Interplanetary Mission Analysis

Interplanetary Mission Analysis Stephen Kemble Senior Expert EADS Astrium stephen.kemble@astrium.eads.net Page 1

Contents 1. Conventional mission design. Advanced mission design options Page

1. Conventional mission design 1.1 Interplanetary transfers and Lambert s problem 1. Spheres of influence and patch conics 1.3 Planetary escape and capture Page 3

.1 Interplanetary transfers and Lambert s problem Page 4

Planet to planet transfers Large energy changes are required to reach another planet Energy is sum of kinetic energy (positive) and gravity potential (negative) Rendez-vous with Mercury requires the greatest energy change Venus and Mars are far less demanding Plot shows energy relative to Earth Specific energy is used (ie per unit spacecraft mass) Energy (J) 6.00E+08 4.00E+08.00E+08 0.00E+00 -.00E+08-4.00E+08-6.00E+08-8.00E+08 Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto V µ E = r Page 5

Properties of planetary orbits sma (AU) sma (km) Energy Period (days) Eccentricity Mu Energy rel Earth Mercury 0.38709893 5.79E+07-1.14E+09 87.97 0.0563069.E+13-6.99E+08 Venus 0.7333199 1.08E+08-6.10E+08 4.70 0.0067733 3.5E+14-1.69E+08 Earth 1.00000011 1.50E+08-4.41E+08 365.6 0.016710 3.99E+14 0.00E+00 Mars 1.536631.8E+08 -.90E+08 686.96 0.0934133 4.8E+13 1.5E+08 Jupiter 5.0336301 7.78E+08-8.48E+07 4335.36 0.0483966 1.7E+17 3.56E+08 Saturn 9.5370703 1.43E+09-4.63E+07 10757.76 0.0541506 3.79E+16 3.95E+08 Uranus 19.191639.87E+09 -.30E+07 30708. 0.04716771 5.83E+15 4.18E+08 Neptune 30.0689635 4.50E+09-1.47E+07 605.0 0.00858587 6.86E+15 4.7E+08 Pluto 39.4816868 5.91E+09-1.1E+07 90613.48 0.4880766 4.4E+13 4.30E+08 Page 6 Rmin V at Rmin Rmax V at Rmax (km) (km/s) (km) Mercury 46001448 58.976 6981746 38.858 Venus 107474994 35.59 10894304 34.784 Earth 14709663 30.87 15099177 9.91 Mars 06655710 6.498 495938 1.97 Jupiter 741096388 13.699 81588797 1.444 Saturn 1348673598 10.193 1508110731 9.115 Uranus 733511855 7.13 991539939 6.509 Neptune 444086960 5.497 4538878194 5.378 Pluto 4433588797 6.106 731604748 3.698

Transfer between circular orbits Planets may be approximated to lie in circular, co-planar orbits,orbit 1, Earth, and orbit, the target The optimum transfer is then a Hohmann transfer Requires two impulses One leaving Earth Second impulse can be used to rendez-vous with the target. V ap r r 1 V pe E 1 = µ r 1 E µ = r E µ = r 1 + r V µ = + E r Page 7 µ µ V1 = + E + E 1 rp 1 rp 1 µ µ V + = + E E rp rp

General transfers In practice: Planet orbits are eccentric Each has a different inclination and so are not co-planar r c θ r 1 The general transfer problem is Lambert s problem Lambert proposed that the time of flight depends upon three quantities 1. The semi-major axis of the connecting ellipse. The chord length, c 3. The sum of the position radii from the focus or the connecting ellipse. Find a transfer between two positions in a given time. s a r1 + r + c = c = r1 + r r cosθ 1 r Page 8

Lambert s problem Difference in Mean anomalies of transfer orbit M µ a ( t t ) = ( t ) M 1 = n 1 3 t1 ( ) ( ( ) ) 1 1 1 t t1 = E E1 e sin E sin E1 = e sin cos µ µ Substitute: Then can derive Substitute Can derive Gives the result Solution depends only on semimajor axis a Page 9 a 3 A = α = a 3 E E 1 E E A + B α sin = E E E cos cos + E B = e 1 3 a µ ( t t ) = ( A sin Acos B) 1 s a 3 a µ β = B A β sin = s c a E + E ( t t ) = (( α sin α ) ( β sin β )) 1

Lambert s problem () Can be applied to hyperbolic orbits Define further variables: α sinh = s a sinh β = s c a Gives the result for hyperbolic orbits Solution depends only on a 3 a µ ( t t ) = (( sinh α α ) ( sinh β β )) 1 In addition to semi-major axis, eccentricity, e can be found, for elliptical or hyperbolic cases p = ar r 4 1 β sin c θ sin α + p = a(1 e ) p = ar r 4 1 β sin c θ sinh α + Page 10

Solving Lamberts problem A transfer from Earth to Mars can be considered. Consider leaving Earth on 3rd October, 011 Transfer durations of 150, 180 and 00 days investigated. Departure and arrival epochs specify two radius vectors, from which c and s may be calculated. An estimate at the value of semimajor axis solving the equation can be made and substituted into the right hand side the equation: A plot of error between left and right hand sides versus semi-major axis can be obtained. The solution lies where the error equals zero, ie crosses the axis error (secs) 8000000 6000000 4000000 000000 0 0.00E+00 1.00E+11.00E+11 3.00E+11 4.00E+11 5.00E+11-000000 -4000000-6000000 -8000000 sma (m) 3 a µ ( t t ) = (( α sinα ) ( β sin β )) 1 00 180 150 Page 11

Solving Lamberts problem () Hyperbolic transfer The spacecraft again leaves Earth on 3rd October, 011 now the transfer duration is only 100 days. The departure and arrival epochs are once again specify the two radius vectors, from which the quantities c and s may be calculated. Now the transfer duration is so short that the connecting orbit must be hyperbolic In this case the semi-major axis at the solution is negative, indicating a solar system escape orbit. error (secs) 400000 300000 00000 100000 0 -.50E+1 -.00E+1-1.50E+1-1.00E+1-5.00E+11 0.00E+00-100000 -00000-300000 -400000 sma (m) Page 1

Transfers to Mars The non-coplanar nature of the orbits is an important aspect The plane change requirement implied results in a ridge in the contour plots Results in the classical pork chop plot. Each synodic period now shows two local minima: short and long transfer types, typically 00 and 350 days durations Page 13 10 11 1 13 14 15 16 17 18 19 0 1 3 4 5 6 7 8 9 30 10 0 30 40 50 60 70 80 90 000 00 0 60 Vinfinity Earth-Mars (m/s) 54700 54775 54850 5495 55000 55075 55150 555 55300 55375 55450 100 140 180 0 60 300 340 380 40 460 500 Transfer time (days) 5555 55600 55675 55750 5585 55900 55975 56050 5615 5600 5675 56350 Launch epoch (MJD) 9000-30000 8000-9000 7000-8000 6000-7000 5000-6000 4000-5000 3000-4000 000-3000 1000-000 0000-1000 19000-0000 18000-19000 17000-18000 16000-17000 15000-16000 14000-15000 13000-14000 1000-13000 11000-1000 10000-11000 9000-10000 8000-9000 7000-8000 6000-7000 5000-6000 4000-5000 3000-4000 000-3000 1000-000 0-1000

Nature of the transfers A particular launch date can be selected The effect of varying transfer duration at that epoch can be examined Use 3 rd October 011. Two minima are expected as the transfer duration evolves. The short and long minima are seen separated by a local maximum at approximately 30 days Vinf total (m/s) 70000 60000 50000 40000 30000 0000 10000 0 0 100 00 300 400 500 600 Duration (days) Page 14

Nature of the transfers Examining the trajectories reveals the nature of the local maximum As the transfer approaches a conjunction type transfer, it becomes increasingly difficult to achieve the out of plane component of the rendez-vous position. The solution is to increase the heliocentric inclination of the transfer orbit. The solutions result in transfers switching from South to North around the local maximum Page 15

The synodic period The synodic period is that between the repetition of a particular relative orbit geometry between the planets in question, such as a particular difference in solar longitude. Such repetitions occur at fixed intervals for two circular, co-planar planet orbits. If the orbits are assumed circular, then the synodic period is calculated as: 360 τ = Where τ p1 is the orbital period of planet 1 and τ p is the orbital period of planet and ω p1 is the orbital period of planet 1 and ω p is the orbital period of planet Page 16 S = 360 360 τ p1 τ p ( ω ω ) p1 360 p

The synodic period The synodic period for the planets, assuming circular orbits are: Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Mercury Venus 0.3958 Earth 0.3173 1.5987 Mars 0.76 0.914.1354 Jupiter 0.458 0.6488 1.090.350 Saturn 0.48 0.683 1.0351.0089 19.8618 Uranus 0.415 0.6198 1.011 1.941 13.834 45.5665 Neptune 0.41 0.6175 1.0061 1.906 1.7945 35.9576 170.5157 Pluto 0.411 0.6167 1.0041 1.8953 1.4719 33.507 16.8006 494.6005 Page 17

Global repeat periods The absolute locations of the planets do not repeat at these intervals, only the relative locations repeat. The time between an exact repetition of an absolute transfer geometry is calculated by: τ = nτ = global S mτ p1 The time is the number of synodic periods required to generate a whole number of orbits of the departure planet, whose period is τ p1. This may in practice be a very long period for a precise repeat It can be approximated, if the repeat geometry departure longitude lies within a few degrees of the original departure longitude. Page 18

Vinfinity Earth-Mars (m/s) Examples of global repeats 5800 5975 53150 5335 53500 53675 Transfer opportunities from Earth show a global repeat to the nearest planets as follows: 53850 5405 5400 54375 54550 5475 54900 Venus: Synodic period 1.6 years, Global repeat 8 years Mars: Synodic period.15 years, Global repeats near 13 and 15 years Jupiter: Synodic period 1.09 years, Global repeats near 1 years Mercury: Synodic period 0.3 years, Global repeats near 1 year 55075 5550 5545 55600 55775 55950 5615 56300 56475 56650 5685 57000 57175 57350 5755 57700 57875 58050 585 58400 58575 Launch epoch (MJD) 8000-30000 6000-8000 4000-6000 000-4000 0000-000 18000-0000 16000-18000 14000-16000 1000-14000 10000-1000 8000-10000 6000-8000 4000-6000 000-4000 0-000 58750 5895 Page 19 10 1 14 16 18 0 4 6 8 30 0 40 60 80 000 00 0 60 100 140 180 0 60 300 340 380 40 460 Transfer time (days) 500 59100 5975

Multi-revolution transfers Many locally optimal transfers are possible when the upper limit on transfer durations is extended. This allows an increased number of heliocentric revolutions. Instead of 0.5 revolutions, it is possible to use 1.5 or in principle, n.5 revolutions. For the example of co-planer, circular planet orbits, the minimum speed change required of the spacecraft is independent of the number of revolutions. The optimal transfer duration is simply incremented by n times the transfer orbit period. Page 0

Multi-revolution transfers () Lambert s problem now has two solutions for the semi-major axis, when n>0 Page 1

Multirevolution transfers to Mars Example of 1.5 revolution case from Earth to Mars Launch dates over two synodic periods are shown Minimum Vinfinities are comparable to 0.5 rev case Launch and arrival dates differ significantly to 0.5 revs Page 10 11 1 13 14 15 16 17 18 19 0 1 3 4 5 6 7 8 9 30 10 0 30 40 50 60 70 80 90 000 00 0 660 Vinfinity Earth-Mars (m/s) 54700 54775 54850 5495 55000 55075 55150 555 55300 55375 55450 700 740 780 80 860 900 940 980 100 1060 1100 Transfer time (days) 5555 55600 55675 55750 5585 55900 55975 56050 5615 5600 5675 56350 Launch epoch (MJD) 9000-30000 8000-9000 7000-8000 6000-7000 5000-6000 4000-5000 3000-4000 000-3000 1000-000 0000-1000 19000-0000 18000-19000 17000-18000 16000-17000 15000-16000 14000-15000 13000-14000 1000-13000 11000-1000 10000-11000 9000-10000 8000-9000 7000-8000 6000-7000 5000-6000 4000-5000 3000-4000 000-3000 1000-000 0-1000

. Spheres of influence and patch conics Page 3

Spheres of influence (1) Consider two major bodies (Sun and planet) Consider the ratio of the perturbing acceleration (from the second body) to the main acceleration (from the first body). Ratio can be evaluated for both the Sun s perturbation on motion about the planet and the planet s perturbation on motion about the Sun. r cs r ps a r pc A surface may then be found defined by r ps where these ratios are equal, in terms of the angular offset from the Sun s direction, a r pc is the distance from planet to Sun Page 4 r ps = r pc µ µ p c 1 ( 1+ 3cos a) 0.5 1 5

Spheres of influence () Hill s sphere of influence can also be considered: includes rotational effect similar to L1 and L distances The radii of the different spheres differ Values for the planets of the solar system differ significantly Classical sphere of Hill's sphere radius Influence Mercury 11837 1387 Venus 61677 1011199 Earth 94648 149669 Mars 577131 1083965 Jupiter 4816966 53150917 Saturn 54615354 654439 Uranus 5174711 70038854 Neptune 86634373 115989405 Pluto 1507953 859593 Page 5

Using spheres of influence Consider a mission from Earth to Mars Keplerian orbits only consider a single gravity field Initially the spacecraft has a Keplerian orbit calculated relative to Earth: Earth s gravity dominates Generally hyperbolic After leaving the Earth s sphere of influence the spacecraft motion can be defined in terms of a Keplerian orbit about the Sun Generally elliptical Finally on reaching Mars the spacecraft has a Keplerian orbit calculated relative to Mars: Martian gravity dominates Generally hyperbolic 3 distinct orbits are used to describe the trajectory. The limitations of applicability are indicated by the sphere of influence of Earth and Mars Method is called patched conics Hyperbola wrt planet Velocity of planet Velocity of planet 1 Hyperbola wrt planet 1 Page 6

.3 Planetary escape and Capture Page 7

Escaping from a planet A hyperbolic orbit about a planet has a non-zero speed at infinite distance from the planet (if no other gravity field than the planet s were considered). When at several millions of kilometres, its speed relative to the planet tends towards the Excess hyperbolic velocity. The excess velocity targeted is that needed to achieve the interplanetary transfer, ie the heliocentric speed increment relative the planet This is approximately equal to the effective departure speed from the planet within the heliocentric domain. This quantity can be calculated from: V = µ a Page 8

Escaping from a planet The direction of the excess hyperbolic departure vector can be chosen by Choosing the appropriate orbit plane Choosing the appropriate pericentre of the initial hyperbolic orbit The asymptotic departure direction within the orbit plane is given by: where θ is the maximum true anomaly in the hyperbolic orbit, ie the velocity vector is asymptotically aligned with the radius vector. The DeltaV needed to reach this orbit is for a general elliptical orbit with apogee, θ cos 1 = 1 e Page 9 a p1 and perigee radius r pl1 µ 1 V = + V µ rpl rpl ( r 1 + r 1 1 pl1 apl1 )

Escaping from a planet The DeltaV needed to reach a given Vinfinity increases more quickly with target Vinfinity as Vinfinity increases In these examples the perigee altitude is 00km DV (m/s) 000 1800 1600 1400 100 1000 800 600 400 00 1000 000 3000 4000 Declinations that can be reached from a given inclination orbit are given by: sin Therefore maximum abs(declination) = inclination 0 0 50000 100000 150000 00000 50000 300000 350000 Apogee (km) DEC = sin isin( ω + θ ) Page 30

Approach to a planet The approach to a planet is characterised by three parameters: Excess hyperbolic speed (magnitude of the approaching, planet relative velocity vector,) Right Ascension and Declination of the of the approaching, planet relative velocity vector. The capture manoeuvre is a large retro-manoeuvre performed at planet pericentre It is the reverse of an escape manoeuvre. Evaluation of the location of the possible pericentre is required to assess the range of possible capture orbits that can be achieved without plane change. Page 31

Approach definitions Three planes are of primary importance Reference plane (eg ecliptic parallel) B plane: perpendicular to approach velocity vector Orbit plane: determined by selection of the Beta angle: locates the position of the pericentre of the orbit Approaching relative velocity vector at Beta angle > 90 deg B plane perpendicular to approach velocity vector Declination (negative as shown) Planet Relative orbit plane β angle measured from reference plane Plane parallel to Ecliptic plane: Reference plane Relative Approach plane Intersection of B plane and ecliptic parallel plane. Defines zero Beta angle Page 3

Capture pericentre The asymptotic approach direction within the orbit plane with respect to pericentre is given by the same relationship as the planetary departure case Two pericentre solutions exist for each approach orbit plane considered (as in the departure case) The approach plane is defined by the Beta angle A 360 degree range of Beta angles can be considered A limit exists on the inclination of the approach orbit that exists sin i sin DEC Page 33

Capture pericentre () Pericentre locations with different approach speeds at Mars 80 Pericentre locations with different approach speeds of 3 km/sec at Mars and a range of approach declinations 60 100 40 80 0 60 Latitude 0-100 -50 0 50 100-0 -40 4000 3000 000 Latitude 40 0 0-100 -50 0 50 100 30 0 10 0-60 -0-40 -80 Longitude -60 Page 34-80 Longitude

Capture orbits example The family of possible capture orbits at Mars are shown, resulting from a typical interplanetary trajectory and approach to Mars Capture orbit has a high apocentre (circa 100000km) Page 35

. Advanced mission design options.1 The three body problem. Designing missions to the Lagrange points.3 Gravity assist.4 Multiple Gravity Assists.5 Plane changing gravity assist.6 Gravitational escape and capture Page 36

.1 The three body problem Page 37

The three body problem The three body problem considers the motion of a spacecraft through a combined gravity field. The first body is the central body, the second is the orbiting, major body and the third is the spacecraft itself, whose mass is considered to be negligible in comparison to the other two. No analytical solutions exist that are comparable to Keplerian orbits for the two body case. Generalisations that describe the overall properties of the motion in certain circumstances exist. Also approximation methods enable certain classes of motion to be described by analytical expressions. The precise motion can only be obtained by numerical integration of the spacecraft state vector derivatives. Page 38

The Jacobi constant A constant of motion can be derived for a special case of the three body problem This is known as the circular, restricted three body problem The two major bodies are assumed to move in circular orbits about a common barycentre A good approximation for many planetary or moon orbits The constant is the Jacobi constant + U C Where V is the speed of the spacecraft relative to a rotating reference frame U is a generalised potential given by: U µ 1 µ r x r Y r1 r ω = + Where rx,ry, rz are the components of position with respect to the barycentre of the two major bodies, r1 and r are the distances from body 1 and body, is the angular velocity of the two bodies in circular orbit µ1 = Gm1, ie the gravitational parameter for mass m1 µ = Gm, ie the gravitational parameter for mass m V = ( ) Page 39

Zero velocity surfaces The limiting regions of the motion can be obtained by considering the case where speed is zero: µ 1 µ ω rx + ry + C = r1 r Zero velocity surfaces show limitations of the possible motion. Can evaluate contours of zero velocity, for different values of the constant, C. Only motion in the ecliptic is considered (ie rz=0) in the following examples Therefore this is now a case of a planar, circular, restricted three body problem. Motion is excluded within a particular shaded area that corresponds to a given value of C. X lies along the Sun to planet vector and Y (vertical axis, in km) lies perpendicular to X in the ecliptic. ( ) 0 Page 40

Earth zero velocity surface Earth lies on the X axis at 1.496e8 km. Series of constant velocity surfaces evaluated over a range of values of the Jacobi constant, C, starting with: CMaximum = 641000000 (m/s)^ (Cmaximum) is a case where escape from Earth is not possible. Further surfaces may be generated with reducing C (ie C increasing): no longer a barrier to an initially Earth neighbouring trajectory transferring into the heliocentric domain. The contours in the figure represents the absolute value of the reduction in the Jacobi constant below the maximum (Cmaximum). Therefore a large value for the contour means that C now lies significantly below Cmaximum As C is reduced inaccessible regions shrink, until motion becomes possible through the locations of the first two colinear Lagrange points, L1 and L, at approximately +/- 1.5 million km along the X axis from Earth Page 41

The Lagrange libration points Points exist where a particle, if placed there, with no velocity in the rotating frame, will experience no resultant acceleration with respect to this rotating frame. Condition is: U = 0 5 points can be found in the orbit plane of the secondary body about the primary There are three, collinear, points, denoted L1, L and L3 Two further points complete an equilateral triangle with the baseline from primary to secondary, L4 and L5 L 3 L 5 d 1 d L 4 L 1 L Page 4

Lagrange points at the planets The L1 and L points are shown for each planet, using the approximation of the circular restricted three body problem Assumes that each planet lies in a circular orbit about the Sun at a value corresponding to its semi-major axis. The semi-major axis used is that from the JPL mean ephemeris at J000 Page 43 m/(m1+m) L1 (km) L (km) Mercury 1.660E-07.04E+05.10E+05 Venus.448E-06 1.008E+06 1.014E+06 Earth 3.003E-06 1.49E+06 1.501E+06 Earth-Moon 3.040E-06 1.498E+06 1.508E+06 Mars 3.7E-07 1.08E+06 1.086E+06 Jupiter 9.537E-04 5.09E+07 5.418E+07 Saturn.857E-04 6.45E+07 6.606E+07 Uranus 4.366E-05 6.954E+07 7.061E+07 Neptune 5.150E-05 1.150E+08 1.169E+08 Pluto 6.607E-09 7.594E+06 7.537E+06

Orbits at the Lagrange points The collinear points are unstable (stable in transverse direction, unstable in radial direction) Assuming a linearisation of the motion allows the derivation of differential equations describing possible motions (similar to Hill s equations for perturbed circular orbits about a single body) Local displacements relative to the Lagrange point are: δr = r Scaled by dividing by the separation between two bodies C11 is a constant d δr' dt' d δr' dt' d δr' dt' x y z dδr' dt' dδr' + dt' = δr' z y x c 11 = δr' = δr' x y ( 1+ c ) ( 1 c ) 11 11 r Ln Page 44

Orbits at the Lagrange points () These differential equations may be solved: δr' δr' δr' x y z = A e 1 1 st' + A z e st' st' = aa1e aae = C cosω t' + C st' + A 3 ba sinω t' cosω t' + A z 4 xy 4 cosω t' + ba xy sinω t' 3 xy sinω t' xy A1 to 4 are constants determined by the initial conditions (also C1 and C) The two angular frequencies differ slightly, depend on c11 Ratios of motion amplitudes in x and y (a and b) also depend on c11 Page 45

Orbit properties at the Lagrange points Initial solutions may be found that remove the exponential components The relative amplitudes of the oscillating motion in the rotating x and y directions is determined by the constant, b. Orbit characteristics can be determined for each planet assuming circular orbits Units of angular velocity are relative to orbit angular frequency Page 46 µ wxy wz Txy Tz b (days) (days) Mercury 1.660E-07.077.006 4 44 3.16 Venus.448E-06.085.014 108 11 3.8 Earth 3.003E-06.086.015 175 181 3.9 Mars 3.7E-07.079.007 330 34 3.18 Jupiter 9.537E-04.170.100 1998 064 3.349 Saturn.857E-04.138.068 503 50 3.303 Uranus 4.366E-05.108.037 14570 15076 3.60 Neptune 5.150E-05.110.039 854 9531 3.63 Pluto 6.607E-09.099.08 43173 44683 3.47

Lissajous orbits 150000 The solution of these equations is a Lissajous figure. It is not a closed orbit Motion projection in xy is closed Initial conditions determine the initial phasing of the motions Figures show examples The origin is a collinear Lagrange point Z Y 100000 50000 0-1000000 -500000 0 500000 1000000-50000 -100000-150000 Y 1000000 800000 600000 400000 00000 0-300000 -00000-100000 -00000 0 100000 00000 300000-400000 -600000-800000 -1000000 X Page 47

Halo orbits The full solution regarding motion about the Lagrange points shows a dependence of orbital frequency on amplitude The dependence differs between in and out of plane components The result is that critical ration of in/out of plane amplitudes yield solutions with identical frequencies This allows closed orbits to be achieved Choosing the appropriate phasing (via initial conditions) allows a Halo orbit to be achieved An example of such a Halo orbit was adopted for ISEE3, with in plane amplitude of 665000km and out of plane amplitude 110000km. 600000 500000 400000 Az 300000 00000 100000 Page 48 0 600000 650000 700000 750000 800000 850000 900000 Ay

Stable and Unstable motions The solutions indicate the possible presence of exponential, time dependent terms. Can be suppressed by suitable selection of the initial states such that the constants A1 and A are zero. If the constant, A1 takes a non-zero value, then an exponentially increasing, time dependent term exists. The result is that a perturbation to an oscillatory solution yielding non-zero A1 will eventually lead to a complete departure from that orbit. Such motions are unstable. This type of perturbation process is one of stepping onto the unstable manifold of the orbit at the Lagrange point. If the constant, A takes a non-zero value, then an exponentially decreasing, time dependent term exists. The result is that a perturbation to an oscillatory solution yielding non-zero A will eventually lead to the perturbation reducing to zero and returning to the oscillatory solution. Such motions are stable. This type of perturbation process is one of stepping onto the stable manifold of the orbit at the Lagrange point Page 49

Stable and Unstable motions () The stable manifold may be used to execute transfers to reach the oscillatory solution. Start at a point on the stable manifold far removed from the target orbit. Evolution of the trajectory with time then results in the spacecraft approaching the orbit described by the oscillatory solution, as the exponential term tends to zero. The conditions for the constant A1 remaining at zero in the presence of a velocity perturbation to an oscillatory solution can be found in terms of a direction in which this velocity perturbation may be applied. This direction lies in the ecliptic plane. A perturbation that has a component that is perpendicular to this direction in the ecliptic leads to an unstable solution where A1 is nonzero. This fact may be used as a feature of orbit generation and maintenance strategies: Station keeping aims to remove the unstable component Page 50

. Designing missions to the Lagrange points Page 51

Transfers Transfers from an initial Earth elliptical orbit to a Lissajous or Halo orbit about the Earth-Sun L1 or L Lagrange points can be achieved without manoeuvre An example can be considered as follows: Initial perigee altitude of 500km and semi-major axis (at Earth perigee) of 700000km. Osculating apogee (at perigee) of nearly 1.4 million km. Solar gravity perturbs the motion as move towards apogee. T Perigee is raised and the energy and angular momentum with respect to Earth are modified. The nature of this perturbation depends on the location of the apogee with respect to the Earth-Sun direction. Particular orientations can be found that enable freely reaching a Lissajous orbit about the L1 Lagrange point. Page 5

Searching for a transfer (1) The orientation of the initial orbit can be obtained by search in the longitude of the line of apses First choose apogee in 1.3 to 1.5 million km range Choose initial longitude to be within, for example, 0 degrees of Sun direction (for transfers to an L1 orbit) or anti-sun direction (for transfers to an L orbit) The figure, seen in a rotating reference frame, shows the effects of a 60 deg variation in perigee longitude. The behaviour includes orbits that return to Earth perigee; enter Lissajous orbits and also a series of orbits that escape Earth s influence Page 53

Searching for a transfer () Having established approximately the required longitude, a much more refined search is required Rule based methods can be used to automatically find such transfers Example shows a variation in longitude mof 0.05 degrees about the required transfer solution Sensitivity to initial conditions is therefore very high Page 54

Target orbit implications The minimum Lissajous orbit amplitude (in ecliptic) depends on the initial perigee altitude For a typical transfer orbit with perigee at 500km, minimum amplitude is 750000km Greater amplitudes can be achieved by appropriate choice of apogee radius and apse longitude Higher enable smaller amplitudes, eg perigee at 36000km altitude can free inject to 500000km amplitude orbit If a lower amplitude than the minimum free injection case is sought then a transfer manoeuvre can be performed Example assumes an initial 500km perigee altitude DeltaV (m/s) 50 00 150 100 50 0 0 100000 00000 300000 400000 500000 600000 700000 800000 Halo SMA (km) Page 55

Transfer characteristics The transfer duration is typically 10 days until entering the Lissajous orbit However the vicinity of the Lagrange point is reached after approximately 30 days In the case of a free injection transfer, final dispersion correction manoeuvres can be performed after this time For injection to smaller amplitude orbits the manoeuvre is made after typically 100-10 days Page 56

Transfer types (1) First consider the motion in the ecliptic Two longitudes give free injection transfers for a given choice of apogee radius (and the same perigee radius) Post injection orbits are similar in amplitude Transfers take slightly different durations Page 57

800000000 600000000 Transfer types () Out of plane motion of the target Lissajous orbit is determined by selecting an appropriate declination of the perigee departure vector (ie determined by the declination of the line of apses) 4 solutions exist for a given out of plane amplitude and left or right in ecliptic transfer type routes transfer to the North routes transfer to the South 1 North and 1 South transfer examples are shown in rotating frame, XZ plane Z rotating (km) Y rotating (km) Z rotating (km) 400000000 00000000 0-1.8E+09-1.6E+09-1.4E+09-1.E+09-1E+09-8E+08-6E+08-4E+08 -E+08 0 E+08-00000000 -400000000-600000000 -800000000 X rotating (km) 1000000000 800000000 600000000 400000000 00000000 0-1.8E+09-1.6E+09-1.4E+09-1.E+09-1E+09-8E+08-6E+08-4E+08 -E+08 0 E+08-00000000 -400000000-600000000 -800000000-1000000000 X rotating (km) 800000000 600000000 400000000 00000000 0-1E+09-8E+08-6E+08-4E+08 -E+08 0 E+08 4E+08 6E+08 8E+08 1E+09-00000000 Type Type 1 Type Type 1 Type Type 1-400000000 Page 58-600000000 -800000000 Y rotating (km)

.3 Gravity Assist Page 59

Gravity assist (1) Gravity assist manoeuvres enable an exchange of energy between a planet and the spacecraft A spacecraft approaches a planet from deep space. Hyperbolic orbit approach They rotate about their common barycentre The spacecraft departs with a modified orbit relative to the Sun Analysis is made using patched conics Hyperbolic orbit with respect to major body Sphere of influence of major body Departure orbit Approach orbit Page 60

Patch Conics Three conic sections are generated to describe the orbital phases. Phase 1: The approach orbit, Expressed in terms relative to the central body. Phase : The fly-by orbit Expressed in terms relative to the major body. This must be a hyperbolic orbit Phase 3: The departure orbit Expressed in terms relative to the central body Page 61

Patch and Link Conics Selection of the radius of the patch sphere is a key aspect The approximation generates a small velocity error compared to the full 3 body solution Typically the classical radius of the Laplace sphere of influence is selected A second approximation is that of Link conics With the link conic method, the fly-by, or gravity assist, is specified in a similar way, generally by ephemeris with respect to the fly-by body. From this ephemeris, it is possible to evaluate the asymptotic approach and departure velocity vectors, ie the relative velocity vector at infinite distance from the planet. Page 6 Limiting true anomaly,, depends only on eccentricity, e θ ± cos 1 = 1 e

Deflection The important feature of a fly-by is that the spacecraft s velocity w.r.t the planet is deflected The deflection only depends on eccentricity It is related to the limiting true anomaly The eccentricity is dependent in the choice of pericentre and the Excess hyperbolic speed α = *sin e = 1 r 1 perirel a 1 e = 1+ r perirel µ Deflection angle, α * V planet True anomaly of asymptotic departure direction, θ Page 63

Link approximations The link conic approximation therefore assumes that the velocity vector change is instantaneous The change takes place at the planet Therefore the time to traverse the sphere of influence is neglected Typically -3 days for the inner planets This is generally small compared to the periods of the interplanetary orbits It is equivalent to the approximation of using excess hyperbolic speed to model the departure or arrival to a planet Page 64

Gravity assist velocity change The gravity assist effect can be described by velocity vector triangles Consider the approach situation: Spacecraft velocity Planet velocity Relative velocity Relative velocity is deflected as the spacecraft swings by the planet Then add the velocity vectors V departure Approach V Γ V p Deflected V rel =V Γ departure V p α Body Approach V rel =V Γ relp Departure Γ relp Page 65

Alternative coplanar solutions Deflection can take place in one of two possible directions, depending on which side of the planet the spacecraft approaches. Determines the sign of alpha (shown as positive in the previous figure). Choice of solution can result in a considerable difference in the post gravity assist heliocentric state Page 66

Gravity assist effects Gravity assist can modify the interplanetary orbit in a variety of ways If co-planar heliocentric orbits are assumed (spacecraft and planet) with fly-by in the plane of the orbits then the spacecraft may fly around either side of the planet by making small targeting manoeuvres when distant from the planet. First the approach vector flight path angle is found V is the initial heliocentric velocity V *sin( Γ Γ planet ) V planet is the velocity of the planet Γ = tan 1 relp V * cos( Γ Γ planet ) V planet The departure flight path is then The departure heliocentric velocity is Page 67 V departure V = V + V V V cos( Γ Γ = V rel planet planet planet ) Γ departure = tan 1 V V planet *sin( α + Γ + V relp ) * cos( α + Γ ( V sin( α + Γ )) + ( V + V *cos( α + Γ )) ) = sqrt * relp planet relp relp )

Total system effect of a gravity assist The deflection of the planet (or major body) relative velocity vector means that the energy and angular momentum of the departure orbit relative to the central body are modified This energy and momentum exchange is matched by a change in these components for the major body The large mass of the major body means that its orbit is negligibly effected Many large spacecraft performing energy raising gravity assists at Earth could slow down the planet! Page 68

Gravity assist example: co-planar assumptions Consider a gravity assist at Venus with a spacecraft in an elitpical orbit Initial Ap Initial Pe Planet Planet Anom RV radius Hyper V Peri h (km) 1.0000 0.5000 10 0.758 11669.1746 400.0000 The possible orbit changes post fly-by are Post GA Ap Post GA Pe Post GA Ap Post GA Pe 1.767 0.6433 0.7973 0.3185 Page 69

General coplanar gravity assist cases Effect of fly-by altitude The previous example of a Venus crossing orbit (perihelion at 0.5AU) can be examined to determine the effect of fly-by altitude (ie pericentre radius) The aphelion raising solution is taken As the altitude increases the effectiveness reduces Ap-Pe (AU 1.8 1.6 1.4 1. 1 0.8 0.6 Ap Pe sma sma-0 0.4 0. 0 0 000 4000 6000 8000 10000 1000 altitude (km) Page 70

.4 Multiple gravity assists Page 71

Multiple gravity assists Repeated gravity assists can be used to perform large modifications to the spacecraft orbit. Repeated gravity assists at the same planetary body or moon can be performed. This is generally a resonant gravity assist sequence. The spacecraft must reach, after the gravity assist, an orbit that is resonant with the body in question. This means that after an integer number of revolutions, the spacecraft will re-encounter the body. Resonance ratios can be: n:1 (n revolutions by the spacecraft, 1 revolution by the body) n:m (m revolutions by the body) 1:m (1 revolution by the spacecraft, m revolutions by the body). In these resonant cases, the excess hyperbolic speed is the same at successive fly-bys Page 7

Multiple gravity assist limits Such repeated gravity assists can be used to progressively raise apohelion Conversely, may be used to lower perihelion Eventually a limiting case is reached where the velocity with respect to the central body cannot be further increased/decreased Ie the relative velocity becomes aligned with fly-by body velocity Page 73 V V planet V + V max min V ( V ) = V flyby ( V ) = V flyby α V rel =V

Multiple gravity assist example Consider a series of resonant gravity assists in the Jovian system. The fly-by body is Ganymede. The initial orbit is chosen to be: Apocentre = 0 million km Pericentre = 900000km Ganymede orbital radius = 1.07 million km At the end of the sequence the orbit is approaching the limiting case with apocentre just above Ganymede Apocentre (km) Pericentre (km) 5000000.0000 0000000.0000 15000000.0000 10000000.0000 5000000.0000 0.0000 900000.0000 800000.0000 700000.0000 600000.0000 500000.0000 400000.0000 300000.0000 00000.0000 100000.0000 0.0000 1 3 4 5 6 7 8 9 10 11 1 13 1 3 4 5 6 7 8 9 10 11 1 13 Page 74

Tisserand s Criterion Tisserand derived a criterion from which it is possible to compare orbits that appear significantly perturbed and deduce whether the relationship between them may be due to a gravity assist at a planet. An example is comets passing close to Jupiter The basis of the method is the circular, restricted three-body problem. The Jacobi constant is used V + U = V is the speed w.r.t the rotating frame When motion is considered at a sufficiently large distance from the fly-by planet, its gravity field may be neglected. Speed in the rotating frame may be converted to speed in an inertial frame, V IC. r c is the distance from the central body. µ VIC ω rc VI zˆ = r Page 75 C ( ) C C

Tisserand s Criterion () The constant may be expressed at distances far from the planet as: µ ωh cos i a In the co-planar case, then the orbit relationships before and after are given by: µ µ ω µ a1(1 e1 ) = ω µ a (1 e a a 1 For a given initial spacecraft orbit the evolution of the possible orbits under repeated gravity assists at the same planet may be evaluated. The locus of the evolution of the orbital elements is given by the above relationship, eg a relationship between a and e. = C ) Page 76

Tisserand s Criterion (3) The change in the orbit at a single gravity assist is determined by: Excess hyperbolic speed w.r.t. the planet (or moon) Gravity constant Fly-by pericentre radius This change therefore determines the size of the step along the locus. The locus can alternatively be expressed in terms of orbit period and pericentre, or apocentre and pericentre Period Step at gravity assist Locus for a given Vinfinity at target body Pericentre Page 77

Tisserand s Criterion Example Consider the example of an Earth crossing orbit and the locus of heliocentric orbit evolution for multiple gravity assists at Earth Both periodpericentre and apocentrepericentre plots are considered Apocentre (km) Period (days) 8.00E+0 7.00E+0 6.00E+0 5.00E+0 4.00E+0 3.00E+0.00E+0 1.00E+0 0.00E+00 8.00E+07 9.00E+07 1.00E+08 1.10E+08 1.0E+08 1.30E+08 1.40E+08 1.50E+08 Pericentre (km) 3.50E+08 3.00E+08.50E+08.00E+08 1.50E+08 3000 4000 5000 3000 4000 5000 1.00E+08 5.00E+07 Page 78 0.00E+00 8.00E+07 9.00E+07 1.00E+08 1.10E+08 1.0E+08 1.30E+08 1.40E+08 1.50E+08 Pericentre (km)

Designing sequences using Tisserand s Criterion Loci of possible orbit sequences at different fly-by bodies can be considered The intersection of loci indicates the possibility to switch from one sequence (at fly-by body 1) to another sequence, at fly-by body This assumes that the required phasing between bodies 1 and can be achieved to execute the transfer. Can always be achieved if wait for long enough (or perform a small phasing manoeuvre) This technique is very relevant to GA sequence design at a planetary moon system, where many possibilities exist for multiple fly-bys (orbit periods are shorter than heliocentric case) Page 79

Designing a gravity assist sequence in the Jovian system The overlapping loci for gravity assists at Callisto, Ganymede and Europa indicate the multitude of options for gravity assist combinations 10000000 9000000 8000000 Apocentre (km) 7000000 6000000 5000000 4000000 3000000 000000 C-7000 C-6000 C-5000 G-7000 G-6000 G-5000 E-4000 E-3000 E-000 E-6000 G-4000 1000000 0 0.00E+00 1.00E+05.00E+05 3.00E+05 4.00E+05 5.00E+05 6.00E+05 7.00E+05 8.00E+05 9.00E+05 1.00E+06 Page 80 Pericentre (km)

Designing a gravity assist sequence in the Jovian system () A sequence of gravity assists at Ganymede and Europa can be used to reach a low approach speed orbit relative to Europa from an initial capture orbit at Jupiter Manoeuvres may also be allowed in Jovian orbits Objective is to minimise the total transfer DeltaV: Manoeuvres in the Jovian system Orbit insertion manoeuvre at Europa V total = i V api + µ r Europa EuropaOrbit + V final r µ EuropaOrbit V api is the V applied at apocentre i to raise pericentre. V final is the excess hyperbolic speed at Europa from which insertion to Europa orbit is made, r Europaorbit is the radius of this circular orbit about Europa. Page 81

Example of gravity assist sequence in the Jovian system This tour uses an initial sequence of gravity assists at Ganymede (4) Then 1 at Europa and the next back to Ganymede to raise pericentre 3 more GA s at Europa then another at Ganymede to raise pericentre Then a sequence of Europa GA s with intermediate DV s to raise pericentre and so progressively reduce approach speed at Europa DeltaV to Europa circular orbit approx 1000 m/s Period (days) 0 18 16 14 1 10 8 6 4 0 3.00E+05 4.00E+05 5.00E+05 6.00E+05 7.00E+05 8.00E+05 9.00E+05 1.00E+06 Pericentre (km) G-7000 G-6000 G-5000 E-4000 E-3000 E-000 E-5000 G-4000 Route Page 8

Example of gravity assist sequence in the Jovian system () The sequence is Page 83 Event Apocentre (km) Pericentre (km) Period (days) V (m/s) Resonance DV (m/s) Capture 018103 900000 31:1 Gany 771518 84748 57 6553 8:1 Gany 4604948 787514 9 6553 4:1 Gany 331 71869 18 6553 5: Gany 31376 61844 1 6553 Europa 187370 600456 9 4969 Gany 14703 664506 11 5883 Europa 1973573 660187 10 310 11:4 Europa 1654999 648966 8 310 9:4 Europa 1319465 6909 6 310 Gany 1360010 66086 6 3733 Europa 1197 65388 6 09 5:3 DV 1197 664000 6 1847-43 Europa 109456 6557 5 1847 3: DV 109456 664000 5 1533-39 Europa 893469 644110 4 1533 5:4 DV 893469 664000 4 945-96 Europa 771907 648514 4 945 11:10 DV 771907 664000 4 479-78

Example of use of gravity assists: Mission to Jupiter A direct transfer to Jupiter needs an Earth departing Vinfinity of approx 9 km/sec A multi-fly-by gravity assist route can substantially reduce this. The Galileo strategy : 1. Earth to Venus. Venus gravity assist to Earth (Vinfinity approx 6 km/sec) 3. Earth gravity assist to Earth via year resonant orbit (Vinfinity approx 9 km/sec) 4. Earth gravity assist to Jupiter Total transfer duration 6. years Page 84

.5 Plane changing gravity assists Page 85

3D Gravity Assist geometry and the B plane The general case considers non co-planar orbits of spacecraft and planet. The B plane is the plane perpendicular to the asymptotic relative velocity vector, or hyperbolic approach vector. For a given, targeted fly-by pericentre radius, the intersection of the forward projection of the approach hyperbola with the B plane can take place at any point on a circle around the planet. A very small deep space manoeuvre can modify the approach to lie anywhere on such a circle A plane may be defined, that contains the approaching asymptotic velocity vector and the velocity vector of the planet the Approach plane. This plane is therefore perpendicular to the B plane. The Beta angle, β, is the angle between the X axis (defined by plane intersection) and the location of the intersection of the approach relative velocity vector with the B plane. Cases 1 and in the figure are therefore at Beta angles of zero and 90 degrees.. Approaching relative velocity vector is offset to achieve an approach that is parallel to the approach plane B plane β = 90 deg 1. Approaching relative velocity vector contained in the approach plane Approach plane Z B Y B X B Page 86

3D deflection It is possible to define a further plane, the fly-by plane, as being the plane containing the approaching velocity vector relative to the major body (the asymptotic approach vector) and also the departing relative velocity vector. This plane is defined by the Beta angle, described previously. The deflection is in the fly-by plane Two deflection angle components can be defined: One in the approach plane and one perpendicular to it α φ α θ α θ is negative in this diagram β α B plane Approaching relative velocity vector Plane containing body velocity vector and spacecraft velocity vector, ie The approach plane Page 87

3D gravity assist example Consider the example of a gravity assist at the Jovian moon, Ganymede. Ganymede lies in a near circular orbit about Jupiter, with semi-major axis at 1.07 million km. Consider a range of initial orbits each crossing Ganymedes s orbit. The initial apocentre is 5 million km, and a range of pericentres from 500000km to 900000km are considered. The result is to generate a range of excess hyperbolic speeds with respect to Ganymede. Page 88 Pericentre (km) Speed at Ganymede (m/s) Vinfinity (m/s) 500000 13803.64 9853.09 43.445 600000 13833.40 8900.53 38.140 700000 1386.06 7954.81 3.89 800000 13889.67 6990.59 7.8 900000 13916.30 5975.76 1.114 Flight path angle (deg)

3D gravity assist example () The Beta angle of the fly-by is varied through a 360 degree range Post gravity assist orbits are then found. In each case it is assumed that the initial spacecraft orbit is coplanar with the orbit of Ganymede. Defines the approach plane The pericentre altitude at Ganymede is assumed to be 300km Page 89

3D gravity assist example (3) 1000000000 A β angle of 90 degrees maximises the inclination change It also yields a small change in the post gravity assist apocentre and pericentre. Inclination change is measured with respect to the initial approach plane Greatest inclination change occurs for the highest initial pericentre (or lowest Vinfinity) case Apocentre (m) Inclination (deg) 10000000000 8000000000 6000000000 4000000000 000000000 0 0 50 100 150 00 50 300 350 400 Beta(deg) 5 4 3 1 0-1 0 50 100 150 00 50 300 350 400 - -3-4 500000 600000 700000 800000 900000 500000 600000 700000 800000 900000-5 Beta(deg) Page 90

Multiple plane changing fly-bys It is possible to find a Beta angle that maintains the relative speed at the fly-by target, after the fly-by (ie the semi-major axis is unchanged). This means that the spacecraft may stay in a resonant orbit (ie same orbital period about the target). This angle is generally close to 90 degrees. Repeated gravity assists at the same planetary body or moon allows a large inclination change to be achieved. Each fly-by is designed such that the effect of the gravity assist is to increase inclination and also to achieve a velocity relative to the central body that yields a resonant orbit with respect to the major body. The spacecraft will then return to the major body after some integer number of revolutions about the central body. Subsequent fly-bys can further increase inclination. The post fly-by velocity relative to the central body an hence resonance can be maintained by choosing the appropriate Beta angle Page 91

Multiple plane changing fly-bys () Repeated gravity assists result in the plane containing planet and spacecraft velocities reaching 90 degrees relative to the initial approach plane (generally close to the fly-by body orbit plane). If the same resonant fly-bys are maintained then this plane will continue to rotate past 90 degrees. Maximum inclination is achieved close to the 90 degree rotation case. Inclination may be further increased by changing the resonant orbit Now the approach plane is unchanged and the approach vector is deflected in this 90 degree rotated plane A maximum inclination can be found Page 9

Maximum achievable inclination Repeated gravity assists result in the plane containing planet and spacecraft velocities reaching 90 degrees relative to the initial approach plane (generally close to the fly-by body orbit plane). Maximum inclination is achieved at the 90 degree rotation case. Plane containing spacecraft and fly-by body velocity vectors V limit V rel =V V 1 V rel =V The maximum inclination (relative to the initial approach plane) in the case of the same resonant orbit being maintained is given by: V i = tan 1 max V planet The maximum achievable inclination (by changing resonance) is given by: Page 93 i max = sinθ + V cosθ sin 1 V V planet i max V planet θ Fly-by body orbit plane ψ=90

Example of multiple plane changing gravity assists A heliocentric example of plane changing with multiple gravity assists cab be considered. Useful for missions that are required to perform out of ecliptic observations. A single gravity assist at Jupiter can achieve a 90 degree inclination with respect to the ecliptic. However considerable time and/or fuel is needed to reach Jupiter. An alternative strategy to reach high inclinations could be considered at planets within the inner solar system. Venus is a good choice Relatively high mass and easily reached from Earth. Page 94

Example of multiple plane changing gravity assists () The maximum, total inclination change that is achievable is dependent on the excess hyperbolic speed with respect to Venus Compare with previous equations Different excess hyperbolic speeds are achieved by orbits with different aphelion/perihelions High inclination requires high Vinfinity Repeated gravity assists will be required to achieve high inclinations. Considered as an option for the Solar Orbiter mission The objective of this mission is to observe the Sun from high latitudes when relatively close to the Sun. Therefore such a sequence is well suited to this type of Solar observing mission. Page 95

35.0000 Example of multiple plane changing gravity assists (3) The initial orbit is: Apocentre = 13 million km, Pericentre = 33 million km. Excess hyperbolic speed with respect to Venus is 0. km/sec A 3: resonance may be reached with Venus. Fly-bys occur approximately every 450 days, orbital period is 150 days. 6 gravity assists are used in reaching a maximum inclination of nearly 35 degrees. This procedure will therefore take approximately 700 days, or approaching 7.5 years. Inclination (deg) Apocentre (km) 30.0000 5.0000 0.0000 15.0000 10.0000 5.0000 0.0000 1 3 4 5 135000000.0000 130000000.0000 Pericentre (km) 15000000.0000 10000000.0000 115000000.0000 110000000.0000 105000000.0000 100000000.0000 1 3 4 5 60000000.0000 50000000.0000 40000000.0000 30000000.0000 0000000.0000 Page 96 10000000.0000 0.0000 1 3 4 5

Example of multiple plane changing gravity assists (5) Inclination increases as the perihelion is progressively raised Orbits of Earth and Venus also shown Page 97

.6 Gravitational escape and capture Page 98

Natural Capture Temporary capture or transit from one heliocentric orbit to another by gravitational perturbation possible for comets and asteroids Jovian neighbourhood shows numerous irregular bodies possible captured via approaching through the Lagrange points. Example is Comet Shoemaker-Levy 9, which eventually impacted Jupiter in 1994. Jupiter studied regarding possible transitions between heliocentric resonant orbits, via Lagrange point passage. Example: Oterma and Gehrels 3 in :3 resonant orbit to 3: resonant orbit References: Carusi, Valsecchi, Koon, Lo, Marsden, Ross Page 99

Application to spacecraft transfers Technique applicable to transfers that include approach or escape speeds that are appropriate for the planet under consideration. Approach with significantly greater speed will not experience any significant advantage from the effect. For the inner planets, these speeds are not high enough to allow a direct transfer to another planet. Therefore methods may be sought that augment this effect to achieve the required transfers. Two techniques that may be used are: Use of additional deep space propulsive manoeuvres. These can be implemented by high specific impulse, low thrust systems. Use of gravity assist manoeuvres at the planet under consideration. This technique also requires intermediate propulsive manoeuvres. Page 100

Escape and capture mechanism Motion under multiple gravity fields can be exploited to achieve a beneficial orbit change In the three body problem initially bound orbits can achieve escape Conversely hyperbolic approach orbits can achieve capture The mechanism can be demonstrated via consideration of the 3 body problem and the Jacobi constant Figure shows energy surface. Assumes planet relative velocity in inertial oriented frame is always radial Earth is at origin, XY ecliptic rotating frame axes Page 101

Escape and capture mechanism () The energy is negative (bound orbit) close to Earth However achieves a positive value at distances >M km (in this example) The key is to find a trajectory that can achieve this Such trajectories require passages in the vicinity of the colinear Lagrange points Initial orbits (in escape case) are similar to free injection transfers to the Lagrange points Typical Earth escape vinfinity achievable 1000 m/s Page 10

Example of BepiColombo: Capture at Mercury A gravitational capture method is used at Mercury This requires a further reduced approach Vinfinity compared with the previous case (typically < 300 m/s). Then after flying close to one of the Mercury-Sun collinear Lagrange libration points, the spacecraft s osculating ephemeris on reaching Mercury pericentre is that of a high apocentre, captured orbit. This is a weakly bound orbit which may then remain captured in a high apocentre orbit for several revolutions. The chemical propulsion DeltaV to reduce apocentre from the weakly captured value (160000 to 00000km) to the target orbit at Mercury is now significantly reduced compared to hyperbolic approach. A net mass fuel mass reduction can be achieved by this strategy. Page 103

Example of BepiColombo: Capture at Mercury () Freely reaching weakly bound orbit significantly improves the robustness of the mission. The spacecraft can miss the nominal Mercury pericentre manoeuvre and remain in the high apocentre orbit and then re-attempt the apocentre reduction manoeuvre at subsequent Mercury pericentre passages. The use of a gravitational capture method means that the spacecraft must spend approximately half a Mercury orbit period longer before reaching pericentre Trajectory is seen in Mercury-Sun rotating frame Page 104