A Kozai-resonating Earth quasi-satellite

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1 MNRASL 437, L85 L89 (2014) Advance Access publication 2013 November 13 doi: /mnrasl/slt147 A Kozai-resonating Earth quasi-satellite M. Connors Athabasca University Observatories, Athabasca AB T9S 3A3, Canada Accepted 2013 October 17. Received 2013 October 16; in original form 2013 October 2 1 INTRODUCTION Co-orbital asteroids share the orbit of a planet at least in an averaged sense. Three general categories of such bodies are known for Earth. Horseshoe objects (Connors et al. 2002) librate in longitude past both triangular Lagrange points and their average longitude avoids that of Earth, which is in the gap. Connors, Wiegert & Veillet (2011) recognized the properties of the single known Earth Trojan asteroid, currently librating around the L 4 triangular point, but whose large libration amplitude should allow transition through the L 3 opposite point to librate around L 5. A more subtle form of co-orbital behaviour is found in quasi-satellites (Connors et al. 2004), whose libration is about the longitude of Earth, in a relative orbit resembling retrograde orbital motion. The first Earth co-orbital found (Wiegert, Innanen & Mikkola 1997, 1998) moves on a complex horseshoe orbit with relatively large inclination. In all of these cases, 1:1 mean motion resonance gives semimajor axis a averaged over a year very close to 1 au, annual epicycles and variation in a, and a longer term libration. Namouni (1999) gives a more analytical approach to co-orbital motion, resulting in differing terminology, which can be extended to high-eccentricity and -inclination cases (Namouni, Christou & Murray 1999) and underlining the importance of the Kozai (1962) mechanism under these conditions. This mechanism (loosely referred to as a resonance) results from the conservation of the component of angular momentum out of the plane of the star and a major planet, with result-coupled changes in eccentricity and inclination of a small third body. Generally, it also results in libration of the argument of perihelion ω, normally around 90 or 270 (with respect to the longitude of the node ). ABSTRACT The recently discovered asteroid 2013 LX 28 is in 1:1 resonance with Earth, but has large eccentricity and inclination. These lead to unusual dynamics in which the Kozai resonance plays a large role on long time-scales, while interaction with the terrestrial planets causes shorter term orbital changes. For the nominal orbit, an interaction with Venus changed the nature of the Kozai resonance and injected the asteroid into resonance with Earth. Despite frequent planetary encounters, the nominal orbit shows remarkable stability. Recovery prospects for this object are excellent, so that investigation of its remarkable properties should be able to proceed on a firmer observational base in future. Key words: celestial mechanics minor planets, asteroids: general minor planets, asteroids: individual: 2013 LX28. Asteroid 2013 LX 28 was discovered 1 on 2013 June 12 by Pan-STARRS, at a magnitude of 20.7, with 18 subsequent observations spanning 24 d. 2 This results in a nominal orbit shown in Fig. 1, establishing the basic dynamics while requiring refinement. The position relative to Earth shows that it was discovered near opposition, when bright and near the ecliptic. It subsequently moved into southern skies and became fainter due to increasing distance and phase angle. The known osculating elements for epoch (2013 Apr 18.0) TDB (Barycentric Dynamical Time), with standard errors, are summarized in Table 1. Co-orbital motion in the sense of mean motion allowing resonant interaction with Earth may be observed for cases where a is within roughly a Hill radius, or about 0.01 au, of 1 au, and this criterion is met within several σ.the eccentricity of roughly 0.45 means that the orbit is very elongated, while at nearly 50, the inclination is large. Having a 1 suggests that the object could be resonant with Earth, while large e and i suggest the possibility of Kozai resonance. The H magnitude of 21.7 indicates a diameter of 130 to 300 m based on an assumed albedo typical of asteroids. Unlike the vast majority of near Earth objects (NEOs), 2013 LX 28 was far from Earth at discovery. Ironically, this greatly enhances the chance of recovery in 2014, when the 1 yr period will result in a very similar apparition to that of Since the object is distant, small errors in the orbit do not translate into large errors in position in the sky, as they do for many NEOs. An observational ephemeris generated at the NASA website shows that the object will in early 2014 June be at slightly southerly declination, near 21st magnitude, and with position known better than 1/2 degree in both coordinates, making it an easy target for recovery. Indeed, there could be value in checking archival images. An added advantage for searching is that at the martinc@athabascau.ca 1 cited cited C 2013 The Author Published by Oxford University Press on behalf of the Royal Astronomical Society

2 L86 M. Connors Figure 1. Orbits of the inner planets and asteroid 2013 LX 28 near the epoch of its discovery in The Sun is in the middle and a grid 1 au 2 is shown in the ecliptic plane with positive X (vernal equinox) to the right. Portions of objects orbits which are below the ecliptic are shown dashed and the nodes are joined by a straight line. Mercury is indicated in orange, Venus in blue, Earth in green, Mars in red and the asteroid in black. The descending node of the asteroid s orbit is now near the orbit of Mars. The position of Earth and the asteroid on the date of discovery are shown by dots on the respective orbits. time of maximum brightness, the motion is nearly from north to south, so quite distinctive. Having inferred that it could have interesting orbital features and that it should be recuperable so that the behaviour may in the near future be investigated in more detail, we proceed to study the orbit using numerical integration illuminated by theory. 2 LONG-TERM BEHAVIOUR OF THE NOMINAL ORBIT As expected for a highly eccentric orbit in the inner Solar system, that of 2013 LX 28 features interaction with the terrestrial planets. Although the object is classified as an NEO, present close approaches are only to Mars. Noting this, one might initially hypothesize that scattering from Mars injected it into its eccentric, inclined orbit. We will show below that this is not likely the case. When the nominal orbit was integrated over long periods of time, correlated changes in e and i, characteristic of the Kozai mechanism, were found. The dynamics of the orbit are extremely interesting and likely representative of highly inclined and eccentric inner Solar system orbits. The errors on the nominal orbit are small enough that the current resonant and Kozai behaviours are certain. Once, after approximately a year, the asteroid can be recovered observationally, clone studies (Mikkola et al. 2004) will become useful in relation to the real object. In the Kozai mechanism, e and i vary in antiphase and in this case, with large amplitude. It conserves the Kozai invariant H K = 1 e2 cos(i), shown in Fig. 2, arising from conservation of the perpendicular component of orbital angular momentum and not involving a directly. The mechanism persists despite changes in the nominal orbit shown through variation in a in Fig. 3, and is certainly operative at the present time. In Fig. 2, it may be seen that there were two distinct domains of Kozai oscillation with a transition between them near the present. In the earlier regime, very large changes in eccentricity, between about 0.15 and 0.75, took place with a period of roughly yr. Later, e was bounded by about 0.45 and 0.75, that is to say on average higher, with a period of about yr. The inclination was initially bounded by about 30 and 55, changing to a range of (Note that Fig. 2 shows cos (i), which is in phase with e, while i itself is in antiphase). H K changed little despite overall change in the e i oscillation. The argument of perihelion ω circulated during the entire interval shown, while the node regressed at an approximately equal rate. The period of variation of e and i is approximately half that of ω, consistent with the results of Kinoshita & Nakai (2007). Theoretical values based on this paper are shown, and agree with numerical results at least as well as their results for asteroid 3040 Kozai (whose values I also calculated to verify implementation of their formulas). The agreement, indicated by boxes in Fig. 2, is better in the later epoch of large e with lower variation, than in the earlier one with more variation. Theoretical values come from for the three-body case of Kinoshita & Nakai (2007), with the disturber being Jupiter. The averaged (*) period of the argument of perihelion ω is P ω = 2π n, ω where the rate n ω = 3 6π α2 α 8K 0 γ. K is the complete elliptic integral of the first kind with argument k = α1 α 0 α 2 α 0. The units of the rate are determined by those of Jupiter, n d,since γ = m d m d +m c (1 ed 2) 3/2 n2 d n,wherem is mass, e the eccentricity, n the rate, a d subscript indicates the disturber and a c subscript the central body. The α parameters are as discussed in Kinoshita & Nakai (2007). From the discussion of the Kozai potential there, it is clear that 2013 LX 28 is expected to circulate. Further, cos (i) should vary between H K α0 and H K α1 in phase with e varying between 1 α1 and 1 α 0. Resulting values are indicated with boxes in Fig. 2, centred on times, ca. 190 kyr and +60 kyr, that are representative of the Kozai behaviour before and after the recent change. The change in Kozai behaviour took place ca BC, the time of a close encounter ( au) with Venus, indicated by a blue vertical line. 3 PLANETARY ENCOUNTERS AND INTERACTIONS Although the long time-scale dynamics is dominated by the Kozai interaction, short-term changes in orbital elements take place due to close encounters with planets. Throughout the time period studied, close approaches took place to Mercury, Venus, Earth and Mars. These are shown in two ways in Fig. 3. The top panel shows all approaches to these planets closer than 0.1 au. What is somewhat Table 1. Osculating orbital elements from cited Element Name Value Error Unit Epoch a Semimajor axis au e Eccentricity e 05 i Inclination deg q Perihelion distance au ω Argument of perihelion deg Longitude of node deg M Mean anomaly deg

3 A Kozai-resonating Earth quasi-satellite L87 Figure 2. Kozai dynamics of the nominal orbit of 2013 LX 28 over approximately yr. Bottom panel: eccentricity, cosine of inclination and Kozai parameter are shown in black, red and blue, respectively. The top panel shows the argument of perihelion (black) and longitude of the node (red). Boxes indicate the theoretical Kozai limits and periods. Figure 3. Semimajor axis and close approaches to planets of 2013 LX 28 on its nominal orbit over approximately yr. In the bottom panel, the semimajor axis a is shown in black. During the present era (grey emphasis), the value is near 1 and the quasi-satellite interaction with Earth results in regular small changes and a wider trace. Interactions with planets (colour key: Mercury orange, Venus blue, Earth green, Mars red) are indicated by the relative potential energy term (see the text) for interactions with all but Mercury. The top panel shows small circles at the appropriate a for any planetary approach closer than 0.1 au. remarkable in view of the general tendency to regard close approaches as stochastic is the large degree of regularity shown. Comparison to Fig. 2 shows that the larger the eccentricity, the more likely are planetary encounters, an unsurprising result. The more surprising repetition of detailed structure appears related to the sweeping of the asteroidal orbit s nodes as the Kozai mechanism changes its shape, as will be discussed below. In the bottom panel, changes in a are related to close approaches, shown now as a potential energy term M/R,whereM is the mass of the planet and R the distance of approach. Roughly, the top of the scale is 400 Earth

4 L88 M. Connors masses per au. Several of the interactions go off scale, but the one at yr, with Venus, causes transition into Earth resonance at a = 1 au. It is also at this time that the Kozai behaviour changed, as indicated by the vertical bar in Fig. 2. Although this was not the strongest interaction (in terms of M/R, a simplistic indicator), it took place near the extreme eccentricity phase of the Kozai oscillations, which may explain why it was effective in changing the Kozai pattern. 4 KOZAI QUASI-SATELLITE STATE The semimajor axis being 1, with regular oscillations (epicycle) is not enough to show that an asteroid resonantly interacts with Earth in a significant way. A resonant state requires libration in the long term. This libration is now centred on Earth (relative longitude 0 )withan amplitude of 180, which classifies 2013 LX 28 as a quasi-satellite. Fig. 4 shows the daily longitude of the asteroid relative to Earth averaged over one year periods. Near the present, the longitude cycles between 90 (270 ) and 90, centred on Earth, with a varying between and au, indicative of low-amplitude quasi-satellite motion. Prior to about 3000 BC, the relative longitude remained near 90, which is one of the characteristic longitudes of Kozai libration. However, normally that libration takes place with respect to the dominant planet, which is Jupiter, not Earth. There appears to be a hybrid Kozai-quasi-satellite state in effect with transition to a quasi-satellite state. The transition appears to be spontaneous in the sense that no close encounter took place near that time. Injection to the hybrid state appeared to take place due to Venus close encounter at BC. The apparent quasi-satellite state before the encounter is in disagreement with the results of Fig. 3 which show a = The close encounter is exquisitely sensitive to numeric effects, so that even the nominal orbit cannot be uniquely traced back beyond it. Further, a high-inclination quasisatellite state has been shown to be unstable by both numerical (Stacey & Connors 2009) and theoretical (Mikkola et al. 2006) studies. The short time (50 kyr) spent in this state is not surprising. Figure 4. Variation of the averaged relative longitude of 2013 LX 28 with respect to Earth (bottom) and of its semimajor axis (top). The change at ca BC is due to an interaction with Venus. This initially injected the asteroid into a previously unobserved Kozai-resonant state librating about relative longitude 90. Subsequently, possibly due to a series of Mars encounters, the state changed to quasi-satellite libration about relative longitude 0 (i.e. centred on Earth), which is the current state. Figure 5. Orbits of the inner planets and asteroid 2013 LX 28 near the epoch of close approach to Venus ca. year The view and colours are as in Fig. 1. The descending node of the asteroid s orbit is now very near the orbit of Venus allowing repeated close encounters near this date. As the orbit changes in eccentricity and inclination due to the Kozai mechanism, similar geometry prevails at other times for combinations of the inner planet orbits and the asteroid nodes, leading to sequences of close encounters. Exit from the state in about 40 kyr does not appear to be through a single close encounter. 5 INTERACTION WITH VENUS As noted above, the Kozai mechanism sweeps the nodes of the asteroid through the orbits of the inner planets in such a way that encounters happen in sequence. Fig. 5 shows the details of the orbits at the time of Venus encounter in BC. Infact,asequenceof encounters took place during this favourable geometry. The discovery orbit of the present epoch favours close approach to Mars in a similar fashion, as shown in Fig. 1. Sweeping of the node through inner planet orbits sets up similar geometry repeatedly, and is an important outcome from the Kozai mechanism. 6 CONCLUSIONS Asteroid 2013 LX 28 has numerous notable features, including the following: (i) at present being a quasi-satellite resonant with Earth, (ii) a highly inclined, eccentric orbit subject to the Kozai oscillation, (iii) a nominal orbit which features long-lasting quasi-regular sequences of planetary interaction, (iv) changes in the Kozai parameters and Earth-resonant state through interaction with Venus, and since it should be observationally recoverable, is an excellent testbed for many aspects of asteroid dynamics. Studies have been done here on the current nominal orbit, and long-term behaviour may differ in detail from what is presented. The general characteristics of the behaviour, which are in the ensemble remarkable, should be a robust result. ACKNOWLEDGEMENTS The author thanks the University of Pisa and NASA JPL for facilities facilitating the study of near Earth asteroids, and Christopher T. Russell, Paul Wiegert and Brian Martin for useful discussions.

5 A Kozai-resonating Earth quasi-satellite L89 REFERENCES Connors M., Chodas P., Mikkola S., Wiegert P., Veillet C., Innanen K., 2002, Meteorit. Planet. Sci., 37, 1435 Connors M., Veillet C., Brasser R., Wiegert P., Chodas P., Mikkola S., Innanen K., 2004, Meteorit. Planet. Sci., 39, 1251 Connors M., Wiegert P., Veillet C., 2011, Nat, 475, 481 Kinoshita H., Nakai H., 2007, Celest. Mech. Dyn. Astron., 98, 67 Kozai Y., 1962, AJ, 67, 591 Mikkola S., Brasser R., Innanen K., Wiegert P., 2004, MNRAS, 351, L63 Mikkola S., Innanen K., Wiegert P., Connors M., Brasser R., 2006, MNRAS, 369, 16 Namouni F., 1999, Icarus, 137, 293 Namouni F., Christou A., Murray C. D., 1999, Phys. Rev. Lett., 83, 2506 Stacey R. G., Connors M., 2009, Planet. Space Sci., 57, 822 Wiegert P., Innanen K., Mikkola S., 1997, Nat, 387, 685 Wiegert P., Innanen K., Mikkola S., 1998, AJ, 115, 2604 This paper has been typeset from a TEX/LATEX file prepared by the author.

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