The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again.
TEMPORARY SATELLITE CAPTURE OF SHORT-PERIOD JUPITER FAMILY COMETSOF SHORT PERIOD JUPITER FAMILY COMETS
FROM THE PERSPECTIVE OF DYNAMICAL SYSTEMS
K.C. Howell, B.G. Marchand, M.W. Lo
AAS/AIAA Space Flight Mechanics MeetingClearwater, FloridaJanuary 23-26, 2000
School of Aeronautics and Astronautics
Capture of Comets by Η
10
p yHelin-Roman-Crockett
0
5
[10
8km
]
Sun
Temporary Satellite Capture (1966-1985)
cent
ric
l Fra
me
1.0
-10
-5
y[1
-10 -5 0 5 10
Hel
ioc
Iner
tial
2019
)
0.5
1.0
m]
4
6
8]
-10 -5 0 5 10
x [108 km]
ric
ame
(194
3-2
-0.5
0.0
y[1
08km
]
L1 L2
-4
-2
0
2
y[1
08km
]
ovio
cent
rno
dic
Fra
-1.0
-0.5
-1.0 -0.5 0.0 0.5 1.0
8
School of Aeronautics and Astronautics
-8
-6
-14 -12 -10 -8 -6 -4 -2 0 2
x [108 km]
Jo Syn x [108 km]
Modeling TSC• Modeling Approach• Modeling Approach
– Initially use CR3BPTh Di i l R i f E l i• Three-Dimensional Regions of Exclusion
• Types of Solutions Available (3D-Periodic)• Application of DS PerspectiveApplication of DS Perspective
– Compute Trajectories on Stable/Unstable Manifold – Numerical Analysis Insight into geometry of phase space
• Analytical Symmetry of Solutions• Numerically Observed Symmetries
School of Aeronautics and Astronautics
3D Regions of ExclusionSun-Jupiter System: Zero Velocity Surface for C (Jacobi Constant) = 3.0058
School of Aeronautics and Astronautics
Periodic SolutionsPeriodic Solutions0.4
Jupiter Sun-Jupiter
-0.2
0.0
0.2
y[1
08km
]
L1L2
L1 and L2 Familiesof Periodic Halo Orbits
-0.4
-0.2
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
x [108 km]
6E+07
8E+07
Jupiter
0.2
0.48
km]
Jupiter
2E+07
4E+07
Z[k
m]
-0.4
-0.2
0.0
z[1
08k
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
L1L2
-4E+07
-2E+07
0
L1L2
School of Aeronautics and Astronautics
-0.4-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
x [108 km]
-4E+07-6E+07 -4E+07 -2E+07 0 2E+07 4E+07 6E+07
X [km]
Northern/Southern SymmetryT
.)](),(),(),(),(),([ form theofsolution second aexiststhere)](),(),(),(),(),([solution every For
T
T
tztytxtztytxtztytxtztytx
−−
0.5
1.0
SouthernJupiter0.5
1.0
Jupiter
0.0
z[1
08km
]
L1 L2
SouthernSolution
0.0
y[1
08
km]
L1 L2
-1.0
-0.5NorthernSolution
-1.0
-0.5
School of Aeronautics and Astronautics
-1.0-1.0 -0.5 0.0 0.5 1.0
x [108 km]
-1.0-1.0 -0.5 0.0 0.5 1.0
x [108 km]
Symmetry Due toSymmetry Due to Time Invariance
T
orbit. on the ],,,,,[ state with theassociated manifold theof plane about the imagemirror a is orbit,
haloaon ],,,,,[ statewith theassociatedmanifold, The
T
T
zyxzyxXZ
zyxzyx
−−−unstable
stable
0.42
3
+Z
(x, -y, z) (x, y, z)
-0.2
0
0.2
y[1
08
km]
L1L20
1z
[108
km]
+Y
(x, -y, z) (x, y, z)
-0.4
-0.2
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
x [108 km]-2
-1
z
L2 Northern Halo
School of Aeronautics and Astronautics
-3 -2 -1 0 1 2 3
y [108 km]
-3
Numerical Near Symmetry (I)
0.2
0.4
m]
X1 0.2
0.4
m]
X1
-0.2
0.0
z[1
08km
X2 -0.2
0.0
z[1
08km
X2
-0.4-0.4 -0.2 0.0 0.2 0.4
y [108 km]
-0.4-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
x [108 km]
1 1 1 1 1 1 1
1
The manifold associated with a state [ ]on a northern L halo exhibits features that are similar to those
of the manifold associated with a state [ ]
TX x y z x y z
X x y z x y z
=
=
stable
unstable T
School of Aeronautics and Astronautics
2 2 2 2 2 2 2of the manifold associated with a state [ ]X x y z x y z=unstable
2 1 2 1 2on a southern L halo for and .y y z z≈ ≈ −
Numerical Near Symmetry (II)Numerical Near Symmetry (II)A numerical inverse (Z) near mirror (XY-plane) symmetry exists between
th t bl / t bl if ld l ti i t d ith Lthe stable/unstable manifold solutions associated with L1northern/southern halo orbits and the unstable/stable manifold solutions
associated with L2 southern/northern halo orbits.
0.5
1.0
Jupiter 0.5
1.0
Jupiter
0.0y
[108
km]
L1 L20.0
z[1
08
km]
L1 L2
-1.0
-0.5
-1.0
-0.5
School of Aeronautics and Astronautics
-1.0-1.0 -0.5 0.0 0.5 1.0
x [108 km]
-1.0-1.0 -0.5 0.0 0.5 1.0
x [108 km]
Ephemeris Model Solution
Periodic Solutions
Quasi-PeriodicSolutions
PeriodicityPeriodicity Assumption
DS P i
Re-establishC M h
School of Aeronautics and Astronautics
Perspective Comet Match
Numerically Integrated 3-D Trajectories
Dynamic Behavior of Oterma (OTR) Follows Stable (S) Manifold
Numerically Integrated 3 D Trajectories3BP Ephemeris Model
Trajectory Associated with a Sun-Jupiter L1 Northern Quasi-Periodic Orbit.
8x 10
7
8x 10
7
FORBIDDEN REGION CR3BP ONLY
2
4
6
]
OTR
Lkm]2
4
6
]
CR3BP ONLY
OTR
km]
−4
−2
0
Y [
km] L
1S
L1
L2Jupiter
Z [k
−4
−2
0
Y [
km]
L1S
L1
L2
Jupiter
Y [k
−8 −6 −4 −2 0 2 4 6 8
x 107
−8
−6
X [km]X [km]
−8 −6 −4 −2 0 2 4 6 8
x 107
−8
−6
X [km]
FORBIDDEN REGION CR3BP ONLY
X [km]
School of Aeronautics and Astronautics
TSC Trajectory of Comet Oterma
X [km]X [km]
Numerically Integrated 3-D Trajectories
Dynamic Behavior of Helin-Roman-Crockett (HRC) Follows St bl (S) d U t bl (U) M if ld T j t i A i t d
Numerically Integrated 3 D Trajectories3BP Ephemeris Model
Stable (S) and Unstable (U) Manifold Trajectories Associated with a Sun-Jupiter L1 Southern Quasi-Periodic Orbit.
x 107
FORBIDDEN REGION
x 107
2
4
6
HRCL
1S
FORBIDDEN REGION (CR3BP ONLY)
L1U
2
4
6
HRC L1S
−2
0L
1L
2Y [
km]
Y [k
m]
−2
0L
1L
2
Z [
km]
Z [k
m]
−6 −4 −2 0 2 4 6
x 107
−6
−4
FORBIDDEN REGION (CR3BP ONLY)
X [km]−6 −4 −2 0 2 4 6
x 107
−6
−4
X [km]
School of Aeronautics and Astronautics
TSC Trajectory of Comet Helin-Roman-Crockett
x 107X [km]X [km] x 10
7X [km]X [km]
SummarySummary
• Modeling TSC w/ R3BP (Sun-Jupiter-Comet)Modeling TSC w/ R3BP (Sun Jupiter Comet)• Characterizing the solution space of the CR3BP
– 3D Regions of Exclusiong– Types of Solutions– DS Stable/Unstable Manifold Flow
• Anal tical s mmetries
• Analytical symmetries• Numerical near symmetries
• Applications– Modeling of natural bodies– S/C Mission Design
School of Aeronautics and Astronautics