A Dynamical Model—A Dynamical Model—Co-orbit Co-orbit Restricted ProblemRestricted Problem,and its ,and its Application in Astronomy and Application in Astronomy and

Astronautics Astronautics

Zhaohua Yi 1,2, Guangyu Li 1

Gerhard Heinzel 3, Oliver Jennrich 4

1. Purple Mountain Observatory ,CAS, Nanjing

2. Nanjing University

3. Max Planck Institute for Gravitational Physics

4. European Space Research and Technology Center

July 14-16, 2006 BeijingZhaohua Yi ： A Dynamical Model—A Dynamical Model—Co-orbit Restricted ProblemCo-orbit Restricted Problem,,

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Astronomical BackgroundAstronomical Background

Asteroid Family; pointed out by Japanese astronomer Hirayama in 1940s.It is composed by some asteroids almost located in same orbit. In 1980s,Liesk and Williams(JPL) pointed out that there were more than 70 asteroid families in main band of minor planet.

July 14-16, 2006 BeijingZhaohua Yi ： A Dynamical Model—A Dynamical Model—Co-orbit Restricted ProblemCo-orbit Restricted Problem,,

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AstronomicalAstronomical Background Background

Trojians and Greeks group in Jupiter’s orbit.Co-orbit phenomena in KBO (Kuiper Band Objects).Co-orbit satellites around major planets.

Arms in Galaxy.

July 14-16, 2006 BeijingZhaohua Yi ： A Dynamical Model—A Dynamical Model—Co-orbit Restricted ProblemCo-orbit Restricted Problem,,

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Astronautical BackgroundAstronautical Background Co-orbit satellite constellations of Earth.LISA ( Laser Interferometer Space Antenna), 3 spacecraft formed as an equilateral triangle whose center of

mass locates on earth’s orbit and moves on same orbit with

earth.

July 14-16, 2006 BeijingZhaohua Yi ： A Dynamical Model—A Dynamical Model—Co-orbit Restricted ProblemCo-orbit Restricted Problem,,

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Restricted ProblemRestricted ProblemClassical restricted problem Classical restricted problem

1 Restricted 3-body problem Two large bodies moves on an orbit of 2-body problem (circle, ellipse, parabola, hyperbola), to study a massless body’s motion under the gravitation of 2 large bodies.

2 Problem of two fixed centers To study a body’s motion under the gravitation of two fixed bodies.

3 Hill’s problem.4 Fatou’s problem.

July 14-16, 2006 BeijingZhaohua Yi ： A Dynamical Model—A Dynamical Model—Co-orbit Restricted ProblemCo-orbit Restricted Problem,,

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Modern restricted problemModern restricted problem

In 1983,USA astronomer V.Szybehely pointed out restricted N+K problem. There are N large bodies and K small bodies;To study any large body’s motion only consider the gravitation of all other large bodies,and to study any small body’s motion must consider the gravitation of all large bodies ;the gravitation of other small bodies may be considered according to the real situation.

To study the orbit of LISA may be looking as a 2+3 co-orbit restricted problem.

July 14-16, 2006 BeijingZhaohua Yi ： A Dynamical Model—A Dynamical Model—Co-orbit Restricted ProblemCo-orbit Restricted Problem,,

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Co-orbit circular restricted Co-orbit circular restricted 3-body problem3-body problem

Let the earth (plus moon) E moves around the sun S on a circular orbit; and the orbital plane denotes xy-plane with x-axis located from origin S to E. This is a rotating coordinate system. At the origin of time, C, the center of mass of 3 spacecraft locates on earth’s orbit, and co-moves with earth. The motion of C is a planar problem. The equations of motion are:

0

2

0

0

1

11

33

2 x

y

nrr

nep

ep

rrr

July 14-16, 2006 BeijingZhaohua Yi ： A Dynamical Model—A Dynamical Model—Co-orbit Restricted ProblemCo-orbit Restricted Problem,,

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Jacobi integralJacobi integral

C22r

222222

2222

1

1

2

1

zyxzyxzyxn

July 14-16, 2006 BeijingZhaohua Yi ： A Dynamical Model—A Dynamical Model—Co-orbit Restricted ProblemCo-orbit Restricted Problem,,

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Transforming to polar coordinate, Transforming to polar coordinate,

nrdt

dr

nrrr

2

2 2

10

1cos2

11cos

1

2

1

2

21

220

rrrrn

rrn

0.00000304

es

e

mm

m

July 14-16, 2006 BeijingZhaohua Yi ： A Dynamical Model—A Dynamical Model—Co-orbit Restricted ProblemCo-orbit Restricted Problem,,

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The undisturbing solution is co-orbitalThe undisturbing solution is co-orbital

02

2

22

nrdt

dr

nnrr

)constantarbitrary ( , 1 0 r

July 14-16, 2006 BeijingZhaohua Yi ： A Dynamical Model—A Dynamical Model—Co-orbit Restricted ProblemCo-orbit Restricted Problem,,

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The approximating disturbed The approximating disturbed solution:solution:

hnr 2

)cos(1

2

BntA

nhr

413222

256

9

2

5

2

3ntkkntkntk

ktnktnkntr 2

1442

122

32768

9

64

312

July 14-16, 2006 BeijingZhaohua Yi ： A Dynamical Model—A Dynamical Model—Co-orbit Restricted ProblemCo-orbit Restricted Problem,,

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The approximating disturbed The approximating disturbed solution:solution:

2sin

2sin4

1

2sin2

2cos

00

00

k

2sin

2cos1

cos803

02

01

k

July 14-16, 2006 BeijingZhaohua Yi ： A Dynamical Model—A Dynamical Model—Co-orbit Restricted ProblemCo-orbit Restricted Problem,,

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The comparative result with prThe comparative result with précised numerical integration écised numerical integration

July 14-16, 2006 BeijingZhaohua Yi ： A Dynamical Model—A Dynamical Model—Co-orbit Restricted ProblemCo-orbit Restricted Problem,,

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The comparative result with précised numThe comparative result with précised numerical integration erical integration

0 2 4 6 8 10

0.0

0.1

0.2

0.3

0.4

Numerical Integrating formula

r-r 0

(106 km

)

Mission days (year)

r0= 1.0 AU

θ0

-20 degree＝

July 14-16, 2006 BeijingZhaohua Yi ： A Dynamical Model—A Dynamical Model—Co-orbit Restricted ProblemCo-orbit Restricted Problem,,

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The configuration of 3 spacecraft The configuration of 3 spacecraft

At the origin of time, let Sc1,Sc2,Sc3 denote 3 spacecraft, and the original positions of them are shown in figure

July 14-16, 2006 BeijingZhaohua Yi ： A Dynamical Model—A Dynamical Model—Co-orbit Restricted ProblemCo-orbit Restricted Problem,,

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cos3

2

311

22 lle

1cos3

2

31

2

lle

sin)1(3

sine

li

)1(3

cos3cos

e

li

1010 ME

The orbiter elements of them can The orbiter elements of them can be calculated as: be calculated as:

July 14-16, 2006 BeijingZhaohua Yi ： A Dynamical Model—A Dynamical Model—Co-orbit Restricted ProblemCo-orbit Restricted Problem,,

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Orbital elements of 3 spacecraftOrbital elements of 3 spacecraft

  a M0

SC1 1 270 270 180

SC2 1 31.17427

SC3 1 146.90162

July 14-16, 2006 BeijingZhaohua Yi ： A Dynamical Model—A Dynamical Model—Co-orbit Restricted ProblemCo-orbit Restricted Problem,,

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At first, the variation of armlengths be discuAt first, the variation of armlengths be discussed in 2-body problem. The positon vectorssed in 2-body problem. The positon vectors of them are: s of them are:

QPr kkkkkEeeE sin1)(cos 2

QPr kkkkkkkfrfr sincos

i

i

i

k

kkkk

kkkk

k

sinsin

cossincoscossin

cossinsincoscos

P

i

i

i

k

kkkk

kkkk

k

sincos

coscoscossinsin

coscossinsincos

Q

July 14-16, 2006 BeijingZhaohua Yi ： A Dynamical Model—A Dynamical Model—Co-orbit Restricted ProblemCo-orbit Restricted Problem,,

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The square of distance of Sc1 and Sc2: The square of distance of Sc1 and Sc2:

)60(cos313sin)60(sin3 1222

122 MeiM

The approximated development to 2 degrees of The approximated development to 2 degrees of ee and sin and sin ii are: are:

2

212 rr

July 14-16, 2006 BeijingZhaohua Yi ： A Dynamical Model—A Dynamical Model—Co-orbit Restricted ProblemCo-orbit Restricted Problem,,

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The optimal inclination angle The optimal inclination angle can be calculated as can be calculated as

602sin3sin3 122

1

2

MeiM

46445.60

July 14-16, 2006 BeijingZhaohua Yi ： A Dynamical Model—A Dynamical Model—Co-orbit Restricted ProblemCo-orbit Restricted Problem,,

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The next work The next work

• To discuss more precise solution;• To study it on elliptic restricted 3-b

ody problem;• To study stability of co-orbit soluti

on;• To explore the application to astron

omical and other astronautical problems

July 14-16, 2006 BeijingZhaohua Yi ： A Dynamical Model—A Dynamical Model—Co-orbit Restricted ProblemCo-orbit Restricted Problem,,

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THANK YOUTHANK YOU ！！