MNRASL 437, L85–L89 (2014)
doi:10.1093/mnrasl/slt147
Advance Access publication 2013 November 13
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
ABSTRACT
The recently discovered asteroid 2013 LX28 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.
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 L4 triangular point,
but whose large libration amplitude should allow transition through
the L3 opposite point to librate around L5 . 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 ).
Asteroid 2013 LX28 was discovered1 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 2456400.5 (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 LX28 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 2013. 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
E-mail: [email protected]
1 http://www.minorplanetcenter.net/mpec/K13/K13L72.html, cited 30-092013.
2 http://newton.dm.unipi.it/neodys, cited 02-10-2013.
1 I N T RO D U C T I O N
C 2013 The Author
Published by Oxford University Press on behalf of the Royal Astronomical Society
L86
M. Connors
Figure 1. Orbits of the inner planets and asteroid 2013 LX28 near the epoch
of its discovery in 2013. The Sun is in the middle and a grid 1 au2 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 L O N G - T E R M B E H AV I O U R O F
THE NOMINAL ORBIT
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 100 000 yr. Later, e was bounded by about
0.45 and 0.75, that is to say on average higher, with a period of about
50 000 yr. The inclination was initially bounded by about 30◦ and
55◦ , changing to a range of 25◦ –63◦ . (Note that Fig. 2 shows cos (i),
which is in phase with e, while i itself is in antiphase). HK 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 Pω∗ = n2π∗ ,
ω
√ √
where the rate nω∗ = 3 8K6π α2 − α0 γ ∗ . K is the complete ellip
α0
tic integral of the first kind with argument k = αα12 −
. The
− α0
units of the rate are determined by those of Jupiter, nd , since
n2
As expected for a highly eccentric orbit in the inner Solar system,
that of 2013 LX28 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’ HK =
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
d
γ ∗ = mdm+m
(1 − ed2 )−3/2 nd , where m is mass, e the eccentricity,
c
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 LX28 is expected to circulate. Further, cos (i)
should vary between √HαK0 and √HαK1 in phase with e varying between
√
√
1 − α1 and 1 − α0 . 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. 13 400 BC,
the time of a close encounter (0.0023 au) with Venus, indicated by
a blue vertical line.
3 P L A N E TA RY E N C O U N T E R S
A N D I N T E R AC T I O N S
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 http://neo.jpl.nasa.gov/, cited 30-08-2013.
Element
Name
Epoch
a
e
i
q
ω
M
2456400.5
Semimajor axis
Eccentricity
Inclination
Perihelion distance
Argument of perihelion
Longitude of node
Mean anomaly
Value
Error
1.001 595 626 814 997
.452 102 184 263 4294
49.980 371 930 568 39
.548 772 056 183 2383
76.682 026 906 842 05
345.780 518 852 4863
158.184 825 778 7053
0.000 2713
9.737 2e−05
0.048 378
0.000 246 11
0.002 4873
0.006 0517
0.029 093
Unit
au
deg
au
deg
deg
deg
A Kozai-resonating Earth quasi-satellite
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Figure 2. Kozai dynamics of the nominal orbit of 2013 LX28 over approximately 700 000 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 LX28 on its nominal orbit over approximately 700 000 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, where M is the mass of the planet and R
the distance of approach. Roughly, the top of the scale is 400 Earth
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M. Connors
masses per au. Several of the interactions go off scale, but the one
at −13 400 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 KO Z A I Q UA S I - S AT E L L I T E S TAT E
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◦ ) with an
amplitude of 180◦ , which classifies 2013 LX28 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 0.998 and 1.002 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 −13 400 BC. The apparent quasi-satellite
state before the encounter is in disagreement with the results of
Fig. 3 which show a = 0.997. 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 5. Orbits of the inner planets and asteroid 2013 LX28 near the epoch
of close approach to Venus ca. year −13 400. 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 I N T E R AC T I O N W I T H V E N U S
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 13 400 BC. In fact, a sequence of
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 LX28 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.
Figure 4. Variation of the averaged relative longitude of 2013 LX28 with
respect to Earth (bottom) and of its semimajor axis (top). The change at ca.
13 400 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.
AC K N OW L E D G E M E N T S
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.
A Kozai-resonating Earth quasi-satellite
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This paper has been typeset from a TEX/LATEX file prepared by the author.