A CLOSE APPROACH BETWEEN A PLANET AND A PARTICLE: SUN-JUPITER SYSTEM
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Recent Researches in Power Systems and Systems Science
A CLOSE APPROACH BETWEEN A PLANET AND A PARTICLE:
SUN-JUPITER SYSTEM
JORGE KENNETY and ANTONIO F B A PRADO
National Institute for Space Research -INPE
Technology Faculty-FATEC-SJC
Av dos Astronautas 1758, SJC-SP
BRAZIL
jkennety@yahoo.com.br
prado@dem.inpe.br
Abstract: - This paper presents an investigation of a close approach between a planet and a particle. It
is assumed that the dynamical system is given by two main bodies that are in circular orbits around
their center of mass and the particle that is moving under the gravitational attraction of the two
primaries. This method has been under study for a long time by several authors, where the dynamical
system given by the “patched-conics” is used and the motion is assumed to be planar. A series of two-
body problems is used to generate analytical equations that describe the problem. Two solutions are
considered for the Swing-By (clock-wise and counter-clock-wise orbit), to take into account the
possibility that the particles crosses the line Sun-Planet between the primaries. The goal is to study the
orbital change (energy, angular momentum, orbital elements) of the particle after some maneuvers
with the planet desired and to know those values after the close approach. In particular, we are looking
for geometries that allows multiple Swing-Bys without correction maneuvers. Finally, numerical
simulations are performed for the Sun-Jupiter system.
Key-Words: - Swing-By, orbital maneuver, astrodynamics, celestial mechanics, orbital dynamics.
1 Introduction Swing-By trajectories around Jupiter [3]; the design
In aerospace engineer, the spacecraft trajectories can of missions using Swing-Bys [4]; optimization of
be controlled by thrusters and several other physical multiple swing-bys around the Moon [5]; the
forces. The determination of these trajectories in the numerical study of the Swing-By in three
solar system, considering gravitational effects, is dimensions [6], the study of a close approach
performed by several techniques. This paper will considering a planet and a cloud of particles [7]; a
use the Swing-By maneuver (gravity-assist) to classification of trajectories making a Swing-By
analyze missions involving celestial bodies and with the Moon [8]; the study of transfer orbits using
spacecrafts (particles) or celestial bodies and a cloud those close approaches to gain energy [9][10][11].
of particles. The maneuver uses a close approach This technique can be combined with gravitational
with a celestial body to modify the energy, angular capture (se references [12] and [13] for more
momentum and velocity of the spacecraft with details) to generate economical trajectories to the
respect to the Sun. The dynamical system given by Moon.
the “patched-conics” is used and the motion is It is known that, when in the neighborhood of a
assumed to be planar. An important example of planet, a spacecraft in a orbit around the Sun
gravitational assist occurred in December 1973, experiences perturbations which depend on the
with the encounter of the Pioneer 10 spacecraft with relative velocity between the spacecraft and the
the planet Jupiter. The description of this encounter planet and the distance separating the two at the
are shown in the detailed ephemerides prepared by point of the closest approach. If only the
NASA’s Jet Propulsion Laboratory and Ames gravitational field of the planet affected the motion
Research Center [1]. In this problem, solar gravity of the spacecraft, the vehicle would make it is
may be ignored, assuming that its effect will not approach along a hyperbolic path.
modify the results [2]. The literature shows several In this paper the maneuver is assumed to be
applications of the Swing-By technique: the study of performed in the Sun-Jupiter System. The motion
ISBN: 978-1-61804-041-1 32Recent Researches in Power Systems and Systems Science
of the spacecraft near the close encounter with the is modified. It is like having a series of single
planet will be studied. The spacecraft leaves the impulses with zero-cost to modify the orbit of the
point A, passes by the point P (Fig. 1) and goes to spacecraft. Fig. 1 explains the geometry involved in
the point B. Theses points are chosen in a such way the close approach.
that the influence of the Sun at those two points can
be neglected and, consequently, the energy can be
assumed to remain constant after B and before A.
Thus, a series of two-body problems is used to
generate analytical equations that describe the
problem. In particular, the energy and the angular
momentum of the spacecraft before and after this
close encounter are calculated, to detect the changes
in the trajectory during the close approach.
Finally, some numerical simulations are
performed with several initial conditions. Two
Fig. 1- Swing-By variables
solutions are considered for the Swing-By: when the
identify one Swing-By trajectory,: (velocity of
maneuver is performed behind the planet (solution Based in Fig.1 a set of variables can be used to
M3 with respect to M1),
(velocity of M3
1) and when the maneuver is performed in front the
planet (solution 2), to take into account the
the referential frame), (velocity of M3
possibility that the particles crosses the line Sun- with respect to M2, before and after the maneuver in
Planet between the primaries. The goal is to study
the orbital change (energy, angular momentum, with respect to M1, before and after the maneuver in
orbital elements) of the particle after some referential frame), δ (half angle of the curvature), rap
maneuvers with the desired planet and to know (the distance from the spacecraft to the center of M2
those values after the close approach in order to at the closest approach) and ψ (angle of approach).
decrease the fuel expense in space missions. In The velocity and orbital elements of M3 are
particular, we are looking for geometries that allows changed when it has a close approach with M2. The
multiple Swing-Bys without corrections maneuvers, orbital elements and energy before the encounter
where it is possible to study the lunar or planetary with the planet are obtained from the equations
environment by having a spacecraft travelling in
different areas without fuel consumption.
(1)
2 Mathematical model (2)
The patched-conic approximation offers an efficient
method for describing interplanetary orbits. By
(3)
partitioning the overall orbit into a series of two-
body orbits, it greatly simplifies mission analysis. (4)
The baseline of the patched conic approximation is
E=energy, =Gms=1,33.1011km3/s2.
that, in any space domain, the trajectory of a where a =semi-major axis, e = eccentricity,
spacecraft is determined by the gravitational field
that dominates the motion. The patched conic theory It is possible to determine the velocity of the
assumes that the dynamical system is given by two particle with respect to the Sun in the moment of the
main bodies that are in circular orbits around their crossing with the planet’s orbit and the true anomaly
center of mass and the particle that is moving under
of that point.
the gravitational attraction of the two primaries. So,
!
in this approach, this problem can be studied (5)
assuming a system formed by three bodies: the Sun
" #$%& ' !(
as the main massive primary (M1), a planet as
secondary mass (M2), that is orbiting the M1 body, (6)
and a particle with infinitesimal mass (M3) that
remains orbiting the primary and makes a close
approach with M2. When M3 enters in the sphere of where the parameter rsp is the distance between the
influence of M2, the orbital motion of M3 around M1 Sun and the planet. Eq. (6) given us two solutions
(θA and θB). In this study we will consider the angle
ISBN: 978-1-61804-041-1 33Recent Researches in Power Systems and Systems Science
θA. The next procedure is to calculate the angle momentum, the orbits are classified by: elliptic
between the inertial velocity of the particle and the direct (negative energy and positive angular
velocity of the planet: momentum), elliptic retrograde (negative energy
%- "
and angular momentum), hyperbolic direct (positive
) *& +, ./
#$% "
energy and angular momentum) and hyperbolic
(7)
retrograde (positive energy and negative angular
momentum).
and the magnitude of the particle velocity with Finally, to obtain the semi-major axis and the
respect to the planet in the moment that the eccentricity after the Swing-By, it is possible to use
approach starts, the equations
0 #$% )
(14)
(8)
@
(15)
This paper considers two solutions assuming a close
approach behind the planet (rotation of the velocity
vector in the counter-clock-wise sense- ψ1) and a
close approach in front of planet (clock-wise sense- 3 Simulations and Numerical Results
ψ2) for the spacecraft around the Sun (Fig. 2). These In this study some simulations will be performed to
two values are obtained from analyze the orbital variation of the spacecraft
1& 23 4 5
subject to a close approach with Jupiter under the
1 673 4 5
“patched conics” model. It is assumed that the
(9)
spacecraft is in orbit around the Sun with a given
semi-major axis and eccentricity where the periapse
distance (rp) and the apoapse (ra) of the orbit are
assumed to be known. All the physical elements of
the planets can be seen in Table 1. The patched
conic model was implemented using the software
Fortran. The energy, angular momentum and orbital
elements have been analyzed for multiple Swing-
Bys without correction maneuvers. The simulations
are stopped when the energy (Eq.12) becomes
Fig. 2- Possible rotations of the velocity vector positive.
Table 1- Physical elements of the Planet
8
where
4 #$% & ' (
Average
distance
Equatorial Orbital µ=Gm
to the
(106
Planet Radius velocity
Sun
5 %-& 9 <
(10)
:;
(km) (km/s) km3/s2)
(106
8 km)
is the gravitational constant of the planet. The
Jupiter 71370 778 13,1 126,0
µsol =1,33 . 1011 km3/s2
next step is to determine the variations in energy
and angular momentum from the equations [4]:
= > > %- 5
All simulations are performed with the following
characteristics:
= %- 5 %- 1
(11)
=
(12) i) The close approach will be at the point A
=
(Fig. 1);
?
(13) ii) The Sun (or the other perturbations) does
not affect the motion of the particle;
where ω is the angular velocity between the
iii) The energy can be assumed to remain
primaries, δ is the angle of deflection and E-, E+ are
constant after B and before A;
the energy before and after maneuver. After
calculating the variations in energy and angular
ISBN: 978-1-61804-041-1 34Recent Researches in Power Systems and Systems Science
iv) The energy and angular momentum will be characteristics of the particle before the Swing-By.
analyzed after and before the maneuver for several It is visible a strong influence of the rp and ra in the
situations; whole mission. Some of these characteristics will be
maneuver behind the planet (considering 1& as the
v) Solution 1 will be performed with the first studied with detail in this paper.
angle of approach) and solution 2 considers the
front the planet (considering 1 as the angle of
situation where the first maneuver is performed in 300
rp=1 Rj
rp=3 Rj
250
rp=10 Rj
approach). 200
rp=30 Rj
The initial conditions can be seen in Table 2 and
Manuevers
150
3, and it is composed by: ra (apoapsis distance), rp
(periapsis distance), a (semi-major axis), e 100
(eccentricity), v (velocity), E (energy) and C 50
(angular momentum). They are obtained from the 0
initial orbit of the spacecraft around the Sun. The rap 1 2 3 4 5 6 7 8
is the distance of the close approach between the 8
rp x 10 (km)
particle and the planet. With the numerical
algorithm available, the given initial conditions are
varied in any desired range and the number of Fig. 3- Maneuvers performed vs. periapsis distance
maneuver and their respective effects in the close (rp), ra=109 km and angle approach ψ1.
approach in the orbit of the spacecraft are studied.
The numerical results will be obtained in the point 200
A (see Fig. 1) with rotation of the velocity vector in 180
counter-clock-wise sense (solution ψ1) and clock- 160
wise sense (solution ψ2). 140
120 rp=1 Rj
Maneuvers
rp=3 Rj
Table 2- Initial conditions: Sun-Jupiter System 100
rp=10 Rj
80 rp=30 Rj
ra rp rap a 60
v
Simulation (108 (108 (108 (108 e 40
(km/s) 20
km) km) km) km)
0
1ª 10 3,5 1,5 6,75 0,48 12,03 8 10 12 14 16 18 20
8
ra x 10 (km)
3.1 Sun-Jupiter System
In the results, the first column shows the results for Fig. 4- Maneuvers performed vs. apoapsis distance
the angle of approach ψ1 (solution 1) and the second (ra), rp=5,5.108 km and angle approach ψ1.
column shows the results for the angle of approach
ψ2 (solution 2). Figures 5 and 6 show the evolution of the amplitude
Fig. 3 shows the number of maneuvers as a of the energy, angular momentum, semi-major axis,
function of the periapsis distance (rp), considering eccentricity and velocity. The results show two
ra=109 km and the angle of approach (ψ1) in the types of maneuvers: the ones performed behind
elliptic initial orbit of the spacecraft around the Sun. Jupiter (solution 1) and the ones performed in front
In Fig. 4 the ra distance was fixed. To understand of Jupiter (solution 2). The energy variation for
better the importance of this parameter, we made these solutions is enough to increase the orbital
simulations using several values for rp and ra. After elements of the particle. The moment that the
performing a large number of simulations using particle escapes of the orbit can be easily seen in
different values for the periapsis and apoapsis Fig. 6, in the plot of semi-major axis vs maneuver.
distances, it is possible to study the effects of this The change of energy and angular momentum
parameter in the whole maneuver. Those results causes the existence of hyperbolic orbits (∆E > 0
(Fig. 3) allow us to analyze the increase in the and ∆C > 0) and the semi-major axis variation leads
number of maneuvers for the range 3.109 km ≤ rp ≤ to eccentricities that reaches a maximum value.
7.109 km. This range implies in regions with 0,1764
≤ e ≤ 0,5385 and 6,5.108 km ≤ a ≤ 8,5.108 km. This
information is essential to analyze the orbital
ISBN: 978-1-61804-041-1 35Recent Researches in Power Systems and Systems Science
0
-10
-70
In the solution 2 (Fig. 5), there is a minimal
-20
-75
amplitude variation of energy that leads to the
-30
-80
accomplishment of several maneuvers. This solution
E ( km / s )
-40
E ( km / s )
2
2
2
-50 -85
is important when we want the particle to remain for
2
-60
-70
-80
-90
a long time in an elliptic orbit. Fig. 7 shows the
-95
-90
-100
energy variation as a function of the angle of
-100
1 2 3 4 5 6
maneuver
7 8 9 10 11 12
5 10 15 20
maneuver
25 30 35 40 45 50
approach that leads to several maneuvers due to the
Energy vs maneuver for Energy vs maneuver for energy gain and energy loss involved. Near the
ψ1 . ψ2. values of 100 km2/s2 (for angle ψ1) and -97,5 km2/s2
10 9
1.4
x 10
9.8
x 10
(for angle ψ2) there are maneuvers with energy loss.
1.3
9.6
9.4
In the vicinity of -2 km2/s2 (ψ1) and -75 km2/s2 (ψ2),
1.2
9.2
there are maneuvers with energy gain. For those
C ( km2 / s )
C ( km /s )
1.1
cases, when ψ1 is around 405 degrees, there is the
2
9
1
0.9
8.8
8.6
existence of parabolic orbits, because the angular
0.8 8.4
momentum is positive and the eccentricity is equal
1 2 3 4 5 6
maneuver
7 8 9 10 11 12 8.2
0 5 10 15 20 25 30 35 40 45 50
to 1,0. An overview of the simulations for the
maneuver
Angular momentum vs Angular momentum vs
maximum values of the amplitude can be seen in
maneuver for ψ1. maneuver for ψ2. Table 3.
Fig. 5- Energy and angular momentum of the spacecraft
after the Swing-Bys for ra=109 km, rp=3,5.108 km, rap=1,5
Rj, considering solution 1 (ψ1) and solution 2 (ψ2). 0
-70
-75
-20
-80
11 8 -40
x 10 x 10
E ( km / s )
12
E ( km / s )
2
2
-85
2
-60
2
9
10
-90
-80
8.5
8 -95
-100
a ( km )
a ( km )
6 8 -100
-120
300 320 340 360 380 400 420
350 355 360 365 370
4 angle of approach ψ1 (deg)
7.5 angle of approach ψ2 (deg)
2
0
7
Fig. 7- Energy variation vs. angle of approach of the
2 4 6
maneuver
8 10 12
6.5
0 5 10 15 20
maneuver
25 30 35 40 45 50 spacecraft for several orbits after the Swing-Bys for
ra=109 km, rp=3,5. 108 km, rap=1,5 Rj.
Semi-major axis vs Semi-major axis vs
maneuver for ψ1. maneuver for ψ2.
1
0.95
0.485
0.48
Table 3- The maximum value of amplitude: energy,
0.9
0.85
0.475
momentum, semi-major axis, eccentricity and velocity
0.8 0.47 after the maneuver (Sun-Jupiter System).
eccentricity
eccentricity
0.75
0.7
0.465
Solution 1 (ψ1)
0.65 0.46
simulation ∆a ∆C
0.6
0.455
∆E ∆v
0.55
0.45 (108 ∆e 2 2 (109-
0.5
(km /s ) 2 (km/s)
0.45
1 2 3 4 5 6
maneuver
7 8 9 10 11 12
0.445
0 5 10 15 20
maneuver
25 30 35 40 45 50 km) km /s)
1ª 5,5 0,51 102 0,57 0,07
Eccentricity vs maneuver Eccentricity vs maneuver Solution 2 (ψ2)
for ψ1. for ψ2. 1ª 2,7 0,03 21 0,15 0,020
11.758
11.76
11.756
11.75
11.754 The second simulation was performed considering
11.74 11.752
the distance of close approach rap = 30 Rj (radius of
V ( km / s )
V ( km / s)
11.75
11.73
11.72
11.748
11.746
Jupiter). These results can be seen in Fig. 8, where it
11.71
11.744 shows that there are a strong dependence with rap
11.7
11.69
11.742
11.74
and the initial conditions. Also in Fig. 8, for solution
2 4 6 8 10 12 0 10 20 30 40 50
maneuver maneuver
1, it is visible the large orbital change after the first
Velocity vs maneuver for Velocity vs maneuver for Swing-By maneuver.
ψ1 . ψ2 .
Fig. 6- Semi-major axis, eccentricity and velocity of the
spacecraft after the Swing-Bys for ra=109 km, rp=3,5. 108
km, rap=1,5 Rj, considering solution 1 (ψ1) and solution 2
(ψ2).
ISBN: 978-1-61804-041-1 36Recent Researches in Power Systems and Systems Science
0
-90
-95
• It is possible to find useful sequences of Swing-
-10
-20
-100
-105
Bys to study the space around a specific body.
-30 -110
References:
E ( km / s )
E ( km / s )
2
2
-40 -115
[1] Allen, J. A. V. Gravitational assist in celestial
2
2
-50 -120
-125
-60
-70
-130
-135
mechanics—a tutorial, American Association of
-80
-90
-140
-145
Physics Teachers.Vol. 5, pp.448-4515, May 2003.
-100
20 40 60 80 100
maneuver
120 140 160 180 200
-150
0 10 20 30 40 50
maneuver
60 70 80 90 100
[2] Battin, Richard H. An introduction to the
Energy vs maneuver for Energy vs maneuver for mathematics and models of astrodynamics,
ψ1 . ψ2. American Institute of Aeronautics and Astronautics,
10 9
1.4
x 10
8.5
x 10
New York, New York, 1987.
1.3
8
[3] Broucke, R.A. & A.F.B.A. Prado (1993).
1.2
7.5
"Jupiter Swing-By Trajectories Passing Near the
C ( km /s )
C ( Km2 / s )
Earth", Advances in the Astronautical Sciences,
2
7
1.1
1
6.5
Vol. 82, No 2, pp. 1159-1176.
0.9 6
[4] Broucke, R. A. The Celestial mechanics of
0.8
0 50 100
maneuver
150 200
5.5
0 10 20 30 40 50
maneuver
60 70 80 90 100
gravity assist, In:AIAA/AAS Astrodynamics
Angular momentum VS Angular momentum VS Conference, Minneapolis, MN, August, 1988.
maneuver for ψ1. maneuver for ψ2. (AIAA paper 88-4220).
[5] Dunham, D. & S. Davis (1985). "Optimization
Fig. 8- Energy and angular momentum of the spacecraft of a Multiple Lunar- Swing by Trajectory
orbits after the Swing-Bys for ra=109 km, rp=3,5.108 km, Sequence," Journal of Astronautical Sciences, Vol.
rap=30 Rj, considering solution 1 (ψ1) and solution 2 (ψ2). 33, No. 3, pp. 275-288.
[6] Felipe, G. & A.F.B.A. Prado (2001). “Numerical
study of three-dimensional close approach”. SBA
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amplitudes, the strong influence of the angle of Maneuvers for a Cloud of Particles with Planets of
approach and the choice of the periapsis position. A the Solar System. WSEAS Transactions on applied
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the Swing-By was developed, based in the "patched- [8] Prado, A.F.B.A. & R.A. Broucke (1995). "A
conic" approximation. Some numerical simulations Classification of Swing-By Trajectories using The
are calculated for the Sun-Jupiter system, Moon". Applied Mechanics Reviews, Vol. 48, No.
considering several initial conditions. A set of 11, Part 2, November, pp. 138-142.
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By in two dimensions and to evaluate the variation orbits in the Earth-Moon system using a regularized
in the orbital elements of the orbit of a spacecraft model. Journal of Guidance, Control and
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compare two solutions to make an orbital maneuver, [10] Prado AFBA, Broucke R Transfer Orbits in
considering two angle of approaches: ψ1 and ψ2. The Restricted Problem, Journal of Guidance Control
results showed that: and Dynamics, v. 18, n. 3, p. 593-598, May-Jun.
• The periapsis of the close approach has a strong 1995
influence in the maneuver and it can be used to [11] Sukhanov AA, Prado AFBA, Constant
obtain lower fuel consumption; tangential low-thrust trajectories near an oblate
• The two solutions considered (ψ1 and ψ2), planet, Journal of Guidance Control and Dynamics,
depending on the geometry of the encounter, v. 24, n. 4, p. 723-731, Jul-Aug. 2001.
have different behavior; [12] Prado, AFBA, Numerical and analytical study
• Considering other cases, one has an increase in of the gravitational capture in the bicircular
energy in the majority of the cases, generating problem, Advances in Space Research, v. 36, n. 3, p.
hyperbolic orbits after the passage; 578-584, 2005.
• One of them has a decrease in energy due to the [13] Prado AFBA Numerical study and analytic
passage in most of the cases (generating only estimation of forces acting in ballistic gravitational
elliptic orbits); capture, Journal of Guidance Control and
• The Solution 1 (ψ1 ) has larger amplitudes than Dynamics, v. 25, n. 2, p. 368-375, Mar-Apr. 2002.
the Solution 2(ψ2);
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