ࡱ> UWb6Tklmnopqrstuvwxyz{|}~ufghijklmnopqrstuvwxyz{|}~4_555@Jbjbj222XXB a  8 < JY|N $ r*r*r*r*999XXXXXXX$ZR]X 9M5999Xr*r* Y}F}F}F9*^r* r*X}F9X}Fb}FFV|N ]Xr*B Ü;@F9W}XLY0JYMW^C^(]X ^ ]X 99}F99999XX %FX NUMERICAL STUDY OF THE GRAVITATIONAL CAPTURE IN THE BI-CIRCULAR FOUR-BODY PROBLEM. Alexandre Lacerda Machuy Francisco1, Teresinha de Jesus Stuchi2, Antonio Fernando Bertachini de Almeida Prado1 1National Institute for Space Research (INPE) So Jos Dos Campos-SP-12227-010-Brazil Phone(55-12)3945-6201-Fax(55-12)3945-6226 2Departament of Mathematical Physics, UFRJ, P. O. Box 68528 21945-970. Rio de Janeiro, RJ - Brazil  HYPERLINK "mailto:machuy@dem.inpe.br" machuy@dem.inpe.br, tstuchi@if.ufrj.br,  HYPERLINK mailto:prado@dem.inpe.br prado@dem.inpe.br ABSTRACT: A gravitational capture occurs when a spacecraft (or any particle with negligible mass) change from a hyperbolic orbit with a small positive energy around a celestial body into an elliptic orbit with a small negative energy without the use of any propulsive system. The force responsible for this modification in the orbit of the spacecraft is the gravitational force of the third and the fourth bodies involved in the dynamics. In this way, those forces are used as a zero cost control, equivalent to a continuous thrust applied in the spacecraft. One of the most important applications of this property is the construction of trajectories to the Moon. The concept of gravitational capture is used, together with the basic ideas of the gravity-assisted maneuver and the bi-elliptic transfer orbit, to generate a trajectory that requires a fuel consumption smaller than the one required by the Hohmann transfer. The goal of the present paper is to study the energy required for the ballistic gravitational capture in a dynamical model that has the presence of four bodies. In particular, the Earth-Moon-Sun-Spacecraft system is considered. 1. INTRODUCTION The bi-circular problem is a particular case of the problem of four bodies, where one of the masses, let us say  EMBED Equation.3 , is supposed to be infinitely smaller than the other three masses. With that hypothesis,  EMBED Equation.3  moves under the gravitational forces of  EMBED Equation.3 ,  EMBED Equation.3  and  EMBED Equation.3 , but it doesn't disturb the motion of the three bodies with significant mass. In the bi-circular problem, the motion of  EMBED Equation.3 ,  EMBED Equation.3  and  EMBED Equation.3  around the center of mass is considered as formed by circular orbits and the motion of  EMBED Equation.3  has to be a certain function of the initial conditions. We can consider the bi-circular problem as a disturbance of the restricted problem of three bodies. This problem can be used as a model for the motion of a space vehicle in the Sun-Earth-Moon system. In the first part of the paper we supplied the equations of motion of the model and we defined gravitational capture. The second part is used for the calculation of some numerical results for the bi-circular problem, such as direct orbits, retrograde orbits, capture orbits, etc. 2. MATHEMATICAL MODELS  EMBED MSPhotoEd.3  Figure 1 Bi-circular problem. The problem of four bodies with the two hypotheses shown below is called bi-circular problem and is shown in Fig.1. First hypothesis: It is considered two bodies with significant mass moving in circular orbits around the mutual center of mass. Those two bodies are called primaries. Second hypothesis: The third body with significant mass is in a circular orbit around the center of mass of the system formed by the two first primaries and its orbit is coplanar with the orbits of those primaries. We will study the motion of this fourth body under the gravitational attractions of the three bodies with significant mass. 2.1 Planar equations of motion We will calculate the planar equations of motion of the space vehicle in the sidereal and synodical systems. We will use the canonical system of units by dividing all the distances by the distance between the two primaries and dividing the masses by the total mass of the two primaries. It will also be defined that the angular speed of the system is unitary. The masses and distances of the Earth, Moon and Sun are: Mass of the Earth,  EMBED Equation.3 ; Mass of the Moon,  EMBED Equation.3 ; Mass of the Sun,  EMBED Equation.3 . Earth-Moon distance  EMBED Equation.3 ; Earth-Sun distance  EMBED Equation.3 . Then, the masses of the Earth, Moon and Sun in the canonical system are: Mass of the Earth = EMBED Equation.3 ; Mass of the Moon = EMBED Equation.3 ; Mass of the Sun  EMBED Equation.3 . The circumferences described by the Moon and the Earth has radius  EMBED Equation.3  and  EMBED Equation.3 , respectively.  EMBED Equation.3  and  EMBED Equation.3  are the coordinates of the space vehicle, the Earth, the Moon and the Sun, respectively. Below are the equations of motion of the Earth, Moon and Sun:  EMBED Equation.3 , EMBED Equation.3 ,  EMBED Equation.3 , EMBED Equation.3 ,  EMBED Equation.3 , EMBED Equation.3  and  EMBED Equation.3 . Where  EMBED Equation.3  is the distance between the Sun and the center of the system and  EMBED Equation.3  is the angular speed of the Sun. We observed that the positions of the Moon, Earth and Sun are:  EMBED Equation.3 ,  EMBED Equation.3  and  EMBED Equation.3 . The distance of the space vehicle to the Earth is  EMBED Equation.3 ; to the Moon is  EMBED Equation.3 ; to the Sun is  EMBED Equation.3 . Therefore, we have the equations of motion of the space vehicle in the inertial system:  EMBED Equation.3 ,  EMBED Equation.3   EMBED Equation.3 .  EMBED Equation.3  We will introduce a system of rotating coordinates on the center of mass of the Earth-Moon system with the same angular speed of the primaries. Be  EMBED Equation.3  the coordinates of the particle in this synodical system. The equations that convert the coordinates of the fixed system to the rotating system are:  EMBED Equation.3   EMBED Equation.3  If we now differentiate each component in equation  EMBED Equation.3  twice we obtain  EMBED Equation.3 , and  EMBED Equation.3   EMBED Equation.3 . To position of the four bodies are: Moon  EMBED Equation.3 , Earth  EMBED Equation.3 , Space Vehicle  EMBED Equation.3 , Sun  EMBED Equation.3 . It is clear that  EMBED Equation.3  is the angular speed of the Sun in the synodical system. The coordinates  EMBED Equation.3  are called synodical and the coordinates  EMBED Equation.3  are called sidereal. The three distances in the synodical system are shown below. From the space vehicle to the Earth is  EMBED Equation.3 ; from the space vehicle to the Moon is  EMBED Equation.3  and from the space vehicle to the Sun is  EMBED Equation.3 . The equations of motion of the space vehicle in the new system are:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  3.0 GRAVITATIONAL CAPTURE  EMBED MSPhotoEd.3  Figure 2 Initial conditions. Figure 2 shows a trajectory that ends in gravitational capture. To define gravitational capture is necessary to use some basic concepts of the problem of two bodies. We will call  EMBED Equation.3  the double of the sum of the kinetic and potential energy of the problem of two bodies, the space vehicle and Moon, that is given by:  EMBED Equation.3 , where  EMBED Equation.3  and  EMBED Equation.3  are, respectively, the distance and the velocity of the space vehicle with respect to the Moon, and  EMBED Equation.3  is the gravitational parameter of the Moon. If we consider only two bodies (the Moon and the space vehicle),  EMBED Equation.3  is constant, if only gravitational forces are considered. We will describe the orbits of the space vehicle for values of  EMBED Equation.3  according to the classification:  EMBED Equation.3  If  EMBED Equation.3 , we have hyperbolic orbits,  EMBED Equation.3  If  EMBED Equation.3 , we have parabolic orbits,  EMBED Equation.3  If  EMBED Equation.3 , we have elliptic orbits. We defined  EMBED Equation.3  as being of the double of the energy of the system Moon-vehicle. Unlike what happens in the problem of two bodies,  EMBED Equation.3  is not constant in the bi-circular problem. Then, for some initial conditions, the space vehicle can alter the sign of the energy from positive to negative or from negative to positive. When the variation is from positive to negative it is called a gravitational capture orbit. The opposite situation, when the energy changes from negative to positive, is called gravitational escape. We describe the numeric methodology below.  EMBED Equation.3  A Runge-Kutta of fourth order integrator was used, programmed in the FORTRAN language.  EMBED Equation.3  We integrated the equations of motion of the space vehicle in the sidereal system.  EMBED Equation.3  The initial conditions are obtained in the following way. We consider the Moon in the origin of the  EMBED Equation.3  system and the Earth with coordinates  EMBED Equation.3 . The starting point of each trajectory is at a distance of 100 km from the surface of the Moon ( EMBED Equation.3  km, starting from the center of the Moon). To specify the initial position completely it is necessary to know the value of one more variable. The variable used is the angle  EMBED Equation.3 , that is the position of the Moon. This angle is measured starting from the Earth-moon line, in the counterclockwise sense, starting from the opposite side of the Earth. The magnitude of the initial velocity  EMBED Equation.3  is calculated from the initial value of  EMBED Equation.3 . The direction of the velocity vector of the vehicle is chosen as being perpendicular to the line that links the space vehicle to the center of the Moon, appearing in the counterclockwise direction for the direct orbits and in the clockwise direction for the retrograde orbits. The orbit is considered of capture when the particle reaches the distance of 100000 km (0.26 canonical units) from the center of the moon in a time smaller than 50 days (approximately 12 canonical units). The sphere with radius 100000 km centered in the Moon is defined as the sphere of influence of the Moon. Figure 2 shows the point P, where the space vehicle escapes from the sphere of influence. The angle that defines this point is called the angle of the entrance position and it is described by the Greek letter  EMBED Equation.3 . During the numeric integration the step of time is negative, therefore the initial conditions are really the final conditions of the orbit after the capture. 4.0 EFFECTS OF THE ANGLE  EMBED Equation.3 . We now show some results obtained. Figure 3 shows direct orbits and figure 4 shows retrograde orbits. In both situations the angle  EMBED Equation.3  is constant and equal to  EMBED Equation.3 , and  EMBED Equation.3 . The angle  EMBED Equation.3  assumes the values:  EMBED Equation.3  (6),  EMBED Equation.3  (5),  EMBED Equation.3  (4),  EMBED Equation.3  (3),  EMBED Equation.3  (2),  EMBED Equation.3  (1). The coordinates of the position vector for the case of direct or retrograde orbits are:  EMBED Equation.3  and  EMBED Equation.3 . The coordinates of the velocity vector for the case of direct orbits are:  EMBED Equation.3  and  EMBED Equation.3 . For the case of retrograde orbits the coordinates of the velocity vector are:  EMBED Equation.3  and  EMBED Equation.3 . We can clearly see the effect of the Sun pushing the trajectories to the right. The direct orbits has a direct path, while the retrograde orbits start their motion to the left and then feel the effects of the Sun and turn to the right.  EMBED PBrush   EMBED PBrush  Figure 3 Direct orbits. Figure 4 Retrograde orbits. 5. VARIATION OF THE ENERGY We now turn our attention to study the effects of the initial value of the energy  EMBED Equation.3 . We consider  EMBED Equation.3  and make  EMBED Equation.3  to assume the values -0.1 (1), -0.2 (2), -0.3 (3) and -0.4 (4). In figure 5 we have direct orbits and in figure 6 we have retrograde orbits. We see that the reduction of the energy increase the length of the trajectories and, in some cases, it includes loops. Trajectories with energy close to zero escape faster from the primary. 6. VARIATION OF THE ANGLE  EMBED Equation.3  We now study the effects of the position of the Sun. We considered  EMBED Equation.3  and  EMBED Equation.3  in figures 7 and 8. The angle  EMBED Equation.3  will assume the values  EMBED Equation.3  (5),  EMBED Equation.3  (4),  EMBED Equation.3  (3),  EMBED Equation.3  (2) and  EMBED Equation.3  (1). In figure 7 we have direct orbits and in figure 8 we have retrograde orbits. It is clear that Sun attracts the trajectories. The results show this fact. The retrograde trajectories have this characteristic more visible.  EMBED PBrush   EMBED PBrush  Figure 5 Direct trajectories varying  EMBED Equation.3 . Figure 6 Retrograde trajectories varying  EMBED Equation.3 .  EMBED PBrush   EMBED PBrush  Figure 7 Direct orbits. Figure 8 Retrograde orbits. 7. MINIMUM VALUE OF  EMBED Equation.3  The goal now is to calculate the smallest value of the energy that allows a gravitational capture. It will be made a variation of the angle  EMBED Equation.3  from  EMBED Equation.3  up to  EMBED Equation.3 , in steps of  EMBED Equation.3 . Figures 9 and 10 show the results. The initial value of  EMBED Equation.3  is of -0.65, and the final value is -0.01, with variations of -0.01. In figure 9 we have the angles  EMBED Equation.3 ,  EMBED Equation.3  and  EMBED Equation.3  and in figure 10 we have the angles  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3  and  EMBED Equation.3 . The results show sinusoidal oscillations with strong variations. It shows several points of maximum and minimum savings and it is very important to take into account the initial conditions to obtain good results for the maneuver.  EMBED PBrush   EMBED PBrush  Figure 9 Minimum  EMBED Equation.3  Figure 10 Minimum  EMBED Equation.3  8. CONCLUSONS This paper studied the problem of gravitational capture under the bi-circular four-body problem. The approach is to perform numerical simulations, in order to know the main characteristics of the problem. In particular, the effects of the initial position and the energy of the spacecraft is considered, as well as the position of the Sun. The results shown here can be to help mission designers to get the most of the gravitational forces involved in the problem. ACKNOWLEDGMENTS The authors are grateful to CAPES-Brazil (Coordenao de Aperfeioamento de Pessoal de Nvel Superior) for the scholarship given to the first author, to the National Council for Scientific and Technological development (CNPq), Brazil, for the research grant receive under Contract 300828/2003-9 and to the Foundation to Support Research in the So Paulo State (FAPESP) for the research grant received under Contract 2006/00187-0. REFERENCES. 1.Vieira Neto, E. Estudo Numrico da Captura Gravitacional Temporria Utilizando o Problema Restrito de Trs Corpos. Ph.D. Dissertation, Instituto Nacional de Pesquisas Espaciais, Brazil, 1999. 2.Prado, A F B A. Numerical study and analytic estimation of forces acting in ballistic gravitational capture. Journal of Guidance, Control and Dynamics 25 (2) 368-375, 2002. 3.Simo, C. Gomez, G., Jorba, A., Masdemont, J. The bi-circular model near the triangular libration points of the R T B P. In: Roy, A. E., Steves, B.A.(Eds.), From Newton to Chaos. Plenum Press, New York, pp. 3453-370, 1995. 4.Castella, E., Jorba, A. On the vertical families of two-dimensional tori near the triangular points of the bi-circular problem. Celestial Mechanics and Dynamical Astronomy 76 (1), 35-54, 2000. 5.Yamakava, H. On Earth-Moon transfer trajectory with gravitational capture. Ph.D. Dissertation, University of Tokyo,1992. 6.Belbruno, E. A., Miller, J.K. Sun-perturbed Earth-to-Moon transfers with ballistic capture. 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r 3 = ObjInfoEquation Native _1215341207F`3LÜ`3LÜOle (x"x S ) 2 +(y"y S ) 2 FMicrosoft Equation 3.0 DS Equation Equation.39q} 2x=" E (x"x E )r 1CompObjfObjInfoEquation Native 8_1218541914F`3LÜ`3LÜ3 " M (x"x M )r 23 " S (x"x S )r 33 FMicrosoft Equation 3.0 DS Equation Equation.39ql (1)Ole CompObjfObjInfoEquation Native 1_1189887033F`3LÜ`3LÜOle CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q XZT 2y=" E (y"y E )r 13 " M (y"y M )r 23 " S (Equation Native 8_1218541926F`3LÜ`3LÜOle CompObjfy"y S )r 33 FMicrosoft Equation 3.0 DS Equation Equation.39qpf$ (2) FMicrosoft Equation 3.0 DS EqObjInfoEquation Native 1_1192624352F`3LÜ`3LÜOle       !"#$%&'()*+,-./012345649;<d>=?A@BCDEFHGIKJLMNOPSQRTUVXWYZ[\]_^`abec4769~WU2zɞ>`SSgxV|O&(L7 AgA 9,/ լEM8;qEzv.OO3D'e3Pzzgo^|coL$ٟ[*侻(G\ϛqAT e Y7T:) ˇs 噹yor)R%dcg؞o\R{ Z)I)H` CbnR,%`-FjڕI9{4ۆd" " 6z5;'z{{XF]GqTJԄh$~3gHA/wyyx sX0ѕtf_5F"uwtRж޶4pǎ1"Pk/̐]_'ɠ)7yiB7Fu{FR'{6W]7 F2Śӑ??RR&2I`=\Ur+,a:yVQ6JN|Ț[^jX6rZq0.t* Tl5 >}pV:]coEs۠]hID@D$ 4%ޮReAʰD(+Pb#M % o1F4Cy–aR؀ɓgI}\[[j.\);_T9{@ogW&˗x]|F UT\7Z " "p_\f( M#yzM+$aIzh:%ݻyuH|l^?7_6F -uL۫o6" "p ܠ 4?[6O$l```z2%PVT$0f4,NWg7VrCwbrrL93 x~eN,a6O5Vơl[B5@`x.U*HEGOwgwOgoιk 5jP8T' v9}2Ԃ(*!j, 7WJjJF cr. =o<HKKqˤ(ðsm⥒xD:|WrμECB$IZOw:ʹ94 C;.x.]I8[K"Q H޶rU3+" "bxsNPaR0B#b$C+=Xre|ёp%7oHhd`#ҕ*"M$o*" M` ZfV#$DScJ gbz&X6ɰ{5$I"9uL*ܪ"Fw\pUr˫㋀>F{k@@u{3ƃU?[0X$ߙ1c[x HW+du\nSD@D% l=5IEy>0!,=AGldީJ68BsZ˙kxfk1uyt7ꬴ@+j7oMkN;3vu"_}fxW[mӜD@D`3 c M5q<M1-H҄*S4mXݵM`˩+FnWIZ> 0 m^u==D@D@*tqH,ZK 2!vM zeQ8)C*&~oM> ^?m."  -'  Ԛo vuם۪רtG^JTK%@m},6o~y|WAbyhS9Tj(o[%" " lEv1]1 kp{k2kk*"ۛ# ͖dm[֟8 !QӁa`C\7=h WD@H,2Yhs |4xH=p\!`"K1O s]In6qkgBȎ<ۭR ɏlևv 5Iɯֶx9Ԅ4`p.?tS_JL@.D@D@ s%Yf<.bâ('T ʍASJҶ滳I>2rMu8-0Dx$4~iZ$7S6kl"Fb3(7gq6H;O_ ޙKlisAą1xAbZuD12acE\k`&88۳/j iInle$M SX?f4=ېLAg1XBh~QJcK3L¦oLҴcux̷K-H_B37kȽgۀ9}n=N*" "0*:WYo~k?}˔[sG͠XV`f\LO9S Y4dS5PLU͗m`!M tv(S_vw ldv?CWwdaQ:7[srwlFj#Ytj\| z+" "٭cz17yLfkGTKZ+fТM4E;7kf&f3vֱ6`6{eqoI_[/|Xl&Ofg4یbIٯ~ {_C5l_hμ4On+B H wcc/_U0߂ M àpf:v|bbۛ12tl6>wvu[XX7:v4I;{|oYs޷iMFhVr<8 ;nӶ,w[r1v߂"y :fl-aJz_e3ɳyudC~>L>ڌ_no)S&~@}dL LXS>iK~%ߗu`.ٞ~cc3.(o#4eOBS3~HV%c}8)o٤Lڱ}ץ4>εӾ" " G u[6w[\lUfkf7_z0 *YêD@D@6y|WqR?z۸ fy~?&d 7}uh(<7ޱ` dbSEC3=&-k$7; 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