Optimal thrust program for deviating an Earth-pointing asteroid

Acta Asfmnaurica Vol. 44, Nos. 7-12. pp. 809-814, 1999 0 1998 International Astronautical Federation Published by Elsevier Science Ltd All rights reserved. Printed in Great Britain 0094-5765/99 $ - see front matter PII: SOO94-5765(99)00057-O

Pergamon

Optimal Thrust Program for Deviating an Earth-Pointing Asteroid Jeng-Shing Chern’ Chung Shan Institute of Science and Technology P.O. Box 90008-6-7, Lungtan, Taiwan 32526 Republic of China and Donglong Sheut Institute of Aeronautics and Astronautics National Cheng Kung University Tainan, Taiwan 70101 Republic of China

E

Abstract The purpose of this study is to investigate the optimal thrust program for deviating an asteroid when it is flying directly toward or crossing the Earth. Under some assumptions, the problem can be considered as a two-body problem of Earthasteroid system. The initial relative speed and distance are specified to be 10 km/s and 150 times of Earth’s radius, respectively. We have about 1 day (or, 93681 seconds exactly) to take action. If a single impulse is applied to the asteroid at the specified initial point, the required impulse to obtain a miss distance of 2 times of Earth’s radius is 169.5 m/s per kg mass. For an asteroid of 10 m in diameter, the total impulse required is 3.02 x lOa m/s. It needs a typical large launching rocket to provide the total impulse. When the asteroid is larger or the initial distance is shorter, the number of launching rockets required increases rapidly. For further analysis with physical and engineering constraints imposed, we shall have to use the variational formulation method. 0 1998 International Astronautical Science Ltd.

Federation

;

= acceleration = 14 = integration

: g 9 H J L m R

= = = = = =

T

=

Published by Elsevier

= =

V

=

V Z,Y

= Iv = coordinate

B

= initial

Y 4

constant

CL 0

=

aAt: or

EO9

J

tf

a dt

= flight path0 angle = adjoint variables, (i = = gravitational constant = true anomaly

SubSWiptS A

system of the two-body

=

problem

5ight path angle of the deviated orbit

in time equation

‘Senior Scientist. Associate Fellow AIAA. ‘Associate Professor. Member AI.\:!.

vector

Igl Hamiltonian performance index initial asteroid-Earth distance mass of asteroid mean radius of orbit radial distance from Earth’s center minimum allowable radius perigee distance time velocity

1’ tP

AV

vector of asteroid

total energy per unit msss eccentricity of deviated orbit control force control force per unit mass, F/m

= lfl = gravity acceleration

Tmin =

Nomenclature a

= = = =

asteroid

T, 8,

V, y)

810

E

f i 1M s 1

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= Earth = final time = initial time = Moon = sun = some reference point

Introduction By virture of discovery, the number of Earthcrossing objects is significant and still increasing continuously. As of 1994, 163 Earth-crossing asteroids are known.‘?’ Also, the spectacular collision of the Shoemaker-Levy 9 asteroid with Jupiter happened in the same year.3 Although the possibility of such massive collisions is very low, smaller objects collide with the Earth regularly and do damage that would be intolerable to human being. For examples, the Tunguska (Siberia) event of 1908 is estimated to have involved a 60 m object exploding at a height of 8 km, and the famous 1 km Meteor Crater in Arizona was made by the impact of an even smaller body only 30 m in diameter. Scientists have predicted that an impact anywhere in either the Atlantic or the Pacific Ocean by a 400 m asteroid will produce tidal waves over 60 m high.‘s4 Consequently, the international community is developing a consensus that although the probability for such a collision is low, some resources should be devoted to threat detection and possible interdiction. The American Institute of Aeronautics and Astronautics (AIAA) has also issued a position paper which declared that “Earth orbit-crossing asteroids clearly present a danger to the Earth and its inhabitants.“5 There are four major areas that merit further study about the asteroids: detection, orbit determination and prediction, deflection, and exploration.6 Several papers have been published in the area of deflection (as well as interception). 1,7-g Actually, with the spectacular collision of the Shoemaker-Levy 9 asteroid with Jupiter in July 1994, it is another choice for us to deviate a potentially dangerous asteroid to a nearby planet other than Earth. It could also be possible to deviate the asteroid to the Moon in some cases. The main purpose of this paper is to investigate the delta-velocity (AV) required to obtain a miss distance of two Earth’s radius. In Refs. 1 and 7, interception of Earth-crossiug asteroids are considered. It is proposed that we intercept the dangerous object at first, and thcu deflect or detroy it. The analysis of deflection is

further studied in Refs. 8 and 9. In Ref. 8, a formulation of the fundamental dynamics and control which clears the coupling between the nearEarth object and the approaching spacecraft has been developed. Then in Ref. 9, the strategy of perihelion impulse and the minimum AV are obtained. In this paper, we use s single impulse which is perpendicular to the relative velocity vector between asteroid and Earth. The required AV depends on the initial distance and the mass of asteroid strongly. In particular, we think about how much we have to do in about one day of time period. The variational formulation is required only when the physical and/or engineering constraints are imposed. Problem Formulation For the scenario as shown in Figs. 1 and 2, the initial position of the asteroid considered is assumed to be about 1 day (24 hours) away from the Earth-crossing point C. In Fig. 2, the geometrical relationships among the Earth (point E), asteroid (point A), and the Earth-crossing point are depicted. The velocity vector of the asteroid VA can be decomposed into two components, VAT and VAT, where VA, 11VE and VE is the Earth’s velocity vector. It is obvious that when VAT = VE, the asteroid will impact the Earth at point C with a relative velocity VAT. Therefore, in the problem of deviating an Earth-crossing asteroid, we shall simplify it to an Earth-asteroid two-body problem. It would be enough to consider only the relative velocity component V,+ (to be denoted by V) and the deviation vector AV produced by the control force F. The meaning of the assumption of 1 day is that, we want to investigate what can happen in one day and to understand what we can do in the same time to prevent the impact. The initial relative speed between the asteroid and the Earth will be assumed to be 10 km/s. Refer to Fig. 3, when there is no control force, the equations of motion of the asteroid are jl=-v

(la)

Q=!%

(lb)

Y2 where ,UE is given in Table 1. The analytic tion for Eqs. (1) are V2

---

2

PE

Y

= E,

(a constant)

solu-

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orbit of Astemid

Figure

1: Supposed

Earth-crossing

scenario.

Let the initial distance of the asteroid be 156rE (TE is given in Table 1) from the Earth. Then we have the initial conditions at ti = 0:

Figure 2: Geometry Earth-crossing point.

of Earth,

asteroid,

and

Vi = 10 km/s Yi = 9.56723 x 10’ km E = 4.95834 x 10 km2/s2

(3)

where Ct = 8.98483 x lo4 s in Eq. (2b). When the asteroid impacts the Earth’s surface, we have the final conditions at tf = 9.36703 x lo4 s: Vf = 1.49718 x 10 km/s

of = 6.37815 x lo3 km

InitialPesition y

(4)

The flight time is 9.36703 x lo4 s or 26.0195 h or 1.08415 days. The impact speed is as high as 14971.8 m/s. In the following sections, we shall find a single minimum impulse or an op timal thrust program, to deviate the asteroid and obtain a minimum fly-by distance of 21’~ from the Earth’s center.

Two-Body

System

Refer to Fig. 3, the motion of the asteroid will be influenced mainly by the Earth’s gravitation gE, the Moon’s gravitation gM, the Sun’s gravitation gs, and the control force F. Let the mass of the asteroid be m. we have the dynamical equation: ma=m(gE+gM+gs)+F or

Figure 3: Undeviated teroid.

and deviated

orbits of as-

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Table 1: Some physical Moon, and Sun.

Parameter Gravitational constant of Earth, FE Mean radius of Earth’s orbit, RE Equatorial radius of Earth, TE Gravitational constant of Moon, P,M Mean radius of Moon’s orbit, RM Gravitational constant of Sun, PS

parameters

3.98599

of Earth,

Data 10’ kmJ/s’

x

1.495 x 10s km 6.37815 x lo3 km 4.90282

x

lo3 km3/s’

3.844 x 10’ km 1.32906 x 10”

km3/s”

Figure 4: Geometry

of single-impulse

the moon. The dynamic simplified to the form:

a=gE+m+gs+f

(5)

where f = F/m, the control force per unit mass. The physical parameters of the Earth, Moon, and Sun to be referred are listed in Table 1. In Eq. (5), the term gs can always be neglected as compared with f. For example, when the distance of the asteroid Gem the Sun is varying from lo6 km + RE to RE, the magnitude of gs varies from 5.87 x lob3 m/s2 to 5.95 x 10m3 m/s2. Furthermore, both gE and gM can also be neglected at least at the initial phase of the deviated flight path. Actually, we have IgEl = 3.99 x 10m4 m/s2 at an initial distance of lo6 km and (gM1 = 1.29 x lo-’ m/s2 at the distance of lo6 km- RM. When the asteroid is in the vicinity of the Earth, it may happen that both gE and g.v must be considered. In order to simplify the analysis, we shall propose two separate strategies for solving the problem. In other words. whruever the term gM is dominant, the asteroid will be deviated to have an rendezvous with the Moon. Otherwise, the term gE is dominant and the asteroid will be deviated to pass the Earth with a minimum allowable distance. In this paper, we shall consider the case in which the term gE is dominant and the term gM is negligible. We can also consider the neglect of g,v as an assumption. When the asteroid is crossing the ?vIoon’s orbit, lglwj will be 10-l of IgEl if the Moon’s position is 20” of arc length beyond. Therefore, this assumption will be valid in most. of the s(‘eililrios. Of course: it is another interesting prnl~l~~i~~ to deviat,e the asteroid to have a rentlezvous with

equation

deviation.

(5) can now be

a=gE+f

(6)

Deviation with an Impulse A single impulse can be applied laterally to the asteroid at its initial position to deviate the flight path. The deviated flight path will be a part of one of the conic sections. Let AV be the impulse per unit mass. Then Vi will be deviated with an angle of AVIV;, where Vi 2 IVil. As shown in Fig: 4, the eccentricity of the devizited orbit, denoted by e, can be expressed aslo

e2=

2

( ) LV? t - 1

(7)

cos’ fi + sin2 p

PE

The energy E per unit mass is

and the perigee distance

The required impulse per unit mass is Av

=

K(90° - 4

(10)

57.29578

For numerical computation, use Eq. (9) to calculate

MKI

we

Set

TV

=

T,,,*,,

e with E obtained

813

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t,.,l,,,I,.,I.‘.l, OO 200

400

Required

ml

ml impulse

Figure 5: Single impulse function of y.

the numerical

2TE

=

Asteroid

1000

AV (m/s)

4V

requirement

1.27563 X lo4 km

computation

c

,I

as a

from Eq. (8). Then the angle p can be solved from Eq. (7) and 4V is finally calculated by using Eq. (10). As an example, for rP = T,in =

. .

(II)

results are

e = 4.17362 ,L?= 89.0286” 4Vi = 169.541 m/s

(12) (13) (14)

It is seen from the eccentricity that the deviated flight path is a part of a hyperbolic orbit. If we applied the impulse at a point which is closer to the Earth, for example, at y1 = 100~~ = 6.37815x lo5 km, then we have t1 = 3.18672 x lo4 s v, = 1.00208 x 10 km/s 4Vl = 253.876 m/s

(15)

The variation of the required single impulse as a function of the initial distance is shown in Fig. 5. When the initial distance is less than 50~~. the 4V requirement increases rapidly. In other words. it would be much easier for us to deviate the asteroid as early as possible. But to discover the asteroid at a farther distance is another big problem. In Fig. 5. the single impulse requirement is fox per unit mass. Now. let us assume that there is an asteroid of 10 m in diameter. The tleusity is. for example. 3.4 x 10” kg/m”. The mass of the asteroid is calculated to be l.iSO24 x 10” kg.

Figure 6: Geometry tion.

of continuous

thrust

devia-

At the initial distance of 150T~, the total impulse required is 3.02 x lo8 m-kg/s. This is equivalent to the total impulse generated by a large launching vehicle. When the initial distance is as short as 25~~, the required total AV will increase to 1.81 x 10’ m-kg/s. We would need 6 large launching vehicles now. For larger asteroid, the required total impulse will be tremendous.

Deviation

with

Continuous

Refer to Fig. 6, the equations scalar form are

Thrust

of motion in the

i = -Vsiny

06a)

8, = -;co,y

(16b)

V = gEsiny

(16~)

,=$[,-(,,-q)cos,]

(16d)

We have assumed that f is always perpendicular to V, to produce the maximum increment in the deviation angle. The Hamiltonian for variational formulation is

A,? The performance

(17)

index will be

.J = min

(18)

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For an extremum, 65 must be zero for arbitrary sf (t) and we have

ax

A7

af=v=O’

o
09)

In the previous section, we have shown that a single impulse can be minimized if the distance of the application point can be maximized. This introduces, at least, two engineering problems: to find the asteroid at the maximum range, and to deliver a thrust generator to the asteroid to produce the required impulse. Therefore, the single impulse method is one of the optimal solutions. When the physical or engineering constraints are imposed and to be considered, we may need to use the variational formulation method.

Conclusions For an Earth pointing asteroid with 10 km/s relative speed and 150 times of Earth’s radius initial distance, the flight time to the Earth to have an impact is 9.36703 x lo4 seconds (aboll one day). The impact speed with the Earth is 14971.8 m/s (about Mach 45). The required delta-velocity to deviate the asteroid, to obtain a miss distance of 2 times of the Earth’s radius, is 169.541 m/s per unit mass. For an asteroid with 10 m in diameter and 3.4 x lo3 kg/m3 in density, the total impulse required is 2.96 x log N-s or 3.02 x lo8 m-kg/s. This is about the total impulse generated by a large launching vehicle. When the initial distance is decreased to 25 times of the Earth’s radius, we may need 6 launching vehicles. We believe that from these data, the aerospace scientists and engineers will have some ideas about the question: How difficult is it to deviate an asteroid in about one day? The method of using a single-impulse has been studied in detail. We have to use the variational formulation method when physical and engineering constraints are imposed.

References 1. Conway, B. A., “Optimal ception of Earth-Crossing nal of Guidance,

Control,

Low-thrust Asteroids,”

Inter-

Jourand Dynamics,

Vol. 20, No. 5, 199i, pp, 995-1002. 2. Rabinowitz, D. L., et al., “The Population of Earth-Crossing Asteroids,” Hazards DUV to Comets and Asteroids, edited by T. G~hrels. University of Arizona Press. Tucson. AZ. 1994, pp. 285-312. 3. Shoemaker, E. M., ‘Comet Shoemaker-Levy 9 at Jupiter,” Geophysical Research Letter, Vol. 22, No. 12, 1995, pp. 15-55. 4. Gehrels, T. (editor), Hazurds Due to Comets and Asteroids, University of Arizona Press, Tucson, AZ, 1994. 5. AIAA Space Systems Technical Committee, “Dealing with the Threat of an Asteroid Striking the Earth,” AIAA Position paper, April 1990. 6. Anonymous, Co&ion of Asteroids and Comets with the Earth: Physical and Human Consequences, Report of a Workshop Held at Snowmass, Colorado, July 1981. 7. Solem, J. C., “Interception of Comets and Asteroids on Collision Course with Earth,” Joumaf of Spacecrufi and Rockets, Vol. 30, No. 2, March-April 1993. 8. Hall, C. D., and Ross, I. M., “Dynamics and Control Problems in the Deflection of NearEarth Objects,” AAS Paper 97-640, Sun Valley, Idaho, Aug. 1997. 9. Park, S. Y., Elder, J. T., and Ross, I.. M., “Minimum Delta-V for Deflecting EarthCrossing Asteroids,” AAS Paper 97-727, Sun Valley, Idaho, Aug. 1997. W. T., Introduction to Space DyJohn Wiley and Sons, Inc., New York, 1961, Chapter 4.

10. Thomson, namics,