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Mon. Not. R. Astron. Soc. 000, 1­6 (2011)

Printed 1 April 2011

A (MN L TEX style file v2.2)

A long-lived horseshoe companion to the Earth
A. A. Christou1, D. J. Asher 1
1
Armagh Observatory, Col lege Hil l, Armagh BT61 9DG, Northern Ireland, UK

Accepted 2011 February 23; Received 2011 February 9; in original form 2010 December 18

ABSTRACT

We present a dynamical investigation of a newly found asteroid, 2010 SO16, and the discovery that it is a horseshoe companion of the Earth. The ob ject's absolute magnitude (H = 20.7) makes this the largest ob ject of its type known to-date. By carrying out numerical integrations of dynamical clones, we find that (a) its status as a horseshoe is secure given the current accuracy of its ephemeris, and (b) the time spent in horseshoe libration with the Earth is several times 105 yr, two orders of magnitude longer than determined for other horseshoe asteroids of the Earth. Further, using a model based on Hill's approximation to the three-body problem, we show that, apart from the low eccentricity which prevents close encounters with other planets or the Earth itself, its stability can be attributed to the value of its Jacobi constant far from the regime that allows transitions into other coorbital modes or escape from the resonance altogether. We provide evidence that the eventual escape of the asteroid from horseshoe libration is caused by the action of planetary secular perturbations and the stochastic evolution of the eccentricity. The questions of its origin and the existence of as-yet-undiscovered co-orbital companions of the Earth are discussed. Key words: celestial mechanics ­ minor planets, asteroids: general ­ minor planets, asteroids: individual: 2010 SO16

1

INTRODUCTION

Table 1. Orbital elements of 2010 SO16 at JD2455400.5. Element a (au) e i (deg) (deg) (deg) M (deg) Value 1.00039 0.075188 14.536 108.283 40.523 137.831 1- uncert. 9.961e-6 3.284e-6 0.0008793 0.003205 0.001306 0.004636

Horseshoes and tadp oles comprise the two classical solutions to the sp ecial case of coorbital motion in the context of the restricted three-b ody problem. Both typ es of motion have b een studied extensively, b oth analytically and numerically, partly due to their wide applicability to quasi-stable configurations in the real solar system. Horseshoe-typ e orbits app ear to b e less prevalent than tadp oles, owing p erhaps to their different stability prop erties. An often-cited example of ob jects horseshoeing with each other are the Saturnian satellites Janus and Epimetheus. In addition, numerical integrations of the motion of Near-Earth Asteroids (NEAs) 54509 YORP and 2002 AA29 show that they are currently horseshoeing with the Earth (Connors et al 2002; Brasser et al 2004). Confirmation of the status of NEA 2001 GO2 as the third Earth horseshoe will have to await further refinement of its orbit (Brasser et al 2004). The libration p eriods and lifetimes of these Earth horseshoes are typically a few hundred and several thousand years resp ectively. Here we demonstrate the horseshoe dynamics of a newly discovered NEA, 2010 SO16. We find that, in contrast to the previous cases, the lifetime of horseshoe libration b etween 2010 SO16 and the Earth is substantially higher, of order 105 years. We attribute this to two factors: its low eccenE-mail: aac@arm.ac.uk (AAC) c 2011 RAS

tricity, moderate inclination orbit and its energy state deep into the horseshoe regime. We discuss the issue of its origin and briefly comment on the p ossible existence of additional ob jects of this typ e.

2

THE ASTEROID

2010 SO16, hereafter referred to as "SO16", was discovered on 2010 Septemb er 17 by the WISE Earth-orbiting observatory (Obs. Code C51) and subsequently followed up by ground-based telescop es. As of 2010/12/03, its orbit has b een determined by 40 observations spanning a data arc of 75 days (MPEC 2010-X02). The orbital elements and their 1- statistical uncertainties as given by the NEODYS orbital information service at that date


2

Christou & Asher
1.0 (a) 0.5 0.4 0.2 S E ar (x 102) 0.0

0.0 -0.5 -1.0 -1.0 -0.5

-0.2 -0.4 0.0 0.5 X (AU) 1.0 30 20 r (deg) 10 0 -8000 -7000 -6000 time (yr) -5000

0.086 0.084 0.082 er 0.080 0.078 0.076 -8000 -7000 -6000 time (yr) -5000

-10 -20 -30 -8000 -7000 -6000 time (yr) -5000

Figure 1. Panel (a): The guiding centre tra jectory of NEO 2010 SO16 in a cartesian ecliptic heliocentric frame co-rotating with the Earth from an integration of the asteroid's nominal orbit solution (Table 1) and for an interval of 2 â 105 yr centred on 2000 Jan 1.5 UT. The radial extent of the horseshoe has been exaggerated by a factor of 20 for clarity. The positions of the Sun and the Earth are denoted by the letters `S' and `E' respectively. The diamond indicates the position of the asteroid at 2010 Jan 1..5 UT. Panel (b): Relative semima jor axis showing the alternation between two values, one interior and the other exterior to the Earth's, typical of horseshoe libration. Panels (c) and (d): Relative eccentricity and argument of pericentre respectively. Note the stepwise behaviour owing to the ob ject's encounter with the Earth every 175 yr.

the solar system used in the integrations included all eight ma jor planets ( initial state vectors for which were taken from the HORIZONS ephemeris service; Giorgini et al 1996) with the gravity of the Moon treated by adding its mass to that of the Earth. In b oth integrations, the ob ject p ersisted as a horseshoe librator of the Earth. Figure 1 (a) shows the cartesian motion of the asteroid for a p eriod of 2 â 105 yr, averaged over one orbital p eriod, on the ecliptic plane. The op ening angle of the horseshoe is 25 and the halfamplitude of variation in a is 4 â 10-3 au. The p eriod of the horseshoe libration is 350 yr. SO16 is currently at the turning p oint of the horseshoe that is lagging b ehind the Earth in its orbital motion, approaching it at 0.13 au in mid-May every year and at < 0.2 au until 2016. It will remain as an evening ob ject in the sky for several decades to come. To test this result further, and explore the likely lifetime of the ob ject in the horseshoe state, we integrated dynamical clones of SO16. Starting conditions for these were generated by varying the asteroid's semima jor axis a, mean anomaly M , eccentricity e and argument of p ericentre by increments equal to their 1- (a) and 1.5- (M , e, ) ephemeris uncertainties resp ectively. Nine different values were used for the semima jor axis and three for the other elements (inclusive of the nominal values), resulting in 243 clones which were integrated with MERCURY for ±105 yr with resp ect to the present. All clones remained in horseshoe libration with resp ect to the Earth.

Y (AU)

4 4.1

ANALYSIS OF THE DYNAMICS Theoretical Framework

(http://newton.dm.unipi.it/neodys/) are shown in Table 1. Its Minimum Orbit Intersection Distance (MOID) with resp ect to the Earth is 0.027 au and its absolute magnitude H is 20.7 implying a diameter of 200-400 m dep ending on the ob ject's unknown alb edo.

3

VERIFYING THE OBJECT'S DYNAMICAL STATE

Following astrometry of SO16 by the Spacewatch I I telescop e (Obs. Code 291) on 18 Nov, and while the obs. arc length was 62 days, the nominal orbit solution as given by the NEODYS ephemeris information system had a semimajor axis (a) uncertainty of a = 1.6 â 10-5 au while the difference b etween the value of this orbital element and the corresp onding value for the Earth was a 4 â 10-4 au. Both are small compared to the range of p ossible values for a that allow co-orbital motion (amax 10-2 au) suggesting that SO16 is a 1:1 resonant librator. We numerically integrated this nominal orbit, as well as the one subsequently published following additional observations obtained on the 1st Decemb er (Table 1), for ±105 yr using the RADAU algorithm implemented within the MERCURY integration package (Chamb ers 1999). The model of

The existence of horseshoe tra jectories within the circular restricted three b ody problem was first demonstrated by Brown (1911). The realisation that two recently discovered satellites of Saturn, Janus and Epimetheus, were engaged in mutual horseshoe libration (Dermott & Murray 1981b), provided new imp etus for theoretical modelling of the dynamics. Dermott & Murray (1981a) investigated quasi-circular symmetric horseshoe orbits. In the symmetric regime, the initial separation in semima jor axis b etween the orbits (the impact parameter) and the eccentricity are adiabatic invariants of the motion, recovering their initial values following two consecutive encounters. In that work, it was also demonstrated numerically that the assumption of symmetry is valid up to |a0 | 0.74. Here, is the mass parameter, a fundamental scaling constant in Hill's approximation ­ where the indirect p erturbation is ignored and only the interaction p otential is considered during the encounter ­ equal to (µ/3)1/3 where µ is the mass ratio b etween the secondary (or secondaries) and the primary. The quantity a0 is the relative semima jor axis far from encounter, equivalent to the Jacobi constant for planar, circular orbits. As the symmetry is not exactly preserved, the horseshoe configuration has a finite lifetime of T /µ5/3 where T is the orbital p eriod of the secondary (eg for the Earth, this expression evaluates to 1.6 â 109 yr). H´non & Petit (1986) formally e demonstrated that the eccentricity receives an impulse of 3 order exp -8 µ/3a0 during an encounter and hence is an
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A long-lived horseshoe companion to the Earth
adiabatic invariant of the motion (see also Namouni 1999). For small to moderate eccentricities ( ) the assumption of symmetry is valid if min a0 where min , related to the impact parameter by
mi n

3

-10 (a) -12 2lr (deg) -14 -16 -18 -20 -10000 -5000

0.10 0.08 0.06 er 0.04 0.02 0.00 10000 8 6 S (x105)

(b)

= 8µ/3a2 , 0

(1)

is the angular separation b etween the two b odies at encounter. In the context of Hill's three-b ody problem (Hill 1878), the evolution of the asteroid's orbit can b e studied by averaging over the fast epicyclic motion (Dermott & Murray 1981a; H´non & Petit 1986; Namouni 1999). Namouni (1999) dee rived an expression that describ es the slow variation of the guiding centre: a
2 r

0 5000 time (yr)

-100

0 (deg)

100

0.44 0.42 ar (x102) 0.40 0.38 0.36 0.34

(c)

(d)

=

a2 - 0

4µ 3

-

(er cos u + ar )2 + 4(lr - er sin u)
2 -1/2

2

4

2 +Ir sin (u + r )

du

(2)

2 0 40 30 20 10 0 10 20 30 40 2lr/er (deg)

where the subscript "r" denotes the relative elements as defined in that work and lr = /2. The constant a0 is related to the Jacobi constant Hr and the conserved Poincar´ action e 2 Kr = 1 - 1 - er cos Ir : 8 a = (Kr - Hr ) 3
2 0

40000

20000 0 time (yr)

20000

(3)

assuming a planetary mean motion of unity. In our case, ar er and we can write: a
2 r

=

a2 - 0

4µ 3 e



r

-

cos2 u + 4 ¯ - sin u l
2 -1/2

2

¯ + K - 1 sin (u + r )

du

(4)

2 ¯ where K = 1 + Ir /e2 and ¯ = lr /er . The second term on l r the right-hand-side of this equation is the p onderomotive p otential S scaled by 8µ/3er (Eq. 29 of Namouni 1999). It l is -p eriodic in r and symmetric with resp ect to ¯ = 0. It was also shown in that pap er that averaging the co-orbital p otential over the phase ¯ (or lr ) allows one to obtain analytl ically the secular evolution of er , Ir , r and r if the orbits are non-collisional. In fact, the regularity of the secular dynamics p ersists at large eccentricity and inclination if Hill's equations are replaced with those of the circular restricted three b ody problem.

Figure 2. Evolution of the nominal orbit of 2010 SO16 (Table 1) for timescales of order 104 yr showing key features of the dynamics. Panel (a): The Earth-trailing branch of the relative mean longitude 2lr between SO16 and the planet. Note the 104 yr periodicity. The rectangle indicates the value of 2lr at JD 2455197.5. Panel (b): Evolution of the relative eccentricity (er ) as a function of the relative argument of pericentre (r ). The diamonds in this and the previous panel indicate the extreme states referred to in Section 4.2. Panel (c): The evolution of the libration width of the relative semima jor axis ar (for clarity, only values +0.41 are shown) through a jump, indicated by the arrow, at t = -6.3 â 103 yr. The time evolution of er and r is superimposed on the plot as the oscillating and slanted curves respectively. Panel (d): The form of the scaled ponderomotive potential (8µ/3er ) S corresponding to the two snapshots in er -r evolution highlighted in panel (b) and Section 4.2. The horizontal dashed line indicates the current state of SO16's horseshoe tra jectory as quantified by the constant a2 . This is to be compared with the 0 two values of the extremum at ¯ = 0 referred to in the text. l

4.2

Short Term Evolution

In the case of SO16, it is exp ected that the orbital elements will evolve in the short term due to the coorbital resonance and in the long term due to secular p erturbations. To understand the dynamical evolution of SO16 we must first test the assumptions under which the results of the previous section are valid. The condition that min a0 is satisfied as |a0 | , however er and Ir are large relative to the mass parameter. In Fig. 1 we show the variation of ar , er and r for several coorbital cycles. We observe impulsive stepwise changes in er at the same time that ar changes sign up on encounter with the Earth. These step changes cancel each other out over a full libration cycle (two consecutive encounters) so that the original eccentricity is recovered. This b ehaviour of impulsive changes, also seen in the plot for r as well as the
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evolution of Ir and r (not shown here), is similar to that observed in the integrations of Dermott & Murray (1981a) for symmetric orbits. It is sup erp osed on a slower, gradual evolution of these orbital elements. These observations indicate that SO16 currently occupies the symmetric regime of coorbital motion as discussed in Section 4.1 and that the secular motion is regular at these timescales. Therefore, the formulation presented in Section 4.1 may b e used as a starting p oint in our effort to explain the p ertinent characteristics of 2010 SO16's long-term orbital evolution. Note that, as the Earth's orbital eccentricity b ecomes significant within the timescales considered in our integrations (Quinn, Tremaine & Duncan 1991), its contribution to er cannot b e neglected. By evaluating Eq. 4 at JD 2455197.5 (2010 Jan 1.0 UT) when lr = -13 .8/2, er = 0.0645, Ir = 14 .5, r = 118 .8 and setting ar = 0, we find a0 = 0.544. The corresp onding angle from Eq. 1 is 15 .5, in good agreement given that the true orbit has a non-negligible eccentricity and inclination. In panel (a) of Fig. 2 we observe an oscillation of min


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Christou & Asher
b ehaviour of the eccentricity remains regular, staying well b elow the collision threshold (dashed horizontal line; see b elow for explanation) despite subtle changes (eg the 5 â 104 yr modulation that sets in towards the middle of the integration). Interestingly, Kr also evolves in a regular fashion over this p eriod, reaching a maximum value of 0.05 near the middle of the time interval. The timescale of variation implies that this evolution is due to secular forcing by the planets, acting to change the asteroid's (and the Earth's) orbital elements over > 5 â 104 yr timescales. On the other hand, the b ehaviour of ar and er of the clone shown in panel (b) changes markedly at t = -2.8 â 105 yr and t = -4.0 â 105 yr. In the first instance, we observe an increase of b oth the average eccentricity and the amplitude of the secular cycle while the width of the horseshoe decreases slightly by 0.04. In the second instance, a further increase of the amplitude of the eccentricity cycle leads to a transition into a passing orbit. In b oth of these instances, the change occurs over 1-2 eccentricity cycles so no clear correlation with particular phases of the cycle - of the sort evident in Fig. 2(c) - can b e made. Establishing such a correlation would require a thorough statistical analysis that is outside the scop e of the pap er. However, Kr evolves smoothly through the change (ie no jumps are observed), suggesting that they are dynamical features of the restricted Sun-Earth-Asteroid problem. On the other hand, the timescale over which these events occur, similar to that governing the evolution of Kr , implies that planetary secular p erturbations also play a role. The transition itself occurs b ecause the maximum of the scaled p otential S is reduced to the p oint of reaching parity with a2 . In physical terms, the 0 eccentricity b ecomes large enough ( 0.12; indicated by the dashed horizontal line) to allow close encounters (< 0.01 au) with Earth while the inclination acts to stabilise the transition. Afterwards, the eccentricity b ecomes high enough to allow close encounters with Venus and the evolution is similar to that of other high-e, high-I coorbitals of the Earth (Christou 2000).

with a p eriod of 104 yr and b etween 14 .3 (at t = -2 â 103 yr) and 12 .5 (at t = +3 â 103 yr). This is due to the contribution of er in Eq. 4 which oscillates with the same p eriod. This element, together with r , form a conjugate pair of variables which, as our integrations show, oscillate b etween two states; one where er = er,max = 0.083 and r = k and the other where er = er,min = 0.061 and r = k + /2 (panel (b) of Fig. 2). This cycle, hereafter referred to as the eccentricity cycle, by itself defines the ob ject's closest p ossible distance to the Earth as 0.12 au. Two such cycles are completed for every full circulation of r . This b ehaviour is characteristic of the horseshoe state (cf Fig. 12 of Namouni 1999) and demonstrates the regularity of the secular evolution of this asteroid's orbit over at least 2 â 104 yr or 102 horseshoe encounters with the Earth. Over the same p eriod, the inclination Ir and action Kr remain essentially constant and equal to 14 and 0.0340. If we substitute these values in Eq. 4 we find min = 14 .1 and min = 12 .4 for the low-e and high-e states resp ectively, in excellent agreement with the simulations. The horseshoe tra jectory of the guiding centre app ears to widen at t -6.3 â 103 yr (panel (c) of Fig. 2). This coincides with er reaching its maximum. However, similar b ehaviour is not observed at the next er maximum (t = +3 â 103 yr) even though the same value is attained. Insp ection of our simulations shows that this increase takes place for 1015% of our clones, indicating that it is not deterministic. Its size is ar 0.01. Nothing similar is observed in the time history of the other elements. Its impulsive nature indicates that this asteroid's orbit evolves towards the b oundary of the symmetric horseshoe regime over time. The ultimate fate of the horseshoe dep ends on the relative evolution of a0 and the p onderomotive p otential S . Since er < Ir the extremum of the scaled p onderomotive p otential at ¯ = 0 changes from a maximum to a minimum l ¯ when r > 0 (K ) (Namouni 1999) (see panel (d) of Fig. 2). This threshold evaluates to 30 for the average er and Ir of this asteroid's orbit. In any case, the horseshoe will transition to, or form a comp ound orbit with, a quasi-satellite orbit or a passing orbit when a2 (8µ/3er )S . For example, 0 and for the er -r b ehaviour observed in the integrations, the value of the extremum varies from (0.661)2 to (0.678)2 (see panel (d) of Fig. 2). Hence, an increase in a0 of 0.12 is required in order for the horseshoe to b e disrupted. 4.3 Long Term Evolution

5

COMPARISON WITH OTHER EARTH HORSESHOE ASTEROIDS

To explore further the question of the horseshoe's likely survival lifetime and the mechanisms which might limit it, we generated additional clones by sampling the semima jor axis with a step of 0.1- out to ±1.4 and then at ±2, ±3 and ±4 . This new set of 35 clones was integrated for ±2 â 106 yr. We regard the integrations in the past and in the future as separate trials. Out of this set of 70 trials, 50 p ersisted as horseshoes for 2 - 5 â 105 yr, 4 for < 2 â 105 yr and the remaining 16 for > 5 â 105 . Eight clones p ersisted for > 106 yr while two remained in horseshoe libration until the end of the integration. The shortest lifetime observed was 1.2 â 105 yr. In Fig. 3 we show two typical cases of a clone that is stable and one that escap es within 106 yr. In panel (a), the width of the libration in ar remains essentially constant over this p eriod. At the same time, the

There are currently 3 near-Earth asteroids known to follow horseshoe tra jectories with resp ect to the Earth (Table 2; also Brasser et al 2004). Of those, 2002 AA29 is the most similar to 2010 SO16, mainly due to its low eccentricity and moderate inclination. Its value of H implies a likely size of a few tens of metres. Hence, it is an order of magnitude smaller than SO16. The orbital eccentricities of 54509 YORP and 2001 GO2 allow close encounters with the Earth. These do not necessarily eject the asteroids from the co-orbital resonance but instead they effect their transition into another mode of libration or circulation. This state is shared by other objects in so-called comp ound and/or transition orbits such as 3753 Cruithne, the first of its class to b e recognised as such (Wiegert et al. 1997, 1998; Namouni 1999; Namouni et al 1999; Christou 2000). The op ening angles 2min of GO2 and 54509 are wider than those of SO16 implying an energy state deep er into the horseshoe domain. However, the contribution of their higher eccentricities in Eq. 4 places them
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A long-lived horseshoe companion to the Earth
Table 2. Known Earth horseshoe asteroids. Data (2004). Information for 3753 Cruithne (Wiegert et of the Earth. Parameters of 54509's horseshoe libr Additional information was taken from Margot &

5

for 2002 AA29 and 2001 GO2 were taken from Connors et al (2002) and Brasser et al al. 1998) is also listed as this ob ject is the archetype of the class of transient coorbitals ation were determined by numerical integration of 9 dynamical clones of that asteroid. Nicholson (2003).

ar Designation 2010 SO16 2002 AA29 2001 GO2b 54509 YORP 3753 Cruithnec
a b c

,max

() 0. 0. 0. 0. 0. 42 70 68 59 23 0. 0. 0. 0. 0.

e 07 01 17 23 51

I ( ) 14.5 10.7 4.6 1.6 19.8

Obs. Data Arca (days) 75 735 5 1100 4200

Lifetime (yr) >120,000 2000 2500 2200 5500

2

mi n ( )

Libration Period (yr) 350 190 200 200 770

H 20. 24. 23. 22. 15. 7 1 5 6 5

25 8 36 54 N/A

At time of demonstration of horseshoe state Obs. arc length 5 days Compound orbit

(a)

0.20 0.15 0.10

6

DISCUSSION

0.5
ar (x 102)

0.05 0.00

0

-0.5 (b) 0.20 0.15 0.10 0.5
ar (x 102)

0.05 0.00

0

-0.5 -1000 -800 -600 -400 time (x103 yr) -200 0

Figure 3. Evolution of two dynamical clones of 2010 SO16 for 106 yr in the past. Panel (a) shows a clone that persists as a horseshoe of the Earth for this period. Panel (b) shows a horseshoe that is disrupted at t = -4 â 105 yr following two impulsive changes in er (indicated by the arrows). The dashed horizontal line indicates the critical value of the eccentricity ( 0.12) as discussed in the text. The evolution of the Poincar´ action Kr , in the same scale e as er , is denoted by the dashed curve.

closer SO16 times Earth

to the transition regime (see panel (d) of Fig 2) than and accounts, at least in part, for their shorter life- a few times 103 yr - as horseshoe companions of the .

The existence of this long-lived horseshoe raises the twin questions of its origin and whether ob jects in similar orbits are yet to b e found. The ob ject's Earth-like orbit makes a direct or indirect origin in the main b elt an unlikely, although not imp ossible, prop osition (Bottke et al 2000; Brasser & Wiegert 2008). Another plausible source is the Earth-Moon system. Margot & Nicholson (2003) have suggested that 54509 may have originated within the Earth-Moon system. SO16's current orbit does not provide such a direct dynamical pathway to the Earth-Moon system as in that case, although the situation is likely to change within a timescale of several times 105 yr. A third p ossibility is that the ob ject originated near the Asteroid-Earth-Sun L4 or L5 equilibrium p oint as a tadp ole librator. Tabachnik & Evans (2000) showed that Earth tadp oles can p ersist for 5 â 107 yr in b oth high (24 < I < 34 ) and low (I < 16 ) inclination orbits and at typical eccentricities of 0.06. Extrap olating their results to 5 â 109 yr timescales, they found it conceivable that a small fraction of a p ostulated initial p opulation of Earth tadp oles can survive to the present. In order for this scenario to apply to SO16, it must have b een a tadp ole until very recently. However, the work of Tabachnik & Evans does not take into account the action of the Yarkovsky effect, which mobilises the semima jor axes of NEAs at typical rates of 10-9 au yr-1 (Chesley et al, Ast. Comets Met. Conf., 2008). As the radial half-width of Earth's tadp ole region is 8µ/3 2.8 â 10-3 au (Murray & Dermott 1999), it may b e difficult for SO16 to survive as a tadp ole companion of the Earth for more than a few times 106 yr. On the other hand, a study of the stability of Tro jans of Mars has shown that the Yarkovsky acceleration does not necessarily destabilise large Tro jan companions of the planets (Scholl et al 2005) and this result may also apply here. In any case, observational determination of the asteroid's basic prop erties - size, sp ectral class and spin state - will b e extremely useful in clarifying the situation. Sub ject to the successful determination of said parameters, SO16 may b e a suitable test target for the direct detection of the Yarkovsky acceleration as an ob ject a few hundred metres across that makes frequent close encounters with the Earth during the next decade (Vokrouhlicky et al. 2005). ´

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er

er


6

Christou & Asher

Finally, regarding the p ossibility of existence of other b odies in similar orbits, the case for Earth-based reconnaissance of the Tro jan regions of the Earth has already b een made (Evans & Tabachnik 2000; Wiegert et al. 2000). For relatively bright targets such as SO16, the main limiting factor is likely to b e the minimum solar elongation that can b e reached by an observational survey. As typical horseshoe libration p eriods are of order hundreds of years, completing the census of large (H < 21) terrestrial horseshoe or tadp ole companions would likely require a space-based platform in a heliocentric orbit interior to the Earth's.

ACKNOWLEDGMENTS The authors would like to thank Ramon Brasser for his insightful comments which improved the manuscript. Astronomical research at the Armagh Observatory is funded by the Northern Ireland Department of Culture, Arts and Leisure (DCAL).

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A This pap er has b een typ eset from a TEX/ L TEX file prepared by the author.

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