Comparative Analysis of Low Thrust Deflection Strategies for Asteroid Collision Mitigation

P.D.Keskara and J.P.Sanchezb

aMSc candidatebSupervisor

Space Research Centre, Cranfield University, UK

AbstractOver the past three and half decades, there has been an unprecedented development in the field of study of extra-terrestrial impacts , in the wake of an improved understanding of the collision threat from Near Earth Asteroids and its hazardous implications to the life on Earth. Amongst all recognised impact mitigation strategies contemplated, the least explored is the category of a low thrust and slow push approach represented by the Gravity Tractor. Currently, there has been an increased interest in this category amongst the scientific community. A comprehensive system level analysis of these recently proposed, alternative low thrust approaches was lacking in current literature. Hence, three of the most promising low thrust methods were identified viz. Gravity Tractor, Ion Beam Shepherd and Scheduled Thrust Pushboat and such analysis was attempted through a system level comparison of their deflection potential, in terms of the Δv imparted to the asteroid by each method. Firstly, a model collision scenario was defined to establish a common platform for comparing performances of these three methods. Sophisticated mathematical models were devised, to suitably modify the core concepts of these methods to enhance their efficacy. Building on this foundation, a detailed comparative analysis was performed, investigating effects of two sets of variable input parameters on Δv. These included the parameters characterising the impact scenario and quantifying the spacecraft (mission) capability. As an outcome of the analysis, Ion Beam Shepherd emerged as the most productive method in a typical collision scenario, considering the impact probability based on asteroid size. Pushboat performed second best in such a case where as for the most destructive case of asteroids larger than 1-2km diameter, Gravity Tractor was found most favourable. Also considering the time window available for impact mitigation before impact (warning time), Gravity Tractor was a preferred choice for warning time higher than 20 yrs. while the other two produced better results for shorter warning times.

Introduction The notion of a possible celestial impact posing any potential threat to the life on Earth was always treated with a considerable if not categorical derision and disregard, amongst the scientific community and even more so amongst the common public. But year 1980 brought a paradigm shift in such perception, as a revolutionary conjecture made in a publication in ‘Science’ (L.W. Alvarez et.al, 1980) claimed that the famously known K-T extinction, that took place 65 million yrs. ago was triggered by an impacting asteroid, sizing approximately 10km in diameter. At the present day, with the launch of second phase of the NEOshield project in early March 2015, impact study has come a long way to be recognised as a separate branch of space sciences and technology. The establishment of Spaceguard Foundation in 1996, was one of the pioneering international collaborative initiatives dedicated to study of Near Earth Objects (NEOs) and impact avoidance. Today, the term Spaceguard is used as an umbrella term for all the international efforts in this direction. Over the past two decades numerous search initiatives world over like NEOWISE, Spacewatch, LINEAR etc. have been hunting for Near Earth Asteroids (NEAs). With each new discovery, their trajectories are being continuously monitored by JPL NASA’s NEO program in US and NEOyds program in Europe, to determine if any of them enter the category of Potentially Hazardous Asteroids (PHAs).As these have a diameter greater than 140m and are likely candidates to cause an impact hazard. NEA statistics show that, the search for asteroids with diameter larger than 1km is 90% over and now the focus will shift on the remaining asteroids with diameters between 140m to a kilometre. So far in total 12826 (and counting) NEAs have been discovered, as of Aug-2015 and the tally of PHAs has reached 1602 ,whereas the number of NEAs found with size over 1km are 873 .[7]The extent of damage that such NEAs can inflict is subject to their size described by diameter. (A.W.Harris, 2015 et.al) [12] presents an elaborate explanation on the relationship between NEA size ,the estimated impact frequency and the subsequent destruction it may cause. The NEAs with diameters between 50 to100m may lead to a violent air blast, posing a damage similar as in case of the Tunguska event in Siberia in 1908. Such events may occur once, every 1000 yrs. causing several deaths and it is believed that there are total 500,000 of such NEAs existing, with diameters greater than 50m. Asteroids having diameters larger than 100m up to 500m are

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estimated to strike every 10,000-70,000 yrs. and these could possibly lead to the formation of a crater 1-2 to several kilometres in diameter depending upon the impact site and also cause a local to national destruction. An impact near the shore may even trigger tsunamis that could claim several to millions of lives. This relatively sinister category of NEAs may have a population numbering as large as 60,000. About 90% of which have diameters under 300m. NEAs even bigger than 500m diameter are relatively fewer in number (3500), but threaten to inflict a much severe devastation of international proportions, as it may dig a crater as large as 10km in diameter or pose a very high risk of tsunami that could harm several millions. Such a catastrophe could occur every 140,000 yrs. Finally ,the asteroids with over 1km diameter which are as few as just 1000 and might visit only once in 5000 yrs., but could have disruptive worldwide implications to Mankind. This indicates that the mitigation strategies should specifically focus on the range of 100-500m, where the destruction is significantly large and so is the probability of collision. NASA recognises four broad categories for collision mitigation. Civil defence measures are required in almost any case of collision, even for diameters below 50m. For asteroids in diameter range of 50-100m slow pushing Gravity Tractor or Kinetic Impactor are considered suitable. Kinetic Impactor is preferred, especially in case where the time window available from the time of detection of a PHA to the estimated time of impact (warning time) is shorter than 20yrs. For asteroids having diameters higher than 100m up to 300m, either a standalone slow push approach or a combination of Gravity Tractor and Kinetic Impactor is considered suitable depending on whether the warning time is larger or shorter than 20 yrs. Same options can be deemed fit, even for the range of diameters between 300 to 500m with more warning time available.[6] Clearly in this range of 100-500m Gravity Tractor has the limitation of waring time, also its not particularly effective on its own and has to be used as a complementary method. Thus this category of low thrust and slow push needs to be explored further by building more robust and reliable alternatives to the classical Gravity tractor. The assessment of low thrust methods in this paper aims to do exactly the same.

BackgroundThis is an attempt to present a condensed review of the fairly diverse literature reviewed in the scope of research and writing this paper.(Sanchez,2009) [1] served as a primary reference for this research as it presented one of the most comprehensive works on the comparison and analysis of impact mitigation strategies. (ESA, NEOshield, 2015)[6] and (NEO,JPL-NASA,2014) [7] elucidated the concept and background of collision threat and also presented some of the key NEA statistics up to date. (Chapman et.al, 2001)[5] gave a compact summary of history of impact threat, a more detailed account was found in (Sanchez, 2009) [1]. (ESA, NEOshield, 2015) [6] also provided a good rationale behind the focus of past as well as current international efforts in the area of NEA monitoring and impact mitigation. It prepared a good background for this research by describing the mitigation strategies that are presently considered pertinent in different collision scenarios. (Lu, Love, 2005) [2] contained the concept of Gravity Tractor in its original published form however (Sanchez, 2009) [1] shade more light on the details and intricacies of the mathematical model. (Lu, Love, 2005) [2] also briefed over the certain limitations of Kinetic impactor that the Gravity Tractor looks to overcome. The inherent drawbacks and limitations of Gravity Tractor were in turn highlighted by (McLeens, 2007) [3] and (Bombardelli, 2013) [4].While the prior superficially described a possible modification to Gravity Tractor, the latter introduced a fresh concept of Ion Beam Shepherd method. The analysis however was more conservative as the effect of gravity was not considered. (Sanchez, 2009) [1] put forth the Pushboat method in a concise manner, yet providing enough details to develop a mathematical model. It also employed a clever idea of scattering factor for quantifying the efficiency of the thrusting manoeuvre. All in all, this rich variety of literature gave a very useful insight into the area of impact mitigation,

Low Thrust Strategies.In the recent years there has been an increasing interest in low thrust deflection strategies for asteroid collision mitigation. This theme involves slow deflection of the asteroid away from its potentially threatening trajectory over a period of several months to years. The spacecraft propulsion system is ingeniously used to either push or pull the asteroid away from its collision course. Amongst several different low thrust strategies proposed so far the methods using gravity, aim to deviate the asteroid by exploiting the gravitational pull of the spacecraft. Whereas some of the other approaches rely on impingement of thruster plume on the asteroid surface to push it away. This slow and consistent deflection can be spanned across a period as long as 20 yrs. or more. The duration of push or pull however is directly governed by the warning time available in the collision scenario as well as the propulsion capacities available and practically achievable at present . Shortly Describing three of these methods, next few sections will lay a foundation for the comparative analysis to follow.

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Establishing the Framework

To facilitate the further comparison of three methods considered, some common grounds must be established. While structuring a framework, all the input parameters must assume identical values for the three methods so that output of the models will give a realistic idea of their relative strengths. The methods will be compared in terms of the cumulative values of Δv imparted to the asteroid, for simplicity of mathematical models and ease of comparison. These models will accept a set of input parameters, perform the calculations and output the Δv values. Asteroid size, density, spacecraft mass, thrusting capacity and time of deflection manoeuvre are some of the most fundamental parameters to be specified to characterise a model collision mitigation scenario. Below mentioned are some reasonable assumptions that quantify these parameters without loss of generality.

1. The asteroid is assumed to be a spherical rigid body of diameter 400m and density 20000kg/m3

2. The ion beam thrusters, typically having a thrust efficiency of 70% (or a power specific thrust of 47.57 mN/KW) and an inverse specific power of 25kg/KW and capable of providing a specific impulse of 3000 sec3. Time of deflection manoeuvre is assumed to be 10 yrs.4. Mass of the power subsystem that provides the abovementioned thrust is assumed to be half of the total dry mass (Initial spacecraft mass minus the propellant mass) of the spacecraft5. Beam divergence of the conical thruster exhaust plume (Half cone angle) is assumed to be 200. [2].

Gravity Tractor

Figure 1 Gravity Tractor Method The concept was first proposed by (Lu, 2005 )[2] in ‘Nature’. It uses the mutual gravitation pull between the spacecraft and the asteroid for deflection. The spacecraft simply hovers over the asteroid thus avoiding all the complexities involved in landing. The spacecraft employs ion thrusters for balancing the gravitational pull to maintain a constant hovering distance over the asteroid. The thrusters must be canted through a certain minimum angle to avert the impinging of their plume on asteroid surface, as it would in turn negate a part of gravitational pull and undesirably setoff the dust and ions on the surface. The major advantage is that it is independent of the poorly understood surface properties and the rotational rate of the asteroid. The mathematical model for Gravity tractor sets up the equation for dynamic equilibrium of spacecraft between the gravitational pull and component of thrust in the form of reaction of canted thrusters as follows,

Where, Fg is the mutual gravitational pull between spacecraft and asteroid, counterbalanced by the component of thrust, Fhover. Expanding,

Where, Ma, mi are masses of asteroid and spacecraft, Tn is the thrust generated by the exhaust plume, Ra is asteroid radius and is the thruster half cone angle. The thrust of the spacecraft can be computed in terms of

spacecraft dry mass md , power specific Thrust and inverse specific power τ as,

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Substituting the thrust value from (3) in equilibrium equation (2) and solving it in an iterative manner, the value of hovering distance d is obtained for which this equilibrium equation is satisfied. Knowing the thrust imparted to the asteroid is equal in magnitude to the mutual gravitational pull, acceleration imparted to the asteroid can be evaluated and then the total Δv imparted over the deflection period can be obtained by integrating the acceleration over the pushtime. Typically a 10 years of push time will provide a net Δv of 0.24cm/s.

Modified Gravity Tractor

In order to use the spacecraft thrust more effectively a modification was suggested in the original gravity tractor concept by (McInnes, 2007) [3].The suggestion was to fly the spacecraft in a displaced, highly non Keplearian orbit about a point in space proximal to the asteroid and lying in its equatorial plane. The gravitational pull plays the same role in deflecting the asteroid as in case of the Gravity Tractor, the only difference is in the way it is balanced. As seen in Fig. 2 (a), only a component of mutual gravitational pull, F g. along equatorial plane (local horizontal) is balanced by a component of thrust, Fhover and the component Fc (of Fg and thrust Tn put together) perpendicular to the equatorial plane, points towards the centre of orbit and provides the spacecraft the centripetal force helping it maintain the non-Keplearian orbit. The radius of this orbit ρ is kept less than asteroid radius so that the thrusters have to be canted to avoid the plume impingement as shown in Fig.2 (a)

(a) (b)

Figure 2: Model and Results for Modified Gravity Tractor Method:

Imposing the modified equilibrium condition the calculations were performed exactly on similar lines to Gravity Tractor. The orbiting radius ρ was used as a variable design parameter .For different choices of ρ, Δv was obtained at corresponding hovering distances d. The results (Fig. 2 (b)) revealed that Δv must be compromised to maintain a safer flying distance (25-30% of Asteroid Radius of 200m) from the asteroid. It turned out to be a trade-off between the two contrasting demands. Typically at an orbiting radius of 60m a considerable Δv rise is obtained (0.32cm/s) over the classical Gravity tractor (0.24cm/s) while still maintaining a safe distance. One of the pitfalls of this modification however is the inherent complexity of thruster pointing .As the thruster is canted, in order to keep it pointing in the desired direction tangential to asteroid surface; it has to maintain a constant slew at the angular rate of orbit. The range of slew rates required spanned from 0.1 to 0.3deg/s

Ion Beam Shepherd

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No.Orbiting Radius ρ (m)

Spacecraft Distance d (m)

ΔV

(cm/s)

1.2.3.4.

060120158

286.1255.0220.2200.0

0.240.320.581.31

Figure3 Maximum Spacecraft Equilibrium Distance in IBS Scenario

The Gravity Tractor relies on a reasonably heavy spacecraft to exert a gravitational pull on the asteroid, also on a precise control for maintaining the equilibrium hovering distance. Ion Beam Shepherd (IBS) is an alternative designed to avert these drawbacks. IBS, unlike Gravity Tractor allows the thruster plume to impinge upon the asteroid surface and uses the momentum transferred by this impingement of propellant ions to slowly deflect the asteroid away. This method too involves hovering over the asteroid instead of actually landing on the surface which makes it independent of surface as well as structural properties of the asteroid. The method however, has an advantage over the Gravity tractor as it is dependent on neither the asteroid nor the spacecraft mass for its slow pushing strategy. The method’s effectiveness is solely dependent on the strength of power subsystem, which in turn depends on the propulsion system employed. Space proven Ion thruster would suffice the requirement. The maximum possible hovering distance in this case will be determined by the thruster half cone angle. As seen in Fig 3, for a given half cone angle øex, the spacecraft cannot hover farther beyond a certain distance d, if the thruster plume must entirely (tangentially) impinge upon the asteroid surface without missing. Another thruster is required on the opposite face to maintain the spacecraft’s equilibrium hovering position. The dynamics of the system reveal that when spacecraft and asteroid together are considered as one system, the total Δv imparted depends on the exhaust of thruster 1 shown in Fig 3.In the mathematical model, the total thrust of both the thrusters put together was maintained constant. The classical IBS deems the gravitational effects as negligible. However, the assumed half cone angle of 200. here, mandates that spacecraft must hover at a closer distance, where gravity cannot be neglected. Thus the spacecraft equilibrium equation could be given as,

Where T1, T2 are thrusts and Fg is gravitational pull. Both thrusts in turn depend on the mass flow rates of thrusters. The resultant equilibrium equation turned out to be a differential equation in these mass flow rates and it did not have a closed form solution. Thus a special case was considered by applying boundary conditions. A new factor λ was introduced to define the percentage of initial thrust contributed by the thruster 2. The resultant equilibrium equation for spacecraft reduces to equation (5)

Where, mpower is the mass of power subsystem and all other parameters assume same notations. This equation is then solved for hovering distance d. Considering dynamic equilibrium, net thrust and the subsequent acceleration ( ) imparted to the asteroid can be found using equation (6).

This acceleration can be integrated over the thrusting time to obtain the Δv. For λ closer to 0.5, the two thrusters would almost balance each other out and the case will degenerate to classical IBS where d is large enough to neglect gravity. However practically the geometry of IBS configuration imposes a constraint on equilibrium hovering distance that spacecraft can station itself at. Beyond a certain distance the divergent plume of thruster would miss the asteroid as explained above, giving fallacious results for Δv.The limiting spacecraft distance can be obtained from the geometry of configuration. For the asteroid radius of 200m and half cone angle of 20 deg. The maximum attainable Δv is 0.39 cm/s at a spacecraft distance of 585m.

Scheduled Low Thrust Pushboat

This method suggests a deflection strategy slightly different from all the others described above. Instead of making the spacecraft hover, here the spacecraft would land on the asteroid’s surface pushing it with the reaction force generated from expulsion of thruster plume. Two spacecrafts are landed on the opposite sides of the asteroid’s equator to ensure continuous thrusting and deflection throughout the push time. The firing periods and timings of the thrusters are regulated appropriately (hence the term scheduled thrust.) so that there is always one thruster firing at a time.

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Ideally all the thruster reaction must be directed in the orbital velocity direction of asteroid to harness maximum deflection in case of a slow push method. [11] Such direction can be termed as an optimal thrusting direction. However in a practical scenario factors like asteroid rotation rate, the landing site of spacecraft with respect to the equatorial plane (co-latitude,’β' ) and the obliquity of the asteroid equator ‘ø’ with the orbital plane, affect the thrust alignment reducing its efficiency. This effect is accounted for by what is termed as a Scattering Factor [1] which can be given as,

(7)

Where, in the numerator the dot product of the thrust vector with the unit vector in optimal thrusting direction, takes into account the offset of thrusting direction from the optimal direction. It is a function of co-latitude of the spacecraft landing location with respect to equatorial frame, the obliquity of equator and the angle for which thrusters are fired (δΨ = Ψf – Ψi). It is then integrated over the complete rotation of the asteroid. To express this as an efficiency term, the expression is divided by the continuous thrust in the optimal direction. The optimal direction vector itself completes a 3600 rotation with respect to the asteroid’s inertial frame during one complete revolution. Thus on integration the denominator gives a value of 2л.Due to identical assumptions, the combined thrust of the two spacecrafts is given by (3). The value of dry mass can in turn be calculated in terms of initial mass as

Where tpush is the total time of deflection manoeuvre. To maximise the Δv output the two thrusters are alternately fired for half rotation of asteroid (δΨ = 1800) .It ensures that the dot product of the unit force vector with the unit vector in optimal direction, always stays positive, giving a continuous rise in momentum transferred to the asteroid. Considering the actual thrust imparted, the acceleration and total Δv amassed over the pushtime can be calculated using (9)

(Sanchez, 2009) [1], claims that for any obliquity it is always possible to attain a minimum scattering factor of 0.25 by choosing an appropriate landing site (co-latitude,β) .Considering this worst case scenario the Δv value obtained over 10 yrs of push time is 0.28cm/s .Given the fact that this is the worst case value and maximum scattering factor could reach as high as 0.32, the method holds a lot of promiseDetailed mathematical treatment of the models can be found in (Keskar et.al, 2015) [10].

Comparison and Analysis For a comprehensive system level analysis that would draw out the design space, it was necessary to investigate the effect of variation of all input parameters on the performance of three methods, to determine their suitability for different collision scenarios. These parameters were divided into two categories. Asteroid size, density and time of push characterised the collision scenario whereas the spacecraft initial mass, specific impulse and the thruster half cone angle were the spacecraft parameters that quantified the mission capabilities. Now these parameters were varied over a feasible range of values and resultant Δv for three methods were plotted. For Pushboat, two curves were plotted for the best and the worst case values of the scattering factor.

Variable Asteroid Size

Considering the NEA statistics explained earlier, the asteroid size was varied between 50 to 2000m diameters .An expected decline is noticed in Fig 4 (a) for all the three methods with increasing asteroid sizes, as increase in size reflects in mass gain and subsequent lowering of acceleration and Δv imparted, for a constant

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thrusting capacity. For diameters under 560m IBS performs the best, whereas Gravity Tractor furnishes the lowest Δv .Since for lower asteroid masses, the spacecraft has to fly closer, in order to generate sufficient gravitational force to balance the thruster reactions. Flying closer also lowers the efficiency of the thrust, as the thrusters have to be canted more to avoid the plume impingement. However, for diameters greater than 560m, performance of Gravity tractor exceeds that of Pushboat. IBS still produces the largest Δv up to asteroid diameters as high as 1570m, beyond which Gravity Tractor tops the performance of the other two methods.

(a) (b)

(c) (d)

(e) (f)

Figure 4 Analysis Results for Comparison of Impact Mitigation Models

Variable Asteroid Density

Asteroid density typically varies between 1300 to 5300kg/m^3.As seen in Fig 4(b) again the Ion beam shepherd performs the best over the entire range of NEA densities. Gravity Tractor produces relatively inferior results to both others for the lower densities due to poorer thrust efficiency just as in previous case. The Gravity Tractor

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catches up with Pushboat for more dense and hence heavier asteroids and yields a higher Δv for asteroid densities higher than3640 kg/m^3. It is also remarkable that the Δv becomes a little insensitive to both size and density variations for higher values, so that all the three methods are observed to yield somewhat closer results towards the end of the range.

Variable Pushtime

The pushtime considerations solely depend on the warning time available in a collision scenario. For all three strategies being compared, the approach is to opt for lower thrust values thus utilise the majority of available pushtime to harness a maximum possible Δv. But in case of a limited asteroid mass and hence a modest Δv requirement ,it may be more appropriate to thrust for only a part of the total warning time by consuming limited amount of fuel at a faster rate. Low thrust methods were only deemed suitable for warning times larger than 20 yrs. thus it was worth exploring effect of variation of pushtime over a range of 1 to 20 yrs. While the curves in Fig 4(c) for IBS and Pushboat rise steeply and then saturate for larger pushtimes, Gravity Tractor is not observed to saturate over the considered range of pushtimes, although it increases gradually. IBS continues to perform best and by some margin here, while the Pushboat performs better than Gravity Tractor up to 17 yrs. It is only for pushtimes higher than 20 yrs., that Gravity Tractor emerges as second best.

Variable Initial Spacecraft Mass

Spacecraft initial mass is the total mass of spacecraft that it carries to the asteroid before the start of the deflection manoeuvre. The initial mass has two principle components, viz. spacecraft dry mass and the mass of propellant required for the deflection manoeuvre. The initial mass of the spacecraft was set to vary between 0 to 100 tons. It can be noted from Fig. 4(d) that all the three methods show a linearly increasing Δv profile with increasing initial mass. The reason for this can be attributed to dependence of thrust on the dry mass of the spacecraft. For all the three methods, the Δv imparted directly depends on the thrust imparted to the asteroid. The expressions for the thrust reveal that they in turn are directly proportional to the dry mass of the spacecraft, which has been shown to bear a direct proportion with the initial mass. IBS again emerges as the most effective method. The Pushboat gives an intermediate performance closely followed by the Gravity Tractor over the range of masses considered.

Variable Specific Impulse

The ion thrusters conventionally known for higher specific impulses and lower thrusts are considered as the obvious choice for propulsion. Typical specific impulses are of the order of 3000s and higher. Still it would be worth examining the range of Δv obtained over a much wider range of specific impulses. Thus the specific impulse is varied from 300 to 9000s, which has been attained in testingA very peculiar variation of Δv with the specific impulse can be seen in Fig 4(e) for all the three methods. For IBS, the Δv climbs up steeply in the start, then slope of the curve progressively decreases till it attains a maxima, then Δv gradually reduces towards the end .In case of Pushboat, a curve mostly similar to IBS is observed, only the Δv performance at all stages is inferior. Gravity Tractor performs better, even than IBS up till 330s, continues to generate higher Δv than Pushboat, for higher specific impulses up to about 1230s. Beyond this point the Pushboat yields a better Δv for a wide range of specific impulses. However in the end the Gravity tractor surpasses the Pushboat again, beyond 7750s and performs better for higher specific impulses till 9000s. The Δv for Gravity Tractor unlike others, shows an increasing profile over the considered range specific impulses. But the overall slope of the curve increases only gradually, indicating that the gain in Δv per unit gain in specific impulse is quite low as compared to the other two methods. In other words Gravity Tractor could be a preferred choice for very low Δv requirements that can be met at lowest specific impulses. To amass a higher Δv comparable to Pushboat, Gravity tractor must employ very high specific impulse closer to 9000s. The reason for these peculiar behaviours is hidden in the mathematics of the problem. The complex Δv expressions for these methods have multiple dependencies on the specific impulse. A detailed reasoning of this behaviour can be found in (Keskar et.al, 2015) [10]

Variable Thruster Half Cone Angle

The half cone angle of a thruster parametrises the divergence of exhaust plume .To determine how Δv is influenced by change in the thruster geometry, the half cone angle was varied from 50 to 200. It is clear from Fig. 4(f) that Pushboat is not affected by the changing beam divergence, as the Δv only depends on the rate and amount of material being exhausted. For IBS there is a small increase in Δv ,with dropping half cone angles, as for lower beam divergences spacecraft can hover at a farther distance, where the influence of gravity reduces and case approaches classical IBS as explained earlier yielding progressively larger Δv. In case of Gravity

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Tractor also the Δv lowers with increasing half cone angle. The reason is attributed to the complex geometry and is elaborated in (Keskar et.al, 2015) [10] It must be noted that for all the descriptions given above, the worst case of Pushboat method with minimal guaranteed Δv was considered. However if the best possible value of scattering factor (0.32) is considered then the method performs better than Gravity tractor over a longer range of varying input parameter or in some cases delivers a superior performance over the entire range as evident from the plots

ConclusionHaving carried out an elaborate multidimensional comparison of all the methods, a series of inferences can be drawn regarding their relative strengths and suitability for different collision scenarios. For asteroid sizes up to about 1600m, IBS excels the performance of the other two and does so by a significant margin for asteroids with sizes under 1km diameter. An estimate made by extrapolating the trends in the discovered population of PHAs indicates that there are about 10,000 of PHAs having diameters larger than 100m, but the number of asteroids larger than 1km in size are fewer than 1000 and hence pose a probabilistically much lower risk of collision. This underlines the significance of Ion beam shepherd method as the most productive low thrust method during an ordinary collision scenario, which is more likely to occur. The results also indicate that in case of a large asteroid, sizing in over a kilometre, Gravity Tractor will turn out to be the most effective option amongst all the others. The Pushboat emerges as the second best option for asteroids in the lower range of sizes under 1 km diameter. For warning times up to 20 yrs., IBS would again be a preferred choice closely followed by the Pushboat, if best case is considered. However for pushtimes even higher than that the Gravity Tractor turns up with a superior Δv. For typical space proven specific impulses for ion thrusters (3000s), the IBS delivers the best performance followed by Pushboat and then the Gravity Tractor. Pushboat reaches a higher peak at Δv at lower specific impulse of 7700s than Gravity tractor and hence is more effective for a given capacity of propulsion system. Finally, the inferior performance of Gravity Tractor is enhanced up to 50% using non Keplearian orbit modification, however this enhancement comes at the cost of added complexity in the control system.

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Aerospace Engineering, Faculty of Engineering University of Glasgow, 2009.[2] Edward T. Lu, Stanley G. Love. Gravitational tractor for towing asteroids, nature, Vol 438, 10 November

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Control, and Dynamics Vol. 30, No. 3, May–June 2007[4] C. Bombardelli et.al ,The ion beam shepherd: A new concept for asteroid deflection, Acta Astronautica 90 ,

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of Space Studies Southwest Research Institute Boulder CO 80302, 24 February 2001[6] www.neoshield.net [7] http://neo.jpl.nasa.gov/ [8] A.W. Harris, Deflection and fragmentation of near-Earth asteroids, Nature, v. 360, 1992 pp. 429-433.[9] S. R. Chesley and P. W. Chodas et.al, Quantifying the Risk Posed by Potential Earth Impacts, Elsevier

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[12] A.W.Harris et.al, Asteroid Impacts and Modern Civilization: Can we Prevent a Catastrophe? ,Asteroids IV, Univ. Arizona Press, 2015

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