A&A 615, A153 (2018) https://doi.org/10.1051/0004-6361/201732349 Astronomy c ESO 2018 & Astrophysics

Solar barycentric dynamics from a new solar-planetary ephemeris Rodolfo G. Cionco1 and Dmitry A. Pavlov2

1 Comisión de Investigaciones Científicas, Provincia de Buenos Aires (CIC) – Universidad Tecnológica Nacional (UTN), Colón 332, San Nicolás, Buenos Aires, Argentina e-mail: [email protected] 2 Institute of Applied Astronomy of the Russian Academy of Sciences (IAA RAS), nab. Kutuzova 10, St. Petersburg 191187, Russia e-mail: [email protected]

Received 23 November 2017 / Accepted 18 April 2018

ABSTRACT

Aims. The barycentric dynamics of the Sun has increasingly been attracting the attention of researchers from several fields, due to the idea that interactions between the Sun’s orbital motion and solar internal functioning could be possible. Existing high-precision ephemerides that have been used for that purpose do not include the effects of trans-Neptunian bodies, which cause a significant offset in the definition of the solar system’s barycentre. In addition, the majority of the dynamical parameters of the solar barycentric orbit are not routinely calculated according to these ephemerides or are not publicly available. Methods. We developed a special version of the IAA RAS lunar–solar–planetary ephemerides, EPM2017H, to cover the whole Holocene and 1 kyr into the future. We studied the basic and derived (e.g., orbital torque) barycentric dynamical quantities of the Sun for that time span. A harmonic analysis (which involves an application of VSOP2013 and TOP2013 planetary theories) was performed on these parameters to obtain a physics-based interpretation of the main periodicities present in the solar barycentric movement. Results. We present a high-precision solar barycentric orbit and derived dynamical parameters (using the solar system’s invariable plane as the reference plane), widely accessible for the whole Holocene and 1 kyr in the future. Several particularities and barycentric phenomena are presented and explained on dynamical bases. A comparison with the Jet Propulsion Laboratory DE431 ephemeris, whose main differences arise from the modelling of trans-Neptunian bodies, shows significant discrepancies in several parameters (i.e., not only limited to angular elements) related to the solar barycentric dynamics. In addition, we identify the main periodicities of the Sun’s barycentric movement and the main giant planets perturbations related to them. Key words. astrometry – ephemerides – planets and satellites: general – Sun: general – solar-terrestrial relations

1. Introduction and Clemence published in 1951. Jose analysed the solar motion in the period 1610–2010, and described the orbital loops and Extrasolar planetary systems are natural laboratories for learning some dynamical parameters of the Sun as its barycentric range r about new features and provide insights on stars and planetary (the magnitude of the barycentric position) and the variation T interactions. These new traits include evidence of photospheric of the modulus L of the orbital angular momentum (T = dL/dt). signatures associated with a possible magnetic star–planet in- He noted basic periodicities coming from Jupiter and Saturn, and teraction (Lanza 2013), and axial tilt variations in a young star the pacemaker effect of the alignment of the giant planets on the due to magnetic torques (Chang et al. 2012; both mechanisms barycentric dynamical parameters studied with a certain period- involve close-in hot-Jupiter exoplanets. Interestingly, our solar icity that we call the Jose-period, that is, the series of dynami- system might also provide empirical evidence (whose interpre- cal parameters that seem to repeat with a periodicity estimated tation has been framed in theoretical explanations) on a possible by Jose of 179 yr. Nevertheless, the period of this repetition is subtle interaction between the Sun and the solar system bodies. variable and∼ we identify this quantity just with a nominal Jose- We refer to the hypothetical modulating effect of planetary grav- period. Fairbridge & Shirley(1987) and Charvátová(1988) also itation on the solar magnetic activity (the solar cycles), which found that very definite patterns in solar inertial motion are re- has been discussed since long ago and recently has also been peated every Jose-period, in which the Sun alternates between a taken into account in the context of the extrasolar planetary trefoliar-like (quasi-symmetric) orbit and a lapse of “disordered” systems (Perryman & Schulze-Hartung 2011). From the field motion. The first orbital phase was arbitrarily determined over an of astronomy and astrophysics, but fundamentally from geo- 50 yr period, which is dominated by the first harmonic of the sciences, several authors have taken into consideration some Jupiter–Saturn∼ synodic period (JS period), that is, JS/2, 9.93 yr, dynamical parameters of the Sun’s orbit around the solar sys- with the remaining portion with a period of about∼ 120 yr tem barycentre. Diverse orbital dynamical parameters of the Sun (Charvátová 2009). These findings (directly or indirectly) have have been used for this task, but most were obtained from very motivated the wide use of solar barycentric dynamics, especially different sources or methods, including synthetic planetary theo- (but not limited to) the analysis of the long-term solar ries, two-body calculations, numerical simulations, unpublished magnetic cycle variability and the development of theoretical data and ephemerides, and standard high-precision ephemerides. mechanisms capable of explaining them. Whatever the under- The first description of what is known as solar inertial motion lying physical mechanism present in this issue, if any, it seems was reported by Jose(1965; see also Wood & Wood 1965), clear that it must be looked for in relation to a dynamical theory using orbital integrations of giant planets by Eckert, Brouwer which should be based on solar barycentric dynamical quantities.

Article published by EDP Sciences A153, page 1 of 11 A&A 615, A153 (2018)

Solar barycentric position, r, velocity, u, and acceleration, of the solar barycentric dynamics for the whole Holocene, a time a, are routinely calculated in high-precision ephemerides: the period of great geophysical importance. Our aim is thus twofold: JPL DE series (Folkner et al. 2014), the Institut de Mécanique first, to produce a tool for the full Holocene, which is publicly Céleste et de Calcul des Éphéméides INPOP series (Fienga et al. available, analysed in terms of the dynamical astronomy and 2008), and the Institute of Applied Astronomy EPM series second, to present and describe the results for a wide scientific (Pitjeva & Pitjev 2014). Modern versions of the DE, INPOP, and audience. EPM ephemerides apply similar dynamical models of the Sun’s barycentric motion that treat the Sun as a regular body in the 2. EPM2017H, a new long-term high-precision solar system. They use the relativistic definition of the barycen- ephemeris tre, take into account not only Newtonian forces between major planets, but also relativistic terms, the non-uniform gravitational Until recently, the last ephemeris released by IAA RAS potential of the Sun (J2 term only), the Earth and Moon (main was EPM2015 (see for details Pitjeva & Pitjev 2013, 2014). harmonics), the influence of asteroids and dwarf planets, etc. EPM2015 covers the period from 1787 to 2014. A new ver- However, there is no 100% correspondence between the formu- sion of EPM has recently been released: EPM20172. The most lations used at different institutions. Because of the inclusion of important differences with previous versions are the inclusion these more complete models, we prefer to name the solar move- of the Lense–Thirring acceleration; definition of the relativistic ment around the solar system’s barycentre, solar barycentric mo- barycentre; modelling of the acceleration of the Sun as a regular tion (SBM), to describe a more accurate modelling of the solar body (in previous versions the Sun was explicitly placed to keep movement than just the inertial reaction to punctual planetary the barycentre at the origin); models of the asteroid belt and the N-body forces. Kuiper belt not as a uniform annulus and ring, but as 180 and The Sun’s barycentric r and u are easily obtained through 160 uniformly distributed point-masses, respectively (which in different means such as web services associated with these turn allows their influence on the orbit of the Sun to be mod- ephemerides; however, other solar dynamical parameters such elled). Also, some recent observations of spacecraft and lunar as acceleration or orbital angular momentum, with all high- laser ranging normal points that appeared after EPM2015 were precision ephemerides, L, are not. In addition, the time span included in the EPM2017 solution. We develop a special version covered by high-precision ephemerides is, in general, short of EPM2017 to cover the whole Holocene and 1 kyr into the fu- on geophysical timescales. As an exception, the JPL DE431 ture. We call it EPM2017H, and it covers the time span from ephemeris covers about 30 kyr as a whole, from the year 13201 10107 BC to AD 3036. Two exclusive modifications were made BC to the year 17191 AD, hence it includes all of the Holocene to the EPM2017H model. First, the latest formulation for the (the recent post-glacial geological period from about 12 kyr precessional quantities of the Earth valid over thousands of years to the present). To date, DE431 is the only high-precision− was included (Vondrák et al. 2011). This formulation, in addition ephemeris officially published that covers such a long time to the standard IAU theory of nutation IAU2000A, enables us to span. While the above-mentioned high-precision ephemerides also obtain the Milankovic´ astro-climatic elements for the Earth are in good agreement with each other, the EPM series has taking into account short-term perturbations (Cionco & Soon an important difference from DE and INPOP: the EPM model 2017). Second, the lunar model of EPM2017H does not have includes the 30 largest trans-Neptunian objects (TNOs) in addi- friction between crust and core in order to avoid exponential tion to Pluto, which cause a significant offset in the definition growing of the angular velocity of the lunar core in the past. of the solar system barycentre (Pitjeva 2010; Pitjeva & Pitjev The lunar parameters of the solution were then re-fitted back to 2013, 2014). The drift induced by the TNOs in the calculation the EPM2017H model. of the solar system barycentre is about 6 km in just one decade As usual, we date the Holocene from about 12 kyr ago, so (Pitjeva 2010). This difference with respect to the other existing we start our study of SBM from about 10 000 BC; exactly, high-precision ephemerides can be seen (on timescales covering from 1/1/ 9999, JD = 1931076.0 (in barycentric dynamical − − 100–150 yr) as a quasi-constant offset in the location of the time, TDB); approximate Julian epoch 9998.9624, to AD 3000, − solar system barycentre in the ephemeris comparison performed JD = 2816795.0; with a time step of 1 d. The date expressed in by the IAU WG41. In addition to the 30 largest TNOs, the EPM years, as a real number, for a corresponding Julian day (JD) is model includes a uniform TNO ring that causes a smaller change JD 2451545.0 t = 2000.0 + − (1) in the Sun’s position with respect to the solar system barycentre 365.25 · (hereafter the barycentre). In what follows, when we mention “time in years” we refer to The impact of the differences in the barycentre determination this Julian epoch (in TDB timescale) given by Eq. (1) as the in- on solar movement is an important issue that must be addressed. dependent variable of the dynamics. This barycentric offset may produce important differences in The longest solar-planetary high-precision ephemeris is SBM angular quantities and orientation changes in vectorial DE431, which has been available since mid-20133. As men- quantities regarding inertial and non-inertial directions. In addi- tioned above, DE431 is in good agreement, inside the corre- tion, changes in the magnitudes of the derived solar dynamical sponding temporal lapse, with all high-precision ephemerides, parameters (e.g., orbital angular momentum), are also expected that is, INPOP4 and also the EPM ephemerides. A first compar- because TNO bodies produce significant torque on SBM due ison DE431-EPM2017H is performed defining the percentage to their heliocentric distances. We are also aware that accessing difference, in position, velocity, and barycentric acceleration of long-term planetary ephemerides and the posterior calculations the Sun of associated solar dynamical parameters can be a difficult task x x for users who are not experts in dynamical astronomy. A stan- ∆x = | D − E| 100, (2) | | x × dardized tool for this kind of study for widespread use in solar | D| physics and geosciences is lacking, specifically a reconstruction 2 http://iaaras.ru/en/dept/ephemeris/epm/2017/ 3 ftp://ssd.jpl.nasa.gov/pub/eph/planets/Linux/de431 1 http://iaucom4.org/ephemerides.html 4 http://www.imcce.fr

A153, page 2 of 11 R. G. CioncoCionco and & D. Pavlov: A. Pavlov: Solar Solar barycentric barycentric dynamics dynamics Cionco & Pavlov: Solar barycentric dynamics AD 1723.4716 - AD 1783.2745 0.004 AD 1723.4716 - AD 1783.2745 0.004 |∆v| ∆v -10 -8 -6 -4 -2 0 2 4 6 8 10 0.0035 | | 10 -10 -8 -6 -4 -2 0 2 4 6 8 1010 0.0035 10 10 0.003 8 8 8 8 0.003 6 6 6 6 0.0025 4 4 0.0025 4 4 2 2 0.002

au 2 2 0 0

0.002 -3 au 0 0 -3 0.0015 10 -2 -2

0.0015 10 -2 -2 -4 -4 0.001 -4 -4 0.001 -6 -6 -6 -6 -8 -8 0.0005 -3 -8 Solar radius ≈ 5x10 au -8 Difference in barycentric velocity (%) 0.0005 -10 Solar radius ≈ 5x10-3 au -10 Difference in barycentric velocity (%) 0 -10-10 -8 -6 -4 -2 0 2 4 6 8 10 -10 −10000 0 −9500 −9000 −8500 −8000 −7500 −7000 −6500 −6000 -10 -8 -6 -4 -2-3 0 2 4 6 8 10 −10000 −9500 −9000 −8500 −8000 −7500 −7000 −6500 −6000 10 au Time [yr] 10-3 au Time [yr] Fig.Fig. 1. 1.PercentagePercentage di difffferenceserences in in the the solar solar barycentric barycentric velocity velocity vectors vectors Fig.Fig. 2. 2.Trefoliar,Trefoliar, particularly particularly quasi-closed quasi-closed 6060 yr yr pattern pattern of of SBM SBM for for Fig. 1. Percentage differences in the solar barycentric velocity vectors Fig. 2. Trefoliar, particularly quasi-closed∼∼ 60 yr pattern of SBM for betweenbetween DE431 DE431 and and EPM2017H EPM2017H (Eq. (Eq. (2)) 2) for for the the indicated indicated years years (one (one thethe indicated indicated interval. interval. The The first first JS JS period period is is marked marked∼ as as a a thick thick line. line. The The datumdatumbetween per per Julian DE431Julian year). year). and The EPM2017H The550550 yr oscillation (Eq.yr oscillation 2) for due the to due indicated dwarf to dwarf planet years planet Eris (oneintervalintervalthe indicated corresponds corresponds interval. to to the the The closest closest first JS approximation approximation period is marked to to theA the as& initialA initiala thickproofs: position position line.manuscript The no. 32349_corr datum per Julian year).∼ The∼ 550 yr oscillation due to dwarf planet interval corresponds to the closest approximation to the initial position canEris be can clearly be clearly seen. seen. ∼ onon the the IP. IP. The Thex-axis x-axis of of the the figure figure coincides coincides with with the the nodal nodal line line equator- equator- Eris can be clearly seen. IPIP (seeon (see the Appendix Appendix IP. TheA x-axis Afor for further of further thefigure information). information). coincides with the nodal line equator- IP (see Appendix A for further information). whereDE431 vector is in goodx agreement,to the set insider, u, a theand corresponding sub-indexes temporal D and DE431 is in good∈ agreement,{ inside} the corresponding4 temporal 9 <60−yr> <60−yr> Elapse, refer with to each all high-precision ephemerides,respectively. ephemerides, In i.e. addition, INPOP weand also also 4 stead, we adopted a 60 yr period, which is more meaning- 7 exploredthelapse, EPM with the ephemerides. percentage all high-precision A di firstfference comparison ephemerides, of magnitude, DE431-EPM2017H i.e. INPOP in position,and also is fulstead, from we a dynamical adopted a∼ point60 of yr view, period, because which it is corresponds more meaning- to −3 the EPM ephemerides. A first comparison DE431-EPM2017H is ∼ 5 velocity,performed and defining barycentric the percentage acceleration, diff whereerence, the in magnitudesposition, veloc- of ful from a dynamical point of view, because it corresponds to / 10 a quasi-complete Jupiter–Saturn synodic cycle of oppositions- r performed defining the percentage difference, in position, veloc- 3 theity, corresponding and barycentric vectors acceleration were used of thein turn. Sun The result of this set conjunctionsa quasi-complete (each Jupiter–Saturn synodic configuration synodic is cycle advanced of oppositions- by ap- 1 ity, and barycentric acceleration of the Sun conjunctions (each synodic configuration is advanced by ap- ∆ λ of comparisons shows differences smaller than 1.7%, 0.0039%, proximately 120◦ at each JS period). More specifically, this pe- UN xD xE 180 proximately 120◦ at each JS period). More specifically, this pe- and∆x 0.0036%= | xD− inx|E∆r 100, ∆u, , and ∆a , respectively. No significant(2) riod corresponds to the second most important perturbation over 135 | ∆|x = | xD− | ×| | |100|, | | (2) riod corresponds to the second most important perturbation over CMs secular| | growth| | can be× seen on these comparisons. As expected, orbital longitudes in the Jupiter–Saturn pair, produced by the 90 xD ∆ λ thewhere differences vector| inx| magnitudeto the set valuesr, v, a areand smaller. sub-indexes The mostD inter-and E trigonometricorbital longitudes argument in theλ Jupiter–Saturn2 λ , where pair,λ and producedλ represent by the 45 ∈ { } J S J S estingreferwhere toresult each vector is ephemerides, onx velocityto the where, respectively. set r, exceptv, a and In for addition, sub-indexes a minute we drift, alsoD and the ex-E thetrigonometric corresponding argument mean orbitalλ−J longitudes2 λS, where ofλ bothJ and planets,λS represent or in refer to each ephemerides,∈ respectively.{ } In addition, we also ex- the corresponding mean orbital− longitudes of both planets, or in 180 plored550 yr theperiod percentage of dwarf di planetfference Eris of can magnitude, be detected in position, by simple ve- short, the (1, 2) ‘inequality’ (see Sect. 4). Each inner equilateral JS 135 ∼ inspectionlocity,plored and the (Fig. barycentric percentage1). acceleration, difference of where magnitude, the magnitudes in position, of the ve- short, the (1−, 2) ‘inequality’ (see Sect. 4). Each inner/ equilateral 90

closed loop in Fig. 2 corresponds to Jupiter–Saturn JS 2 period. ∆ λ locity, and barycentric acceleration, where the magnitudes of the closed loop in− Fig. 2 corresponds to Jupiter–Saturn JS/2 period. 45 corresponding vectors were used in turn. The result of this set The first complete loop of JS period, 19.86 yr, is marked with a 0 corresponding vectorsff were used in turn. The result of this set The first complete loop of JS period,∼ 19.86 yr, is marked with a 1400 1450 1500 1550 1600 1650 1700 3.of SBM comparisons from EPM2017H shows di erences smaller than 1.7 %, 0.0039 %, thick line. In Fig. 2, after a 60 yr period∼ (between the years andof 0.0036comparisons % in shows∆r , ∆ div ,ff anderences∆a , smaller respectively. than 1.7 No %, significant 0.0039 %, 1723.4716thick line. and In Fig. 1783.2745) 2, after athe∼ projection60 yr period of (between the Sun’s the move- years Time [yr] and 0.0036 % in| ∆|r|, ∆|v , and| ∆|a , respectively. No significant 1723.4716 and 1783.2745) the∼ projection of the Sun’s move- 3.1.secular General growth description can| be| seen| | on these| | comparisons. As expected, ment on the IP tends to occupy the same position after that time thesecular differences growth in can magnitude be seen values on these are comparisons. smaller. The As most expected, inter- Fig.span,Fig.ment 3. 3.Basic withBasic on thedynamical a dynamical magnitude IP tends parameters parameters to of occupy the for di for theff twoerence two same Jose-periods Jose-periods inposition spatial of of after 177 position 177 yr that yr (con- (con- time of Fig. 4. Relative position of icy and gaseous giant planets and their ef- tinuoustinuous line). line). The The intervals intervals of of trefoliar trefoliar 60 60 yr yr orbits orbits are are indicated indicated (dotted (dotted fect on r over two Jose-periods. The disordered orbits begin when Sat- Dueestingthe to di resultthefferences mass is on and in velocity magnitude semi-major where, values axis except are of Jupiter,smaller. for a minute SBM The most drift, is per- inter- the 3.span,4 10 with5 au, ai.e. magnitude less than of 51 the 000 diff km.erence Not inall spatial60 yr position trefoliar of 3 lines).lines). All All− the the5 quantities quantities are are reckoned reckoned in in astronomical astronomical units units and and days. days. urn and the Uranus–Neptune pair tend to be aligned (near conjunction) formedesting550 atyr result a period mean isof on barycentric dwarf velocity planet where, range Eris of except can5 be for10 detected− a minuteau, as by can drift, simple be the orbits3.4× are10 so− closed,au, i.e. lesshowever. than 51 000 km. Not all∼ 60 yr trefoliar seen∼ in Fig.2, where we have selected a∼ period× of time related TheThe “form” ‘form’× of of the the series series seems seems to to be be repeated, repeated, although although more∼ more important important being Jupiter near quadrature. All the quantities are reckoned in astro- inspection550 yr (Fig. period 1). of dwarf planet Eris can be detected by simple orbitsIn Fig. areso 3, closed, we plot however. the magnitudes of barycentric position to a∼ trefoliar-like solar orbit after Charvátová(2009). To present changeschanges can can appear appear in in the the kinematics kinematics series series trough trough time. time. For For example, example, nomical units, days, and sexagesimal degrees. Hollow circles around inspection (Fig. 1). rthe, velocity doubleIn Fig. peaksv, 3, and in wer accelerationseries plot inside the magnitudes ‘disordered’a, for two of motion Jose-periods barycentric intervals starting position change 1632 indicate a particular planetary quasi-alignment which produces a our results, we use the solar system invariable plane (IP) as refer- the double peaks in r series inside “disordered” motion intervals change toaroundto twor two, velocity valleys, valleys, theyear forv e.g., example, and1366 between acceleration after between the yearsa Jupiter–Saturn thea 2100, years for and 2100two 2500 Jose-periods and opposition (result 2500 (resultnot shown). (when starting not solar retrograde motion (see Sect. 3.2). ence3. SBM plane. from This plane EPM2017H is the most natural reference for long-term shown).thearound difference the year in orbital 1366 longitudes after a Jupiter–Saturn between Jupiter opposition and Saturn, (when dynamics3. SBM and from it isEPM2017H normal to the total angular momentum ff 3.1. General description ∆λthe, di iserence 0◦). The in corresponding orbital longitudes trefoliar between (quasi-closed) Jupiter and 60Saturn, yr of the solar system. SBM is performed in the proximities of JS 3.1. General description periodstion∆λJS (coincident, is of 0 each◦). The withJose-period, corresponding the Sun are entering indicated trefoliar in periods (dotted (quasi-closed) of lines). disordered Each 60 yr AD 1831.3457 - AD 1877.0212 the IP, following the movement of the giant planets, which or in short, the (1 2) “inequality” (see Sect.4). Each inner Due to the mass and semi-major axis of Jupiter, SBM is per- Jupiter–Saturnmotionperiods after of each the opposition Charvátová Jose-period− corresponds 2009 are indicated nomenclature), to peaks (dotted in r forand lines). example,v plots. Each Due to the mass and semi-major axis of Jupiter,3 SBM is per-equilateral closed loop in Fig.2 corresponds to Jupiter–Saturn -10 -8 -6 -4 -2 0 2 4 6 8 10 mostlyformed produces at a mean the barycentric total solar system’s range of angular5 10 momentum− au, as can (see be Jupiter–Saturn opposition corresponds to peaks in r and v plots. ∼ × 3 JSThen,double/2 period. after peaked The a Jose-period variations first complete in ofr 177series. loop yr of We(the JS canperiod, timing explain in19.86 the these repeti-yr, fea- is 10 10 Appendixseenformed in Fig.A at for a 2, mean further where barycentric details).we have selected range of a period5 10 of− timeau, as related can be Then, after a Jose-period of 177 yr (the timing∼ in the repeti- seen in Fig. 2, where we have selected∼ a period× of time relatedmarkedtiontures of as with the follows: kinematics a thick when line. parameters theIn Fig. Sun2, ends after can a a be trefoliar highly60 yr period 60 variable yr (betweenmovement regard- 8 8 toCharváitová a trefoliar-like identified solar orbit these after Charvátovátrefoliar orbits (2009). over To present50 yr (aftertion aof Jupiter–Saturn the kinematics opposition) parameters Jupiter can∼ be is highly rapidly variable situated regard- near to a trefoliar-like solar orbit after Charvátová (2009). To∼ presenttheing years the mean 1723.4716179 and yr period 1783.2745) found the by projectionJose), the trefoliar of the Sun’s orbit 6 6 periodour results, as the we most use symmetricalthe solar system patterns invariable trying plane to find (IP) a as dis- ref- quadratureing the mean with∼ Saturn179 yr (∆ periodλ = foundλ λ by Jose),90 ). the In addition, trefoliarorbit the our results, we use the solar system invariable plane (IP) as5 ref-movementand the series on the of∼ IP kinematics tends toJS occupy elementsJ theS seemsame positionto◦ be repeated, after that al- 4 4 tinctiveerence plane. solar orbit This plane related is the to solarmost natural Grand reference Minima events for long-; Uranus–Neptuneand the series of centre kinematics of mass elements| (CM)− tends|seem ∼ to tobe be aligned repeated, and in al- erence plane. This plane is the most natural reference for long-timethough span, more with important a magnitude changes of the can diff appearerence in in the spatial kinematics position se- 2 2 instead,term dynamics we adopted and it a is normal60 yr period, to the total which angular is more momentum meaning- of though more5 important changes can appear in the kinematics se- λ λ = ∆λ au ∼ ofriesconjunction 3.4 trough10− time.au, with that We Saturn, is, note less i.e. that than theCM 51 non-Keplerian 000S km. NotCMs all nature18060◦ of yr(where SBM tre- 0 0 fultheterm from solar dynamics a system. dynamical and SBM it point is is normal performed of view, to the inbecause total the proximities angular it corresponds momentum of the to IP, of | − | ∼ -3 isλCMries evidenced×is trough the orbital time. by the We longitude fact note that that ofthe the the longer non-Keplerian Uranus–Neptune/shorter the nature∼ barycentric centre of SBM of foliar orbits are so closed, however. 10 afollowing quasi-completethe solar the system. movement Jupiter–Saturn SBM isof performed the giant synodic planets, in the cycle proximities which of oppositions- mostly of the pro- IP, -2 -2 range,mass),isIn evidenced Fig. the producing3, higher we by plot/lower anthe the important fact the magnitudes that orbital the redistribution speed. longer of barycentric/ Nevertheless,shorter of mass, the position barycentric which therer, is conjunctionsducesfollowing the total the (each solar movement system’s synodic of angularthe configuration giant momentum planets, is which advanced (see mostlyAppendix by pro- -4 -4 velocityaexpressedrange, lag betweenv the,as and higher a the variation acceleration extrema/lower of the in thear orbital,and orbital forv twoseries; speed. torque Jose-periodsfor Nevertheless, for example, which starting the the there Sun big is approximatelyduces the total 120 solarat system’s each JS angularperiod). momentum More specifically, (see Appendix this -6 -6 A for further details).◦ dropreactsa lag in to between solar keep velocity the the barycentre extrema around inthe atr rest. yearand Nevertheless,v 1672series; occurs for example, about a sine one qua the year non big A for further details). around the year 1366 after a Jupiter–Saturn opposition (when the -8 -8 periodCharvátová corresponds identified to the second these trefoliar most important orbits over perturbation50 yr drop in solar velocity around the year 1672 occurs about one year -3 ∼ diafterconditionfference the drop in for orbital thesein barycentric longitudes changes distance.is between that both TheJupiter icy shortest giant and Saturn, planets recognizable∆ areλJS, at Solar radius ≈ 5x10 au overperiod orbitalCharvátová as the longitudes most identified symmetrical in the these Jupiter–Saturn patterns trefoliar trying orbits pair, producedto over find a50 dis-by yr after the drop in barycentric distance. The shortest recognizable -10 -10 ∼ 5 isperiodicitynear 0◦). conjunction The corresponding in velocity with and Saturn, trefoliar acceleration which (quasi-closed) is parameters, expressed 60 in yr 1.5–1.6 the periods addi- yr, -10 -8 -6 -4 -2 0 2 4 6 8 10 thetinctiveperiod trigonometric solar as the orbit most argument related symmetrical toλJ solar2 λ GrandpatternsS, where Minima tryingλJ and eventstoλS findrepre-; a in- dis- 5 istionalperiodicity due tocondition the Venus–Earth in velocity∆λ . and synodic90◦ acceleration, where period.∆λ parameters, Althoughis the di theff 1.5–1.6erence succes- in yr, -3 senttinctive the corresponding solar orbit related mean orbital to− solar longitudes Grand Minima of both events planets,; in-of each Jose-period areUN indicated (dottedUN lines). Each Jupiter– 10 au 4 http://www.imcce.fr Saturnsiveorbitalis due JS oppositionlongitudes periods to the Venus–Earth of corresponds conjunction-oppositionsof both planets. synodic to peaks These period. in giantr rulesand Although planetv theplots. cyclicity configura- the Then, succes- of 4 5 5 Periodhttp://www.imcce.fr of time when sunspots almost disappear from the solar sur- afterextremationssive a canJose-period JS values periods be seen of of of inthese conjunction-oppositions177 Fig. time yr (4) (the where,series, timing after in additionin the the rules finalization repetition to ther series, cyclicity of of the dif- the of Period5 of time when sunspots almost disappear from the solar surface.face.Period of time when sunspots almost disappear from the solar sur-kinematics60ferencesextrema yr period in parameters values orbital the series of Jupiter–Saturn these can show be time highly an series, apparent longitudes, variable after change regardingthe and finalization in di theirff theerences mean evolu- of in the Fig. 5. Quite symmetric and quasi-closed projected (on the IP) pattern face. 60 yr period the series show an apparent change in their evolu- of SBM for SU period, associated disordered motion interval. The longitudes between the Uranus–Neptune centre of mass and the ∼ Saturn position are plotted. The filled dotsArticle mark number,A153, the pagedi pagefference 3 3 of of 11 11 of x-axis of the figure coincides with the nodal line equator-IP. An approx- Article number,∆λ page 3 of 11 imate value for the Sun’s radius is indicated. After that time span the longitudes between Uranus and Neptune when CMs 180◦. It Sun returns to less than 15 000 km of its initial position. can also be noted that the ‘disturbances’ in r series occur' when ∆λ 180 , but only if ∆λ . 90 . Indeed, a disturbance CMs ∼ ◦ UN ◦ in r series begins to occur when ∆λUN is near 90◦ as in the years 1430 or 1610. Then the ordered motion changes to disordered tern is repeated for several Jose-periods, and it is shown in Fig. when the Sun-Jupiter-Saturn movement is highly perturbed by (5) for the indicated temporal interval. Then, the pattern is mod- the Saturn-Uranus-Neptune alignment. Uranus and especially ified, when the inner ‘lemon-like’ loop begins progressively to Neptune produce significant torque over the solar motion, ow- rotate. In the case of the SBO shown in Fig. (5), the Sun passes ing to their distance from the barycentre. In particular, Neptune at a distance shorter than 15000 km from its initial position (less produces very distinctive features in solar orbital torque, as can than 295 km in the x y plane). Close approaches of the Sun be seen in Cionco & Abuin (2016). The double peaked effect in to the same barycentric− position occur continuously, following r series persists over one Saturn–Uranus synodic period (SU pe- the basic periodicities involved in SBM (JS, JS/2, SU, etc.). The riod), 45.36 yr. Then, we wanted to determine whether another most closed orbit occurs at the year 7654.3272, after a Saturn– quasi-closed∼ or quite symmetrical (projected) orbital pattern re- Neptune synodic period (SN period),− 35.87 yr, with a differ- 7 ∼ lated to these repeated features could be expected. We find, after ence in position of 6.6 10− au, about 99 km. A very close the first double-peaked pair and lasting for approximately one approach to the barycentre× around the year 1632 can be seen SU period, a quite symmetrical and quasi-closed orbital pattern, in r panel (Fig. 4, open dots). This is produced by a superior between the years 1474.245 and 1519.0856 (being the magni- quasi-conjunction of Jupiter with the other giants. As a result, the tude of difference in position 8000 km). This particular pat- Sun fails to loop the barycentre, entering in a retrograde motion ∼ Article number, page 4 of 11 A&A proofs: manuscript no. 32349_corr

9 <60−yr> <60−yr> 7 −3 5 / 10

r 3 1 ∆ λ 180 UN 135 CMs 90 ∆ λ 45

180

JS 135 90 ∆ λ 45 0 1400 1450 1500 1550 1600 1650 1700 Time [yr]

Fig. 3. Basic dynamical parameters for two Jose-periods of 177 yr (con- Fig. 4. Relative position of icy and gaseous giant planets and their ef- tinuous line). The intervals of trefoliar 60 yr orbits are indicated (dotted fect on r over two Jose-periods. The disordered orbits begin when Sat- lines). All the quantities are reckoned in astronomical units and days. urn and the Uranus–Neptune pair tend to be aligned (near conjunction) The ‘form’ of the series seems to be repeated, although more important being Jupiter near quadrature. All the quantities are reckoned in astro- changes can appear in the kinematics series trough time. For example, nomical units, days, and sexagesimal degrees. Hollow circles around A&A proofs: manuscriptthe no. double 32349_corr peaks in r series inside ‘disordered’ motion intervals change 1632 indicate a particular planetary quasi-alignment which produces a to two valleys, e.g. between the years 2100 and 2500 (resultA&A not shown). 615, A153solar (2018) retrograde motion (see Sect. 3.2).

AD 1831.3457 - AD 1877.0212 tion (coincident9 <60−yr> with the Sun entering <60−yr> in periods of disordered motion after7 the Charvátová 2009 nomenclature), for example, −3 -10 -8 -6 -4 -2 0 2 4 6 8 10 double peaked5 variations in r series. We can explain these fea- 10 10 / 10

r 3 tures as follows: when the Sun ends a trefoliar 60 yr movement 8 8 1 (after a Jupiter–Saturn opposition) Jupiter is rapidly situated near 6 6 ∆ λ 180 UN quadrature with Saturn (∆λJS = λJ λS 90◦). In addition, the 4 4 Uranus–Neptune135 centre of mass| (CM)− tends| ∼ to be aligned and in CMs 2 2 90

λ λ = ∆λ au conjunction∆ λ with Saturn, i.e. CM S CMs 180◦ (where 0 0 45 | − | ∼ -3 λCM is the orbital longitude of the Uranus–Neptune centre of 10 -2 -2 mass),180 producing an important redistribution of mass, which is -4 -4 expressedJS 135 as a variation of the orbital torque for which the Sun 90 -6 -6 reacts ∆ λ to45 keep the barycentre at rest. Nevertheless, a sine qua non -8 -8 condition0 for these changes is that both icy giant planets are at Solar radius ≈ 5x10-3 au near conjunction1400 with1450 Saturn,1500 which1550 1600 is expressed1650 1700 in the addi- -10 -10 Time [yr] -10 -8 -6 -4 -2 0 2 4 6 8 10 tional condition ∆λ . 90◦, where ∆λ is the difference in UN UN 10-3 au Fig. 3. Basic dynamical parameters for two Jose-periods of 177 yr (con- Fig.Fig.orbital 4. 4. Relative longitudes position of of ofboth icy icy planets.and and gaseous gaseous These giant giant giant planets planets planet and and configura- their their ef- tinuous line). The intervals of trefoliar 60 yr orbits are indicated (dotted efectffecttions on onr canroverover be two two seen Jose-periods. Jose-periods. in Fig. (4) The Thewhere, disordered disordered in addition orbits orbits begin to beginr whenseries, when Sat- dif-Fig. 5. Quite symmetric and quasi-closed projected (on the IP) pattern lines). All the quantities are reckoned in astronomical units and days. Saturnurnferences and and the the Uranus–Neptune in Uranus–Neptune orbital Jupiter–Saturn pair pair tend tend to longitudes, beto bealigned aligned (near and (near conjunction)diff conjunc-erences inof SBMFig. 5. forQuiteSU symmetric period, associated and quasi-closed disordered projected motion (on interval. the IP) The pattern of SBM for∼ SU period, associated disordered motion interval. The The ‘form’ of the series seems to be repeated, although more important tion)beinglongitudes being Jupiter Jupiter near between nearquadrature. quadrature. the Uranus–Neptune All the All quantities the quantities centre are reckoned are of reckoned mass in astro-and in thex-axis of the figure∼ coincides with the nodal line equator-IP. An ap- changes can appear in the kinematics series trough time. For example, astronomicalnomicalSaturn units, position units, days, are days, and plotted. sexagesimal and sexagesimal The filled degrees. dots degrees. Hollow mark Hollow the circles difference circles around ofproximatex-axis of value the figure for the coincides Sun’s radius with theis indicated. nodal line After equator-IP. that time An spanapprox- the double peaks in r series inside ‘disordered’ motion intervals change around1632 indicate 1632 indicate a particular a particular planetary planetary quasi-alignment quasi-alignment which∆λ which produces pro- a theimate Sun returns value to for less the than Sun’s 15 radius 000 km is of indicated. its initial After position. that time span the longitudes between Uranus and Neptune when CMs 180◦. It Sun returns to less than 15 000 km of its initial position. to two valleys, e.g. between the years 2100 and 2500 (result not shown). ducessolarcan retrogradea alsosolar beretrograde noted motion that motion (see the Sect. (see ‘disturbances’ 3.2). Sect. 3.2). in r series occur' when ∆λCMs 180◦, but only if ∆λUN . 90◦. Indeed, a disturbancemovement is highly perturbed by the Saturn–Uranus–Neptune 179 yr period∼ found by Jose), the∆λ trefoliar orbit and the se- ∼ in r series beginsAD to 1831.3457 occur when - AD 1877.0212UN is near 90◦ as in the yearsalignment. Uranus and especially Neptune produce significant tion (coincident with the Sun entering in periods of disordered ries1430 of kinematics or 1610.Then elements the seemordered to bemotion repeated, changes although to disordered more torquetern over is repeated the solar for motion, several Jose-periods, owing to their and distance it is shown from in the Fig. motion after the Charvátová 2009 nomenclature), for example, important changes can appear in the kinematics series trough when the Sun-Jupiter-Saturn-10 -8 -6 -4 -2 0 movement2 4 6 8 is10 highly perturbed bybarycentre.(5) for the In indicated particular, temporal Neptune interval. produces Then, very the distinctive pattern is fea- mod- double peaked variations in r series. We can explain these fea- time.the We Saturn-Uranus-Neptune note10 that the non-Keplerian alignment. nature Uranus10 of SBM and especially is evi- turesified, in solar when orbital the inner torque, ‘lemon-like’ as can be loop seen begins in Cionco progressively & Abuin to tures as follows: when the Sun ends a trefoliar 60 yr movement dencedNeptune by the produce fact8 that significant the longer torque/shorter over the the barycentric8 solar motion, range, ow-(2016rotate.). The In the double case peaked of the SBO effect shown in r series in Fig. persists (5), the over Sun one passes (after a Jupiter–Saturn opposition) Jupiter is rapidly situated near theing higher to their/lower distance6 the orbital from speed. the barycentre. Nevertheless, In particular,6 there is Neptune a lag Saturn–Uranusat a distance synodic shorter than period 15000 (SU km period), from its45.36 initial yr. position Then, we (less quadrature with Saturn (∆λ = λ λ 90 ). In addition, the ∼ JS J S ◦ betweenproduces the extremavery4 distinctive in r and featuresv series; in for solar example, orbital4 the torque, big drop as canwantedthan to 295 determine km in the whetherx y anotherplane). Closequasi-closed approaches or quite of sym-the Sun Uranus–Neptune centre of mass| (CM)− tends| ∼ to be aligned and in − in solarbe seen velocity in Cionco2 around & Abuinthe year (2016). 1672 occurs The double about2 peaked one year eff af-ect inmetricalto the (projected) same barycentric orbitalpattern position related occur to continuously, these repeated following fea- conjunction with Saturn, i.e. λ λ = ∆λ 180 (where au CM S CMs ◦ terr the drop in0 barycentric distance. The shortest0 recognizable / | − | ∼ series-3 persists over one Saturn–Uranus synodic period (SU pe-turesthe could basic be periodicities expected. involved We find, in after SBM the (JS, first JS double-peaked2, SU, etc.). The λCM is the orbital longitude of the Uranus–Neptune centre of periodicity in velocity and acceleration parameters, 1.5–1.6 yr, riod), 45.3610 -2 yr. Then, we wanted to determine-2 whether anotherpairmost and closed lasting orbit for approximately occurs at the year one SU7654 period,.3272, a after quite a sym- Saturn– mass), producing an important redistribution of mass, which is is due to the∼ Venus–Earth synodic period. − quasi-closed-4 or quite symmetrical (projected)-4 orbital pattern re-metricalNeptune and synodic quasi-closed period orbital(SN period), pattern,35.87 between yr, with the yearsa differ- expressed as a variation of the orbital torque for which the Sun Although the successive JS periods of conjunction- 7 ∼ lated to these-6 repeated features could be expected.-6 We find, after1474.245ence in and position 1519.0856 of 6.6 (being10− theau, magnitude about 99 of km. diff Aerence very closein reacts to keep the barycentre at rest. Nevertheless, a sine qua non oppositions rules the cyclicity of extrema values of these time × the first double-peaked-8 pair and lasting for-8 approximately onepositionapproach8000 to km). the barycentre This particular around pattern the year is repeated 1632 can for be sev- seen -3 condition for these changes is that both icy giant planets are at series,SU period, after the a quitefinalizationSolar symmetrical radius of ≈ the5x10 and 60 au yr quasi-closed period the seriesorbital show pattern,eralin Jose-periods,r panel∼ (Fig. and 4, open it is dots). shown This in Fig. is produced5 for the by indicated a superior near conjunction with Saturn, which is expressed in the addi- -10 -10 anbetween apparent the change years-10 -8 in 1474.245-6 their-4 -2 evolution0 and2 1519.08564 (coincident6 8 10 (being with the magni-Sun temporalquasi-conjunction interval. Then, of Jupiter the pattern with the is other modified, giants. when As a theresult, in- the ∆λ ∆λ ff tional condition UN . 90◦, where UN is the di erence in enteringtude of in di periodsfference of in disordered position-3 motion8000 km). after This the Charvátová particular pat-nerSun “lemon-like” fails to loop loop the begins barycentre, progressively entering to in rotate. a retrograde In the case motion 10 au∼ orbital longitudes of both planets. These giant planet configura- 2009 nomenclature), for example, double peaked variations in of the SBO shown in Fig.5, the Sun passes at a distance shorter tions can be seen in Fig. (4) where, in addition to r series, dif- r series.Article We number, can explain page 4 of these 11 features as follows: when the Sun than 15 000 km from its initial position (less than 295 km in the ferences in orbital Jupiter–Saturn longitudes, and differences in endsFig. 5. aQuite trefoliar symmetric 60 yr movement and quasi-closed (after projected a Jupiter–Saturn (on the IP) oppo- pattern of SBM for SU period, associated disordered motion interval. The x–y plane). Close approaches of the Sun to the same barycen- longitudes between the Uranus–Neptune centre of mass and the sition) Jupiter∼ is rapidly situated near quadrature with Saturn tric position occur continuously, following the basic periodic- Saturn position are plotted. The filled dots mark the difference of (x-axis∆λ = ofλ the figureλ coincides90 ). In with addition, the nodal the lineUranus–Neptune equator-IP. An approx-centre imateJS value| J − forS the| ∼ Sun’s◦ radius is indicated. After that time span the ities involved in SBM (JS, JS/2, SU, etc.). The most closed longitudes between Uranus and Neptune when ∆λCMs 180◦. It of mass (CM) tends to be aligned and in conjunction with Saturn, ' Sun returns to less than 15 000 km of its initial position. orbit occurs at the year 7654.3272, after a Saturn–Neptune can also be noted that the ‘disturbances’ in r series occur when i.e. λCM λS = ∆λCMs 180◦ (where λCM is the orbital synodic period (SN period),− 35.87 yr, with a difference in posi- ∆λ ∆λ | − | ∼ CMs 180◦, but only if UN . 90◦. Indeed, a disturbance longitude of the Uranus–Neptune centre of mass), producing tion of 6.6 10 7 au, about 99∼ km. A very close approach to the ∼ ∆λ − in r series begins to occur when UN is near 90◦ as in the years an important redistribution of mass, which is expressed as a vari- barycentre× around the year 1632 can be seen in r panel (Fig.4, 1430 or 1610. Then the ordered motion changes to disordered ationtern is of repeated the orbital for torque several for Jose-periods, which the Sun and reacts it is shown to keep in Fig.the open dots). This is produced by a superior quasi-conjunction of when the Sun-Jupiter-Saturn movement is highly perturbed by barycentre(5) for the indicatedat rest. Nevertheless, temporal interval. a sine Then, qua nonthe pattern condition is mod- for Jupiter with the other giants. As a result, the Sun fails to loop the Saturn-Uranus-Neptune alignment. Uranus and especially theseified, changes when the is inner that both ‘lemon-like’ icy giant loopplanets begins are at progressively near conjunc- to the barycentre, entering in a retrograde motion mode, and this Neptune produce significant torque over the solar motion, ow- tionrotate. with In Saturn, the case which of the is SBO expressed shown in in the Fig. additional (5), the Suncondition passes change is consistent with a momentary inversion of the SBO. ing to their distance from the barycentre. In particular, Neptune ∆atλUN a distance. 90◦, whereshorter∆ thanλUN 15000is the km di fromfference its initial in orbital position longi- (less produces very distinctive features in solar orbital torque, as can tudesthan 295 of both km planets. in the x Thesey plane). giant Closeplanet approaches configurations of the can Sun be − 3.2. Retrograde solar motion be seen in Cionco & Abuin (2016). The double peaked effect in seento the in same Fig.4 barycentricwhere, in addition position to occurr series, continuously, differences following in or- r series persists over one Saturn–Uranus synodic period (SU pe- bitalthe basic Jupiter–Saturn periodicities longitudes, involved in and SBM diff (JS,erences JS/2, in SU, longitudes etc.). The Lapses of retrograde solar motion around the barycentre were riod), 45.36 yr. Then, we wanted to determine whether another betweenmost closed the orbit Uranus–Neptune occurs at the yearcentre7654 of mass.3272, and after the a Saturn– Saturn detected by the changing sign of the z-component, L , of ∼ − z quasi-closed or quite symmetrical (projected) orbital pattern re- positionNeptune are synodic plotted. period The filled (SN dots period), mark the35.87 diff yr,erence with of a longi- differ- the orbital angular momentum (Jose 1965; Javaraiah 2005), 7 ∼ lated to these repeated features could be expected. We find, after tudesence betweenin position Uranus of 6.6 and10 Neptune− au, aboutwhen ∆ 99λ km. A180 very. It close can which produces a brief clockwise SBM. Retrograde solar mo- × CMs ◦ the first double-peaked pair and lasting for approximately one alsoapproach be noted to the that barycentre the “disturbances” around the in yearr series 1632' occur can be when seen tion and also variations in angular momentum were linked to SU period, a quite symmetrical and quasi-closed orbital pattern, ∆inλ r panel180 (Fig., but 4, only open if ∆ dots).λ . This90 . is produced by a superior possible solar cycle mechanisms (see e.g., Landscheidt 1999; CMs ∼ ◦ UN ◦ between the years 1474.245 and 1519.0856 (being the magni- quasi-conjunctionIndeed, a disturbance of Jupiter in r withseries the begins other giants. to occur As when a result,∆λUN the Juckett 2003; Javaraiah 2005; Perryman & Schulze-Hartung tude of difference in position 8000 km). This particular pat- isSun near fails 90 to loopas in the the barycentre, years 1430 entering or 1610. in a Then retrograde the ordered motion 2011; Cionco & Soon 2015), especially Grand Minima events ∼ ◦ motion changes to disordered when the Sun–Jupiter–Saturn and their possible periodicity over the time. Article number, page 4 of 11 A153, page 4 of 11 R. G. Cionco and D. A. Pavlov: Solar barycentric dynamics

Table 1. Retrograde solar motion intervals (Lz < 0) regarding the solar barycentric phenomena can be produced by the simple fact that system’s invariable plane. the planetary periods are not commensurable among themselves, that is “the planets are not trapped in mean motion resonances”. Initial time (yr) Duration (d) Initial time (yr) Duration (d) This bimillennial wave of orbital inversions (whose possi- ble origin is related to the long-term dynamics of Uranus and –9129.5578 203 –2637.4292 528 Neptune; see Sect.4) can be seen quite clearly by the simple in- –8950.7132 230 –2458.4422 466 spection of almost all the dynamical quantities analysed, for ex- –2318.4641 352 ample the SBO inclination (Fig.6). The above-mentioned SBO –7621.5305 139 –2279.2033 209 was calculated following Cionco & Compagnucci(2012), deter- –7443.1923 492 –2139.8220 437 mining the inclination, iS, and the ascending node, ΩS, of the –7264.4244 552 –1960.8515 314 SBO regarding the IP reference system. In addition, we also de- –7085.5003 506 termined the Sun’s positional angle on its SBO, uS (the argument –6945.4538 360 –54.7406 352 of latitude), reckoned from the nodal line equator-IP. The SBO –6906.4230 333 124.0575 432 allows us to describe the Sun’s acceleration from a co-orbital –6766.8364 473 302.9979 382 reference system, owing to the projection of the Sun’s acceler- –6587.9562 413 482.1821 96 ation onto the solar orbital plane (see AppendixA for graph- –6408.5749 49 ical descriptions). This coplanar acceleration, aπ, shows the 1632.2930 250 impulsive changes in acceleration due to the giant planets –4681.5661 200 1810.8255 453 quasi-alignments, a dynamical feature that cannot be seen in –4502.8008 357 1989.6783 475 the usual Cartesian representation of a. The magnitude of –4323.8960 331 2168.7584 374 aπ, aπ, is also plotted in Fig.6. Consistent with the retro- 2308.5749 308 grade motion intervals, it is observed that the Sun’s orbital –2994.6995 245 2487.2936 357 inclination overpasses 90◦ at these moments and the copla- –2816.2300 486 2666.4668 119 nar acceleration increases impulsively its magnitude. These impulses are only produced by the giant planets. It is inter- Notes. The empty cells/rows indicate the absence of orbital inversion esting to note that planetary alignments have also been con- near most initial times. sidered regarding the solar cycles, and these extrema values could be relevant in this regard (McCracken et al. 2014). We searched for all the retrograde motion lapses (on a The coplanar acceleration is obtained by the decomposition daily basis) in this studied period and their durations (Table1). of the Cartesian acceleration in a co-orbital reference sys- 6 Negative Lz values depend on the reference system used (see tem, following the radial, transversal, and normal directions 2 2 1/2 AppendixA for graphical descriptions). Nevertheless, an esti- associated with the SBO, that is, aπ = (ar + at ) , being ar and mation using the equinox-ecliptic of J2000.0 as reference plane at the components of the radial and transversal accelerations, yields few days of difference in the duration of these events. respectively (see Figs. A.2, A.3 and A.4). The positive radial Clearly, orbital inversions are not periodic phenomena in terms direction is taken from the barycentre pointing towards the of the Jose-period, they appear in irregular ways, in groups of Sun. The most common situation in this N-body system is thus three to nine events. In turn, these retrograde motion intervals that the “orbital centre of force” is near the barycentre, and appear with a bimillennial cycle (six cycles inside the stud- in its “normal” prograde motion the Sun loops the barycentre ied period) of, on average, 13 kyr/6 2.2 kyr. Moreover, inside with negative values of ar (Figs.2 and5). As was shown in these bimillennial periods of retrograde∼ motion, they do not hap- Cionco & Compagnucci(2012), the impulsive changes induced pen regularly at the Jose-period. Some intervals of retrograde by the giant planet alignments produce positive values of radial motion are separated by about 140 yr, for example between the acceleration. Nevertheless, analysing the fine structure of these years 2168 and 2308 or between 2458 and 2318. In this last impulses at this level of precision, we can see that radial accel- case, the retrograde intervals of the− bimillennial− “wave” begin- eration is not always positive at retrograde lapses (Fig.7). The ning at the year 2994, appear at each Jose-period until the process can be explained as follows: when an alignment of gi- year 2458. Although− the Jose-period beginning at 2458 is ant planets is produced, the transversal acceleration oscillates maintained,− with another orbital inversion occurring at− 2279, rapidly, causing a variation in the orbital torque, and the Sun fails an intermediate lapse of retrograde motion takes place− at the to loop the barycentre (see e.g., Perryman & Schulze-Hartung year 2318, that is, 39.26 yr before the expected event of the 2011, Fig. 2); hence, the centre of curvature of the SBO leaves year −2279. Notably, before the retrograde interval of the year the barycentre “outside” the orbit, and positive values of ra- 2994−, all orbital inversions inside each bimillennial cycle occur dial acceleration (i.e. in the barycentre–Sun sense) are expected. at− the Jose-period. The explanation of these shorter periods Nevertheless, the effect of the Earth–Venus pair in the accelera- tion (evidenced in the 1.6 yr peaks also visible in Fig.7) causes is simple. At each Jose-period, which is strongly related to ∼ the Uranus–Neptune synodic period (UN period) of 171 yr, a a momentary change in the sign of ar: the expected positive val- quasi-conjunction between Jupiter and the other giants∼ tends to ues change, momentarily, to negative. This can be seen in the occur. These quasi-alignments are not always able to produce embedded plots of Fig.7, when a lapse of ar < 0 can be ob- a solar retrograde motion because of the ever changing plane- served in addition to Lz < 0 values. tary positions and velocities at these particular events. One Jose- period (as already noted by Jose) is 9 JS 179 yr; sometimes, 3.3. Differences in derived dynamical quantities 2 JS 40 yr before a Jose-period,∼ × that∼ is, 7 JS 139 yr ∼after× an∼ earlier inversion event, giant planets are∼ already× ∼ quasi- To evaluate the differences between EPM2017H and DE431 in aligned and able to produce a retrograde motion. In brief, JS pe- the SBM derived dynamical parameters, we compare the values riod, as will be also shown in Sect.4, is the fundamental period in this problem. Nevertheless, significant differences in periodic 6 That is, in the direction of the solar orbital angular momentum.

A153, page 5 of 11 AA&&AA proofs: proofs:manuscriptmanuscript no. no. 32349_corr 32349_corr A&A 615, A153 (2018)

Fig.Fig.Fig. 6. 6. 6.ExtremaExtremaExtrema in inL inz Lvalues,Lz zvalues,values, coplanar coplanar coplanar accel- ac- ac- erationcelerationceleration magnitude, magnitude, magnitude, and andSBO and SBO SBOinclination inclination inclination for the for for wholethethe whole whole studied studied studied period period period showing showing showing the thebimillennial the bimillen- bimillen- wavenialnial waveof wave impulsive of of impulsive impulsive effects e offf effects giantects of of planets. giant giant planets. Whenplanets. theWhenWhen SBM the the is retrograde,SBM SBM is is retrograde, retrograde, the iS is larger the theiSi thanSisis larger 90larger◦. Allthanthan the 90 90 quantities◦.◦. All All the the are quantities quantities reckoned are are in reckoned reckoned astronomical in in as- as- units,tronomicaltronomical days, and units, units, degrees. days, days, and and degrees. degrees. Cionco & Pavlov: Solar barycentric dynamics

2 2 8,8, where where only only the the di difffferenceserences whose whose absolute absolute values values are are larger larger effect) in the system under study. Moreover, the periodicity of thanthan 1 1◦ ◦areare plotted. plotted. For ForiSiSthethe di difffferenceserences (not (not shown) shown) are are much much that supposed ‘perturbation’ to the SBM must be highly variable 1.51.5 smallersmaller (about (about 6 6◦ ◦atat most) most) and and are are simply simply related related to to retrograde retrograde in order to be easily evidenced in the corresponding harmonic motionmotion epochs epochs (di (difffferenceserences in in orbital orbital inclinations inclinations larger larger than than 1 1◦ ◦ analysis. In this work, we have also been identifying particular 1 1 occuroccur just just 172 172 times times in in the the whole whole period). period). It It seems seems clear clear that that periodicities with meaningful combinations of integers and plan- -8 -8 thesethese di difffferenceserences in iniSiSareare explained explained by by the the impulsive impulsive nature nature of of etary mean motions that do produce significant effects in terms 0.50.5 thesethese retrograding retrograding events events when when the the Sun Sun is is very very near near the the barycen- barycen- of perturbative methods, and we are going to clarify this rela- tre.tre. The The same same applies applies to toθθangle,angle, for for which which the the larger larger di difffferenceserences tionship. It is also worth noting that basic planetary dynamics 0 0 areare all all associated associated with with orbital orbital inversion inversion intervals. intervals. In In turn, turn, the the (heliocentric distance, velocity, etc.) depend on non-linear com- Units of 10 Units of 10 largerlarger di difffferenceserences in inuuS Sangleangle are are not not related related to to orbital orbital inversion inversion binations of the orbital elements. As a result, a certain spectral -0.5-0.5 epochs,epochs, neither neither related related to to the the largest largest values values in inrrororv.v. Neverthe- Neverthe- peak in SBM could be a direct measure of the involved periodic less,less, the the big big di difffferenceserences seen seen around around 1672.6, 1672.6, for for example, example, seem seem planetary perturbations, and also the result of a non-linear mix- -1-1 aar Lr Lz z toto be be associated associated with with very very small small values values of of orbital orbital speed speed (see (see Fig. Fig. ing of several perturbing periodicities present in the variations of a at t -1.5-1.5 3).3). In In addition addition to to these these angular angular quantities, quantities, we we also also compare compareLL, the, the these elements. In addition, if a periodic perturbation is impor- 19801980 19851985 19901990 19951995 20002000 torquetorqueTT,, and and the the norm norm of of the the orbital orbital (vectorial) (vectorial) torque torqueΓΓ, ,ΓΓ.. The The tant (i.e. of significant amplitude) for certain orbital elements, it TimeTime [yr] [yr] didifffferenceserences in inLLareare in in general general less less than than 6%, 6%, with with their their largest largest is not necessarily relevant for other dynamical parameters. Our valuesvalues related related to to the the close close approaches approaches to to the the barycentre. barycentre. The The ab- ab- search for periodicities in SBM and their relationship with plan- Fig. 7.EvolutionEvolutionEvolution of of of radial radial radial and and and transversal transversal transversal accelerations accelerations accelerations and and andLLLaroundzaroundaround Fig.Fig. 7. 7. zz Fig.solutesoluteFig. 8. 8. valueDi valueDifferencesff oferences of the the in percentage percentage inuSuandS andθ angles diθ diffanglesfferenceserences determined determined DE431-EPM2017H DE431-EPM2017H from from DE431 DE431 and in and in etary perturbations is far from exhaustive, but provides a basic the retrograde interval of the year 1990. Contrary to what was expected, thethe retrograde retrograde interval interval of of the the year year 1990. 1990. Contrary Contrary to to what what was was expected, expected, EPM2017HTTEPM2017HandandΓΓquantitiesquantities ephemerides. ephemerides. larger larger Just thanJust than the the 5%di 5%ff dierences (0.32%ff (0.32%erences whose of whoseof the the absolute total) absolute total) are values are values shown shown are are guideline on how this topic should be addressed. radialradialradial acceleration acceleration acceleration points points points towards towards towards the the the barycentre barycentre barycentre (i.e. (i.e. (i.e. it it itis is is negative) negative) negative) dur- dur- dur- largerininlarger Fig. Fig. than 9. than 9. Of 1 Of◦ 1are these,◦ these,are plotted. plotted. the the biggest They biggest They are valuesare valuesa small a small are are fraction fractionall all related related of of the the to total to orbital total orbital (0.45% (0.45% in- in- inginging brief brief brief moments moments moments when when whenLLLz<<<0,0,0, marked marked marked with with with arrows arrows arrows and and and depicted depicted depicted in in in zz andversionversionand 0.02%, 0.02%, epochs. epochs. respectively), respectively), Furthermore, Furthermore, but but are are theynot they not necessarily seem necessarilyseem to to be relatedbe clumpedrelated clumped to to the theevery everyimpul- impul- thethethe embedded embedded embedded plots. plots. plots. They They They are are are produced produced produced by by by Venus–Earth Venus–Earth Venus–Earth synodic synodic synodic period period period sive events of retrograde motion (see Sect.3.3). The largestff differences∼ sive500500 events yr, yr, which which of retrograde is is produced produced motion by by (see dwarf dwarf Sect. planet 3.3 planet). The Eris, Eris, largest more more di massiveerences massive∼ 4.1. Theoretical estimates perturbations.perturbations.perturbations. All All All the the the quantities quantities quantities are are are reckoned reckoned reckoned in in in astronomical astronomical astronomical units units units and and and in inu u(e.g.,S (e.g. around around 1672.6) 1672.6) are are related related to to low low values values of of solar solar velocity. velocity. thanthanS Pluto, Pluto, orbiting orbiting between between 38.838.8 and and 97.4 97.4 au au (Santos-Sanz (Santos-Sanz days.days.days. ∼∼ The physical analysis of the main giant planets perturbations can etet al. al. 2012). 2012). Orbital Orbital simulations simulations by by numerical numerical integrations integrations using using be very easily performed using VSOP2013 planetary theory (Si- 10000 of key angular and orbital torque related quantities. First, we cal- example,justjust% the the major major seem planets planetsto be associated and and the the major major with planets planets very small plus plus Eris ErisvaluesT (result (result ofor- not not mon et al. 2013), where planetary perturbations are expressed as shown)shown) clearly clearly show show this this oscillation oscillation and and approximately approximatelyΓ the the same same culateinin the the the sign sign di offf oferencesaar:r: the the in expected expecteduS and positiveiS positiveangles. values values Also, change, we change, estimate momen- momen- the bital speed (see Fig.3). In addition to these angular quantities, Poisson series. In that development, the purely periodic part is ditarily,tarily,fferences to to negative. negative. in the positional This This can can be angle be seen seenθ in, in which the the embedded embedded we define plots plots as the of of Fig. an- Fig. wedispersiondispersion also compare in in the the correspondingL corresponding, the torque T torque, torque and the di difffferences normerences of as theas observed observed orbital obtained as a trigonometric series in terms of planetary orbital inin Fig. Fig. 9. 9. Then, Then, as as expected, expected, Eris Eris produces produces a a detectable detectable di diffffer-er- gular7,7, when when distance a a lapse lapse (reckoned of ofaar r<<0 in0 can canthe be SBObe observed observed plane) in between in addition additionaπ to toandLLz z< the<00 (vectorial) torque Γ, Γ. The differences in L are in general less longitudes. Then, for the giant planets, the periodic variations in enceence in in the 1000 the solar solar orbital orbital torque torque quantities. quantities. The The di difffferenceserences can can radialvalues.values. direction (the barycentre–Sun direction, see Fig. A.4). The than 6%, with their largest values related to the close approaches the orbital element X along time, ∆Xper(t), have the form bebe very very large, large, especially especially in inTT,, one one of of the the most most commonly commonly used used results for uS and θ angles are shown in Fig.8, absolute values to the barycentre. The absolute value of the percentage differ- parametersparameters in in the the applications applications to to solar-terrestrial solar-terrestrial relation relation stud- stud- N are larger than 1◦ are plotted. For iS the differences (not shown) ences DE431-EPM2017H in T and Γ quantities larger than 5% 3.3.3.3. Differences Differences in in derived derived dynamical dynamical quantities quantities ies.ies. These TheseTTvaluevalue di difffferenceserences are are sensibly sensibly larger larger than thanΓΓdidiffffer-er- are much smaller (about 6◦ at most) and are simply related to ret- (0.32% of the total) are shown in Fig.9. Of these, the biggest val- ∆Xper(t) = S k sin[Λ pk] + Ck cos[Λ pk], (3) ences,ences, which which 100 is is easily easily explained explained by by the the fact fact that thatTT== Γ Γcos(cos(LL, Γ, Γ),), · · rograde motion epochsff (differences in orbital inclinations larger ues are all related to orbital inversion epochs. Furthermore, they k=1 ToTo evaluate evaluate the the di differenceserences between between EPM2017H EPM2017H and and DE431 DE431 in in hencehence this this parameter parameter also also takes takes into into account account the the angular angular di diffffer-er- X thanthethe SBM 1 SBM◦ occur derived derived just dynamical172 dynamical times in parameters, parameters, the whole weperiod). we compare compare It seems the the valuesclear values seem to be clumped every 500 yr, which is produced by dwarf encesences between betweenLLandand the the vectorial vectorial∼ planetary planetary torque. torque. These These direc- direc- where Λ = (λ5, λ6, λ7, λ8) is the vector of mean orbital lon- thatof key these angular differences and orbital in iS torqueare explained related quantities. by the impulsive First, we na- cal- planet Eris, more massive than Pluto, orbiting between 38.8 of key angular and orbital torque related quantities. First, we cal- tionaltional di difffferenceserences are are amplified amplified at at the the retrograde retrograde motion motion∼ inter- inter- gitudes ranging from Jupiter (5) to Neptune (8), and pk = tureculate of these the diff retrogradingfferences inuu eventsS andi wheniS angles. the Sun Also, is we very estimate near the the and 97.4 au 10 (Santos-Sanz et al. 2012). Orbital simulations by nu- culate the di erences in S and S angles. Also, we estimate the vals,vals, when when the the Sun’s Sun’s orbital orbital angular angular momentum momentum rotates rotates rapidly rapidly (p5, p6, p7, p8) is a particular combination of integers. The mean barycentre.difffferences The in the same positional applies angle to θθangle,θ, which for we which define the as larger the an- merical integrations using just the major planets and the major di erences in the positional angle , which we define as the an- Abs. value of percentage differences > 5 λ = + λ ff byby about about 180 180◦.◦. orbital longitude for planet i is written as i ni (t t0) i 0, digularerences distance are all (reckoned associated in withthe SBO orbital plane) inversion between intervals.π and In the planets plus−10000 Eris (result−8000 not−6000 shown)−4000 clearly−2000 show 0 this 2000 oscilla- − gular distance (reckoned in the SBO plane) between aaπ and the where ni is the mean motion of the planet and λi 0 is an initial turn,radial the direction larger di (thefferences barycentre–Sun in uS angle direction, are not seerelated Fig. to A.4 or- in tion and approximately the same dispersion in the corresponding radial direction (the barycentre–Sun direction, see Fig. A.4 in Time [yr] phase related to the reference time t0. Therefore, the period of bitalAppendix inversion A). epochs, The results neither for relatedu andθ toθ angles the largest are shown values in in Fig.r torque differences as observed in Fig.9. Then, as expected, Appendix A). The results for uS Sand angles are shown in Fig. the k-perturbation is 2π/(n pk), where n is the vector of plane- or v. Nevertheless, the big differences seen around 1672.6, for ErisFig. produces 9. Absolute a detectable values of diff percentageerence in di thefferences solar orbital larger torque than 5% tary mean motions. For example,· a particular periodic perturba- ArticleArticle number, number, page page 6 6 of of 11 11 in T and Γ magnitudes determined from DE431 and EPM2017H A153, page 6 of 11 ephemerides (see Sect. 3.3 for details). The differences are clumped tion on orbital longitudes will produce a difference or ‘inequal- every 500 yr and related to dwarf planet Eris, which produces a de- ity’ regarding the nominal Keplerian mean motion. Hence, with tectable∼ torque because of its mass and mean distance to the Sun. a slight abuse of the language, we are referring to the pertur- bation by its cause: the (1, 2, 0, 0) term for Jupiter’s longitude corresponds to the perturbing− periodic effect given by Eq. (3) 4. Harmonic analysis of SBM (with the corresponding amplitudes for Jupiter’s longitude) with the argument λ 2 λ , or in short, the (1, 2) inequality. Then, The determination of the main periodicities present in SBM was J − S − another issue attracting attention, regarding the different short- the strength or ‘importance’ of that perturbation can be obtained and long-term modulating periods detected in the solar cycles by the combination of both amplitudes (see e.g. Soon et al. 2014; Beer et al. 2018). Several authors have 2 2 determined orbital periodicities from the SBM dynamical pa- Pk = S k + Ck , (4) rameters and further identification of these spectral peaks came q from the evident synodic and orbital periods, or simple combina- which permits the establishment of the relative intensity of the tions of planetary mean motions, trying to find numeric relation- thousands of perturbations considered here. Following this pro- ships as multiples or ‘resonant harmonics’ of more basic periods cedure, not only are we able to arrange the perturbations over (see e.g. McCracken et al. 2014; Scafetta et al. 2016). It is im- a particular planetary orbital element from the strongest to the portant to note that we can find an arbitrary number of combina- weakest, but also to identify the mean motion combination which tions between integers and mean motions very close to zero, but produces such a physical perturbation. Hence, taking into ac- it is not an indication of resonance, as a well-defined and strong count the reordered series, the k-order of a perturbation corre- dynamical phenomenon. An arbitrary combination of integers sponds now to its position in the ranking (1st, 2nd, etc.). In Table and mean motions can be identified with a certain periodicity, 2 we present (based on VSOP2013 constants) the nominal peri- but this does not assure its dynamical relevance (as a periodic ods of some inequalities discussed in the text, characterized by

Article number, page 7 of 11 Cionco & Pavlov: Solar barycentric dynamics

effect) in the system under study. Moreover, the periodicity of that supposed ‘perturbation’ to the SBM must be highly variable in order to be easily evidenced in the corresponding harmonic analysis. In this work, we have also been identifying particular periodicities with meaningful combinations of integers and plan- etary mean motions that do produce significant effects in terms of perturbative methods, and we are going to clarify this rela- tionship. It is also worth noting that basic planetary dynamics (heliocentric distance, velocity, etc.) depend on non-linear com- binations of the orbital elements. As a result, a certain spectral peak in SBM could be a direct measure of the involved periodic planetary perturbations, and also the result of a non-linear mix- ing of several perturbing periodicities present in the variations of these elements. In addition, if a periodic perturbation is impor- tant (i.e. of significant amplitude) for certain orbital elements, it is not necessarily relevant for other dynamical parameters. Our search for periodicities in SBM and their relationship with plan- Fig. 8. Differences in uS and θ angles determined from DE431 and etary perturbations is far from exhaustive, but provides a basic EPM2017H ephemerides. Just the differences whose absolute values are guideline on how this topic should be addressed. larger than 1◦ are plotted. They are a small fraction of the total (0.45% and 0.02%, respectively), but are not necessarily related to the impul- sive events of retrograde motion (see Sect.3.3). The largest differences 4.1. Theoretical estimates in uS (e.g. around 1672.6) are related to lowR. G. values Cionco of andsolar D. velocity. A. Pavlov: Solar barycentric dynamics The physical analysis of the main giant planets perturbations can be very easily performed using VSOP2013 planetary theory (Si- 10000 In addition, if a periodic perturbation is important (i.e., of signif- % T mon et al. 2013), where planetary perturbations are expressed as Γ icant amplitude) for certain orbital elements, it is not necessarily relevantPoisson for series. other Indynamical that development, parameters. the Our purely search periodic for period- part is icitiesobtained in SBM as aand trigonometric their relationship series with in terms planetary of planetary perturbations orbital longitudes. Then, for the giant planets, the periodic variations in 1000 is far from exhaustive, but provides a basic guideline on how this topicthe should orbital beelement addressed.X along time, ∆Xper(t), have the form

N 4.1.∆X Theoreticalper(t) = estimatesS k sin[Λ pk] + Ck cos[Λ pk], (3) 100 · · The physicalXk= analysis1 of the main giant planets perturbations

canwhere be veryΛ easily= (λ5 performed, λ6, λ7, λ8) using is the VSOP2013 vector of planetary mean orbital theory lon- Simongitudes et al. ranging(2013), from where Jupiter planetary (5) perturbations to Neptune are (8), expressed and pk = 10 as( Poissonp5, p6, p series.7, p8) is In a particularthat development, combination the purely of integers. periodic The part mean

Abs. value of percentage differences > 5 isorbital obtained longitude as a trigonometric for planet seriesi is written in terms as λ ofi planetary= ni (t t orbital0) + λi 0, −10000 −8000 −6000 −4000 −2000 0 2000 − longitudes.where ni Then,is the for mean the motion giant planets, of the theplanet periodic and λ variationsi 0 is an initial in ∆ Time [yr] thephase orbital related element to theX along reference time, timeXpert(0t.), Therefore, have the form the period of the k-perturbationN is 2π/(n pk), where n is the vector of plane- Fig. 9. Absolute values of percentageff differences larger than 5% Fig. 9. Absolute values of percentage di erences larger than 5% ∆ tary mean= motions. ForΛ example,·+ a particularΛ , periodic perturba- in inT TandandΓ Γmagnitudesmagnitudes determined determined from from DE431 DE431 and and EPM2017H EPM2017H Xper(t) S k sin[ pk] Ck cos[ pk] (3) tion on orbital longitudes· will produce a· difference or ‘inequal- ephemeridesephemerides (see (see Sect. Sect. 3.3 3.3 for for details). details). The The diff dierencesfferences are are clumped clumped Xk=1 every 500 yr and related to dwarf planet Eris, which produces a de- whereity’ regardingΛ = (λ the, λ , nominal λ , λ ) is Keplerian the vector mean of motion. mean orbital Hence, lon- with every 500∼ yr and related to dwarf planet Eris, which produces a de- 5 6 7 8 tectabletectable∼ torque torque because because of of its its mass mass and and mean mean distance distance to tothe the Sun. Sun. gitudesa slight ranging abuse fromof the Jupiter language, (5) we to areNeptune referring (8), to and thep pertur-= bation by its cause: the (1, 2, 0, 0) term for Jupiter’s longitudek (p5, p6, p7, p8) is a particular combination of integers. The mean corresponds to the perturbing− periodic effect given by Eq. (3) orbital longitude for planet i is written as λi = ni (t t0) + λi 0, quantities.4. Harmonic The di analysisfferences can of SBM be very large, especially in T, (with the corresponding amplitudes for Jupiter’s longitude)− with where ni is the mean motion of the planet and λi 0 is an initial one of the most commonly used parameters in the applications the argument λ 2 λ , or in short, the (1, 2) inequality. Then, The determination of the main periodicities present in SBM was phase related to theJ referenceS time t0. Therefore, the period of the to solar-terrestrial relation studies. These T value differences are − − another issue attracting attention, regarding the different short- k-perturbationthe strength or is ‘importance’ 2π/(n p ), where of thatn perturbationis the vector can of beplanetary obtained sensibly larger than Γ differences, which is easily explained by · k and long-term modulating periods detected in the solar cycles meanby the motions. combination For example, of both amplitudes a particular periodic perturbation the fact that T = Γ cos(L, Γ), hence this parameter also takes (see e.g. Soon et al. 2014; Beer et al. 2018). Several authors have on orbital longitudes will produce a difference or “inequality” into account the angular differences between L and the vectorial 2 2 determined orbital periodicities from the SBM dynamical pa- regardingPk = S the+ nominalC , Keplerian mean motion. Hence, with a(4) planetary torque. These directional differences are amplified at k k rameters and further identification of these spectral peaks came slight abuse of the language, we are referring to the perturbation the retrograde motion intervals, when the Sun’s orbital angular q from the evident synodic and orbital periods, or simple combina- bywhich its cause: permits the (1the, establishment2, 0, 0) term for of Jupiter’sthe relative longitude intensity corre- of the momentum rotates rapidly by about 180 . − tions of planetary mean motions, trying◦ to find numeric relation- spondsthousands to the of perturbing perturbations periodic considered effect here.given Following by Eq. (3) this (with pro- ships as multiples or ‘resonant harmonics’ of more basic periods thecedure, corresponding not only amplitudes are we able for to Jupiter’s arrange the longitude) perturbations with the over 4.(see Harmonic e.g. McCracken analysis et al.of 2014;SBM Scafetta et al. 2016). It is im- argumenta particularλJ planetary2 λS, or in orbital short, elementthe (1, 2) from inequality. the strongest Then, to the the strength or “importance”− of that perturbation− can be obtained by Theportant determination to note that of we the can main find anperiodicities arbitrary number present of in combina- SBM weakest, but also to identify the mean motion combination which the combination of both amplitudes wastions another between issue integers attracting and mean attention, motions regarding very close the to di zero,fferent but produces such a physical perturbation. Hence, taking into ac- it is not an indication of resonance, as a well-defined and strong count the reordered series, the k-order of a perturbation corre- short- and long-term modulating periods detected in the solar 2 2 Pk = S + C , (4) cyclesdynamical (see e.g., phenomenon. Soon et al. An 2014 arbitrary; Beer combination et al. 2018). of Several integers spondsk now tok its position in the ranking (1st, 2nd, etc.). In Table authorsand mean have motions determined can orbital be identified periodicities with a from certain the SBMperiodicity, dy- which2 weq permits present (based the establishment on VSOP2013 of constants) the relative the intensity nominal peri- of namicalbut this parameters does not assure and further its dynamical identification relevance of these (as aspectral periodic theods thousands of some of inequalities perturbations discussed considered in the here. text, Following characterized this by procedure, not only are we able to arrange the perturbations peaks came from the evident synodic and orbital periods, or sim- Article number, page 7 of 11 ple combinations of planetary mean motions, trying to find nu- over a particular planetary orbital element from the strongest meric relationships as multiples or “resonant harmonics” of more to the weakest, but also to identify the mean motion combi- basic periods (see e.g., McCracken et al. 2014; Scafetta et al. nation which produces such a physical perturbation. Hence, 2016). It is important to note that we can find an arbitrary number taking into account the reordered series, the k-order of a of combinations between integers and mean motions very close to perturbation corresponds now to its position in the ranking zero, but it is not an indication of resonance, as a well-defined and (1st, 2nd, etc.). In Table2 we present (based on VSOP2013 con- strong dynamical phenomenon. An arbitrary combination of in- stants) the nominal periods of some inequalities discussed in the tegers and mean motions can be identified with a certain periodic- text, characterized by its respective k-order for the semi-major ity, but this does not assure its dynamical relevance (as a periodic axis and longitude of Jupiter and Neptune (kJ or kN), respec- effect) in the system under study. Moreover, the periodicity of that tively. We have only considered the giant planet perturbations. supposed “perturbation” to the SBM must be highly variable in Nevertheless, the Theory of the Outer Planets, TOP2013 order to be easily evidenced in the corresponding harmonic anal- (Simon et al. 2013), is more accurate than VSOP2013 over ysis. In this work, we have also been identifying particular peri- several thousands of years, with a different Fourier series rep- odicities with meaningful combinations of integers and planetary resentation in Poisson’s expansion. Using this representation, as mean motions that do produce significant effects in terms of per- suggested by the reviewer, we can also estimate the possible vari- turbative methods, and we are going to clarify this relationship. It ations in the periodicities of these inequalities, comparing their is also worth noting that basic planetary dynamics (heliocentric corresponding TOP2013/VSOP2013 values. distance, velocity, etc.) depend on non-linear combinations of the orbital elements. As a result, a certain spectral peak in SBM could 4.2. Spectral analysis be a direct measure of the involved periodic planetary perturba- tions, and also the result of a non-linear mixing of several per- In order to determine the main periodicities present in our mod- turbing periodicities present in the variations of these elements. elling of SBM, we use the multi-taper method (MTM) and

A153, page 7 of 11 A&A 615, A153 (2018)

Table 2. Selected inequalities and their periods from the VSOP2013 theory.

Inequality Period (yr) kJ kN Pk (au) (J) Pk (au) (N) kJ kN Pk (rad) (J) Pk (rad) (N) 5 5 5 5 (1, 0, 0, 0) 11.86 17 43 2.1 10− 4.9 10− 17 10 1.8 10− 4.7 10− × 7 × 1 × 7 × 3 (1, 0, 0, 1) 12.78 123 1 2.0 10− 1.5 10− 117 2 1.8 10− 4.4 10− − × 4 × 4 × 4 × 7 (1, 1, 0, 0) 19.86 6 23 2.1 10− 2.5 10− 4 153 2.4 10− 1.1 10− − × 10 × 2 × 9 × 4 (0, 1, 0, 1) 35.87 2641 2 1.2 10− 3.6 10− 573 3 2.6 10− 9.3 10− − × 5 × 6 × 5 × 9 (1, 3, 0, 0) 57.01 12 236 5.5 10− 1.1 10− 10 404 5.0 10− 9.7 10− − × 4 × 5 × 4 × 8 (1, 2, 0, 0) 60.95 2 75 3.2 10− 1.3 10− 2 281 6.2 10− 2.4 10− − × 9 × 7 × 9 × 8 (0, 1, 2, 0) 98.58 510 359 5.3 10− 3.6 10− 428 242 5.4 10− 3.6 10− − × 7 × 9 × 7 × 10 (1, 3, 2, 0) 159.65 130 2381 1.7 10− 2.7 10− 67 1033 7.7 10− 7.6 10− − × 12 × 3 × 9 × 4 (0, 0, 1, 1) 171.44 8616 5 5.2 10− 4.6 10− 475 6 4.0 10− 1.6 10− − × 13 × 4 × 10 × 5 (0, 0, 2, 3) 178.70 17636 13 6.4 10− 5.1 10− 876 9 8.7 10− 4.7 10− − × 8 × 6 × 8 × 7 (1, 3, 0, 2) 185.12 299 156 2.0 10− 2.7 10− 152 104 7.9 10− 2.7 10− − × 10 × 8 × 9 × 9 (2, 6, 0, 5) 211.22 1906 1020 2.6 10− 2.5 10− 753 652 1.3 10− 2.7 10− − × 10 × 9 × 9 × 11 (1, 4, 5, 2) 242.94 1356 2494 5.6 10− 2.3 10− 531 3840 3.0 10− 2.1 10− − − × 9 × 10 × 8 × 11 (4, 10, 1, 2) 493.08 1042 4585 1.0 10− 4.3 10− 313 2331 1.2 10− 8.6 10− − − × 10 × 7 × 8 × 8 (0, 1, 2, 2) 501.44 1223 341 7.0 10− 4.1 10− 329 263 1.1 10− 3.0 10− − − × 9 × 6 × 8 × 7 (0, 1, 5, 4) 777.88 879 141 1.5 10− 3.4 10− 227 71 2.7 10− 9.2 10− − × 8 × 8 × 7 × 8 (2, 5, 1, 2) 1116.17 282 734 2.2 10− 5.4 10− 75 297 6.3 10− 2.1 10− − − × 8 × 6 × 7 × 7 (1, 3, 0, 3) 1498.63 327 193 1.7 10− 1.8 10− 74 76 6.3 10− 7.6 10− − × 6 × 8 × 5 × 8 (2, 6, 3, 0) 1597.80 72 1185 1.3 10− 1.7 10− 13 379 4.6 10− 1.1 10− − × 7 × 8 × 5 × 9 (4, 11, 3, 0) 1975.15 109 1412 3.4 10− 1.0 10− 16 445 2.1 10− 7.2 10− − × 10 × 4 × 8 × 4 (0, 0, 2, 4) 2116.57 2289 11 1.7 10− 6.9 10− 260 4 1.8 10− 4.2 10− − × 9 × 7 × 8 × 8 (1, 3, 1, 1) 2319.94 995 590 1.1 10− 1.0 10− 163 202 6.4 10− 6.2 10− − × 8 × 6 × 6 × 6 (2, 6, 4, 2) 2566.51 178 208 6.5 10− 1.5 10− 33 60 3.8 10− 1.2 10− (0, 0−, 1, 2)− 4233.14 619 3 3.3 × 10 9 8.3 × 10 3 78 1 5.9 × 10 7 1.0 × 10 2 − × − × − × − × − Notes. The order and the Pk amplitude for the semi-major axis and for the longitude of Jupiter and Neptune, respectively, are detailed. Cionco & Pavlov: Solar barycentric dynamics Lomb (normalized) periodogram (Ghil et al. 2002; Press et al. 10000 1 JS 1992). We find the Lomb periodogram to be the best choice be- JN 99% confidence level 90% confidence level 1000 95% confidence level UN 95% confidence level cause it gives a sharper determination of the longer periodicities JU J SN 0.1 99% confidence level (>200 yr). The time span considered comprises all 13 063 yr over 100 SU S 60.9 UN which the SBM analysis was performed. We have used one 10 N 0.01 value per year, hence we cannot resolve periods shorter than U 1 214 2 yr, according to the Nyquist theorem. First of all, we found the 0.001 0.1 240 basic and shorter periodicities, which are easier to determine. (0,1,−2) 516 4300 0.01 767 1500 The results for the magnitude of the velocity and barycentric 0.0001 2300 range correspond well with each other and are shown in Fig. 10. 0.001 The giant planet periods, and the periodicities of several inequal- 0.0001 1e−05 ities already shown in Table2, are present. Neptune’s orbital 1e−05 (1,−3) period, N, is clearly involved in the majority of the main pe- 1e−06 riodicities present in the corresponding spectral bands (e.g., 1e−06 Averaged Lomb periodogram [adim] decadal, centennial), such that the JN periodicity, for example, 1e−07 1e−07 Lomb periodogram for barycentric range [adim] 10 100 1000 is stronger than Jupiter’s orbital period, J. Those periods are im- printed even more strongly than the Jupiter–Saturn (1, 2) in- Period [yr] Period [yr] − equality, detected at 60.9 yr, which is virtually coincident with its Fig.Fig. 10. 10.MainMain short-term short-term periodicities periodicities detected detected in in barycentric barycentric range, range,r r Fig. 12. Main long-term periodicities detected in the averaged raw nominal value. The percentage difference TOP2013/VSOP2013 (raw(raw periodogram periodogram over over 13 13036 036 yr). years). The (1 The, 2) (1 inequality, 2) inequality of periodicity of period- power spectra of L, T, and Γ (over 13036 years). Numerical values of − (hereafter T/V) of this periodicity is just 0.09% (i.e., its oficity 60.9 ofyr 60.9is clearly yr is seenclearly (indicated seen (indicated with its− with numerical its numerical value). value).We note We several periods detected are indicated and explained in the text. TOP2013 period is 60.89 yr). The periods of the− (0, 1, 0, 1) and thenote participation the participation of Neptune of Neptune on important on important peaks. J, peaks. S, U, and J, S, N U, are and the N (0, 1, 2) inequalities, which appeared with relatively− low in- giantare planets’ the giant orbital planets’ periods. orbital This periods. result This is similar result to is the similar corresponding to the corre- tensity− for Jupiter in Table2, are also clearly detected. In the velocitysponding spectrum. velocity spectrum. these three parameters. The result in Fig. 12 shows the 2.3 kyr and 4.3 kyr spectral bands. The bimillennial band is detected∼ at spectral band related to the Jose-period, the largest power cor- ∼ responds to the N and UN periods. A spectral peak coincident low confidence levels in these parameters. Other bands around with the nominal Jose-period, or related to the (0, 0, 2, 3) in- The MTM andUN the periodogram also show, for solar r, wide 1.0–1.5 kyr can be seen. There are many inequalities related to − 0.0001 N 2266 equality which is also an important perturbation for Neptune, bands in the range186−yr,2–5 (1,−3, kyr, 0, 2), significant(0, 0, 3,−5) at the 95–99% con- these bands in the VSOP2013 theory, especially (2, 5, 1, 2), ∼ (1, 3, 0, 3), and (2, 6, 3). In addition, (2, 6, 3) presents− − a was not clearly identified. Around the UN period there are fidence level; nevertheless, these longer periodicities can be − − − 4300Γ 9% T/V difference in its periodicity. Also, the second har- many side-lobes associated with it. We think that these peri- more easily specified in other parameters such as aπ, L, , etc. ∼ a monic of the Uranus and Neptune great inequality, (0, 0, 3, 6), odicities could be present, but masked by these spectral power In particular, 1e−05 the acceleration π, shows clear long-term peri- − acceleration [adim] kN = 13 for Neptune’s longitude, has a 1411 yr periodic- variations. ods (butπ with relatively low power and low confidence level a ity. The Jupiter–Saturn great inequality (GI),∼ (2, 5) of 883.3 A153, page 8 of 11 yr (VSOP2013) or 919.8 yr (TOP2013) of nominal− periodicities 1e−06 never appears with distinctive power on these raw spectra. The GI is the most important perturbation for the Jupiter longitude 50% confidence level and it is kJ = 4 for Jupiter’s semi-major axis. Nevertheless, as 90% confidence level was pointed out by the reviewer, following the TOP2013 theory, 1e−07 95% confidence level 99% confidence level the contribution of the GI to solar r is negligible: it is the 2841th Lomb periodogram for 1000 term. Period [yr] A notable spectral band around 516 yr (also visible in Fig. 11) is found, which is virtually coincident with the most impor- Fig. 11. Main long-term periodicities detected in aπ acceleration. Spec- tant band detected in the calculation of the modelled torque over tral bands coincident with the bimillennial wave and the great inequality the solar tachocline zone, performed by Abreu et al. (2012). The (0, 0, 1, 2) between Uranus and Neptune are detected at low confidence , , , − (0 1 2 2) inequality, has a quite stable periodicity of around level in this parameter. Other indicated inequalities are discussed in the 500 yr,− but− far from the 516 yr period. TOP2013 just detects one text. perturbation around that periodicity, of 513.99 yr, equivalent to the period of (4, 10, 1, 2) of 493.08 yr for VSOP2013 (see Ta- ble 2). Another band− /peak− around 767 yr (Fig. 12) is quite promi- kyr, the VSOP2013 and TOP2013 theories show us the exis- nent in its spectral zone, and it is also present in the spectrum of tence of some inequalities with significant relative amplitudes, Fig. 11, but with low relative power. The (0, 1, 5, 4) inequality especially in orbital longitudes, for both Jupiter and Neptune: is especially related to this band, with a period− of 760 yr, fol- , , , , , ff (4 11 3), (2 6 4 2), both exhibit important di erences in lowing the TOP2013 theory. We also find two notable∼ spectral − − − . / their periodicities of 26.4% and 14 9% (2496.54 yr 1975.15 peaks at 214 yr and 240 yr. We find (2, 6, 0, 5), with a period of / − yr and 2184.48 yr 2566.51 yr for the TOP2013 and VSOP2013 211.22 yr and 213.12 yr for VSOP2013− and TOP2013, respec- , , , theories, respectively). The (1 3 1 1) inequality is also present tively. Also, (1, 4, 5, 2), the 243 yr period of VSOP2013, −. in this spectral zone, with a 5 8% (2184.5 yr and 2319.93 yr) but with 246 yr− in TOP2013.− ∼ The inequality (4,-10,-1,3), with / ff − T V di erence. Other shorter periodicities that can be identified quite low∼ amplitudes in VSOP2013, has a 243 yr period in are near the N and UN peaks, namely the 186 yr period, which TOP2013. ∼ closely coincides with the period of the inequality (1, 3, 0, 2). This period is also coincident with an important perturbation− for Neptune’s longitudes, (0, 0, 3, 5), k = 29. In addition, a sharp N 5. Conclusions peak is observed at 159 yr that− is coincident with the period of the (1, 3, 2) inequality. − To date, EPM2017H is the most complete high-precision solar- Power spectra of L, T, and Γ are quite similar and we present planetary ephemeris accounting for the SBM during the whole them as the averaged (previously normalized) periodogram of Holocene. The positioning of the SBM was done using a ref-

Article number, page 9 of 11 Cionco & Pavlov: Solar barycentric dynamics

10000 1 JS JN 99% confidence level 90% confidence level 1000 95% confidence level UN 95% confidence level JU J SN 0.1 99% confidence level 100 SU S 60.9 UN 10 U N 0.01 1 214 0.001 0.1 240 (0,1,−2) 516 4300 0.01 767 1500 0.0001 0.001 2300

0.0001 1e−05

1e−05 (1,−3) 1e−06

1e−06 Averaged Lomb periodogram [adim]

1e−07 1e−07

Lomb periodogram for barycentric range [adim] 10 100 1000 Period [yr] Period [yr]

Fig. 10. Main short-term periodicities detected in barycentric range, r Fig. 12. Main long-term periodicities detected in the averaged raw (raw periodogram over 13036 years). The (1, 2) inequality of period- power spectra of L, T, and Γ (over 13036 years). Numerical values of icity of 60.9 yr is clearly seen (indicated with− its numerical value). We several periods detected are indicated and explained in the text. note the participation of Neptune on important peaks. J, S, U, and N are the giant planets’ orbital periods. This result is similar to the corre- sponding velocity spectrum. R. G. CioncoCionco and & D. Pavlov: A. Pavlov: Solar Solar barycentricthese barycentric three dynamics parameters. dynamics The result in Fig. 12 shows the 2.3 kyr and 4.3 kyr spectral bands. The bimillennial band is detected∼ at ∼ 10000 low confidence 1 levels in these parameters. Other bands around JS JN 99% confidence level 1.0–1.5 kyr can be seen. There are90% many confidence inequalities level related to 1000 UN 95% confidence level UN 95% confidence level 0.0001 N JU 2266 J186−yr, (1,−3, 0, 2), (0, 0,SN 3,−5) these bands 0.1 in the VSOP2013 theory,99% confidence especially level (2, 5, 1, 2), 100 SU (1, 3, 0, 3), and (2, 6, 3). In addition, (2, 6, 3) presents− − a S 60.9 UN − − − 10 4300N 9% T 0.01/V difference in its periodicity. Also, the second har- U ∼ 1 monic of the Uranus and Neptune great inequality, (0, 0, 3, 6), 1e−05 214 − acceleration [adim] kN = 13 0.001 for Neptune’s longitude, has a 1411 yr periodic- π 0.1 (0,1,−2) 240 4300 a ity. The Jupiter–Saturn great516 inequality (GI),∼ (2, 5) of 883.3 0.01 767 1500 yr (VSOP2013) 0.0001 or 919.8 yr (TOP2013) of nominal− periodicities 0.001 2300 1e−06 never appears with distinctive power on these raw spectra. The 0.0001 GI is the 1e−05 most important perturbation for the Jupiter longitude 1e−05 (1,−3) and it is kJ = 4 for Jupiter’s semi-major axis. Nevertheless, as 50% confidence level 1e−06 90% confidence level was pointed out by the reviewer, following the TOP2013 theory,

1e−06 Averaged Lomb periodogram [adim] 1e−07 95% confidence level the contribution of the GI to solar r is negligible: it is the 2841th 1e−07 99% confidence level 1e−07 Lomb periodogram for Lomb periodogram for barycentric range [adim] 10 1000 100 term. 1000 PeriodPeriod [yr][yr] A notable spectral band aroundPeriod 516 [yr] yr (also visible in Fig. 11) is found, which is virtually coincident with the most impor- Main short-term periodicities detected in barycentric range, r Fig. 12. MainMain long-term long-term periodicities periodicities detected detected in in the the averaged averaged raw raw Fig.Fig.Fig. 11. 10.11.MainMain long-term long-term periodicities periodicities detected detected in a inπ aacceleration.π acceleration. Spec- Spec- Fig.tant band12. detected in the calculation of the modelled torque over (raw periodogram over 13036 years). The (1, 2) inequality of period- powerpower spectra spectra of ofL, TL,,T and, andΓ (overΓ (over 13 036 13036 yr). years). Numerical Numerical values valuesof sev- of traltral bands bands coincident coincident with with the the bimillennial bimillennial wave wave and− and the the great great inequality inequality the solar tachocline zone, performed by Abreu et al. (2012). The (0,icity(00,,10,, of1,2) 60.9 between2) between yr is clearly Uranus Uranus seen and and Neptune (indicated Neptune are with are detected detected its numerical at low at low confidence value). confidence We eralseveral(0 periods, 1, periods2, detected2) inequality,detected are indicated are has indicated aand quite explained and stable explained in periodicity the in text. the text. of around − − levelnotelevel in the thisin this participation parameter. parameter. Other of Other Neptune indicated indicated on inequalities important inequalities peaks. are are discussed discussed J, S, U, in and inthe the N 500 yr,− but− far from the 516 yr period. TOP2013 just detects one are the giant planets’ orbital periods. This result is similar to the corre- text.text. perturbation around that periodicity, of 513.99 yr, equivalent to sponding velocity spectrum. these three parameters. The result in Fig. 12 shows the 2.3 kyr 919.8the periodyr (TOP2013) of (4, 10 of, nominal1, 2) of periodicities 493.08 yr for never VSOP2013 appears∼ (see with Ta- in the Lomb-periodogram), even in the raw power spectrum and 4.3 kyr spectral− bands.− The bimillennial band is detected at distinctiveble 2).∼ Another power band on these/peak raw around spectra. 767 yrThe (Fig. GI 12) is the is quite most promi- im- (Fig.kyr, 11 the). Notably, VSOP2013 spectral and TOP2013 bands around theories 2.2–2.3 show kyr us (with the exis- a low confidence levels in these parameters. Other bands around portantnent in perturbation its spectral for zone, the and Jupiter it is longitude also present and in it the is k spectrumJ = 4 for of measurable peak at 2266 yr) and around 4.3 kyr are easily found 1.0–1.5 kyr can be seen. There are many inequalities related to tence of someUN inequalities with significant relative amplitudes, Jupiter’sFig. 11, semi-major but with low axis. relative Nevertheless, power. The as (0was, 1, pointed5, 4) inequality out by withespecially any 0.0001 kind inNof orbital filtering longitudes, (no sampling). for both The Jupiter2266 MTM and applied Neptune: these bands in the VSOP2013 theory, especially− (2, 5, 1, 2), 186−yr, (1,−3, 0, 2), (0, 0, 3,−5) theis reviewer, especially following related to the this TOP2013 band, with theory, a period the contribution of 760− yr,− of fol- to(4 this, 11 parameter, 3), (2, shows6, 4, 2), a both2 kyr exhibit band significant important diatff theerences 95% in (1, 3, 0, 3), and (2, 6, 3). In addition, (2, 6, 3)∼ presents a ∼ thelowing GI− to solar the TOP2013r is negligible:− theory. it Weis the also 2841th find term.two− notable spectral confidencetheir− periodicities level.− We of relate− 26.4% the and first14 mentioned.9% (2496.54 band yrwith4300/1975.15 the 9% T/V difference in its periodicity. Also, the second har- ∼peaksA notable at 214 yr spectral and 240 band yr. We around find (2 516, 6, yr0, 5), (also with visible a period in of bimillennialyr and 2184.48 wave yrdiscussed/2566.51 in yr Sect. for the−3.2. TOP2013 These bands, and weVSOP2013 think, monic of the Uranus and Neptune great− inequality, (0, 0, 3, 6), 1e−05 Fig.211.22 11) is yr found, and 213.12 which is yr virtually for VSOP2013 coincident and with TOP2013, the most respec- im-− aretheories,mainly acceleration [adim] respectively). produced by Thethe most (1, 3 important, 1, 1) inequality inequalities is also found present kN = 13 for Neptune’s longitude, has a 1411 yr periodic- π portant band detected, , in, the calculation of the modelled torque a tively. Also, (1 4 5 2), the 243 yr period∼ of VSOP2013, atin these this frequencies: spectral zone, the with Uranus–Neptune a 5−.8% (2184.5 (0, 0 yr, 1, and2) 2319.93 great in- yr) ity. The Jupiter–Saturn− − great inequality∼ (GI), (2, 5) of 883.3 − overbut the with solar246 tachocline yr in TOP2013. zone, performed The inequality by Abreu (4,-10,-1,3), et− al.(2012 with). equality/ Simonff et al.(1992; kN −= 1 for Neptune longitude), and yr (VSOP2013) or 919.8 yr (TOP2013) of nominal periodicities T V di erence. Other shorter periodicities that can be identified Thequite (0, 1 low, ∼2 amplitudes, 2) inequality, in VSOP2013, has a quite has stable a 243 periodicity yr period of in its first harmonic, 1e−06 (0, 0, 2, 4), with periodicities of 4233.14 yr never appears− − with distinctive power on these raw spectra. The are near the N and UN peaks,− namely the 186 yr period, which aroundTOP2013. 500 yr, but far from the 516 yr period.∼ TOP2013 just andclosely 2116.57 coincides yr, respectively, with the for period VSOP2013. of the inequality The T/V di (1ff, erence3, 0, 2). GI is the most important perturbation for the Jupiter longitude − detects one perturbation around that periodicity, of 513.99 yr, ofThis these period inequalities is also is coincident3%. In particular,with an important TOP2013 perturbation shows that for and it is kJ = 4 for Jupiter’s semi-major axis. Nevertheless, as , , , ∼ 50% confidence level equivalent to the period of (4, 10, 1, 2) of 493.08 yr for (0Neptune’s0 1 2) is longitudes, the only inequality (0, 0, 3, 5), withk90%N period= confidence29. In4.3 addition,level kyr. Around a sharp was pointed out by the reviewer,− following− the TOP2013 theory, 2–2.5 kyr,− 1e−07 the VSOP2013 and TOP2013− 95% confidence theories∼ level show us the VSOP20135. Conclusions (see Table2). Another band /peak around 767 yr peak is observed at 159 yr that is coincident99% confidence with level the period of (Fig.the 12 contribution) is quite of prominent the GI to solar in itsr is spectral negligible: zone, it is and the 2841th it is existencetheLomb periodogram for (1, of3, 2) some inequality. inequalities with significant relative am- term. − 1000 alsoTo present date, EPM2017H in the spectrum is the most of Fig. complete 11, but high-precision with low relative solar- plitudes,Power especially spectra of inL, orbitalT, and Γ longitudes,are quite similar for both and we Jupiter present planetaryA notable ephemeris spectral accounting band around for 516 the SBM yr (also during visible the in whole Fig. , , , Period, , [yr] power. The (0, 1, 5, 4) inequality is especially related to this andthem Neptune: as the averaged (4 11 3), (previously (2 6 4 normalized)2), both periodogram exhibit im- of 11)Holocene. is found, The which− positioning is virtually of the coincident SBM was with done the most using impor- a ref- portant differences− in their− periodicities− of 26.4% and band, with a period of 760 yr, following the TOP2013 theory. Fig. 11. Main long-term periodicities detected in aπ acceleration. Spec- tant band detected in the∼ calculation of the modelled torque over 14.9% (2496.54 yr/1975.15 yr and 2184.48 yr/2566.51 yr for We also find two notable spectral peaks at 214 yr and 240 yr. − tral bands coincident with the bimillennial wave and the great inequality Wethe find solar (2 tachocline, 6, 0, 5), zone,with a performed period of by 211.22Article Abreu yrnumber, et and al. (2012). 213.12 page 9 ofyr The 11 the, TOP2013, , and VSOP2013 theories, respectively). The (0 0 1 2) between Uranus and Neptune are detected at low confidence (0, 1, 2, −2) inequality, has a quite stable periodicity, of, , around (1level, 3, in1,− this1) inequality parameter. is Other also indicated present in inequalities this spectral are discussedzone, with in a the for VSOP2013− − and TOP2013, respectively. Also, (1 4 5 2), − the500243 yr, yrbut period far from of VSOP2013, the 516 yr period. but with TOP2013246 yr just in− TOP2013. detects− one 5text..8% (2184.5 yr and 2319.93 yr) T/V difference. Other shorter perturbation∼ around that periodicity, of 513.99∼ yr, equivalent to −periodicities that can be identified are near the N and UN peaks, The inequality (4, 10, 1, 3), with quite low amplitudes in VSOP2013,the period ofhas (4 a, −24310, yr−1, period2) of 493.08 in TOP2013. yr for VSOP2013 (see Ta- namely the 186 yr period, which closely coincides with the pe- ble 2). Another band∼− /peak− around 767 yr (Fig. 12) is quite promi- riodkyr, of the the VSOP2013 inequality (1and, TOP20133, 0, 2). This theories period show is also us coinci-the exis- − nent in its spectral zone, and it is also present in the spectrum of denttence with of an some important inequalities perturbation with significant for Neptune’s relative longitudes, amplitudes, Fig. 11, but with low relative power. The (0, 1, 5, 4) inequality especially in orbital longitudes, for both Jupiter and Neptune: 5. Conclusions (0, 0, 3, 5), kN = 29. In addition, a sharp peak is observed is especially related to this band, with a period− of 760 yr, fol- , −, , , , ff at(4 15911 yr3), that (2 is coincident6 4 2), both with exhibit the period important of the di (1erences, 3, 2) in Tolowing date, EPM2017H the TOP2013 isthe theory. most We complete also find high-precision two notable∼ spectralsolar- − − − . /− inequality.their periodicities of 26.4% and 14 9% (2496.54 yr 1975.15 planetarypeaks at ephemeris 214 yr and accounting 240 yr. We forfind the (2, SBM6, 0, during5), with the a period whole of / − yrPower and 2184.48 spectra yrof 2566.51L, T, and yrΓ forare thequite TOP2013 similar and and we VSOP2013 present Holocene.211.22 yr The and positioning 213.12 yr for of VSOP2013the SBM− was and done TOP2013, using a respec- ref- , , , themastheories, the respectively). averaged (previously The (1 3 normalized)1 1) inequality periodogram is also present of erencetively. system Also, (1 defined, 4, 5, by2), the the solar243 system’s yr period IP. Theof VSOP2013, released in this spectral zone, with a 5−.8% (2184.5 yr and 2319.93 yr) − − ∼ these three parameters. The result− in Fig. 12 shows the 2.3 kyr filesbut with with the246 SBM yr data in TOP2013. are stored The in the inequality repository (4,-10,-1,3), PANGAEA with andT/V4.3 diff kyrerence. spectral Other bands. shorter The periodicities bimillennial that band can isdetected be∼ identified at data publisher.∼ The improvement in the solar orbit calculation is ∼ quite low amplitudes in VSOP2013, has a 243 yr period in loware confidence near the N levels and UN in these peaks, parameters. namely the Other 186 yr bands period, around which evidentTOP2013. when comparing the SBM in EPM2017H∼ and DE431 closely coincides with the period of the inequality (1, 3, 0, 2). 1.0–1.5 kyr can be seen. There are many inequalities related− to ephemerides. Although they are very consistent ephemerides, theseThis bands period in is the also VSOP2013 coincident theory,with an especially important perturbation(2, 5, 1, 2), for the differences in certain important dynamical quantities of Neptune’s, , , , longitudes,, , (0, 0, 3, 5), kN =, 29., In addition,− − a sharp (1 3 0 3) and (2 6 3). In addition,− (2 6 3) presents a 9% SBM5. Conclusions (and not only in angular quantities) obtained from both T/peakV− diff iserence observed in its at− periodicity. 159 yr that Also, is coincident the− second with harmonic the period∼ of of ephemerides can be very significant. The differences in torque the (1, 3, 2) inequality. , , , = the Uranus− and Neptune great inequality, (0 0 3 6), kN 13 relatedTo date, quantities EPM2017H are clearly is the imprinted most complete by thehigh-precision dynamics of dwarf solar- for Neptune’sPower spectra longitude, of L, hasT, anda 1411Γ are yr quite periodicity. similar− and The we Jupiter– present planetplanetary Eris, ephemeris which is included accounting in for the the modelling SBM during of the the TNOs whole Saturnthem great as the inequality averaged (GI), (previously (2∼, 5) of normalized) 883.3 yr (VSOP2013) periodogram or of regionHolocene. performed The positioning in EPM2017H. of the In SBM this sense, was done it is using clear thata ref- − ArticleA153, number, page page 9 of 9 of11 11 A&A 615, A153 (2018) undiscovered massive bodies beyond the Kuiper belt would have Jose-period or that belonging to the TNO bodies, such as the a non-negligible effect on the SBM, even on the inclination of the dwarf planet Eris, are not detected (even at a low confidence solar spin axis with respect to the IP (Bailey et al. 2016). There- level) in the raw power spectra presented here. fore, significant differences in SBM parameters could also be expected in EPM2017H regarding a more accurate model that Acknowledgements. Part of this work was supported by the grant PID-UTN includes massive planets that are farther away. 4362 (2017–2018) of UTN. The authors acknowledge the reviewer, Jean Louis The periodicities present in the SBM were analysed for the Simon, for his comments that improved this work. RGC especially acknowledges first time in terms of the main physical planetary perturbations him for correcting an early claim that the observed variations in the periodicities involving the giant planets. We emphasize that a simple com- were the main cause of the lack of some inequalities in SBM. The authors are indebted to Elena Pitjeva for useful discussions regarding the new EPM2017 bination of mean motions does not produce a “resonance” in planetary model. RGC thanks William Folkner for an open discussion of the the system, and does not guarantee the dynamical importance or DE431 ephemeris at an earlier stage of this work, and for encouraging him to strength of that inequality7 in the SBM. This is particularly per- address several topics. RGC especially thanks Willie Soon for the very important tinent in the periodicity of the solar retrograde motion intervals, discussions, and José Valentini and Luciana Andrín for their help in the editing which seem to be far from a regular metric. Particular long-term process. periods in the solar cycles, such as the 2.2–2.3 kyr cycle (known as the Hallstatt cycle, see e.g., Beer et al. 2018) were also related References in the past to SBM using different arguments. We show cyclic variations in key dynamical parameters of SBM with that bimil- Abreu, J. A., Beer, J., Ferriz-Mas, A., McCracken, K. G., & Steinhilber, F. 2012, lennial periodicity, and confirm the existence of a 2.3 kyr spec- A&A, 548, A88 ∼ Bailey, E., Batygin, K., & Brown, M. E. 2016, AJ, 152, 126 tral band in SBM, although with varying spectral power (i.e., Beer, J., Tobias, S. M., & Weiss, N. O. 2018, MNRAS, 473, 1596 at different confidence levels). We are not arguing that the Chang, Y.-L., Bodenheimer, P. H., & Gu, P.-G. 2012, ApJ, 757, 118 Hallsttat cycle is produced by or is connected to this bimillen- Charvátová, I. 1988, Adv. Space Res., 8, 147 nial period. We propose the origin of this spectral band in SBM, Charvátová, I. 2009, New Ast., 14, 25 Cionco, R. G., & Abuin, P. 2016, Adv. Space Res., 57, 1411 taking into account the important periodic perturbation, is the Cionco, R. G., & Compagnucci, R. H. 2012, Adv. Space Res., 50, 1434 inequality (0, 0, 2, 4). In this sense, statements such as those of − Cionco, R. G., & Soon, W.-H. 2015, New Ast., 34, 164 Scafetta et al.(2016) regarding a “stable resonance” produced by Cionco, R. G., & Soon, W.-H. 2017, Earth-Sci. Rev., 166, 206 the inequality (1, 3, 1, 1) as the specific origin of this bimillen- Fairbridge, R. W., & Shirley, J. H. 1987, Sol. Phys., 110, 191 nial periodicity, just− because of the periodicity coincidence, are Fienga, A., Manche, H., Laskar, J., & Gastineau, M. 2008, A&A, 477, 315 Folkner, W. M., Williams, J. G., Boggs, D. H., Park, R. S., & Kuchynka, P. 2014, very vulnerable to criticism. In addition, the TOP2013 period of Interplanet. Network Prog. Rep., 196, 1 this inequality is 2184.5 yr. Ghil, M., Allen, M. R., Dettinger, M. D., et al. 2002, Rev. Geophys., 40, 3 The most important periods are detected with different con- Javaraiah, J. 2005, MNRAS, 362, 1311 fidence levels according to each parameter studied. We have Jose, P. D. 1965, AJ, 70, 193 Juckett, D. 2003, A&A, 399, 731 not used stringent criteria of statistical confidence levels to re- Landscheidt, T. 1999, Sol. Phys., 189, 415 ject low power periodicities because we are analysing a theo- Lanza, A. F. 2013, A&A, 557, A31 retical model of the solar system. Although it is not a purely McCracken, K. G., Beer, J., & Steinhilber, F. 2014, Sol. Phys., 289, 3207 periodic model, the use of high-level semi-analytical theories Perryman, M., & Schulze-Hartung, T. 2011, A&A, 525, A65 (VSOP2013/TOP2013) enables us to determine a physical root Pitjeva, E. V. 2010, IAU Symp., 263, 93 Pitjeva, E. V., & Pitjev, N. P. 2013, MNRAS, 432, 3431 to the bands/peaks, even if they are of small amplitude. The ap- Pitjeva, E. V., & Pitjev, N. P. 2014, Celest. Mech. Dyn. Astron., 119, 237 plication of very precise semi-analytical theories is fundamental Press, W. H., Flannery, B. P., Teukolsky, S. A., & Vetterling, W. T. 1992 , in order to identify possible perturbations contributing to some Numerical recipes in Fortran (Cambridge: Cambridge University Press) spectral band of the SBM. Moreover, a full treatment of this Santos-Sanz, P., Lellouch, E., Fornasier, S., et al. 2012, A&A, 541, A92 Scafetta, N., Milani, F., Bianchini, A., & Ortolani, S. 2016, Earth-Sci. Rev., problem implies the application of these theories to obtain the 162, 24 orders of the specific inequalities in each dynamical parameter Simon, J. -L., Joutel, F., & Bretagnon, P. 1992, A&A, 265, 308 studied. Another important long-term inequality clearly present Simon, J.-L., Francou, G., Fienga, A., & Manche, H. 2013, A&A, 557, A49 (but with low confidence levels in some analysed parameters) is Soon, W., Herrera, V. M. V., Selvaraj, K., et al. 2014, Earth-Sci. Rev., the great inequality (0, 0, 1, 2) between Uranus and Neptune. 134, 1 − Souami, D., & Souchay, J. 2012, A&A, 543, A133 The relevance of Neptune (with more than 30% of the contri- Vondrák, J., Capitaine, N., & Wallace, P. 2011, A&A, 534, A22 bution of Uranus to the solar system’s angular momentum) is Wilson, C. 1985, Arch. Hist. Exact Sci., 33, 15 evident in the spectral analysis. Periodicities like the nominal Wood, R. M., & Wood, K. D. 1965, Nature, 208, 129

7 A word intensively used by Laplace to address the departures of the observed planetary motions from the Keplerian elliptic motion; see e.g., Wilson(1985).

A153, page 10 of 11 R. G. CioncoCionco and & Pavlov: D. A. Pavlov: Solar barycentric Solar barycentric dynamics dynamics Cionco & Pavlov: Solar barycentric dynamics

Appendix A: AdditionalZ information L BB Z L B The SBO for thezz whole studied time interval (4747872 Julian days) and its dynamicalz parameters are stored in PANGAEA web8. There are six self-explanatory files with labels over each nˇ aar nˇ ar saved quantity andH aH README file. The EPM ephemerides,H including EPM2017H, are oriented SBOSBO to the International Conventional Reference System (ICRS) SBO (with accuracy of 0.2 milliarcseconds; Pitjeva 2017, priv. rˇ ˇtˇ comm.) which uses the≈ equator as reference plane. As explained rˇ tˇ ICRS equator B Y in Sect.ICRS2, we equator transformB the solar coordinates obtainedY from Fig. A.2. Non-inertialNon-inertial co-orbital co-orbital system system defined defined by by the the radial radial ˘,r˘, Fig.Fig. A.2. Non-inertialNon-inertial˘ co-orbital co-orbital system system defined defined by the by radial ther˘ radial, transver-r˘, 00.9 transversal˘t, and normal direction n˘ to the SBO. The symbol marks EPM2017H to a reference frame attached to the3100.9 solar system’s transversal t˘, and normal direction n˘ to the SBO. The symbol marks 003100.9 transversalsalthe Sun’s˘t, and positiont normal, and normal on direction its orbit,directionn˘ Bto isn the˘ theto SBO. barycentre. the SBO. The The symbol The symbol solar movementmarks marks the IP. We take into account the estimated orientation◦00 31 between the the Sun’s position on its orbit, B is the barycentre. The solar movement 23◦ 0 theSun’s Sun’s position position on on its its orbit, orbit, B B is is the the barycentre. The The solar solar movement movement I==23 is anti-clockwise, prograde, regarding B. L is the Sun’s orbital angu- IP frame and the materializing frameI = of ICRS, ICRF, follow- is anti-clockwise, prograde, regarding B. LLis the Sun’s orbital angu- I islar anti-clockwise, momentum, is parallelprograde, to regardingn˘. The vector B. arisis the the Sun’s Sun’s orbital acceleration angu- ing Souami & Souchay(2012), i.e. using two orientation angles, lar momentum, is parallel to n˘. The vector ar is the Sun’s acceleration lardecomposed momentum, in the is parallel radial direction to n˘. The (a vectorr < 0).ar r is the Sun’s acceleration ω = 3◦510900.4 decomposed in the radial direction (ar < 0). namely, the inclination I = 23ω◦0=0313◦00.5190 and900.4 the ascending node decomposeddecomposed in in the the radial radial direction direction ( (ar r <0).0). ω = 3◦5109.004 (see Fig. A.1) X xx X x VI = Rx(I) Rz(ω) VE, (A.1) Fig. A.1. Reference systems associated with the ICRS and IP. The an- aar Fig. A.1. Reference systems associated with the ICRS and IP. The an- nˇ ar Fig.wheregles A.1.I and theω vectorare usedVE tois transform defined in the the solar EPM2017H coordinates reference from the ICRS sys- nˇ gles I and ω are used to transform the solar coordinates from the ICRS rˇ temto the and IP system.VI is the With correspondingH we depict the vector solar withsystem’s components total angular in mo- the ˇtˇ rˇ to the IP system. With H we depict the solar system’s total angular mo- tˇ SBO mentumIPmentum system, (always (always i.e. normal normal defined to to the with the IP). IP). the B B isx is-axis the the solar solar towards system’s system’s the barycentre. barycentre. nodal line SBO mentumequator-IP (always and normal the z-axis to the coincident IP). B is the with solar the system’s direction barycentre. of the to- tal angular momentum of the solar system. Cionco & Pavlov: Solar barycentric dynamics

AppendixAppendix A: A: Additional Additional information information Appendix A: AdditionalZ information BB B The SBO for the whole studied time interval (4747872 Julian B L The SBO for the whole studied time interval (4747872 Julian LL Thedays) SBO and for its the dynamical wholez studied parameters time areinterval stored (4747872 in PANGAEA Julian L days)8 and its dynamical parameters are stored in PANGAEA Fig. A.3. Non-inertial co-orbital system defined (as in Fig.A.2) for a web8 . There are six self-explanatory files with labels over each Fig.Fig. A.3. A.3. Non-inertial co-orbital system defined (as in Fig.A.2)Fig. A.2) for for a a web 8. There are six self-explanatory files with labels over each Fig.retrograde A.3. Non-inertial Sun. Now co-orbitalis anti-parallel system todefined˘. The (as solar in Fig.A.2) movement for a is web . There are six self-explanatory files with labels over each retrograderetrograde Sun. Sun. Now NowLLisis anti-parallel anti-parallel to ton˘n˘.. The The solar solar movement movement is is saved quantity and a README file. retrograde Sun. Now L isnˇ> anti-parallelar to n˘. The solar movement is saved quantity and a README file. clockwiseclockwise regarding regarding B B (a (ar>>0).0). The EPM ephemerides,H including EPM2017H, are oriented clockwiseclockwise regarding regarding B B (a (ar r> 0).0). The EPM ephemerides, including EPM2017H, are oriented toto the the International International Conventional Conventional Reference Reference System System (ICRS) (ICRS) SBO to(with the accuracy International of Conventional0.2 milliarcseconds; Reference Pitjeva System 2017, (ICRS) private (with accuracy of ≈0.2 milliarcseconds; Pitjeva 2017, private zz communication)communication) which which≈ uses uses the the equator equator as as reference reference plane. plane. As As rˇ z tˇ communication) which uses the equator as reference plane. As LL explainedexplainedICRS in in Sect. Sect. equator 2, 2, we we transform transformB the the solar solar coordinates coordinates obtained obtainedY L explainedfrom EPM2017H in Sect. 2, to we a reference transform frame the solar attached coordinates to the obtainedsolar sys- Fig. A.2. Non-inertial co-orbital system defined by the radial r˘, from EPM2017H to a reference frame attached to the.9 solar sys- transversal ˘, and normal directionLLz ˘ to the SBO. The symbol marks from EPM2017H to a reference frame attached to00 the solar sys- t L Lz n tem’s IP. We take into account the estimated orientation0 31 between L nˇ tem’s IP. We take into account the estimated orientation◦ 0 between the Sun’s position on itsL orbit, B is the barycentre.ˇ Thenˇ solar movement the IP frame and the materializing frame of23 ICRS, ICRF, follow- tˇt nˇ the IP frame and the materializing frameI of= ICRS, ICRF, follow- is anti-clockwise, prograde, regarding B. Laisπ t the Sun’s orbital angu- ing Souami & Souchay (2012), i.e. using two orientation angles, aπ rˇ ing Souami & Souchay (2012), i.e. using two orientation angles, lar momentum, is parallel to n˘. The vector180◦arθis the Sun’srˇ acceleration ing Souami & Souchay (2012), i.e. using two orientation angles, 180◦ −θ rˇ namely, the inclination I = 23 0 31.9 and the ascending node 180◦ θ = ◦ 0 .00 decomposed in the radial direction (ar < 0).− namely, the inclination I = 23◦ω0031= 003.9◦51 and0900. the4 ascending node ωω==3351◦51909.004.4 (see (see Fig.A.1) Fig.A.1) B ω = 3◦510900.4 (see Fig.A.1) B y = ω , X x y VI= RR(xI()I)RR(zω( )) VE, (A.1)(A.1) us VI = Rx(I) Rz(ω) VE , (A.1) us Ω s Fig. A.1. Reference systems associated with the ICRS and IP. The an- x Ω s ar wherewhereFig. A.1. the the vector vector VEisis defined defined in in the the EPM2017H EPM2017H reference reference sys- sys- x Ωs ˇ is wheregles I and the vectorω are usedVE tois transform defined in the the solar EPM2017H coordinates reference from the ICRSsys- n is temgles I andandVωI areis the used corresponding to transform the vector solar coordinates with components from the in ICRS the s tem and VI is the correspondingH vector with components in the rˇ temtoIPto the the system, and IP IPV system. system.I is i.e. the With With defined correspondingH we with depict the the vectorx-axis solar with system’s towards components total the angular nodal in the linemo- tˇ IPmentum system, (always i.e. defined normal to with the IP). the Bx is-axis the solar towards system’s thebarycentre.nodal line SBO equator-IPmentum (always and thenormalz-axis to the coincident IP). B is the with solar the system’s direction barycentre. of the to- RelationshipRelationship between between the the IP IP system system and and the the co-orbital co-orbital system sys- equator-IP and the z-axis coincident with the direction of the to- Fig.Fig. A.4. A.4. Relationship between the IP system and the co-orbital sys- equator-IP and the z-axis coincident with the direction of the to- Fig. A.4. Relationship between the IP system and the co-orbitalΩ sys- tal angular momentum of the solar system. tem(progradetem (prograde (prograde Sun). Sun). Sun). It is Itmaterialized It is is materialized materialized by the by by three the the angles three three anglesu anglesS, iS, anduuS, ,i iS,., and The and tal angularThe geometry momentum of of the the co-orbital solar system. system can be seen in tem (prograde Sun). It is materialized by the three angles uS , iS , and The geometry of the co-orbital system can be seen in ΩxyS-plane. The xy is-plane the IP. is The thea IP.π vector The a isπ vector the projection is the projection of the Sun’s of the acceler- Sun’s The geometry of the co-orbital system can be seen in ΩS . The xy-plane is the IP. The aπ vector is the projection of the Sun’s Figs.The A.2 geometry and A.3; of the the geometry co-orbital of system the θ angle can beis shown seen in in ΩS . The xy-plane is the IP. The aπ vector is theθ projectionθ of the Sun’s Fig.(A.2) and Fig.(A.3); the geometry of the θ angle is shown accelerationationacceleration onto the onto onto instantaneous the the instantaneous instantaneous SBO plane.SBO SBO plane. plane.The Theangle Theθ angle isangle the is polaris the the polar angle polar Fig.(A.2)Fig.Appendix A.4. andThe A: transformationFig.(A.3); Additional the geometry from information the IP of system the θ angle to the is co-orbital shown acceleration onto thea instantaneous SBO plane. The θ angle is the polar in Fig. (A.4). The transformation from the IP system to the co- whichangle which positions positionsπBfromaπ from the radial the radial direction. direction. The Thez-componentzz-component of the of in Fig. (A.4). The transformation from the IP system to the co- angle which positions aπ from the radial direction. The z-component of insystem Fig. (A.4). is performed The transformation by applying three from rotation the IP system matrices, to thenamely co- orbitalthe orbital solar solar angular angular momentum momentum vector, vector,L =LL=n˘,L isn˘L, isz =LzL=cosL cosiS. iS . orbitalThe SBO system for theis performed whole studied by applying time interval three rotation (4747872 matrices, Julian the orbital solar angular momentumL vector, L = L n˘, is Lz = L cos iS . orbital system is performed by applying three rotation matrices, the orbital solar angular momentum vector, L = L n˘, is Lz = L cos iS . namelyVnamelydays)= R and(u its) R dynamical(i ) R (Ω ) parametersV , are stored in PANGAEA(A.2) namelywebO 8. Therez S arex S sixz self-explanatoryS I files with labels over each Fig. A.3. Non-inertial co-orbital system defined (as in Fig.A.2) for a retrograde Sun. Now L is anti-parallel to n˘. The solar movement is VsavedO== R quantityz(uSV) Rx and(iS ) aR READMEzΩ(ΩS ) V,I, file. (A.2) VwhereO = Rz now(uS ) ROx(isiS ) theRz( correspondingΩS ) VI , vector with defined compo-(A.2) clockwise regarding B (ar > 0). nentsThe in theEPM co-orbital, ephemerides, non-inertial including system. EPM2017H, are oriented where now VO is the corresponding vector with defined compo- whereto the now InternationalVO is the corresponding Conventional vector Reference with defined System compo- (ICRS) wherenents nowin theV co-orbital,O is the corresponding non-inertial system.vector with defined compo- nents(with in accuracy the co-orbital, of 0 non-inertial.2 milliarcseconds; system. Pitjeva 2017, private z communication) which≈ uses the equator as reference plane. As L explained in Sect. 2, we transform the solar coordinates obtained from EPM2017H to a reference frame attached to the solar sys- Lz tem’s IP. We take into account the estimated orientation between L nˇ tˇ 8the8 IP frame and the materializing frame of ICRS, ICRF, follow- 8 https://doi.pangaea.de/10.1594/PANGAEA.882534 aπ 8 https://doi.pangaea.de/10.1594/PANGAEA.882534 ˇ ing Souami & Souchay (2012), i.e. using two orientation angles, 180 θ r ◦− namely, the inclination I = 23◦003100.9 and the ascending node Article number,A153, page 11 of 11 Article number, page 11 of 11 ω = 3◦510900.4 (see Fig.A.1) B y VI = Rx(I) Rz(ω) VE, (A.1) us x Ωs where the vector VE is defined in the EPM2017H reference sys- is tem and VI is the corresponding vector with components in the IP system, i.e. defined with the x-axis towards the nodal line equator-IP and the z-axis coincident with the direction of the to- Fig. A.4. Relationship between the IP system and the co-orbital sys- tal angular momentum of the solar system. tem (prograde Sun). It is materialized by the three angles uS , iS , and The geometry of the co-orbital system can be seen in ΩS . The xy-plane is the IP. The aπ vector is the projection of the Sun’s Fig.(A.2) and Fig.(A.3); the geometry of the θ angle is shown acceleration onto the instantaneous SBO plane. The θ angle is the polar in Fig. (A.4). The transformation from the IP system to the co- angle which positions aπ from the radial direction. The z-component of orbital system is performed by applying three rotation matrices, the orbital solar angular momentum vector, L = L n˘, is Lz = L cos iS . namely

VO = Rz(uS ) Rx(iS ) Rz(ΩS ) VI, (A.2)

where now VO is the corresponding vector with defined compo- nents in the co-orbital, non-inertial system.

8 https://doi.pangaea.de/10.1594/PANGAEA.882534

Article number, page 11 of 11