Change in General Relativistic Precession Rates due to Lidov-Kozai oscillations in Solar System
Aswin Sekhar, David Asher, Stephanie Werner, Jeremie Vaubaillon, Gongjie Li
aa r X i v : . [ a s t r o - ph . E P ] F e b Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 6 March 2017 (MN L A TEX style file v2.2)
Change in General Relativistic Precession Rates due toLidov-Kozai oscillations in Solar System
A. Sekhar , ∗ , D. J. Asher , S. C. Werner , J. Vaubaillon , G. Li Centre for Earth Evolution and Dynamics, Faculty of Mathematics and Natural Sciences, University of Oslo, Blindern N-0315, Norway Armagh Observatory and Planetarium, College Hill, Armagh BT61 9DG, United Kingdom IMCCE, Observatoire de Paris, 77 Avenue Denfert Rochereau, F-75014 Paris, France Harvard-Smithsonian Center for Astrophysics, Cambridge, MA, USA ∗ E-mail: [email protected], [email protected]
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MNRAS
ABSTRACT
Both General Relativistic (GR) precession and the Lidov-Kozai mechanism, sepa-rately, are known to play an important role in the orbital evolution of solar systembodies. Previous works have studied these two mechanisms independently in great de-tail. However, both these phenomena occurring at the same time in real solar systembodies have rarely been explored. In this work, we find a continuum connecting theGR precession dominant and Lidov-Kozai like mechanism dominant regimes, i.e. anintermediate regime where the competing effects of GR precession and Lidov-Kozailike oscillations co-exist simultaneously. We find some real examples in the solar sys-tem in this intermediate regime. Moreover we identify a rare example amongst them,comet 96P/Machholz 1, which shows significant changes in the rates of GR precession(an order of magnitude higher than Mercury’s GR precession rate) due to sungrazingand sun colliding phases induced by Lidov-Kozai like oscillations. This comet’s com-bination of orbital elements and initial conditions (at the present epoch) favour thismeasurable rapid change in GR precession (at some points peaking up to 60 timesMercury’s GR precession rate) along with prograde-retrograde inclination flip (due toLidov-Kozai like oscillations). Similar tests are performed for hundreds of bodies lyingin the moderately low perihelion distance and moderately low semi-major axis phasespace in the solar system, the present lowest perihelion distance asteroid 322P/SOHO1, and further examples connected with 96P/Machholz 1 namely, the Marsden andKracht families of sungrazing comets plus low perihelion meteoroid streams like Day-time Arietids (ARI) and Southern Delta Aquariids (SDA).
Key words:
General relativistic precession, Lidov-Kozai oscillations, comets, aster-oids, meteoroids, satellites
Two well known phenomena associated with low periheliondistance bodies in orbital dynamics are general relativistic(GR) precession and Lidov-Kozai oscillations.The accurate prediction of the perihelion shift of Mer-cury in accord with real observations is one of the signifi-cant triumphs of the general theory of relativity developedby Einstein (1915). Past works have looked into the GRprecession in perihelion in different types of solar systembodies like planets (Weinberg 1972; Brumberg 1991; Quinn,Tremaine & Duncan 1991; Iorio 2005), asteroids (Sitarski1992), comets (Shahed-Saless & Yeomans 1994) and mete-oroid streams (Fox, Williams & Hughes 1982; Sekhar 2013;Galushina, Ryabova & Skripnichenko 2015). More recently some works have explored the cases of GR precession in ex-oplanetary systems (Naoz et al. 2011, 2013; Li et al. 2015).A dynamical mechanism first found by Lidov (1962),and further studied by Kozai (1962) who applied it to sun-Jupiter-asteroid 3-body systems, explains the periodic ex-change between eccentricities e and inclinations i therebyincreasing or decreasing the perihelion distance q secularlyin the orbiting body. In its purest form the Lidov-Kozaimechanism involves three bodies, namely a central body,test particle and perturber. In real situations such as the so-lar system, where the perturber may be interior or exterior(page 154, Morbidelli 2011) to the test particle and/or theremay be higher order multiplicity effects (due to perturba-tions from other giant planets than Jupiter, lesser in strength A. Sekhar, D. J. Asher, S. C. Werner, J. Vaubaillon, G. Li compared to Jovian perturbations), pure Lidov-Kozai stilloffers a convenient way to understand the dynamical be-haviour. We shall use the term Lidov-Kozai like mechanismto cover these situations.This mechanism has been related to the rapid change inorbits of artificial satellites (Lidov 1962) around the Earth.Past works have shown that the Lidov-Kozai like mechanismcan lead to a flip in orbits i.e. inclinations switching fromprograde to retrograde or vice-versa (Naoz et al. 2011; Lith-wick & Naoz 2011; Naoz et al. 2013; Li et al. 2014b) duringthe body’s secular evolution. Naoz et al. (2011) discussedthe possibility of orbital flips in hierarchical triple body sys-tems for the first time in the context of exoplanet systems.Bailey, Chambers & Hahn (1992) found that Lidov-Kozai(here it is not pure Lidov-Kozai mechanism in strict termsbut higher order multiplicity effects) is an efficient way bywhich asteroidal and cometary orbits could land up in sun-skirting or sungrazing orbits. The mechanism is known tohave an important role in the long term evolution of differ-ent classes of small bodies (Vaubaillon, Lamy & Jorda 2006;Granvik, Vaubaillon & Jedicke 2012) in the context of im-pact studies (Werner & Ivanov 2015) due to the Lidov-Kozailike cycles in orbital elements which lead to complications inusing analytical and numerical techniques (Manley, Miglior-ini & Bailey 1998) to compute impact probabilities fromsmall bodies on planets. More recently there are examplesfound in the exoplanetary systems (Naoz et al. 2013; Li etal. 2014a; Naoz 2016) which undergo Lidov-Kozai like oscil-lations around the central body thus showing the generalityof this phenomenon in any suitable dynamical system.Of the past studies mentioned above, the ones that per-tain to the solar system have generally related to either ofthe two phenomena separately, not GR and Lidov-Kozai likeeffects happening at the same time. Moreover previous worksin exoplanet systems have indicated that GR precession cansuppress Kozai like oscillations in some cases (Naoz et al.2013) depending on the orbital elements phase space. Henceit has been shown that one phenomenon can compete ordominate over another in exoplanet systems. Again, thisparticular idea has not been well explored in the contextof real solar system bodies. Naoz et al. (2013) showed thatinclusion of GR precession can lead to re-triggering eccen-tricity excitations and orbital flips thereby pointing to thesituation of both GR precession and Kozai effect co-existingat the same time in exoplanet systems.In this work, we find real examples of solar system bod-ies where significant (to be quantified later) GR precessionand Lidov-Kozai like behaviour can co-exist and complementeach other, forming a continuum between the two regimeswhere the respective separate effects dominate. The Lidov-Kozai like mechanism leads to secular lowering of q which inturn leads to a huge increase in GR precession of argumentof pericentre ω . This in turn gives feedback to the Lidov-Kozai like mechanism as the e , i and ω cycles are closelycorrelated.A real solar system body, 96P/Machholz 1, exhibitingthese trends by combining the two dynamical effects duringpresent times is identified out of the hundreds of small bodieslying in the moderately low q and moderately low a (semi-major axis) phase space. This particular effect would bemore pronounced especially when a body gradually evolvesinto a sungrazing and sun colliding orbit. The dynamical evolution becomes even more interesting when inclinationflips occur, the secular Lidov-Kozai like reduction in q ap-plying in both prograde and retrograde cases.For all calculations in this paper, Newtonian n-body in-tegrations were done using Chambers’ (1999) MERCURYpackage incorporating the RADAU (Everhart 1985) andMVS (Mixed-Variable Symplectic: Wisdom, Holman &Touma 1996) algorithms. For GR cases, MERCURY inte-grations included an additional sub-routine (provided by G.Li) incorporating the GR corrections (Benitez & Gallardo2008) in the n-body code; the MVS algorithm was used. TheRADAU accuracy parameter was set to 10 − and the MVStime step was chosen as one day. Each asteroid, comet andmeteoroid particle was treated as a test particle (i.e. zeromass) and integrated in the combined presence of the sunand eight planets Mercury to Neptune. Non-gravitationalforces were not considered in comets, nor radiative forces forasteroids and meteoroid streams, so that we can distinguishGR active and Kozai like dynamics active spaces withoutadditional complicating effects. In the GR-included integra-tions, GR effects were taken into account for the evolutionof all the big bodies (i.e. planets) and small bodies involved.In the Newtonian integrations, GR did not act on either bigor small bodies. For all integrations the initial orbital ele-ments (taken from JPL-Horizons: Giorgini et al. 1996) andinitial epoch (JD 2451000.5) of the eight planets remainedthe same, for uniformity in comparisons. Hence every GRcase presented here, and likewise every Newtonian case, isbased on an identical solar system model. Cross-checks be-tween RADAU and MVS Newtonian integrations confirmedthat either integrator could reproduce the same orbital evo-lution. Searches for small bodies with various ranges of or-bital elements used IAU-MPC (Minor Planet Center) andinitial elements of specific bodies to be integrated were thelatest epoch available from JPL-Horizons. Long term Lidov-Kozai like oscillations can be found in dif-ferent examples of sungrazing comets (Bailey, Chambers &Hahn 1992), bringing these comets closer and closer to thesun after every perihelion passage and q being lowered sec-ularly. Although the perihelion distances stay in the rangeof GR-active space and can, in principle, increase the GRprecession rates (due to decrease in q ), the orbital periods P (and a ) mostly stay way beyond the significant GR-activespace. For GR effects happening near the perihelion pas-sage trajectory to accumulate efficiently over time, the bodyneeds a relatively small orbital period so that perihelion pas-sages happen frequently. Hence for most known bodies thisGR precession increase is insignificant and difficult to sepa-rate and confirm from observations. In this section, we talkabout single orbit timescales of the order of ∼ − yr,so that even if GR precession per orbit is significant, theaccumulated effect over these timeframes is low.In this work we quantify GR precession as significant ifit is comparable to that of Earth. For Earth’s present orbit∆ ω ∼ ω ∼ hange in General Relativistic Precession Rates due to Lidov-Kozai oscillations in Solar System Table 1.
Initial conditions for the bodies presented as plots in this work. All taken from JPL-Horizons except for meteoroid streamDaytime Arietids (ARI) from IAU-Meteor Data Center.Epoch JDT a e i ω Ω M Fig.C/1932 G1 2426760.5 45.00523804 0.972127 74.277600 303.5157 213.4835 359.980648 12011 CL50 2455606.5 0.88303406 0.147614 0.173164 285.543879 22.791201 201.817484 296P 2456541.5 3.03393972 0.959211 58.312214 14.757748 94.323236 77.992760 3ARI 2457546.5 2.67 0.974 27.7 28.7 79.1 0.0 4322P 2457179.5 2.51626146 0.978676 12.589235 49.049476 359.524487 337.760658 52008 KP 2455509.5 1.10063074 0.789847 59.835824 344.973090 62.464197 293.337766 6Mercury 2451000.5 0.38709895 0.205620 7.005045 29.121849 48.332314 106.519879 12(a)(b)
Figure 1.
Orbital evolution of (a) eccentricity (b) inclinationof C/1932 G1 (Houghton-Ensor) forward for 7 kyr from present.This body gets ejected out of solar system in about 6.3 kyr in oursimulations.
GR precession in ω can be computed using this closedform expression (page 197, Weinberg 1972):∆ ω = 6 πGM/a (1 − e ) = 6 πGM/q (1 + e ) (1)in radians/revolution. It should be noted that the directionof the precession of pericentre is always in the same direc-tion of the motion of the orbiting body (page 197, Weinberg1972).In the case of Halley-type comets (with relatively low q ≤ perihelion distance of Mercury), the periods (order of ∼ yr) are high enough that it is practically difficultto distinguish GR precession effects (from other effects orchanges in orbits) after each perihelion passage and more-over, the number of perihelion passages to accumulate GR Table 2.
GR precession in argument of pericentre ∆ ω per peri-helion passage for examples of highly eccentric sungrazing comets(orbital elements taken from sungrazing comets section of Mars-den & Williams 2008). For comparison, the value for Mercury is0.103 arc seconds/revolution.Body q e ∆ ω P (AU) (arc seconds (yr)per q passage)C/1979 Q1 0.005 1 4.0 NAC/1965 S1-A 0.008 1 2.5 900C/2008 E7 0.055 1 0.35 NAC/1997 H2 0.136 1 0.14 NA precession (during every near q passage) is small per unitof time. However Kozai like oscillations can be easily dis-tinguished in these bodies during their long term evolutionand the Kozai like mechanism stays dominant (over GRprecession) in these bodies. This point about insignificantGR precession accumulation applies to other minor bod-ies such as Edgeworth-Kuiper Objects which have relativelyhigh P > ∼ yr.Figure 1 shows the e and i evolution of C/1932 G1 (arepresentative Halley-type low- q comet). Kozai like oscilla-tions can be seen in plots (a) and (b). Tests were repeatedfor Halley-type comets taken from the cometary cataloguecompiled by Marsden & Williams (2008) and they show in-significant GR precession as expected, Kozai like mechanismdominant over GR precession being the typical behaviourbecause of the higher P .Long period sungrazers like the Kreutz family of comets(Kreutz 1888) and Meyer group come incredibly close to thesun. In these cases, although the GR perihelion precessionper single perihelion passage is at the highest (see Table 2), P is of the range ∼ − yr (mostly members of theKreutz family) and hence GR precession accumulating overtime is out of the question for the timeframes we discussin this work. Moreover because of the perturbations fromgalactic tides and passing stars when these bodies reach nearaphelion, it would be difficult to imagine any of these bod-ies showing a consistent or periodic pattern of GR preces-sion accumulation in comparable magnitude over long timescales.The examples presented in Table 2 are representativesshowing low, intermediate and high q amongst the sungraz-ing candidates ( e ∼
1) in the Marsden & Williams (2008)comet catalogue. Table 2 shows that lowest q candidate canhave GR precession per revolution as much as 40 times that A. Sekhar, D. J. Asher, S. C. Werner, J. Vaubaillon, G. Li
Table 3.
GR precession in argument of pericentre ∆ ω per centuryfor some examples of low q , low a and low i bodies in solar system.For comparison, the value for Mercury is 43 arc seconds/century.Body q a ∆ ω ∆ ω (AU) (AU) (arcseconds (arcsecondsper century) per century)(analytic) (integrations)2015 KE 0.85 0.97 4.2 4.62013 BS45 0.91 0.99 4.0 4.52010 VQ 0.69 0.86 5.8 6.22011 CL50 0.76 0.89 5.3 5.9 of Mercury’s GR precession per revolution but because oflarge P of typical sungrazing comets, this precession accu-mulating efficiently over time is unrealistic for future jour-neys into the inner solar system. Moreover some extremesungrazers do not survive for another perihelion passage dueto their disruption near the sun or collision with the sun.However in some particular cases in the solar system,the GR precession can be significant and measurable, whileLidov-Kozai like cycles still survive during the same time(Section 4). GR precession in pericentre was first studied for the classicexample of Mercury’s orbit. More recently there are works(Kerr 1992; Bou´e, Laskar & Farago 2012; Lithwick & Wu2014) which looked into the long term dynamical behaviourof Mercury and its stability within the solar system. Theselatest works have indicated that GR precession along withsecular resonances are the dominating effects in Mercury’slong term evolution ruling out an important role from anyKozai like mechanism. In this section, we talk about GRaccumulation with single orbit timescales of the order of ≤ e ≤ ∼ eccentricity of Mercury), i ≤ ∼ inclination of Mer-cury) and a ≤ . ∼ semi-major axis of Earth). Thisrange enlists 115 objects in the IAU-MPC database. Nu-merical integrations of these bodies show measurable GRprecession (which can be tested using comparison of ana-lytical calculations and numerical integrations; see Table 3)and Lidov-Kozai like oscillations at some point in the fu-ture if the evolution is followed for a few 10 yr. However allthe subset of these bodies with i ≤ ◦ (61 objects) do notshow (see the example in Figure 2) consistent and long termLidov-Kozai like oscillations for the near future ( ∼ yr).The bodies shown in Table 3 (the last of which is plottedin Figure 2) are representative of these 61 objects. Becauseboth e and i are low at the same time in these cases com-pared to bodies discussed in Section 2, the Lidov-Kozai likeoscillations do not dominate over other effects like GR pre-cession or secular resonances. It should be noted that orbitaltimescales depend only on semi-major axis and eccentricityhere. But inclination is mentioned and discussed becauseanti-correlation of e and i is a crucial signature of Kozai like (a)(b) Figure 2.
Evolution of e and i of 2011 CL50 for 10 kyr intofuture. oscillation and the nature of inclination evolution helps toidentify Kozai like behaviour in the solar system.The e and i evolution of 2011 CL50 (Figure 2) showsno consistent Kozai like oscillations whereas the GR preces-sion is significant (Table 3). This trend is typical for otherlow q , low a and low i bodies from the 61 bodies enlistedand discussed in this section. The cases showing absence ofconsistent Lidov-Kozai like oscillations (or negative results)are presented here to highlight the fact that there are areasin the solar system where significant GR precession domi-nates over Kozai like mechanism (similar to the exoplanetarycases presented in Naoz et al. 2013), in contrast to examplesin Section 2. The overlap of these phenomena (GR preces-sion and Kozai like oscillations), complementing each otherthereby leading to unique dynamical behaviour, is discussedin detail in Section 4. In this section, we consider Lidov-Kozai like cycle timescalesof the order of some kyr which contributes to GR enhance-ment due to sungrazing phases induced by a Kozai like mech-anism. We consider bodies with q ≤ ∼ periheliondistance of Mercury) and a ≤ ∼
10 times the semi-major axis of Mercury), thus single orbit timescales of theorder of ≤ hange in General Relativistic Precession Rates due to Lidov-Kozai oscillations in Solar System the IAU-MPC database. The same condition was applied tothe list of established meteor showers from the IAU-MDC(Meteor Data Center) database. Some representative bodieswith relatively high GR precession rates from these lists andtheir GR precession calculated using analytical formulae andnumerical integrations are shown in Table 4.We were interested to identify bodies evolving in thenear future ( ∼ thousands of years) into rapid sungrazingand sun colliding phases and undergoing inclination flips,due to Lidov-Kozai like oscillations and being GR active atthe same time. Of all the bodies we checked from the IAU-MPC, and Marsden plus Kracht families from the comet cat-alogue (Marsden & Williams 2008), 96P/Machholz 1 standsout because it shows all these trends in the near future.The uniqueness of 96P has been reported before in differentcontexts, its dynamical behaviour having octuple crossingpossibilities (Babadzhanov & Obrubov 1987, 1992a, 1992b,1992c, similar to the case of another periodic comet Mach-holz, 141P, discussed in Asher & Steel 1996). The linkageof the orbit of 96P with orbits of Extreme Trans NeptunianObjects (ETNOs) has been explored by de la Fuente Mar-cos, de la Fuente Marcos & Aarseth (2015). 96P has beenlinked with two families of sungrazing comets and two me-teoroid streams. Because of its previously established con-nection (Ohtsuka, Nakano & Yoshikawa 2003, Sekanina &Chodas 2005) with the Marsden and Kracht sungrazers andlow q meteoroid streams like Daytime Arietids (ARI) andSouthern Delta Aquariids (SDA), our tests were repeatedon all these related objects as well.Figures 3a, 3b and 3c, 3d show the near future e and i evolution of 96P for GR-included and Newtonian-only casesrespectively. The test particle undergoes Kozai like oscilla-tion (cf. Abedin et al. 2017) and near the final phase of about120 yr, inclination flip occurs from prograde to retrograde.By about 9 kyr the particle falls into the sun due to rapiddecrease in q due to Lidov-Kozai like mechanism and even-tually reaches near-ecliptic inclination i ∼ ◦ close to thetimeframe of collision with the sun. The general behaviourevident in Figure 3 of GR-included and Newtonian-only dy-namical evolution being very similar is confirmed by inte-gration of clones (see later, where the small but significantdifference will also be discussed). This trend of general dy-namical behaviour in e and i being nearly identical betweenNewtonian-only and GR-included cases holds true for otherbodies discussed in this work.Figure 4 shows the future orbital evolution of ARI.Kozai like oscillations are apparent and the same patternapplies to SDA (not shown here).Among the highest i set from the 244 objects mentionedat the start of this section are the lowest perihelion distanceasteroid 322P (Figure 5), designated a comet but whosecharacteristics such as composition, density and lack of ac-tivity point to asteroidal origin (Knight et al. 2016), andhigh i asteroids 2008 KP (Figure 6), 385402, 333889 and2010 KY127. All these undergo Kozai like oscillations andGR precession at the same time. However we are unable tofind any inclination flips or sun colliding phase in these bod-ies in the near future other than the unique example of 96P.Hence 96P stands out in terms of its dynamical behaviourcompared to all other bodies discussed in this section.To confirm our conclusions about 96P, we made tenclones by varying a with ± ∆ a in equal steps where (a)(b)(c)(d) Figure 3.
Orbital evolution of (a,c) eccentricity (b,d) inclina-tion of 96P for 9 kyr for GR-included and Newtonian-only casesrespectively. Kozai like oscillations in e and i can be seen. A. Sekhar, D. J. Asher, S. C. Werner, J. Vaubaillon, G. Li
Table 4.
GR precession in argument of pericentre ∆ ω per cen-tury. For comparison, the GR value for Mercury is 43 arc sec-onds/century. The orbital elements for meteoroid streams andother minor bodies are taken from IAU-Meteor Data Center andJPL-Horizons respectively. Amongst the bodies listed here, thelong term orbital evolutions for 96P, Daytime Arietids and 322Pare shown in Figures 3, 4 and 5 respectively. The short term evo-lutions of GR precession rates for 96P are shown in Figures 9, 10and 11.Body q a ∆ ω ∆ ω (au) (au) (arcseconds (arcsecondsper century) per century)(analytic) (integrations)96P/Machholz 1 0.124 3.035 3.0 3.7322P/SOHO 0.054 2.516 9.0 9.61566 Icarus 0.187 1.078 10.0 10.63200 Phaethon 0.140 1.271 10.1 10.7Geminids 0.2 1.3 9.3 9.9Daytime Arietids 0.08 2.7 5.7 6.5S. Delta Aquariids 0.07 0.98 6.7 7.3(a)(b) Figure 4.
Evolution in e and i of Daytime Arietids (ARI) 10kyr into future. Initial conditions are given in Table 1 which weretaken from IAU-MDC referring to observations reported in Jen-niskens et al. 2016. (a)(b) Figure 5.
Orbital evolution of (a) eccentricity (b) inclination inlowest perihelion distance asteroid 322P. a =3.03393972 and ∆ a =0.0001 au. Both Newtonian-only aswell as GR-included n-body numerical tests were conductedto check how both sets of bodies dynamically evolve forwardin time.The general dynamical behaviour of any 96P clone ispractically identical in the GR-included and Newtonian-onlycases (Figure 3), looking different only for the final sungraz-ing and sun colliding phase (about 100 yr for 96P which isshown in Figure 7). The rate of change of e and i is differenttowards the end of the bodies’ evolution in the Newtonian-only and GR-included cases, with e reaching 1 later (typi-cally 5–30 yr for different clones), and i reaching 180 ◦ simi-larly later, in the GR case.The dynamical behaviour in forward integrations stayssimilar for both models (which agrees with de la FuenteMarcos et al. 2015) in terms of inclination flips and sun-grazing plus sun colliding phases due to Lidov-Kozai likeoscillations occurring in both sets of integrations. We findthat all the ten 96P clones survive longer ( ∼ hange in General Relativistic Precession Rates due to Lidov-Kozai oscillations in Solar System (a)(b) Figure 6.
Evolution in e and i of high i asteroid 2008 KP. phases in 96P are longer in GR-included simulations com-pared to Newtonian-only simulations. In our simulations, theretrograde phase happens only in the final ∼
100 yr of 96P(during the same Lidov-Kozai like cycle for all clones) andwe do not see any notable change in its duration betweenNewtonian-only and GR-included integrations.Our integrations of 96P clones give an indication to thelikely timeframes when possible inclination flips and suncolliding episodes could occur in future. We find that thetimeframes are different from previous studies (Gonczi et al.1992, Levison & Dones 2014, de la Fuente Marcos et al. 2015)focused on 96P. This could be due to the relatively strong(compared to strength of close approaches with other plan-ets like Venus, Earth, Mars and Jupiter, which we see in ourintegrations) close encounters with Mercury during the ini-tial phase of our integration. The different initial conditionsand orbital uncertainties have led to differences in results inprevious works as well; this point has been mentioned in dela Fuente Marcos et al. (2015). Hence it is evident that exactinclination flip and sun colliding timings depend on the ex-act initial conditions, the type of algorithm/integrator andnumber of gravitational perturbers (which is different in thecase of de la Fuente Marcos et al. 2015 compared to previ-ous works) considered in the integrations. In our work, initialconditions for 96P were taken from JPL-Horizons for epochJD 2456541.5 (which is same as in de la Fuente Marcos et al.2015) but the particles were integrated including the grav-itational effects of eight planets only (whereas de la Fuente (a)(b)
Figure 7.
Difference in dynamical behaviour of (a) eccentric-ity and (b) inclination between Newtonian-only (black cross) andGR-included (red diamonds) cases, for final ∼
100 yr of 96P evo-lution. The particle falls into the sun earlier in the Newtoniancases compared to the GR cases.
Figure 8.
Survival times of both sets of Newtonian-only and GR-included cases for all the 10 clones of 96P. The GR-included casessurvive some extra years longer than Newtonian-only cases for allthe clones in our integration.
A. Sekhar, D. J. Asher, S. C. Werner, J. Vaubaillon, G. Li
Marcos et al. 2015 included the effects of dwarf planet Pluto-Charon system and the 10 most massive main belt asteroids;their initial conditions were taken in the barycentric framefrom JPL-Horizons for JD 2457000.5 and used a Hermiteintegrator for their calculations). In our work, initial condi-tions for eight planets were taken in the heliocentric framefor JD 2451000.5 and we employed Mixed-Variable Symplec-tic or RADAU algorithms. Hence the difference in results isnot surprising.Having said this, it should be noted that irrespectiveof the small changes in initial conditions, all these clonesundergo inclination flip and sungrazing plus sun collid-ing phases at some point in time (different clones showingchange in times of the order of few hundreds of years typi-cally; Figure 8), while staying within the orbital phase spacewhere GR effects are measurable. This is the central pointrelevant to this work.Because of sungrazing behaviour during the final phaseof 96P’s evolution, GR precession per revolution drasticallyincreases in comparison to the initial phase of its evolution.Figures 9 and 10 show the GR precession per century andper revolution respectively; P stays approximately constant.Figure 11 shows the change in GR precession for each revo-lution ( ∼ ∼ ∼ ≥ ∼ × Mercury) with peakvalue > ≥ ∼ × Mercury). These extreme points in GRprecession rate are due to direct effects of Lidov-Kozai likemechanism.Our simulations show that the combination of Lidov-Kozai like oscillations and GR precession lead to these strik-ing sudden changes in the rate of GR precession at differentpoints in time. In contrast with 96P, the change in GR pre-cession rate of Mercury itself (see Figure 12) remains almostconstant throughout the same period. These are two extremesituations in the solar system in terms of this behaviour re-lated to change in GR precession rates.We find that Lidov-Kozai like oscillations, evolution ofKeplerian orbital elements and the general dynamical be-haviour remain the same in Newtonian and GR models forthe bodies discussed in this work. However we were par-ticularly interested in the rate of change of GR precessionfor individual bodies due to rapid changes in perihelion dis-tances induced by Kozai oscillations and this particular pat-tern (due to combination of GR and Kozai) is the highlightof this study.When Lidov-Kozai like oscillations and GR precessionco-exist for the same particle, the secular decrease in q caused by the former can lead to sungrazing and sun col-liding phases which in turn can lead to drastically higherGR precession than at the starting time, during the finalphase of these orbits (see Figures 9, 10 and 11) before theyfall into the sun. This phenomenon becomes even more dy-namically interesting when inclination flips occur during thefinal sungrazing or sun colliding phase for a body. Such aunique example in the solar system is provided by comet Figure 9.
Change in GR precession (in arc seconds) per centuryfor 96P/Machholz 1 for 9 kyr from present. For comparison, GRprecession of Mercury is 43 arc seconds per century.
Figure 10.
Change in GR precession (in arc seconds) per revolu-tion for 96P/Machholz 1 for 9 kyr from present. Drastic changesin GR precession rate occur during some intervals. For compari-son, GR precession of Mercury is 0.104 arcsec/rev. q rapidly and hence is not taken into accountwhen computing values analytically using Equation 1. On hange in General Relativistic Precession Rates due to Lidov-Kozai oscillations in Solar System Figure 11.
Change in GR precession (arcsec/rev) for96P/Machholz 1 for about final 120 yr of same clone shown inFigure 8. There is a steady increase in GR precession rate duringthe final phase of a sungrazing and sun colliding orbit.
Figure 12.
Planet Mercury’s GR precession (arcsec/century) for10 kyr from present. the other hand, in numerical integrations the rapid changesin q values at different time steps are correctly accounted forwhile calculating GR precession and hence this value is morereliable in the final sungrazing phase. Although the magni-tude of Newtonian precession always exceeds GR precessionat any instant of time, the Newtonian and GR precessiondisplay opposite trends during the sungrazing phase (Table5). This behaviour is typical for all the 96P clones in oursimulations.Although some areas of the solar system can be ex-tremely chaotic in general, the test particles evolving in thegravitational models we present here appear to follow a rea-sonably predictable pattern, that is, the evolution of orbitalelements into the future is predictable to a good degree asa function of initial epoch and initial orbital elements. Thisis supported by the different evolution of clones with ini-tial separation ∆ a = 0 . a = 10 − au (not plotted here) then the clones have prac-tically identical dynamical evolution. Table 5.
Change in GR and Newtonian precession in argument ofpericentre per century (∆ ω/ ∆ t = ω t /t − ω t /t ) for a typicalclone of 96P during initial and final hundred years of its evo-lution. For comparison, the GR and Newtonian precession val-ues for Mercury are 43 and 5557 arc seconds/century (page 199,Weinberg 1972). Initial orbital elements are from JPL-Horizonsand final elements from two independent numerical integrations(Newtonian-only and GR-included) performed using MERCURYpackage to compute the differences in ω which gives the numericalGR and Newtonian precession. a ∼ .
034 au during both initialand final phases of dynamical evolution discussed here.Body q ∆ ω/ ∆ t ∆ ω/ ∆ t ∆ ω/ ∆ t (au) (arcsec/ (arcseconds (arcsecondscentury) per century) per century)(analytic) (integrations) (integrations)(GR) (GR) (Newtonian)96P/Machholz 1 0.124 3.0 3.6 3533.0(for first 100 yrin the simulation)96P/Machholz 1 0.022 16.9 53.3 285.2(for last 100 yrin the simulation) We have shown that there are bodies in the solar system inwhich both GR precession and Lidov-Kozai like mechanismcan co-exist and be comparable in their effects and for whichthese complementary effects can be measured and identi-fied using analytical and numerical techniques. Thus thereis a continuum of bodies encompassing, firstly GR preces-sion dominant, secondly GR precession plus Lidov-Kozai likemechanism co-existing and finally Lidov-Kozai like mecha-nism dominant states which are all permissible in nature. Areal solar system body in this intermediate state is identi-fied using compiled observational records from IAU-MPC,Cometary Catalogue, IAU-MDC and performing analyticalplus numerical tests on them. This intermediate state bringsup the interesting possibility of drastic changes in GR pre-cession rates (at some points peaking to about 60 times thatof Mercury’s GR precession) during orbital evolution due tosungrazing and sun colliding phases induced by the Lidov-Kozai like mechanism, thus combining both these importanteffects in a unique and dynamically interesting way. Comet96P/Machholz 1 stands out as the only real body identi-fied (from our simulations) to be exhibiting these interestingtraits, as well as inclination flips, in the near future.This study’s purpose was to focus on the real smallbodies in the solar system (including the planetary pertur-bations during the present epoch) exhibiting these two dy-namical phenomena at the same time. For future work, itwould be instructive to do a detailed abstract study, withonly Jupiter so as to induce the Kozai mechanism (in itspure form) and exclude perturbations from other planets,mapping the entire Keplerian elements phase space to findthe boundaries between three regions namely, GR preces-sion dominant regime, GR precession plus Kozai mechanismco-existing regime and Kozai mechanism dominant regime.One could then compare the stability and chaotic levels A. Sekhar, D. J. Asher, S. C. Werner, J. Vaubaillon, G. Li between these three regions. This would enable us to un-derstand the various patterns in change of GR precessionrates and the maximum rates of GR precession possible ina short time for bodies in our solar system due to sungraz-ing phases driven by strong Lidov-Kozai like oscillations.Independently, a study of this nature could tell us the exactphase spaces which can contribute to possible sudden peaksin rates of GR precession due to Kozai like oscillations in thecontext of stability of artificial satellite orbits (Rosengren etal. 2015) so that precise measurements can be made for theconfirmation of this combined phenomenon in future longterm satellites depending on the phase space traversed.
ACKNOWLEDGMENTS
The authors thank Smadar Naoz for numerous valuable sug-gestions and improvements. Sekhar and Werner acknowl-edge the Crater Clock project (235058/F20) based at Centrefor Earth Evolution and Dynamics (through the Centres ofExcellence scheme project number 223272 (CEED) fundedby the Research Council of Norway) and USIT UNINETTSigma2 computational resource allocation through the Stallocluster with project accounts Notur NN9010K, NN9283K,NorStore NS9010K and NS9029K. Research at Armagh Ob-servatory and Planetarium is funded by the Department forCommunities for N. Ireland. Vaubaillon thanks the CINESsupercomputing facility of France. Li acknowledges MattPayne for developing and providing the GR sub-routine forMERCURY.