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doi: 10.3934/dcdsb.2022073
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Using normal forms to study Oterma's transition in the Planar RTBP

Gladston Duarte 1,2,, and  Àngel Jorba 1,

1. 

Departament de Matemàtiques i Informàtica, Universitat de Barcelona & Barcelona Graduate School of Mathematics, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
2. 

Faculty of Applied Mathematics, AGH University of Science and Technology, Aleja Adama Mickiewicza 30, 30-059, Kraków, Poland

* Corresponding author: Gladston Duarte

Received  September 2021 Revised  February 2022 Early access April 2022

Comet 39P/Oterma is known to make fast transitions between heliocentric orbits outside and inside the orbit of Jupiter. In this note the dynamics of Oterma is quantitatively studied via an explicit computation of high order Birkhoff normal forms at the points $ L_1 $ and $ L_2 $ of the Planar Restricted Three-Body Problem. A previous work [14] has shown the existence of heteroclinic connections between the neigbourhood of $ L_1 $ and $ L_2 $ which provide paths for this transition. Here we combine real data on the motion of Oterma with normal forms to compute the invariant objects that are responsible for this transition.

Keywords: Lagrangian points, Normal form computation, Comet 39P/Oterma.
Mathematics Subject Classification: Primary: 70K45, 37G05; Secondary: 70F07, 37N05.
Citation: Gladston Duarte, Àngel Jorba. Using normal forms to study Oterma's transition in the Planar RTBP. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022073
References:
[1]

A. Deprit, Canonical transformations depending on a small parameter, Celestial Mech., 1 (1969/1970), 12-30.  doi: 10.1007/BF01230629.

[2]

G. Duarte and À. Jorba, Invariant manifolds of tori near ${L}_1$ and ${L}_2$ in the Planar Elliptic RTBP, In preparation, 2022.

[3]

G. Duarte and À. Jorba, Modelling Oterma's transition using the Planar Elliptic RTBP, In preparation, 2022.

[4]

E. J. Doedel, R. C. Paenroth, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. E. Oldeman, B. Sandstede and X. Wang, Auto 2000: Continuation and bifurcation software for ordinary differential equations (with homcont), 1997.

[5]

L. Dieci and J. Rebaza, Point-to-periodic and periodic-to-periodic connections, BIT Numerical Mathematics, 44 (2004), 41-62.  doi: 10.1023/B:BITN.0000025093.38710.f6.

[6]

G. Gómez, J. Llibre, R. Martínez and C. Simó, Dynamics and Mission Design Near Libration Points. Vol. I, Fundamentals: The Case of Collinear Libration Points, World Scientific Monograph Series in Mathematics, 2. World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812810632_bmatter.

[7]

N. W. Harris and M. E. Bailey, Dynamical evolution of cometary asteroids, Mon. Not. R. Astron. Soc., 297 (1998), 1227-1236.  doi: 10.1046/j.1365-8711.1998.01683.x.

[8]

G. Hori, Theory of general perturbations with unspecified canonical variables, Publications of the Astronomical Society of Japan, 18 (1966), 287-296. 

[9]

À. Jorba, A methodology for the numerical computation of normal forms, centre manifolds and first integrals of {H}amiltonian systems, Exp. Math., 8 (1999), 155-195.  doi: 10.1080/10586458.1999.10504397.

[10]

À. Jorba and J. Masdemont, Dynamics in the centre manifold of the collinear points of the restricted three body problem,, Phys. D, 132 (1999), 189-213.  doi: 10.1016/S0167-2789(99)00042-1.

[11]

À. Jorba and B. Nicolás, Transport and invariant manifolds near ${L}_3$ in the Earth-Moon bicircular model, Commun. Nonlinear Sci. Numer. Simul., 89 (2020), 105327, 19 pp. doi: 10.1016/j.cnsns.2020.105327.

[12]

À. Jorba and M. Zou, A software package for the numerical integration of ODEs by means of high-order Taylor methods, Exp. Math., 14 (2005), 99-117.  doi: 10.1080/10586458.2005.10128904.

[13]

http://ssd.jpl.nasa.gov/horizons.html-.

[14]

W. S. Koon, M. W. Lo, J. E. Marsden and S. D. Ross, Resonance and capture of Jupiter comets, Celest. Mech. Dyn. Astron., 81 (2001), 27-38.  doi: 10.1023/A:1013398801813.

[15]

B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763-791.  doi: 10.1142/S0218127405012533.

[16]

K. R. Meyer and D. Offin, Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem, 3$^{rd}$ edition, Applied Mathematical Sciences, 90. Springer, Cham, 2017. doi: 10.1007/978-3-319-53691-0.

[17]

J. Moser, On the generalization of a theorem of A. Liapounoff, Comm. Pure Appl. Math., 11 (1958), 257-271.  doi: 10.1002/cpa.3160110208.

[18]

K. Ohtsuka, T. Ito, M. Yoshikawa, D. J. Asher and H. Arakida, Quasi-Hilda comet 147P/Kushida-Muramatsu - Another long temporary satellite capture by Jupiter, Astron. Astrophys., 489 (2008), 1355-1362. 

[19]

D. L. Richardson, A note on a Lagrangian formulation for motion about the collinear points, Celestial Mech., 22 (1980), 231-236.  doi: 10.1007/BF01229509.

[20]

A. Souza and M. Tao, Metastable transitions in inertial Langevin systems: What can be different from the overdamped case?, European J. Appl. Math., 30 (2019), 830-852.  doi: 10.1017/S0956792518000414.

[21] V. Szebehely, Theory of Orbits, Academic Press, 1967. 
[22]

M. Tao, Hyperbolic periodic orbits in nongradient systems and small-noise-induced metastable transitions, Phys. D: Nonlinear Phenomena, 363 (2018), 1-17.  doi: 10.1016/j.physd.2017.10.001.

show all references

References:
[1]

A. Deprit, Canonical transformations depending on a small parameter, Celestial Mech., 1 (1969/1970), 12-30.  doi: 10.1007/BF01230629.

[2]

G. Duarte and À. Jorba, Invariant manifolds of tori near ${L}_1$ and ${L}_2$ in the Planar Elliptic RTBP, In preparation, 2022.

[3]

G. Duarte and À. Jorba, Modelling Oterma's transition using the Planar Elliptic RTBP, In preparation, 2022.

[4]

E. J. Doedel, R. C. Paenroth, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. E. Oldeman, B. Sandstede and X. Wang, Auto 2000: Continuation and bifurcation software for ordinary differential equations (with homcont), 1997.

[5]

L. Dieci and J. Rebaza, Point-to-periodic and periodic-to-periodic connections, BIT Numerical Mathematics, 44 (2004), 41-62.  doi: 10.1023/B:BITN.0000025093.38710.f6.

[6]

G. Gómez, J. Llibre, R. Martínez and C. Simó, Dynamics and Mission Design Near Libration Points. Vol. I, Fundamentals: The Case of Collinear Libration Points, World Scientific Monograph Series in Mathematics, 2. World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812810632_bmatter.

[7]

N. W. Harris and M. E. Bailey, Dynamical evolution of cometary asteroids, Mon. Not. R. Astron. Soc., 297 (1998), 1227-1236.  doi: 10.1046/j.1365-8711.1998.01683.x.

[8]

G. Hori, Theory of general perturbations with unspecified canonical variables, Publications of the Astronomical Society of Japan, 18 (1966), 287-296. 

[9]

À. Jorba, A methodology for the numerical computation of normal forms, centre manifolds and first integrals of {H}amiltonian systems, Exp. Math., 8 (1999), 155-195.  doi: 10.1080/10586458.1999.10504397.

[10]

À. Jorba and J. Masdemont, Dynamics in the centre manifold of the collinear points of the restricted three body problem,, Phys. D, 132 (1999), 189-213.  doi: 10.1016/S0167-2789(99)00042-1.

[11]

À. Jorba and B. Nicolás, Transport and invariant manifolds near ${L}_3$ in the Earth-Moon bicircular model, Commun. Nonlinear Sci. Numer. Simul., 89 (2020), 105327, 19 pp. doi: 10.1016/j.cnsns.2020.105327.

[12]

À. Jorba and M. Zou, A software package for the numerical integration of ODEs by means of high-order Taylor methods, Exp. Math., 14 (2005), 99-117.  doi: 10.1080/10586458.2005.10128904.

[13]

http://ssd.jpl.nasa.gov/horizons.html-.

[14]

W. S. Koon, M. W. Lo, J. E. Marsden and S. D. Ross, Resonance and capture of Jupiter comets, Celest. Mech. Dyn. Astron., 81 (2001), 27-38.  doi: 10.1023/A:1013398801813.

[15]

B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763-791.  doi: 10.1142/S0218127405012533.

[16]

K. R. Meyer and D. Offin, Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem, 3$^{rd}$ edition, Applied Mathematical Sciences, 90. Springer, Cham, 2017. doi: 10.1007/978-3-319-53691-0.

[17]

J. Moser, On the generalization of a theorem of A. Liapounoff, Comm. Pure Appl. Math., 11 (1958), 257-271.  doi: 10.1002/cpa.3160110208.

[18]

K. Ohtsuka, T. Ito, M. Yoshikawa, D. J. Asher and H. Arakida, Quasi-Hilda comet 147P/Kushida-Muramatsu - Another long temporary satellite capture by Jupiter, Astron. Astrophys., 489 (2008), 1355-1362. 

[19]

D. L. Richardson, A note on a Lagrangian formulation for motion about the collinear points, Celestial Mech., 22 (1980), 231-236.  doi: 10.1007/BF01229509.

[20]

A. Souza and M. Tao, Metastable transitions in inertial Langevin systems: What can be different from the overdamped case?, European J. Appl. Math., 30 (2019), 830-852.  doi: 10.1017/S0956792518000414.

[21] V. Szebehely, Theory of Orbits, Academic Press, 1967. 
[22]

M. Tao, Hyperbolic periodic orbits in nongradient systems and small-noise-induced metastable transitions, Phys. D: Nonlinear Phenomena, 363 (2018), 1-17.  doi: 10.1016/j.physd.2017.10.001.

Figure 1.  Transition in Oterma's trajectory shown in sidereal coordinates (left) and in synodical ones (right), both in purple. On the left, Jupiter's orbit is plotted in green. On the right, Jupiter's and Sun's positions are plotted in red
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Figure 2.  The five equilibrium points of the RTBP
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Figure 3.  ZVC (in black) related to $ C = 3.035 $, equilibrium points (in purple) and periodic orbits around $ L_1 $ and $ L_2 $ (in blue) in the RTBP
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Figure 4.  Oterma's orbit using the data in Table 1 as initial conditions
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Figure 5.  Oterma's orbit in RTBP synodical $ (x,y) $ positions, near $ L_1 $ and $ L_2 $
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Figure 6.  Distance between Oterma and (a) $ L_1 $ and (b) $ L_2 $ in configuration space. The time (horizontal axis) is the physical one, measured in days, being January 1$ ^{\mathrm{st}} $ the day 0
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Figure 7.  $ x $ (purple) and $ y $ (green) components of the distances from Oterma and (a) $ L_1 $ and (b) $ L_2 $ points
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Table 1.  Oterma's positions and momenta used as initial conditions in the RTBP
$ x $ -1.0952439413131636$ \times 10^{0} $
$ y $ -2.9918455882452549$ \times 10^{-2} $
$ p_x $ -8.0698975794308611$ \times 10^{-1} $
$ p_y $ -1.0358302646539692$ \times 10^{0} $
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$ x $ -1.0952439413131636$ \times 10^{0} $
$ y $ -2.9918455882452549$ \times 10^{-2} $
$ p_x $ -8.0698975794308611$ \times 10^{-1} $
$ p_y $ -1.0358302646539692$ \times 10^{0} $
Table 2.  $ I_1 $, $ I_2 $ and $ h_{30} $ computed at $ L_1 $
$ t $ $ I_1=q_1p_1 $ $ I_2=(q_2^2+p_2^2)/2 $ $ h_{30} $
-23498.68 0.23735585604354004 0.42878140057480135 1.127$ \times 10^{-20} $
-23498.675 0.23735585604354054 0.42878140057480163 1.127$ \times 10^{-20} $
-23498.67 0.23735585604354045 0.42878140057480196 1.127$ \times 10^{-20} $
-23498.665 0.23735585604354040 0.42878140057480207 1.127$ \times 10^{-20} $
-23498.66 0.23735585604354045 0.42878140057480213 1.127$ \times 10^{-20} $
-23498.655 0.23735585604354040 0.42878140057480219 1.127$ \times 10^{-20} $
-23498.65 0.23735585604354040 0.42878140057480213 1.127$ \times 10^{-20} $
-23498.645 0.23735585604354062 0.42878140057480230 1.127$ \times 10^{-20} $
-23498.64 0.23735585604354009 0.42878140057480185 1.127$ \times 10^{-20} $
-23498.635 0.23735585604354056 0.42878140057480202 1.127$ \times 10^{-20} $
-23498.63 0.23735585604354006 0.42878140057480268 1.127$ \times 10^{-20} $
-23498.625 0.23735585604353993 0.42878140057480224 1.127$ \times 10^{-20} $
-23498.62 0.23735585604354126 0.42878140057480219 1.127$ \times 10^{-20} $
-23498.615 0.23735585604354120 0.42878140057480119 1.127$ \times 10^{-20} $
-23498.61 0.23735585604354051 0.42878140057480241 1.127$ \times 10^{-20} $
-23498.605 0.23735585604354129 0.42878140057480252 1.127$ \times 10^{-20} $
-23498.6 0.23735585604353923 0.42878140057480257 1.127$ \times 10^{-20} $
-23498.595 0.23735585604354037 0.42878140057480124 1.127$ \times 10^{-20} $
-23498.59 0.23735585604354004 0.42878140057480252 1.127$ \times 10^{-20} $
-23498.585 0.23735585604354048 0.42878140057480130 1.127$ \times 10^{-20} $
-23498.58 0.23735585604354170 0.42878140057480124 1.127$ \times 10^{-20} $
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$ t $ $ I_1=q_1p_1 $ $ I_2=(q_2^2+p_2^2)/2 $ $ h_{30} $
-23498.68 0.23735585604354004 0.42878140057480135 1.127$ \times 10^{-20} $
-23498.675 0.23735585604354054 0.42878140057480163 1.127$ \times 10^{-20} $
-23498.67 0.23735585604354045 0.42878140057480196 1.127$ \times 10^{-20} $
-23498.665 0.23735585604354040 0.42878140057480207 1.127$ \times 10^{-20} $
-23498.66 0.23735585604354045 0.42878140057480213 1.127$ \times 10^{-20} $
-23498.655 0.23735585604354040 0.42878140057480219 1.127$ \times 10^{-20} $
-23498.65 0.23735585604354040 0.42878140057480213 1.127$ \times 10^{-20} $
-23498.645 0.23735585604354062 0.42878140057480230 1.127$ \times 10^{-20} $
-23498.64 0.23735585604354009 0.42878140057480185 1.127$ \times 10^{-20} $
-23498.635 0.23735585604354056 0.42878140057480202 1.127$ \times 10^{-20} $
-23498.63 0.23735585604354006 0.42878140057480268 1.127$ \times 10^{-20} $
-23498.625 0.23735585604353993 0.42878140057480224 1.127$ \times 10^{-20} $
-23498.62 0.23735585604354126 0.42878140057480219 1.127$ \times 10^{-20} $
-23498.615 0.23735585604354120 0.42878140057480119 1.127$ \times 10^{-20} $
-23498.61 0.23735585604354051 0.42878140057480241 1.127$ \times 10^{-20} $
-23498.605 0.23735585604354129 0.42878140057480252 1.127$ \times 10^{-20} $
-23498.6 0.23735585604353923 0.42878140057480257 1.127$ \times 10^{-20} $
-23498.595 0.23735585604354037 0.42878140057480124 1.127$ \times 10^{-20} $
-23498.59 0.23735585604354004 0.42878140057480252 1.127$ \times 10^{-20} $
-23498.585 0.23735585604354048 0.42878140057480130 1.127$ \times 10^{-20} $
-23498.58 0.23735585604354170 0.42878140057480124 1.127$ \times 10^{-20} $
Table 3.  $ I_1 $, $ I_2 $ and $ h_{30} $ computed at $ L_2 $
$ t $ $ I_1=q_1p_1 $ $ I_2=(q_2^2+p_2^2)/2 $ $ h_{30} $
-23499.67 0.19332991772006497 0.41263461076678953 3.045$ \times 10^{-22} $
-23499.665 0.19332991772006392 0.41263461076678948 3.045$ \times 10^{-22} $
-23499.66 0.19332991772006430 0.41263461076678759 3.045$ \times 10^{-22} $
-23499.655 0.19332991772006414 0.41263461076678970 3.045$ \times 10^{-22} $
-23499.65 0.19332991772006439 0.41263461076678870 3.045$ \times 10^{-22} $
-23499.645 0.19332991772006375 0.41263461076678903 3.045$ \times 10^{-22} $
-23499.64 0.19332991772006444 0.41263461076678820 3.045$ \times 10^{-22} $
-23499.635 0.19332991772006430 0.41263461076678898 3.045$ \times 10^{-22} $
-23499.63 0.19332991772006386 0.41263461076678859 3.045$ \times 10^{-22} $
-23499.625 0.19332991772006433 0.41263461076678837 3.045$ \times 10^{-22} $
-23499.62 0.19332991772006486 0.41263461076678953 3.045$ \times 10^{-22} $
-23499.615 0.19332991772006428 0.41263461076678931 3.045$ \times 10^{-22} $
-23499.61 0.19332991772006441 0.41263461076678837 3.045$ \times 10^{-22} $
-23499.605 0.19332991772006436 0.41263461076678865 3.045$ \times 10^{-22} $
-23499.60 0.19332991772006472 0.41263461076678876 3.045$ \times 10^{-22} $
-23499.595 0.19332991772006516 0.41263461076678876 3.045$ \times 10^{-22} $
-23499.59 0.19332991772006516 0.41263461076678898 3.045$ \times 10^{-22} $
-23499.585 0.19332991772006580 0.41263461076678898 3.045$ \times 10^{-22} $
-23499.58 0.19332991772006619 0.41263461076678942 3.045$ \times 10^{-22} $
-23499.575 0.19332991772006589 0.41263461076678914 3.045$ \times 10^{-22} $
-23499.57 0.19332991772006602 0.41263461076678926 3.045$ \times 10^{-22} $
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$ t $ $ I_1=q_1p_1 $ $ I_2=(q_2^2+p_2^2)/2 $ $ h_{30} $
-23499.67 0.19332991772006497 0.41263461076678953 3.045$ \times 10^{-22} $
-23499.665 0.19332991772006392 0.41263461076678948 3.045$ \times 10^{-22} $
-23499.66 0.19332991772006430 0.41263461076678759 3.045$ \times 10^{-22} $
-23499.655 0.19332991772006414 0.41263461076678970 3.045$ \times 10^{-22} $
-23499.65 0.19332991772006439 0.41263461076678870 3.045$ \times 10^{-22} $
-23499.645 0.19332991772006375 0.41263461076678903 3.045$ \times 10^{-22} $
-23499.64 0.19332991772006444 0.41263461076678820 3.045$ \times 10^{-22} $
-23499.635 0.19332991772006430 0.41263461076678898 3.045$ \times 10^{-22} $
-23499.63 0.19332991772006386 0.41263461076678859 3.045$ \times 10^{-22} $
-23499.625 0.19332991772006433 0.41263461076678837 3.045$ \times 10^{-22} $
-23499.62 0.19332991772006486 0.41263461076678953 3.045$ \times 10^{-22} $
-23499.615 0.19332991772006428 0.41263461076678931 3.045$ \times 10^{-22} $
-23499.61 0.19332991772006441 0.41263461076678837 3.045$ \times 10^{-22} $
-23499.605 0.19332991772006436 0.41263461076678865 3.045$ \times 10^{-22} $
-23499.60 0.19332991772006472 0.41263461076678876 3.045$ \times 10^{-22} $
-23499.595 0.19332991772006516 0.41263461076678876 3.045$ \times 10^{-22} $
-23499.59 0.19332991772006516 0.41263461076678898 3.045$ \times 10^{-22} $
-23499.585 0.19332991772006580 0.41263461076678898 3.045$ \times 10^{-22} $
-23499.58 0.19332991772006619 0.41263461076678942 3.045$ \times 10^{-22} $
-23499.575 0.19332991772006589 0.41263461076678914 3.045$ \times 10^{-22} $
-23499.57 0.19332991772006602 0.41263461076678926 3.045$ \times 10^{-22} $
Table 4.  Comparison between evaluations of the maximum degree monomials of different degrees. For the ones around $ L_1 $ the chosen point is the one with $ t = -23498.63 $, and for the ones around $ L_2 $ it is the one with $ t = -23499.62 $
$ L_1 $ $ L_2 $
Order 24 7.9760239699283809$ \times 10^{-10} $ 2.7765943768364664$ \times 10^{-10} $
Order 32 2.2427370346101517$ \times 10^{-12} $ 3.9594494889226486$ \times 10^{-13} $
Order 40 8.1359645807725896$ \times 10^{-15} $ 1.0198945467998282$ \times 10^{-15} $
Order 48 2.3883565010998806$ \times 10^{-17} $ 1.4307930312682510$ \times 10^{-18} $
Order 56 1.1230484307642622$ \times 10^{-19} $ 8.3245178575158536$ \times 10^{-21} $
Order 60 1.1272938164516016$ \times 10^{-20} $ 3.0458619961877403$ \times 10^{-22} $
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$ L_1 $ $ L_2 $
Order 24 7.9760239699283809$ \times 10^{-10} $ 2.7765943768364664$ \times 10^{-10} $
Order 32 2.2427370346101517$ \times 10^{-12} $ 3.9594494889226486$ \times 10^{-13} $
Order 40 8.1359645807725896$ \times 10^{-15} $ 1.0198945467998282$ \times 10^{-15} $
Order 48 2.3883565010998806$ \times 10^{-17} $ 1.4307930312682510$ \times 10^{-18} $
Order 56 1.1230484307642622$ \times 10^{-19} $ 8.3245178575158536$ \times 10^{-21} $
Order 60 1.1272938164516016$ \times 10^{-20} $ 3.0458619961877403$ \times 10^{-22} $
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