interplanetary mission analysis

Interplanetary Mission Analysis

Stephen KembleSenior ExpertEADS [email protected]

Contents

1. Conventional mission design

2. Advanced mission design options

1. Conventional mission design

1.1 Interplanetary transfers and Lambert’s problem1.2 Spheres of influence and patch conics1.3 Planetary escape and capture

2.1 Interplanetary transfers and Lambert’s problem

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)

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0.00E+00

2.00E+08

4.00E+08

6.00E+08

Ene

rgy

(J)

Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

rV

Eµ−=

2

2

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 0.20563069 2.22E+13 -6.99E+08Venus 0.72333199 1.08E+08 -6.10E+08 224.70 0.00677323 3.25E+14 -1.69E+08Earth 1.00000011 1.50E+08 -4.41E+08 365.26 0.01671022 3.99E+14 0.00E+00Mars 1.52366231 2.28E+08 -2.90E+08 686.96 0.09341233 4.28E+13 1.52E+08Jupiter 5.20336301 7.78E+08 -8.48E+07 4335.36 0.04839266 1.27E+17 3.56E+08Saturn 9.53707032 1.43E+09 -4.63E+07 10757.76 0.0541506 3.79E+16 3.95E+08Uranus 19.1912639 2.87E+09 -2.30E+07 30708.22 0.04716771 5.83E+15 4.18E+08Neptune 30.0689635 4.50E+09 -1.47E+07 60225.02 0.00858587 6.86E+15 4.27E+08Pluto 39.4816868 5.91E+09 -1.12E+07 90613.48 0.24880766 4.42E+13 4.30E+08

Rmin (km)

V at Rmin (km/s)

Rmax (km)

V at Rmax

Mercury 46001448 58.976 69817246 38.858 Venus 107474994 35.259 108942304 34.784 Earth 147096623 30.287 152099177 29.291 Mars 206655710 26.498 249225938 21.972 Jupiter 741096388 13.699 815828797 12.444 Saturn 1348673598 10.193 1508110731 9.115 Uranus 2733511855 7.123 2991539939 6.509 Neptune 4440286960 5.497 4538878194 5.378 Pluto 4433588797 6.106 7321604748 3.698

Transfer between circular orbits

• Planets may be approximated to lie in circular, co-planar orbits,orbit 1, Earth, and orbit 2, 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.

Vpe

Vap

r1

r2

11 2r

Eµ−=

22 2r

Eµ−=

21 rrE

+−= µ

��

�

�

��

�

�+−

��

�

�

��

�

�+=∆ 1

111 22 E

rE

rV

pp

µµ��

�

�

��

�

�+−

��

�

�

��

�

�+=∆ E

rE

rV

pp 22

22 22

µµ

��

�

�

��

�

�+= E

rV

µ2

General transfers

• In practice:– Planet orbits are eccentric– Each has a different inclination

and so are not co-planar

• 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

2. The chord length, c3. The sum of the position radii

from the focus or the connecting ellipse.

• Find a transfer between two positions in a given time.

θ r1

r2 c

2a

221 crr

s++

= θcos2 212

22

12 rrrrc −+=

Lambert’s problem

• Difference in Mean anomalies of transfer orbit

• Substitute:

• Then can derive

• Substitute

• Can derive

• Gives the result– Solution depends only on semi-

major axis ‘a’

( ) ( )1231212 tta

ttnMM −=−=− µ

( ) ( )( ) ��

��

��

��

��

��

� +��

��

� −−��

��

� −=−−−=−2

cos2

sin2

2sinsin 1212123

1212

3

12

EEEEe

EEaEEeEE

att

µµ

212 EE

A−

=2

coscos 12 EEeB

+=

( ) ( )BAAa

tt cossin23

12 −=−µ

BA +=α AB −=β

as

22sin 2 =α

acs

22sin 2 −=β

( ) ( ) ( )( )ββααµ

sinsin3

12 −−−=− att

Lambert’s problem (2)

• Can be applied to hyperbolic orbits

• Define further variables:

• Gives the result for hyperbolic orbits– Solution depends only on ‘a’

• In addition to semi-major axis, eccentricity, e can be found, for elliptical or hyperbolic cases

as

22sinh 2 −=α

acs

22sinh 2 −−=β

( ) ( ) ( )( )ββααµ

−−−−=− sinhsinh3

12

att

��

��

� +=2

sin2

sin4 22

221 βαθ

crar

p

��

��

� +−=2

sinh2

sin4 22

221 βαθ

crar

p)1( 2eap −=

Solving Lamberts problem

• A transfer from Earth to Mars can be considered.

• Consider leaving Earth on 23rd October, 2011

• Transfer durations of 150, 180 and 200 days investigated.

• Departure and arrival epochs specify two radius vectors, from which ‘c’ and ‘s’ may be calculated.

• An estimate at the value of semi-major 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

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0.00E+00 1.00E+11 2.00E+11 3.00E+11 4.00E+11 5.00E+11

sma (m)

erro

r (s

ecs) 200

180

150

( ) ( ) ( )( )ββααµ

sinsin3

12 −−−=− att

Solving Lamberts problem (2)Hyperbolic transfer• The spacecraft again leaves Earth

on 23rd October, 2011– 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.

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0

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400000

-2.50E+12 -2.00E+12 -1.50E+12 -1.00E+12 -5.00E+11 0.00E+00

sma (m)

erro

r (s

ecs)

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 200 and 350 days durations

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Transfer time (days)

Vinfinity Earth-Mars (m/s)

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Nature of the transfers

• A particular launch date can be selected

• The effect of varying transfer duration at that epoch can be examined

• Use 23rd October 2011. • Two minima are expected as

the transfer duration evolves.• The short and long minima

are seen separated by a local maximum at approximately 230 days

0

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0 100 200 300 400 500 600

Duration (days)

Vin

f tot

al (m

/s)

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-vousposition.

• 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

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 synodicperiod is calculated as:

• Where τp1 is the orbital period of planet 1 and τp2 is the orbital period of planet 2 and ωp1 is the orbital period of planet 1 and ωp2 is the orbital period of planet 2

( )21

21

360

360360

360

pp

pp

S ωωττ

τ−

=

��

�

�

��

�

�−

=

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 0.2762 0.9142 2.1354 Jupiter 0.2458 0.6488 1.0920 2.2350 Saturn 0.2428 0.6283 1.0351 2.0089 19.8618 Uranus 0.2415 0.6198 1.0121 1.9241 13.8324 45.5665 Neptune 0.2412 0.6175 1.0061 1.9026 12.7945 35.9576 170.5157 Pluto 0.2411 0.6167 1.0041 1.8953 12.4719 33.5207 126.8006 494.6005

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:

• The time is the number of synodic periods required to generate a whole number of orbits of the departure planet, whose period is τp1.

• 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.

1pSglobal mn τττ ==

Examples of global repeats• Transfer opportunities from

Earth show a global repeat to the nearest planets as follows:

• Venus: Synodic period 1.6 years, Global repeat 8 years

• Mars: Synodic period 2.15 years, Global repeats near 13 and 15 years

• Jupiter: Synodic period 1.09 years, Global repeats near 12 years

• Mercury: Synodic period 0.32 years, Global repeats near 1 year

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)



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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.

Multi-revolution transfers (2)

• Lambert’s problem now has two solutions for the semi-major axis, when n>0

Multi-revolution transfers to Mars• Example of 1.5

revolution case from Earth to Mars

• Launch dates over two synodic periods are shown

• Minimum Vinfinitiesare comparable to ‘0.5’ rev case

• Launch and arrival dates differ significantly to ‘0.5’revs

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2.2 Spheres of influence and patch conics

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.

• A surface may then be found defined by rpswhere these ratios are equal, in terms of the angular offset from the Sun’s direction, ‘a’

• rpc is the distance from planet to Sun

rpc

rps

rcs

a

( )51

5.022

2

cos31

1��

�

�

��

�

�

+=

arr

c

ppcps µ

µ

Spheres of influence (2)

• Hill’s sphere of influence can also be considered: includes rotational effect – similar to L1 and L2 distances

• The radii of the different spheres differ • Values for the planets of the solar system differ significantly

Classical sphere of Influence

Hill's sphere radius

Mercury 112837 221387 Venus 616277 1011199 Earth 924648 1496629 Mars 577131 1083965 Jupiter 48216966 53150917 Saturn 54615354 65244392 Uranus 51747112 70038854 Neptune 86634373 115989405 Pluto 15079253 28259593

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 1

Hyperbola wrt planet 2

Velocity of planet 1

Velocity of planet 2

2.3 Planetary escape and Capture

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:

aV

µ−=∞


• 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,

ap1 and perigee radius rpl1

��

��

�−= −

e1

cos 1θ

��

�

�

��

�

�

+−−

��

�

�

��

�

�+=∆ ∞ )(

112

2

111

2

1 aplplplpl rrrV

rV µµ


• The DeltaV needed to reach a given Vinfinity increases more quickly with target Vinfinity as Vinfinity increases

• In these examples the perigee altitude is 200km

• Declinations that can be reached from a given inclination orbit are given by:

• Therefore maximum abs(Declination) = inclination

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Apogee (km)

DV

(m/s

) 1000

2000

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4000

)sin(sinsin θω += iDEC

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 achievedwithout plane change.

Approach definitions

• Three planes are of primary importance

• Reference plane (egecliptic 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

β angle measured from reference plane

B plane perpendicular to approach velocity vector

Approaching relative velocity vector at Beta angle > 90 deg

Plane parallel to Ecliptic plane: Reference plane

Intersection of B plane and ecliptic parallel plane. Defines zero Beta angle

Relative Approach plane

Declination (negative as shown)

Planet Relative orbit plane

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

DECi sinsin ≥

Capture pericentre (2)

• Pericentre locations with different approach speeds at Mars

• Pericentre locations with different approach speeds of 3 km/sec at Mars and a range of approach declinations

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Latit

ude 4000

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ude

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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)

2. Advanced mission design options

2.1 The three body problem2.2 Designing missions to the Lagrange points2.3 Gravity assist2.4 Multiple Gravity Assists2.5 Plane changing gravity assist2.6 Gravitational escape and capture

2.1 The three body problem

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 Keplerianorbits 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 integrationof the spacecraft state vector derivatives.

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

• Where V is the speed of the spacecraft relative to a rotating reference frame

• U is a generalised potential given by:

• Where rx,ry, rz are the components of position with respect to the barycentre of the two major bodies, r1 and r2 are the distances from body 1 and body 2, � is the angular velocity of the two bodies in circular orbit

• µ1 = Gm1, ie the gravitational parameter for mass m1 • µ2 = Gm2, ie the gravitational parameter for mass m2

CUV −=+ 22

( )222

2

2

1

1

2 Yx rrrr

U +−��

��

�−−= ωµµ

Zero velocity surfaces

• The limiting regions of the motion can be obtained by considering the case where speed is zero:

• 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.

( ) 02 222

2

2

1

1 =++−��

��

�−− Crr

rr Yxωµµ

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 = 2641000000 (m/s)^2

• (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 co-linear Lagrange points, L1 and L2, at approximately +/- 1.5 million km along the X axis from Earth

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:

• 5 points can be found in the orbit plane of the secondary body about the primary

• There are three, collinear, points, denoted L1, L2 and L3

• Two further points complete an equilateral triangle with the baseline from primary to secondary, L4 and L5

0=∇U

d1 d2

L4

L5

L1 L2 L3

Lagrange points at the planets

• The L1 and L2 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 J2000

m2/(m1+m2) L1 (km) L2 (km) Mercury 1.660E-07 2.204E+05 2.210E+05 Venus 2.448E-06 1.008E+06 1.014E+06 Earth 3.003E-06 1.492E+06 1.501E+06 Earth-Moon 3.040E-06 1.498E+06 1.508E+06 Mars 3.227E-07 1.082E+06 1.086E+06 Jupiter 9.537E-04 5.209E+07 5.418E+07 Saturn 2.857E-04 6.425E+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:– Scaled by dividing by the separation between two bodies – C11 is a constant

( )

( )

112

2

112

2

112

2

''

'

1'''

2'

'

21''

'2

''

crdt

rd

crdtrd

dt

rd

crdt

rd

dtrd

zz

yxy

xyx

δδ

δδδ

δδδ

−=

−=+

+=−

Lnrrr −=δ

Orbits at the Lagrange points (2)

• These differential equations may be solved:

• A1 to 4 are constants determined by the initial conditions (also C1 and C2)

• The two angular frequencies differ slightly, depend on c11• Ratios of motion amplitudes in x and y (a and b) also depend on

c11

'sin'cos'

'sin'cos'

'sin'cos'

21

34'

2'

1

43'

2'

1

tCtCr

tbAtbAeaAeaAr

tAtAeAeAr

zzz

xyxystst

y

xyxystst

x

ωωδωωδ

ωωδ

+=

+−−=

+++=−

−

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

µµµµ wxy wz Txy (days)

Tz (days)

b

Mercury 1.660E-07 2.077 2.006 42 44 3.216 Venus 2.448E-06 2.085 2.014 108 112 3.228 Earth 3.003E-06 2.086 2.015 175 181 3.229 Mars 3.227E-07 2.079 2.007 330 342 3.218 Jupiter 9.537E-04 2.170 2.100 1998 2064 3.349 Saturn 2.857E-04 2.138 2.068 5032 5202 3.303 Uranus 4.366E-05 2.108 2.037 14570 15076 3.260 Neptune 5.150E-05 2.110 2.039 28542 29531 3.263 Pluto 6.607E-09 2.099 2.028 43173 44683 3.247

Lissajous orbits

• 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

-150000

-100000

-50000

0

50000

100000

150000

-1000000 -500000 0 500000 1000000

Y

Z

-1000000

-800000

-600000

-400000

-200000

0

200000

400000

600000

800000

1000000

-300000 -200000 -100000 0 100000 200000 300000

X

Y

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 amplitudesyield 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.

0

100000

200000

300000

400000

500000

600000

600000 650000 700000 750000 800000 850000 900000

Ay

Az

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 A2 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, A2 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

A2 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

Stable and Unstable motions (2)

• 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 non-zero.

– This fact may be used as a feature of orbit generation and maintenance strategies: Station keeping aims to remove the unstable component

2.2 Designing missions to the Lagrange points

Transfers

• Transfers from an initial Earth elliptical orbit to a Lissajous or Halo orbit about the Earth-Sun L1 or L2 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.

Searching fora 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, 20 degrees of Sun direction (for transfers to an L1 orbit) or anti-Sun direction (for transfers to an L2 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

Searching fora transfer (2)• 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

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, egperigee 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

0

50

100

150

200

250

0 100000 200000 300000 400000 500000 600000 700000 800000

Halo SMA (km)

Del

taV

(m/s

)

Transfer characteristics

• The transfer duration is typically 120 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 100-120 days

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

Transfer types (2)

• 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

• 2 routes transfer to the North• 2 routes transfer to the South• 1 North and 1 South transfer

examples are shown in rotating frame, XZ plane

-1000000000

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-200000000

0

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400000000

600000000

800000000

1000000000

-1.8E+09 -1.6E+09 -1.4E+09 -1.2E+09 -1E+09 -8E+08 -6E+08 -4E+08 -2E+08 0 2E+08

X rotating (km)

Y ro

tatin

g (k

m)

Type 2Type 1

-800000000

-600000000

-400000000

-200000000

0

200000000

400000000

600000000

800000000

-1E+09 -8E+08 -6E+08 -4E+08 -2E+08 0 2E+08 4E+08 6E+08 8E+08 1E+09

Y rotating (km)

Z ro

tatin

g (k

m)

Type 2Type 1

-800000000

-600000000

-400000000

-200000000

0

200000000

400000000

600000000

800000000

-1.8E+09 -1.6E+09 -1.4E+09 -1.2E+09 -1E+09 -8E+08 -6E+08 -4E+08 -2E+08 0 2E+08

X rotating (km)

Z ro

tatin

g (k

m)

Type 2Type 1

2.3 Gravity Assist

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’

Sphere of influence of major body

Hyperbolic orbit with respect to major body

Approach orbit

Departure orbit

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 2: 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

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.– Limiting true anomaly, �, depends only on eccentricity, e

��

��

�−±= −

e1

cos 1θ

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

True anomaly of asymptotic departure direction, θ

Deflection angle, α

��

��

�= −

e1

sin*2 1α

planet

perirelperirel Vr

a

re

µ

2*11 ∞+=−=

Link approximations

• The link conic approximation therefore assumes that the velocityvector change is instantaneous

• The change takes place at the planet• Therefore the time to traverse the sphere of influence is neglected

– Typically 2-3 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

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

Vp

Approach V Approach Vrel=V∞

Γrelp

Body

Γ

α

Deflected Vrel=V∞

Vdeparture

Departure Γrelp

Γdeparture

Vp

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

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• Vplanet is the velocity of the planet

• The departure flight path is then

• The departure heliocentric velocity is

��

�

�

��

�

�

−Γ−ΓΓ−Γ

=Γ −

planetplanet

planetrelp VV

V

)cos(*

)sin(*tan 1

��

�

�

��

�

�

Γ++Γ+

=Γ∞

∞−

)cos(*

)sin(*tan 1

relpplanet

relpdeparture VV

V

αα

( ) ( )( )22 )cos(*)sin(* relpplanetrelpdeparture VVVsqrtV Γ+++Γ+= ∞∞ αα

∞=Γ−Γ−+= VVVVVV planetplanetplanetrel )cos(222

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!

Gravity assist example: co-planar assumptions• Consider a gravity assist at Venus with a

spacecraft in an elitpical orbit

• The possible orbit changes post fly-by are

Initial Ap Initial Pe Planet Planet Anom RV radius Hyper V Peri h (km)1.0000 0.5000 2 120 0.7258 11669.1746 400.0000

Post GA Ap Post GA Pe Post GA Ap Post GA Pe1.7267 0.6433 0.7973 0.3185

General coplanar gravity assist casesEffect 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

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 2000 4000 6000 8000 10000 12000

altitude (km)

Ap-

Pe

(AU

) Ap

Pe

sma

sma-0

2.4 Multiple gravity assists

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 sameat successive fly-bys

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

Vplanet

V

Vrel=V∞

α

V

( )( )flybyVVV += ∞max

( )( )∞−= VVV flybymin

Multiple gravity assistexample• 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 = 20 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

0.0000

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10000000.0000

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25000000.0000

Apo

cent

re (k

m)

1 2 3 4 5 6 7 8 9 10 11 12 13

0.0000

100000.0000

200000.0000

300000.0000

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500000.0000

600000.0000

700000.0000

800000.0000

900000.0000

Per

icen

tre

(km

)

1 2 3 4 5 6 7 8 9 10 11 12 13

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 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, VIC. rc is the distance from the central body.

CUV −=+ 22

( ) Cr

zVrVC

ICIC −=−•∧− µω 2ˆ22

Tisserand’s Criterion (2)

• The constant may be expressed at distances far from the planet as:

• In the co-planar case, then the orbit relationships before and after are given by:

• 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’.

Ciha

−=−− cos2ωµ

)1(2)1(2 222

2

211

1

eaa

eaa

−−−=−−− µωµµωµ

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

Pericentre

Period

Locus for a given Vinfinity at target body

Step at gravity assist

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 period-pericentre and apocentre-pericentre plots are considered

0.00E+00

1.00E+02

2.00E+02

3.00E+02

4.00E+02

5.00E+02

6.00E+02

7.00E+02

8.00E+02

8.00E+07 9.00E+07 1.00E+08 1.10E+08 1.20E+08 1.30E+08 1.40E+08 1.50E+08

Pericentre (km)

Per

iod

(day

s) 300040005000

0.00E+00

5.00E+07

1.00E+08

1.50E+08

2.00E+08

2.50E+08

3.00E+08

3.50E+08

8.00E+07 9.00E+07 1.00E+08 1.10E+08 1.20E+08 1.30E+08 1.40E+08 1.50E+08

Pericentre (km)

Apo

cent

re (k

m)

300040005000

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 2• This assumes that the required phasing between bodies 1 and 2

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)

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

0

1000000

2000000

3000000

4000000

5000000

6000000

7000000

8000000

9000000

10000000

0.00E+00 1.00E+05 2.00E+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

Pericentre (km)

Ap

oce

ntr

e (k

m)

C-7000

C-6000

C-5000

G-7000

G-6000

G-5000

E-4000

E-3000

E-2000

E-6000

G-4000

Designing a gravity assist sequence in the Jovian system (2)• 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

– ∆Vapi 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, – rEuropaorbit is the radius of this circular orbit about Europa.

tEuropaOrbi

final

tEuropaOrbi

Europa

iapitotal r

Vr

VVµµ

−++∆=∆ ∞� 22

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 EuropaGA’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

0

2

4

6

8

10

12

14

16

18

20

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)

Per

iod

(day

s)

G-7000

G-6000

G-5000

E-4000

E-3000

E-2000

E-5000

G-4000

Route

Example of gravity assist sequence in the Jovian system (2)• The sequence is

Event Apocentre (km)

Pericentre (km)

Period (days)

V∞∞∞∞ (m/s)

Resonance DV (m/s)

Capture 20218103 900000 222 31:1 Gany 7712518 847482 57 6553 8:1 Gany 4604948 787514 29 6553 4:1 Gany 3223221 718692 18 6553 5:2 Gany 2321376 618442 12 6553 Europa 1873702 600456 9 4969 Gany 2147032 664506 11 5883 Europa 1973573 660187 10 3210 11:4 Europa 1654999 648966 8 3210 9:4 Europa 1319465 629092 6 3210 Gany 1360010 662086 6 3733 Europa 1222197 653882 6 2209 5:3 DV 1222197 664000 6 1847 -43 Europa 1094256 655272 5 1847 3:2 DV 1094256 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 Venus2. Venus gravity assist to Earth

(Vinfinity approx 6 km/sec)3. Earth gravity assist to Earth via

2 year resonant orbit (Vinfinityapprox 9 km/sec)

4. Earth gravity assist to Jupiter• Total transfer duration 6.2 years

2.5 Plane changing gravity assists

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 2 in the figure are therefore at Beta angles

of zero and 90 degrees.

1. Approaching relative velocity vector contained in the approach plane

β = 90 deg

B plane

2. Approaching relative velocity vector is offset to achieve an approach that is parallel to the approach plane

Approach plane

XB

YB

ZB

3D deflection

Approaching relative velocity vector

β

B plane

Plane containing body velocity vector and spacecraft velocity vector, ie The approach plane α

αθ

αφ

αθ is negative in this diagram

• 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

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.

Pericentre (km)

Speed at Ganymede (m/s)

Vinfinity (m/s)

Flight path angle (deg)

500000 13803.64 9853.09 43.445 600000 13833.40 8900.53 38.140 700000 13862.06 7954.81 32.829 800000 13889.67 6990.59 27.282 900000 13916.30 5975.76 21.114

3D gravity assist example (2)

• 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 co-

planar with the orbit of Ganymede.– Defines the approach plane

• The pericentre altitude at Ganymede is assumed to be 300km

3D gravity assist example (3)

• A β angle of 90 degrees maximises the inclination change

• It also yields a small change in the post gravity assist apocentreand 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

0

2000000000

4000000000

6000000000

8000000000

10000000000

12000000000

0 50 100 150 200 250 300 350 400

Beta(deg)

Apo

cent

re (m

) 500000

600000

700000

800000

900000

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 50 100 150 200 250 300 350 400

Beta(deg)

Incl

inat

ion

(deg

) 500000

600000

700000

800000

900000

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

Multiple plane changing fly-bys (2)

• 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

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.

• The maximum inclination (relative to the initial approach plane) in the case of the same resonant orbit being maintained is given by:

• The maximum achievable inclination (by changing resonance) is given by:

��

�

�

��

�

�

+=

∞

∞−

θθcos

sintan 1

max VVV

iplanet

��

�

�

��

�

�= ∞−

planetVV

i 1max sin

θ ψ=90

Vplanet

V1 Vrel=V∞

Fly-by body orbit plane

Plane containing spacecraft and fly-by body velocity vectors

imax

Vlimit Vrel=V∞

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.

Example of multiple plane changing gravity assists (2)• 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.

Example of multiple plane changing gravity assists (3)• The initial orbit is:

– Apocentre = 132 million km, – Pericentre = 33 million km. – Excess hyperbolic speed with

respect to Venus is 20.2 km/sec• A 3:2 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 2700 days, or approaching 7.5 years.

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

2.6 Gravitational escape and capture

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 2:3 resonant orbit to 3:2 resonant orbit• References: Carusi, Valsecchi, Koon, Lo, Marsden, Ross

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 beimplemented by high specific impulse, low thrust systems.Use of gravity assist manoeuvres at the planet under consideration. This technique also requires intermediate propulsive manoeuvres.

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

Escape and capture mechanism (2)

• The energy is negative (bound orbit) close to Earth

• However achieves a positive value at distances >2M 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 vinfinityachievable 1000 m/s

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 200000km) 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.

Example of BepiColombo: Capture at Mercury (2)• 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 apocentrereduction 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

Summary

• Designing optimal interplanetary trajectories utilises a range methods in Mathematics and Astrodynamics

• The field of interplanetary trajectory design is constantly evolving as new methods in astrodynamicsare applied and new spacecraft propulsion concepts are considered.

• Efficient design methods are required to both make preliminary problem assessments and detailed mission designs

• Mission/trajectory designs must be integrated with spacecraft capabilities, to ensure an optimal end to end system design

interplanetary mission analysis

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