Rotational and translational considerations in kinetic impact deflection of potentially hazardous asteroids

Accepted Manuscript Rotational and translational considerations in kinetic impact deflection of potentially hazardous asteroids Fei Zhang, Bo Xu, Christian Circi, Lei Zhang PII: DOI: Reference:

S0273-1177(17)30016-9 http://dx.doi.org/10.1016/j.asr.2017.01.003 JASR 13042

To appear in:

Advances in Space Research

Please cite this article as: Zhang, F., Xu, B., Circi, C., Zhang, L., Rotational and translational considerations in kinetic impact deflection of potentially hazardous asteroids, Advances in Space Research (2017), doi: http:// dx.doi.org/10.1016/j.asr.2017.01.003

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Rotational and translational considerations in kinetic impact deflection of potentially hazardous asteroids Fei Zhanga , Bo Xua,∗, Christian Circib , Lei Zhangc a

b

School of Astronomy and Space Science, Nanjing University, Nanjing 210023, China Department of Astronautical, Electrical and Energy Engineering, Sapienza University of Rome, Via Salaria 851, I-00138 Rome, Italy c Satellite Communication and Navigation Collaborative Innovation Center, Nanjing 210007, China

Abstract Kinetic impact may be the most reliable and easily implemented method to deflect hazardous asteroids using current technology. Depending on warning time, it can be effective on asteroids with diameters of a few hundred meters. Current impact deflection research often focuses on the orbital dynamics of asteroids. In this paper, we use the ejection outcome of a general oblique impact to calculate how an asteroid’s rotational and translational state changes after impact. The results demonstrate how small impactors affect the dynamical state of small asteroids having a diameter of about 100 m. According to these consequences, we propose using several small impactors to hit an asteroid continuously and gently, making the deflection mission relatively flexible. After calculating the rotational variation, we find that the rotational state, especially of slender non-porous asteroids, can be changed significantly. This gives the possibility of using multiple small impactors to mitigate a potentially hazardous asteroid by spinning it up into pieces, or to despin one for future in-situ investigation (e.g., asteroid retrieval or mining). Keywords: celestial mechanics, asteroid deflection, kinetic impactors, rotation and translation

∗

Corresponding author Email address: [email protected] (Bo Xu)

Preprint submitted to Advances in Space Research

December 31, 2016

1. Introduction Among near-Earth asteroids, there exist populations that have the potential to impact the Earth. Since most asteroids have great speed relative to the Earth, even some small meteorites can cause significant damage. The Chelyabinsk airburst, which injured more than 1500 people, was caused by only a ∼15 m meteorite (Brumfiel, 2013). A 75 meter diameter asteroid that impacts Earth could destroy a city as big as Washington,D.C. or Moscow (Morrison et al., 1994). The impacting frequency of a hazard asteroid decreases with the increase of its impact energy relative to the Earth, and they approximately have a power-law relationship on average (Brown et al., 2002). A ∼60 m impact event as happened in Tunguska (Chyba et al., 1993) may occur once in about 500 years or even less (Harris and DAbramo, 2015). Although such events occur very rarely, the regional consequences can be devastating (Chapman and Morrison, 1994). To protect the Earth from a potential impact, many strategies have been proposed, including nuclear explosions (Ahrens and Harris, 1992), mass drivers (Melosh et al., 1994), solar sails (Melosh et al., 1994), solar collector (Melosh et al., 1994), gravity tractors (Love and Lu, 2005), ion-beam shepherds (Bombardelli and Pelez, 2011) and many others. Deflection by non-nuclear kinetic impactors is a relatively mature approach and could be very effective in a range of scenarios. As an impulsive technique, kinetic impactors may be less complicated in operation, faster to effect and have fewer long-term reliability issues than slow-push strategies (Dearborn and Miller, 2014). Compared to an approach involving a nuclear device, kinetic impactors have fewer political complications (Koenig and Chyba, 2007) and are less likely to cause unintentional disruption. Kinetic impact deflection requires one or more impactors to be delivered to intercept the threatening asteroid. The dissipation of an impactor’s kinetic energy produces a crater as the asteroid material is ejected into space. The impact and cratering process change the shape, rotational state, and the momentum of target asteroid. Many studies on kinetic impact deflection of potentially hazardous asteroids have focused on the orbital dynamics aspects of the problem. Koenig and Chyba (2007) studied the use of multiple intercepts to deflect asteroids as large as 1km in diameter by considering current rocket capacities. Sanchez and Colombo (2013) considered the epistemic uncertainty of the asteroid impact threat and generated a very large sample of impacting ephemerides 2

to explore the protection efficiency of a small impactor. McInnes (2004), Dachwald et al. (2007) and Dachwald and Wie (2007) respectively designed different kinetic impact deflection missions. The impact process itself is also important in deflection as it could tell us how an asteroid responds after being impacted. Many studies have been proposed in both simulative and experimental ways. Asphaug et al. (1998) studied the collisional evolution of asteroids having different internal structures through numerical simulation. Herbold et al. (2015) simulated deflection strategies for asteroid Bennu by focusing on standoff explosion and hypervelocity impact. Owen et al. (2015) successfully predicted some common features of an asteroid’s response to impulsive deflection approaches in their simulations. Yanagisawa and Hasegawa (2000) and Yanagisawa et al. (1996) studied the normal and tangential components of momentum transfer efficiency through a series of impact experiments. Holsapple and Housen (2012) defined the momentum multiplication factor for kinetic impact deflection and used point-source scaling relations to determine its functional relationship with the initial impact conditions, including radius, velocity, mass density of impactor and the surface strength, mass density of target asteroid. In addition to momentum transfer, it is also important to consider change in the rotational state of asteroids after impact. Errors in real missions make impact velocities and points uncertain. When ejecta escapes from an asteroid, the velocity of ejecta contains a component from the rotation of the asteroid. Part of the rotational kinetic energy is transferred to translational kinetic energy and the rotation of the asteroid may be changed. If the impact is oblique, the tangential component of impact velocity also changes the rotation. If an impactor has a great impact velocity, the asteroid may undergo a great velocity change, but the uncertainty of the consequence is also great. Some unexpected debris may be produced and move towards the Earth and could be big enough to cause damage but too small to track and intercept, which would make things worse. Based on the above considerations, we propose a relatively gentle impact strategy: several relatively small impactors are equipped on a mother ship which accompanies target asteroid hitting the target gently and continuously. Thus one would be able to observe the change in state after each impact and plan a suitable condition for next impact. It could be more efficient and reliable and the relatively low impact speed would decrease the burdens in navigation and control in the mission. In such 3

a multiple impact scenario, it is necessary to calculate the translational and rotational variation of asteroid after each impact so that the efficiency of the strategy could be known. The paper aims to understand the capability of multiple gentle impact in our proposed story, and we consider both the momentum and angular momentum transfer between the impactor and asteroid, and analyse the state change of asteroid after a general oblique impact. Each impactor’s mass is set as 500 kg, impact speed is not greater than 10 km s−1 . Considering a small size asteroid has high possibility of hitting the Earth, the diameter of asteroid is set at around 100 m in the simulation. As the mass of these asteroids is also relatively small, their responses after impacts would be more obvious. The paper is organized as follows: Sec.2 describes the basic deflection dynamic theory. Sec.3 describes the impact model used to study the momentum and angular momentum transfer in impact between impactor and target asteroid. Sec.4 displays the results of different impact calculations and some discussions. Sec.5 makes a brief summary of this paper. 2. Asteroid deflection theory For the keplerian motion, a small velocity perturbation at the initial time will cause huge drift after several orbital periods. The trajectory of the target asteroid is modeled as a two-body problem. The state transition matrix (STM), denoted as Φ(t, t0 ), is used to calculate the final state shift, which is (∆r(tf ), ∆va (tf )) = Φ(t, t0 )∆x(t0 ),

(1)

where t0 , tf denote the initial and final times respectively, ∆r(tf ) is the final deflection distance, ∆va (tf ) is the final velocity shift, ∆x(t0 )=(0,0,0,∆v1 ,∆v2 ,∆v3 ) is the initial state change, ∆v = (∆v1 , ∆v2 , ∆v3 ) is the velocity perturbation, and the initial position change is set as zero. The state shift could be calculated in heliocentric inertial frame such as the J2000 heliocentric equatorial reference frame. Assuming a hazard asteroid would impact the Earth aiming at center of the Earth, the two objects have a relative distance of zero when encountering. After a deflection manoeuvre is applied on the asteroid, the achieved position shift at the asteroid-Earth encounter moment can be obtained by computing the relative distance between the two objects orbiting the Sun. During the 4

final Earth approach, the asteroid’s orbit is hyperbolic with the Earth at the focus of the hyperbola because of the gravitational focus of the Earth, and the minimum position shift to avoid the impact of the asteroid with the Earth would clearly be larger than one Earth radiusSanchez Cuartielles (2009). The minimum position shift could be expressed as s 2µ⊕ (2) d ⊕ = r⊕ 1 + 2 , v∞ r⊕ where r⊕ , µ⊕ are the radius and gravitational constant of the Earth respectively, v∞ is the hyperbolic excess velocity of asteroid, which also is the relative velocity between the two objects at the encounter moment. 3. Kinetic impact model We investigate the effect of impact issues with the following assumptions: (1). the target asteroid is a homogeneous monolithic body, (2). the melt and vaporization of target material is ignored, (3). the collision completely destroys the impactor and makes a rounded simple crater, (4). the impact angle, measured from vertical, is no more than 60 degrees. Then, the dynamical change of the target asteroid is calculated. Most NEAs with diameters D . 150 m rotate with periods of less than 2 h (Pravec et al., 2002). Their intact structure can only be bound by an amount of cohesive strength. Holsapple and Housen (2012) pointed out that ejecta produced by a collision would not return to the surface for less porous asteroids with diameters less than a few kilometers. In the analysis, we choose relatively small targets, and assume that all the ejecta escapes from the asteroid, and the material strength of asteroid surface would determine the cratering mechanics. In our proposed story of multiple gentle impact, the mothership accompanies target asteroid, impactors have a similar relative velocity to the asteroid, each impactor needs a high-efficiency rocket to be accelerated. Hence, the impact velocity would be limited and we choose impactor velocities < 10 km s−1 . The melting and vaporization is also negligible with the chosen impact velocity (Holsapple, 1993). 5

The impact crater is considered in our analysis. Excavating a crater will change the centre of mass of the asteroid and the rotation state of asteroid. Experimental studies of Gault and Wedekind (1978) indicate craters remain circular when impact angles θ (measured from vertical) are less than 60 degrees, and that the total mass excavated in crystalline targets (strength regime) varies as cos2 θ whereas the variation for particulate targets (gravity regime) is proportional to cos θ (Pierazzo and Melosh, 2000). Holsapple and Housen (2012) mentioned that any impact with impact angle less than 70 degrees or so will give the same crater and dominant ejecta as a vertical impact having the same normal velocity component. A ≤ 60◦ impact could excavate a simple crater and the crater dimensions depend only on the vertical component of the impact velocity(Pierazzo and Melosh, 2000). Moreover, the final state of the impactor is strongly dependent on the impact. Ricochet is imminent as the impact angle approaches 60◦ , and the ricocheted fragments of the impactor can retain a significant portion of the impact velocity (Pierazzo and Melosh, 2000; Gault and Wedekind, 1978), which is not an efficient impact for deflection. Therefore, we only consider oblique impact with impact angle θ ≤ 60◦ . 3.1. The conservation law Momentum and angular momentum conservation of the system containing both the impactor and target asteroid are used. We use a reference frame Ri (O, X, Y, Z) so that the origin O is the centre of mass of target asteroid, the X-axis coincides with the direction of motion of the asteroid when the impactor impacts the asteroid, the Z-axis is directed along the normal direction of the orbital plane of the target asteroid and Y-axis completes the right-handed reference frame. The frame Ri also has a uniform linear motion in the heliocentric inertial frame during the impact simulation, and the velocity is the orbital velocity of the target asteroid right before the impact. Obviously, the frame Ri is considered to be inertial through a single impact simulation. When the linear momentum of an impactor is low, it’s absolute value of incident momentum may approximate to the momentum value of the Sun’s pull during the impact. But the momentum conservation could still be used in the impact simulation (see Appendix A). The conservation law is written as below respectively mimp vimp = m∗ v ∗ + Pejecta , 6

(3)

La,rot + Limp,rot + La,orb + Limp,orb = L∗ + Lejecta ,

(4)

where mimp is the mass of the impactor, vimp is the velocity of the impactor relative to the target asteroid, m∗ is the mass of target asteroid after the impact, v ∗ is the velocity change of the target asteroid after impact, Pejecta is the relative momentum of the ejecta (see Appendix A). La,rot and Limp,rot are the rotational angular momenta of the asteroid and impactor before collision, respectively, La,orb and Limp,orb are the orbital angular momenta of the asteroid and impactor before collision, respectively, L∗ is the angular momentum of the asteroid after the impact, and Lejecta is the angular momentum of the ejecta after impact. Since the formation of crater has been assumed to depend only on the vertical incident momentum, the momentum and angular momentum can be calculated respectively in normal and tangential components to the local impact surface of the target asteroid like Yanagisawa et al. (1996) did. The momentum multiplication factors in both directions are introduced to calculate the momentum and angular momentum transfer efficiencies. The two factors are defined as η=

∆pn ∆pt ,ζ = , mimp vimp,n mimp vimp,t

(5)

where ∆p is the momentum increment of the target asteroid, subscripts ‘n’ and ‘t’ denote the normal and tangential components of the target’s local surface at the impact point, respectively. Then the momentum of asteroid obtained from impact can be expressed as ∆Pasteroid = m∗ v ∗ − ma va = ηmimp vimp,n + ζmimp vimp,t ,

(6)

where va is the velocity of target asteroid before impact and is obviously zero in frame Ri , ma is the mass of target asteroid after the impact. The momentum of ejecta is Pejecta = (1 − η)mimp vimp,n + (1 − ζ)mimp vimp,t .

(7)

The orbital angular momenta of asteroid and impactor are calculated respectively as La,orb = ma (ra × va ), (8) Limp,orb = mimp (rimp × vimp ),

7

where rimp and ra are the radius vectors of the impactor and the asteroid before impact, respectively. In frame Ri , La,orb is obviously zero. The rotational angular momenta of asteroid and impactor are respectively calculated as La,rot = Ia ωa , (9) Limp,rot = Iimp ωimp , where Iimp , Ia are the inertia tensors of the impactor and the asteroid before collision, respectively. ωimp , ωa are the angular velocities of the impactor and the asteroid before collision, respectively. The angular momentum of asteroid after impact is L∗ = m∗ (r ∗ × v ∗ ) + I∗a ωa∗ ,

(10)

r ∗ is the radius vector of centre of mass of asteroid after collision, Ia ∗ is the inertia tensor of asteroid after collision, and ωa∗ is the rotational angular velocity relative to the centre of mass of asteroid after collision. When calculating the momentum of the ejecta, the rotation of asteroid should also be considered, since escape velocity of ejecta should contain the rotational velocity of impact point. Then the momentum of the ejecta turns into Pejecta = (1 − η)mimp vimp,n + (1 − ζ)mimp vimp,t (11) + mejecta ωa × rcrater , where mejecta is the mass of ejecta, which could be substituted by the excavated mass of asteroid material mcrater i.e. mcrater = mejecta , and rcrater is by the position vector of the centre of mass of the impact crater. The duration time of single impact can be expressed by the crater formation time tc (Holsapple and Housen, 2007), which is Rcrater tc = C1 p , Y /ρ

(12)

where constant C1 relates to the component material of target asteroid, Y is the material strength of the asteroid surface, Rcrater is the radius of the crater, ρ is the mass density of the target. Obviously, the equation consists with the assumption of strength-regime cratering process. Referring to Tab.1, the formation time of a 3-m crater, which corresponds to the largest crater in the simulation, is only about 0.2 s for non-porous asteroid. The time is so short that we ignore the position and rotation change of asteroid during impact 8

calculation in frame Ri and the position vector of ejecta after impact could be approximated linearly as rejecta = rcrater +

1 Pejecta tc . 2 mejecta

(13)

3.2. Momentum and angular momentum transfer efficiency The transfer efficiencies are calculated through the momentum multiplication factors η and ζ defined in Eqn.(5). Yanagisawa and Hasegawa (2000) obtained values of the factors through different impact experiments and found the factors depend on the impact angle, then they gave simple relations between transfer efficiencies and impact angle, but the results were only suitable in specific impact case. Holsapple and Housen (2012) achieved a comparatively complete relation between the efficiencies and the initial impact condition by assuming that the impact is normal and the ejecta have a cone-shaped axisymmetric distribution. They also gave formulas of different porous asteroid materials. In a general oblique impact scenario, the latter relations can be used to calculate the momentum transfer in normal direction. Since ejecta move in the opposite direction to the impactor in normal direction, the normal efficiency could be expressed by η= 1+

Pejecta,n , mimp vimp,n

(14)

where Pejecta,n and vimp,n are the scalars of the corresponding vectors respectively. Considering the strength-regime cratering process and referring to the study of Holsapple and Housen (2012), the normal factor of asteroid consisting of non-porous basalt is 0.65 η = 1 + (0.13)vimp,n Y −0.325 ρ0.125 δ 0.2 ,

(15)

where ρ and δ are mass density of the target and the impactor respectively, Y is the strength of the asteroid surface, and the cgs units are used for all variables. When the material is weakly cemented basalt (about 23% porosity), the normal factor is 0.2 η = 1 + (0.083)vimp,n Y −0.1 ρ−0.1 δ 0.2 .

(16)

As to the tangential factor ζ, Yanagisawa and Hasegawa (2000) found ζ positively correlated with the penetration depth for basaltic targets in their 9

impact experiments. An impactor with greater normal velocity impacting on a target with less strength would obtain a larger ζ and the value of ζ would decrease with the increasing incident angle. That means ζ is correlated with η, and may also have a functional relationship: ζ = f (vimp , θ, Y, ρ, δ),

(17)

which is similar to Eqn.(16). The specific expression of the above function may need to be obtained by a sufficient number of impact experiments. Yanagisawa and Hasegawa (2000) gave several values of ζ obtained from impact experiments and an expression of ζ only relating with impact angle ζ = ζ0 cos2 θ.

(18)

The expressions in Eqn.18 do not contain all the related parameters and value of ζ0 is obtained experimentally with specific impact case. The expression of ζ can be changed to adapt to a different impact case unless a general formalism is being modified. More impact experiments and simulations concerning oblique impact are needed to obtain a more accurate model of the tangential transfer efficiencies, which is beyond the scope of this paper. For the current first stage, the simulation is based on the experimental results of Yanagisawa and Hasegawa (2000). According to their experimental results on basalt targets, we work out the value of ζ0 is 0.178, 0.269 and 0.409, respectively corresponding to impact velocity 2 km s−1 , 3 km s−1 and 4 km s−1 . Then ζ0 is roughly fitted into a simple linear expression ζ0 = 0.09vimp (1 + pr),

(19)

where vimp is in the unit of km s−1 , pr is the porosity of the target asteroid. According to Yanagisawa and Hasegawa (2000), ζ is less than unity in all experiments. Since the simulation is based on their experimental results, the range of ζ is set to be [0,1). Since the range of ζ is very small, a rough value of ζ would not influence the results too much. Eqn.19 simply shows ζ positively correlates with impact velocity and porosity of target, which could be enough in preliminary impact simulation. More impact experiments are needed to construct a more accurate model of ζ, and find the possibility of ζ > 1. 3.3. The impact crater The formation of the crater will change the position of the centre of mass and the inertia tensor of the target asteroid, thus the rotational direction 10

may change correspondingly. Scaling laws presented by Holsapple (1993) are used to calculate the scale of a crater. The crater’s volume Vcrater is given by Vcrater =

mimp πV 4πa3 δ = πV , ρ 3 ρ

(20)

where a is the impactor’s radius, and πv is given by πV = K1 π2 =



ga

2 vimp,n

π2 ρδ


6υ−2−µ 3µ

, π3 =

Y

h + K2 π3

2 ρvimp,n

,

ρ δ

i 2+µ
6υ−2 2 3µ

3µ − 2+µ

,

(21)

where g is the surface gravity of the asteroid and could be neglected in the strength-regime cratering, constants K1 and K2 ,exponents µ and υ come from experiments and database (Holsapple, 2015). Based on the bowl-shaped (Hargitai and Watters, 2014) character of a simple crater, we model the impact crater as paraboloid of revolution. Since the size of the impactor is much smaller than that of the target, the surface is assumed to be flat at the impact location. The radius and depth of the crater are given by 1/3

1/3

Rcrater = Kr Vcrater , Dcrater = Kd Vcrater ,

(22)

where Kr and Kd are the shape constants specific for a material. In the simulations, non-porous and weakly cemented basalt material are chosen. Corresponding parameter values are listed in Table 1. For low strength materials, which comprise highly porous asteroids, its hard to make meaningful impact experiments at Earths surface that are still within our assumed strength regime because the much more compelling gravity would determine the cratering mechanics (Holsapple and Housen, 2012). Therefore, cases of low-strength, highly-porous asteroids are ignored in the simulations.

Porosity 0 23%

Table 1: Constants for basaltic material. C1 K1 K2 µ υ Y (dynes cm−2 ) ρ (g cm−3 ) 0.44 0.095 0.257 0.55 0.33 1.0 × 107 3.2 0.91 0.095 0.215 0.41 0.4 1.0 × 106 2.1

11

Kr Kd 1.1 0.6 1.1 0.6

3.4. Calculating the inertia tensor The initial shape of the asteroid is set as sphere or ellipsoid, the inertia tensor can be calculated analytically. The inertia tensor after the impact is changed because of the crater. Since the crater has a simple shape, the analytical form of inertia tensor of the crater can also be obtained. The calculation is in the inertia frame Ri during the impact process. The inertia tensor relative to the centre of mass of the target can be calculated in body fixed frame first and transformed into the inertia frame. The transformational matrix from body fixed frame into inertia frame is denoted as Mtran . Then the inertia tensor in the inertia frame is given by Ia = Mtran Ia,body MTtran ,

(23)

where Ia,body is the inertia tensor in body fixed frame B, superscript ‘T ’ denotes the transposition of the matrix. The inertia tensor of the portion that turns into crater, denoted as Icrater , is calculated in the same way. Then the inertia tensor of the asteroid after collision is given by I∗a = Ia − Icrater .

(24)

Since the centre of mass(CM) changed after collision, the inertia tensor in body fixed frame also needs to be updated to the new CM according to parallel axis theorem. 3.5. Tumbling motion The rotational angular velocity vector after impact is ωa∗ = I∗a −1 (Ia ωa + mimp rimp × vimp − m∗ r ∗ × v ∗ − Lejecta )

(25)

When ωa∗ is not aligned with the principal axis of maximal or minimal inertia, asteroid will start tumbling motion. Since the time of impact process is very short, the Euler angles are assumed to stay the same right during the impact. Before the next impact happens, the asteroid would keep tumbling motion. The Euler angles could be calculated by integrating the Euler kinematics equations, while at the same time the angular velocity in body-fixed frame (ωa∗)B could be calculated by integrating the Euler dynamics equations. The body-fixed frame B(o, x, y, z) is chosen so that the origin o is the centre of mass of target asteroid, the z-axis is aligned with the spin axis of 12

asteroid before the first impact, x-axis and y-axis can be arbitrarily set on the equator to complete the right-handed reference frame. Once the frame B is set, the orientation of coordinate axes remain the same in the following impact simulation, hence the CM shift would not influence the value of Euler angles. In addition, the frame Ri is different in each impact calculation of the multiple impact simulation. Transformation of coordinates between different Ri is needed, the orbital velocity change of asteroid is very small and the orbital plane is barely changed. The transform can be achieved by a rotation around the Z-axis and the rotational angle is just the angular variation of orbital velocity during the impact interval. 4. Results and discussions Fictitious target asteroids with spherical and ellipsoidal shapes are chosen respectively. The spherical asteroid’s diameter is 100 metres, the ellipsoidal asteroid has the same mass as the spherical one and its three principal axes have a ratio of 5:3:2, this ratio makes the asteroid slender. The impactor is a spherical copper ball, and its mass is 500 kg, density is 8.92 g cm−3 . For most asteroids with diameter less than 150 m, the spin period is less than 2 hours (Pravec et al., 2002). The spin period is chosen as 2 h before the first impact and the spin axis is aligned with the principal axis of maximal inertia. For ellipsoidal asteroid, the spin axis is aligned with the shortest axis. The impactor is very small, its rotational angular momentum can be ignored. The orbit of fictitious asteroid is generated by making the asteroid have same position as the Earth at MJD=62502.5 and a proper approach velocity. The orbital element in J2000 heliocentric equatorial reference frame is listed in Table 2 and the unit of angle is radian. Table 2: The keplerian element of fictitious asteroid at MJD62502.5.

a e i Ω ω M 0.861629 0.155806 0.403786 0.286781 4.252747 3.642686

4.1. The translational variation For simplification, the transformational matrix is set to be the identity matrix, i.e. the spin axis parallel to orbital angular momentum of target

13

(a)

1.1

(b)

3.5

1

3

0.9 2.5

∆v(cm s -1)

∆v(cm s -1)

0.8 0.7

2

1.5

0.6 1 0.5 0.5

0.4 0.3

0 0

10

20

30

40

50

60

0

2

4

6

8

10

v imp(km s-1)

θ imp(°)

Figure 1: The speed variation of the target asteroid versus (a) the impact angle with 5-km s−1 impact speed (b) the impact speed with 0 degrees impact angle. All the impact points are (0◦ , 0◦ ).

asteroid and the velocity parallel to the longest axis of equatorial ellipse, which is also the x-axis of the frame B. For both shapes, the velocity changes are almost the same when the impact situations are identical. Fig.1 shows the speed variation of the target impacted at (0◦ , 0◦ ), which is also the angular position of the vertex of the longest axis of the equatorial ellipse for ellipsoid. It is clear that a normal impact is the most efficient to transfer the momentum for both shapes. This is because the net ejecta momentum and the spacecraft momentum vector are aligned when impact is normal. The needed minimal deflection distance is 2.7r⊕ with the chosen asteroid’s approaching speed. Fig.2 shows the speed change needed to eliminate the threat for different warning times. It is known that the velocity changes along the motion of asteroid are the most efficient to change the orbital energy. The valleys in the curve correspond to periapsis. Therefore, impact deflection should be implemented when the asteroid is near periapsis for a longer warning time than one orbital period of target asteroid (Vasile and Colombo, 2008). After the asteroid is impacted, the needed velocity change is significantly reduced if the following impacts are implemented in a much shorter warning time. Fig.3 shows the directional variation of the deflection distance vector between two successive impacts, the asteroid’s orbital perturbation and flyby with other planets are ignored. The deflection distance vector is the vector distance of the asteroid from the Earth at the asteroid-Earth encounter. Since the change in orbital velocity of asteroid is very small, the deflection 14

32 (a) (b) (c)

30 28 26

−1

speed change (cm s )

24 22 20 18 16 14 12 10 8 6 4 2 0

0

2

4

6 warning time (yr)

8

10

Figure 2: (a) The minimal speed change for different warning times,(b) the variation of curve (a) after the asteroid has 1-cm s−1 velocity change, (c) the curve (b) moves up 1 cm s−1 . The right end of curve (b) is the moment that asteroid is impacted.

distance vector is nearly parallel even if the two impacts have a time interval of two orbital periods of asteroid and it can be predicted that the parallel is still fit when orbital perturbation of asteroid is considered. This means the deflection distance could be added linearly in a preliminary multiple impact deflection analysis. Therefore, if the time intervals of multiple impacts are very short relative to one asteroid orbital period or very close to one orbital period (intersection between curve (a) and (c) in fig.2), the translational change in multiple impact can be treated linearly. 4.2. The rotational variation 4.2.1. Spherical asteroid Obviously, an oblique impact lying in the equatorial plane of the spherical asteroid would change the rotational period more efficiently. Similarly, an oblique impact lying in the longitudinal plane is the most efficient to change the orientation of rotational angular velocity. Fig.4 shows the variation of spherical asteroid impacted in both situations. In the figure, ∆T = Ta∗ − Ta , 15

2 1.8 1.6 1.4

α (°)

1.2 1 0.8 0.6 0.4 0.2 0

0

0.5

1 ∆ timp (Torb)

1.5

2

Figure 3: The directional change of the asteroid’s deflection distance vector versus different time intervals of impact in two successive impacts. ∆timp is the time interval between two continuous impacts. α is the included angle between the two deflection distance vectors. The velocity change of asteroid after each impact is 1cm s−1 .

16

(a)

0.01

(b)

×10 -5

1

0.015

0

0.8

-1

0.6

-2

0.4

-3

0.2

-4

∆T β

0.8 0.7 0.6 0.5

-5 0.4

-6

β(°)

β(°)

0

0

∆T / Ta

∆T / Ta

0.005

-0.2 0.3

-7

-0.005

-0.4 -8 -0.6

-0.01 ∆T β

-60

-40

-20

0

20

40

-0.8 -1 60

0.2

-9 0.1 -10 0

θ imp(°)

10

20

30

40

50

0 60

θ imp(°)

Figure 4: The variation of rotational state of spherical asteroid, including the variation of spin period and direction of angular velocity, versus different impact angle. The impact point is (0◦ ,0◦ ), the impact speed is 5 km s−1 , and the pre-impact spin period is 2 h. The direction vector of the impact velocity is chosen as: (a) lying in the equatorial plane, (b)lying in longitudinal plane.

where Ta , Ta∗ are the rotational periods before and after the impact respectively, Ta = 2π/|ωa|, β is the angle between the rotational angular velocity before and after the impact. The negative impact angles in Fig.4(a) denote the impact velocity is along the rotational velocity of the target (the same below). The units for ∆T are in the spin periods of the asteroid before impact, same for the following figures in the paper. Considering the symmetry of sphere, impact point is chosen to be (0◦ ,0◦ ). It can be seen that the optimal impact angle to change the period the most is around 37 degrees. Since impacts on the equator will not change the rotational axis, asteroid can continue a stable rotation. In order to deflect a spherical asteroid efficiently and stably, the impact velocity should lie in equator plane and be normal. From Fig.4(b), it can be seen that the period is barely changed, which means the rotational velocities contained in the ejecta are not the main factor of the change of the rotational state. The rotational state of spherical asteroid changes very little, overall. 4.2.2. Ellipsoidal asteroid For ellipsoidal asteroid, its rotational change would be very different. Similar to spherical asteroid, an oblique impact lying on the equator would be the most efficient way to change the rotational period since it would not change the orientation of rotational axis. Fig.5 shows the variation of rotational period when ellipsoid is impacted at different points on the equator 17

Figure 5: The variation of the rotational period at different impact points on the equator of ellipsoid when the impact velocities lie in the equatorial plane, the impact speed is 5 km s−1 , the pre-impact spin period is 2 h.

with different impact angles. In order to change the rotational period efficiently, two pairs of symmetric sections, which relate to the eccentricity of equator, are the best option. When an impact is on the areas that are in the first and third quadrant of the equatorial ellipse, named despinning areas, the rotation always slows down, while the other two quadrants, named spinup areas, always have spin-up effects. The increase in the value of period is larger than the decrease of the value, which indicates it may be easier to despin the asteroid than to spin it up. The impact angle that changes period the most is around 6 degrees, which is approximately normal. Obviously, that angle will change if the eccentricity of equator changes. In order to know about rotational change of asteroid in different impact cases, the orbital velocity’s constraint on the impact direction is ignored in single impact simulation. Fig.6 shows the rotational response of ellipsoidal asteroid when impact is normal at every point on the surface. Both the rotational period and angular momentum change a lot at some specific locations. The despinning areas and spin-up areas have similar longitude distributions 18

Figure 6: The response of the rotational state of ellipsoid at different locations when the impact is normal, (a) the rotational period, (b) the rotational angular velocity. The impact speed is 5 km s−1 , the pre-impact rotational period is 2 h.

to Fig.5. When the impact velocities are in the equatorial ellipse or aligned with the two poles, the symmetry remains the same, the rotational angular velocity and the principal axis of maximum inertia are unchanged, and the asteroid would continue the stable spin. For the other impact case, the spin axis is not aligned with the principal axis, the asteroid starts a tumbling motion, which would lead to a more complicated rotational change in multiple impact. Since the spin period could decrease significantly for slender shape, there exists the possibility that the target disintegrates or tosses out some large pieces after several collisions when rotational angular speed increases to a critical value. Bombardelli (2009) gave a formula for the value. For a smallsize asteroid dominant for material strength, the flaws’ distribution in the rock plays an important role in disruption of the asteroid. Impacts may promote the growth of flaws and decrease the critical value of rotational angular speed, beyond which centrifugal force may disrupt the asteroid. As to the slender ellipsoidal asteroid, there are spin-up areas which are close to the longest end, where the centrifugal force is large. After several impacts are implemented on the spin-up area, both the centrifugal force and flaws near the longest end increase, then parts near the longest end may be tossed out, and the remaining parts could be spun up further. 19

Impact could also be a very simple and efficient way to despin an asteroid, which would make some asteroid exploring missions easier, like sampling, landing or mining on a small size asteroid. For asteroid redirect mission, kinetic impactors may be used to despin the target or change the orientation of rotation for ease of retrieval. Considering the target is no larger than 10 m (Brophy and Muirhead, 2013), much smaller impactors should be chosen. The target is so small that more advanced GNC technologies are needed to improve the current GNC accuracy (Hawkins et al., 2012), and the efficiencies of kinetic impactors used in such missions need further demonstration. As for porous asteroids, the rock strength is lower than for the non-porous ones, and the crater dimension is larger with the same impact velocity. Fig.7 shows the rotational period variation of porous asteroid impacted at different points on the equator and the velocity variation of porous asteroid impacted by impactors with different speeds. Asteroid has the same dimension as the former ellipsoidal one and a porosity about 23%. Pores would dissipate the kinetic energy preventing the production of ejecta, the rotational variation is much less than the non-porous target, and so is the speed variation, even if the porous asteroid has less mass. In addition, the impact angle leading to the largest variation of period is larger than non-porous ellipsoid. In the simulation, we also notice the angle only correlates with the eccentricity of the equator for the same porosity(including non-porosity). Generally, the variation of speed and rotation of porous asteroid is much less than nonporous one for the same shape. The formation of crater alters slightly the moment of inertia tensor which would also influence the rotational states. The crater’s influences on the rotational state are also shown. Fig.8 shows the variation of the crater’s size and centre of mass(CM) of asteroid versus normal impact speed. When the impact velocity is lower than 4 km s−1 , there would be a larger crater produced in porous asteroid. But the mass of ejecta produced in non-porous asteroid is always more than the porous one. Moreover, the CM shift is about 10−3 m, the crater’s volume is about a ten thousandth of the asteroid’s. A single impact just causes regional damage to the asteroid, and the CM shift can also be ignored in a continuous collisions calculation. Fig.9(a) indicates that the difference of single impact is tiny, while Fig.9(b) shows that tiny different could cause a huge difference in rotational state when asteroid has tumbling motion. Hence, in single impact simulation or an alltime nontumbling motion in multiple impact simulation, the crater could be neglected. And the tiny difference also indicates rotational variation is mainly because 20

Figure 7: (a)Variations of rotational period of porous asteroid impacted at different points on the equator. The impact velocities lie in equatorial plane. The impact velocity is 5 km s−1 and the pre-impact spin period is 2 h. (b) Variations of speed of porous asteroid versus different impact speeds, the impacts are normal at (0◦ ,0◦ ).

of the impact torque. Target asteroids with other slender irregular shapes would have a similar change. 4.3. Multiple impact As mentioned in Sec.4.1, if continuous impacts happened in a very short time compared to a whole orbital period of asteroid, each variation could be added linearly to obtain the total change. If the asteroid keeps non-tumbling rotation, the rotational change can also be added linearly. In order to keep the stability of rotation, the two locations should best be centrosymmetry on the equator. Referring to Fig.2, asteroid needs 2 cm s−1 velocity change for 10 yr warning time. In order to get such a speed change of a non-porous asteroid, we need a single impact with 8-km s−1 impact speed or two impacts each with 5.5-km s−1 impact speed. For porous asteroids, the needed impact numbers are about 3 times more. Although the fuel consumption may be much more expensive to launch several impactors with lower impact speed than to launch one impactor with higher impact speed, it is much easier to make impactors aim at certain regions for lower impact speed considering the target asteroid is also small. Moreover, it is easier to experiment with lower impact speed on ground, which may be helpful to learn about the impact consequence.

21

(a)

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Figure 8: (a) the size of impact crater in different normal impact speeds, (b) the variation of the centre of mass of asteroid in different normal impact speeds.

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Figure 9: The comparison of simulating results of rotational state between considering and ignoring the crater. (a)The difference of angular velocity after impact versus different impact speeds. The impact is normal at (37◦ , 0◦ ). (b) The included angle of the two angular velocities versus time after impact leading to tumbling. The impact speed is 5 km s−1 , impact point is (30◦ , 30◦ ).

22

(a)

7 Sphere Ellipsoid

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Figure 10: The variation of rotational period under different rotational periods with impact speed of 5 km s−1 . (a) Despin of both shapes, (b) Spin-up of both shapes. For spherical asteroid, impact point is (0◦ ,0◦ ), impact angle is 35 degrees, ellipsoidal asteroid is impacted normally at (37◦ ,0◦ ).

As to rotational variation, both stable spin and tumble could happen in multiple impact. For stable spin, it is mainly related to the rotational period of asteroid. Under the same impact condition, the longer period will have a larger change in value(see Fig.10(a)), which also means despin may be easier than spin up(see Fig.10(b)). For different spin angular velocity, the rotational kinetic energy change of the asteroid is almost the same with the same impact condition (e.g. the velocity, angle and location of the impact), then the relative change of the spin period is larger for a slower spin asteroid, and the it becomes easier and easier to despin the target asteroid. It is pointed out that the rotational period of a spherical asteroid with hard rocks is not easy to be changed while the rotational period of an ellipsoidal asteroid can be changed much more easily. Fig.11(a) is generated by enlarging the range of Ta in Fig.10(a) about ellipsoids. It can be seen that the impact almost makes the asteroid stop spinning when the rotational period is around 9.3 h. The yellow section in Fig.11(b) indicates the peak regions such as Fig.11(a) of different impact speeds. For a specific rotational period, there exists a corresponding impact speed that could nearly stop the rotation. Obviously, a faster rotation needs larger impact speed. Therefore, multiple impacts could not only slow down the rotation efficiently, but they may also despin the slender asteroid almost completely. 23

Figure 11: (a) The extension of the ellipsoid curve in Fig. 11(a) by enlarging the range of Ta , and the impact velocity is still 5 km s−1 (b) The variation of rotational period of ellipsoidal asteroids with different rotational periods impacted by different speeds. The impact is normal at (37◦ ,0◦ ).

When asteroid’s rotation becomes slow, the despinning areas also decline. Fig.12 shows the variation of rotation when the asteroid’s rotational period before impact is 8.5 h. In order to show the impact response of slow rotator, the constraint of orbital velocity on the impact angle is also ignored just like Fig.6. The bright sections, which mean an impact at those regions would cause period to be changed much more than in other regions. In Fig.12(a), it is much smaller than that in Fig.6(a). When the rotational period is close to 9.3 h for a 5-km s−1 impact, the bright areas almost reduce to a single point, which corresponds to the peak value in Fig.11(a). The small despinning areas restrict the sequence of time interval in multiple impact and also mean more accurate controls are needed to despin the asteroid efficiently. The orientation of angular velocity is also changed much more than that in Fig.6(b), which indicates that an efficient way to change rotational orientation may be to slow down the rotation first. In conclusion, it is much easier to change the rotational state when asteroid’s rotation slows down. As to tumbling motion, the impact consequence could be very complicated in multiple impact. Fig.13 shows consequence of impact during tumbling motion. The tumbling motion is caused by a 5-km s−1 impact at (30◦ ,30◦ ). The spin period before the first impact is still set as 2 h. The interval time of the second is 0.05 days. The impact velocity is parallel to the orbital velocity of asteroid, only the impact points that have < 60 degree impact angle are shown in Fig.13(a). Since the value of β does not have a large range 24

Figure 12: The response of the rotational state of ellipsoid at different locations when the impact is normal, (a) the rotational period, (b) the rotational angular velocity. The impact speed is 5 km s−1 , the pre-impact rotational period is 8.5 h.

of variation, the colored areas in Fig.13(b) are small. In the simulation, if the spin period before the first impact is set to be a larger value, then β would have a larger range of variation. The corresponding figure as Fig.13 would be very different. Since the orientation of spin axis keeps changing in inertial space, different time intervals of impact would also have different rotational change. Considering the great uncertainty in tumbling motion, the simulation of multiple impact in tumbling motion needs to combine specific mission demands. In general, just like the stable rotation, the rotational state will change more if the asteroid is in slower rotation. 5. Summary The translational variations of asteroid in the kinetic impact indicate that it is enough to use several small impactors to deflect potential hazard asteroids with a diameter around one hundred meters in more than 10 years warning time. Because there are more ejecta produced, it is more efficient to deflect non-porous asteroids. In a multiple impact scenario, the deflection distances after each impact are almost parallel to another, the deflection distances could be added linearly. When the time interval of each impact is very short or approximates to one orbital period, the total velocity changes along the direction of velocity of the target asteroid can also be added linearly. Moreover, the rotational variations of the asteroid are also shown. It is clear that single impact could change the rotational states a lot, especially for 25

Figure 13: (a) The variation of angular velocity after impact with different impact points. (b) The orientation distribution of angular velocity after impact in body-fixed frame, λ and φ are the angular coordinates of the angular velocity vector, and the red pentagram in the figure denotes the angular velocity before impact. The impact velocity is parallel to the orbital velocity of asteroid, the impact speed is 5 km s−1 , and the asteroid has a tumbling motion before impact.

non-porous asteroids with slender shapes, and in the case of multiple impact, it is also shown that to despin the asteroid is much easier than to spin it up, which means the kinetic impact may be used in small asteroid exploration which needs the slowing down of the rotation like asteroid mining and asteroid redirect mission. In addition, a normal impact at sections near equator may be appropriate to keep the asteroid rotating as stably as possible in a multiple impact deflection scenario. Since the impact momentum is relatively low, the formation of the crater could be neglected in single impact simulation, but once the asteroid begins to have a tumbling motion after the impact, the small crater would significantly affect the rotational state in the following time. In the future, a multiple impact scenario having large interval time between each impact will be under consideration. The minimum fuel consumption performance may affect the tradeoffs between choosing to impact near perihelion or another orbital position. Furthermore, the smaller component of a binary asteroid may also be a suitable target in following research.

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Acknowledgments The authors wish to thank the anonymous reviewers for their valuable suggestions. We thank Mike Owen for kindly explaining their simulation results. This research was partially supported by the National Key Basic Research Program of China or 973 Program (2015CB857100), the National Basic Research Program 973 of China (2013CB834103) and the Satellite Communication, Navigation Collaborative Innovation Center (SatCN201409), and the Base of National Defense Scientific Research Fund (No. 2016110C019). References Ahrens, T. J., Harris, A. W., Dec 3 1992. Deflection and fragmentation of near-earth asteroids. Nature 360 (6403), 429–433. Asphaug, E., Ostro, S. J., Hudson, R. S., Scheeres, D. J., Benz, W., Jun 4 1998. Disruption of kilometre-sized asteroids by energetic collisions. Nature 393 (6684), 437–440. Bombardelli, C., Oct-Nov 2009. Artificial spin-up and fragmentation of subkilometre asteroids. Acta Astronautica 65 (7-8), 1162–1167. Bombardelli, C., Pelez, J., 2011. Ion beam shepherd for asteroid deflection. Journal of Guidance, Control, and Dynamics 34 (4), 1270–1272. Brophy, J. R., Muirhead, B., 2013. Near-earth asteroid retrieval mission (arm) study. Vol. 33 of International Electric Propulsion Conference. Brown, P., Spalding, R., ReVelle, D., Tagliaferri, E., Worden, S., 2002. The flux of small near-earth objects colliding with the earth. Nature 420 (6913), 294–296. Brumfiel, G., 2013. Russian meteor largest in a century. Nature 12438. Chapman, C. R., Morrison, D., 1994. Impacts on the earth by asteroids and comets: assessing the hazard. Nature 367 (6458), 33–40. Chyba, C. F., Thomas, P. J., Zahnle, K. J., 1993. The 1908 tunguska explosion: atmospheric disruption of a stony asteroid. Nature 361 (6407), 40–44. 27

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Appendix A. Momentum conservation in the small impact The momentum change of the system containing both the impactor and target asteroid after single impact is R tc Fsun dt = (m∗ (v ∗ + va1 ) + P ejecta )− 0 (A.1) (ma va0 + mimp (vimp + va0 )), where Fsun is the solar gravity of the system, tc is the crater formation time i.e. the duration time of the impact process, va0 is the orbital velocity of the target asteroid right before the impact, va1 is the orbital velocity of asteroid at time tc if impact does not happen, v ∗ is the velocity change of asteroid, ma and m∗ are the masses of target asteroid before and after the impact respectively, mimp is the mass of the impactor, vimp is the velocity of the impactor relative to the target asteroid, P ejecta is the momentum of ejecta when the impact process is ended, which is Z tc Fes dt, (A.2) P ejecta = mejectava1 + Pejecta + 0

where mejecta is the mass of all the ejecta, Fes is the solar gravity of all the ejecta, Pejecta is the relative momentum of all the ejecta, which is Pejecta =

Z

tc

me (t)ve (t)dt,

(A.3)

0

where me (t) is mass of ejecta R tc which is just leaving the surface of the asteroid at time t and mejecta = 0 me (t)dt, ve (t) is velocity of ejecta relative to the asteroid when the ejecta just leaves the surface of the asteroid. Moreover, when target is still before an impact, Pejecta is just the momentum of ejecta after the impact. This impact situation is similar to the impact experiment 30

on ground, hence Pejecta can be calculated by referring to the experimental results. The mass of the system is mc = mimp + ma , and m∗ = mc − mejecta . Substituting Eqn.A.2 into Eqn.A.1, the momentum change of the system becomes R tc R tc ∗ ∗ Fes dt)− F dt = (m v + m v + P + sun c a1 ejecta 0 0 (A.4) (mc va0 + mimp vimp ). Since the solar gravity mainly changes the orbital velocity of the system, the linear momentum from the Sun pull is Z tc Z tc Fes dt. (A.5) Fsun dt = mc (va1 − va0 ) + 0

0

Subtracting Eqn.A.5 from Eqn.A.4, then the momentum change of target asteroid caused by the single impact could be written as m∗ v ∗ = mimp vimp − Pejecta .

(A.6)

It should be pointed out that the velocity change v ∗ is relative to the orbital velocity of asteroid va1 . Since the crater formation time tc is very much shorter than the orbital period of asteroid, va1 ≈ va0 . The velocity change can be added to va0 in the orbital calculation after impact. This means the Sun’s pull barely influence the momentum change during the impact even when the incident momentum approximates to the linear momentum from the Sun pull. The momentum conservation can still be used in the gentle impact.

31