1. The document describes modeling the temporary satellite capture (TSC) of comets by Jupiter using the restricted three-body problem framework of the Sun-Jupiter-comet system. 2. It characterizes the solution space using dynamical systems concepts like stable and unstable manifolds to understand the geometry of phase space and identify periodic and quasi-periodic solutions. 3. Symmetries found both analytically and numerically provide insight into the dynamics and are applied to model the trajectories of observed TSC comets.

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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 Meeting
Clearwater, Florida
January 23-26, 2000
School of Aeronautics and Astronautics

2. Capture of Comets by Η
10
p y
Helin-Roman-Crockett
0
5
[108
km]
Sun
Temporary Satellite Capture
(1966-1985)
centric
lFrame
1.0
-10
-5
y[1
-10 -5 0 5 10
Helioc
Inertial
2019)
0.5
1.0
m]
4
6
8
]
-10 -5 0 5 10
x [10
8
km]
ric
ame
(1943-2
-0.5
0.0
y[108
km]
L1
L2
-4
-2
0
2
y[10
8
km]
oviocentr
nodicFra
-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 [10
8
km]
Jo
Syn
x [10
8
km]

3. Modeling TSC
• Modeling Approach• Modeling Approach
– Initially use CR3BP
Th 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

4. 3D Regions of Exclusion
Sun-Jupiter System: Zero Velocity Surface for C (Jacobi Constant) = 3.0058
School of Aeronautics and Astronautics

5. Periodic SolutionsPeriodic Solutions
0.4
Jupiter Sun-Jupiter
-0.2
0.0
0.2
y[10
8
km]
L1
L2
L1 and L2 Families
of 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 [10
8
km]
6E+07
8E+07
Jupiter
0.2
0.4
8
km]
Jupiter
2E+07
4E+07
Z[km]
-0.4
-0.2
0.0
z[10
8
k
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
L1
L2
-4E+07
-2E+07
0
L1
L2
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 [10
8
km]
-4E+07
-6E+07 -4E+07 -2E+07 0 2E+07 4E+07 6E+07
X [km]

6. Northern/Southern Symmetry
T
.)](),(),(),(),(),([formtheofsolutionseconda
existsthere)](),(),(),(),(),([solutioneveryFor
T
T
tztytxtztytx
tztytxtztytx


−−
0.5
1.0
Southern
Jupiter0.5
1.0
Jupiter
0.0
z[10
8
km]
L1 L2
Southern
Solution
0.0
y[10
8
km]
L1 L2
-1.0
-0.5
Northern
Solution
-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 [10
8
km]

7. Symmetry Due toSymmetry Due to
Time Invariance
T
orbit.on the],,,,,[statewith theassociated
manifoldtheofplaneabout theimagemirroraisorbit,
haloaon],,,,,[statewith theassociatedmanifold,The
T
T
zyxzyx
XZ
zyxzyx


−−−
unstable
stable
0.42
3
+Z
(x, -y, z) (x, y, z)
-0.2
0
0.2
y[108
km]
L1
L20
1
z[10
8
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 [10
8
km]
-2
-1
z
L2 Northern Halo
School of Aeronautics and Astronautics
-3 -2 -1 0 1 2 3
y [10
8
km]
-3

8. Numerical Near Symmetry (I)
0.2
0.4
m]
X1 0.2
0.4
m]
X1
-0.2
0.0
z[108
km
X2 -0.2
0.0
z[10
8
km
X2
-0.4
-0.4 -0.2 0.0 0.2 0.4
y [10
8
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 [10
8
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 [ ]
T
X 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≈ ≈ −

9. 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 L1
northern/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
0.0
y[10
8
km]
L1 L2
0.0
z[10
8
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 [10
8
km]
-1.0
-1.0 -0.5 0.0 0.5 1.0
x [108
km]

10. Ephemeris Model Solution
Periodic
Solutions
Quasi-Periodic
Solutions
PeriodicityPeriodicity
Assumption
DS
P i
Re-establish
C M h
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Perspective Comet Match

11. Numerically Integrated 3-D Trajectories
Dynamic Behavior of Oterma (OTR) Follows Stable (S) Manifold
Numerically Integrated 3 D Trajectories
3BP Ephemeris Model
Trajectory Associated with a Sun-Jupiter L1 Northern Quasi-Periodic Orbit.
8
x 10
7
8
x 10
7
FORBIDDEN REGION
CR3BP ONLY
2
4
6
]
OTR
L
km]
2
4
6
]
CR3BP ONLY
OTR
km]
−4
−2
0
Y[km]
L
1S
L
1
L
2Jupiter
Z[k
−4
−2
0
Y[km]
L
1S
L
1
L2
Jupiter
Y[k
−8 −6 −4 −2 0 2 4 6 8
x 10
7
−8
−6
X [km]
X [km]
−8 −6 −4 −2 0 2 4 6 8
x 10
7
−8
−6
X [km]
FORBIDDEN REGION
CR3BP ONLY
X [km]
School of Aeronautics and Astronautics
TSC Trajectory of Comet Oterma
X [km]X [km]

12. 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 Trajectories
3BP Ephemeris Model
Stable (S) and Unstable (U) Manifold Trajectories Associated
with a Sun-Jupiter L1 Southern Quasi-Periodic Orbit.
x 10
7
FORBIDDEN REGION
x 10
7
2
4
6
HRC
L
1S
FORBIDDEN REGION
(CR3BP ONLY)
L
1U
2
4
6
HRC L1S
−2
0
L
1
L
2
Y[km]Y[km]
−2
0
L1
L2
Z[km]
Z[km]
−6 −4 −2 0 2 4 6
x 10
7
−6
−4
FORBIDDEN REGION
(CR3BP ONLY)
X [km]
−6 −4 −2 0 2 4 6
x 10
7
−6
−4
X [km]
School of Aeronautics and Astronautics
TSC Trajectory of Comet Helin-Roman-Crockett
x 10
7X [km]
X [km] x 10
7X [km]
X [km]

13. 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
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