The dynamics around the collinear equilibrium points of the RTBP
Abstract
This paper is devoted to the analysis of an extended neighborhood of the collinear equilibrium points of the Restricted Three Body Problem. The analysis is done using numerical tools for the determination of periodic orbits and invariant 2D tori. All the relevant information of the neutrally stable behavior of the dynamics in the vicinity of the three libration points is given. The results in this paper extend those given in \cite{jm-rtbp}, since they avoid the convergence restrictions of the semi--analytical approach used in this reference.
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-1.6 -1.58 -1.56 -1.54 -1.52 -1.5
Ro a ion numbe
Ene gy
L1
1
2
3
α
β
γ
>100
<100
<50
<25
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-1.6 -1.58 -1.56 -1.54 -1.52 -1.5 -1.48
Ro a ion numbe
Ene gy
L2
1
2
3
α
β
γ
>100
<100
<50
<25
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
Ro a ion numbe
Ene gy
L3
1
2
3
α
β
γ
>100
<100
<50
<25
? D=
9 - &
!
%
%&
&&
&&
+
$ . $ "
$ ' +
' . "
5)
1 G "
0
0)1 0%1
.
01 " 9
!! #( & " ".
+ 6 6
" 6
?
" 6
"
%<D <"
)$<$$
6 0 ? &1 ?
"
%D(&(
%&D($ 0
? $1
DD 6
" 6 9
09 F
1 "
6 6 0 &&1 + ? <
6
6 + F"
&
0 D1
;)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-1.6 -1.58 -1.56 -1.54 -1.52 -1.5 -1.48
Ro a ion numbe
Ene gy
L1
<25
<100
>100
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-1.58 -1.56 -1.54 -1.52 -1.5 -1.48
Ro a ion numbe
Ene gy
L2
<25 <100 >100
0
0.02
0.04
0.06
0.08
0.1
0.12
-1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6
Ro a ion numbe
Ene gy
L3
<100 >100
? <=
9 - &
: +
5%
6 ? D 6#
? D
6 +
"
&" 6 01 <!
# F
6 6" H 6
6# " 6
9
" )
%"
5"
) 0 1 F 9 =5 =
0 = "
;&" 1
# 6 "
" 6 F "
#
F $ &
0
0.5
1
1.5
2
2.5
3
3.5
-1.53 -1.52 -1.51 -1.5 -1.49 -1.48
Ro a ion numbe
Ene gy
L1
0
0.5
1
1.5
2
2.5
3
3.5
-1.53 -1.52 -1.51 -1.5 -1.49 -1.48
Ro a ion numbe
Ene gy
L2
? $=
9 - &
;
.22 ! .22
0
0.5
1
1.5
2
2.5
3
3.5
-1.52 -1.51 -1.5 -1.49 -1.48 -1.47
Ro a ion numbe
Ene gy
L1
0
0.5
1
1.5
2
2.5
3
3.5
-1.52 -1.51 -1.5 -1.49 -1.48
Ro a ion numbe
Ene gy
L2
? &=
9 - &
!
+ .5
N " ? 6
5(
L1, pe iod duplica ion
-1 -0.95 -0.9 -0.85 -0.8 -0.4-0.200.20.4
-0.2
-0.1
0
0.1
0.2
0.3
L1, pe iod iplica ion
-1 -0.95 -0.9 -0.85-0.8 -0.75-0.4-0.200.20.4
-0.2
-0.1
0
0.1
0.2
0.3
L2, pe iod duplica ion
-1.15 -1.1 -1.05 -1 -0.95-0.2
0
0.2
-0.1
0
0.1
0.2
0.3
L2, pe iod iplica ion
-1.2-1.15 -1.1 -1.05 -1 -0.95-0.4-0.200.20.4
-0.2
-0.1
0
0.1
0.2
0.3
? =
! & !
;
%&
; &
%)&<
; &
%$ D%
!
;
%&D
;&
5%((5
;&
55D)<
+ 6 69
0%!" !1" 666 06
1 E
;&"
&" :
? " 5 ) 6
"
" + F 6
#
6
;&"
&
-
"
" 6
- B " 6
+ F
-
? " " 6
F
- . . " 6
5D
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
-0.855 -0.85 -0.845 -0.84 -0.835 -0.83 -0.825
Ene gy -1.59
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
-0.87 -0.86 -0.85 -0.84 -0.83 -0.82 -0.81
Ene gy -1.575
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
-0.925 -0.9 -0.875 -0.85 -0.825 -0.8 -0.775
Ene gy -1.537
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
-0.96 -0.93 -0.9 -0.87 -0.84 -0.81 -0.78
Ene gy -1.529
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-1 -0.96 -0.92 -0.88 -0.84 -0.8 -0.76
Ene gy -1.507
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-1 -0.95 -0.9 -0.85 -0.8 -0.75
Ene gy -1.4991
? =
;&
&
5<
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
-1.18 -1.17 -1.16 -1.15 -1.14 -1.13 -1.12
Ene gy -1.58
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
-1.2 -1.18 -1.16 -1.14 -1.12 -1.1 -1.08
Ene gy -1.565
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
-1.24 -1.2 -1.16 -1.12 -1.08 -1.04 -1
Ene gy -1.532
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-1.25 -1.2 -1.15 -1.1 -1.05 -1 -0.95
Ene gy -1.512
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-1.25 -1.2 -1.15 -1.1 -1.05 -1 -0.95
Ene gy -1.51
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-1.25 -1.2 -1.15 -1.1 -1.05 -1 -0.95
Ene gy -1.508
? 5=
;&
&
5$
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.85 0.9 0.95 1 1.05 1.1 1.15
Ene gy -1.50
-1.5
-1
-0.5
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Ene gy -1.20
-1.5
-1
-0.5
0
0.5
1
1.5
-0.5 0 0.5 1 1.5 2
Ene gy -1.06
? )=
;&
&
- 6 F
( F - 0F
51 E
" 6 F
+ " 6 "
6 "
E
6 " 6
6 " 6
2 4 6# #
" F % ( 6 F
6 -
" F" " " "
" -
6 9 6
# + "
6 9 F 6
# + "
G 6 " 6
<!
" 6 6
)&
0.11
0.115
0.12
0.125
0.13
-0.904 -0.9 -0.896 -0.892
L1, ene gy -1.529
0.08
0.1
0.12
0.14
0.16
0.18
-0.96 -0.95 -0.94 -0.93 -0.92 -0.91 -0.9
L1, ene gy -1.507
? %=
*+ + "" &
! %
M
;&
&
-0.22
-0.2
-0.18
-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
-1.1 -1.08 -1.06 -1.04 -1.02 -1 -0.98
L2, ene gy -1.512
-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
-1.05 -1.04 -1.03 -1.02 -1.01 -1 -0.99
L2, ene gy -1.51
? (=
*+ + "' &
; &
&
2 4 " " 6
6 F - "
6 6 6 6
+ ? 6
%&D ? ? " 6B 6
- "
" "
;&
F + 6 " 6 F "
6 6
6 : 9
?6 3 ? &"
"
;
)$<$ "
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;
%&& ? 6
%&&
)$<$ 6 " F
)