

Grazing

trajectory

Rectilinear

trajectory

STP

Target

point

Boundary of

impact

trajectories

Edge

of SOI

Figure 8: SOI target plane (STP) and gravitational focusing

However, aiming for the points that lie on this circle and invoking the Lambert methoddoes not result in the same velocity vector at each of these points as at the center asassumed in (2). To investigate the true gravitational focusing, the plane at SOI, orthe SOI target plane (STP), was defined with the x-axis along the intersection ofthe heliocentric orbital plane and the STP, the z-axis positive in the direction of therelative velocity, and the y-axis completing a right-handed coordinate frame.

Investigating the arrival velocities on the STP at increments of θ (measured from +x-axis) around the circle of radiusb, it became apparent that each component could

be approximated using a first order Fourier series. The residual errors in this ap-proximation were nearly sinusoidal inθ, such that a Fourier series converges quickly.Varying the radius led to the conclusion that the coefficients of the first order Fourierseries vary nearly linearly withr, and the velocity about which they oscillated was aconstant plus a term proportional tor2. Similar to the angular dependence, the resid-ual errors in the secular growth of the center-line could be reduced several orders ofmagnitude by adding a term proportional tor4.

To compute the point of closest approach, or radius at periapsis of each orbit, eachvelocity component was approximated by the form,

vi = air cosθ+bir sinθ+cir2 +di (3)

By evaluating only 10 points on the STP, the above constants were determined usinglinear least squares. Errors in the velocity components were found to be of the orderof 10−4 km/s, and the periapsis radius was computed to within±0.003 Earth radii.

The points on the STP where the periapsis radius equaled the radius of the Earth de-fine the true boundary of gravitational focusing. The results are shown in Figure 9(a)which shows both the circle of radiusb and the true gravitational focusing.

−2 0 2

−4

−2

0

2

4

x (R⊕)

y (R

⊕)

Target point

Standard gravitationalfocusing (allvelocities equal)

True gravitationalfocusing

(a) Gravitational focusing region on STP

0 2 4 618.1

18.15

18.2

θ (rads)

v x (km

/s)

0 2 4 6−24

−23.9

−23.8

θ (rads)

v y (km

/s)

0 2 4 6−9

−8.9

−8.8

θ (rads)

v z (km

/s)

(b) Departure velocity for constant radius

Figure 9: Investigating the gravitational focus effect

Further analysis showed that the departure velocities in 2029 could also be approxi-mated with the same form as (3), as seen in Figure 9(b). Again,errors in each velocitycomponent with this first order approximation were of the order of 10−4 km/s withonly 10 points on the target plane searched out.

Thus, for a given point in the covariance position sub-spacein 2029 and a given arrivaltime in 2036, the entire subset of velocities leading to Earth impact can be definedusing the above expression whose constants can be derived bya very small numberof Lambert algorithm solutions. Thus, iterating over the impact time provides all ofthe relevant information for any specific point in the 2029 uncertainty region.

Keyhole size and shape

After identifying the location within the position uncertainty sub-space with the high-est probability of impact, the region surrounding that point must be investigated toprovide a preliminary size and shape of the keyhole. This investigation can be in-formed through the methods described above to quickly and easily define the veloc-ities required for each point to impact the Earth. The entirety of this information isthen used to inform a search of the space using a higher fidelity model.

Conclusions

Two favorable launch windows have been found around 2011 and2019 to visit ordeflect 99942 Apophis. Methods for locating and defining the size and shape in 6-D space of the 2029 keyhole that leads to an impact in 2036 havebeen developedusing patched-conic approximations. These preliminary numbers must then be usedto inform a search using higher fidelity models. Once the keyhole has been identifiedutilizing full n-body code, the probability density function can be integrated over that6-D region to produce a more accurate probability estimate.

References

[1] S.R. Chesley. Potential impact detection for Near-Earth asteroids:the case of 99942Apophis. InProc. IAU Sym. No. 229, August 2005.

[2] R.L. Schweickart. A call to (considered) action. InInternational Space DevelopmentConference, Washingtion, DC, May 2005.

[3] A. Milani, S.R. Chesley, P.W. Chodas, and G.B. Valsecchi. Asteroid Close Approaches:Analysis and Potential Impact Detection. In W. Bottke and et al., editors,Asteroids III,pages 55–70, 2002.

[4] A. Milani. Virtual asteroids and virtual impactors. InProc. IAU Colloq. No. 197, 2005.

[5] J. Junkins, P. Singla, D. Mortari, W. Bottke, and D. Durda. A Study of Six Near-EarthAsteroids. InInternational Conference on Computational & Experimental Engineeringand Sciences, Chennai, India, 2005.

[6] A. Galvez. ESA selects targets for asteroid-deflecting mission Don Quijote.http://www.esa.int/esaCP/SEML9B8X9DEindex 0.html, 2005. ESA.

[7] R.H. Battin. An Intro to Mathematics and Methods of Astrodynamics. AIAA EducationSeries, Reston, VA, 1999.

[8] A. Milani, S.R. Chesley, and G.B. Valsecchi. Asteroid close encounters with Earth: riskassessment.Planetary and Space Sci., 48:945–954, August 2000.

[9] G.B. Valsecchi, A. Milani, G.F. Gronchi, and S.R. Chesley. Resonant returns to closeapproaches: Analytical theory.A&A, 408:1179–1196, 2003.

[10] JPL. HORIZONS System. http://ssd.jpl.nasa.gov/?horizons.

[11] R.M. Baker and M.W. Makemson.An Introduction to Astrodynamics. Academic Press,New York, 1960.