Temporary Satellite Capture Of Short-Period Jupiter Family Comets From The Perspective Of Dynamical Systems

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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 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


-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


3D Regions of ExclusionSun-Jupiter System: Zero Velocity Surface for C (Jacobi Constant) = 3.0058


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


-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


-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


-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


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


-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


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]


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]


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


Technology

Temporary Satellite Capture Of Short-Period Jupiter Family Comets From The Perspective Of Dynamical Systems