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Mon. Not. R. Astron. Soc. 319, 63±79 (2000)
Asteroids in the inner Solar system ± I. Existence
S. A. Tabachnik1,2 and N. W. Evans1w
1
Theoretical Physics, 1 Keble Road, Oxford OX1 3NP
2
Princeton University Observatory, Princeton, NJ 08544±1001, USA
Accepted 2000 June 13. Received 2000 June 13; in original form 1999 December 29
A B S T R AC T
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Ensembles of in-plane and inclined orbits in the vicinity of the Lagrange points of the
terrestrial planets are integrated for up to 100 Myr. The integrations incorporate the gravi-
tational effects of the Sun and the eight planets (Pluto is neglected). Mercury is the least
promising planet, as it is unable to retain tadpole orbits over 100-Myr time-scales. Mercurian
Trojans probably do not exist, although there is evidence for long-lived, corotating horseshoe
orbits with small inclinations. Both Venus and the Earth are much more promising, as they
possess rich families of stable tadpole and horseshoe orbits. Our survey of Trojans in the
orbital plane of Venus is undertaken for 25 Myr. Some 40 per cent of the survivors are on
tadpole orbits. For the Earth, the integrations are pursued for 50 Myr. The stable zones in the
orbital plane are larger for the Earth than for Venus, but fewer of the survivors (,20 per
cent) are tadpoles. Both Venus and the Earth also have regions in which inclined test
particles can endure near the Lagrange points. For Venus, only test particles close to the
orbital plane i & 168 are stable. For the Earth, there are two bands of stability, one at low
inclinations i & 168 and one at moderate inclinations 248 & i & 348: The inclined test
particles that evade close encounters are primarily moving on tadpole orbits. Two Martian
Trojans (5261 Eureka and 1998 VF31) have been discovered over the last decade and both
have orbits moderately inclined to the ecliptic (208: 3 and 318: 3 respectively). Our survey of
in-plane test particles near the Martian Lagrange points shows no survivors after 60 Myr.
Low-inclination test particles do not persist, as their inclinations are quickly increased until
the effects of a secular resonance with Jupiter cause destabilization. Numerical integrations
of inclined test particles for time-spans of 25 Myr show stable zones for inclinations between
148 and 408. However, there is a strong linear resonance with Jupiter that destabilizes a
narrow band of inclinations at ,298. Both 5261 Eureka and 1998 VF31 lie deep within the
stable zones, which suggests that they may be of primordial origin.
Key words: Earth ± minor planets, asteroids ± planets and satellites: individual: Mercury ±
planets and satellites: individual: Venus ± planets and satellites: individual: Mars ± Solar
system: general.
edu/iau/lists/Trojans.html), although the total popu-
1 INTRODUCTION
lation exceeding 15 km in diameter may be as high as ,2500
Lagrange's triangular solution of the three-body problem was long (French et al. 1989; Shoemaker, Shoemaker & Wolfe 1989).
thought to be just an elegant mathematical curiosity. The three Roughly 80 per cent of the known Trojans are in the L4 swarm.
bodies occupy the vertices of an equilateral triangle. Any two of The remaining 20 per cent librate about the L5 Lagrange point,
the bodies trace out elliptical paths with the same eccentricity which trails 608 behind the mean orbital longitude of Jupiter.
about the third body as a focus (see e.g. Whittaker 1904; Pars There are also Trojan configurations amongst the Saturnian
1965). The detection of 588 Achilles near Jupiter's Lagrange point moons. The Pioneer 11 and the Voyager 1 and 2 flybys of Saturn
in 1906 by Wolf changed matters. This object librates about the discovered five small moons on tadpole or horseshoe orbits. The
Sun±Jupiter L4 Lagrange point, which is 608 ahead of the mean small moon Helene librates about the Saturn±Dione L4 point. The
orbital longitude of Jupiter (e.g. EÂrdi 1997). About 470 Jovian large moon Tethys has two smaller Trojan moons called Telesto
Trojans are now known (see http://cfa-www.harvard. and Calypso, one of which librates about the Saturn±Tethys L4
and the other about the Saturn±Tethys L5 points. Finally, the two
w
E-mail: w.evans1@physics.oxford.ac.uk small moons Janus and Epimetheus follow horseshoe orbits
q 2000 RAS64 S. A. Tabachnik and N. W. Evans
coorbiting with Saturn (see e.g. Smith et al. 1982; Yoder et al. introduction to the dynamics of coorbital satellites within the
1983; Yoder, Synnott & Salo 1989). Another example of a Trojan framework of the elliptic restricted three-body problem. Our fully
configuration closer to home is provided by the extensive dust numerical survey is discussed in Section 3 and takes into account
clouds in the neighbourhood of the L5 point of the Earth±Moon the effects of all of the planets (excepting Pluto), as well as the
system claimed by Winiarski (1989). most important post-Newtonian corrections and the quadrupole
This paper is concerned with the existence of coorbiting moment of the Moon. The results of the survey are presented for
asteroids near the triangular Lagrange points of the four terrestrial each of the terrestrial planets in Sections 4±7 (Mercury, Venus, the
planets. For non-Jovian Trojans, the disturbing forces caused by Earth and Mars). Finally, a companion paper in this issue of
the other planets are typically larger than those caused by the Monthly Notices discusses the observable properties of the
primary planet itself. For this reason, it was formerly considered asteroids (Evans & Tabachnik 2000).
unlikely that long-lived Trojans of the terrestrial planets could
survive. Over the past decade, two lines of evidence have
suggested that this reasoning is incorrect. The first is the direct 2 THEORETICAL FRAMEWORK
discovery of inclined asteroids librating about the Sun±Mars L5 Consider the restricted three-body problem, defined by a system
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Lagrange point. The second is numerical integrations, which have of two massive bodies on Keplerian orbits attracting a massless
steadily increased in duration and sophistication. test particle that does not perturb them in return. The five
The first non-Jovian Trojan asteroid, 5261 Eureka, was Lagrange points are the stationary points of the effective potential
discovered by Holt & Levy (1990) near the L5 point of Mars. (e.g. Danby 1988). The collinear Lagrange points L1, L2 and L3 are
Surprisingly, the orbit of 5261 Eureka is inclined to the plane of unstable, whereas the triangular Lagrange points L4 and L5 form
the ecliptic by 208: 3. The determination of the orbital elements of equilateral triangles with the two massive bodies, as illustrated in
Eureka and a preliminary analysis of its orbit were published in Fig. 1. Motion around L4 and L5 can be stable. The planar orbits
Mikkola et al. (1994). Numerical integrations were performed for that corotate with a planet are classified as either tadpoles or
several dozen Trojan test particles with different initial inclina- horseshoes. Tadpole orbits perform a simple libration about either
tions for ,4 Myr by Mikkola & Innanen (1994), who claimed that of the Lagrange points L4 or L5, whereas the horseshoes perform a
long-term stability of Martian Trojans was possible only in well- double libration about both L4 and L5. The tadpole and horseshoe
defined inclination windows, namely 158 < i < 308 and 328 < orbits are divided by a separatrix, which corresponds to an orbit of
i < 448 with respect to Jupiter's orbit. The discovery of a second infinite period that passes through L3. On average, both tadpoles
Mars Trojan, 1998 VF31, soon followed (see e.g. Minor Planet and horseshoes move about the Sun at the same rate as the parent
Circular 33763; Tabachnik & Evans 1999). Perhaps the most planet.
important point to take from the observational discoveries is that if It is helpful to develop an approximate treatment of coorbital
a comparatively puny body like Mars possesses Trojans, it is motion. Within the framework of the elliptic restricted problem of
indeed quite likely that the more massive planets also harbour three bodies (Sun, planet and asteroid), the Hamiltonian of the
such satellites. asteroid in a non-rotating heliocentric coordinate system reads:
The main argument for the possible existence of Trojans of the
Earth, Venus and Mercury comes from numerical test particle k2
H2 2 mp k2 R; 1
surveys. Much of the credit for reviving modern interest in the 2a
problem belongs to Zhang & Innanen (1988a,b,c). Using the where mp is the mass of the planet in Solar masses and k is the
framework of the planar elliptic restricted four- and five-body Gaussian gravitational constant. The disturbing function R is
problems, their integrations extended from 2000 to 105 yr, defined by (e.g. Danby 1988)
although their model was not entirely self-consistent as mutual
interactions between the planets were not taken into account. 1 r cos S
R 2 : 2
Further important results on the stability of the Trojans of the r 2 1 r 2p 2 2rr p cos S1=2 r 2p
terrestrial planets were obtained in a series of papers by Mikkola
& Innanen (1990, 1992, 1994, 1995). The orbits of the planets Here, the radius vectors r and rp refer to the asteroid and the planet
from Venus to Jupiter were computed at first using a Bulirsch± respectively, S being the elongation of the asteroid from the parent
Stoer integration and later with a Wisdom±Holman (1991) planet. Following Brouwer & Clemence (1961), cos S can be
symplectic integrator. In the case of Mercury, test particles placed expressed in function of the mutual inclination of the two orbits i,
at the Lagrange point exhibited a strongly unstable behaviour the true anomalies n and n p, the argument of perihelion v and the
rather rapidly. Conversely, the mean librational motion of all the longitude of the ascending node V of the asteroid:
Trojan test particles near the Lagrange points of Venus and the i
Earth appeared extremely stable. Thus far, the longest available cos S cos2 cos n 2 np 1 v 1 V
2
integrations are the 6-Myr survey of test particles near the
i
triangular L4 points of Venus and the Earth (Mikkola & Innanen 1 sin2 cos n 1 np 1 v 2 V: 3
1995). The initial inclinations ranged from 08 to 408 with respect 2
to the orbital plane of the primary planet. The Trojans of Venus The disturbing function is then expressed in terms of the mean
and the Earth persisted on stable orbits for small inclinations synodic longitudes lp M p and l M 1 v 1 V: Here, Mp and
(i < 188 and i < 118, respectively). M are the mean anomalies of the planet and the asteroid. In other
These lines of reasoning suggest that a complete survey of the words, we are using a coordinate system for which (x, y) lies in the
Lagrange points of the terrestrial planets is warranted. It is the orbital plane of Mars and the x-axis points towards Mars'
purpose of the present paper to map out the zones in which perihelion. The disturbing function is expanded to second order in
coorbital asteroids of the terrestrial planets can survive for time- the eccentricities and to fourth order in the inclination. Finally, we
scales of up to 100 Myr. Section 2 provides a theoretical make a change of variables f lp 2 l and average R over the
q 2000 RAS, MNRAS 319, 63±79Asteroids in the inner Solar system ± I 65
mean anomaly of the planet Mp: For small and moderate inclinations, kRl has a double-welled
2p
structure. The libration about the Lagrange points can be viewed as
1 the trapping of the test particle in the well, with the tadpole orbits
kRl R dM p U 0 1 U 2 1 O e3 ; e3p ; i5 : 4
2p 0 located in the deeper part of the well and the horseshoe orbits in the
The zeroth order term in the averaged disturbing function is shallower part. The two orbital families are divided by a separatrix.
The slight eccentricities and inclinations of the planets cause small
1 a 2 i deviations of the Lagrange points from their classical values (c.f.
U0 1=2 2 2 cos cos f: 5
ap 2 Namouni & Murray 1999). These deviations can be calculated by
a2 1 a2p 2 2aap cos2 2i cos f
finding the local minima of the secular potential (4). The results for
each terrestrial planet are recorded in Table 1.
The term that is second order in the eccentricities and inclinations
The position of the separatrix is found by locating the local
has three parts:
maximum of the secular potential (4). The separatrix terminates at
U 2;3 a differential longitude from the planet f p. This angle is the
U 2 U 2;0 1 3=2 closest any tadpole orbit can come to the planet. In p the
restricted
a2 1 a2p 2 2aap cos2 2i cos f
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circular problem, this angle is fp 2 asin 1= 2 2 1=2
U 2;5 238: 906 (see e.g. Brown & Shook 1933). The deviation from
1 5=2 ; 6 this classical value caused by the eccentricity and inclination of
a2 1 a2p 2 2aap cos2 2i cos f the planets is also recorded in Table 1.
The secular potential (4) can also be used to work out the
where U2,0, U2,3 and U2,5 are given in Appendix A. Still lengthier approximate period P of small angle librations about the Lagrange
expressions accurate to the fourth order in the eccentricities and points. For simplicity, we take only the zeroth order term (5) with
inclinations are given in Tabachnik (1999). It is interesting to note a ap and obtain the minimum of the secular potential at
that the second order expressions depend only on the longitude of
perihelion, whereas the fourth order contains terms depending on !
1
both the longitude of perihelion and the longitude of the ascending f acos : 7
node. Some of these averaged disturbing functions are shown in 2 cos2 2i
Fig. 2 and are compared with the numerically computed and
averaged disturbing function kRlNum. The period of small librations about this minimum is
2 Pp
P q ; 8
1.5
3 m 4 cos4 ÿ i 2 1
p 2
1.0 where Pp is the orbital period of the planet. The period of libration
increases with increasing inclination.
Expansion of the disturbing function is a good guide to the
0.5 dynamics provided that the inclinations and eccentricities are low
or moderate. At high inclinations and eccentricities, the classi-
fication of the corotating orbits has only recently been undertaken
by Namouni (1999).
Y
0.0
-0.5 3 NUMERICAL METHOD
For each of the terrestrial planets, we carry out numerical surveys
of in-plane and inclined corotating orbits. Section 3.1 introduces
-1.0
the integration method, while Section 3.2 discusses the numerical
procedure.
-1.5 Table 1. The angular positions of the Lagrange
-1.5 -1.0 -0.5 0.0
X
0.5 1.0 1.5 points as inferred from (4) for each of the
terrestrial planets. The asteroidal orbit is assumed
Figure 1. The positions of the five Lagrange points for the three-body to have vanishing inclination and argument of
pericentre. Note that the Lagrange points are
system in a corotating frame. There are two types of planar, coorbiting
displaced only slightly from their classical values.
orbits ± the tadpoles and the horseshoes. Tadpole orbits are shown in green Also recorded is the angular position of the
and librate about either the leading L4 or trailing L5 Lagrange points. separatrix, which physically represents the mini-
Horseshoe orbits are shown in blue and perform a double libration about mum differential longitude that can be reached by
both L4 and L5. The orbits are based on numerical data for the case of the the tadpole orbits.
Sun, the Earth and an asteroid, but the difference in semimajor axis of
the asteroid and the planet has been magnified by a factor of 40 for Planet L4 L5 fp
clarity. The dividing separatrix or orbit of infinite period is shown in
red. The boundaries of the coorbital region are shown as full lines. Mercury 598: 822 3008: 178 258: 736
They are schematic and correspond to bounding positions limited by Venus 608: 000 3008: 000 238: 908
Earth 598: 999 3008: 001 238: 917
L1 and L2, which are themselves arbitrarily located. The satellite
Mars 598: 964 3008: 036 248: 266
regime occupies a sphere around the planet.
q 2000 RAS, MNRAS 319, 63±7966 S. A. Tabachnik and N. W. Evans
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Figure 2. The averaged disturbing function (4) plotted as a function of differential longitude f . The figures shows approximations carried out to second order
in inclination and eccentricities kRl22, second order in inclination and fourth order in eccentricity kRl24 and fourth order in both quantities kRl44. They are
compared with the numerically evaluated averaged disturbing function kRlNum.
Here, the test particle is given the first Jacobi index and the planets
3.1 Mixed variable symplectic integrators
carry the higher Jacobi indices. As Wisdom & Holman (1991)
Our model includes all of the planets (except Pluto whose point out, this is an advantageous choice as it gives the simplest
contribution is negligible in the inner Solar system). The asteroids interaction Hamiltonian. Denoting the Jacobi position of the test
are represented as test particles with infinitesimal mass. They are particle as x1 and its velocity as v1, then
perturbed by the planets but they do not perturb them in return.
The orbits of the planets are integrated using a mixed variable v2 k 2 X x1 ´ xi
N11
1
H kep 1 2 and H int k2 mi 2 ; 10
symplectic integrator scheme (Kinoshita, Yoshida & Nakai 1991; 2 r1 i2
r 3i r 1i
Wisdom & Holman 1991) with individual time-steps (Saha &
Tremaine 1994), which takes into account post-Newtonian correc- where k is the Gaussian gravitational constant and N is the number
tions and the quadrupole moment of the Sun's attraction on the of planets included in the model N 8 for our calculations). We
barycentre of the Earth±Moon system (Quinn, Tremaine & Duncan have used the notation r 1i jx1 2 xi j as the distance from the test
1991). The two latter contributions must be included for particle to the ith body. As mentioned by Wisdom & Holman
calculations in the inner Solar system, whereas the cumulative (1991), the intuitive interpretation of Hint is to note that the
effect of the asteroids, satellites, galactic tidal acceleration, passing attraction of the Sun on the test particle equals the difference
stars, solar-mass loss and oblateness is believed to be smaller than between the direct acceleration of the massive planets on the test
,10210 and is neglected (Quinn et al. 1991). Mixed variable particle and the gravitational pull of the planets on the Sun.
symplectic integrators exploit the fact that the Hamiltonian written The most important general relativistic effects can be included
in Jacobi coordinates (Plummer 1960; Wisdom & Holman 1991) by modifying the test particle Hamiltonian to
is dominated by a nearly Keplerian term. The mixed variable H tp H kep 1 H int 1 H PN : 11
symplectic integrators are so-called because they evaluate the
planetary disturbing forces in Cartesian coordinates while using A clever device for incorporating the most important post-
the elements to advance the orbits. Fast algorithms, like Gauss' f Newtonian effects into mixed variable symplectic integrators is
and g functions, exist to perform the latter task (e.g. Danby 1988; given by Saha & Tremaine (1994). The post-Newtonian Hamil-
Wisdom & Holman 1991). tonian is recast as
The orbit of any of the test particles is derived from the
1 3 2 k4 v4
Hamiltonian: H PN 2 H kep 2 2 2 1 : 12
c 2 r1 2
H tp H kep 1 H int : 9 The last expression contains three terms that are each integrable
q 2000 RAS, MNRAS 319, 63±79Asteroids in the inner Solar system ± I 67
Table 2. Statistics for test particles in the plane of each parent planet. The second
column lists the number of test particles removed from the integration because the
orbit becomes hyperbolic. The following columns give the number of close
encounters with each named planet. The number of survivors on tadpole and
horseshoe orbits, respectively, are also recorded.
Hyp. Mer. Ven. Ear. Mar. Jup. Tad. Hor. Total
Mercury 6 572 157 1 0 3 0 53 792
Venus 0 0 385 0 0 0 168 239 792
Earth 1 0 2 280 0 0 95 414 792
Mars 1 0 0 27 763 1 0 0 792
Table 3. As for Table 2, but for test particles inclined with respect to the plane of the parent planet. There are
1104 test particles initially. The numbers terminated because their orbits become hyperbolic or because they
enter the sphere of influence of any of the planets are reported. The number of survivors on tadpole and
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horseshoe orbits, respectively, are also recorded. There is some uncertainty as to the number of surviving
highly inclined test particles around Mercury, as explained in the text.
Hyp. Mer. Ven. Ear. Mar. Jup. Sat. Ura. Nep. Tad. Hor. Tot
Mercury 320 558 132 40 1 33 6 1 0 ??(0) ??(13) 1104
Venus 32 19 695 209 1 6 5 0 0 129 8 1104
Earth 44 0 171 669 9 7 3 0 1 182 18 1104
Mars 99 0 75 311 445 12 4 0 1 157 0 1104
Table 4. This table lists the average eccentricity
kel and inclination kil of the surviving test 3.2 The numerical procedure
particles at the end of the simulation. Also
recorded is the mean of the maxima of the For each of the terrestrial planets, we carry out two surveys. The
eccentricities kemaxl and inclination kimaxl during first is restricted to test particles in the orbital plane of the planet.
the course of the simulation. The test particles are given the same eccentricity e, inclination i,
longitude of the ascending node V and mean anomaly M, as the
Planet kel kil kemaxl kimaxl planet. The argument of pericentre v is varied from 08 to 3608 in
In-plane survey steps of 58. The initial semimajor axis is equal to the semimajor
Mercury 0.186 68: 968 0.352 138: 307 axis of the planet multiplied by a semimajor axis factor (c.f. the
Venus 0.027 08: 706 0.079 38: 518 investigation of Saturnian Trojans by Innanen & Mikkola 1989).
Earth 0.038 18: 349 0.086 38: 213 The second survey is restricted to test particles with the same
Mars ± ± ± ±
semimajor axis as the planet. The initial inclinations of the test
Inclined survey particles (with respect to the plane of the planet's orbit) are spaced
Mercury 0.400 348: 069 0.626 438: 945 every 28 and the initial arguments of pericentre are spaced every
Venus 0.041 68: 941 0.122 108: 084 158. The eccentricities of the asteroids are inherited from the
Earth 0.064 158: 778 0.129 178: 936 parent planet.
Mars 0.103 258: 858 0.173 248: 267
The initial conditions come from the Jet Propulsion Laboratory
(JPL) Planetary and Lunar Ephemerides DE405, which is
individually. The Keplerian part of equation (12) can be incor- available at http://ssd.jpl.nasa.gov/. The starting
porated into the usual Keplerian orbital advance using epoch of the integration is JED 244 0400.5 (1969 June 28). The
standard units used for the integration are the astronomical unit,
3 the day and the Gaussian gravitational constant k2 G M( : The
exp t ; H kep 1 2 H 2kep ; 13
2c Earth±Moon mass ratio is M% =ML 81:3: For most of the
where the curly brackets are Poisson brackets. The Keplerian computations described below, the time-step for Mercury is
Hamiltonian is conserved with time and equals 2 1=2k2 =atp ; atp 14.27 d. The time-steps of the planets are in the ratio
being the semimajor axis of the test particle. After some straight- 1:2:2:4:8:8:64:64 for Mercury moving outwards, so that Neptune
forward algebra, equation (13) becomes has a time-step of 2.5 yr. The test particles all have the same time-
step as Mercury. These values were chosen after some experi-
3k2 mentation to ensure the relative energy error has a peak amplitude
exp t 0 { ; H kep }; with t 0 1 2 2 t: 14
2c atp of68 S. A. Tabachnik and N. W. Evans
Stability Zones for Mercury (100 Myr) the planet in solar mass units. As the algorithm we use does not
1.0016
allow the variable stepsize necessary for the treatment of close
1.0012 encounters, the exact size of the sphere of influence is not of great
Initial Population
importance. Furthermore, test particles that enter the sphere of
1.0008
influence are typically ejected from the Solar system in another
Semimajor Axis Factor
1.0004 1±10 Myr (e.g. Holman 1997). In the case of the Sun, a close
encounter is defined to be passage within 10 solar radii
1.0000 (.0.005 au). This general procedure is inherited from a number
0.9996
of recent studies on the stability of test particles in the Solar
system (see e.g. Gladman & Duncan 1990; Holman & Wisdom
0.9992 1 Myr 1993; Holman 1997; Evans & Tabachnik 1999)
5 Myr
20 Myr
We discuss the detailed results for each of the terrestrial planets
0.9988
100 Myr in turn. A broad overview of the results is given in three tables.
0.9984 The fates of the test particles in the in-plane and inclined surveys
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0 60 120 180 240 300 360
Initial Longitude [degrees] are summarized in Tables 2 and 3, respectively. These provide the
number of survivors on tadpole and horseshoe orbits, as well as
Figure 3. This shows the gradual erosion of test particles in the vicinity of the numbers suffering close encounters with each planet. Table 4
the Lagrange points of Mercury. The initial population of test particles provides the average eccentricities and inclinations of the
covers every 58 in longitude and every 0.0004 in semimajor axis factor.
survivors at the end of the simulations.
Red, green, blue and yellow mark the positions of the survivors after 1, 5,
20 and 100 Myr.
Stability Zones for Mercury (100 Myr)
1.0040 4 M E R C U R I A N S U RV E Y S
1.0030 For the Mercurian in-plane survey, the semimajor axis factor is
chosen between 0.998 and 1.002 in steps of 0.0004. If the
1.0020
semimajor axis factor is exactly unity, and the argument of
Semimajor Axis Factor
1.0010 pericentre is displaced by 608 or 3008, then the test particle is at
the classical Lagrange point and can remain there on a stable orbit
1.0000 if the perturbations from the rest of the Solar system are neglected.
0.9990
Fig. 3 shows the time evolution of this array of test particles. The
regions occupied by the test particles remaining after 1, 5, 20
0.9980 and 100 Myr are shown in red, green, blue and yellow,
respectively. After 100 Myr, only 53 out of the original 792
0.9970
test particles remain. The locations of the survivors are shown in
0.9960 close-up in Fig. 4. The most striking point to notice is that the
-40 -30 -20 -10 0 10 20 30 40
Initial Longitude [degrees] stable zones do not include the classical Lagrange points
themselves. In fact, all the survivors follow horseshoe orbits
Figure 4. This shows the surviving test particles in the plane of initial and there are no surviving tadpole orbits. There are no long-lived
longitude and semimajor axis factor marked as circles. The surviving Mercurian Trojans. There are two possible reasons for this. First,
orbits are all horseshoes. The length of integration is 100 Myr. Unstable
Mercury is the least massive of the terrestrial planets and therefore
test particles in the elliptic restricted three-body problem (comprising the
the potential wells in which any long-lived Trojans inhabit are less
Sun, Mercury and test particle) are shown as diamonds. Everything within
the inner boundary of diamonds is stable at the level of the elliptic deep than for the other terrestrial planets. Secondly, Mercury is the
restricted three-body problem, and so this indicates the damaging effect of most eccentric of the terrestrial planets and this also encourages
perturbations from the rest of the Solar system. the erosion of the test particles. During the course of the 100-Myr
simulation, Mercury's eccentricity fluctuates between ,0.1 and
challenges for long-term integrations, and there is some evidence ,0.3.
that round-off errors may be affecting our results for the highly The top three panels of Fig. 5 show the evolution of the
inclined test particles around Mercury. semimajor axis, inclination and eccentricity for Mercury, together
As the orbits of the test particles are integrated, they are with a stable and an unstable test particle. The distributions of the
examined at each time-step. If their trajectories become parabolic orbital elements of the survivors suggest that they belong to a low-
or hyperbolic orbits, they are removed from the survey. In inclination, low-eccentricity family of horseshoe orbits. At the end
addition, test particles that experience close encounters with a of the simulation, the average eccentricity of the survivors is 0.186
massive planet or the Sun are also terminated. The sphere of and their average inclination is 68: 968. The lowest panel shows the
influence is defined as the surface around a planet at which the evolution of the amplitude of libration D (in degrees) about the
perturbation of the planet on the two-body heliocentric orbit is Lagrange point for the two test particles. The stable particle starts
equal to that of the Sun on the two-body planetocentric orbit (e.g. off at a semimajor axis factor of 1.001 and an initial longitude of
Roy 1988). If the mass of the planet is much less than that of the 158. The unstable particle starts off at exactly the L5 Lagrange
Sun, this surface is roughly spherical with radius: point. Notice that the eccentricity and inclination variations of the
stable particles closely follow those of Mercury. The unstable
r s ap m2=5
p ; 16 test particle maintains its tadpole character only for some
,1:2 107 yr; before passing through a brief horseshoe phase
where ap is the semimajor axis of the planet and mp is the mass of for ,0:5 107 yr: After this, its semimajor axis increases to
q 2000 RAS, MNRAS 319, 63±79Asteroids in the inner Solar system ± I 69
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Figure 5. The first three panels of the figure show the evolution of the semimajor axis, eccentricity and inclination for Mercury (blue), a stable test particle
(red) and an unstable test particle (green). The final panel shows the amplitude of libration D about the Lagrange point. Note that the stable test particle
follows a horseshoe orbit, as evidenced by D , 3208: The unstable test particle is initially on a tadpole orbit, then passes through a brief horseshoe phase
before entering the sphere of influence of Mercury.
1.012
Stability Zones for Venus (25 Myr) ,0.42 au and finally decreases to ,0.35 au before entering the
sphere of influence of Mercury. The stable test particle follows a
1.008 horseshoe orbit. This is obvious on examining the lowest panel,
which shows that its angular amplitude of libration about the
Lagrange point is ,3208.
Semimajor Axis Factor
1.004
For the Mercurian inclined survey, the orbits of 1104 test
particles around Mercury are followed for 100 Myr. Only thirteen
1.000
test particles survived until the end of the 100-Myr integration.
Seven of these were low inclination i , 68 test particles that
0.996
started off at arguments of pericentre v 158 or 3458) very
close to that of Mercury. The remaining six survivors were all
0.992
high-inclination objects. On repeating these calculations on
similar machines with identical roundoff, but an updated
ephemerides, all six of these highly inclined orbits were
0.988
0 60 120 180 240 300 360
Initial Longitude [degrees] terminated before the end of the 100 Myr (mostly because they
Figure 6. This shows the surviving test particles near Venus. The survivors
entered the sphere of influence of Mercury), although the low
are plotted as circles in the plane of initial longitude and semimajor axis inclination results were reproduced successfully. Clearly, it is not
factor. Filled circles are tadpole orbits, open circles are horseshoe orbits. possible to draw definitive conclusions about the longevity of
Unstable test particles in the restricted three-body problem comprising the highly inclined asteroids near Mercury, although it seems likely
Sun, Venus and asteroid are shown as diamonds. The length of integration that any stable zones must be small and depend sensitively on
is 25 Myr. initial conditions.
q 2000 RAS, MNRAS 319, 63±7970 S. A. Tabachnik and N. W. Evans
Extrema for Remaining Test Particles near Venus (25 Myr)
0.7270 5 V E N U S I A N S U RV E Y S
0.7263
Perhaps one of the likeliest planets in the inner Solar system to
0.7256 harbour undiscovered Trojans is Venus. Fig. 6 shows the results of
0.7249 the in-plane survey of Venusian test particles. The starting
Semimajor Axis [AU]
semimajor axes are scaled by a fraction of the semimajor axis
0.7242
of the planet in steps of 0.0012, whereas the argument of
0.7235
pericentre is offset from that of the planet by 58 steps. The test
0.7228 particles are integrated for 25 Myr and the survivors recorded in
0.7221 Fig. 6. A filled circle represents a tadpole orbit, an open circle
represents a horseshoe orbit. Notice that there are some long-lived
0.7214
survivors on horseshoe orbits even around the conjunction point.
0.7207 The tadpole orbits survive around L4 for starting longitudes
0.7200 between 158 and 1608 and around L5 for starting longitudes
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0 60 120 180 240 300 360
Initial Longitude [degrees] between 1958 and 3458. The offset in the semimajor axes of the
survivors Da compared with the parent planet ap satisfy Da=ap &
Figure 7. The extrema of the semimajor axes of the surviving test particles 0:72 per cent. There are 792 test particles at the beginning of the
near Venus are plotted against their initial longitude. The extrema of
simulation, but only 407 persist till the end, of which 168 are on
tadpole orbits are shown as filled circles, the extrema of horseshoes as
open circles. The extrema of survivors starting with the same semimajor
tadpole orbits. Horseshoes and tadpoles are of course divided by a
axis factor fall on the same curve (equation 19), which is drawn as a full separatrix in phase space (see Section 2). The break-up of the
line. This is really a consequence of the fact that the motion is derivable separatrix is associated with a chaotic layer, and this is responsible
from an integrable Hamiltonian to an excellent approximation. for erosion between the filled and open circles in the stable zones.
At the edge of the figure, we show the stability boundary inferred
0.15 0.15
Eccentricity
Eccentricity
0.10 0.10
0.05 0.05
0.00 0.00
0.996 0.998 1.000 1.002 1.004 0 60 120 180 240 300 360
Initial Semimajor Axis Factor Initial Differential Longitude [deg.]
14 14
12 12
Inclination [deg.]
Inclination [deg.]
10 10
8 8
6 6
4 4
2 2
0 0
0.996 0.998 1.000 1.002 1.004 0 60 120 180 240 300 360
Initial Semimajor Axis Factor Initial Differential Longitude [deg.]
Figure 8. This shows the distributions of eccentricities and inclinations of the surviving test particles of Fig. 6 against their initial semimajor axis and
longitude from Venus. In all the panels, the maximum values attained during the course of the orbit integrations are marked by crosses, the instantaneous
values after 25 Myr are marked by circles. The mean maximum eccentricity is 0.079, while the mean maximum inclination is 3.5188. At the end of the
simulation, the average eccentricity and inclination of the sample is 0.027 and 0.7068, respectively. The survivors are members of a low-eccentricity and low-
inclination family.
q 2000 RAS, MNRAS 319, 63±79Asteroids in the inner Solar system ± I 71
from the elliptic restricted three-body problem. The diamonds points are still connected by some surviving test particles at
represent unstable test particles in the elliptic restricted three-body conjunction. Bodies trapped around the Lagrange points should be
problem (comprising the Sun, Venus and the massless asteroid). sought at all longitudes close to the orbital plane. Fig. 10 shows a
Everything within this outer boundary of diamonds is stable at the close-up of the stable zones, with tadpole orbits represented by
level of the restricted three-body problem. Fig. 7 shows an at first filled circles and horseshoe orbits by open circles. Most of the
sight surprising regularity of the orbits of the survivors. The objects that do survive are true Trojans, in that their orbits are
extrema of the semimajor axis aext of the Trojan test particles are recognisably of a tadpole character. There are just eight surviving
plotted against the initial longitude and fall on a one-parameter horseshoe orbits. The ensemble has a mean eccentricity of 0.041
family of curves according to the semimajor axis factor. The and a mean inclination of 68: 941.
heliocentric Hamiltonian for a test particle in the frame rotating Assuming that they are primordial, we can estimate the number
with the mean motion of the planet may be written of coorbiting Venusian satellites by extrapolating from the number
k2 p of Main Belt asteroids (c.f. Holman 1997; Evans & Tabachnik
H2 2 mp k2 R 2 np 1 1 mp a: 17 1999). The number of Main Belt asteroids NMB is N MB &
2a p
SMB AMB f ; where AMB is the area of the Main Belt, SMB is the
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Here, np 1 1 mp =a3p is the mean motion of the planet using surface density of the protoplanetary disc and f is the fraction of
Kepler's third law, and R is approximated by the zeroth order term primordial objects that survive ejection (which we assume to be a
in the disturbing function: universal constant). Let us take the Main Belt to be centred on
2 3 2.75 au with a width of 1.5 au. Fig. 6 suggests that the belt of
1 6 1 7 Venusian Trojans is centred on 0.723 au and has a width of
6 2 cos f7
R
ap 4 f 5; 18 &0.008 au. If the primordial surface density falls off inversely
2 sin proportional to distance, then the number of coorbiting Venusian
2
asteroids NV is
where f is the difference in longitude between the planet and the
asteroid. This follows from setting a ap in equation (5). This 2:75 0:723 0:008
NV & N MB < 0:0053N MB : 20
Hamiltonian depends on time only through the slow variation of 0:723 2:75 1:5
the orbital elements of the planet. These take place on a time-scale
much longer than the libration of the Trojan, and so the The number of Main Belt asteroids with diameters *1 km is
Hamiltonian is effectively constant. Expanding in the difference ,40 000, which suggests that the number of Venusian Trojans is
between the semimajor axis of the planet and the Trojan, we ,100 with perhaps a further ,100 coorbiting companions on
readily deduce that a test particle with a semimajor axis factor f horseshoe orbits.
and that starts at a differential longitude u has extrema aext
satisfying
6 T E R R E S T R I A L S U RV E Y S
" !#1=2
aext 2 8m 1 1 The Earth is slightly more massive than Venus, and this augurs
1 1 f 2 1 1 2 cos u 2 ; 19
ap 3 2 sin 12 u 2 well for the existence of coorbiting satellite companions. The
Earth has no known Trojans (Whiteley & Tholen 1998), but it
where m mp = 1 1 mp is the reduced mass. The extrema of the does possess the asteroidal companion 3753 Cruithne, which
semimajor axis of the Trojans fall on this one-parameter family of moves on a temporary horseshoe orbit (Wiegert, Innanen &
curves, as depicted in Fig. 7. Mikkola 1997; Namouni, Christou & Murray 1999). The asteroid
Fig. 8 shows distributions of the eccentricities and inclinations will persist on this horseshoe orbit for a few thousand years.
of the survivors plotted against their initial semimajor axis and Fig. 11 shows the surviving in-plane test particles near the Earth
longitude from Venus. In all the panels, the maximum values after their orbits have been integrated for 50 Myr. Again, filled
attained during the course of the 25-Myr orbit integrations are circles represent the tadpole orbits, open circles the horseshoe
marked by crosses, the instantaneous values are marked by circles. orbits. On comparison with Fig. 6, we see that the stable zones of
There are two striking features in these diagrams. First, the the Earth are more extensive and the number of survivors is
eccentricities and the inclinations of the survivors remain very low greater than is the case for Venus. Tadpole orbits survive for
indeed. The mean eccentricity of the sample after 25 Myr is 0.027, Da=ap & 0:48 per cent and horseshoes for Da=ap & 1:20 per cent.
while the mean inclination is 08: 706. This is a very stable family. Although the number of survivors is greater, the number of true
Secondly, most of the survivors seem to occupy very nearly the Trojans on tadpole orbits is much less than the number for Venus.
same regions of the plots. The striking horizontal line of crosses in Specifically, of the 792 original test particles, only 509 persist till
the rightmost panels of Fig. 8 suggest that these test particles do the end of the simulation and of these just 95 are on tadpole orbits.
lie on similar orbits belonging to similar families, and exploring In the case of the Earth, just 19 per cent of the survivors are
similar regions of phase space. tadpoles, as opposed to 41 per cent for Venus. The visual
Fig. 9 shows the results of the Venusian inclined survey. Here, consequence of this is that the holes in the stable regions for the
the orbits of 1104 test particles around Venus are integrated for Earth are more pronounced than for Venus (compare also with the
75 Myr. The initial inclinations of the test particles (with respect to equivalent diagram for Saturn provided by Holman & Wisdom
the plane of Venus' orbit) are spaced every 28 and the initial 1993). The survivors have a mean eccentricity of 0.038 and a mean
longitudes (again with respect to Venus) are spaced every 158. The inclination of 18: 349, consistent with a long-lived population.
test particles are colour-coded according to whether they survive Fig. 12 shows the erosion of an ensemble of 1104 inclined test
till the end of the 5-, 20-, 50- and 75-Myr integration time-spans. particles positioned at the same semimajor axis as the Earth but
The 137 survivors after 75 Myr all have smallish inclinations varying in longitude. Again, the initial inclinations of the test
i , 168. Notice, too, that the stable zones around the Lagrange particles (with respect to the plane of the Earth's orbit) are spaced
q 2000 RAS, MNRAS 319, 63±7972 S. A. Tabachnik and N. W. Evans
Stability Zones for Venus (75 Myr) Stability Zones for the Earth (50 Myr)
90 1.012
80
1 Myr 1.008
70 5 Myr
20 Myr
Semimajor Axis Factor
60 50 Myr 1.004
Inclination [degrees]
75 Myr
50
1.000
40
30 0.996
20
0.992
10
0 0.988
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0 60 120 180 240 300 360 0 60 120 180 240 300 360
Initial Longitude [degrees] Initial Longitude [degrees]
Figure 9. This shows the erosion of an ensemble of inclined test particles Figure 11. This shows the surviving in-plane test particles near the Earth.
positioned at the same semimajor axis as the Lagrange point of Venus but Filled circles are tadpole orbits, open circles are horseshoe orbits. Unstable
varying in longitude. The test particles surviving after 5, 20, 50 and 75 Myr test particles in the restricted three-body problem are shown as diamonds.
are shown in green, blue, yellow and red, respectively. Unlike the case of The length of integration is 50 Myr.
Mercury, there are zones of long-lived stability that hug the the line of
vanishing inclination (to the plane of Venus' orbit).
Stability Zones for the Earth (25 Myr)
90
Remaining Test Particles near Venus (75 Myr)
90
80
1 Myr
80
70 5 Myr
10 Myr
70
60 25 Myr
Inclination [degrees]
60
Inclination [degrees]
50
50
40
40
30
30
20
20
10
10
0
0 60 120 180 240 300 360
0 Initial Longitude [degrees]
0 60 120 180 240 300 360
Initial Longitude [degrees]
Figure 12. This shows the erosion of an ensemble of inclined test particles
Figure 10. This shows a close-up of the stability zones of the inclined positioned at the same semimajor axis as the Lagrange point of the Earth
Venusian survey. Only test particles surviving the full integration time of but varying in longitude. The test particles surviving after 1, 5, 10 and
75 Myr are plotted. The filled circles are tadpole orbits, while the open 25 Myr are shown in green, blue, yellow and red, respectively.
circles are horseshoe orbits. Moderate or highly inclined i * 168
Venusian asteroids are unstable, but this figure provides evidence in Let us remark that the Earth has more surviving test particles in
favour of a stable population of low inclination Venusian asteroids. both of our surveys than does Venus, as well as more extensive
stable zones. This suggests that the asteroid 3753 Cruithne may
every 28 and the initial longitudes are spaced every 158. The test not be unique, but that it might be the first member of a larger
particles are colour-coded according to their survival times ± the class of coorbiting terrestrial companions (Namouni et al. 1999).
particles surviving after 1, 5, 10 and 25 Myr are shown in green, Making the same assumptions as in (20), our estimate for the
blue, yellow and red, respectively. The ensemble after 25 Myr is number of terrestrial companions is
shown in Fig. 13 with tadpoles represented as filled circles and
horseshoes as open circles. Again, the first thing to note is the 2:75 1:00 0:01
NE & N MB < 260: 21
number of survivors ± there are 200 in total, almost all of which 1:00 2:75 1:5
are moving on tadpole orbits. There seem to be two bands of
stability, one at low starting inclinations i & 168 and one at Here, we have used Fig. 11 to set the width of the stable zone
moderate starting inclinations 248 & i & 348: On careful inspec- around the Earth asAsteroids in the inner Solar system ± I 73
example, Mikkola et al. 1994; Tabachnik & Evans 1999, and the particles are colour-coded according to their survival times. Most
references therein). Both have moderate inclinations to the eclip- of the particles are already swept out after 5 Myr. After 50 Myr,
tic, namely 208: 3 and 318: 3, respectively. there is only one test particle remaining and it too is removed
The in-plane Martian survey is presented in Fig. 15. The test shortly afterwards. There are no survivors after 60 Myr. This
confirms the earlier suspicions of Mikkola & Innanen (1994) that
90
Remaining Test Particles near the Earth (25 Myr)
Trojans in the orbital plane of Mars are not long-lived. As is
80
Stability Zones for Mars (60 Myr)
70 1.0015
0.1 Myr
60 1.0012 0.5 Myr
Inclination [degrees]
1 Myr
1.0009 5 Myr
50
10 Myr
1.0006 20 Myr
Semimajor Axis Factor
40 50 Myr
1.0003
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30
1.0000
20
0.9997
10
0.9994
0
0.9991
0 60 120 180 240 300 360
Initial Longitude [degrees]
0.9988
Figure 13. This shows a close-up of the stability zones of the inclined 0.9985
0 60 120 180 240 300 360
survey of terrestrial test particles. Only particles surviving the full Initial Longitude [degrees]
integration time of 25 Myr are plotted. The filled circles are tadpole orbits,
while the open circles are horseshoe orbits. There seem to be two bands of Figure 15. This shows the erosion of the in-plane test particles near Mars.
stability, one at low starting inclinations i & 168 and one at moderate There are no surviving test particles after 60 Myr. The test particles are
starting inclinations 248 & i & 348: colour-coded according to their survival times.
0.30 0.30
0.25 0.25
0.20 0.20
Eccentricity
Eccentricity
0.15 0.15
0.10 0.10
0.05 0.05
0.00 0.00
0 10 20 30 40 0 60 120 180 240 300 360
Initial Inclination [deg.] Initial Differential Longitude [deg.]
40 40
30 30
Inclination [deg.]
Inclination [deg.]
20 20
10 10
0 0
0 10 20 30 40 0 60 120 180 240 300 360
Initial Inclination [deg.] Initial Differential Longitude [deg.]
Figure 14. This shows the distributions of eccentricities and inclinations of the surviving test particles of Fig. 13 against their initial semimajor axis and
longitude from the Earth. In all the panels, the maximum values attained during the course of the orbit integrations are marked by crosses, the instantaneous
values after 25 Myr are marked by circles. The bimodality of the inclination distribution of the survivors is manifest.
q 2000 RAS, MNRAS 319, 63±7974 S. A. Tabachnik and N. W. Evans
Remaining Test Particles near Mars (25 Myr) of pericentre k4Ç l or longitude of node kVl _ for the asteroid
90
becomes nearly equal to an eigenfrequency of the planetary
80 system (e.g. Brouwer & Clemence 1961; Williams & Faulkner
Eureka
70 1998 VF31 1981; Scholl et al. 1989). The secular precession frequencies in
1998 QH56 linear theory are usually labelled gj and sj j 1¼8 for Mercury±
Neptune) for the longitude of pericentre and longitude of node,
60 1998 SD4
Inclination [degrees]
50 respectively. Their mean values computed over 200 Myr are listed
40
in Laskar (1990).
Fig. 18 can be used to infer the positions of some of the
30 principal secular resonances as a function of inclination in the
20 vicinity of each Lagrange point. The vertical axis is the rate of
variation of various angles, the horizontal axis is the inclination. A
10
resonance occurs whenever the rate of variation vanishes. The blue
0 curve shows the frequency k4_ l 2 g5 ; which vanishes at inclina-
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0 60 120 180 240 300 360
Initial Longitude [degrees] tions ,28±308 for the L4 point and 298 at the L5 point. Notice that
the resonance is much broader at the L4 point. The yellow curves
Figure 16. This shows the results of the survey of inclined test particles show k3V _ 1 2v_ l 2 g : This frequency vanishes at a range of
5
near Mars. The horizontal axis marks the longitude measured from Mars
inclinations between ,6±118, again with slight differences
and the vertical axis the inclination with respect to Mars of the starting
positions of test particles. Only the particles surviving till the end of the
noticeable at the two Lagrange points. As these are resonances
25-Myr integration are marked. Also shown are the instantaneous positions with Jupiter, they are expected to be the most substantial. At this
of the two Martian Trojans, namely 5261 Eureka and 1998 VF31, as well semimajor axis, there is just one resonance with Saturn. The black
as the asteroids 1998 QH56 and 1998 SD4. The latter two have been curves show the frequency k2V _ 1 3v_ l 2 g : This resonance
6
suggested as possible Trojans, although this now seems unlikely. occurs at inclinations of ,158. Lastly, there are two weaker
resonances with the Earth. These may be tracked down using the
red curve, which shows k4_ l 2 g3 ; and the green curve, which
evident from Table 2, the most common fate of the test particles shows k6V _ 1 v_ l 2 s3 : This completes the list of the main
is to enter the sphere of influence of Mars. The inclined survey is resonances in the vicinity of Mars. The importance of the Jovian
shown in Fig. 16, together with the positions of the two Trojans, resonances, in particular, has been pointed out previously by
5261 Eureka (marked by a square) and 1998 VF31 (marked by an Mikkola & Innanen (1994).
asterisk). Two further asteroids ± 1998 QH56 (triangle) and 1998 Fig. 19 illustrates the evolution of a few arbitrarily selected
SD4 (diamond) ± have been suggested as Trojan candidates, orbits. The left-hand panels refer to the L4 Lagrange point, the
although improved orbital elements together with detailed right-hand panels to the L5 point. The panels are labelled
numerical simulations (Tabachnik & Evans 1999) now make according to the initial inclination with respect to Mars' orbit.
this seem rather unlikely. The result of the inclined survey is to They plot the evolution of the eccentricity e and the inclination
show stable zones for inclinations between 148 and 408 for time- with respect to Jupiter ijup for a typical test particle (full curve)
spans of 25 Myr. The stable zones are strongly eroded at ,298. and Mars (broken curve). The inclinations over which the two
Following this, we conducted another experiment in which Jovian resonances operate are shown as shaded bands. The
inclined Martian Trojans are simulated for 100 Myr. The initial uppermost two panels refer to orbits that start out in the orbital
conditions are inherited from Mars, except for the argument of plane of Mars. Their inclination is initially increased, and this
pericentre, which is offset by 608 (L4) and 3008 (L5), and the takes the orbits into the regime in which one of the Jovian
inclinations that are selected in the range 08±408 from Mars' resonances is dominant. The eccentricity of the orbit is pumped
orbital plane. To examine the effects of the perturbations of the whenever it lingers in the shaded inclination band. This makes the
other planets in more detail, a 18 step in inclination is chosen. orbit Mars-crossing and the test particle is terminated. The middle
Fig. 17 shows the results of this exercise. The upper panel two panels show the fates of orbits at intermediate inclinations of
expresses the termination time versus the initial inclinations of the 128 at the L4 point and 58 at the L5 point. Although the behaviour
test particles at both Lagrange points. Not surprisingly, low is quite complex, the final increase in eccentricity in both cases
inclination Trojans i , 58 enter the sphere of influence of Mars coincides with prolonged stays in the resonant region. We
on a 10-Myr time-scale. The stable inclination windows are also conclude that the low inclinations test particles are destabilized
recovered with a strong disturbing mechanism at 298 for L5 and by this secular resonance with Jupiter. The bottom two panels
308 for L4. The four lower panels give the eccentricities and show fates of test particles starting off at 308 at the L4 point and
inclinations of the remaining test particles at the end of the 298 at the L5 point. In both cases, there is a rapid and pronounced
integration. Crosses identify the maximum quantities over the increase in the eccentricity, which takes it onto a Mars-crossing
entire time-span, while open circles show the instantaneous values path. This destabilization occurs only for a very narrow range of
at 100 Myr. The general trend is to have stable orbits emax , 0:2 inclinations. This is manifest in the erosion in Fig. 16, especially
in the range 158±348 in the case of L4 and 98±368 in the case of L5. at the L4 point near inclinations of 308.
Interestingly, the two securely known Trojans, namely 5261 Mikkola & Innanen (1994) suggested that the instability of low
Eureka and 1998 VF31, occupy positions at L5 corresponding to inclination Martian Trojans was the result of a secular resonance
the two local minima of the maximum-eccentricity curve. with Mars driving the inclination upwards. In their picture, this
The results for the inclined Martian Trojans are unusual, and it continues until a critical inclination of ,128 is reached when
is natural to seek an explanation in terms of secular resonances. 3V 1 2v resonates with Jupiter. The difficulty with this is that it is
Large disturbances can occur when there is a secular resonance, not clear whether the claimed Martian secular resonance ± which
that is, when the averaged precession frequency of the longitude is really just equivalent to the statement that the test particle is
q 2000 RAS, MNRAS 319, 63±79Asteroids in the inner Solar system ± I 75
1•108
L4
8•107 L5
Survival Time [yr]
6•107
4•107
2•107
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0
0 10
10 20
20 30
30 40
40
Initial Inclination [deg]
0.40 45
L4 40 L4
Inclination [deg]
0.30 35
Eccentricity
30
0.20
25
0.10 20
15
0.00 10
L5 40 L5
Inclination [deg]
0.3 35
Eccentricity
30
0.2
25
0.1 20
15
0.0 10
10 15 20 25 30 35 40 10 15 20 25 30 35 40
Initial inclination [deg] Initial Inclination [deg]
Figure 17. This upper panel shows the termination time of the 100-Myr survey of the Martian Lagrange points. Test particles at L4 are identified by triangles
while squares mark test particles at L5. The instantaneous (open circles) and the maximum (crosses) values of the eccentricity and inclination are illustrated in
the four lower panels.
coorbiting ± is responsible for the inclination increase. Our Fig. 18 most to 100 Myr, which is a small fraction of the age of the Solar
seems to show that k3V _ 1 2v_ l 2 g nearly vanishes over a broad system (,5 Gyr). To gain an idea of the possible effects of longer
5
range of inclinations and we suspect it may be able to cause the integration times, we can use the approximate device of fitting our
damage on its own. existing data and extrapolating. Graphs of the number of surviving
particles against time are shown in Figs 20±21 for the in-plane
and inclined surveys, respectively. In the former case, the data are
generally fitted well by a logarithmic decay law of the form
8 CONCLUSIONS
The possible existence of long-lived coorbiting satellites of the N t a 1 b log10 t yr: 22
terrestrial planets has been examined using numerical simulations
of the Solar system. Of course, integrations in the inner Solar Table 5 shows the best-fitting values of a and b. It also gives the
system are laborious as much smaller time-steps are required to extrapolated number of test particles after 1 and 5 Gyr. For both
follow the orbits of the satellites of Mercury as opposed to the Venus and the Earth, this suggests that several hundred test
giant planets like Jupiter. Our numerical surveys have been particles remain, even if the simulations are run for the age of the
pursued for time-scales of up to 100 Myr ± typically an order of Solar system. In the inclined case, the data is not well-fitted by
magnitude greater than previous computations in the inner Solar logarithmic decay laws. Instead, Table 6 gives the results of fitting
system. The numerical algorithm is a symplectic integrator with the data between 10 and 100 Myr to a power-law decay of the
individual time-steps that incorporates the most important post- form
Newtonian corrections (Wisdom & Holman 1991; Saha &
Tremaine 1994). 10c
N t : 23
One worry concerning our integrations is that they extend at t yrd
q 2000 RAS, MNRAS 319, 63±79You can also read
