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.
Full text
½ ¾ ! ""#!" " $ % &#' () * +"# " , -.)/ 0. )+ ( ( 12 -.)/ 0. )+ + +. !" # $% &'" &(&'" &)&* + " , " - . " " / 01 234 0 )!" 5!1 " " 6 " 7" 8" 9 6 / " " " : " ; & & %&%<%" 8#/ + 6 6 6 6 6 6 6 6
" " 6 ," 6 = 0 " 1 - " 06 0 (!11 > 6 " 6 6 6 ?" ?" 6 " 6 ' 6 @/ 0 !1 " ,6" 6 6 " ) 0 1 A . 5 6 2 4 A " 6 8B7 0 1 6 >" : C 0 (!"D!1 ? " B 6 6 : A , " 6 E 0 %! !1 96 > " " " -BE 0 %!" D!" $!" %!1 + 9 " , F " 6 + 6 G " 6 6 B %! ! 6 " 6 " > 6 F " 6 ?6 5!" 6 F " 6 ? 6 + 6 6 " 6 7 5 6 ? 6 6 3 6 = " 6 B
B:B " = G 6 6 ; ;5 6 " 6 6 ;& & + 6 6 %! ! ? 6 " ) F 2E 4 . : F 6 0 " 1 " &" : " 6 '" F 6 ;&" 6 H F 6 ;& - A ; & F = & $ & D$ 6 " : - 0 5 1 F & <( - 6 0 55 1 96 - . ." + 6 F F " 6 96 0 51 0 1 0 551 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.98 -0.96 -0.94 -0.92 -0.9 -0.88 -0.86 -0.84 -0.82 -0.8 -0.78 -0.76 Energy -1.507 ? = ; & & ; %&D 5
6 96" 6 " " 6 9 6 " 6 & !" F 0 & &1 0 & &1" I " 6 B : 0 $!1 J K ; L J MK ; L J ; L 6 L; 0 M 1M M M 0 1 ; 0 1 M M " ; 0 M 1 M M : F " N " 6 F 0 1;L K K K F " 6 " " 6 0 M 5 &1" " 6 ; ;& + ; 5" 6 6 ¾ ½ ¿ ? " ¾ ½ 5 M 0 1 ¿ ; M 0 1 ; K " ; K M ; K " 6 , 6 , 0 1; 0 M M 1 M : K ; M K ; 0 1 0 M 1 K ; K ; K ; K ; 0 1 6 G 6 , 24" N " " ; 0 1 )
3 ," 6 6 ; M M 0 1M M 6 0 " 1 + ? 0 %!1 1 2 3 4 5 6 7 8 0 0.1 0.2 0.3 0.4 0.5 c2 µ L1 L2 L3 ? = 0 1 & & %! ? " " " 6 ; 0 1; M0 1 M0 M 1 ; "6 0 1 ; & ; $ < " " 6 & & 6 ; - 0 1 H6 , 0 1 B B H6 " 6 ; 0 1 = 0 1; M 0 1M 0 1 %
" 7 ; " F ; M ; " 6 9= = " + = " + ; 6 0 ;" 1 = + " ; + " " " ; + " " 6 # # + " # 0 1 3 0 " M 0 11" 6 ? " + 6 6" 6 6 + ; " ? " 6 # 6 6 6 9 F = " " 6 9 6 9 " 6 G 6 0 <!1" Æ G6 01 6 : (
' , " 0 1" 6 BF 0 1H6 0 1 0 1 ; 0 0 11 0 1 ; - ! 0 1 ; & 0 1 ! 0 1;& " ! 0 1" " F 6 0 1 ; & 0 1 ; & 0 1 ; & ;& Ü ½ 0 1 ; & 01 6 0 1 ; & ; ( M ( M 96= " > " F 0 1" ! 0 1;& + >6O " 0 1" 0 1 0 1 0 1 F " 0 1 : ! 0 Ü 0 11 &" 6 0 1; ! 0 Ü 0 11 Ü 0 1 ! 0 Ü 0 11 0 Ü 0 11 + ! 0 1;& " ! 0 Ü 0 11 6 7 01 + 6 F " 0 1 6 " 6 A 9 96 01 6 : " 6 6 ? 6" 2 964 6 6 9 96 F 6 96 01 6 6 0 1 7 = F = 6 96 " F = 6 F 96 " F 6 = 6 96 D
96" 6 >6O - # 0 1 ; &" F 6 # 0 1 $ ; # 0 1 051 ; $ F % # 0 1 G 9 " F" 051 #9 # " 6 69 # " P 3 07* )!1 G 7* 6 Æ" :" 6 " 6 6 ! " , " A 6 " 6 6 D! A 96 01 6 6 7 = ' 6 = 6 " ' 6 = 6 - # 0 1 ; & Q " 6 6" 6 7 # 0 1 ; &" " " @ # 0 1 " 6 6 = " G $ ; M Æ& 0)1 Æ " & @ # 0 1 & ; G F $ ;= $ >6O = # 0 $ 1 ' ; # 0 $ 1 0%1 $ ; $ ' G 9 6 " " F $ # 0 $ 1 $ $ <
" Æ " 9 F > " # 0 1 ; & " 0%1 # 9 G 0%1 6 8 " F 24 + 6 7*" 9 # 0 1 & B 0%1 6 0 1 96 " " 6 9 F" " " 6 " # 0 1 ; & F 9 ? 3 " 9 B 9 5! '" 0 1" 0 1 0 1 G 9 3 B ( = ; 0 ) ; ( 1" ( 0 * M + 1; 0 ( 0 * 11 * + 0(1 6 ;0 1 - " ; " * ;0 , 1 + " 9 3 6 " 6 9 3 ; 0 , ; , 1 " 6 ¾ " ¾ 0 ( 0 , 11 ; ( 0 , M 1 , 0D1 " 6 9 3 - = - 0 , M 1; Æ 0 - 0 , 11 , 0<1 6 Æ ; ; Æ > 6 9 6# 3 6 3 ; 0(1 6 " ( 0 , 1; ¼ ¾ Æ - 0 , 1 0$1 + " 3 - 0<1" 0$1 F 3 - 0 , 1 ½ 0(1" 6 6 $
! 0 , &1 ½ ? 0 1" 0 $1 0 <1" 6 6 9 ; 0 1 / ; Æ ; / ; 6 0 ( , M ( , 1 ; 0 2 ; 6 0 ( , ( , 1 / 2 ; & 0 1 051 + " ? Æ 9 Ü ¼ 0 0 , 11 ; 0 1M 6 0 , M , 1 , M 6 0 , , 1 , 6 ; 0 1 ( > , + 3 66 Æ +" F 051" 6 9 3 / 2 " 6 6 2 94 " 6 + " 96 Æ " 6 6 " 6 F 96 0 1 6 6 ! & ,(""- 6 9 24 " 9 Æ 0<1 G 6 9 ! 0& 1 " ½ 0 " 1" 6 6 ? Æ / / 2 ? ! 0& 1; ¾ 0 0 1 6 Æ ?? ! 0& 1 3 + " 6 9? Æ - ? Æ 0 ! 0& 11 Æ 9 F 01 Æ ; 0 M 0)1 " 6 0<1 0D1 0 6 1" 9 Æ 01" ; Æ ; 0 1 0 M 0%1 (
+ 6 6 G 6 " ' . " F " 6 5 6 & " 6 H6 " F H6 6 ) A 6 "6 Æ 6= G 6 6 % G % + " 6 5 G9 0 6 9 " Æ " ? Æ 31 + 06 % 1" 6 + " F ) G 9 0 D1 " F" 6 3 " 3 6 ? Æ % G Æ 9 >6 6 6 = 3 6 + )" 3 6 6 0 D1 + " 5 %" > 6 9 ' / & 0 : F 9= D
0 (1 : M 3 H6 : 0 1 F 6 : 0 (1 B F F 0 )1 F # 0 %1 0 D1 0 M 3 1 H6 0 1 3 6" + F 0 )1" >6O 0 M 0 M 3 1 ) 1 0 0 M 3 1 ) 1 B9 " 6 P 0)!1 + 0 %1" 0 M 0 M 3 1 ) 1 0 M 0 M 3 11" >6O 6 # 0 M 0 M 3 1 ) 1 0 M 3 1 9#F G P 6 )! 6 7* " P 6 '" 3 9 7 " 7* ) 3 M< 3 06 )!" 6 6 1" 6 P 6 ) 37 7 0 M 3 1M) 7 5 ? ; 3 ; 7 " ( % " 6 6 6 ; 3 ; && 06 1" 6 >6 ) 5 ) 8 >6 9 ( 5 + + + + %&& /,3 '! 1" . & 2 " (& - / ; 8 ) 6 ) 9 / ; ) G @ / # / ; 8 6 6 6 + 6 P 3 6 / )!" 6 6 9 / T; 7 7 & & 6 T ) ) " 7 7 0 ) 1 " 6 T ;0 1 9 8 ;0 . 1 " # / ; 8 0 )!" %$1 T ; = 7 ; 7 0(1 <
" 3 0)!1 6 " 6 @ / 6 6 : - # / ; 8 " 6 ; & 0(1" : : # @ / " 6 ; . ; & ; # 0(1" 6 # # 6 T ;T $ ; 6 ; ) 0 ) 1 6 : : # # # ; ;T 7 B #9" P 6 '' """ & ? " 6 + F " " 69 : + " 69 F ) % # 29 4 3 + B 6 ) & ) % Q " + 6" 6 0 ) % ) & 1 6 " 6 ) % ) & F " 2 4 6 9 69 */ # 6 ? " 6 P 6 6 -'@ !" Q8P?" Q8P?" A/P 7 " " 69 . G " 3 ) " = ? 0 (1 :" M 3 + " . 6 6 " 6 7-'@ " " $
3 P 6 6 A 7-'@ !" Q8P?" Q8P?" A/P 7 0 D1 3 . 0 M 3 1 F9" 6 !" -O " 6 G 6 - - " 6 " 8#/ " B %$) D B %<(&< B %&(&D = ;& & %&%<% ? " F ? " 01 " 6 7O ? 001 1" 001 1 + ? 5" 6 0 1 " 6 G 0 1" 6 0 1 + ? ) 6 6 " G 6 9 U )$%$& )<5%) & & %5D & ) 5$ & %%<)$ & 5& 5 & %)55& = ? - " "6 01" 01 01 7O F" 6 6 &
-10 -2 2 0 10 100 1000 1e4 -1.5 -1 -0.5 0 0.52.5 3 3.5 4 4.5 5 5.5 6 6.5 -1.4959 0.41391 Stability parameters Period Energy -10 -2 2 0 10 100 1000 1e4 -1.5 -1 -0.5 0 0.5 2.5 3 3.5 4 4.5 5 5.5 6 6.5 -1.48354 0.55849 Stability parameters Period Energy 1 1.5 2 2.5 3 3.5 -1.5 -1 -0.5 0 0.5 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7 -0.01537 0.32201 0.5433 Stability parameters Period Energy ? 5= ! " -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 L1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 L2 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 L3 ? )= ! # $ ! %
6 6 " 0 ?(1 ? 9 6 " 6 6 + ? % 6 -100 -10 -2 2 0 10 100 1000 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 -1.58718 -1.47464 Stability parameters Period Energy -2 2 0 10 100 1000 -1.6 -1.55 -1.5 -1.45 -1.4 3 4 5 6 7 8 9 10 -1.57606 -1.50688 -1.47786 -1.41765 Stability parameters Period Energy 0 0.5 1 1.5 2 2.5 3 3.5 -1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6 6.21 6.22 6.23 6.24 6.25 6.26 6.27 6.28 6.29 6.3 6.31 -1.21177 -0.89598 Stability parameters Period Energy ? %= & ! ' ,E 0 &!1" 6 5 A 6 ,E # 0 1 ; 1 5 ' ## " V VK ; 1 8 . V VK : 9 0 &!1= = 7 6 # 6 1 ; " ;&
-1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 L1 -1.5 -1 -0.5 0 0.5 1 1.5 -2 -1.5 -1 -0.5 0 0.5 1 L2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 L3 ? (= # $ ! % = 7 6 #" 6 1 ; " 8 ;& '= 7 6 # #" 6 1 ; ;& 1 ; 8 ;& + 5 6 ? &! ; & % 6 6 + " F # 5 6 ;& ? " 6- 6B U U 0 1 8 " 6 ;& " + ? D N " 6 6 ? < #$" " - 5 6 6 ;& 96 6 0 1 6 " " " 9 0 51 6 6 ;& 96 " 6 = 6 01 01 6B96 0&!1 ? $ " " 6 5
2 10 100 1000 -1.512 -1.51 -1.508 -1.506 -1.504 -1.502 -1.5 -1.498 -1.496 -1.494 3.94 3.96 3.98 4 4.02 4.04 4.06 4.08 Stability parameters Period Energy L1 1 2 3 4 2 10 100 1000 -1.51 -1.505 -1.5 -1.495 -1.49 -1.485 -1.48 4.3 4.32 4.34 4.36 4.38 4.4 4.42 4.44 Stability parameters Period Energy L2 1.95 2 2.05 2.1 2.15 2.2 2.25 2.3 -1 -0.8 -0.6 -0.4 -0.2 0 6.261 6.262 6.263 6.264 6.265 6.266 6.267 Stability parameters Period Energy L3 ? D= U 8 8 8 B %<D < B %D(&( B DD B % &D& B %&(<< B&$$%) 5B )D)() 'B )DD<( ' B&<$%$< )B ) D(% ' 5= )
Planar orbit -0.96-0.92-0.88-0.84-0.8 -0.2 -0.1 00.1 0.2 -0.2 -0.1 0 0.1 0.2 Orbit 1, lane 1 -0.96-0.92-0.88-0.84-0.8 -0.2 -0.1 00.1 0.2 -0.2 -0.1 0 0.1 0.2 Orbit 1, lane 2 -0.96-0.92-0.88-0.84-0.8 -0.2 -0.1 00.1 0.2 -0.2 -0.1 0 0.1 0.2 Orbit 2, lane 1 -0.96-0.92-0.88-0.84-0.8 -0.2 -0.1 00.1 0.2 -0.2 -0.1 0 0.1 0.2 Orbit 2, lane 2 -0.96-0.92-0.88-0.84-0.8 -0.2 -0.1 00.1 0.2 -0.2 -0.1 0 0.1 0.2 Vertical orbit -0.96-0.92-0.88-0.84-0.8 -0.2 -0.1 00.1 0.2 -0.2 -0.1 0 0.1 0.2 ? <= '( ) ) ! ! %
7 6 ? ( - 6 !! #( & (" + ? D 6 " " " # " # - 8 = 6 5 0 1 - . F 6 3 ; ¾ " 6 0 < 0 51 - 0 1 6 > 6 0 : " " 6 C 0 " 1 C 01" B 6 0 5 1" ;&" 6 + 6 - + " 6 0 # 1 6 " 6 : 6 6 G 6 A 9 F 6 9 F + " " 2 4 " 6 6 " 6 0 1 + " " " 6 6 6 3 6" 6 && - 9 N :" 6 ; & D( 5 'I + +++ %&&/,3" && 0 % 6 1 ? " 9 " 0 < 1 0 5 5
-0.85 -0.84 -0.83 x -0.04 -0.02 00.02 0.04 y -0.04 -0.02 0 0.02 0.04 z -0.85 -0.84 -0.83 x -0.04 -0.02 00.02 0.04 y -0.04 -0.02 0 0.02 0.04 z -0.85 -0.84 -0.83 x -0.04 -0.02 00.02 0.04 y -0.04 -0.02 0 0.02 0.04 z -0.85 -0.84 -0.83 x -0.04 -0.02 00.02 0.04 y -0.04 -0.02 0 0.02 0.04 z -0.85 -0.84 -0.83 x -0.04 -0.02 00.02 0.04 y -0.04 -0.02 0 0.02 0.04 z -0.85 -0.84 -0.83 x -0.04 -0.02 00.02 0.04 y -0.04 -0.02 0 0.02 0.04 z -0.85 -0.84 -0.83 x -0.04 -0.02 00.02 0.04 y -0.04 -0.02 0 0.02 0.04 z -0.85 -0.84 -0.83 x -0.04 -0.02 00.02 0.04 y -0.04 -0.02 0 0.02 0.04 z ? (= 8 "( / %$ / 55
0 0.1 0.2 0.3 0.4 0.5 0.6 -1.6 -1.58 -1.56 -1.54 -1.52 -1.5 Rotation number Energy L1 1 2 3 α β γ >100 <100 <50 <25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -1.6 -1.58 -1.56 -1.54 -1.52 -1.5 -1.48 Rotation number Energy L2 1 2 3 α β γ >100 <100 <50 <25 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 Rotation number Energy L3 1 2 3 α β γ >100 <100 <50 <25 ? D= 9 - & ! % %& && && + $ . $ " $ ' + ' . " 5)
1 G " 0 0)1 0%1 . 01 " 9 !! #( & " ". + 6 6 " 6 ? " 6 " %<D <" )$<$$ 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 Rotation number Energy 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 Rotation number Energy 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 Rotation number Energy 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 Rotation number Energy 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 Rotation number Energy L2 ? $= 9 - & ; .22 ! .22 0 0.5 1 1.5 2 2.5 3 3.5 -1.52 -1.51 -1.5 -1.49 -1.48 -1.47 Rotation number Energy L1 0 0.5 1 1.5 2 2.5 3 3.5 -1.52 -1.51 -1.5 -1.49 -1.48 Rotation number Energy L2 ? &= 9 - & ! + .5 N " ? 6 5(
L1, period duplication -1 -0.95 -0.9 -0.85 -0.8 -0.4-0.200.20.4 -0.2 -0.1 0 0.1 0.2 0.3 L1, period triplication -1 -0.95 -0.9 -0.85-0.8 -0.75-0.4-0.200.20.4 -0.2 -0.1 0 0.1 0.2 0.3 L2, period duplication -1.15 -1.1 -1.05 -1 -0.95-0.2 0 0.2 -0.1 0 0.1 0.2 0.3 L2, period triplication -1.2-1.15 -1.1 -1.05 -1 -0.95-0.4-0.200.20.4 -0.2 -0.1 0 0.1 0.2 0.3 ? = ! & ! ; %& ; & %)&< ; & %$ 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 Energy -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 Energy -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 Energy -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 Energy -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 Energy -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 Energy -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 Energy -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 Energy -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 Energy -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 Energy -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 Energy -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 Energy -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 Energy -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 Energy -1.20 -1.5 -1 -0.5 0 0.5 1 1.5 -0.5 0 0.5 1 1.5 2 Energy -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, energy -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, energy -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, energy -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, energy -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 ? &" " ; )$<$ " " 6 ; %&& ? 6 %&& )$<$ 6 " F )