Astronomy & Astrophysics manuscript no. CDFFSS ©ESO 2021 June 2, 2021

Chaotic diffusion of the fundamental frequencies in the Solar System Nam H. Hoang, Federico Mogavero and Jacques Laskar

Astronomie et Systèmes Dynamiques, Institut de Mécanique Céleste et de Calcul des Éphémérides CNRS UMR 8028, Observatoire de Paris, Université PSL, Sorbonne Université, 77 Avenue Denfert-Rochereau, 75014 Paris, France e-mail: [email protected]

June 2, 2021

ABSTRACT

The long-term variations of the orbit of the Earth govern the insolation on its surface and hence its climate. The use of the astronomical signal, whose imprint has been recovered in the geological records, has revolutionized the determination of the geological time scales (e.g. Gradstein & Ogg 2020). However, the orbital variations beyond 60 Myr cannot be reliably predicted because of the chaotic dynamics of the planetary orbits in the Solar System (Laskar 1989). Taking into account this dynamical uncertainty is necessary for a complete astronomical calibration of geological records. Our work addresses this problem with a statistical analysis of 120 000 orbital solutions of the secular model of the Solar System ranging from 500 Myr to 5 Gyr. We obtain the marginal probability density functions of the fundamental secular frequencies using kernel density estimation. The uncertainty of the density estimation is also obtained here in the form of confidence intervals determined by the moving block bootstrap method. The results of the secular model are shown to be in good agreement with those of the direct integrations of a comprehensive model of the Solar System. Application of our work is illustrated on two geological data: the Newark-Hartford records and the Libsack core. Key words. Chaos – Diffusion - Celestial mechanics – Methods: statistical – Solar System – Milankovitch cycles

1. Introduction tical approach should be adopted. Geological constraints like- wise should be retrieved in a statistical setting. In this spirit, a Milankovitch (1941) hypothesized that some of the past large recent Bayesian Markov Chain Monte Carlo approach has been changes of climate on the Earth originate from the long-term proposed to provide geological constraints on the fundamental variations of its orbital and rotational elements. These variations frequencies (Meyers & Malinverno 2018). For such a Bayesian are imprinted along the stratigraphic sequences of the sediments. approach to give any meaningful constraint, proper prior distri- Using their correlations with an orbital solution (Laskar et al. butions of the fundamental secular frequencies are required and 2004, 2011a; Laskar 2020), some of the geological records can therefore a statistical study of the orbital motion of the Solar be dated with precision. This method, named astrochronology, System planets is needed. This constitutes the motivation for the has becomes a standard practice in the stratigraphic community, present study. and proves to be a powerful tool to reconstruct the geological Laskar (2008) performed the first statistical analysis of the timescale (e.g Gradstein et al. 2004, 2012; Gradstein & Ogg long-term chaotic behavior of the planetary eccentricities and 2020). inclinations in the Solar System. Mogavero (2017) reconsidered The climate rhythms found in the geological records are di- the problem from the perspective of statistical mechanics. Our rectly related to the Earth’s precession constant and to the fun- study is a follow-up of (Laskar 2008). We study fundamental damental secular frequencies of the Solar System: the preces- frequencies instead of orbital elements because they are more sion frequencies (gi)i=1,8 of the planet perihelia and the preces- robust and are closer to the proxies that can be traced in the geo- sion frequencies (si)i=1,8 of their ascending nodes. The evolu- logical records. This study is based on the numerical integrations tion of these fundamental frequencies is accurately determined of 120 000 different solutions of the averaged equations of the up to 60 Myr (Laskar et al. 2004, 2011a; Laskar 2020). Beyond Solar System over 500 Myr, among which 40 000 solutions were this limit, even with the current highest precision ephemerides, arXiv:2106.00584v1 [astro-ph.EP] 28 May 2021 integrated up to 5 Gyr. The initial conditions of the solutions it is hopeless to obtain a precise past history of the Solar Sys- are sampled closely around a reference value, that is compatible tem simply by numerical integration. This limit does not lie in with our present knowledge of the planetary motion. the precision of determination of the initial conditions but orig- inates in the chaotic nature of the Solar System (Laskar 1989, 1990; Laskar et al. 2011b). However, because the astronomi- 2. Dynamical model cal signal is recorded in the geological data, it appears to be 2.1. Secular equations possible to trace back the orbital variations in the geological record and thus to constrain the astronomical solutions beyond We use the secular equations of motions of (Laskar 1985, 1990, their predictability horizon (Olsen & Kent 1999; Ma et al. 2017; 2008, and references therein). They are obtained by series ex- Olsen et al. 2019). Nevertheless, a deterministic view of the So- pansion in planetary masses, eccentricities and inclinations, and lar System is no longer reliable beyond 60 Myr, and a statis- through analytical averaging of second order over the rapidly-

Article number, page 1 of 18 A&A proofs: manuscript no. CDFFSS changing mean longitudes of the planets. The expansion is trun- 3. Estimation of probability density functions cated at second order with respect to the masses and degree 5 in eccentricities and inclinations. The equations include corrections The samples of this study consist of the secular frequencies of from general relativity and Earth-Moon gravitational interaction. the astronomical solutions which are obtained by integrating the This leads to the following system of ordinary differential equa- secular equations (Eq. 1) from very close initial conditions. Due tions: to this initial proximity, the correlation of the solutions in the samples lasts for a long period of time, but will eventually dimin- dω √ = −1{Γ + Φ3(ω, ω¯ ) + Φ5(ω, ω¯ )}, (1) ish. Our objective is to obtain a robust estimation of the marginal dt probability density functions (PDFs) from these correlated sam- where ω = (z1,..., z8, ζ1, . . . , ζ8) with zk = ek exp($k), ζk = ples. In fact, this correlation is the main motivation for our use sin(ik/2) exp(Ωk). The variable $k is the longitude of the peri- of the estimation methods in this section. Details of our samples helion, Ωk is the longitude of the ascending node, ek is eccen- are described in the first part of Sect. 4. tricity and ik is inclination. Φ3(ω, ω¯ ) and Φ5(ω, ω¯ ) are the terms We use Kernel Density Estimation (KDE) to estimate the of degree 3 and 5. The 16 × 16 matrix Γ is the linear Laplace- time-evolving marginal PDFs of the fundamental frequencies of Lagrange system, which is slightly modified to make up for the the Solar System. In addition, the statistical uncertainty of our higher order terms in the outer Solar System. With adapted ini- density estimations (i.e. PDF estimations) is measured by mov- tial condition, the secular solution is very close to the solution ing block bootstrap (MBB). To our knowledge, this application of direct integration over 35 Myr (Laskar et al. 2004). The major of MBB for KDE of a time-evolving sample, whose correlation advantage of the secular system over direct integration is speed. changes overtime, is new. Therefore, in order to ensure the va- Numerically integrating averaged equations is 2000 times faster lidity of the method, we have carried out several tests in Sect. than non-averaged ones, due to the much larger step size: 250 3.3. years instead of 1.8265 days. It is thus desirable to employ the secular equations to study the statistics of the Solar System. Yet, we will also compare their predictions to those of a non-averaged 3.1. Kernel Density Estimation comprehensive dynamical model in Sect. 4. We chose KDE, also known as the Parzen–Rosenblatt window method, as our preferred non-parametric estimator of the PDFs 2.2. Frequency Analysis of the fundamental frequencies of the Solar System because of its high convergence rate and its smoothing property (Rosenblatt We employed the frequency analysis (FA) technique proposed 1956; Parzen 1962). by Laskar (1988, 1993) to extract the fundamental secular fre- We briefly present the method here. Let X = {X1, X2,..., Xn} quencies from the integrated solutions. The method find a quasi- be a univariate independent and identically distributed (i.i.d.) 0 PN iνkt periodic approximation f (t) = k=1 ake of a function f (t) sample drawn from an unknown distribution P with density func- over a time span interval [0, T]. It first finds the strongest mode, tion p(x). Then the KDE of the sample is defined as: which corresponding to the maximum of the function: n 1 X  x − Xi  Z T pˆh(x|X) = K , (3) iσt 1 −iσt nh h φ(σ) = h f (t)|e i = χ(t) f (t)e dt, (2) i=1 T 0 where χ(t) is a weight function which improves the preci- where K is a non-negative kernel function, and h is bandwith of sion of the maximum determination; it is chosen to be the Han- the KDE. The choice of bandwidth h is much more important ning window filter, that is χ(t) = 1 + cos(πt/T). The next step is than the choice of kernel K, which is chosen to be the standard the Gram-Schmidt orthogonalization. The complex amplitude a normal distribution in this paper. In this work we shall consider 1 bandwidths of the following form: of the first frequency ν1 is calculated by orthogonal projection of iν t function f (t) on e 1 . This mode is then subtracted from the func-  IQR νt −β tion f (t) to get a new function f (t) = f (t) − a e . The process BWβ = 0.9 min σ,ˆ n , (4) 1 1 1.34 is then repeated with this newly-obtained function until desired N strongest modes are obtained. This technique works very well whereσ ˆ is the standard deviation of the sample, IQR is its in- for weakly-chaotic systems like the Solar System when variables terquartile range and β is a constant of choice. The bandwidth can be decomposed into quasi-periodic modes over a sufficiently with β = 1/5 corresponds to the Silverman’s rule of thumb (Sil- short period of time. It has been proven that this algorithm con- verman 1986). BWβ=1/5 is a slightly modified version of the opti- verges towards the true frequencies much faster than the clas- mal choice of bandwidth for Gaussian distributed data for better sical Fast Fourier Transform (Laskar 2005). Therefore, it is a adaption to non-Gaussian data. Bias error and variance error of good tool to study the chaotic diffusion of the fundamental fre- KDE with this bandwidth will be in the same order of magni- quencies. In this work, we used a routine naftab written in the tude. Undersmoothing, i.e choosing a smaller bandwidth, shrink publicly-available computer algebra software TRIP (Gastineau the bias so that the total error is dominated by the variance er- & Laskar 2011), developed at IMCCE, to apply directly the fre- ror, which can then be estimated by bootstrap method (Hall et al. quency analysis. 1995); the common value of β for undersmoothing is 1/3. To extract the fundamental secular frequencies of the So- When the sample is identically distributed but correlated, • • lar System, we apply the FA to the proper variables (zi , ζi )i=1,8 kernel density estimation is still valid under some mixing con- of the secular equations (Laskar 1990). Each fundamental fre- ditions (Robinson 1983; Hart 1996). Indeed, in case of observa- quency is obtained as the frequency of the strongest Fourier com- tions that are not too highly correlated, the dependence among ponent of the corresponding proper variable. To track the time the data that fall in the support of the kernel function K can evolution of the frequencies, the FA is applied with a moving in- be actually much weaker than it is among the entire sample. terval whose sliding step is 1 Myr. The interval sizes are 10 Myr, This principle is known as whitening by windowing (Hart 1996). 20 Myr and 50 Myr. Therefore, the correlation in the sample does not invalidate the

Article number, page 2 of 18 Nam H. Hoang, Federico Mogavero and Jacques Laskar: Chaotic diffusion of the fundamental frequencies in the Solar System

∗ employ of kernel density estimation, and only impacts the vari- and its expectation E[p ˆk,l] = pˆk(x|B1,l,..., Bn−l+1,l). Let’s define ability of the estimation (see Sect. 3.2). With regard to the choice r of bandwidth, Hall et al. (1995) suggested that using the asymp- k δ∗ (x) = (p ˆ∗ − E[p ˆ∗ ]), (5) totically optimal bandwidth for the independent data is still a k,l h k,l k,l good choice, even for some strongly dependent data sequences. The samples generated by a chaotic measure-preserving dynam- so that if h is chosen properly to reduce the bias to be asymp- ical system (as in the case of the numerical integration of the totically negligible with respect to the stochastic variation, then ∗ Solar System) resembles those of mixing stochastic processes, the MBB distribution P(δk,l(x)|X) is a consistent estimator of the hence the theory of KDE should also be applicable for dynam- error distribution P(δ(x)) when h → 0, nh → ∞, k → 0, lk → ∞ ical systems, although the formulation might be different (Bosq and n/l → ∞. Note that if l = 1, then k = h and MBB reverts & Guégan 1995; Maume-Deschamps 2006; Hang et al. 2018). to the undersmoothing procedure for i.i.d sample studied by Hall et al. (1995). The efficiency of this estimator depends sensitively on two tuning parameters l and k. We are interested in the un- certainty of the KDE, which is characterised by the confidence 3.2. Moving Block Bootstrap interval CI1−α(x) and the confidence band CB1−α, which are de- fined as: Since the seminal paper by Efron (1979), bootstrap has become P(|δ(x)| < CI1−α(x)) = 1 − α, (6) a standard resampling technique to evaluate the uncertainty of a statistical estimation. Bias and variance errors of KDE with the choice of bandwidth BW1/5 (Eq. 4) are in the same order of magnitude, hence one should either undersmooth, i.e. choose P(|δ(x)| < CB1−α∀x ∈ Ω) = 1 − α, (7) a smaller bandwidth, to minimize the bias (Hall 1995) or use where α denote the level of uncertainty; for example, α = 0.05 an explicit bias correction with appropriate studentizing (Cheng denotes 95% CI. et al. 2019; Calonico et al. 2018). However, applying naively the They can all be estimated by the MBB distribution i.i.d. bootstrap procedure on dependent data could underestimate P(δ∗ (x)|X) as: the true uncertainty, because all the dependence structure would k,l be lost just by “scrambling” data together (Hart 1996). To rem- P(|δ∗ (x)| < CIc (x)) = 1 − α, (8) edy the problem, Kunsch (1989) and Liu et al. (1992) indepen- k,l 1−α ∗ dently devised the moving block bootstrap (MBB), which later P(|δk,l(x)| < CBc 1−α∀x ∈ Ω) = 1 − α, (9) becomes standard practice to evaluate the uncertainty in depen- dent data (See Kreiss & Lahiri (2012) for a review). Although the In practice, we will also use CB and CI without the hat overhead MBB for smooth functional has been intensively studied, the lit- to denote the estimated value for short. erature on MBB for KDE of dependent data is very limited. Re- Our choice of the parameters of the MBB procedure l and k cently, Kuffner et al. (2019) has formulated an optimality theory is based on the effective sample size neff, defined as (Kass et al. of the block bootstrap for KDE under weak dependence assump- 1998): tion. They proposed both undersmoothing and explicit bias cor- n n = , (10) rection scheme to obtain the sampling distribution of the KDE. eff 1 + 2 P∞ ρ(k) However, good tuning parameters, which are generally difficult k=1 to find if the data are from an unknown distribution, is required where ρ(k) is the sample autocorrelation of lag k. The block to provide a decent result. In this paper, we propose to overcome length l is chosen by the sample correlation size lcorr as this problem with an inductive approach: the optimal tuning pa- n rameters obtained in a known model are tested on different mod- l = lcorr := , (11) els, and then extrapolated to our study - the Solar System. neff and the bootstrap bandwidth is parametrized as k = h(c + (1 − c )l−γ ), (12) Procedure of MBB We briefly describe the undersmooth 0 0 corr MBB for the kernel density estimation method (the detailed where γ and c0 are two optimizing constants. The reason for this version can be found in (Kuffner et al. 2019)). Suppose X = choice of parametrization is two folds. First, when lcorr → 1, the {X1, X2,..., Xn} is a dependent sample of a mixing process with sample become independent, then k → h. Secondly, the rate of underlying density function p(x). The KDE of the sample is change of k with respect to l should be greater when lcorr is small pˆh = pˆh(x|X); the hat overhead a given quantity denotes its than when it is large. Therefore when lcorr  1, the optimal value estimated value. We use MBB to estimate the distribution of of k should be quite stable. We also observe experimentally that δ(x) = pˆh(x) − p(x), where x ∈ Ω, and Ω is the domain the optimal value of k is indeed relatively stable at around 2h as of interest. Let l be an integer satisfying 1 ≤ l ≤ n. Then long as l = lcorr  1. So we simply choose γ = 1 and c0 = 2. Bi,l = {Xi, Xi+1,..., Xi+l−1} with i ∈ {1,..., n − l + 1} denotes This choice of parameters turns out to be quite robust as it will all the possible overlapping block of size l of the sample. Sup- be demonstrated by the two numerical experiments in Sect. 3.3. pose l divides n for simplicity, then b = n/l. MBB samples The literature on KDE and MBB focuses on stationary, are obtained by selecting b blocks randomly with replacement weakly dependent sequences. The data in our case, however, are from {B1,l,..., Bn−l+1,l}. Serial concatenation of b blocks will different: they are not strictly stationary; the formulation of mix- ∗ ∗ ∗ give n bootstrap observations: Xl = {B1,l,..., Bb,l}. By choos- ing condition might be different (Hang et al. 2018); the correla- ing l large enough (preferably larger than the correlation length), tion in the sample is not constant but evolving with time; finally MBB sample can retain the structure of the sample dependence. and most importantly, the data structure is different. Our sample ∗ ∗ For k > 0, the KDE of the bootstrap sample isp ˆk,l = pˆk(x|Xl ) units, which are the orbital solutions, are ordered based on their Article number, page 3 of 18 A&A proofs: manuscript no. CDFFSS

DGMM n = 104, Density Estimation, 90% CI DGMM n = 104, Estimation of 90% CI 0.35 KDE, lcorr = 140 lcorr = 500 0.10 True Density l = 140 0.30 corr lcorr = 40

lcorr = 20 0.25 0.08 lcorr = 10

lcorr = 1 0.20 0.06

0.15 0.04 0.10

0.02 0.05

0.00 0.00 4 2 0 2 4 6 8 60 80 100 120 140 x x

Fig. 1. Kernel density estimation (red line) with bandwidth h = BW1/3 Fig. 2. Estimation of the confidence interval (90% CI) of KDEs from and its 90% pointwise confidence interval (red band) from a MCMC different samples with different correlation lengths. Solid lines are the 4 sample of n = 10 units with lcorr = 140 of double Gaussian mixture true values and the bands denote the distribution of MBB estimations model (Eq. 13 - black line). (mean ± standard deviation), both are computed from 103 different sam- ples, each composes of n = 104 units generated from the double Gaus- sian mixture model by MCMC algorithm with the same lcorr. initial distances in phase space. The solutions evolve over time but their order remains unchanged. Therefore statistical notions such as correlation and stationarity should be considered within On each MCMC sequence, we apply the KDE/MBB proce- this framework, for the sample at fixed values of time. Because dure with 500 MBB samples and the parameters specified above of the differences presented, an optimality theory, which is not (Eqs. 11-12) with the undersmoothing bandwidth h = BW1/3, currently available, might be needed for this case. However, we to get the uncertainty estimation (standard error, confidence in- assume it would not differ significantly from the orthodox analy- terval and confidence band). Figure 1 shows an example of the sis, and that a decent working MBB procedure could be obtained density estimation of a MCMC sequence of 104 units, with cor- with some good choice of parameters. This will be tested in the relation length lcorr = 140; the sequence is generated by choosing section below. σp = 0.25. It is clear in the figure that the true PDF (in black) lies inside the range of the 90% CI estimated by MBB. To assess the precision of the MBB uncertainty estimation, at 3.3. Numerical experiments each value of lcorr, we sample 1000 different MCMC sequences, We perform the KDE/MBB procedure on two numerical experi- and apply the KDE/MBB process on each of them to obtain their ments to ensure its validity. The double Gaussian mixture model estimation of the confidence interval. We thus obtain a distri- (DGMM) with different degrees of correlation is used to cali- bution of confidence interval estimated by MBB, which are de- brate the algorithm of KDE/MBB method, that is to calibrate picted by the bands of mean ± standard deviation (Fig. 2). With and test the tuning parameters (Eqs. 11-12). Then the resulting the knowledge of the true PDF (Eq. 13), the true values of CI algorithm is applied on the Fermi Map as an example of a real and CB are obtained from this collection of KDEs at each corre- dynamical system for assessment. lation length (Eqs. 6-7). Figure 2 compares the true values of the pointwise 90% CI with the distribution of its estimated values by MBB at various correlation lengths. We see that the 90% confi- Double Gaussian Mixture Model The calibration of dence interval of KDE estimated by MBB is accurate across a KDE/MBB was done on Markov chain Monte Carlo (MCMC) very wide range of lcorr, that is the true values almost always lie sequences that sample a double Gaussian distribution. Our inside the estimation bands. particular choice of the DGMM, inspired by Cheng et al. (2019), is Fermi Map The second test for the MBB scheme is a 2- fDG(x) = 0.6φ(x) + 0.4φ(x − 4), (13) dimensional chaotic Hamiltonian system: The Fermi map (Fermi 1949; Murray et al. 1985). This simple toy model, originally de- where φ(x) is the standard normal distribution. The MCMC se- signed to model the movement of cosmic particles, describes the quence X ,..., X is obtained by the Metropolis-Hasting algo- 1 n motion of a ball bouncing between two walls, one of which is rithm, with a Gaussian proposal distribution whose standard de- fixed and the other oscillating sinusoidally. The equations of the viation σ characterises the correlation length of the sequence p Fermi map reads: (Metropolis et al. 1953; Hastings 1970). We can then either vary σp or perform thinning to obtain a sequence of a desired corre- ut+1 = ut + sin ψ, 4 lation length. The size of each MCMC sequence is n = 10 . The 2πM (14) initial state of the sequence is directly sampled from the distri- ψt+1 = ψt + (mod 2π), ut+1 bution fDG itself, so that burn-in is not necessary and the whole sequence is usable. where ut and ψt are the normalized velocity of the ball and the

Article number, page 4 of 18 Nam H. Hoang, Federico Mogavero and Jacques Laskar: Chaotic diffusion of the fundamental frequencies in the Solar System

0.150 0.30 BW1/3, 95% CB true value, BW1/3 BW1/5 true value, BW1/5 0.125 True Density 0.25 estimation, BW1/3

0.100 0.20

0.075 B C 0.15 % 5

0.050 9

0.10 0.025

0.000 0.05

0.025 60 80 100 120 140 10 20 30 40 50 u Time

Fig. 3. Density estimation of Fermi Map. The black line (“true” density) Fig. 4. Comparison of the confidence band estimation with its true is the KDE with bandwidth h = BW1/5 from a sample of the velocity u value. The blue region represents the mean ± 3 standard deviations of n = 106 solutions of Fermi Map at t = 30. Red line denotes the KDE of the confidence bands estimated by MBB of KDEs with bandwidth 3 with bandwidth h = BW1/3 with its 95% confidence band estimated by h = BW1/3 from 300 different samples, each consisting in n = 10 so- MBB (red band) from a sample of n = 103 solutions at the same time. lutions of the Fermi Map; the true values of confidence band are also Blue-dashed line represents the KDE with bandwidth h = BW1/5 from calculated from the KDEs with bandwidth h = BW1/5 (red dotted line) the same sample. and KDEs with bandwidth h = BW1/3 (blue dotted line) from the same 300 different samples. phase of the moving wall right before the tth collision, respec- tively; the stochastic parameter of the system M is chosen here to with h = BW1/3 will represent an upper bound of a 95% CB of be 104. The system is studied in the region of large-scale chaos. a KDE with h = BW1/5 (Fig. 4). The same also applies for the Our sample is obtained by evolving Eq. 14 from n = 103 initial confidence interval. Also from Fig. 4, we can see that the estimation of the confi- conditions. The initial conditions u0, ψ0 are drawn from uniform distributions on [90 − 10−5, 90 + 10−5] and [0, 2π], respectively; dence band follows well the true value. This is quite remarkable because, first we can extend the use of MBB to measure the un- then they sorted by ascending value of ψ0. The sorting step is imperative to quantify the sample dependence as the initially- certainty of density estimation from solutions of chaotic dynami- ordered neighboring solutions still tend to be correlated after- cal system, where the correlation of the sample defined by initial ward and the autocorrelation function can be calculated from or- distances in phase space changes over time, and secondly, our dered samples. Large initial phase variation will guarantee that simple choice of MBB parameters appears to work well across chaotic diffusion will be immediately perceptible and the PDF very different models. of the velocity will center around its initial distribution. At each time t, we compute the KDE of the sample with the bandwidth 3.4. Combining samples BW1/3. The KDE uncertainty is estimated by applying the MBB 400 times with the same parameters specified above (Eqs. 11- After having obtained a well-tested uncertainty estimator of the 12). The whole process is then repeated 300 times with differ- KDE for correlated data, we proceed with a practical ques- ent sets of initial conditions. In the Fermi Map experiment, the tion: given several samples of different correlation length lcorr, true analytical form of the density is not available, a numerical how can we efficiently combine them? Let’s assume we have m { 1 1 1} “true” density, which is obtained by calculating the KDE with samples of the same size n: X1 = X1 , X2 ,..., Xn ,..., Xm = 6 {Xm, Xm,..., Xm}, where the correlation within each sample is bandwidth BW1/5 from a n = 10 sample, is used instead to as- 1 2 n sess the validity of the MBB uncertainty estimation. From this different. The KDE of these m samples arep ˆ1,..., pˆm, and “true” density and the KDEs from 300 samples, we are able to their pointwise standard error can be estimated by MBB as determine the true confidence band (Eq. 6-7). σˆ 1,..., σˆ m. Figure 3 shows an example of KDE and its confidence band If all the samples conform to the same probability density estimated by MBB at t = 30. Although the estimated confi- p, we can simply use the inverse variance weighting to get our dence band is valid, as the true curve lies completely in the band, combined KDE: P −2 the KDE itself looks quite jagged. The jagged KDE is the re- i σˆ i pˆi pˆwa = , (15) sult of the undersmooth bandwidth h = BW1/3 so the bias is P σˆ −2 dominated by the variance error. In fact, with the rule-of-thumb i i 2 bandwidth h = BW1/5, we can have a smoother KDE (cf. blue- wherep ˆi andσ ˆ i are the individual KDE and its pointwise esti- dashed line of Fig. 3). This choice is valid because its uncertainty mated variance, respectively. The variance of the weigted aver- 2 P −2 −1 is always smaller than that estimated with h = BW1/3. There- age will beσ ˆ wa = ( i σˆ i ) . With this choice, the variance of fore we choose to use the uncertainty estimation computed with the weighted averagep ˆwa will be minimized and we can thereby h = BW1/3 as an upper bound of the uncertainty of a KDE with have the most accurate estimation of the density p (Hartung et al. the rule of thumb bandwidth. For example, a 95% CB of a KDE 2011).

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Density Estimation, 95% CI

10 10 10 Z1 150 Myr 60 8 Z2 8 8

Z3 6 Z 6 6 4 40

4 4 4

20 2 2 2

0 0 0 0 5.00 5.25 5.50 5.75 6.00 7.40 7.45 7.50 17.0 17.2 17.4 17.6 17.50 17.75 18.00 18.25 18.50

g1 [00/yr] g2 [00/yr] g3 [00/yr] g4 [00/yr]

Fig. 5. Density estimation with 95% pointwise confidence interval of the fundamental frequencies of the four sets of the batch {Zi}i=1,4. Frequency values are obtained by FA over an interval of 20 Myr centering at 150 Myr in the future.

This inverse variance weighting is only applicable if we as- value, then numerically integrate the secular equations (Eqs. 1) sume that all our samples follow the same underlying distribu- from those initial conditions to obtain a sample of orbital solu- tion, as for example in the double Gaussian mixture model. In the tions. KDE is then used to estimate the marginal PDF of the fre- case of the Solar System, different samples come from different quencies of the sample at fixed value of time and then the MBB sets of initial conditions. The densities evolving from different method is applied to estimate the uncertainty of the density esti- initial conditions will be different. Although they might converge mation. towards a common density function after a large period of time, The evolution of our sample can be divided into two stages: the assumption that different samples follow the same distribu- Lyapunov divergence and chaotic diffusion. In the first stage, be- tion is not generally true. However, if the differences between cause the initial density is extremely localized around the ref- the density estimations from different samples are small com- erence value, all solutions essentially follow the reference tra- pared to their in-sample density uncertainty, then we can assume jectory, the difference between the solutions is very small but a common true density that all the sample share, and the inverse diverges exponentially with a characteristic Lyapunov exponent variance weighted mean and its minimized variance will in this of ∼ 1/5 Myr−1 (Laskar 1989). The solutions in the first stage case capture this assumed density function. are almost indistinguishable and the correlation between them A more conservative approach consists in not trying to es- is so great that regardless of how many solutions in the sample timate the common true density, but rather capture the variabil- we integrated, the effective size of the sample is close to one. ity stemming from different samples and combine it with their The second stage begins when the differences between the solu- in-sample uncertainty. In this case, a combined KDE may be tions are large enough to become macroscopically visible. The introduced as a pointwise random variable, whose distribution Lyapunov divergence saturates and gives place to chaotic diffu- function is given by: sion. The correlation between the solutions starts to decrease,

m the distribution of the sample settles and the memory of the ini- X tial conditions fades. It can take several hundreds of millions of pˆ ∼ m−1 P(p ˆ (x) − δ∗(x)|X ), (16) c i i i years for the sample to forget its initial configuration. Contrary i=1 to the exponential growth in the first stage, the dispersion of the ∗ ∗ samples expands slowly with a power law in time (see Fig. 10). where δi (x) is defined in Eq. (5) and P(p ˆi(x) − δi (x)|Xi) is the MBB estimation of the distribution of the pointwise KDE from The time boundary between the two stages depends on the dis- sample Xi. If the m samples are good representatives of any other persion of the initial conditions, the wider they are, the faster the sample taken from a certain population, thenp ˆc is a reasonable second stages come and vice versa. If they are chosen to repre- choice for the KDE from an arbitrary sample generated from that sent the uncertainty of the current ephemeris, the second stage same population. We samplep ˆc in a pointwise manner. For each should take place around 60 Myr in the complete model of the value of x, the same number of realizations are drawn from each Solar System (Laskar et al. 2011a,b). of the m MBB distributions, which are assumed here to be Gaus- In this section, we focus on the statistical description of the sian for practical purpose. It should be noticed that a realization fundamental frequencies of the Solar System in the second stage. ofp ˆc(x) is not a continuous curve. When needed, we can choose The aim is to have a valid estimation of time-evolving PDF of the the pointwise median ofp ˆc as the nominal continuous combined frequencies beyond 60 Myrs. However, the PDF evolution gen- KDE. erally depends on the choice of initial conditions. Moreover, the simplification of the secular equations compared to the complete model of the Solar System could, a priori, provide results which 4. Application to the Solar System are not sufficiently accurate. The goal of this paper is to have a consistent statistical descrip- tion of the propagation of the dynamical uncertainty on the fun- 4.1. Choice of initial conditions damental secular frequencies of the Solar System induced by its chaotic behavior; that is, simply put, to obtain their time- For a complete model of the Solar System, the initial conditions evolving marginal PDF. We first sample the initial orbital ele- should be sampled in such a way that they are representative of ments of the Solar System planets closely around the reference the current planetary ephemeris uncertainty. Nevertheless, for a

Article number, page 6 of 18 Nam H. Hoang, Federico Mogavero and Jacques Laskar: Chaotic diffusion of the fundamental frequencies in the Solar System

Density Estimation, 95% CI

20 100 20 20 Y 100 Myr 1 4 X1 4 80 15 15 15 Z1 4 60 10 10 10 40

5 5 5 20

0 0 0 0 5.00 5.25 5.50 5.75 6.00 7.40 7.45 7.50 17.00 17.25 17.50 17.50 17.75 18.00 18.25 18.50 20 100 20 20 200 Myr 80 15 15 15

60 10 10 10 40

5 5 5 20

0 0 0 0 5.00 5.25 5.50 5.75 6.00 7.40 7.45 7.50 17.00 17.25 17.50 17.50 17.75 18.00 18.25 18.50

g1 [00/yr] g2 [00/yr] g3 [00/yr] g4 [00/yr] 20 10 25 Y1 4 20 100 Myr X1 4 8 20 15 Z 1 4 15 6 15 10 10 4 10

5 2 5 5

0 0 0 0 6.0 5.5 5.0 8.0 7.5 7.0 6.5 19.2 19.0 18.8 18.6 18.0 17.8 17.6 20 10 25 20 200 Myr 8 20 15 15 6 15 10 10 4 10

5 2 5 5

0 0 0 0 6.0 5.5 5.0 8.0 7.5 7.0 6.5 19.2 19.0 18.8 18.6 18.0 17.8 17.6

s1 [00/yr] s2 [00/yr] s3 [00/yr] s4 [00/yr]

Fig. 6. Density estimation with 95% pointwise confidence interval of the fundamental frequencies of the 12 sets coming from the 3 batches {Xi} (yellow colors), {Yi} (blue colors) an {Zi} (green colors). Estimations of sets from the same batch have the same color. Frequencies are obtained by FA over an interval of 20 Myr centering at 100 Myr in the past (first and third row) and 200 Myr in the past (second and fourth row). The time reference is the time of {Zi}, while the solutions of {Xi}, {Yi} are shifted ahead 30 Myr and 20 Myr respectively. simplified secular model (Eq. 1), the difference with the com- to study first the effect of sampling the initial conditions on the plete model is greater than the ephemeris uncertainty. An opti- PDF estimation. mized secular solution follows the complete solution initially but Initial conditions can be sampled in many ways, especially in will depart from it long before 60 Myr (Laskar et al. 2004). A a high-dimensional system like the Solar System. Our choice of direct adaptation of the current planetary ephemeris uncertainty initial conditions is quite particular, but they encompass different into the initial conditions of the secular model can thus be mis- possible ways of sampling initial conditions (Table 1). There are leading. Therefore, we adopt a more cautious approach, that is 3 batches: {Xi}, {Yi} and {Zi}, which corresponds to 3 different

Article number, page 7 of 18 A&A proofs: manuscript no. CDFFSS

Density Estimation, 95% CI

12 100 12 12 Y 150 Myr 1 4 10 10 10 X1 4 80 Z1 4 8 8 8 La09 60 6 6 6 40 4 4 4 20 2 2 2

0 0 0 0 5.00 5.25 5.50 5.75 6.00 7.40 7.45 7.50 17.00 17.25 17.50 17.50 17.75 18.00 18.25 18.50 12 100 12 12 350 Myr 10 80 10 10

8 8 8 60 6 6 6 40 4 4 4 20 2 2 2

0 0 0 0 5.00 5.25 5.50 5.75 6.00 7.40 7.45 7.50 17.00 17.25 17.50 17.50 17.75 18.00 18.25 18.50

g1 [00/yr] g2 [00/yr] g3 [00/yr] g4 [00/yr] 6 20 15.0 Y 15.0 150 Myr 1 4 X1 4 5 12.5 15 12.5 Z1 4 4 10.0 La09 10.0

7.5 3 10 7.5

5.0 2 5.0 5 2.5 1 2.5

0.0 0 0 0.0 6.0 5.5 5.0 8.0 7.5 7.0 6.5 19.2 19.0 18.8 18.6 18.0 17.8 17.6 6 20 15.0 15.0 350 Myr 5 12.5 15 12.5 4 10.0 10.0

7.5 3 10 7.5

5.0 2 5.0 5 2.5 1 2.5

0.0 0 0 0.0 6.0 5.5 5.0 8.0 7.5 7.0 6.5 19.2 19.0 18.8 18.6 18.0 17.8 17.6

s1 [00/yr] s2 [00/yr] s3 [00/yr] s4 [00/yr]

Fig. 7. Density estimation with 95% pointwise confidence interval of the fundamental frequencies of the 12 sets coming from the 3 batches {Xi} (yellow colors), {Yi} (blue colors) an {Zi} (green colors) and from the 2 500 solutions of complete model of Laskar & Gastineau (2009) (red colors), which is denoted as La09. Estimations of sets from the same batch have the same color. Frequencies are obtained by FA over an interval of 20 Myr centering at 150 Myr (first and third rows) and 350 Myr (second and fourth rows) in the future. The time reference is the time of {Zi}, while the solutions of {Xi}, {Yi}, La09 are shifted 30 Myr, 20 Myr and -70 Myr respectively.

variation sizes  of initial conditions. Each batch composes of past and 500 Myr to the future. For the batch {Zi}, the integration 4 different sets of samples. Each set contains 10 000 initial con- time is 5 billion years in both directions. The frequencies are ditions, where eccentricity of the associated planet is linearly- then extracted using frequency analysis (section 2.2). It should spaced from the reference value, with the spacing  correspond- be noted that we do not aim to obtain the joint probability dis- ing to the batch it belongs to. We then integrated the secular tribution of all the fundamental frequencies, but rather their in- equations (Eqs. 1) from these initial conditions 500 Myr to the dividual marginal PDFs, i.e. the PDF of one frequency at a time.

Article number, page 8 of 18 Nam H. Hoang, Federico Mogavero and Jacques Laskar: Chaotic diffusion of the fundamental frequencies in the Solar System

Solutions Offsets  rates of convergence are higher, and the overlap generally hap- −10 Xi −5000  to 5000  10 pens at around −100 Myr (see Fig. 6). −8 Yi −5000  to 5000  10 Z −5000  to 5000  10−11 i 4.3. Second test: different samples of different variation sizes Table 1. Offsets of the initial eccentricities of the 4 planets: {Mercury, Venus, Earth, Mars}, which corresponds to i = {1, 2, 3, 4}. Different in- Comparing the density estimation of the 3 batches {Xi}, {Yi} and tegrations correspond to offsets of N in eccentricity of a single planet {Z } is our second test of robustness. Although the initial con- − × 3 × 3 i for N = 5 10 ,..., +5 10 , while other variables are kept to their ditions of the 3 batches are varied around the same reference nominal values. values, the ways they are sampled are different from each other, since the variation sizes are different. Differences in the initial variation sizes mean that the batches will enter the diffusion The marginal PDFs of the frequencies are estimated by the KDE stage at different times and also at different points in the phase with rule-of-thumb bandwidth (BW1/5), upper bounds of their space, so that the convergence between batches, if exists, would 95% confidence intervals (denoted for short as 95% CI) are mea- take longer. The result of our test is summarised by the density sured by the MBB method with the bandwidth h = BW1/3 and estimation of the frequencies at two times in the past, −100 Myr the optimized parameters (Eqs. 11-12) from 1 000 MBB sam- and −200 Myr (Fig. 6). At −100 Myr, the density estimations ples. from different sets of each batch cluster around each other as de- We first compare the evolution of the density of the four scribed in the previous test. Each batch forms a cluster of density sets in each batch in section 4.2. In section 4.3, the second estimations and the differences between 3 clusters are notice- test is performed to compare the statistics between the batches. able. Moreover, the estimation uncertainty, depicted by the col- Robustness of secular statistics will be assessed by these two ored band, is quite large. Fast forward 100 Myrs of chaotic mix- tests, which will additionally shed light on the initial-condition- ing, at −200 Myr, density estimations of the frequencies spread dependence aspect of the statistics. All the density estimations out, estimation uncertainty shrinks, and more importantly, dif- from these sets will then be compared with that of 2 500 com- ferences between the 3 batches are much smaller, and they will plete solutions obtained in the previous work of (Laskar & continue to diminish even further as time goes. In the opposite Gastineau 2009), to test the accuracy of the secular statistics. time direction, the same phenomenon is observed but the rate of It should be reminded that this numerical experiment needed 8 convergence between the batches is slower (Fig. 7). At 150 Myr millions hours of CPU time, the output of which was saved and in the future, the density estimation of frequencies of {Xi} and could thus be used in the present study. {Zi} has practically converged when those of {Yi} is still trying When comparing two sets of different sizes, because the rates to, yet differences between 3 batches are noticeable. However, of divergence in the first stage are similar, the wider set reaches the estimations of all 12 sets from 3 batches nearly overlap each the chaotic diffusion phase faster than the more compact set, other at 350 Myr, which demonstrates that the effect of different hence it is essentially diffusing ahead for a certain time in the initial samplings vanishes by chaotic mixing. It should be noted second stage. Therefore, in order to have a relevant comparison, that when looking at some specific properties of the PDF such as a proper time shift is introduced to compensate for this effect. means and variances, the differences between the sets are small. For example, the differences between the means of the PDF es- We shift {Xi} and {Yi} ahead 30 Myr and 20 Myr respectively, timation of the 12 sets are generally smaller than 0.1 00/yr for while keeping the time of {Zi} as reference. This choice is moti- most of the fundamental frequencies at −100 Myr; at −200 Myr, vated by the fact that the transition to the chaotic diffusion of {Zi} is around 50 - 60 Myr, which is indicated by direct integration of the differences diminish to be twice as small. the Solar System (Laskar et al. 2011a,b). 4.4. Final test: comparison with the complete model 4.2. First test: different samples of the same variation size  The convergence of the density estimations from different sets of Comparing the density estimation of different sets in the same initial conditions (seen in the first two tests) does not guarantee batch is the first test of prediction robustness from our secular convergence between the secular model and the complete model model. The evolution of the density estimations will be sensitive of the Solar System. As a result, whether the secular statistics to how initial conditions are sampled and therefore this initial- will resemble that of the complete model is still not clear. In this condition sensitivity must be quantified for a valid prediction. final test, we respond to this question by comparing the density In the first test, we compare the time-evolving PDFs whose ini- estimation from our secular solutions with those obtained from tial conditions are sampled with the same variation size  but in the 2 500 solutions of the complete model integrated in the fu- different variables. ture (Laskar & Gastineau 2009). The initial conditions of the The result is quite clear. The different sets of the same batch complete solutions are sampled in a way similar to the present slowly lose the memory of their initial differences due to chaos work, that is one variable (the semi-major axis of Mercury) is and then converge towards the same distribution. This conver- linearly-spaced with a spacing of 0.38 mm and a range of about gence is illustrated by Fig. 5, which shows that the density es- one meter. We shift the complete solution backward 70 Myrs to timation of (g)i=1,4 of the 4 sets of the batch {Zi} nearly overlap adjust for the difference in the initial variation sizes and also for each other at 150 Myr in the future. The rates of convergence the difference between the initial divergence rates. of different batches are different. So that although {Xi} and {Yi} The density estimations of the frequencies of the 2 500 com- exhibit the same behavior as {Zi}, they converge differently with plete solutions are shown along with the estimations from the disparate rates: At around 100-150 Myr in the future, the density secular model in Fig. 7. The estimations of the complete model estimations of the frequencies of {Xi} nearly overlap each other, are either close to or overlap those of the secular model even at this occurs at 150-200 Myr for {Yi}, depending on the frequency. 150 Myr. Both models even predict the same minor features in Interestingly, for the samples that are integrated in the past, the the frequency density, for example, the second peaks of g3 or the

Article number, page 9 of 18 A&A proofs: manuscript no. CDFFSS

Density Estimation, 95% CI, sample Z1

10 80 8 8 10 Myr 150 Myr 8 20 Myr 50 Myr 60 6 6 6 40 4 4 4

20 2 2 2

0 0 0 0 10 80 8 8 350 Myr 8 60 6 6

6 40 4 4 4

20 2 2 2

0 0 0 0 5.00 5.25 5.50 5.75 6.00 7.40 7.45 7.50 17.00 17.25 17.50 17.50 17.75 18.00 18.25 18.50

g1 [00/yr] g2 [00/yr] g3 [00/yr] g4 [00/yr]

Fig. 8. Density estimation with 95% pointwise confidence interval of the fundamental frequencies of the set Z1, which are obtained by FA over an interval of 10 Myr (blue colors), 20 Myr (yellow colors) and 50 Myr (green colors) centering at −150 Myr (top row) and −350 Myr (bottom row).

tails of g4. The origins of these features are related to the reso- is too large, the obtained frequency will tend to be the average nances associating with g3 and g4, so having these same features of its variation over the same period. We chose ∆t = 20 Myr indicates that the secular model could capture the resonance dy- as the standard FA interval. In some circumstances that requires namics of the complete model. At 350 Myr, for most of the fre- detection of rapid changes in frequencies such as the resonance quencies, the differences between the results of the two models transition, a smaller ∆t is more favorable (see section 6.3). are very small, especially for some frequencies like g1 where it is difficult to distinguish the two models. If we look at some The extracted frequencies are sensitive to the choice of specific properties of the PDF like its mean for example, the dif- ∆t, yet, their density estimation is relatively robust. Figure 8 ferences between the secular model and the complete model are compares the density estimation of the eccentricity frequencies at the same order with the variability of results from the same (gi)1,4, which are extracted by FA with 3 different ∆t at two dif- secular model. These differences of means of the PDFs are gen- ferent times. The differences are generally small but still notable erally smaller than 0.05 00/yr at 350 Myr. Although some minor for g2 and g4 at 150 Myr, and they will diminish as time goes. differences between the two models are still visible, especially in s1 for example, these differences diminish as time goes. This convergence between the two models strongly suggests the com- patibility of the secular system with the direct integration of the realistic model of the Solar System used in (Laskar & Gastineau 5. Parametric fitting 2009). From Sect. 4, we see that the long-term PDFs of the secular fre- 4.5. A complementary test on frequency analysis quencies possess a distinct Gaussian-like shape that flattens as time passes. It is interesting to approximate these densities by a The fundamental frequencies of the Solar System are central in simple parametric model, so that its parameters can characterise our work, and the method to obtain them is thus essential. In this the shape of the density and summarize its evolution. The model section, we briefly examine the frequency analysis method: FA with its fitting parameters can also be used as an approxima- (Sect. 2.2). FA searches a quasi-periodic approximation of the tion of the numerical densities for later application. For this pur- solution of the Solar System with constant frequencies over a pose, we will use the density estimation of the secular frequen- time window ∆t. So unique frequencies at time t are extracted cies of the Solar System from the batch {Zi}, which composes from an oscillating sequence in the time interval [t − ∆t/2, t + of 40 000 different orbits over 5 Gyr. The inner fundamental fre- ∆t/2]. For a quasi-periodic solution, the longer we choose the quencies (i.e. the frequencies of the inner planets (gi, si)i=1,4) are time interval ∆t, the more accurate the extracted frequencies are. obtained by FA over an interval of 20 Myr. For the outer fun- Nevertheless, the fundamental frequencies of the Solar System damental frequencies (i.e. the frequencies of the outer planets are expected to vary over a few Lyapunov times, that is 5 Myr. (gi)i=5,8,(si)i=6,8), the FA interval is 50 Myr. The frequency s5 is Therefore we have a trade-off between the extraction accuracy zero due the conservation of the total angular momentum of the and time variation of the frequencies when choosing ∆t: when ∆t Solar System.

Article number, page 10 of 18 Nam H. Hoang, Federico Mogavero and Jacques Laskar: Chaotic diffusion of the fundamental frequencies in the Solar System

8 60 8 8 500 Myr 1000 Myr 50 6 6 6 4000 Myr 40

4 30 4 4

20 2 2 2 10

0 0 0 0 5.00 5.25 5.50 5.75 6.00 7.40 7.45 7.50 17.0 17.2 17.4 17.6 17.50 17.75 18.00 18.25 18.50

g1 [00/yr] g2 [00/yr] g3 [00/yr] g4 [00/yr]

8 4 8 8

6 3 6 6

4 2 4 4

2 1 2 2

0 0 0 0 6.0 5.5 5.0 8.0 7.5 7.0 6.5 19.5 19.0 18.5 18.0 18.2 18.0 17.8 17.6

s1 [00/yr] s2 [00/yr] s3 [00/yr] s4 [00/yr]

×104 ×102 ×104 ×104 5 3.0 8 2.5

4 2.5 2.0 6 2.0 1.5 3 1.5 4 1.0 2 1.0 2 1 0.5 0.5

0.0 0 0.0 0 3 4 5 6 0.5 0.0 0.5 1.0 1.5 2 0 2 1 0 1 2 4 2 1 4 4 1 g5 [00/yr] ×10 +4.257 g6 [00/yr] ×10 +2.824×10 g7 [00/yr] ×10 +3.088 g8 [00/yr] ×10 +6.73×10

×103 ×104 ×105 2.5 1.4 5 1.2 2.0 4 1.0

3 1.5 0.8

0.6 2 1.0 0.4 1 0.5 0.2

0 0.0 0.0 1 0 1 0.00 0.25 0.50 0.75 1.00 0.0 0.5 1.0 1.5 3 1 3 4 1 s6 [00/yr] ×10 2.6348×10 s7 [00/yr] ×10 2.993 s8 [00/yr] ×10 6.918×10

Fig. 9. Density estimation with 95% pointwise confidence interval of the fundamental frequencies of the solar System from the {Zi} solutions in the past at 500 Myr (red band), 1 Gyr (green band) and 4 Gyr (blue band); the dashed curves with the corresponding colors denote their fitting distribution (Eqs. 19 and Table 2). The frequencies are obtained by FA over an interval of 20 Myr.

5.1. Skew Gaussian mixture model quencies of the Solar System resemble Gaussian distributions, but many of them get skewed as time passes, this is especially Laskar (2008) found that the 250 Myr-averaged marginal PDF of true for the inner frequencies. To account for this skewness, we eccentricity and inclination of the inner Solar System planets are propose the skew normal distribution as the fitting distribution to described quite accurately by the Rice distribution, which is es- the density estimation of the frequencies: sentially the distribution of the length of a 2D vector when its in- ! !! dividual components follow independent Gaussian distributions. 2 x − µ0 x − µ0 fµ0,σ0,α0 (x) = φ Φ α , (17) In in our case, the density estimations of the fundamental fre- σ0 σ0 σ0 Article number, page 11 of 18 A&A proofs: manuscript no. CDFFSS

g1 g2 g3 g4 17.925 7.45 17.28 5.75 Past 0 Fututre 7.44 17.900 5.70 17.27 Fit 7.43 17.875

0.015 0.075 0.015 0.0010 2 0 0.050 0.010 0.010 0.0005 0.025 0.005 0.005

0 2.5 2

5.0 1

s1 s2 s3 s4 5.6 17.65 18.75

0 6.75 17.70 7.00 18.80 5.8 17.75

0.1 0.03 0.05 2 0 0.25 0.02 0.01 0.0 0.00 0.00

0 1 2 2 0 2 5 3 0 4

×10 5+4.2574 g5 ×10 3+2.824×101 g6 ×10 5+3.0879 g7 ×10 5+6.73×10 1 g8 6 3 5 5.0 0 2 4 4.5 4 1 ×10 9 ×10 6 ×10 9 ×10 9 2 5

2 1 0 2

0 0 0 0 0 2 4 time [Gyr] ×10 4 2.6347×101 s6 ×10 5 2.992 s7 ×10 5 6.917×10 1 s8 50.0 3 8.0 0 8.5 52.5 4

×10 8 ×10 8 ×10 10 5 1 2 0 2.5

0 0 0.0 0 2 4 0 2 4 0 2 4 time [Gyr] time [Gyr] time [Gyr]

Fig. 10. Evolution of the parameters of the skew Gaussian distribution in the fitting model (Eqs. 19) for density estimation of the fundamental frequencies of the Solar System of {Zi} solutions in the past (blue line) and in the future (green line). The bands denote ±2 standard error. The 2 dashed orange lines denote the power-law fits for σ0 and the linear fits for µ0 and α0. where α is the parameter characterising the skewness; φ(x) de- emerge at the beginning of the diffusion stage and disappear notes the standard normal probability density distribution with quickly after. Therefore they are not included in this paramet- mean µ0 and standard deviation σ0, and Φ(x) is its cumulative ric model, whose aim is to fit the long-term PDF of the fun- distribution function given by damental frequencies. However, some persist for a long time " !# and have a small but non-negligible amplitude. To account for Z x 1 x Φ(x) = φ(t) dt = 1 + erf √ , (18) these secondary modes, we simply add Gaussian functions to −∞ 2 2 the skew Gaussian distribution and adjust for their amplitudes so that the fitting distribution is ultimately the skew Gaussian where erf denotes the error function. mixture model: Some of the frequencies, interestingly, has several secondary modes in their density estimation apart from their primary one. Xm Most of the secondary modes, if exists, are quite small compared 2 f (x) = A0 fµ0,σ0,α0 (x) + AiN(µi, σi ), (19) to the primary one. They are also often short-lived, most of them i=1 Article number, page 12 of 18 Nam H. Hoang, Federico Mogavero and Jacques Laskar: Chaotic diffusion of the fundamental frequencies in the Solar System

00 00 2 00 2 00 2 µ0 [ /yr] a [ /yr] b α0 µ1 [ /yr] σ1 [ /yr] A1 −2 g1 5.759 + 0.006 T 3.37 · 10 0.52 − 2.25 − 0.50 T −4 g2 7.448 − 0.004 T 4.17 · 10 0.70 1.38 + 0.21 T −3 g3 17.269 + 0.002 T 6.63 · 10 0.43 −3 g4 17.896 + 0.005 T 6.88 · 10 0.41 17.6755 0.0034 0.110 − 0.012 T −2 s1 − 5.652 − 0.032 T 2.68 · 10 0.83 1.12 + 0.16 T −1 s2 − 6.709 + 0.030 T 1.20 · 10 0.76 − 2.94 − 1.23 T −2 s3 − 18.773 + 0.009 T 2.86 · 10 0.56 − 3.40 − 0.08 T − 18.5256 0.0028 0.023 −2 s4 − 17.707 + 0.013 T 1.19 · 10 0.68 − 1.73 − 0.28 T −6 −10 g5 4.257454 − 2.1 · 10 T 4.63 · 10 0.88 −4 −6 g6 2.824523 − 1.4 · 10 T 1.40 · 10 0.84 −6 −10 g7 3.087957 − 1.2 · 10 T 4.80 · 10 1.11 −11 g8 6.730237 9.89 · 10 1.49 −5 −8 s6 − 2.634787 + 1.5 · 10 T 1.21 · 10 0.85 −10 s7 − 2.992527 8.31 · 10 1.39 −11 s8 − 6.917366 2.93 · 10 1.47 Table 2. Linear and power law fits for the time evolution of the parameters (Fig. 10) of the skew Gaussian mixture model (Eq. 19) for the fundamental frequencies of the Solar System. Col. 1 contains the considered secular frequencies. In col. 2, we show the linear fits of µ0 which 2 2 b represent the center of the distribution. The power law fit of σ0 has the form σ0(t) = aT (Eq. 20), where T = t/(1 Gyr), a and b are given in col. 3 and col. 4 respectively. Linear fits of the skewness parameter α0 are given in col. 5. The last three columns show linear fits of the secondary mode 2 of g4 and s3 (Eq. 19). The parameter σ0 is fitted from 200 Myr to 5 Gyr in the past, while all the other ones are fitted from 500 Myr.

Pm where i=0 Ai = 1, and m is the number of secondary modes. (mean and standard deviation) of the parameters of a given The secondary modes are much smaller than the primary mode: model, we implement a bootstrap approach based on Eq. (16), Pm A0  i=1 Ai. In our case here, m = 1 for g4 and s3, while with the assumption that pointwise standard errors of the KDE the asymptotic secondary modes for the other frequencies can estimated by MBB are independent. We remark that, with such be considered negligible. an assumption, the variance of the fitting parameters tends to be For the outer fundamental frequencies, we do not observe underestimated. significant skewness or secondary modes in their density esti- The time evolution of the mean of the parameters of the fit- mations. Therefore, the density estimations of the outer funda- ting models is shown in Fig. 10, along with their ±2 standard mental frequencies can be approximated by a simple Gaussian error, for both past and future. It turns out that the evolution of 2 distribution. σ0 is robustly fitted from 200 Myr to 5 Gyr in the past by a power The density estimations of the frequencies are shown at 3 law function well separated times in Fig. 9. The diffusion is clearly visible as σ2 t a T b, the density estimations get more and more disperse over time. 0( ) = (20) Moreover, the density estimations get more skew as time goes. where T = t/(1 Gyr), as shown in Fig. 10. For all the other pa- For g3 and g4, the skewness of the density estimations is small, rameters, we perform a linear fit. All these fits are summarised so we assume α0 = 0 (Eq. 17) for these frequencies for simplic- in Table 2. ity. The fundamental frequencies of the outer planets are very The differences between past and future evolutions are small stable. Their variations are much smaller then those of the in- and generally tend to decrease with time. Therefore, the fit for ner frequencies. Taking as an example the most unstable outer the parameters in the past given in Table 2 should also be repre- frequencies g6, its standard deviation at −4 Gyr is only about sentative of their future evolution. In general, the parameters fol- −3 00 2 · 10 /yr, while that of the most stable inner frequency g2 low relatively smooth curves with distinct tendencies. The skew- is 15 times larger. Therefore, when considering a combination ness parameters α0, increasing in absolute value, show that the involving an inner fundamental frequency and an outer one, the PDFs of the inner fundamental frequencies get more and more latter can be effectively regarded as a constant. skew over time. The center of the distributions, indicated by µ0, The result of our fitting model is also plotted in Fig. 9. It 2 does not change significantly compared to α0 and σ0 (Fig. 10). is remarkable that the density estimations of the frequencies are The secondary modes of g4 and s3 are also quite stable. well approximated by the fitting curve over 3 different epochs. The diffusion of the frequencies is quantified by the increas- It should be noticed that the base of our fitting model - the skew 2 ing σ0, which is closely linked to their distribution variance. As Gaussian distribution only has 3 parameters. Only for some fre- the exponents b of the power laws in Table 2 are different from quencies, g4, s3 for examples, more parameters are needed. Nev- unity, the chaotic diffusion of the fundamental frequencies turns ertheless, the additional parameters only account for the minor out to be an anomalous diffusion process. Interestingly, all the features and 3 parameters are sufficient to represent the bulk of inner frequencies clearly undergo subdiffusion, that is the expo- the density estimations over a long time scale. nent of the power law b is smaller than 1 (b = 1 corresponding to Brownian diffusion). Therefore, an extrapolation of the vari- 5.2. Evolution of the parameters ance of the inner frequencies based on the assumption of linear diffusion over a short time interval would generally lead to its The parameters of our fitting models are extracted by the method overestimation over larger times. On the contrary, the exponents of least squares, implemented by the routine curve_fit in the b are either smaller or larger than unity for the outer frequen- scipy package in Python. To retrieve the statistical distribution cies. It should be noted that, because the variations of the outer

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Density Estimation, 95% CI

80 100 Myr 210 Myr 400 Myr 60

40

20

0 400 405 410 400 405 410 400 405 410

Pg2 g5 [kyr] Pg2 g5 [kyr] Pg2 g5 [kyr]

25

20

15

10

5

0 170 175 180 170 175 180 170 175 180

Ps3 s6 [kyr] Ps3 s6 [kyr] Ps3 s6 [kyr]

Fig. 11. Density estimation with 95% pointwise confidence interval of the period of g2 − g5 (top row) and s3 − s6 (bottom row) from the combining solutions from {Xi}, {Yi} an {Zi}. Frequencies are obtained by FA over an interval of 20 Myr centering at −100 Myr (left column), −210 Myr (middle column) and −400 Myr (right column).

frequencies are very small, the value of the corresponding ex- s3 − s6 can be used as astronomical metronomes is their stability. ponents b might be overestimated due to the finite precision of Indeed, their uncertainty has to be small to be reliably used for frequency analysis. practical application. In previous work, the uncertainty of the frequency combi- nation was derived from the analysis of a few solutions (Laskar 6. Geological application et al. 2004, 2011a; Laskar 2020). Here we use a much larger The aim of this project is to have a reliable statistical picture of number of solutions and the KDE/MBB method to derive both the secular frequencies of the Solar System beyond 60 Myr. It the PDF of the frequencies and their statistical errors. Starting is interesting to put recent geological results in this astronomi- from the results of all the sets of orbital solutions (Sect. 4), we cal framework. First, this can be used as a geological test of our produce the compound density estimation of the fundamental study, and secondly the application provides a glimpse of how frequencies and their relevant combinations following the con- astronomical data could be used in a cyclostratigraphy study. We servative approach in Eq. (16). These data are archived with the first show how the uncertainty of a widely-used dating tool in as- paper. trochronology can be quantified, then we apply our work on two With this result, we can reliably estimate the uncertainty recent geological results, which are from the Newark-Hartford of the astronomical metronomes. The uncertainty of the cycles data (Olsen et al. 2019) and Libsack core (Ma et al. 2017). g2 − g5 and s3 − s6 is in fact the density width of the period of these frequency combinations, which is shown in Fig. 11. As time goes, the two metronomes get more uncertain as their den- 6.1. Astronomical metronomes sity spreads due to the chaotic diffusion, but they are still reliable. Although it is not possible to recover the precise planetary or- At 400 Myr in the past, the relative standard deviations (ratio of − − bital motion beyond 60 Myr, some astronomical forcing compo- standard deviation to mean) of g2 g5 and s3 s6 metronomes nents are stable and prominent enough so that they can be used to are approximately 0.4% and 1.26%, respectively. calibrate geological records in the Mesozoic Era or beyond (see (Laskar 2020) for a review). The most widely-used is the 405- 6.2. Newark-Hartford data kyr eccentricity cycle g2 − g5, which is the strongest component of the evolution of Earth’s eccentricity. Recently the inclination In (Olsen et al. 2019), the astronomical frequencies were re- cycle s3 − s6 has also been suggested for the time calibration of trieved from a long and well-reserved lacustrine deposit. The stratigraphic sequences (Boulila et al. 2018; Charbonnier et al. frequency signals of the Neward-Hartford data are very sim- 2018). Although s3 − s6 is not the strongest among the obliquity ilar to that of the astronomical solution La2010d, which was cycles, it is quite isolated from other cycles and thus easy to be taken among 13 available astronomical solutions (Laskar et al. identified (Laskar 2020). The main reason that g2 −g5 or possibly 2011a). Having 120 000 astronomical solutions at hand, we can

Article number, page 14 of 18 Nam H. Hoang, Federico Mogavero and Jacques Laskar: Chaotic diffusion of the fundamental frequencies in the Solar System

Density Estimation, 95% CI

10 10 210 Myr NH La2010d 5 5 Current Estimation

0 0 10.0 10.2 10.4 10.6 10.8 11.0 5.0 5.2 5.4 5.6 5.8 6.0

g4 g2 [00/yr] g1[00/yr] 10 60

40 5 20

0 0 9.5 9.6 9.7 9.8 9.9 10.0 10.1 10.2 7.40 7.42 7.44 7.46 7.48 7.50 7.52

g3 g2 [00/yr] g2[00/yr] 10 10

5 5

0 0 0.4 0.6 0.8 1.0 1.2 16.9 17.0 17.1 17.2 17.3 17.4 17.5 17.6 17.7

g4 g3 [00/yr] g3[00/yr] 10 10

5 5

0 0 1.4 1.6 1.8 2.0 2.2 2.4 17.6 17.8 18.0 18.2 18.4

g2 g1 [00/yr] g4[00/yr]

Fig. 12. Density estimation with 95% pointwise confidence interval (blue colors) of the secular frequencies (right column) and their combinations (left column) of the combined solutions from {Xi}, {Yi} and {Zi}, along with the corresponding values extracted from from the La2010d solution (yellow lines) and Newark-Hartford data (red lines) (Olsen et al. 2019) and the current values (green lines). The frequencies are obtained by FA over an interval of 20 Myr centering at 210 Myr in the past. The time reference is the time of {Zi}. derive a more precise statistical analysis of this result. The fun- and their combinations (left) extracted from NH data with our damental frequencies from the geological record are obtained in density estimation around the same time in the past. For (g)i=1,3, the following way. The data were dated by the relatively sta- both the geological and La2010d lie relatively close to the main ble 405-kyr cycle (g2 − g5) and additionally verified by a zircon peak of their corresponding density while their values for g4 are U–Pb-based age model (Olsen et al. 2019). A frequency analysis near the tail. The frequency combinations tell the same story, as (Sect. 2.2) was performed on the geological data as well as on the geological values involving g4 are off from the main peak. the eccentricity of the astronomical solution La2010d to retrieve Yet, they are all consistent with our density estimation as there their strongest frequencies. The frequency analysis of geologi- is a non-negligible possibility of finding a secular solution that cal data are cross-checked with that of astronomical solution to agrees with the geological data. It should be noted that certain identify its orbital counterpart. For example, the third strongest frequencies are significantly correlated, g3 and g4 for example. frequency 12.989 00/yr from Newark data is identified with the We cannot assume that they are independent, therefore we have second strongest frequency 12.978 00/yr from La2010d, which is to calculate the density estimation of their combination directly. g3 − g5. The frequency g3 is then derived by summing the ob- Given the unavailability of the uncertainty in geological fre- tained combination with g5, which is nearly constant. The same quencies, the probability of finding the geological frequencies in goes for g1, g2 and g4. Definite values of astronomical funda- a numerical orbital solution cannot be directly obtained. How- mental frequencies are thus determined. Unfortunately, the un- ever, we can use La2010d, which is the solution from the com- certainty estimation of the geological frequencies is yet not avail- plete model of the Solar System that matches the best with able. NH data, as the benchmark for our secular statistics. There Our density estimation of the frequencies is obtained by are several criteria to determine how good a solution is, i.e. combining all our samples in a conservative approach specified how well it could match with the geological data. A simple in section 3.4. Figure 12 compares the secular frequencies (right) and rather straightforward criterion that we will use is δ =

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Density Estimation, 95% CI, batch {Zi}

10 90 Myr 0.2 6 0.2 8 0.0 0.0 0.9 1.0 1.1 18.1 18.2 18.3 6 4

4 2 2

0 0

10 300 Myr 0.2 6 0.2 8 0.0 0.0 0.9 1.0 1.1 18.1 18.2 18.3 6 4

4 2 2

0 0 0.4 0.6 0.8 1.0 1.2 17.6 17.8 18.0 18.2 18.4

g4 g3 [00/yr] g4[00/yr]

Fig. 13. Density estimation with 95% pointwise confidence interval (red colors) of g4 − g3 (left column) and g4 (right column) at −90 Myr (first row) and −300 Myr (second row) of {Zi}, overlaid by their histograms (gray blocks) for better visualisation. Frequencies values are obtained by frequency analysis over an interval of 10 Myr.

7.0% 20.0% 80Myr-90Myr 80Myr-90Myr 17.5%

6.0% ) r ) y / r 0

0 15.0% y

/ 5.0% 0 0 9 . 9 0

. 12.5% 0 4.0% > 3 >

g 10.0% 3

g 3.0% 4 7.5% g 4 ( r g y

( 2.0% M 5.0% 0

1 1.0% 2.5%

0.0% 0.0% 50 100 150 200 250 300 350 400 450 50 100 150 200 250 300 350 400 450 time [Myr] time [Myr]

Fig. 14. Percentage of the solutions (black line) whose g4 − g3 > 0.9 at each time (left) and over a 10 Myr interval (right) and their 90% CI estimation uncertainty (larger gray band) and their standard errors (smaller gray band). The time interval of Libsack record, which is 80 Myr −90 Myr, is denoted by the red band in the left plot and by a red line in the right plot. q 1 P4 ∗ 2 ∗ the complete model of the Solar System is thus (1.37%, 33.31%) (gi − g ) , where gi, g are the frequencies from as- 4 i=1 i i (Wilson 1927). Therefore, with the criterium δ, our result is sta- tronomical solution and geological data respectively. A better- tistically compatible with that of (Olsen et al. 2019). suited solution will have smaller δ and vice versa. We found that in the range from −200 Myr to −220 Myr, out of the 120 000 so- lutions, there are around 5000 solutions (accounting for roughly 4.2% of the total number) having smaller s than that of La2010d 6.3. Libsack core at -210 Myr, which is the value originally used to compared with the geological data. It should be noted that La2010d is one of 13 available complete solutions. The 95% confidence interval of Laskar (1990, 1992) presented several secular resonances to ex- the probability of obtaining such a good matching solutions from plain the origin of chaos in the Solar System. In particular, the argument of the resonance (s4 − s3) − 2(g4 − g3) is currently in a

Article number, page 16 of 18 Nam H. Hoang, Federico Mogavero and Jacques Laskar: Chaotic diffusion of the fundamental frequencies in the Solar System librational state, that is We have benchmarked the secular model by sampling the initial conditions in different ways, then compare the density es- (s4 − s3) − 2(g4 − g3) = 0 (21) timation of their solutions between themselves and finally with the complete model. The results are two folds. First, regardless and moves out to the rotational state around −50 Myr. The dy- of how initial conditions are sampled, their density estimation namics can even switch to the librational state of a new reso- will converge towards a single PDF; after this overlap, a robust nance: estimation is guaranteed. Secondly, the density estimation of the secular model is compatible with that of the complete model of (s4 − s3) − (g4 − g3) = 0. (22) the Solar System. This agreement means that results of the sec- ular model, with superior computational simplicity, can be used This transition corresponds to a change from the 2:1 resonance for application on geological data. − − to the 1:1 resonance of two secular terms g4 g3 and s4 s3. We observe that the density estimations of the fundamental Because of this resonance, the frequencies can be well fitted by skew Gaussian mixture models. Ma et al. (2017) found a sudden change of the period of a 2 The time evolution of the parameters σ0, related to the frequency long cycle from 2.4 Myr to 1.2 Myr in the Libsack core of the variances, follows power-law functions. Interestingly for the in- Cretaceous basin from around −90 Myr to −83 Myr. This change ner fundamental frequencies, the exponents of such power laws was also visible in La2004 astronomical solution, and the long are all smaller than 1, which indicates that they undergo subdif- cycle was attributed to frequency combination g4 − g3, visible fusion processes. from the spectrum of the eccentricity of the Earth. Although the We show several examples of how this result can be used for exact value before and especially after the transition is not clear, geological applications. First, the uncertainty of any astronom- the change of the period is visible from the band power of the ical frequency signal is fully quantified, so that, for example, a core (Fig. 1 of Ma et al. (2017)). proper quantitative response can be now made for the question of This change of g4 − g3 observed in the Libsack core corre- how stable are the astronomical metronomes. With this statisti- sponds to a transition from the resonance (s4 − s3) − 2(g4 − g3), cal framework, previous results from geological records beyond which is the resonance that the Solar System is currently at, to 60 Myr can also be interpreted with a more comprehensive ap- the resonance (s4 − s3) − (g4 − g3). With a large number of astro- proach. A more quantitative answer, not only about the possibil- nomical solutions, we could understand better this phenomenon. ity but also about the probability of occurrence of an astronomi- The transition is usually very fast: the frequency changes quickly cal signal in geological data can be made. Apart from these direct to another value and then reverts back also as quickly. Therefore, applications, a more systematic approach could fully make use to study the transition we use a smaller window for the frequency of the density estimation of frequencies. The method TimeOpt analysis, 10 Myr instead of 20 Myr. Figure 13 shows the density by Meyers & Malinverno (2018), for example, shows that it is estimation of g4 − g3 at 90 Myr as well as 300 Myr in the past. 00 possible to combine both the uncertainty from astronomical sig- Both has a principle population in the range [0.4, 0.8] /yr, and a nals and geological records to derive an effective constraint for small but non-insignificant one centered around 1.0 00/yr, which 00 both. In fact, any similar Bayesian method could use the density corresponds to the small chunk of g4 centered at 18.2 /yr. The estimation of frequencies as proper priors. transition observed is a sudden jump of frequency from the main population to the secondary one of g , and therefore g − g as Acknowledgements. NH is supported by the PhD scholarship of CFM Founda- 4 4 3 tion for Research. This project has received funding from the European Research well. Council (ERC) under the European Union’s Horizon 2020 research and inno- The size of the secondary population is defined as the pro- vation programme (Advanced Grant AstroGeo-885250) and from the French 00 portion of the solutions whose g4 − g3 > 0.9 /yr and denoted Agence Nationale de la Recherche (ANR) (Grant AstroMeso ANR-19-CE31- 00 0002-01). as P(g4 − g3 > 0.9 /yr). The rate of transition is defined as the 00 proportion of the solutions whose g4 − g3 > 0.9 /yr over 10 00 Myr and denoted as P10Myr(g4 − g3 > 0.9 /yr). Both are shown in Fig 14. During the predictable period that is from now to −50 References Myr, no transition is observed. After −50 Myr, a transition can occur, its rate rises until 100 Myr, when the percentage of sec- Bosq, D. & Guégan, D. 1995, Statistics & probability letters, 25, 201 Boulila, S., Vahlenkamp, M., De Vleeschouwer, D., et al. 2018, Earth and Plan- ondary population stay relatively stable at around 2.1% ± 0.5% etary Science Letters, 486, 94 at a time, and the rate of the transition could also be determined Calonico, S., Cattaneo, M. D., & Farrell, M. H. 2018, Journal of the American to be 8% ± 1% every 10 Myr during this period. At 80-90 Myr, Statistical Association, 113, 767 when the transition was detected in the Libsack core, with our Charbonnier, G., Boulila, S., Spangenberg, J. E., et al. 2018, Earth and Planetary numerical solutions, that rate of transition during this period is Science Letters, 499, 266 Cheng, G., Chen, Y.-C., et al. 2019, Electronic Journal of Statistics, 13, 2194 found to be 7.7% ± 4.5%. Efron, B. 1979, Ann. Statist., 7, 1 Fermi, E. 1949, Physical review, 75, 1169 Gastineau, M. & Laskar, J. 2011, ACM Communications in Computer Algebra, 7. Conclusion 44, 194 Gradstein, F. & Ogg, J. 2020, in Geologic Time Scale 2020 (Elsevier), 21–32 In this work, we give a statistical description of the evolution Gradstein, F. M., Ogg, J. G., Schmitz, M., & Ogg, G. 2012, The geologic time of the fundamental frequencies of the Solar System beyond 60 scale 2012 (elsevier) Gradstein, F. M., Ogg, J. G., Smith, A. G., Bleeker, W., & Lourens, L. J. 2004, Myr, that is beyond the predictability horizon of the planetary Episodes, 27, 83 motion, with the aim to quantify the uncertainty induced by its Hall, P., Horowitz, J. L., & Jing, B.-Y. 1995, Biometrika, 82, 561 chaotic behaviour. The base of our analysis is 120 000 orbital Hang, H., Steinwart, I., Feng, Y., & Suykens, J. A. 2018, The Journal of Machine solutions of the secular model of the Solar System. The PDF Learning Research, 19, 1260 Hart, J. 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