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A preliminary feasibility and costs study about an orbital transfer between Titan and Saturn.
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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
t2 = 4 days 11 hours 39minutes 0.8 seconds
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
t3 = 4 days 10 hours 3minutes 0.7 seconds
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,
t4 = 4 days 3 hours 13minutes 0.6 seconds
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]
Titans orbitFinal orbitPerturbed orbit
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
0.219378eccentricity variation
[]
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
20.270010inclination variation
[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
15.00010pericentre anomaly variation
[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
15.00003ascending node anomaly variation
[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]
Titans orbitFinal orbitPerturbed orbit
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
410268.004semimajor axis variation
[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
0.21936881eccentricity variation
[]
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
20.2700002inclination variation
[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
15.00004pericentre anomaly variation
[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
15.000000ascending node anomaly variation
[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]
Titans orbitFinal orbitPerturbed orbit
20
0 0.5 1 1.5 2 2.5 3 3.5410263
410264
410265
410266
410267
410268
410269semimajor axis variation
[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
0.219388eccentricity variation
[]
days
21
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
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
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
15.0003ascending node anomaly variation
[deg]
days
3.2 Earth revolution orbit period
Revolution year orbit period has been approximated to Gregorian year of 365mean solar days.
3.2.1 Solar wind pressure
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
23
0 50 100 150 200 250 300 350 400410263
410264
410265
410266
410267
410268
410269
410270semimajor axis variation
[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
0.223000eccentricity variation
[]
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
20.305inclination variation
[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
15.025ascending node anomaly variation
[deg]
days
3.2.2 Influence of other bodies
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]
Titans orbitFinal orbitPerturbed orbit
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
410268.8semimajor axis variation
[km]
days
0 50 100 150 200 250 300 350 4000.219200
0.219400
0.219600
0.219800
0.220000
0.220200
0.220400eccentricity variation
[]
days
27
0 50 100 150 200 250 300 350 40020.267500
20.268000
20.268500
20.269000
20.269500
20.270000
20.270500
20.271000inclination variation
[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
15.00500pericentre anomaly variation
[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
15.00100ascending node anomaly variation
[deg]
days
3.2.3 Non uniformity of Saturn mass distribution
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]
Titans orbitFinal orbitPerturbed orbit
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
410268.160semimajor axis variation
[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
0.21936960eccentricity variation
[]
days
30
0 50 100 150 200 250 300 350 40020.2699988
20.2699990
20.2699992
20.2699994
20.2699996
20.2699998
20.2700000
20.2700002inclination variation
[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
15.00400pericentre anomaly variation
[deg]
days
31
0 50 100 150 200 250 300 350 40014.997500
14.998000
14.998500
14.999000
14.999500
15.000000ascending node anomaly variation
[deg]
days
3.2.4 All perturbations
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]
Titans orbitFinal orbitPerturbed orbit
32
0 50 100 150 200 250 300 350 400410263
410264
410265
410266
410267
410268
410269semimajor axis variation
[km]
days
0 50 100 150 200 250 300 350 4000.219000
0.219500
0.220000
0.220500
0.221000
0.221500
0.222000
0.222500eccentricity variation
[]
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
20.305inclination variation
[deg]
days
0 50 100 150 200 250 300 350 40014.4
14.5
14.6
14.7
14.8
14.9
15pericentre anomaly variation
[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
15.035ascending node anomaly variation
[deg]
days
35
3.3 Saturn revolution orbit period
Saturn revolution orbit period has been approximated to 29.7 earth years.
3.3.1 Solar wind pressure
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 5 10 15 20 25 30410263
410264
410265
410266
410267
410268
410269
410270semimajor axis variation
[km]
years
36
0 5 10 15 20 25 300.200000
0.220000
0.240000
0.260000
0.280000
0.300000
0.320000eccentricity variation
[]
years
0 5 10 15 20 25 30
20.4
20.6
20.8
21
21.2
21.4
21.6inclination variation
[deg]
years
37
0 5 10 15 20 25 302
4
6
8
10
12
14
16pericentre anomaly variation
[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
15.45ascending node anomaly variation
[deg]
years
38
3.3.2 Influence of other bodies
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]
Titans orbitFinal orbitPerturbed orbit
0 5 10 15 20 25 30410267.0
410267.2
410267.4
410267.6
410267.8
410268.0
410268.2
410268.4
410268.6
410268.8semimajor axis variation
[km]
years
39
0 5 10 15 20 25 300.215000
0.220000
0.225000
0.230000
0.235000
0.240000
0.245000
0.250000eccentricity variation
[]
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
20.280000inclination variation
[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
15.20000pericentre anomaly variation
[deg]
years
0 5 10 15 20 25 3014.75000
14.80000
14.85000
14.90000
14.95000
15.00000
15.05000
15.10000ascending node anomaly variation
[deg]
years
41
3.3.3 Non uniformity of Saturn mass distribution
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]
Titans orbitFinal orbitPerturbed orbit
0 5 10 15 20 25 30410267.000
410268.000
410269.000
410270.000
410271.000
410272.000
410273.000
410274.000semimajor axis variation
[km]
years
42
0 5 10 15 20 25 300.21936500
0.21937000
0.21937500
0.21938000
0.21938500
0.21939000
0.21939500
0.21940000eccentricity variation
[]
years
0 5 10 15 20 25 3020.2699988
20.2699990
20.2699992
20.2699994
20.2699996
20.2699998
20.2700000
20.2700002inclination variation
[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
15.14000pericentre anomaly variation
[deg]
years
0 5 10 15 20 25 3014.930000
14.940000
14.950000
14.960000
14.970000
14.980000
14.990000
15.000000
15.010000ascending node anomaly variation
[deg]
years
44
3.3.4 All perturbations
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]
Titans orbitFinal orbitPerturbed orbit
0 5 10 15 20 25 30410180
410190
410200
410210
410220
410230
410240
410250
410260
410270semimajor axis variation
[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
0.290000eccentricity variation
[]
years
0 5 10 15 20 25 30
20.4
20.6
20.8
21
21.2
21.4
21.6inclination variation
[deg]
years
46
0 5 10 15 20 25 300
5
10
15pericentre anomaly variation
[deg]
years
0 5 10 15 20 25 3014.9
15
15.1
15.2
15.3
15.4
15.5
15.6ascending node anomaly variation
[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.
48
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
Non uniformity of Saturn mass distribution
Non uniformity of Saturn mass distribution
All perturbations considered
All perturbations considered
49
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.
50