Preliminary mission study: orbital transfer from Titan to Saturn

Politecnico di Milano

Dipartimento di scienze e tecnologieaerospaziali

Orbital mechanics end course project

Preliminary Titan - Saturn interplanetary transfer

Authors:Marco CiarambinoSimone Colciago

Contents

1 Overview of the problem 3

2 Orbital transfer 42.1 Initial orbit and departure hyperbola . . . . . . . . . . . . . . . . 42.2 Plane change maneuver . . . . . . . . . . . . . . . . . . . . . . . 52.3 Pericentre anomaly change . . . . . . . . . . . . . . . . . . . . . . 62.4 Hohmann maneuver . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Perturbation analysis 103.1 Final orbit period . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1.1 Solar wind pressure . . . . . . . . . . . . . . . . . . . . . 103.1.2 Influence of other bodies . . . . . . . . . . . . . . . . . . . 143.1.3 Non uniformity of Saturn mass distribution . . . . . . . . 173.1.4 All perturbations . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Earth revolution orbit period . . . . . . . . . . . . . . . . . . . . 233.2.1 Solar wind pressure . . . . . . . . . . . . . . . . . . . . . 233.2.2 Influence of other bodies . . . . . . . . . . . . . . . . . . . 263.2.3 Non uniformity of Saturn mass distribution . . . . . . . . 293.2.4 All perturbations . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Saturn revolution orbit period . . . . . . . . . . . . . . . . . . . . 363.3.1 Solar wind pressure . . . . . . . . . . . . . . . . . . . . . 363.3.2 Influence of other bodies . . . . . . . . . . . . . . . . . . . 393.3.3 Non uniformity of Saturn mass distribution . . . . . . . . 423.3.4 All perturbations . . . . . . . . . . . . . . . . . . . . . . . 45

4 Considerations about numerical and general issues encountered 48

5 Block diagram 49

1

Abstract

In the following report it will be presented an orbital transfer fromTitan to Saturn, taking into account some forms of perturbations to thecanonical restricted two body problem.The first section deals with the computation of the main parameters ofthe journey, approaching the problem through a pure analytical form theXVIII century-developed restricted two body problems. Through thistheory it has been possible to compute the ideal behaviour of the space-craft from parking orbit around Titan, to the final path around Saturn,that was given as a mission parameter.Secondly perturbation analysis around the final orbit has been performed.In particular solar wind pressure, non uniform mass of Saturn and thegravitational influence of other celestial objects have been taken into ac-count, in order to achieve better accuracy in the spacecraft motion.

2

1 Overview of the problem

At the beginning of the simulation, spacecraft is parked in a circular orbitaround Titan with pericentre height of 150 km, while the rest of parametersfor the initial orbit were left free to choose. The keplerian parameters of Titan,orbiting around Saturn on a very low eccentric orbit, depend on the day chosenas first instant for simulation: May 3, 2013.

Destination characteristics were given by data:

pericentre height [km] 260000apocentre height [km] 440000inclination [ ] 20.27ascending node anomaly [ ] 15pericentre anomaly [ ] 15

Table 1: Final orbit parameters

Graphically, the problem is summarised by the following picture:

1

0.5

0

0.5

1

x 106

1

0.5

0

0.5

1

x 106

101

x 105

Titans orbitCurrent positionFinal orbit

Figure 1: Overview of the problem

3

2 Orbital transfer

2.1 Initial orbit and departure hyperbola

In order to escape from Titans gravitational attraction and to directly putthe spacecraft onto an orbit such that it is equal to Titans one but with finalinclination and ascending node anomaly, velocity at limits of Titans sphere ofinfluence V must be equal to the first variation of velocity V1 = 1.8793 km/s.Once known velocity at limits of Titans sphere of influence and pericentreheight, it is possible to define geometric features of the departure hyperbola:

pericentre height [km] 150eccentricity [] 2.0723semimajor axis [km] -2542.1079deviation angle, 2 [ ] 57.70 [ ] 118.85

Table 2: Departure hyperbola characteristics

In order to reach proper velocity at the boundaries of Titans sphere ofinfluence and consequently to enter in the correct orbit, a tangential V =1.33421 km/s must be performed at the hyperbola pericentre, equal to the dif-ference between velocity on the circular parking orbit and velocity at pericentreof hyperbola.

Figure 2: Departure hyperbola

4

2.2 Plane change maneuver

The final requested orbit has no intersections with initial Titans orbit. Thepurpose of the first manuever is to put the spacecraft onto an orbit geometricallyequal to Titans orbit (same eccentricity and semi-major axis) with ascendingnode anomaly and inclination of the final requested orbit. Thus it is necessaryto start the maneuver in the point of intersection between Titans orbit and theplane containing the final orbit. There are two possible intersections: for sakeof convenience in terms of time it has been selected the point of maneuver closerto the ascending node and Titans current position (May 3, 2013).

Solving the spherical triangle it is possible to compute , the angle betweenvelocities before and after the maneuver equal to 19.97, and , the anglebetween the point of maneuver and the ascending node of the first orbit, equalto 13.25.

The time between Titans current position and point of maneuver is

t1 = 3 days 20 hours 32minutes 0.6 seconds

while the total cost of maneuver is equal to

V1 = 1.8793km

s.

1

0.5

0

0.5

1

x 106

1

0.5

0

0.5

1

x 106

4

2

0

2

4

x 105

Titan orbitcurrent positionfinal orbitAscending nodepoint of maneuverorbit after maneuver

Figure 3: Plane change maneuver

5

2.3 Pericentre anomaly change

The second manuever changes pericentre anomaly keeping all geometrical prop-erties unaltered. This maneuver can be done in the two points of intersectionbetween the current orbit the spacecraft is located on and the orbit with thefinal requested pericentre anomaly; as before, for sake of convenience in terms oftime, the selected point is the one closer to the current position of the satellite.The waiting period in order to perform the maneuver of pericentre anomalychange is


and total cost of maneuver is equal to

V2 = 0.3185km

s.

1

0.5

0

0.5

1

x 106

10.8

0.60.4

0.20

0.20.4

0.60.8

1

x 106

4

2

0

2

4

x 105

orbit after plane changepericenterorbit after change of pericenter anomalypericenterpoint of maneuver

Figure 4: Change of pericenter anomaly

6

2.4 Hohmann maneuver

The orbit the probe is moving on has ascending node anomaly, inclination andpericentre anomaly of the final requested orbit. Now it is necessary to obtainthe desired shape and this operation is done exploiting a Hohmann transferbetween the apocentre of the current orbit and the pericentre of the final one.This maneuver is the most efficient one and the decision to perform it fromapocentre to pericentre and not in the reversed order is justified from the factthat the aim of this maneuver is a reduction of orbit shape (in fact final orbitis smaller than the current one).The Hohmann manuever is a bi-tangent elliptic maneuver represented by twodifferent V , both negative because of velocity decreasing, done respectively atapocentre of current orbit and pericentre of final orbit.The time between the previous maneuver point and the apocentre is


whereas the cost (negative because it is a braking maneuver) is

V3 = 1.9132 kms

The time required to reach the pericentre of the final orbit is equal to thesemi-period of Hohmann transfer orbit,


whereas the cost (negative because it causes a reduction of velocity) is

V4 = 1.7224 kms

7

1

0.5

0

0.5

1

x 106

10.8

0.60.4

0.20

0.20.4

0.60.8

1

x 106

4

2

0

2

4

x 105

final orbitpericenterpoint of previous maneuverpoint of Hohmann maneuverHohmann transfer

Figure 5: Hohmann maneuver

8

2.5 Results

Maneuver Cost [km/s]

Departure hyperbola 1.3662Plane change maneuver 1.8793Pericentre anomaly change 0.3185Hohmann transfer 3.6356Total 7.1996

Table 3: Maneuvers costs

1

0.5

0

0.5

1

x 106

1

0.5

0

0.5

1

x 106

2

0

2

4

x 105

Titan orbitfinal orbit1st burnoutorbit after plane change2nd burnoutorbit after change of pericentre anomaly3rd burnoutHohmann transfer4th burnout

Figure 6: Transfer overview

9

3 Perturbation analysis

Once that nominal orbits have been computed, we will now approach perturba-tion section, that will be performed only over final orbit path. Three kinds ofthem have been modelled:

Solar wind pressure, that in our case has been particularly intense due tothe high exposed surface (22.10 m2);

Influence of masses of Saturn, Titan, Jupiter, Uranus, Sun on the space-craft;

Non uniformity of the Saturn mass distribution, where only the first har-monic of the gravitational potential has been considered.

It has been not considered the perturbation effects due to atmospheric drag.Indeed at final orbit pericentre the spacecraft is 320267 km high on Saturn sur-face and at such a height atmospheric drag effects are completely negligible.Computation has relied on Cowells method, that is actually the direct integra-tion of Keplers equation with addiction of perturbation term. Simulation ofperturbation effects will be performed over three different periods: a final or-bit period, an Earth revolution orbit period and over a Saturn revolution orbitperiod. Proper details will be given on each section.

3.1 Final orbit period

Data relative of the destination orbit has been specified in Table 1. Resultingorbit period amounts to 3 days 2 hours 27 min 36 seconds.

3.1.1 Solar wind pressure

Solar wind perturbing acceleration term to be inserted in Keplers equation hasbeen modelled as:

asw = psunCrAscm

rsunsc (1)

where:

psun is solar pressure; Cr = 1 + takes into account optical reflectivity of the spacecraft that in

this case = 0.7;

Asc = 22.10 m2 is the area of each side of the spacecraft, that is modelledas cube;

m spacecraft mass amounts to 2000 kg;

10

rsunsc is position versor from Sun to spacecraft. It is the sum betweenSun-Saturn and Saturn-spacecraft versors, and because the first was inSun-centred inertial frame it has been rotated in Saturn-centred equatorialinertial frame with a Saturns tilt angle of 26.73 and a north pole rightascension equal to 40.6

It has been neglected eclipses occurrence.In the following pages are shown keplerian orbital parameters variations in time.

10.5

00.5

1

x 106

1

0.5

0

0.5

1

x 106

101

x 105

[km]

Orbit variation due to solar wind pressure

[km]

[km]

Titans orbitFinal orbitPerturbed orbit

0 0.5 1 1.5 2 2.5 3 3.5410263

410264

410265

410266

410267

410268

410269

410270semimajor axis variation

[km]

days

11

0 0.5 1 1.5 2 2.5 3 3.50.219365

0.219370

0.219375

0.219380

0.219385

0.219390

0.219395

0.219400eccentricity variation

[]

days

0 0.5 1 1.5 2 2.5 3 3.520.2699

20.27

20.2701

20.2702

20.2703

20.2704inclination variation

[deg]

days

12

0 0.5 1 1.5 2 2.5 3 3.514.994

14.995

14.996

14.997

14.998

14.999

15

15.001pericentre anomaly variation

[deg]

days

0 0.5 1 1.5 2 2.5 3 3.514.9993

14.9994

14.9995

14.9996

14.9997

14.9998

14.9999

15

15.0001

15.0002

15.0003ascending node anomaly variation

[deg]

days

13

3.1.2 Influence of other bodies

Other celestial bodies considered in the simulation are: Saturn, Titan, Jupiter,Uranus, Sun. They have been modelled as dot masses.

10.5

00.5

1

x 106

1

0.5

0

0.5

1

x 106

101

x 105

[km]

Orbit variation due to the most influencing nearby bodies

[km]

[km]


0 0.5 1 1.5 2 2.5 3 3.5410267.2

410267.4

410267.6

410267.8

410268.0

410268.2

410268.4

410268.6

410268.8semimajor axis variation

[km]

days

14

0 0.5 1 1.5 2 2.5 3 3.50.219368

0.219369

0.219370

0.219371

0.219372

0.219373

0.219374

0.219375

0.219376

0.219377


[]

days

0 0.5 1 1.5 2 2.5 3 3.520.269970

20.269975

20.269980

20.269985

20.269990

20.269995

20.270000

20.270005


[deg]

days

15

0 0.5 1 1.5 2 2.5 3 3.514.99965

14.99970

14.99975

14.99980

14.99985

14.99990

14.99995

15.00000

15.00005


[deg]

days

0 0.5 1 1.5 2 2.5 3 3.514.99994

14.99995

14.99996

14.99997

14.99998

14.99999

15.00000

15.00001

15.00002


[deg]

days

16

3.1.3 Non uniformity of Saturn mass distribution

To model the non uniform mass of Saturn, perturbing acceleration componentson the spacecraft in body reference frame are:

ar = 32J2Saturn

R2Saturnr4

[1 3sin2(i) sin2( + )]

a = 3J2SaturnR2Saturn

r4sin2(i) sin( + ) cos( + )

ah = 3J2SaturnR2Saturn

r4sin(i) cos(i) sin( + )

being J2 = 16298x106 the first term of gravitational potential harmonics. Ob-

viously, to be added into Keplers equation it has been necessary to rotate bodycoordinates onto the Saturn-centred equatorial inertial frame.

10.5

00.5

1

x 106

1

0.5

0

0.5

1

x 106

101

x 105

[km]

Orbit variation due to non uniform mass of Saturn

[km]

[km]


17

0 0.5 1 1.5 2 2.5 3 3.5410267.986

410267.988

410267.990

410267.992

410267.994

410267.996

410267.998

410268.000

410268.002


[km]

days

0 0.5 1 1.5 2 2.5 3 3.50.21936874

0.21936875

0.21936876

0.21936877

0.21936878

0.21936879

0.21936880


[]

days

18

0 0.5 1 1.5 2 2.5 3 3.520.2699988

20.2699990

20.2699992

20.2699994

20.2699996

20.2699998

20.2700000


[deg]

days

0 0.5 1 1.5 2 2.5 3 3.514.99999

15.00000

15.00000

15.00001

15.00001

15.00002

15.00002

15.00003

15.00003


[deg]

days

19

0 0.5 1 1.5 2 2.5 3 3.514.999980

14.999982

14.999984

14.999986

14.999988

14.999990

14.999992

14.999994

14.999996

14.999998


[deg]

days

3.1.4 All perturbations

Hereby the whole set of perturbation is taken into account.

10.5

00.5

1

x 106

1

0.5

0

0.5

1

x 106

101

x 105

[km]

Orbit variation due to the whole perturbation set

[km]

[km]


20

0 0.5 1 1.5 2 2.5 3 3.5410263

410264

410265

410266

410267

410268


[km]

days

0 0.5 1 1.5 2 2.5 3 3.50.219368

0.219370

0.219372

0.219374

0.219376

0.219378

0.219380

0.219382

0.219384

0.219386


[]

days

21

0 0.5 1 1.5 2 2.5 3 3.520.2699

20.27

20.2701

20.2702

20.2703


[deg]

days

0 0.5 1 1.5 2 2.5 3 3.514.994

14.995

14.996

14.997

14.998

14.999

15


[deg]

days

22

0 0.5 1 1.5 2 2.5 3 3.514.9993

14.9994

14.9995

14.9996

14.9997

14.9998

14.9999

15

15.0001

15.0002


[deg]

days

3.2 Earth revolution orbit period

Revolution year orbit period has been approximated to Gregorian year of 365mean solar days.


10.5

00.5

1

x 106

1

0.5

0

0.5

1

x 106

101

x 105

[km]


[km]

[km]


23

0 50 100 150 200 250 300 350 400410263

410264

410265

410266

410267

410268

410269


[km]

days

0 50 100 150 200 250 300 350 4000.219000

0.219500

0.220000

0.220500

0.221000

0.221500

0.222000

0.222500


[]

days

24

0 50 100 150 200 250 300 350 40020.265

20.27

20.275

20.28

20.285

20.29

20.295

20.3


[deg]

days

0 50 100 150 200 250 300 350 40014.3

14.4

14.5

14.6

14.7

14.8

14.9

15pericentre anomaly variation

[deg]

days

25

0 50 100 150 200 250 300 350 40014.995

15

15.005

15.01

15.015

15.02


[deg]

days


10.5

00.5

1

x 106

1

0.5

0

0.5

1

x 106

101

x 105

[km]


[km]

[km]


26

0 50 100 150 200 250 300 350 400410267.2

410267.4

410267.6

410267.8

410268.0

410268.2

410268.4

410268.6


[km]

days

0 50 100 150 200 250 300 350 4000.219200

0.219400

0.219600

0.219800

0.220000

0.220200


[]

days

27

0 50 100 150 200 250 300 350 40020.267500

20.268000

20.268500

20.269000

20.269500

20.270000

20.270500


[deg]

days

0 50 100 150 200 250 300 350 40014.96500

14.97000

14.97500

14.98000

14.98500

14.99000

14.99500

15.00000


[deg]

days

28

0 50 100 150 200 250 300 350 40014.99300

14.99400

14.99500

14.99600

14.99700

14.99800

14.99900

15.00000


[deg]

days


10.5

00.5

1

x 106

1

0.5

0

0.5

1

x 106

101

x 105

[km]


[km]

[km]


29

0 50 100 150 200 250 300 350 400410267.980

410268.000

410268.020

410268.040

410268.060

410268.080

410268.100

410268.120

410268.140


[km]

days

0 50 100 150 200 250 300 350 4000.21936870

0.21936880

0.21936890

0.21936900

0.21936910

0.21936920

0.21936930

0.21936940

0.21936950


[]

days

30

0 50 100 150 200 250 300 350 40020.2699988

20.2699990

20.2699992

20.2699994

20.2699996

20.2699998

20.2700000


[deg]

days

0 50 100 150 200 250 300 350 40014.99950

15.00000

15.00050

15.00100

15.00150

15.00200

15.00250

15.00300

15.00350


[deg]

days

31

0 50 100 150 200 250 300 350 40014.997500

14.998000

14.998500

14.999000

14.999500


[deg]

days


10.5

00.5

1

x 106

1

0.5

0

0.5

1

x 106

101

x 105

[km]


[km]

[km]


32

0 50 100 150 200 250 300 350 400410263

410264

410265

410266

410267

410268


[km]

days

0 50 100 150 200 250 300 350 4000.219000

0.219500

0.220000

0.220500

0.221000

0.221500

0.222000


[]

days

33

0 50 100 150 200 250 300 350 40020.265

20.27

20.275

20.28

20.285

20.29

20.295

20.3


[deg]

days

0 50 100 150 200 250 300 350 40014.4

14.5

14.6

14.7

14.8

14.9


[deg]

days

34

0 50 100 150 200 250 300 350 40014.995

15

15.005

15.01

15.015

15.02

15.025

15.03


[deg]

days

35

3.3 Saturn revolution orbit period

Saturn revolution orbit period has been approximated to 29.7 earth years.


10.5

00.5

1

x 106

1

0.5

0

0.5

1

x 106

101

x 105

[km]


[km]

[km]


0 5 10 15 20 25 30410263

410264

410265

410266

410267

410268

410269


[km]

years

36

0 5 10 15 20 25 300.200000

0.220000

0.240000

0.260000

0.280000

0.300000


[]

years

0 5 10 15 20 25 30

20.4

20.6

20.8

21

21.2

21.4


[deg]

years

37

0 5 10 15 20 25 302

4

6

8

10

12

14


[deg]

years

0 5 10 15 20 25 3014.95

15

15.05

15.1

15.15

15.2

15.25

15.3

15.35

15.4


[deg]

years

38


10.5

00.5

1

x 106

1

0.5

0

0.5

1

x 106

101

x 105

[km]


[km]

[km]


0 5 10 15 20 25 30410267.0

410267.2

410267.4

410267.6

410267.8

410268.0

410268.2

410268.4

410268.6


[km]

years

39

0 5 10 15 20 25 300.215000

0.220000

0.225000

0.230000

0.235000

0.240000

0.245000


[]

years

0 5 10 15 20 25 3020.190000

20.200000

20.210000

20.220000

20.230000

20.240000

20.250000

20.260000

20.270000


[deg]

years

40

0 5 10 15 20 25 3014.20000

14.30000

14.40000

14.50000

14.60000

14.70000

14.80000

14.90000

15.00000

15.10000


[deg]

years

0 5 10 15 20 25 3014.75000

14.80000

14.85000

14.90000

14.95000

15.00000

15.05000


[deg]

years

41


10.5

00.5

1

x 106

1

0.5

0

0.5

1

x 106

101

x 105

[km]


[km]

[km]


0 5 10 15 20 25 30410267.000

410268.000

410269.000

410270.000

410271.000

410272.000

410273.000


[km]

years

42

0 5 10 15 20 25 300.21936500

0.21937000

0.21937500

0.21938000

0.21938500

0.21939000

0.21939500


[]

years

0 5 10 15 20 25 3020.2699988

20.2699990

20.2699992

20.2699994

20.2699996

20.2699998

20.2700000


[deg]

years

43

0 5 10 15 20 25 3014.98000

15.00000

15.02000

15.04000

15.06000

15.08000

15.10000

15.12000


[deg]

years

0 5 10 15 20 25 3014.930000

14.940000

14.950000

14.960000

14.970000

14.980000

14.990000

15.000000


[deg]

years

44


10.5

00.5

1

x 106

1

0.5

0

0.5

1

x 106

101

x 105

[km]


[km]

[km]


0 5 10 15 20 25 30410180

410190

410200

410210

410220

410230

410240

410250

410260


[km]

years

45

0 5 10 15 20 25 300.210000

0.220000

0.230000

0.240000

0.250000

0.260000

0.270000

0.280000


[]

years

0 5 10 15 20 25 30

20.4

20.6

20.8

21

21.2

21.4


[deg]

years

46

0 5 10 15 20 25 300

5

10


[deg]

years

0 5 10 15 20 25 3014.9

15

15.1

15.2

15.3

15.4

15.5


[deg]

years

47

4 Considerations about numerical and generalissues encountered

To perform integration of Keplers equation with perturbation term, it has beenopted for the Matlab R ode113 algorithm, a variable step Adams method, muchmore efficient and precise with respect to the usual Runge-Kutta method inode45. Relative and absolute tolerances has been set to 1010, while maintain-ing Matlab R s default 104 value was not enough neither to grant convergenceof the method.Computational times of course strictly depends on the used machine. In ourcase, for an Intel R Core 2 Duo dual core processor clocked at 2.4 GHz (note thatMatlab R resorts only on a single core processing capability) times requested toperform restricted two body problem orbit, velocities, times and a single caseof perturbation, or all together at the same time, are hereby listed:

final orbit period: around 15 seconds Earth revolution orbit period: around 70 seconds Saturn revolution orbit period: around 1 hour and 45 minutes

A minor note to users relying on a UNIX derived operative system and Matlab R:with version R2013a on both GNU/Linux Ubuntu 13.10 and Mac OS X 10.9,encoding problems for 3D plots using standard OpenGL graphic libraries raised,making Matlab Rto crash. In order to avoid this problem, one should resort tothe implemented zbuffer graphical encoding.

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5 Block diagram

Find the date of departure (May 7, 2013)

Change of inclination and ascending node

anomaly

Departure hyperbola

First maneuver

First intersection (May 11, 2013)

Second intersection

Find intersections

Change of pericentre anomaly

Arrival at apocentre of orbit (May 15, 2013)

Final orbit around Saturn (May 19, 2013)

Perturbation analysis

One period (3 days)

Third bodies effects

Non uniformity of Saturn mass distribution

Solar wind pressure

All perturbations considered

Initial orbit around Titan (May 3, 2013)

Hohmann maneuver

Earth revolution orbit period (365 days)

Saturn revolution orbit period (29.7 Earth years)

Solar wind pressure Solar wind pressure

Third bodies effects Third bodies effects





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References

[1] Bate, Mueller, White (1971), Fundamentals of astrodynamics, Dover Publi-cations Inc., New York.

[2] Curtis (2005), Orbital mechanics for engineering students, Elsevier, Oxford.

[3] Wikipedia website.

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Preliminary mission study: orbital transfer from Titan to Saturn

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