Dynamics of Tethered Asteroid Systems to Support Planetary Defense

Home , Gravity tractor, Janus (spacecraft)

Eur. Phys. J. Special Topics 229, 1463–1477 (2020) c EDP Sciences, Springer-Verlag GmbH Germany, THE EUROPEAN part of Springer Nature, 2020 PHYSICAL JOURNAL https://doi.org/10.1140/epjst/e2020-900183-y SPECIAL TOPICS Regular Article

Dynamics of tethered asteroid systems to 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 Planetary Radar 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 2019 Published 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 main motivations to track and study these objects. This paper presents a tether assisted methodology to deflect a PHA by connecting a smaller asteroid, altering the center of mass of the system, and consequently, moving the PHA to a safer orbit. Some of the advantages of this method are that it does not result in fragmentation, which could lead to another problem, and also the flexibility to change the configuration of the system to optimize the deflection according to the warning time. The dynamics of the PHA-tether-asteroid system is analyzed, and the amount of orbit change is determined for several initial conditions. Only motion in the plane of the orbit of the PHA around the Sun is considered, thus the PHA chosen for the simulations has low orbit inclination. Analysis of the dynamics of the system shows that the method is feasible for planetary defense.

1 Introduction

To date, more than 21 000 near-Earth asteroids (NEAs) have been discovered, including almost 2000 PHAs. Small bodies with perihelion of less than 1.3 AU and having orbits 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 hazardous asteroids (PHAs), which are objects larger than about 140 m and that can get closer than 0.05 AU, or approximately 20 times the distance from the Earth to the Moon. Potential impacts of NEOs are one of the biggest motivations to study and detect these objects. The threat of an asteroid on the collision path with the Earth has encouraged the development of several deﬂection techniques. Based on the warning

a e-mail: [email protected] 1464 The European Physical Journal Special Topics time, the deflection method can be chosen from months to several years, or even decades. Some existing methods in the literature for this purpose are: fragmentation of the asteroid using nuclear explosives or collision with a massive asteroid [1]; using the 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 evaporation of the surface layers, slightly pushing the asteroid [4]; the use of the gravitational pull of a thereby stationary spacecraft or in a “tugging” mode near an asteroid to deflect it slowly, which is called the gravity tractor method [5,6]. In this work, the use of a tether assisted technique is considered. It consists of connecting two asteroids, a PHA and a smaller asteroid nearby, so that the motion of the secondary asteroid could change the initial trajectory of the larger one. The methodology aims to transfer a PHA to a new safer orbit through the displacement of the center of mass. Thus, no unwanted consequences related to fragmentation would happen after the deflection. The applications of this technique are especially important for planetary defense, but could also help in the scientific exploration of these objects. The study of small bodies has been growing fast in the past decade, and there are several 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 valuable information about the formation of the Solar System. Some of the past missions are: NEAR that landed on asteroid Eros [7]; Hayabusa, a mission developed by the Japanese Space Agency with the goal of collecting material from asteroid Itokawa and bring back samples [8]; Dawn orbiting Ceres and Vesta, which are the most massive asteroids in the solar system, respectively [9]; PROCYON and Hayabusa 2 [10], both launched in 2014, meeting their target asteroids in 2016 and 2018, respectively. 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 comet 67P/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 1999 RQ36, or 101955 Bennu, by the end of 2018, and after mapping the surface of the asteroid it will collect a sample to bring back to Earth [13–16]. A mission named Lucy, scheduled to launch in 2021, will have the goal to explore six Trojan asteroids around the orbit of Jupiter [17]. The motivation arises due to the fact that Trojans are thought to be remnants of the primordial material that formed the outer planets, holding important information about the formation of the Solar System. Another mission planned for the future is Psyche, a solar electric propulsion spacecraft concept targeted to be launched in 2022 [18]. Psyche appears to be the exposed nickel-iron core of an early planet, which might help to understand planetary formation. It will be the first time that a metallic asteroid is visited. Another pioneer mission recently selected by NASA is Janus. The goal is to explore two binary asteroids to better understand 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 objects are asteroids, comets, and even some satellites of planets. Most asteroids are located between the orbits of Mars and Jupiter, in the main belt asteroid, but NEAs orbit much closer, and sometimes may come uncomfortably near, even crossing the Earth’s orbit, 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 larger than 140 m is still a work in progress. In addition, some asteroids that were not considered a threat may have their orbits perturbed to a point where it could eventually become dangerous. An example is the thermal radiation driven Yarkovsky/YORP effect [19,20], or even by collision with other objects. Asteroids smaller than 140 m are more challenging to be detected by NEA surveys, but the damage in case it is on the collision course with the Earth is still considerable, like the Chelyabinsk meteor in Russia in 2013 [21]. Before sending a spacecraft to an asteroid, the environment around it must be carefully mapped. The first step is to obtain data of the asteroid by performing observation with ground-based or space telescopes. Characteristics such as shape, mass, rotation, and surface properties are some of the important information that should be known in advance, and can be obtained with ground radar observations at the Arecibo Observatory, in Puerto Rico, or the Goldstone Solar System Radar, in California [22]. Characterization is crucial especially for landing and sample return missions. Practically all missions to small bodies select targets that can be observed with radar prior to the mission. In the history of space missions there is no record of an asteroid mitigation test performed yet. The first proposed mission is the international collaboration AIDA (Asteroid Impact and Deflection Assessment) composed by NASA’s DART, and the European 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 a satellite on the secondary component of the binary system Didymos and analyze the consequences of the impact on the system [23]. One of the studies related to this mission 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 first studies 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 for debris removal [27,28], and using tethers for power and propulsion [29], to name a few. Some projects that make use of tethers are: Tether Physics and Survivability (TiPS), from the US Naval Research Laboratory, with the goal to understand how the libration motion of endmasses affects the motion of the center of mass of the system; formation flying tethers, which are tether systems that can enable groups of satellites to fly in tight formation, for applications such as long baseline interferome- try, like SPHERES (NASA and MIT); electrodynamics tether, such as the Tethered Satellite System Reflight (TSS-1R), with the goal of interacting with the planet’s magnetosphere to generate power or propulsion without consuming propellant. TUI, an electrostatic radiation belt remediation project, for safer manned and unmanned missions in Earth’s orbit.

3.2 PHA-tether-asteroid dynamics

The equations of motion consist of four coupled equations, which include the orbital parameters 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 1466 The European Physical Journal Special Topics

Fig. 1. Configuration of two asteroids connected by a tether. configuration of the system consists of an irregularly shaped body representing the PHA, connected to another asteroid with a tether [30,31]. The smaller asteroid is used as an artifact to perturb the orbit of the main asteroid, and it is modeled as a point mass. In this paper only planar motion is considered, therefore only PHAs with very low orbit inclination are appropriate for this study. Small bodies have spin rate that can range from seconds to hours, and the tether dynamics is strongly affected by the rotation of the main body. Therefore, in order to 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 connected with a tether the rotation is neglected, since its rotational motion would be affected by the orbit transfer to the PHA. In this study, the tether is considered inextensible and massless, since these parameters would depend on the material of the tether, and this is out of the scope of this work. The system described is shown in Figure1 . 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 the distance between the Sun and the smaller asteroid; ` the length of the tether; A is the center of mass of the PHA; rB is the distance between the smaller asteroid and the center of mass of the PHA; P is the point of attachment of the tether; ν is the true anomaly of the PHA; θ is the rotation angle of the PHA; α is the angle that the tether makes with the PHA, which gives the pendulum motion. The equations of motion of the system are derived according to Lagrange’s equation, given by equation (1).

d ∂L ∂L − = 0, qi ≡ R, ν, θ, α, (1) dt ∂q˙i ∂qi where, qi ≡ R, ν, θ, α are the generalized coordinates that describe the dynamics of the system. The total kinetic and potential energy of the system are given by equations (2) and (3), respectively:

2 1 2 1 2 1 ˙ K = mAν + mBν + IA θ +ν ˙ (2) 2 Axy 2 Bxy 2 GMm l GMm U = − B 1 − cos (θ + α) − A − m U , (3) R R R B PHA 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].

Gm C r 2 r 2 U = A 1 − 20 0 + 3C cos (2θ + 2α) 0 . (4) PHA l 2 l 22 0 l

According to the conﬁguration of the system PHA-tether-asteroid in Figure1, the relative velocities can be obtained:

˙ ~νAxy = Raˆ1 + Rν˙aˆ2 (5) h ˙ ˙ i ~νB/Axy = −l sin (α + θ) α˙ + θ +ν ˙ − rP/A sin(θ) θ +ν ˙ aˆ1 h ˙ ˙ i + l cos (α + θ) α˙ + θ +ν ˙ + rP/A cos(θ) θ +ν ˙ aˆ2 (6) h ˙ ˙ ˙ i ~νBxy = R − l sin (α + θ) α˙ + θ +ν ˙ − rP/A sin(θ) θ +ν ˙ aˆ1 h ˙ ˙ i + Rν˙ + l cos (α + θ) α˙ + θ +ν ˙ + rP/A cos(θ) θ +ν ˙ aˆ2. (7)

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

v2 = R˙ 2 + R2ν˙ 2 (8) Axy 2 2 v2 = R˙ 2 + R2ν˙ 2 + l2 θ˙ +α ˙ +ν ˙ + r 2 θ˙ +ν ˙ Bxy P/A ˙ ˙ ˙ + 2lrP/A θ +ν ˙ θ +α ˙ +ν ˙ cos(α) + 2l θ +α ˙ +ν ˙ × −R˙ sin (θ + α) + Rν˙ cos(θ + α) ˙ h ˙ i + 2rP/A θ +ν ˙ −R sin(θ) + Rν˙ cos(θ) . (9)

The distance from the center of the PHA to the point of attachment, represented by 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 in order to simplify the calculations, and only the distance from the center of mass of the PHA is considered. Finally, the Lagrangian of the system represented in Figure1 is obtained using the kinetic and potential energy, and is given by equation (10).

1 1 2 L = K − U = (m + m ) R˙ 2 + R2ν˙ 2 + I θ˙ +ν ˙ 2 A B 2 A 1 2 + m l2 θ˙ +α ˙ +ν ˙ + 2l θ˙ +α ˙ +ν ˙ −R˙ sin(θ + α) 2 B GM Gm m C r 2 + Rν˙ cos(θ + α) + (m + m ) + A B 1 − 20 0 R A B l 2 l r 2 GMl + 3C cos (2θ + 2α) 0 − m cos(θ + α). (10) 22 0 l R2 B

The four equations of motion are derived according to equation (1), and are shown in equations (11)–(14) for α, R, θ, and ν, respectively: 1468 The European Physical Journal Special Topics

¨ α : mBl l θ +α ¨ +ν ¨ − R¨ sin(θ + α) + R˙ ν˙ + Rν¨ cos(θ + α) − R˙ cos(θ + α) + Rν˙ sin(θ + α) θ˙ +α ˙ + R˙ cos(θ + α) + Rν˙ sin(θ + α)

6GJ m r2 sin 2(θ + α) GM × θ˙ +α ˙ +ν ˙ + 22 A 0 0 − sin(θ + α) = 0 (11) l4 R2 GM R :(m + M ) R¨ − Rν˙ 2 + − m l θ¨ +α ¨ +ν ¨ sin(θ + α) A B R2 B 2 GM + θ˙ +α ˙ +ν ˙ + 2 cos(θ + α) = 0 (12) R3 ¨ ¨ θ : IA θ +ν ¨ + mBl l θ +α ¨ +ν ¨ + R˙ ν˙ + Rν¨ cos(θ + α) − R¨ sin(θ + α) − R˙ cos(θ + α) + Rν˙ sin(θ + α) θ˙ +α ˙

GM + θ˙ +α ˙ +ν ˙ R˙ cos(θ + α) + Rν˙ sin(θ + α) − sin(θ + α) = 0 (13) R2 ¨ 2 h ¨ ν : IA θ +ν ¨ + (mA + mB) 2RR˙ ν˙ + R ν¨ + mBl l θ +α ¨ +ν ¨ − R¨ sin(θ + α) i + 2R˙ ν˙ cos(θ + α) + R − sin(θ + α)(θ˙ +α ˙ + 2ν ˙) + cos(θ + α)(θ¨ +α ¨ + 2¨ν) = 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 variables by the mean orbital rate. Next, the four equations of motion will be used to obtain the orbit variation generated by adding the tether system. Note that the parameter R, represented in equation (12), provides the amount of change in the orbit when compared to an unperturbed orbit, and it will be carefully analyzed. Thus, the objective is not to analyze each of the four parameters individually, but how the four coupled equations combined 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 is because it is classiﬁed as a PHA, passing close to Earth about every 6 years, and it has one of the highest impact hazard ratings among PHAs. Another reason is because it has low orbit inclination. Bennu is a B-type (primitive and carbon rich), about 492 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 approximately 0.005 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 the Arecibo Observatory’s S band radar system (2.38 GHz, 12.6 cm) and the Goldstone Solar System Radar (8.56 GHz, 3.5 cm), and again in 2005 and 2011. Delay-Doppler radar images helped to characterize this asteroid, allowing to constrain Bennu’s physical and orbital properties enough to support the mission to come years later [35]. Launched in 2016, the OSIRIS-REx sample return mission approached asteroid Bennu 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 Values Diameter (km) 0.492 Rotational P (h) 4.29 a (AU) 1.1263910259 e 0.20374510 i(◦) 6.034939 Ω(◦) 2.060867 ω(◦) 66.223068 M(◦) 101.7039479 Perihelion dist. (AU) 0.89689436 Aphelion dist. (AU) 1.3558876877 Orbital P (days) 436.64872813 ρ(g/cm3) 1.26 Mass (Kg) 7.327 × 1010 with boulders of dozens of meters in size [36]. Figure2 shows an image of Bennu from the OSIRIS-REx spacecraft. The orbital and physical parameters of Bennu used for the simulations are listed in Table1 [ 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 MATLAB software. Three initial parameters were tested in order to compare the optimal configuration: mass for the smaller asteroid, tether lengths and points of attachment for the tether. The deflection is measured by analyzing the difference between the initial unperturbed orbit and the perturbed orbit (after connecting the smaller asteroid with 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 trajectory between 2020 and 2040, for a hypothetical deﬂection mission before Bennu’s close 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 a spacecraft from Earth to Bennu in order to minimize the fuel consumption, using the 1470 The European Physical Journal Special Topics

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

Fig. 4. Orbit deviation for initial mass ratio of mB /mA = 1/10000, and tether lengths of 1000 km, 2000 km, and 3000 km. patched-conics approach [39–41]. The optimal launch date for the spacecraft obtained is October 28th, 2035. The tether is considered in the simulations at the moment of arrival of the spacecraft to Bennu. Since the focus of this work is to analyze the dynamics of the tethered asteroid system, the logistics of capturing and transfering the smaller asteroid is not included in this study. To analyze the efficacy of the deflection method, the first step was to obtain the orbit deviation compared to the initial unperturbed trajectory (before the tether- small asteroid attachment). To test how the initial parameters might affect the results, it was considered two different mass ratios and three tether lengths. The orbit deviation (∆) is the distance between the unperturbed (Keplerian) and the perturbed orbit (with the tether-asteroid system). The orbit deviation is measured in terms of the Earth’s radius (REARTH = 6371 km). The simulations are performed for a period of 300 years. Figures3 and4 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 tether lengths 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 longer tethers. For tethers three times longer, the deviation values are of an order of 5 times higher than for the shorter tether chosen for the simulations. Comparing diﬀerent mass values for the smaller asteroid attached with a tether, it is evident that, for a faster 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 consider is if the point of attachment for the tether on the PHA would influence the results. In order to analyze the change in the orbit due to the tether attachment angle, it was also tested α(0) equal 45◦ and 90◦. Figures5 and6 show the orbit deviation (∆) for a mass ratio between Bennu and the smaller asteroid of 1/1000, and an initial tether attachment angle of 45◦ and 90◦, respectively. Table2 shows the approximate values for the orbit deviation in terms of the Earth’s radius considering a mass ratio of mB/mA = 1/1000, tether lengths of 1000 km (blue) 2000 km (green), and 3000 km (magenta) and point of attachment for the tether of α(0) = 0◦, 45◦, and 90◦. According to Table2 , it is evident that the deflection increases with the tether length and the angle of attachment between the tether and the PHA. Based on the results the most efficient scenario would be for α(0) = 90◦ and tether length of 3000 km. Next, we will investigate how the orbit change analyzed so far would influence 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 or decreased. For an asteroid on the collision path with the Earth, it would be risky to 1472 The European Physical Journal Special Topics

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

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

α(0) = 0◦ α(0) = 45◦ α(0) = 90◦ l = 1000 km 400 1600 2000 l = 2000 km 800 3200 3800 l = 3000 km 1200 5000 5900 bring the asteroid any closer. However, an exploration mission such as sample return or resource utilization, could benefit from having the asteroid placed on a new closer safe orbit. The parameter δ represents the distance from the PHA to the Earth. Similar to previous cases, it is measured in terms of Earth’s radius. For a mass ratio of mB/mA = 1/1000, Figure7 shows Bennu’s distance from the Earth for a period of 300 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 this study we will use only the case of point of attachment for the tether of α(0) = 0◦. In Figure7 , the lines for different initial parameters overlap. For a better visu- alization, Figure8 shows the same graph, but only for δ less than 2000 REARTH. As a comparison, the black line represents Bennu’s trajectory without any added perturbation. If the lower spike is in black, it would mean that none of the tether lengths considered for the simulations affected the minimum orbit distance for that specific approach date. There are two very close approaches that will be investigated in detail. In Figure9 , the same tether lengths and point of attachment used in Figures7 and8 are considered, except for the mass ratio. Now we will consider the case of a less massive mB, with a mass ratio of mB = mA/10000. In Figures8 and9 , 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 Figure8 , with the close approaches in detail in Figure 10 (left column). 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 lengths of 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 exploration support, in which the object could be brought closer if required, but not a desired configuration for planetary defense purposes. For the second very close approach, around 226.9 years, adding the tether with a smaller asteroid helped to avoid the close approach, altering the center of mass of the PHA in a way that the distance to the Earth at that time increased. In this case the longer tether was the most efficient configuration, avoiding a possible collision with the Earth by adding to the minimum approach distance about 900 REARTH. The same study was performed for a mass ratio of mB/mA = 1/10000, meaning a smaller mass to be attached to the PHA with a tether. Figure9 shows the approxi- mation to Earth closer than 2000 Earth’s radius. It is possible to notice two moments where the PHA-tether-asteroid system gets very close. Figure 10 shows the close approaches in detail (right column). The first significant approach happens near 56.26 years after the tether-asteroid attachment. The black 1474 The European Physical Journal Special Topics

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 of 1000 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 tethers in this case helped to increase the distance of minimum approach to Earth, with the longer tether of 3000 km (magenta) resulting in the largest deﬂection. The second very close approach happens around 226.9 years. The black line, representing the PHA without the tether, shows a dangerous approach of practically less than 20 Earth’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, with the longer tether increasing the close approach from less than 20 REARTH to more than 100 REARTH.

5 Conclusion

The dynamics of a tether assisted technique was presented consisting of four coupled equations of motion for the system, which include the orbital parameters 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 methodology presented proved to be efficient in supporting planetary defense. The ability to change the center of mass of a PHA on the collision course with the Earth can increase the minimum orbit 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, and different mass ratios between the PHA and smaller asteroid to be connected with a tether. The results suggest that, for a faster deflection, longer tethers and a more massive asteroid attached to the PHA would be more effective. The simulations showed that the method is dynamically feasible for asteroid impact mitigation, and it could also be used to facilitate exploration missions, such as sample return and resource utilization missions. The technique studied offers the flexibility to adjust the amount of deflection by changing the different parameters for the system, considering the warning time available. In addition, this method does not result in fragmentation, which could be a potential risk.

The authors wish to express their appreciation for the support provided by NASA through grant #NNX13AQ46G awarded to Universities Space Research Association (USRA) and grants #80NSSC18K1098 and #80NSSC19K0523 awarded to the University of Central Florida (UCF). Grants #140501/2017-7, 301338/2016-7 and 406841/2016-0 from the National Council for Scientiﬁc and Technological Development (CNPq) grants 2014/22295-5 and 2016/24561-0 from Sao Paulo Research Foundation (FAPESP). This work was also par- tially supported by NASA Solar System Exploration Research Virtual Institute contract NNA14AB07A (PI David A. Kring).

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