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A.S. Perminov, E.D. Kuznetsov
Ural Federal University

EXPANSION OF THE HAMILTONIAN OF A PLANETARY SYSTEM
INTO THE POISSON SERIES IN ALL ELEMENTS

JOURNèES 2014, St. Petersburg 22­24 September


2

INTRODUCTION


INTRODUCTION

3

Jacobi coordinate system
z0 z2 z1 Q0, 0 x
0

Q2, 2 , 2


y0
Inertial coordinate system

G x
2

2

G

1

y2 Q1, 1 , 1





x

1

y1

body 0 has mass 0 , body has mass 0 ( = 1 ... 4), ­ small parameter

­ barycenters of subsystems ­ Jacobi radius vectors ­ Inertial radius vectors


INTRODUCTION

4

Converting between Inertial and Jacobi coordinates
From Inertial to Jacobi 1 0 = 0 +


From Jacobi to Inertial


=1


-1 1

0 = 0 -
=1


1

1 = - 0 - -1 -


=1

-1 = 0 + - -
-1

=1



= 1 + ­ sum of masses ­ small parameter
for Solar system = 0.001

Differences
between

- 0 = +
=1


-1

coordinates - = - +

=




INTRODUCTION

5

The Hamiltonian
= 0 + 1
0 = - 2
0 =1





1 =

0 =2
-1

1 1 - - 0
2

-1

-
=1 =1
2

-

Introduce next notations:

=
=1

, =

+ 2 + 2 .

Then: - 0 = + ,


- 0 =
2

and if
-1

1 =

0 0

2 get

2 =
=2

0 2 + +

-
=1 =1

-

Disturbing function


INTRODUCTION

6

Second system of Poincare elements
Poincare orbital elements:
· = · = + + · 1 =

2 1 - 1 -

2

cos( + )
2

eccentric elements

· 1 = - 2 1 - · 2 =

1 -

sin( + )

1 , 1 ~ e

2 1 - 2 1 - cos cos
2

oblique elements

· 2 = - 2 1 -

1 - cos sin

2 ,

2~

i

where a, e, i, , , l are Kepler orbital elements,

2 is gravitational parameter, is normalized mass


INTRODUCTION

7

Expansion of the Hamiltonian
· The Hamiltonian can be written in the Poisson series in the following form:

= 0 +
· 0 ­ undisturbed Hamiltonian · ­ numerical coefficients





· = 1 1 1,1 2 1,1 3 2,1 4 2,1 5 ... 4 16 1,4 17 1,4 18 2,4 19 2,4 · = 1 1 + 2 2 + 3 3 + 4 4

20

· Computer algebra system "Piranha"

[1]

was used for this.

[1]

Biscani F. The Piranha algebraic manipulator. ­ 2009. ­ p.24. arXiv: 0907.2076v1.


8

ALGORITHM


ALGORITHM

9

Expansion of general part of disturbing function
Expansion of -
-1

-1

- = - +
=


1 2 -2

1 -

=

1 -

1+

2 -

+ -
2

2



=

1 = - 3 +

2

1 3 2 - + + 2 3 2 5


= - , = -

, =



2


ALGORITHM

10

Expansion of general part of disturbing function
Expansion of
2 = -
2

1

= -

-1

= 2 + 2 - 2 cos
1 -2

11 = 1 + 2 - 2 cos is

=

1



cos
=0

a generating function of Legendre polynomials
· =


· cos ­ Legendre polynomial of degree n · ­ angle between vectors and


ALGORITHM

11

Expansion of second part of disturbing function
1 =3 + 2 + 2 + 1+ 2 + 2 + 1+ 1+ 2 2
2 3 2 5 2 2

=

=

3

+

1 2 3

+

+
=

2 3 2 5

+

3 5 2 7 2

+...

=



,



· For expansion of scalar products and radius vectors we used classical

expansions of Celestial mechanics for



,



,



,



,



.


ALGORITHM

12

Basic expansions of



,



,




ALGORITHM

13

Basic expansions of



,




14

RESULTS


RESULTS

15

Precision of disturbing function approximation
· Expansion was made to 1st degree of small parameter and to 6th degree of eccentric and

oblique Poincare elements. · For example, precision of the approximation was calculated for Solar system and 47 Uma star system Kepler orbital elements:

Jupiter

Saturn 9.537 0.0539

Uranus 19.189 0.0473

Neptune 30.070 0.0086

47 Uma b 2.10 0.032 0.10 3.76 5.83 0.89

47 Uma c 3.60 0.098 0.10 5.36 5.14 1.70

47 Uma d 11.6 0.16 0.10 1.48 1.92 1.92

a, AU e

5.203 0.0484

i, rad
, rad , rad l, rad

0.0228
1.754 4.801 0.328

0.0434
1.983 5.865 5.591

0.0135
1.291 1.685 2.495

0.0309
2.300 4.636 4.673

Kepler elements for Solar system w.r.t the epoch J2000.0 and correspond to mean ecliptic

red items from www.exoplanet.eu black items have arbitrary values


RESULTS

16

Precision of disturbing function approximation
Solar system Major part of disturbing function:

i

j

accurate formula

series expansion

relative differences

12

6.247339·10-

2

6.247214·10-

2

2·10 1·10 2·10 7·10 1·10 1·10 2·10

-5 -5 -6 -6 -6 -6 -5

13
14 23

2.117572·10

-3
3 4

2.117594·101.598990·105.715304·10-

3
3 4

1.598993·105.715344·10-

24
34

4.429494·10

-4
4 2

4.429499·101.953599·10-

4
4 2

1.953597·10-

6.932077·10-

6.932205·10-


RESULTS

17

Precision of disturbing function approximation
Solar system Second part of disturbing function:

i

accurate formula

series expansion

relative differences

1 2 3

1.583793·109.514375·106.555984·101.593963·10-

2 5 6 2

1.583794·109.514331·106.555954·101.593964·10-

2 5 6 2

4·10 5·10 5·10 3·10

-7 -6 -6 -7

Whole disturbing function:
accurate formula series expansion relative differences

8.526040·10-

2

8.526169·10-

2

2·10-

5


RESULTS

18

Precision of disturbing function approximation
47 Uma star system
i j accurate formula

Major part of disturbing function:

series expansion

relative differences

12

0.265913

0.265903

4·10 5·10 2·10 4·10

-5 -5 -5 -5

13
23

0.310104
0.084960 0.660976

0.310087
0.084985 0.660949


RESULTS

19

Precision of disturbing function approximation
47 Uma star system
i accurate formula

Second part of disturbing function:

series expansion

relative differences

1 2

0.0496716 0.0254878 0.0751594

0.0496752 0.0254881 0.0751633

7·10 1·10 5·10

-5 -5 -5

Whole disturbing function:
accurate formula series expansion relative differences

0.6367925

0.6367620

5·10-

5


20

CONCLUSION


CONCLUSION

21

· We have got expansion of the Hamiltonian for planetary system with 4 · · ·

·

·
·

planets into the Poisson series in all elements. The expansion was made to 6th degree of orbital elements and to 1st degree of small parameter. Legendre polynomials were saved as symbols. Estimation accuracy of disturbing function (h1) expansion is presented. Relative difference between series estimation and accurate formula is about 10-4 ­ 10-5. The Hamiltonian is h0 + h1. So, Poisson series for the Hamiltonian was constructed with precision about 10-7 ­ 10-8. Now we are constructing the expansion for the Hamiltonian to 11th degree of orbital elements and 2nd degree of small parameter.


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