patryk@dragon.kobe-u.ac.jp

Objects orbiting around the Lagrangian L4 and L5 points of a planet’s orbit – called Trojan
objects – lie 60º ahead of and 60º behind that planet in its orbit [1]. These objects are of greatinterest because they can be dynamically stable over billions of years, and may therefore be able
to tell us a great deal about the early solar system. The total population of large Neptunian
Trojans (diameter > 50km) has been estimated to be at least the same as that of Jovian Trojans of
the same size [2]. A total of six Neptune Trojans have been found to date (Fig. 1) [3]. These objects
appear to be primordial, stable over four Gyr [2,4,5]. Surprisingly, two of these objects possesshigh inclinations within the 25–30º range, thus challenging models of Trojan formation [2,6-8].
The currently known Neptunian Trojans and their libration amplitudes

N18AU – fast migration
Horseshoe: 208 L4: 141 L5: 197

INSITU

N0=1000INSITU

Composite of the initialconditions for the variant scenario with Neptune
starting at 18AU. Objects orbiting around L4 andL5 points, and a disk of bodies beyond thelocation of Neptune are shown.

The origin and dynamical evolution of Neptunian Trojans can be used to constrain: i) Models
proposed to explain the orbital architecture of the trans-Neptunian region [9,10]; (ii) Previously
proposed Trojan formation mechanisms [2,6,7]; (iii) Details of the migration of the giant planets.
Outstanding questions: Are the six Neptune Trojans stable over billions of years? Did the Neptunian
Trojans form locally or were they captured during migration, or both? How do we explain the highinclination Trojans? How do Neptunian Trojans relate to other small body populations?

Horseshoe: 37 L4: 23 L5: 29

Horseshoe: 174 L4: 26 L5: 25

DISK

N0=30000

INSITU

N23AU – fast migration

Initial conditions (18AU case)

Horseshoe: 242 L4: 352 L5: 371

Neptunian Trojans. The observational data was taken from the AstDys database
http://tinyurl.com/65op2k on the 24th April 2008. All six Trojans orbit around the L4 point. The libration amplitudes
were averaged over individual values calculated for the nominal object + 100 clones after integrating their orbits ten
Myr into the future. Libration amplitude refers to the maximum angular displacement from the center of libration
during resonant motion.

N18AU – slow migration

N0=1000

DISK

N0=100000

N23AU – slow migration
Horseshoe: 12 L4: 242 L5: 249

Horseshoe: 268 L4: 27 L5: 31

DISK

N0=60000

Horseshoe: 72 L4: 28 L5: 20

N0=30000

INSITU

N0=1000

Horseshoe: 50 L4: 27 L5: 26

DISK

N0=80000

Neptunian Trojans surviving after the giant planets migrated to their current orbits for each of the four variants (Neptune starting at 18 or 23AU, andslow or fast migration). The number of Trojans are given according to the resonant behaviour: horseshoe, L4 and L5 orbits. Objects that were initially in Trojan orbits andin the planetesimal disk beyond Neptune prior to migration are referred as “INSITU” and “DISK”. The initial number of used objects is given by N0.

N18AU – fast migration

Horseshoe
L4 L5

N18AU – slow migration

INSITU

INSITU

INSITU

INSITU

DISK

DISK

DISK

DISK

We performed numerical simulations of several thousand massless objects under thegravitational influence of the four giant planets using the orbital symplectic integrators EVORB[11] and MERCURY [12]. For the automatic detection of Trojans and determination of theirresonant properties, we employed the RESTICK code [13].
Observed Neptune Trojans. We followed the nominal orbit + 100 clones of each body for 4Gyr. • Despite the wide conditions at the end of planet migration, only Trojans with smallFirst stage – planet migration. Firstly, we looked at the dynamics of Trojans with migrating
eccentricities and libration amplitudes less than ~45º (L4/L5 orbits) can survive for 4Gyr.
giant planets. The migration followed an exponential behaviour as described in [7]:, where an is the semimajor axis of the giant planet.

Four variants were considered: Neptune starting at 18 and 23AU (“N18AU “ and “N23AU”), using
two different migration timescales !=1, 10Myr (“fast” and “slow” migration). For each of thesefour variants, a few times 104 objects were placed in a cold disk stretching from 1AU beyond theinitial position of Neptune to 30AU (“DISK”), and a cloud of several thousand objects were placedat the L4/L5 points (“INSITU”). All disks/clouds started with initially e<0.01; i<0.6º. The orbits wereintegrated for a total time of 5!, after which the giant planets acquired their current orbits (Fig. 2) .
Second stage – long-term evolution. From the final Trojan populations obtained at the end ofstage one, we selected 100 surviving seeds for each disk/cloud (800 seeds in total). We then cloned
each seed obtaining 5.8x105 clones. These clones were integrated over 4Gyr.

• With the exception of 2001 QR322, the majority or all clones of other Neptune Trojans survived
over 4Gyr. Overall, these bodies showed no significant changes in orbital or resonant properties• A significant population of Trojans survived in each of the migration scenarios, including
objects in horseshoe, L4 and L5 orbits showing varied properties (Fig. 3).
• While Trojans captured from the cold disk yielded a wide range of orbital elements e=0–0.3
and i=0–50º, those from the pre-Trojan populations survived in general with e=0–0.1 and i=0–5º(except the variant scenario N18AU-slow migration, which showed e=0–0.35 and i=0–40º).

N18AU – fast migration
INSITU

Horseshoe
L4 L5DISK

N18AU – slow migration
INSITUDISKThe six Neptunian Trojans currently known appear to be stable over the age of the solar system
Pre-formed Trojan populations evolved with little dependence on planet migration parameters.
Retention fractions after migration were 20-95%. However, strong disruption induced bydynamical instabilities is possible during a slow and extended migration of Neptune (N18AU)
Trojan populations captured from the planetesimal disk showed wide range of orbital properties.
However, capture efficiency was quite low though: ~0.5-1% (!=1Myr) and ~0.1% (!=10Myr)
No significant i-excitation was found among the Trojan bodies in the simulations over 4Gyr
Neptunian Trojans may represent the remnants of both pre-formed Trojans and bodies
captured from the disk during a slow and extended Neptunian migration[1] Murray & Dermott (1999) [2] Sheppard & Trujillo (2006) [3] http://cfa-www.harvard.edu/iau/lists/Trojans.html [4] Horner & Evans (2006) [5] Nesvorny & Dones (2002) [6] Chiang & Lithwick(2005) [7] Hahn & Malhotra (2005) [8] Li et al. (2007) [9] Lykawka & Mukai (2008) [10] Levison et al. (2008) [11] Brunini & Melita (2002) [12] Chambers (1999) [13] Lykawka & Mukai (2007)