The moon is the first milestone on the road to the stars Arthur C. Clarke

Transcription

1 Lo we n e r g yt r a j e c t o r i e st ot h emo o n b yj e p p esø g a a r dj u u l De c e mb e r S u p e r v i s o r : Po u l Hj o r t h De p a r t me n t o f Ma t h e ma t i c s : Te c h n i c a l Un i v e r s i t yo f De n ma r k I n t e r n a l S u p e r v i s o r : Mo g e n shø g hj e n s e n Ni e l sbo h ri n s t i t u t e Un i v e r s i t yo f Co p e n h a g e n

2 The moon is the first milestone on the road to the stars Arthur C. Clarke i

3 Abstract A new design strategy for finding low energy trajectories from Earth orbit to Moon orbit through the first Lagrange point has been developed. The found trajectories are very close to the theoretical minimum values of v necessary to go from Earth orbit to L1 and from L1 to Moon orbit. However, a large velocity change at L1 gives the most fuel effective trajectory a total v of m/s. As part of the project, seven different one step integrators have been compared, with six of these being symplectic integrators. It was found that the integrator, which approximated the flow of the restricted three body system the fastest and most accurately, was the non-symplectic 4th order Runge-Kutta Method. ii

4 Contents 1 Introduction 1 2 Hamiltonian mechanics 1 3 The Earth-Moon system Equations of motion in a synodic frame Effective potential and the Lagrange points Theoretical minimum for v Existing trajectory designs Direct transfer: Hohmann Weak stability boundaries: Belbruno-Miller L1 hyperbolic transit orbits: Topputo Trajectory design Trajectories from the Earth approaching L Trajectories from L1 approaching the Earth Optimizing the trajectories Trajectory from L1 to Moon orbit Numerical integrators Numerical integrators and orders Partitioned methods and symplectic integrators Symmetric integrators and composition methods Splitting methods Choice of integrator Results Earth-L1 leg L1-Moon leg Total Earth-Moon trajectory Discussion 17 9 Possible improvements on the trajectory designs Conclusion Acknowledgements 20 References 20 A MATLAB files: Finding trajectories 22 A.1 Trajectories from Earth approaching L A.2 Trajectories from L1 approaching the Earth A.3 Optimizing the trajectories A.4 Subfunction: Createsystem A.5 Subfunction: Findmatches A.6 Subfunction: Etadot A.7 Subfunction: Etadotreverse iii

5 B MATLAB files: The best integrator 30 B.1 Main program integrating and comparing B.2 Symplectic Euler integrator B.3 Störmer/Verlet integrator B.4 4th order Runge-Kutta B.5 2nd order splitting method B.6 4th order splitting method B.7 6th order splitting method B.8 8th order splitting method B.9 phi1 and phi2 used to the splitting methods iv

6 1 Introduction The space race, inspired by the Cold War between the Soviet Union and the United States, led to an increased interest in the exploration of the Moon. The landing of the first humans on the Moon in 1969 is generally viewed as the culmination of the space race. After the mid-1970s only few spacecrafts has reached Moon orbit, among these the Japanese Hiten spacecraft (1990) and the European Smart 1 spacecraft (2003). On January 14, 2004, U.S. President George W. Bush called for a plan to return manned missions to the Moon by 2020 [12]. NASA is now planning for the construction of a permanent outpost at one of the lunar poles [13]. Russia and the People s Republic of China have also expressed plans for exploring the Moon. Like NASA, China hopes to land people on the Moon by 2020 [21]. This renewed interest in Moon expeditions calls for new low-energy trajectories going from low Earth orbit to Moon orbit. Especially for unmanned expeditions, where time of flight is not a central issue, much propellant can be saved by using special trajectories that exploit the physical structure of the Earth-Moon system [11]. In this project a new method of finding low energy trajectories between Earth and Moon orbit is developed, using the programming language MATLAB. Before this, basic concepts of Hamiltonian mechanics and the Earth-Moon system are explained and the most important existing trajectory designs are described. In the search for low energy trajectories numerical integration plays a crucial part. The theory of numerical one-step integrators will therefore be described, with special attention being paid to symplectic integrators. The integrator, which approximates the trajectories fastest and most accurately, will be found systematically. I have not previously had an analytical mechanics course, nor worked with numerical integrators or the programming language MATLAB. The project thus gave me an opportunity to work with an area of physics that was new to me and also allowed me to acquire new skills in mathematics and programming. 2 Hamiltonian mechanics In this section the basic concepts of Hamiltonian mechanics will be stated briefly without further proofs. The section is primarily based on [5], [6] and [8]. For a conservative mechanical system the Lagrangian L is defined as the kinetic energy minus the potential energy. If the state of the system can be completely described by the generalized coordinates q and velocities q, the Euler-Lagrange equations d L dt q L q = 0 (1) are equivalent to Newton s second law, and thus determines the flow of the system in phase space. Instead of describing the system by (q, q), one can carry out the Legre transformation (q, q) (q, p) given by p = L q if 2 L q q 0. (2) Here p is called the generalized momentum. In the variables (q, p) the Euler-Lagrange equations are equivalent to Hamilton s equations: ṗ = H q and 1 q = H p, (3)

7 where the Hamiltonian H is defined as Defining the column vector η = ( p q H = p q L. (4) ) we can write the Hamilton s equations (3) compactly η = J 1 H(η) where J = [ 0 I I 0 ]. (5) The matrix J is called the symplectic matrix. If the function η(t) is a solution to (5) with starting value η(0) = η 0 in phase space, we define the flow ϕ t (η 0 ) of the system as the map that associates η 0 with η(t). That is ϕ t (η 0 ) = η(t) if η(0) = η 0. (6) The Hamiltonian will be conserved along the flow of the system, i.e. the quantity H(η(t)) will have the same value for all t. The Jacobian ϕt η of the flow of any Hamiltonian satisfies the relation T ϕ t J ϕ t η η = J, (7) with J given by (5). All maps with Jacobians satisfying (7) are called symplectic maps. 3 The Earth-Moon system 3.1 Equations of motion in a synodic frame If all other celestial objects are neglected, the Moon and the Earth rotate about their center of mass in approximately circular Kepler orbits confined to a two dimensional plane. If the masses of the two objects are denoted m e and m m we can define the Earth-Moon parameter as [11] [20] µ = m m m e + m m = (8) We will observe the rotating system in the non-inertial reference frame [10] with origo at the center of mass and same angular frequency as the system, which we will normalize to 1. If the distance between the Earth and the Moon is also normalized to 1, the two bodies will be located on the x-axis at the points ( µ, 0) and (1 µ, 0) (see figure 1). This is called the synodic frame. Given the Earth-Moon distance and period, one can calculate the unit of length, velocity and time in this frame [20] unit length = 384, 405 km unit velocity = km/s (9) unit time = days. The kinetic energy of a satellite of unit mass moving in this frame will be K(x, y, ẋ, ẏ) = 1 2 ( (ẋ y) 2 + (ẏ + x) 2). (10) 2

8 Figure 1: The synodic frame of the Earth-Moon system has origo in the center of mass, and rotates with normalized angular frequency so both the Earth and the Moon are situated on the x-axis. Using Newton s gravitational potential energy where the gravitational constant is normalized to 1, the Lagrangian will be given by [11] L = K V = 1 2 ( (ẋ y) 2 + (ẏ + x) 2) + 1 µ (x + µ) 2 + y 2 + µ (x + µ 1) 2 + y 2. (11) The generalized momenta can be found using (2) p x = ẋ y, p y = ẏ + x ẋ = p x + y, ẏ = p y x, (12) p x and p y are thus the components of the velocity of the satellite in the inertial frame. Now, the Hamiltonian can be found using (4). H = p2 x + p 2 y 2 + p x y p y x 1 µ (x + µ) 2 + y µ 2 (x + µ 1) 2 + y. (13) 2 The right hand side of (12) can be identified as the equations of motion for x and y. For p x and p y the equations of motion can be found using (3) p x = H x = p (1 µ)(x + µ) y + ((x + µ) 2 + y 2 ) 3/2 + µ(x + µ 1) ((x + µ 1) 2 + y 2 ) 3/2 (14) p y = H y = p (1 µ)y x + ((x + µ) 2 + y 2 ) 3/2 + µy. ((x + µ 1) 2 + y 2 3/2 ) (15) As in all other Hamiltonian systems, the Hamiltonian (13) is conserved along the flow. Therefore C = H/( ) is also conserved. This quantity is called the Jacobi-integral. [19] 3.2 Effective potential and the Lagrange points When the satellite is stationary in the synodic frame we see from (12) that the generalized momenta will simply be given by p x = y and p y = x. If this is inserted in the Hamiltonian (13) we see that the result equals the Hamiltonian minus 1 2 (ẋ2 + ẏ 2 ). The expression is therefore denoted the effective potential (see figure 2) V eff = x2 + y µ (x + µ) 2 + y 2 µ (x + µ 1) 2 + y 2. (16) 3

9 Figure 2: Left: The effective potential with the five Lagrange points. Right: The satellite can not enter regions where the effective potential is higher than the value of the Hamiltonian of the satellite. Figure taken from [11]. As 1 2 (ẋ2 + ẏ 2 ) is always positive, we have H > V eff. If the satellite starts at a point i phase space with Hamiltonian H, this inequality defines the region of space where the satellite can move. The extrema of V eff corresponds to equilibrium points in the synodic frame, i.e. points where the satellite can remain stationary with respect to the Moon and the Earth. Five such points exist, called the Lagrange points. The Lagrange point L 1 is located between the Moon and the Earth, approximately at (x, y) = (1 (µ/3) 1/3, 0). [4] 3.3 Theoretical minimum for v The Hamiltonian of a satellite orbiting the Earth at an altitude of 167 km can be found using (13) and simple Newtonian calculations. If the same calculations are carried out for a satellite orbiting the Moon at an altitude of 100 km, one gets [19] H Earth orbit = H Moon orbit = (17) When the satellite travels from the Earth to the Moon, we see from figure 2 that it must cross a point in phase space with V eff greater or equal to V eff evaluated in the first Lagrange point L 1. Because H V eff the Hamiltonian must at that point satisfy x + µ 1 H L1 V eff L1 = x2 2 1 µ x + µ µ x=1 ( µ 3 )1/3 = 1.594, (18) where (8) has been used. The only way the Hamiltionian can change from in Earth orbit to at L1, is through one or more velocity changes v along the way. From (13) it is seen that such an acceleration will change the Hamiltonian by an amount H H = v 2 + v v (19) where v is the velocity of the satellite in the inertial frame. Thus, the Hamiltonian is changed the most if the acceleration and velocity are parallel, and if the acceleration is carried out when the satellite has the greatest velocity. This is when the satellite is closest to the Earth. Using (17), (18) and (19) the lowest theoretical value of v EL, necessary to get the satellite from Earth orbit to L1, can be calculated. Using the same procedure velocity change v LM 4

10 necessary for the satellite to go from L1 to Moon orbit can be found. denoted v EM. [19] The sum of these is v EL = 3099 m/s v LM = 622 m/s v EM = 3721 m/s. (20) 4 Existing trajectory designs So far it has not been possible to find a trajectory from Earth orbit to Moon orbit, using only the minimum v given by (20). To find such a trajectory many design strategies have been proposed. In this section I will describe some of the more important of these. In table 1 fuel cost and typical flight times for different types of trajectories are shown. 4.1 Direct transfer: Hohmann The simplest and the fastest type of trajectory is the direct transfer, including the Hohmann transfer. A large burn ss a satellite from low Earth orbit to an elliptic trajectory, which crosses the orbit of the Moon. If the timing is right, the satellite will up with a distance to the Moon corresponding to the radius of the desired final orbit and with a velocity tangential to the Moon. Another burn is then performed to slow the satellite down, causing it to enter Moon orbit. [18] 4.2 Weak stability boundaries: Belbruno-Miller Belbruno and Miller have found a class of trajectories where the gravitational pull from the Sun is taken into account along with that of the Moon and the Earth. This system is called the restricted four body problem [2]. Regions in phase space where gravitational attractions of the Sun, the Moon and the Earth t to balance are called weak stability boundaries (WSB) [9]. The second Lagrange point L2 is in such a region. From low Earth orbit the satellite is accelerated, causing it to fly by the Moon and continuing approximately four Earth-Moon distances into the WSB region. Here a small burn is performed to put the satellite into a ballistical lunar captured trajectory. That is, an unstable elliptic orbit with low eccentricity. Then a final maneuver is performed to put the satellite into circular orbit. [3] [1] [18] [20] 4.3 L1 hyperbolic transit orbits: Topputo As oppose to the WSB trajectories, that pass close by L2, Topputo has developed a design strategy where the satellite passes close by the first Lagrange point, L1. A linearization of the equations of motion (12), (14) and (15) shows that the satellite close to L1 will follow hyperbolic transit orbits. The trajectory from Earth orbit to Moon orbit is split into two legs: The Earth-L1 leg and the L1-moon leg. Both legs are found by solving a two-point boundary value problem for the restricted three body problem using the Lambert three-body arc method [20]. The solutions are trajectories with very long flight-time, but especially in the L1-Moon leg much propellant is saved (see table 1). 5 Trajectory design Having studied existing trajectory designs, I have developed my own design and written a MATLAB program to find the trajectories. 5

11 Trajectory Total v EM Earth injection v EL Moon injection v LM Flight time Minimum 3721 m/s 3099 m/s 622 m/s Not found Hohmann 3959 m/s 3140 m/s 819 m/s 5 days Belbruno-Miller 3838 m/s 3187 m/s 651 m/s 3 months Topputo 3895 m/s 3265 m/s 630 m/s 8 months Table 1: Propellant requirements and typical flight times for different types of trajectories going from Earth orbit (167 km altitude) to moon orbit (100 km altitude). All v are in m/s. [19] [20] As in the Topputo method described in subsection 4.3, I let the trajectory pass through the first Lagrange point L1. The trajectory can therefore be split up into an Earth-L1 leg and a L1-Moon leg. As opposed to Topputo, I will not find these legs by solving a two-point boundary value problem. Instead I will find a lot of trajectories going from Earth orbit out in space, and a lot of trajectories ing at the point L1 with a velocity toward the Moon. The trajectories starting in Earth orbit are integrated forward in time and the trajectories that in L1 are integrated backward in time. At the intersections between these two types of trajectories the v transfer necessary to go from one trajectory to the other is calculated (see figure 3). If v transfer is added to the velocity change v Earth the satellite performed in Earth orbit, the total change of velocity v EL necessary to go from Earth orbit to L1 is found. Figure 3: Trajectories starting in Earth orbit are integrated forward in time, and trajectories ing in L1 are integrated backwards in time. At the intersections the v transfer necessary to go from one trajectory to the other is calculated. This change of velocity, along with the v Earth that made the satellite leave Earth orbit, amounts to the total v EL to get from the Earth to L1. In this design it is ensured that the main velocity change, v Earth, takes place when the satellite is orbiting the Earth, which results in the greatest change of the Hamiltonian (see section 3.3). When low v EL trajectories are found, the same procedure is carried out for the L1-Moon leg. This time the trajectories starting at L1 are integrated forward in time, and the trajectories ing in Moon orbit are integrated backwards in time. In the the Earth-L1 leg and the L1-Moon leg are connected to one another via a change of velocity v L1. The MATLAB-program finding trajectories going from Earth orbit to L1 is made up of three parts: 6

12 1. The first gives different satellites orbiting the Earth the burn v Earth, and stores information of the resulting trajectories. 2. The second starts at L1 and integrates backwards in time in different directions and using different initial velocities. For every Earth-trajectory that is crossed, the program calculates v EL, and the matches with the lowest values are stored. 3. The third repeats the two first parts with increased accuracy and resolution, now only using initial values lying close to those yielding the lowest v EL. In the next three sections the three parts of the program are described in detail. The code for the three parts can be found in appix A.1, A.2 and A.3. In section 5.4 it is described how to change the program in order to find trajectories going from L1 to the moon. The discussion of how to integrate the equations of motion (12), (14) and (15) fastest and most accurately is saved for section Trajectories from the Earth approaching L1 The first task in this part of the program is to choose a number of initial values η 0 the satellite can have after the burn v Earth is performed. The satellite can orbit the Earth either clockwise or counterclockwise, and the change of velocity v Earth can be carried out while the angle θ 1 has any value between 0 and 2π (see figure 4). The change of velocity should be v Earth 3099 m/s (see section 4) but can vary slightly in magnitude. The angle θ 2 between the burn and the velocity of the satellite can take any (small) value. This gives four degrees of freedom when constructing the initial values η 0 of the trajectory. Choosing 10 different values of v Earth, 63 values of θ 1 and 5 values of θ 2 a total number of 6300 different initial values are used. These are chosen in such a way that the resulting trajectories differ from each other as much as possible, and so the values H(η 0 ) are close to that of H L1 found in (18). Figure 4: The satellite can start out at any angle θ 1 in the orbit of the Earth and rotate either clockwise or counterclockwise. The velocity change v Earth can vary slightly in magnitude and can be applied at different small angles θ 2. Varying these parameters 6300 different initial values of the satellite were selected. I choose to integrate the trajectories for a period of 100 days, or until the satellite either crashes into the Earth or the Moon, or until the absolute value of the x- or y-component of the trajectory exceeds 1.5, corresponding to 1.5 times the distance between the Earth and the Moon. If this happens the satellite is considered to have escaped the Earth-Moon system, and Sun-perturbations ought to be taken into account. 7

13 Information on the trajectories should be stored in a way that uses as little memory as possible and allows the second part of the program to easily search for the lowest v transfer. By storing the information in different matrices corresponding to different (x, y)-values, only the information of points with the right (x, y)-values has to be searched in the second part of the program. Therefore, for each point {(x, y) x, y Z/200 x, y < 1.5} a matrix is created in the cell array called system. For a fixed number of points nopoints along each trajectory the (x, y)-components of η(t) are rounded to the nearest part in 200, and in the corresponding matrix in system the velocity vector of the satellite is stored along with the time t and a number to identify the initial value η 0 of the trajectory. The number of points nopoints is chosen such that information of only one or two η(t) is stored in each matrix the trajectory passes. The resolution of the cell array system is set to x = 1/200. This resolution is chosen by considering the advantages and disadvantages of a high resolution. With a high resolution (small x) the number of stored points has to be higher, and more memory is required. Using a low resolution, there is an increased risk that two trajectories, that are thought to intersect, actually only pass by each other with a distance d < x. This effect will lead to wrong results for the found v EL and the error will be largest close to the earth, as the effective potential (16) has the largest gradient in this area. I therefore choose to disregard all transfers closer than 0.1 to the Earth. 5.2 Trajectories from L1 approaching the Earth When the trajectory of a satellite s at L1, only two degrees of freedom are available for constructing the final values η final of the trajectory. These are the direction angle and magnitude of the velocity at L1. Choosing 63 different direction angles and 25 different velocities, a total number of 1575 final values are used. The trajectories are integrated 100 days backward in time, or until the satellite either collides with the Earth or the Moon, or the absolute value of the x- or y-component of the trajectory exceeds 1.5. For the number nopoints of points along each trajectory the (x, y)-components of η(t) are rounded to the nearest part in 200, and the corresponding matrix in system is identified. This matrix contains information of all trajectories starting in Earth orbit, that in a point close to (x, y) intersect the trajectory ing in L1. For each set of information the v transfer is calculated and added to the corresponding v Earth to get v EL. The minimum value of v EL is saved in a new matrix called matches, along with information about η 0 and η final of the intersecting trajectories, the point (x, y) and the integration time before and after the intersection. In the the 2000 matches with lowest v EL are picked out and all other matches were deleted. 5.3 Optimizing the trajectories In this part of the program the matches found in section 5.2 are integrated again, using smaller step size h and smaller x. Furthermore the trajectories in the matches are varied to optimize the final result for the velocity change. The program can be split into 6 parts: 1. First the matches that are to be optimized are selected. These are the 20 best matches, which do not share the same initial values both in Earth orbit and at L1. This way it is ensured that the same trajectories are not optimized several times. For each of the 20 matches the following actions are carried out: 8

14 2. The accuracy and resolution are increased by decreasing x and the step size h. 3. The integration times tmaxel and tmaxle are set to be 4.3 days after the intersection occurs in the match initial values in Earth orbit and 9 initial values at L1 are computed in a way, so that they are close to the initial values η 0 used in the original match. Using these in the integration, the optimal initial values are found, and the original match is therefore optimized. 5. From the 27 initial Earth values a new cell array system is generated, and using the 9 initial L1 values the matches with lowest v EL are found and stored. 6. For the best match found under point 5, the steps 3-5 are repeated. This way the optimized trajectory is again optimized, decreasing v EL further. 5.4 Trajectory from L1 to Moon orbit The trajectory going from L1 to Moon orbit using the smallest velocity change v LM is found using the same procedure as the trajectory going from Earth orbit to L1. However, a few changes have to be made. First of all, forward time integration now has to be used for the trajectories starting at L1, and backward time integration is used for the trajectories starting in Moon orbit. Furthermore, the distance between the Moon and L1 is far smaller than the distance between the Earth and L1, and the size of the system should therefore be reduced correspondingly. The satellite is now said to have left the system if the x-component of the trajectory becomes less than 0.8 or more than 1.2, or if the absolute value of the y-component of the trajectory exceeds 0.2. For the same reasons the resolution of the system has to be increased, and x is therefore decreased to instead of Because of the higher resolution and because the gradient of the effective potential is smaller close to the Moon than equally close to the Earth, only the transfers closer than 0.02 to the moon are disregarded. For the Earth-L1 leg all transfers closer than 0.1 to the Earth were disregarded. 6 Numerical integrators 6.1 Numerical integrators and orders Numerical integrators are used to approximate the flow over time of a system of differential equations. In this paper, only systems of first order ordinary differential equations with no explicit time depence will be considered. The equations of motion (3) of a Hamiltonian system will be of this form, if the Hamiltonian has no explicit time depence. A one-step numerical integrator is a map Φ h (η 0 ), which associates the point η 0 in phase space with a point close to ϕ h (η 0 ) defined by (6). Here h is the step size in time t. The smaller h is chosen to be, the closer Φ h (η 0 ) will be to ϕ h (η 0 ). A one-step method is said to have order p if the distance between Φ h (η 0 ) and ϕ h (η 0 ) for small h is of order O(h p+1 ). [5] Φ h (η 0 ) = ϕ h (η 0 ) + O(h p+1 ) for h 0. (21) Typically, the difference between H(Φ h (η 0 )) and H(ϕ h (η 0 )) will also be of O(h p+1 ). The total difference after n iterations will thus be of the order: H(Φ nh (η 0 )) H(ϕ nh (η 0 )) = n O(h p+1 ) = O(nh p+1 ) = O(th p ). [14] (22) 9

15 Most often order considerations are based on Taylor-expansions. Throughout the project all maps will be assumed analytical, thus making Taylor expansions possible to any order Example: Explicit 4th order Runge-Kutta The Runge-Kutta methods is an important class of one-step integrators. For the system η = f(η), an explicit 4th order Runge-Kutta method is given by: Φ h (η) = = η + h 6 (k 1 + 2k 2 + 2k 3 + k 4 ) where k 1 = f(η) = d dt η d 2 k 2 = f(η + h 2 k 1) = d dt η + h 2 dt 2 η (23) k 3 = f(η + h 2 k 2) = d dt η + h d 2 2 dt 2 η + h2 d 3 2 dt 3 η (24) k 4 = f(η + hk 3 ) = d dt η + h d2 dt 2 η + h2 d 3 2 dt 3 η + h3 d 4 η. 4 dt4 (25) Inserting k 1 k 4 and collecting terms of h we get Φ h (η) = η + h d dt η + h2 2 d 2 dt 2 η + h3 6 d 3 dt 3 η + h4 24 d 4 η. (26) dt4 This is exactly the first 5 terms of the Taylor expansion of ϕ h (η). The next leading term will be of order O(h 5 ) making this the difference between ϕ h (η) and Φ h (η). Thus this Runge-Kutta method is indeed of order 4. The method is explicit because all iteration steps can be calculated directly without solving any non-trivial equations. 6.2 Partitioned methods and symplectic integrators A one step integrator is symplectic if it satisfies the symplecticity condition (7) T Φ h J Φ h η η = J (27) Symplectic integrators give rise to long time energy conservation (see figure 5). The proof of this is comprehensive, but can be found in e.g. [14] and [5]. Symplecticity is sometimes obtained using partitioned Runge-Kutta methods, where some of the variables of η are integrated using one Runge-Kutta method, and the rest are treated with another Runge-Kutta method Example: Symplectic Euler method The symplectic Euler method is a partitioned Runge-Kutta method of first order [5]. The p and the q variables of η are integrated using two different Runge-Kutta methods giving the map Φ h (η 0 ) the form Φ h (η 0 ) = η 0 = ( p1 q 1 ) = ( p0 h H q (p 1, q 0 ) q 0 + h H p (p 1, q 0 ) ). (28) 10

16 Figure 5: Long time energy conservation of symplectic integrators. In short periods of time the 4th order Runge Kutta integrator conserves energy better than the first order symplectic Euler, but in longer timescales the latter conserves energy better due to its symplecticy. The plot is generated using the restricted three body problem with a step size in time h = Note that it is an implicit method, as p 1 must be found by solving the equation (28). To show the symplecticity of the method the Jacobian Φ h(η) η = η 1 η must be computed. Differentiating 0 (28) with respect to η 0 yields [ p η 1 p ] [ 1 p 1 = p 0 q 1 hhqp 1 p ] 0 = p hh qq hh 1 qp 0 q 0, (29) η 0 q 1 p 0 q 1 q 0 hh pp p 1 q hh pq + hh pp p 1 q 0 where H ij is the second partial derivative of H with respect to i and j evaluated in (p 1, q 0 ). Solving the four equations the Jacobian is found to be [ ] η hh = qq η 0 (1 + hh qp ) 2 hh pp (1 + hh pq ) 2 h 2. (30) H qq H pp It is now a simple matter to show that the symplecticity condition (27) is satisfied. 6.3 Symmetric integrators and composition methods For a one-step integrator Φ h, the adjoint Φ h is defined as the inverse map with negative timestep: Φ h = Φ 1 h (31) The adjoint defines a new integator that will be of same order as Φ h [5]. An integrator is symmetric if it equals its adjoint Φ h = Φ h. We will show that the order of symmetric integrators always will be even. If the method is symmetric and of order p we have from (21) Φ h (η 0) = Φ h (η 0 ) = ϕ(η 0 ) + C(η 0 )h p+1 + O(h p+2 ). (32) If the local error of Φ h is denoted e, we have e (η 0 ) = Φ h (η 0) ϕ(η 0 ) = C(η 0 )h p+1 + O(h p+2 ). (33) 11

17 If this error is projected back by Φ h the result e(η 0 ) will be given by e(η 0 ) = Φ h (Φ h (η 0)) Φ h (ϕ(η 0 )) = η 0 ( η 0 + C(ϕ(η 0 ))( h) p+1 + O(h p+2 ) ) = ( 1) p C(ϕ(η 0 ))h p+1 + O(h p+2 ) = ( 1) p C(η 0 )h p+1 + O(h p+2 ), (34) where the last equality arises because C(η 0 ) = C(ϕ(η 0 ))(1 + O(h)). Likewise, e and e must have the same leading term [14]. For this to be true we see from (34) and (33) that p must be even. A way to develop symmetric integrators is to use composition methods Ψ h : Ψ h = Φ γsh Φ γs 1 h... Φ γ1 h where γ γ s = 1. (35) If the components Φ i is a method of order p, the local errors at each step (see figure 6) will be e 1 = C(η 0 ) γ p+1 1 h p+1 + O(h p+2 ) e 2 = C(η 1 ) γ p+1 2 h p+1 + O(h p+2 ). e s = C(η s 1 ) γ p+1 s h p+1 + O(h p+2 ). (36) To leading order the error E of Ψ h will be the sum of the local errors. Generally, a composition Figure 6: (a) Composition Ψ h with the three components Φ γ2h, Φ γ2h and Φ γ3h. (b) Each step Φ γih has an error e i, giving the total error E. To leading order E will be the sum of the e i s. Figure reconstructed from [5]. method will therefore be of same order as its components, but if γ p γ p+1 s = 0 (37) we see that the total error will have no term of order O(h p+1 ) and the composition method Ψ h will at least be of order p + 1. A composition of symplectic integrators will also be symplectic. This is easily shown for a composition Φ 2 Φ 1 by inserting the Jacobian into (27) T Φ 2 (Φ 1 (η)) J Φ 2(Φ 1 (η)) η η = ( Φ2 (Φ 1 (η)) = Φ 1(η) η Φ 1 (η) Φ 1 (η) η T Φ 2 (Φ 1 (η)) Φ 1 (η) = Φ T 1(η) J Φ 1(η) η η = J. 12 ) T J Φ 2(Φ 1 (η)) Φ 1 (η) Φ 1 (η) η T J Φ 2(Φ 1 (η)) Φ 1 (η) Φ 1 (η) η (38)

18 6.3.1 Example: Störmer/Verlet The Störmer/Verlet method Ψ h is the composition of the symplectic Euler (28) with its adjoint [5]. It is easy to show that it is a symmetric method: Ψ h = Φ h/2 Φ h/2 = Φ h/2 Φ 1 h/2 (39) ( ) 1 Ψ h = Ψ 1 h = Φ h/2 Φ 1 h/2 = Φh/2 Φ 1 h/2 = Ψ h, (40) and due to its symmetry it has order 2. Because the Störmer/Verlet method is composed of symplectic integrators, it is itself symplectic. 6.4 Splitting methods Another way to approximate the flow of a Hamiltonian system η = J 1 H, is to split it into a sum of two or more systems, η = J 1 ( H [1] H [n]), each of which can be solved explicitly (see figure 7). Figure 7: If the vector field can be split into two or more components for which the flow can be found explicitly, a splitting method can be developed. If the exact flow of η = J 1 (H [i] (η) is called ϕ [i] t, a one step integrator that approximates the flow ϕ h of the total system is Φ h = ϕ [n] h... ϕ[1] h. (41) By Taylor expansion it is found that both ϕ h (η 0 ) and Φ h (η 0 ) can be expressed as η 0 + hj 1 H(η 0 ) + O(h 2 ). Thus they only differ at order O(h 2 ), and a general non-symmetric splitting method is therefore of order 1. [14] However, one can construct a symmetric splitting as follows Φ 2nd h = ϕ [1] h/2... ϕ[n] h/2 ϕ[n] h/2... ϕ[1] h/2. (42) Due to its symmetry it must be of order 2 (see section 6.3). From the second order symmetric splitting method, it is possible to construct integrators of arbitrarily high orders, using composition methods. A fourth order method can be constructed in the following way Φ 4th h = Φ 2nd γ 1 h Φ2nd γ 2 h Φ2nd γ 3 h where γ 1 = γ 3 = /3, γ 2 = 21/ /3. (43) 13

19 Inserting (43) into (37) shows that the method is at least of order 3, but due to its symmetry it must be of order 4. Likewise, a method of order 6 can be constructing by Φ 6th h = Φ 4nd γ 1 h Φ4nd γ 2 h Φ4nd γ 3 h where γ 1 = γ 3 = /5, γ 2 = 21/ /5. (44) Generally, a splitting method of order p + 2 can be constructed by composition of integrators of order p using the γs: γ 1 = γ 3 = p+1, γ 2 = 2 1 p p+1, (45) which satisfies (37). [5] Because splitting methods are compositions of flows of Hamiltonian systems, and because such flows are always symplectic (see (7)), splitting methods are always symplectic. 6.5 Choice of integrator To find the best integrator to use in this project, all of the above mentioned integrators have been constructed in MATLAB (see appix B) Symplectic Euler and Störmer/Verlet are the only implicit integrators described above. Instead of having MATLAB solve the implicit equations at every iteration, I have solved the equations by hand, thereby making the programs consist only of function evaluations. In order to construct the splitting methods, the Hamiltonian (13) must be split into two or more Hamiltonians for which the equations of motion can be solved analytically. This has, to my best knowledge, never been done before for the Hamiltonian (13), and this is the subject of next subsection. In subsection the integrators are compared to each other Splitting the Hamiltonian and solving equations of motion One way of splitting the Hamiltonian into two, resulting in simple analytical expressions for ϕ [1] t and ϕ [2] t is H [1] = p2 x 2 + p2 y 2 + p xy p y x 7 2 x2 + 2xy (46) H [2] = 1 µ (x + µ) 2 + y µ 2 (x + µ 1) 2 + y x2 2xy, (47) yielding the equations of motions: H [1] : η = J 1 H [1] = H [2] : η = J 1 H [2] = (x+µ)(1 µ) ((x+µ) 2 +y 2 ) 3/2 y(1 µ) ((x+µ) 2 +y 2 ) 3/2 η (48) µ(x+µ 1) 7x + 2y ((x+µ 1) 2 +y 2 ) 3/2 yµ + 2x ((x+µ 1) 2 +y 2 ) 3/ (49) The terms of 7 2 x2 + 2xy are added and subtracted in (46) and (47) in order to give the matrix in (48) simple eigenvalues. In this case the values -2, -1, 1 and 2. 14

20 Now the equations of motion may be solved to give [17] t (η 0 ) = C 1 ϕ [1] ϕ [2] t (η 0 ) = [ e2t + C (x(0)+µ)(1 µ) [((x(0)+µ) 2 +y(0) 2 ) 3/2 y(0)(1 µ) ((x(0)+µ) 2 +y(0) 2 ) 3/2 et + C e t + C e 2t ] µ(x(0)+µ 1) 7x(0) + 2y(0) t + p ((x(0)+µ 1) 2 +y(0) 2 ) 3/2 x (0) ] y(0)µ + 2x(0) t + p ((x(0)+µ 1) 2 +y(0) 2 ) 3/2 y (0) x(0) y(0). The constants C 1 -C 4 in ϕ [1] are determined using the initial values η(0) Comparison of integrators Comparing the numerical methods, the satellite was set to start at η 0 (0, , , 0), corresponding to a location 167 km above the surface of the earth, with a value of the Hamiltonian close to (18). For each integrator the trajectory was integrated from t = 0 to t = 23.04, corresponding to 100 days, using time steps ranging from h = to h = The integration time on a 2.4 GHz dual core computer was found for each integration. The final energy was compared to the initial energy, and the position in phase space was compared to a position obtained by integrating the same problem to the limit of double precision. The results are shown in figure 8. Figure 8: The errors in energy (left) and phase space (right) for each integrator as a function of stepsize in time (upper) and total integration time (lower) for each integrator. For very small h the accuracy is limited by rounding errors in MATLAB. For integration times above seven seconds the 4th order Runge-Kutta method is seen to be most accurate. 15

21 It is seen that for relatively large step sizes h, the integrators with higher orders are closest to the right energy and point in phase space. However, for very small h, all integrators have problems getting closer than 10 4 of the correct η. This is due to rounding errors in MATLAB, and could be avoided by enhancing the precision. However, this would increase the integration time drastically, and I therefore choose not to do this. It appears that the 4th order Runge-Kutta method is not affected by the rounding errors to the same extent as the other methods. This is because each time step only involves 5 calculations. In comparison the 8th order splitting method involves 81 calculations. The two lower plots of figure 8 show that the second order splitting method is the most accurate for integration times below approximately seven seconds, while 4th order Runge Kutta is most accurate for integration times of more than seven seconds. This is surprising, as the Runge-Kutta method is the only non-symplectic integrator used. However, figure 5 shows that the long-time energy conservation of the symplectic integrators are relevant only for longer timescales and larger h. Based on the results displayed in figure 8, I chose to use the 4th order Runge-Kutta method with h = 10 4 in the first two parts of the MATLAB program, described in section 5.1 and 5.2. According to the results this integrator gives an integration time of approximately 17 seconds per trajectory and an accuracy in energy and phase space after 100 days of approximately 10 5 and 10 3, respectively. In the third part of the program, described in section 5.3 the accuracy should be enhanced, so I choose to use a step size h = giving an integration time of approximately 70 seconds and an accuracy in energy and phase space of approximately 10 8 and 10 5, respectively. 7 Results 7.1 Earth-L1 leg Of the 6300 different trajectories starting in Earth orbit 2521 were integrated for the full 100 days, 4310 trajectories ed up colliding with the Earth or the Moon and 201 ed up leaving the system. Of the 1575 trajectories starting at L1 927 were integrated for the full 100 days, 443 trajectories ed up colliding with the Earth or the Moon and 205 ed up leaving the system. Before the optimization described in section 5.3, the match with the minimum velocity change had v EL = 3065 m/s. The match with the 20th lowest velocity change had v EL = 3088 m/s. This is to be compared to the theoretical minimum v EL = 3099 m/s. This, and all other results, are discussed in section 8. After the 20 best results had been optimized, higher values for the velocity change were obtained. The velocity changes and flight times of the three best optimized matches are shown in table 2 together with the velocity the satellite s up having at L1. All of the 20 optimized trajectories have the velocity 650 m/s at L1 and angles of direction within 0.3 degrees of each other. Velocity change v EL v Earth v transfer Velocity at L1 Flight time m/s 3080 m/s 16.8 m/s 650 m/s 83.3 days m/s 3080 m/s 26.8 m/s 650 m/s 83.1 days m/s 3076 m/s 39.8 m/s 650 m/s days Table 2: Data of the three trajectories going from Earth orbit to L1 having the lowest velocity change. 16

22 7.2 L1-Moon leg Of the 6300 different trajectories starting in Earth orbit 529 were integrated for the full 100 days, 5385 trajectories ed up colliding with the Moon or the Earth, and 386 ed up leaving the system. Of the 1575 trajectories starting at L1 985 were integrated for the full 100 days, 479 trajectories ed up colliding with the Earth or the Moon, and 111 ed up leaving the system. Before optimization the match with the minimum velocity change had v LM = m/s, with the theoretical minimum being v LM = 622 m/s. The match with the 20th lowest velocity change had v EL = m/s. After the optimization, higher values for the velocity change of the best matches were again obtained. The velocity changes and flight times of the three best optimized matches are shown in table 3 along with the velocity the satellite s up having at L1. For all of the 20 optimized matches the velocities at L1 were between 8.64 m/s and 9.28 m/s, and the angles of direction were within 0.5 degrees of each other. Velocity change v LM v Moon v transfer Velocity at L1 Flight time m/s 616 m/s 4.5 m/s 8.66 m/s 68.4 days m/s 616 m/s 4.5 m/s 8.67 m/s days m/s 615 m/s 6.3 m/s 8.99 m/s days Table 3: Data of the three trajectories going from L1 to Moon orbit having the lowest velocity change. 7.3 Total Earth-Moon trajectory When connecting the Earth-L1 leg and the L1-Moon leg, a velocity change v L1 is needed at L1. The total velocity change v EM = v EL + v L1 + v LM can be minimized by choosing different combinations of Earth-L1 trajectories and L1-Earth trajectories. It is found that the minimum v EM is obtained when the trajectory with the lowest v EL is combined with the trajectory with the lowest v LM. The data for the trajectory going from Earth orbit to Moon orbit with the lowest v EM is listed in table 4 along with the theoretical minimum as found in section 3.3. In figure 9 the trajectory is plotted in the synodic frame. Trajectory Total v EM v EL v L1 v LM Flight time Found m/s m/s m/s m/s days minimum 3721 m/s m/s - Table 4: Data of the full Earth-Moon trajectory with lowest v EM along with the theoretical minimum. 8 Discussion Many of the integrated trajectories ed up colliding with the Earth or the Moon, especially in the case of the L1-Moon leg. This can be understood in the following way: If the Earth is ignored, the satellite would, when it is accelerated from Moon orbit, enter an elliptic orbit with perigee 100 km above the surface of the Moon. The presence of the Earth will, of course, modify this trajectory, and in many cases the satellite will be displaced more than 100 km (

23 Figure 9: The Earth-L1 trajectory with the lowest v EL = m/s is plotted in the synodic frame along with the L1-Moon trajectory with the lowest v LM = m/s. The velocity change at L1 is v L1 = m/s, giving the total v EM = m/s. unit lengths) in the direction of the Moon, and therefore collide with it. The same effect is present, though less prominent, in the Earth-L1 leg. In both the Earth-L1 and the L1-Moon leg, the 20 optimized matches have velocities and angles of direction very similar to each other. This indicates that the found trajectories are close to each other in phase space, and therefore not indepent of each other. This also explains why all combinations of matches from the Earth-L1 leg and the L1-Moon leg had about the same, very high v L1. Both before and after the optimization, v EL and v LM are lower than the theoretical minimum values (see table 4). This is due to the fact that the resolution of the system is finite. When two trajectories are within x from each other, they are thought to intersect, but if they instead pass by each other in parallel at a distance d < δx, a velocity change can be saved proportional to the change of the effective potential (16) between the trajectories. That this is indeed what has happened is shown in figure 10. This also explains why both v EL and v LM increase during the optimization described in section 5.3. Here x is decreased by a factor of 10, and v EL and v LM therefore approach the theoretical minimum values. If the resolution is increased further, the velocity change would exceed the minimum values. 9 Possible improvements on the trajectory designs The most obvious way to improve the trajectory design strategy is to implement a minimization of the velocity change at L1 in the program. This could be done as illustrated in figure 11: The cell-array system is still created for both the Earth and the Moon using the same techniques as described in section 5.1. In the second part of the program, described in section 5.2, the minimum v EL and v LM are found for each initial value at L1. Afterwards, a new 18

24 Figure 10: At the points where the trajectory going from Earth is thought to intersect the trajectory coming from L1 (backward in time), they are actually passing each other in parallel. Since the value of the effective potential changes between the trajectories, a velocity change can be saved causing the calculated v EL and v LM to be lower than the theoretical minimum values. program could pair different trajectories from the two legs, searching for the lowest possible total velocity change v EM = v EL + v L1 + v LM. Another way to ext the project would be to compare more integrators than the seven described in section 6, or one could vary the step size h during the integration, so the smallest step size is used when the gradient of the effective potential is largest. This way fewer integration steps are necessary to obtain the same accuracy on the results. Figure 11: Proposal for a new trajectory design strategy incorporating the minimization of v L1 : For each of the trajectories going from L1 and approaching either the Earth or the Moon, the velocity change to go into orbit ( v EL or v LM ) is found using the same methods as described in section 5.2. Afterwards these results are searched for the minimum v EM = v EL + v L1 + v LM 19

25 10 Conclusion If all other celestial objects than the Earth and the Moon are neglected, the motion of a satellite in the synodic frame will be governed by the equations of motion (12), (14) and (15), with the Hamiltonian (13) conserved along the flow. The equilibrium points of this chaotic, dynamical system are called Lagrange points. The theoretical minimum v EM necessary to get a satellite from Earth orbit (167 km altitude) to Moon orbit (100 km altitude) is 3721 m/s, and the corresponding trajectory passes through the first Lagrange point L1. This trajectory has not been found. In order to find a low energy trajectory going from Earth orbit to L1, 6300 initial values in Earth orbit have been numerically integrated 100 days forward in time and 1575 initial values at L1 have been numerically integrated 100 days backward in time. At the intersections between these two types of trajectories, the total v EL to go from Earth orbit to L1 was calculated. For the best results the calculations were repeated with better accuracy and resolution. The trajectory with the lowest velocity change found this way had v EL = m/s, which is lower than the theoretical minimum of 3099 m/s. The procedure was repeated for a satellite going from L1 to Moon orbit. Here, the trajectory with the lowest velocity change had v LM = m/s, the theoretical minimum being 622 m/s. The found velocity changes are lower than the theoretical minima because a finite resolution was used in the numerical models. When the Earth-L1 and L1-Moon trajectories are put together, an extra velocity change v L1 has to be performed. Due to this, the most fuel efficient trajectory found, going from Earth orbit to Moon orbit, has the total velocity change of v EM = m/s. This is much higher that the lowest theoretical value of 3721 m/s. A future model should therefore implement a minimization of v L1 along with v EL and v LM. To find the one-step integrator, which fastest and most accurately approximates the flow of the equations of motion, seven different one step integrators were considered. Six of these were symplectic integrators, meaning that they satisfy the symplecticity condition (27) and give rise to long time energy conservation. Despite the properties of symplectic integrators, the most efficient integrator proved to be the non-symplectic, 4th order one step Runge-Kutta integrator, which was therefore used in the project. 11 Acknowledgements This project would have been impossible without the help and guidance from my supervisor Poul Hjorth from the Department of Mathematics, Technically University of Denmark. I also wish to thank my internal supervisor Mogens Høgh Jensen from the Niels Bohr Institute, University of Copenhagen. Furthermore a special thanks to Charlotte Strandkvist, University of Cambridge, for invaluable support and profitable discussions of the report and to Art Director Kenneth Dahlin, oprust.dk, for design of the front page. References [1] Belbruno E A and Miller J: Sun-pertubed Earth-to-Moon transfers with ballistic capture; Journal of Guidance, Control, and Dynamics, Vol 16 No 4 p , 1993 [2] Belbruno E A: The dynamical mechanism of ballistic lunar capture transfers in the fourbody problem from the perspective of invariant manifolds and Hill s regions; CRM Research Report 270, Centre de Recerca Matematica, Institute d Estudis Catalans, Barcelona,