arXiv:1410.3727v2 [astro-ph.EP] 2 Feb 2015 rpitsbitdt hsc etr A Letters Physics to submitted Preprint Lages) (J. fobtlmomentum. orbital of J Lages) (J. to mi- used as also such been phenomena physics has astro- atomic Alongside map and describe Kepler dynamics [13–15]. the celestial mat- galaxies physics in dark and application of SS its capture the chaotic by and trajecto- ter 9–12] comet of [7, pri- diffusion ries with chaotic resonances 8], motion [7, maries mean 6], [1, dy- chaotic namics beyond 1P/Halley [2–5], perihelion study radius orbital with to Jupiter used comets been parabolic has map nearly Kepler 1P/Halley the of Then case [1]. realistic dimensional for three constructed the numerically and [2] problem three restricted body dimensional two the the in of Ke- derived framework pre- analytically The area originally was time. dimensional map and two pler energy a involving is map which serving Kepler Such a 2] by [1, described map [1]. be chaos can trajectories dynamical chaotic by be- governed orbital its unpredictable with havior and contrasts irregular (SS) term system long Solar the in ances Introduction 1. o description reliable a gives map potential 10 Halley of dipole fun The gravitational kick barycenter. rotating system system a Solar Solar of the 1P/H and to to potential contribution transfer Keplerian energy planet the Each ie system. function kick Solar 1P/Halle system describing Solar map the symplectic pute two-dimensional the determine We Abstract h hr emrglrt f1/alyappear- 1P/Halley of regularity term short The ntttUIA,Osraor e cecsd ’nvr TH l’Univers de Sciences des Observatoire UTINAM, Institut URL: address: Email 4 rwieo agrsae h aaeeso h a r lwychang slowly are map the of parameters the scales larger a on while yr http://perso.utinam.cnrs.fr/ ypetcmpdsrpino alyscmtdynamics comet Halley’s of description map Symplectic [email protected] .Rli,P ag .Lages J. Haag, P. Rollin, G. ∼ lages Besan¸con, France o+00Jva er rudJ2000.0 around -1000 from years 1) Jovian Fig. in +1000 planets snapshots eight to (see the Sun by the comet constituted and Halley’s SS for the inte- equation orbiting numerical Newton’s exact an of from to gration extract function we the one kick comparing the SS robustness computed term semi-analytically we long Then its time. discuss sojourn an give 1P/Halley and the map of Halley estimate with symplectic the 1P/Halley of of help the il- dynamics We chaotic barycenter. the to SS due lustrate around term movement potential potential Sun Keplerian dipole the rotating a a into and split term kick be SS the can to function contribution planet each We that show [25]. passages and perihelion observed 1P/Halley previously computed from by [1] extracted analysis numerically Sat- Fourier already and were Jupiter which of urn functions particular kick in the and retrieve planet we functions major kick each the to exactly associated compute to (see 20–24]) major integral Melnikov eight [4, use We the SS. and the of Sun planets the dynamics account into 1P/Halley taking describing map symplectic [19]. Ryd- states molecular berg of autoionization chaotic and [16– 18], atoms hydrogen excited of ionization crowave nti okw eiaayial eemn the determine semi-analytically we work this In T,CR,Uiested Franche-Comt´e, Universit´e 25030 de CNRS, ETA, le ln n asg hog the through passage one along alley u oteSnmvmn around movement Sun the to due to per ob h u fa of sum the be to appears ction oe yaiso iescales time on dynamics comet f hoi yais ecom- We dynamics. chaotic y n u oso oscillations slow to due ing etme ,2021 3, September ie from eg about -10 000BC to about 14 000AD. Exact in- tegration over a greater time interval does not provide exact ephemerides since Halley’s comet dynamics is chaotic, see eg [6] where integration of the dynamics of SS constituted by the Sun, Jupiter and Saturn have been computed for 106 years.
2. Symplectic Halley map Orbital elements of the current osculating orbit Figure 1: Two examples of three dimensional view of Hal- of 1P/Halley are [26] ley’s comet trajectory. The left panel presents an ortho- graphic projection and the right panel presents an arbi- e ≃ 0.9671, q ≃ 0.586 au, trary point of view. The red trajectory shows three suc- i ≃ 162.3, Ω ≃ 58.42, cessive passages of Halley’s comet through SS, the other near circular elliptic trajectories are for the eight Solar sys- ω ≃ 111.3, T0 ≃ 2446467.4 JD tem planets, the yellow bright spot gives the Sun position. Along this trajectory (Fig. 1) the comet’s energy At this scale details of the Sun trajectory is not visible. per unit of mass is E0 = −1/2a =(e−1)/2q where a is the semi-major axis of the ellipse. In the fol- lowing we set the gravitational constant G = 1, the total mass of the Solar system (SS) equal to trajectory. The energy wn+1 of the osculating or- 1, and the semi-major axis of Jupiter’s trajectory bit after the nth pericenter passage is given by equal to 1. In such units we have q ≃ 0.1127, a ≃ 3.425 and E0 ≃ −0.146. Halley’s comet peri- center can be written as q = a (1 − e) ≃ ℓ2/2 wn+1 = wn + F (xn) −3/2 (1) where ℓ is the intensity per unit of mass of the xn+1 = xn + wn+1 comet angular momentum vector. Assuming that the latter changes sufficiently slowly in time we can consider the pericenter q as constant for many where F (xn) is the kick function, ie the energy comet’s passages through the SS. We have checked gained by the comet during the nth passage and by direct integration of Newton’s equations that depending on Jupiter phase xn when the comet this is actually the case (∆q ≃ 0.07) at least for is at pericenter. The second row in (1) is the a period of -1000 to +1000 Jovian years around third Kepler’s law giving the Jupiter’s phase at J2000.0. Consequently, Halley’s comet orbit can the (n + 1)th passage from the one at the nth be reasonably characterized by its semi-major axis passage and the energy of the (n+1)th osculating a or equivalently by Halley’s comet energy E. orbit. During each passage through the SS many body The set of equations (1) is a symplectic map interactions with the Sun and the planets mod- which captures in a simple manner the main fea- ify the comet’s energy. The successive changes in tures of Halley’s comet dynamics. This map has energy characterize Halley’s comet dynamics. already been used by Chirikov and Vecheslavov [1] Let us rescale the energy w = −2E such as now to study Halley’s comet dynamics from previously positive energies (w> 0) correspond to elliptic or- observed or computed perihelion passages from bits and negative energies (w < 0) to hyperbolic −1403BC to 1986AD[25]. In [1] Jupiter’s and Sat- orbits. Let us characterize the nth passage at the urn’s contributions to the kick function F (x) had pericenter by the phase xn = tn/TJ mod 1 where been extracted using Fourier analysis. In the next tn is the date of the passage and TJ is Jupiter’s or- section we propose to semi-analytically compute bital period considered as constant. Hence, x rep- the exact contributions of each of the eight SS resents an unique position of Jupiter on its own planets and the Sun. 2 3. Solar system kick function This change in energy depends on the phases (x = t/T mod 1) of the planets when the comet Let us assume a SS constituted by eight plan- i i passes through pericenter. From (4) we see that ets with masses {µi}i=1,...,8 and the Sun with mass 8 each planet contribution ∆Ei (xi) are decoupled 1 − µ = 1 − i=1 µi. The total mass of the from the others and can be computed separately. SS is set to 1 and µ ≪ 1. In the barycentric P The integral (3) is similar to the Melnikov in- reference frame we assume that the eight plan- tegral (see eg [4, 20–24]) which is usually used in ets have nearly circular elliptical trajectories with the vicinity of the separatrix to obtain the energy semi-major axis a . We rank the planets such as i change of the pendulum perturbed by a periodic a1 < a2 <...< a8 so a5 and µ5 are the orbit parametric term. In the case of the restricted 3- semi-major axis and the mass of Jupiter. The cor- body problem the Melnikov integral can be used responding mean planet velocities {v } are i i=1,...,8 to obtain the energy change of the light body in 2 such as vi = 1 − j≥i µj /ai ≃ 1/ai. Here the vicinity of 2-body parabolic orbit (w ≃ 0) [4]. we have set the gravitationalP constant G = 1 We checked that integration (3) along an ellip- and in the following we will take the mean ve- tical osculating orbit or along the parabolic or- locity of Jupiter v5 = 1. The Sun trajectory bit corresponding to the same pericenter give no in the barycentric reference frame is such as noticeable difference as long as the comet semi- 8 major axis is greater than planet semi-major axis. (1 − µ) r⊙ = − i=1 µiri. In the barycentricP reference frame, the potential To be more realistic we adopt integration over an experienced by the comet is consequently elliptical osculating orbit C0 since in the case of 8 1P/Halley slight differences start to appear for 1 − µ µ Φ(r) = − − i Neptune contribution to the kick function. kr − r⊙k kr − r k Xi=1 i After the comet’s passage at the pericenter, when the planet phases are x1,...,x8, the new = Φ0(r) 1 osculating orbit corresponds to the energy E0 + 8 ∆E (x1,...,x8). Knowing the relative positions r · r r + µ −1 − i + of the planets, the knowledge of eg x = x5 is i r2 kr − r k Xi=1 i sufficient to determine all the xi’s. Hence, for 2 Halley map (1) the kick function of the SS is + o µ (2) 8 F (x) = −2∆E (x) = i=1 Fi(xi) where Fi(xi) is where Φ0(r)= −1/r is the gravitational potential the kick function of thePith planet. In the follow- assuming all the mass is located at the barycenter. ing we present results obtained from the compu- Let us define a given osculating orbit C0 with tation of the Melnikov integral (3) using coplanar energy E0 and corresponding to the Φ0(r) poten- circular trajectories for planets. We have checked tial. The change of energy for the comet following the results are quite the same in the case of the the osculating orbit C0 under the influence of the non coplanar nearly circular elliptic trajectories SS potential Φ(r) (2) is given by the integral for the planets taken at J2000.0 (see dashed lines in Fig. 3 right panel). ∆E (x1,...,x8)= ∇ (Φ0(r) − Φ(r)) · dr (3) IC0 Fig. 2 shows contributions for each of the eight which gives at the first order in µ planets to the SS kick function. The two upper- most panels in Fig. 2 right column show contribu- ∆E (x1,...,x8) tions of Jupiter, F5(x), and Saturn, F6(x6), to the 8 r · ri 1 SS kick function. We set x = x5 = 0 when Hal- ≃ µi ∇ − · dr (4) I r3 kr − r k ley’s comet was at perihelion in 1986. We clearly Xi=1 C0 i 8 see that the exact calculus of the Melnikov inte- gral (3) are in agreement with the contributions ≃ ∆Ei (xi) of Jupiter and Saturn extracted by Fourier anal- Xi=1 3 Mercury (x10-6) Jupiter (x10-2) dipole of amplitude µ ≃ M /M similar to the 0.2 i i S
) ) 0.5 one analyzed for Rydberg molecular states [19] 1 5 0.1 that gives additional kick function of sinus form. (x (x 1 0 5 0 In Fig. 2 we clearly see that the saw-tooth shape F F -0.1 used in [1] to model the kick function is only a pe- -0.5 -0.2 culiar characteristic of Jupiter and Saturn contri- Venus (x10-4) 1.5 Saturn (x10-3) butions. Also, the sinus shape analytically found
) 0.5 ) 1 in [2] for large q can only be considered as a 2 6 0.5 crude model for the planet contributions of the (x 0 (x 2 6 0 SS kick function. For Venus, Earth and Mars the F F -0.5 -0.5 kick function is dominated by the Kepler poten- -1 tial term, the dipole potential term being weaker 1 Earth (x10-4) Uranus (x10-4) by an order of magnitude. Uranus contribution to 0.4
) 0.5 ) the SS kick function share the same characteris- 3 7 0.2 0 tics as Jupiter’s (Saturn’s) contributions but two (x (x 0 3 7 (one) orders of magnitude weaker. For Neptune
F -0.5 F -0.2 as its semi-major axis is about 60 times greater -1 -0.4 than Halley’s comet perihelion, the direct gravita- -5 -4 1.5 Mars (x10 ) Neptune (x10 ) tional interaction of Neptune is negligible and the
) 1 ) 0.2 dipole term dominates the kick function. Neptune 4 8 0.5 indirectly interacts on Halley’s comet by influenc-
(x (x 0 4 0 8 ing the Sun’s trajectory. As Mercury semi-major F F -0.5 -0.2 axis is less than perihelion’s comet, Mercury, like -1 the Sun, acts as a second rotating dipole, conse- 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 quently the two potential terms in (4) contribute xi xi equally. For a given osculating orbit, the shape f (x ) Figure 2: Contributions of the eight planets to SS kick i i function F (x). On each panel Fi(xi) is obtained from Mel- of the kick function defined such as Fi(xi) = 2 nikov integral calculation (thick line), the red dashed line µifi(xi)vi has to be only dependent on q/ai. Let (the blue dotted-dashed line) shows the Keplerian contri- us use the case of 1P/Halley to study general bution (dipole contribution) to the Melnikov integral. On features of fi(xi). Fig. 3 left panel shows the Jupiter and Saturn panels, the kick functions extracted peak amplitude f of f (x ). In the region by Fourier analysis from observations and exact numerical imax i i calculations are shown (•, see Fig. 2 in [1]). 0.25 . q/ai . 0.75 the peak amplitude fimax is clearly dominated by the Keplerian potential term and even diverges for close encounters at q ≃ 0.3ai and q ≃ 0.7ai. For q & 1.5ai the Keplerian poten- ysis [1] of previously observed and computed per- tial and the circular dipole potential terms give ihelion passages [25]. As seen in Fig. 2 the kick comparable sine waves almost in phase opposi- function is the sum of two terms (4): the Kepler tion (Fig.2 top left panel and [27]). We clearly −1 potential term −kr − rik (dashed red line in observe for q & 1.5ai an exponential decrease of 3 Fig. 2) and the dipole potential term r · ri/r the peak amplitude, fimax ∼ exp(−2.7q/ai), con- (dot dashed blue line in Fig. 2). These two terms sistent with the two dimensional case studied in are of the same order of magnitude, the dipole [2, 3]. term due to the Sun displacement around the SS The orbital frequency of the planets being only barycenter is therefore not negligible for Jupiter near integer ratio, for a sufficiently long time (Saturn) kick function. The rotation of the Sun randomization occurs and any 8-tuple {xi}i=1,...,8 around SS barycenter creates a rotating circular can represent the planets position in the SS. For 4 103 0.8 0.5 0.5 (x10-2) 102 0.6 0.49 101 a 0.4 b 0.45 0.48 1 0.2 w 10-1 0 0.4 0.47 max i f 10-2 F (x) -0.2 0.46 -3 -0.4 0.35 10 0.45 0 0.25 0.5 0.75 1 10-4 -0.6
Jupiter Mars Earth Venus Mercury 0.3 10-5 -0.8 x 0 1 2 3 4 0 0.25 0.5 0.75 1 q / a i x 0.25 w
Figure 3: Left panel: Peak amplitude of the kick function 0.2 shape fi(xi) (thick black line) as a function of pericenter distance q/ai [27]. The red dashed line (the blue dotted- 0.15 0.32 dashed line) shows the maximum amplitude of the Kep- 0.31 lerian contribution (dipole contribution). Vertical dashed 0.1 0.3 1:6 3:19 3:19 lines show relative positions of planets. On that scale Sat- w 0.29 3:19 2:13 3:20 2:13 3:20 urn, Uranus and Neptune relative positions are not shown. 0.05 0.28 3:20 Right panel: Variation domain of the SS kick function 0.27 1:7 F (x) (light blue shaded area) as a function of Jupiter’s 0 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 phase x = x5. The variation width is ∆F ≃ 0.00227. Data from observations and exact numerical calculations x x (Fig. 1 from [1]) are shown (•). The dashed lines bound Figure 4: Left panel: Poincar´esection of Halley’s map gen- the variation domain of the SS kick function when current erated only by Jupiter’s kick contribution F5(x) (red area). elliptical trajectories for planets are considered. The cross symbol (×)at(x =0, w ≃ 0.2921) gives Halley’s comet state at its last 1986 perihelion passage. An example of orbit generated by the Halley map (1) with the contri- butions of all the planets is shown by black dots. Right x = x5 the SS kick function F (x) is a multivalued top panel: closeup on the invariant KAM curve stopping function for all 0 ≤ x ≤ 1. We can nevertheless chaotic diffusion. Right bottom panel: closeup centered on define a lower and upper bound to the SS kick Halley’s current location. Stability islands are tagged with function which are presented as the boundaries of the corresponding resonance p:n between Halley’s comet the blue shaded region in Fig. 3 right panel. We and Jupiter orbital movements. clearly see that raw data points extracted in [1] from previously observed and computed Halley’s comet passages at perihelion [25] lie in the varia- is a feature of dynamical chaos. In the region tion domain of F (x) deduced from the Melnikov 0 < w . wcr ≃ 0.125 the comet can rapidly dif- integral (3). fuses through a chaotic sea whereas in the sticky region wcr ≃ 0.125 . w . 0.5 the diffusion is 4. Chaotic dynamics of Halley’s comet slowed down by islands of stability. The estimated threshold wcr ≃ 0.125 is the same as the one es- The main contribution to the SS kick func- timated analytically in the saw-tooth shape ap- tion F (x) is F5(x) the one from Jupiter as the proximation in [1]. Stability islands are located other planet contributions are from 1 (Saturn) to far from the separatrix (w = 0) on energies corre- 4 (Mercury) orders of magnitude weaker. The dy- sponding to resonances with Jupiter. The current namics of Halley’s comet is essentially governed position of Halley’s comet (x = 0,w ≃ 0.2921) is by Jupiter’s rotation around the SS barycenter. between two stability islands associated with 1:6 The red shaded area on Fig. 4 shows the sec- and 3:19 resonances with Jupiter orbital move- tion of Poincar´eobtained from Halley map (1) ment (Fig. 4 right bottom panel). As the comet’s taking only into account Jupiter’s contribution dynamics is chaotic the unavoidable imprecision F (x) = F5(x). We clearly see that the accessi- on the current comet energy w allows us only to ble part of the phase space is densely filled which follow its trajectory in a statistical sense. Ac- 5 cording to the Poincar´esection (Fig. 4) associated orbital periods. In accordance with the results with Halley map (1) for F = F5 the motion of the presented in [1] we retrieve for the mean sojourn comet is constrained by a KAM invariant curve time a 10 factor between the only Jupiter contri- around w ≃ 0.5 (Fig. 4 top right panel) consti- bution case and the all planets contribution case tuting an upper bound to the chaotic diffusion. (Jupiter and Saturn only in [1]). But we note Consequently as the comet dynamics is bounded that the mean sojourn times computed here are upwards the comet will be ejected outside SS as 10 times greater than those computed in [1] where soon as w reaches a negative value. Taking 105 only 40 initial conditions have been used. random initial conditions in an elliptically shaped area with semi-major axis ∆x = 5 · 10−3 and −5 5. Robustness of the symplectic map de- ∆w = 5 · 10 centered at the current Halley’s scription comet position (x = 0,w = 0.2921) we find a mean sojourn time of τ ≃ 4 · 108 yr and a mean In order to test the robustness of the kicked number of kicks of N ≃ 4 · 104. A wide dispersion picture for Halley’s comet dynamics we have di- has been observed since 3 · 105 yr . τ . 3 · 1013 rectly integrated Newton’s equations for a period yr and 749 ≤ N . 9 · 107. of -1000 to +1000 Jovian years around J2000.0 in Now let us turn on also the other planets con- the case of a SS constituted by the Sun and the tributions. As shown in [6], where only Jupiter eight planets with coplanar circular orbits (Fig. 5 and Saturn are considered, diffusion inside pre- first row), the Sun and Jupiter with elliptical or- viously depicted stability islands is now allowed bits (Fig. 5 second row), and the Sun and the eight as the other planets act as a perturbation on the planets with elliptical orbits (Fig. 5 third row). Jupiter’s kick contribution. In the example pre- From Fig. 5 right panels we see that our modern sented in Fig. 4 left panel the comet is locked for era is embedded in a time interval −400PJ
-4 butions from each SS planets have been isolated. 0.12 -6 e f In particular, we have retrieved the kick functions 0 0.25 0.5 0.75 1 -1000 -500 0 500 1000 of Jupiter and Saturn previously extracted by x t/PJ Fourier analysis [1]. The Sun movement around SS barycenter induces a rotating gravitational Figure 5: Numerical simulation of Halley’s comet dynam- dipole potential which is non negligible in the en- ics over a time period of −1000 to 1000 Jovian years ergy transfer from the SS to 1P/Halley. The sym- around J2000.0 (t = 0) with SS modeled as (a,b) the Sun and 8 planets with coplanar circular orbits, (c, d) the Sun plectic Halley map allows us to follow the chaotic and Jupiter with non coplanar elliptical orbits, (e,f) the trajectory of the comet during relatively quiet dy- Sun and the eight planets with non coplanar elliptical or- namical periods ∆t . 104yr exempt of closer ap- bits. Left panels: kick function F (x) values (+) extracted proach with major planets. One can expect that a from 290 successive simulated pericenter passages of Hal- higher dimensional symplectic map involving the ley’s comet. The color symbol goes linearly from black for data extracted at time t = 0 to light green for data angular momentum and other orbital elements 3 extracted at time |t| ≃ 10 PJ . We show only points in would allow to follow Halley’s comet dynamics for the range −0.008 < F (x) < 0.008. Data from observa- longer periods taking into account large variation tions and exact numerical calculations (Fig. 1 from [1]) in energy (close planet approach) and precession. are shown (•). Right panels: time evolution of pericen- In spite of the slow time variation of the Halley ter q (black curves, left axis) and of the osculating orbit energy w (red curves, right axis). Numerical simulations map parameters such a symplectic map descrip- have been done time forward and time backwards from tion allows to get a physical understanding of the t = 0. The gray curves show the time evolution of the global properties of comet dynamics giving a local 2 ℓ /2 quantity. structure of phase pace and a diffusive time scale of chaotic escape of the comet from the Solar sys- tem. In Fig. 5 left panels we show only kick function Acknowledgments values in the interval range −0.008
http://arxiv.org/ps/1410.3727v2