Eur. Phys. J. Special Topics 229, 1463–1477 (2020)c© EDP Sciences, Springer-Verlag GmbH Germany,

part of Springer Nature, 2020https://doi.org/10.1140/epjst/e2020-900183-y

THE EUROPEAN

PHYSICAL JOURNALSPECIAL TOPICS

Regular Article

Dynamics of tethered asteroid systemsto support planetary defense

Flaviane C.F. Venditti1,a, Luis O. Marchi2, Arun K. Misra3, Diogo M. Sanchez2,and Antonio F.B.A. Prado2

1 NASA Solar System Exploration Research Virtual Institute (SSERVI) and PlanetaryRadar Department, Arecibo Observatory, University of Central Florida, Arecibo,PR, USA

2 Division of Space Mechanics and Control, National Institute for Space Research,Sao Jose dos Campos, SP, Brazil

3 Mechanical Engineering Department, McGill University, Montreal, QC, Canada

Received 28 August 2019 / Received in final form 23 November 2019Published online 29 May 2020

Abstract. Every year near-Earth object (NEO) surveys discover hun-dreds of new asteroids, including the potentially hazardous asteroids(PHA). The possibility of impact with the Earth is one of the mainmotivations to track and study these objects. This paper presents atether assisted methodology to deflect a PHA by connecting a smallerasteroid, altering the center of mass of the system, and consequently,moving the PHA to a safer orbit. Some of the advantages of thismethod are that it does not result in fragmentation, which could lead toanother problem, and also the flexibility to change the configuration ofthe system to optimize the deflection according to the warning time.The dynamics of the PHA-tether-asteroid system is analyzed, and theamount of orbit change is determined for several initial conditions.Only motion in the plane of the orbit of the PHA around the Sunis considered, thus the PHA chosen for the simulations has low orbitinclination. Analysis of the dynamics of the system shows that themethod is feasible for planetary defense.

1 Introduction

To date, more than 21 000 near-Earth asteroids (NEAs) have been discovered, includ-ing almost 2000 PHAs. Small bodies with perihelion of less than 1.3 AU and havingorbits passing close to the orbit of the Earth are called near-Earth objects (NEOs),which can include asteroids and comets. Among NEAs there are the potentially haz-ardous asteroids (PHAs), which are objects larger than about 140 m and that canget closer than 0.05 AU, or approximately 20 times the distance from the Earth tothe Moon.

Potential impacts of NEOs are one of the biggest motivations to study and detectthese objects. The threat of an asteroid on the collision path with the Earth hasencouraged the development of several deflection techniques. Based on the warning

a e-mail: venditti@naic.edu

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time, the deflection method can be chosen from months to several years, or evendecades. Some existing methods in the literature for this purpose are: fragmentationof the asteroid using nuclear explosives or collision with a massive asteroid [1]; usingthe impulse of a direct collision on the asteroid, called the kinetic impact method [2,3];the use of solar energy with solar sails to cause a boost generated by the evaporationof the surface layers, slightly pushing the asteroid [4]; the use of the gravitational pullof a thereby stationary spacecraft or in a “tugging” mode near an asteroid to deflectit slowly, which is called the gravity tractor method [5,6]. In this work, the use of atether assisted technique is considered. It consists of connecting two asteroids, a PHAand a smaller asteroid nearby, so that the motion of the secondary asteroid couldchange the initial trajectory of the larger one. The methodology aims to transfer aPHA to a new safer orbit through the displacement of the center of mass. Thus, nounwanted consequences related to fragmentation would happen after the deflection.The applications of this technique are especially important for planetary defense, butcould also help in the scientific exploration of these objects.

The study of small bodies has been growing fast in the past decade, and there areseveral missions to explore closely these objects, as well as planned for the future.One of the reasons to study asteroids and comets is that they may carry valuableinformation about the formation of the Solar System. Some of the past missionsare: NEAR that landed on asteroid Eros [7]; Hayabusa, a mission developed by theJapanese Space Agency with the goal of collecting material from asteroid Itokawaand bring back samples [8]; Dawn orbiting Ceres and Vesta, which are the mostmassive asteroids in the solar system, respectively [9]; PROCYON and Hayabusa 2[10], both launched in 2014, meeting their target asteroids in 2016 and 2018, respec-tively. There are also missions that were launched to orbit comets, such as Star-dust [11] in order to collect samples from the tail of comet P/Wild2, and Rosetta,which after performing a flyby on asteroids Steins and Lutetia, landed on comet67P/Churyumov–Gerasimenko in late 2014 [12].

There are also ongoing and planned missions to small bodies. Launched in 2016,the OSIRIS-REx mission (Origins, Spectral Interpretation, Resource Identification,Security, Regolith Explorer) encountered the potentially hazardous asteroid 1999RQ36, or 101955 Bennu, by the end of 2018, and after mapping the surface of theasteroid it will collect a sample to bring back to Earth [13–16]. A mission namedLucy, scheduled to launch in 2021, will have the goal to explore six Trojan asteroidsaround the orbit of Jupiter [17]. The motivation arises due to the fact that Trojansare thought to be remnants of the primordial material that formed the outer planets,holding important information about the formation of the Solar System. Anothermission planned for the future is Psyche, a solar electric propulsion spacecraft concepttargeted to be launched in 2022 [18]. Psyche appears to be the exposed nickel-ironcore of an early planet, which might help to understand planetary formation. It willbe the first time that a metallic asteroid is visited. Another pioneer mission recentlyselected by NASA is Janus. The goal is to explore two binary asteroids to betterunderstand how primitive bodies form and evolve into multiple asteroid systems.

2 Planetary defense – near-Earth asteroids

The Solar System has a large number of irregularly shaped bodies. These objectsare asteroids, comets, and even some satellites of planets. Most asteroids are locatedbetween the orbits of Mars and Jupiter, in the main belt asteroid, but NEAs orbitmuch closer, and sometimes may come uncomfortably near, even crossing the Earth’sorbit, which are NEAs part of the Aten and Apollo groups.

Celestial Mechanics in the XXIst Century 1465

The number of NEAs discovered by optical and infrared surveys grows each year,and most of the PHAs larger than 1 km are known. However, the list of asteroids largerthan 140 m is still a work in progress. In addition, some asteroids that were not con-sidered a threat may have their orbits perturbed to a point where it could eventuallybecome dangerous. An example is the thermal radiation driven Yarkovsky/YORPeffect [19,20], or even by collision with other objects. Asteroids smaller than 140 mare more challenging to be detected by NEA surveys, but the damage in case it is onthe collision course with the Earth is still considerable, like the Chelyabinsk meteorin Russia in 2013 [21].

Before sending a spacecraft to an asteroid, the environment around it must becarefully mapped. The first step is to obtain data of the asteroid by performingobservation with ground-based or space telescopes. Characteristics such as shape,mass, rotation, and surface properties are some of the important information thatshould be known in advance, and can be obtained with ground radar observations atthe Arecibo Observatory, in Puerto Rico, or the Goldstone Solar System Radar, inCalifornia [22]. Characterization is crucial especially for landing and sample returnmissions. Practically all missions to small bodies select targets that can be observedwith radar prior to the mission.

In the history of space missions there is no record of an asteroid mitigation testperformed yet. The first proposed mission is the international collaboration AIDA(Asteroid Impact and Deflection Assessment) composed by NASA’s DART, and theEuropean Space Agency’s Hera. DART stands for Double Asteroid Redirection Test,which will consist of testing the effects of kinetic impact. The goal is to crash asatellite on the secondary component of the binary system Didymos and analyze theconsequences of the impact on the system [23]. One of the studies related to thismission is the effects of the ejecta resulted from the impact.

3 Methodology

3.1 Space tethers

Tethers are long space cables with several different applications. Some of the firststudies using the concept of space tethers started with the space elevator idea [24],and lunar elevator [25]. Also the use of tether satellite systems [26], tether nets fordebris removal [27,28], and using tethers for power and propulsion [29], to name afew. Some projects that make use of tethers are: Tether Physics and Survivability(TiPS), from the US Naval Research Laboratory, with the goal to understand howthe libration motion of endmasses affects the motion of the center of mass of thesystem; formation flying tethers, which are tether systems that can enable groups ofsatellites to fly in tight formation, for applications such as long baseline interferome-try, like SPHERES (NASA and MIT); electrodynamics tether, such as the TetheredSatellite System Reflight (TSS-1R), with the goal of interacting with the planet’smagnetosphere to generate power or propulsion without consuming propellant. TUI,an electrostatic radiation belt remediation project, for safer manned and unmannedmissions in Earth’s orbit.

3.2 PHA-tether-asteroid dynamics

The equations of motion consist of four coupled equations, which include the orbitalparameters for the tethered system relative to the Sun, the rotation of the PHA,and also the pendular motion of the secondary asteroid in relation to the PHA. The

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Fig. 1. Configuration of two asteroids connected by a tether.

configuration of the system consists of an irregularly shaped body representing thePHA, connected to another asteroid with a tether [30,31]. The smaller asteroid isused as an artifact to perturb the orbit of the main asteroid, and it is modeled asa point mass. In this paper only planar motion is considered, therefore only PHAswith very low orbit inclination are appropriate for this study.

Small bodies have spin rate that can range from seconds to hours, and the tetherdynamics is strongly affected by the rotation of the main body. Therefore, in orderto obtain a more accurate model, the pendulum motion of the tether is considered.The rotation period of the PHA is adopted, but for the smaller asteroid connectedwith a tether the rotation is neglected, since its rotational motion would be affectedby the orbit transfer to the PHA. In this study, the tether is considered inextensibleand massless, since these parameters would depend on the material of the tether, andthis is out of the scope of this work. The system described is shown in Figure 1.

The mass of the PHA is represented as mA; mB is the mass of the smaller asteroid;M is the mass of the Sun; R the distance between the Sun and the PHA; RB thedistance between the Sun and the smaller asteroid; ` the length of the tether; A isthe center of mass of the PHA; rB is the distance between the smaller asteroid andthe center of mass of the PHA; P is the point of attachment of the tether; ν is thetrue anomaly of the PHA; θ is the rotation angle of the PHA; α is the angle that thetether makes with the PHA, which gives the pendulum motion.

The equations of motion of the system are derived according to Lagrange’s equa-tion, given by equation (1).

d

dt

(∂L

∂qi

)− ∂L

∂qi= 0, qi ≡ R, ν, θ, α, (1)

where, qi ≡ R, ν, θ, α are the generalized coordinates that describe the dynamics ofthe system.

The total kinetic and potential energy of the system are given by equations (2)and (3), respectively:

K =12mAν

2Axy

+12mBν

2Bxy

+12IA

(θ + ν

)2

(2)

U = −GMmB

R

[1− l

Rcos (θ + α)

]− GMmA

R−mBUPHA, (3)

where IA is the moment of inertia. UAST is the gravitational potential for the PHA,and the general expression considered for a non-spherical body using the spherical

Celestial Mechanics in the XXIst Century 1467

harmonics approach is given by equation (4) [32–34].

UPHA =GmA

l

1− C20

2

(r0l

)2

+ 3C22 cos (2θ0 + 2α)(r0l

)2. (4)

According to the configuration of the system PHA-tether-asteroid in Figure 1, therelative velocities can be obtained:

~νAxy= Ra1 +Rνa2 (5)

~νB/Axy=[−l sin (α+ θ)

(α+ θ + ν

)− rP/A sin(θ)

(θ + ν

)]a1

+[l cos (α+ θ)

(α+ θ + ν

)+ rP/A cos(θ)

(θ + ν

)]a2 (6)

~νBxy=[R− l sin (α+ θ)

(α+ θ + ν

)− rP/A sin(θ)

(θ + ν

)]a1

+[Rν + l cos (α+ θ)

(α+ θ + ν

)+ rP/A cos(θ)

(θ + ν

)]a2. (7)

Using equations (5)–(7), it is possible to obtain the scalar product of the velocityvectors for the kinectic energy, represented by equations (8) and (9).

v2Axy

= R2 +R2ν2 (8)

v2Bxy

= R2 +R2ν2 + l2(θ + α+ ν

)2

+ rP/A2(θ + ν

)2

+ 2lrP/A

(θ + ν

)(θ + α+ ν

)cos(α) + 2l

(θ + α+ ν

)×(−R sin (θ + α) +Rν cos(θ + α)

)+ 2rP/A

(θ + ν

) [−R sin(θ) +Rν cos(θ)

]. (9)

The distance from the center of the PHA to the point of attachment, representedby rP/A in the previous equations, is much smaller compared to the tether lengths,and it is not likely to have significant influence. Thus, it is neglected from now on inorder to simplify the calculations, and only the distance from the center of mass ofthe PHA is considered.

Finally, the Lagrangian of the system represented in Figure 1 is obtained usingthe kinetic and potential energy, and is given by equation (10).

L = K − U =12

(mA +mB)(R2 +R2ν2

)+

12IA

(θ + ν

)2

+12mB

[l2(θ + α+ ν

)2

+ 2l(θ + α+ ν

)(−R sin(θ + α)

+ Rν cos(θ + α))]

+GMR

(mA +mB) +GmAmB

l

1− C20

2

(r0l

)2

+ 3C22 cos (2θ0 + 2α)(r0l

)2− GMl

R2mB cos(θ + α). (10)

The four equations of motion are derived according to equation (1), and are shownin equations (11)–(14) for α, R, θ, and ν, respectively:

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α : mBl

[l(θ + α+ ν

)− R sin(θ + α) +

(Rν +Rν

)cos(θ + α)

−(R cos(θ + α) +Rν sin(θ + α)

)(θ + α

)+(R cos(θ + α) +Rν sin(θ + α)

)×(θ + α+ ν

)+

6GJ22mAr20 sin 2(θ0 + α)l4

− GMR2

sin(θ + α)]

= 0 (11)

R : (mA +MB)[R−Rν2 +

GMR2

]−mBl

[(θ + α+ ν

)sin(θ + α)

+((

θ + α+ ν)2

+ 2GMR3

)cos(θ + α)

]= 0 (12)

θ : IA(θ + ν

)+mBl

[l(θ + α+ ν

)+(Rν +Rν

)cos(θ + α)

− R sin(θ + α)−(R cos(θ + α) +Rν sin(θ + α)

)(θ + α

)+(θ + α+ ν

)(R cos(θ + α) +Rν sin(θ + α)

)− GM

R2sin(θ + α)

]= 0 (13)

ν : IA(θ + ν

)+ (mA +mB)

(2RRν +R2ν

)+mBl

[l(θ + α+ ν

)− R sin(θ + α)

+ 2Rν cos(θ + α) +R(− sin(θ + α)(θ + α+ 2ν) + cos(θ + α)(θ + α+ 2ν)

)]= 0.

(14)To facilitate the analysis, the equations of motion are non-dimensionalized divid-

ing the length parameters by the semi-major axis, and multiplying the time variablesby the mean orbital rate.

Next, the four equations of motion will be used to obtain the orbit variationgenerated by adding the tether system. Note that the parameter R, represented inequation (12), provides the amount of change in the orbit when compared to anunperturbed orbit, and it will be carefully analyzed. Thus, the objective is not toanalyze each of the four parameters individually, but how the four coupled equationscombined affect the orbit of a PHA.

4 Results

4.1 101955 Bennu (1999 RQ36)

Asteroid 101955 Bennu (1999 RQ36) was selected for the simulations. One reason isbecause it is classified as a PHA, passing close to Earth about every 6 years, and ithas one of the highest impact hazard ratings among PHAs. Another reason is becauseit has low orbit inclination. Bennu is a B-type (primitive and carbon rich), about492 m in diameter, and spins every 4.3 h. The next close approach within 2 lunar dis-tances (LD) will happen in 2060, when Bennu will pass at a distance of approximately0.005 AU (∼1.95 LD). Bennu was discovered on September 11th 1999, by the LIN-EAR survey. During its discovery apparation, it was possible to observe it with theArecibo Observatory’s S band radar system (2.38 GHz, 12.6 cm) and the GoldstoneSolar System Radar (8.56 GHz, 3.5 cm), and again in 2005 and 2011. Delay-Dopplerradar images helped to characterize this asteroid, allowing to constrain Bennu’s phys-ical and orbital properties enough to support the mission to come years later [35].

Launched in 2016, the OSIRIS-REx sample return mission approached asteroidBennu by the end of 2018, sending astonishing images. It revealed a surface covered

Celestial Mechanics in the XXIst Century 1469

Fig. 2. Bennu from OSIRIS-REx spacecraft (NASA/Goddard/University of Arizona).

Table 1. Physical and orbital parameters for Bennu.

Bennu parameters ValuesDiameter (km) 0.492Rotational P (h) 4.29a (AU) 1.1263910259e 0.20374510i() 6.034939Ω() 2.060867ω() 66.223068M() 101.7039479Perihelion dist. (AU) 0.89689436Aphelion dist. (AU) 1.3558876877Orbital P (days) 436.64872813

ρ(g/cm3) 1.26Mass (Kg) 7.327× 1010

with boulders of dozens of meters in size [36]. Figure 2 shows an image of Bennufrom the OSIRIS-REx spacecraft.

The orbital and physical parameters of Bennu used for the simulations are listed inTable 1 [37]. The mass was obtained based on radiometric measuments from OSIRIS-REx [38].

4.2 Simulations

Simulations for several different configurations were performed using the MATLABsoftware. Three initial parameters were tested in order to compare the optimal con-figuration: mass for the smaller asteroid, tether lengths and points of attachment forthe tether. The deflection is measured by analyzing the difference between the ini-tial unperturbed orbit and the perturbed orbit (after connecting the smaller asteroidwith the tether). The integration step used for the simulations was 60 s.

4.2.1 Orbit deviation

The initial configuration used for the system was obtained calculating the optimal tra-jectory between 2020 and 2040, for a hypothetical deflection mission before Bennu’sclose approach to Earth of approximately 0.005 AU, in 2060. The trajectory opti-mization FORTRAN code used for this task has been established for transferring aspacecraft from Earth to Bennu in order to minimize the fuel consumption, using the

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Fig. 3. Orbit deviation for initial mass ratio of mB/mA = 1/1000, and tether lengths of1000 km, 2000 km, and 3000 km.

Fig. 4. Orbit deviation for initial mass ratio of mB/mA = 1/10000, and tether lengths of1000 km, 2000 km, and 3000 km.

patched-conics approach [39–41]. The optimal launch date for the spacecraft obtainedis October 28th, 2035. The tether is considered in the simulations at the moment ofarrival of the spacecraft to Bennu. Since the focus of this work is to analyze thedynamics of the tethered asteroid system, the logistics of capturing and transferingthe smaller asteroid is not included in this study.

To analyze the efficacy of the deflection method, the first step was to obtain theorbit deviation compared to the initial unperturbed trajectory (before the tether-small asteroid attachment). To test how the initial parameters might affect theresults, it was considered two different mass ratios and three tether lengths. The orbitdeviation (∆) is the distance between the unperturbed (Keplerian) and the perturbedorbit (with the tether-asteroid system). The orbit deviation is measured in terms ofthe Earth’s radius (REARTH = 6371 km). The simulations are performed for a periodof 300 years. Figures 3 and 4 show the orbit deviation for a mass ratio (mB/mA) of

Celestial Mechanics in the XXIst Century 1471

Fig. 5. Orbit deviation for α(0) = 45 initial mass ratio of mB/mA = 1/1000, and tetherlengths of 1000 km, 2000 km, and 3000 km.

1/1000 and 1/10000, respectively. For both cases tether lengths of 1000 km (blue),2000 km (green), and 3000 km (magenta) were considered.

The results show in both cases a clear increase in the orbit deviation for longertethers. For tethers three times longer, the deviation values are of an order of 5 timeshigher than for the shorter tether chosen for the simulations. Comparing differentmass values for the smaller asteroid attached with a tether, it is evident that, for afaster deviation, larger masses would be required.

4.2.2 Tether point of attachment

In the previous simulations it was considered that α(0) = 0. One situation to con-sider is if the point of attachment for the tether on the PHA would influence theresults. In order to analyze the change in the orbit due to the tether attachmentangle, it was also tested α(0) equal 45 and 90. Figures 5 and 6 show the orbitdeviation (∆) for a mass ratio between Bennu and the smaller asteroid of 1/1000,and an initial tether attachment angle of 45 and 90, respectively.

Table 2 shows the approximate values for the orbit deviation in terms of theEarth’s radius considering a mass ratio of mB/mA = 1/1000, tether lengths of1000 km (blue) 2000 km (green), and 3000 km (magenta) and point of attachmentfor the tether of α(0) = 0, 45, and 90.

According to Table 2, it is evident that the deflection increases with the tetherlength and the angle of attachment between the tether and the PHA. Based onthe results the most efficient scenario would be for α(0) = 90 and tether lengthof 3000 km. Next, we will investigate how the orbit change analyzed so far wouldinfluence Bennu’s proximity to the Earth.

4.2.3 Close approaches to the Earth

Besides knowing how much the orbit of the PHA deviated from its original trajectory,it is important to know if the minimum orbit distance to the Earth increased ordecreased. For an asteroid on the collision path with the Earth, it would be risky to

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Fig. 6. Orbit deviation for α(0) = 90, initial mass ratio of mB/mA = 1/1000, and tetherlengths of 1000 km, 2000 km, and 3000 km.

Table 2. Orbit deviation ∆ (REARTH) for different tether point of attachment angles andtether lengths l for mass ratio mB/mA = 1/1000.

α(0) = 0 α(0) = 45 α(0) = 90

l = 1000 km 400 1600 2000l = 2000 km 800 3200 3800l = 3000 km 1200 5000 5900

bring the asteroid any closer. However, an exploration mission such as sample returnor resource utilization, could benefit from having the asteroid placed on a new closersafe orbit.

The parameter δ represents the distance from the PHA to the Earth. Similarto previous cases, it is measured in terms of Earth’s radius. For a mass ratio ofmB/mA = 1/1000, Figure 7 shows Bennu’s distance from the Earth for a period of300 years, considering the same three tether lengths used for the previous simulations.The closer δ gets to zero, the closer the system would be from the Earth. For thisstudy we will use only the case of point of attachment for the tether of α(0) = 0.

In Figure 7, the lines for different initial parameters overlap. For a better visu-alization, Figure 8 shows the same graph, but only for δ less than 2000REARTH.As a comparison, the black line represents Bennu’s trajectory without any addedperturbation. If the lower spike is in black, it would mean that none of the tetherlengths considered for the simulations affected the minimum orbit distance for thatspecific approach date. There are two very close approaches that will be investigatedin detail.

In Figure 9, the same tether lengths and point of attachment used in Figures 7and 8 are considered, except for the mass ratio. Now we will consider the case of aless massive mB , with a mass ratio of mB = mA/10000.

In Figures 8 and 9, it is possible to see two very close approaches for each graph,around 122.9 years and 226 years, and around 55 years and 230 years, respectively.We will first discuss Figure 8, with the close approaches in detail in Figure 10 (leftcolumn). The first approach was caused by adding a tether of 2000 km (green line),and even the other tether lengths caused the PHA to come closer than if no orbit

Celestial Mechanics in the XXIst Century 1473

Fig. 7. Approximations to Earth for mass ratio of mB/mA = 1/1000, and tether lengthsof 1000 km, 2000 km and 3000 km.

Fig. 8. Approaches to the Earth within 2000 Earth’s radius for mass ratio of mB/mA =1/1000, and tether lengths of 1000 km, 2000 km and 3000 km.

perturbation was added. This is a good example of a case suitable for asteroid explo-ration support, in which the object could be brought closer if required, but nota desired configuration for planetary defense purposes. For the second very closeapproach, around 226.9 years, adding the tether with a smaller asteroid helped toavoid the close approach, altering the center of mass of the PHA in a way that thedistance to the Earth at that time increased. In this case the longer tether was themost efficient configuration, avoiding a possible collision with the Earth by addingto the minimum approach distance about 900REARTH.

The same study was performed for a mass ratio of mB/mA = 1/10000, meaninga smaller mass to be attached to the PHA with a tether. Figure 9 shows the approxi-mation to Earth closer than 2000 Earth’s radius. It is possible to notice two momentswhere the PHA-tether-asteroid system gets very close.

Figure 10 shows the close approaches in detail (right column). The first significantapproach happens near 56.26 years after the tether-asteroid attachment. The black

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Fig. 9. Approaches to the Earth within 2000 Earth’s radius for mass ratio of mB/mA =1/10000, and tether lengths of 1000 km, 2000 km and 3000 km.

Fig. 10. Approaches to the Earth within 2000 Earth’s radius in detail for tether lengths of1000 km, 2000 km and 3000 km, and mass ratios of mB/mA = 1/1000 (left) and mB/mA =1/10000 (right).

Celestial Mechanics in the XXIst Century 1475

line, without the tether-asteroid added, shows a closer approach Adding the tethersin this case helped to increase the distance of minimum approach to Earth, with thelonger tether of 3000 km (magenta) resulting in the largest deflection. The secondvery close approach happens around 226.9 years. The black line, representing thePHA without the tether, shows a dangerous approach of practically less than 20Earth’s radius, or approximately 1/3 of the distance from the Earth to the Moon(∼0.0008 AU). Adding the tether-asteroid system helped to deflect the PHA, withthe longer tether increasing the close approach from less than 20REARTH to morethan 100REARTH.

5 Conclusion

The dynamics of a tether assisted technique was presented consisting of four coupledequations of motion for the system, which include the orbital parameters for thetethered system relative to the Sun, the rotation of the PHA, and also the pendularmotion of the secondary asteroid in relation to the PHA. The methodology presentedproved to be efficient in supporting planetary defense. The ability to change the centerof mass of a PHA on the collision course with the Earth can increase the minimumorbit distance during the PHA’s passage. Several initial configurations were tested,such as different tether lengths, points of attachment for the tether on the PHA, anddifferent mass ratios between the PHA and smaller asteroid to be connected with atether. The results suggest that, for a faster deflection, longer tethers and a moremassive asteroid attached to the PHA would be more effective.

The simulations showed that the method is dynamically feasible for asteroidimpact mitigation, and it could also be used to facilitate exploration missions, suchas sample return and resource utilization missions. The technique studied offers theflexibility to adjust the amount of deflection by changing the different parametersfor the system, considering the warning time available. In addition, this method doesnot result in fragmentation, which could be a potential risk.

The authors wish to express their appreciation for the support provided by NASA throughgrant #NNX13AQ46G awarded to Universities Space Research Association (USRA) andgrants #80NSSC18K1098 and #80NSSC19K0523 awarded to the University of CentralFlorida (UCF). Grants #140501/2017-7, 301338/2016-7 and 406841/2016-0 from the NationalCouncil for Scientific and Technological Development (CNPq) grants 2014/22295-5 and2016/24561-0 from Sao Paulo Research Foundation (FAPESP). This work was also par-tially supported by NASA Solar System Exploration Research Virtual Institute contractNNA14AB07A (PI David A. Kring).

Publisher’s Note The EPJ Publishers remain neutral with regard to jurisdictional claimsin published maps and institutional affiliations.

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