In a protoplanetary disk around a young star, it is believed that regions with higher gas densities tend to concentrate solid material. If these high gas density regions are sufficiently long-lived, the aggregates of solid material within them can gravitationally collapse to form planets. The presence of a gas giant planet like Jupiter can create such long-lived, high gas density regions; especially around the gas giant planet’s leading (L4) and trailing (L5) Lagrangian points. The high gas density regions around L4 and L5 tend to concentrate solid material in a process known as Lagrangian trapping. A paper by Lyra et al. (2008) show that Lagrangian trapping can lead to gravitational collapse of aggregated solid material and form objects with masses in the regime of terrestrial planets (~ 0.1 to 10 Earth masses).
Figure 1: A contour plot showing the 5 Lagrangian points of the Sun-Earth system. The same layout of the 5 Lagrangian points also applies for the Sun-Jupiter system. The regions around the leading (L4) and trailing (L5) Lagrangian points are “islands of stability”.
As a gas giant planet orbits around its infant star, it clears out a wide gap in the protoplanetary disk. The environment within the gap is depleted of gas and other materials. Nevertheless, the regions around L4 and L5 serve as “islands of stability” where high gas densities are retained. The high gas densities create drag and damp the motion of solid particles. This allows Lagrangian trapping to occur where solid particles become trapped in the high gas density regions around L4 and L5. Simulations were conducted by Lyra et al. (2008) to determine the effectiveness of Lagrangian trapping in forming terrestrial planets with masses ranging from sub-Earth-mass to super-Earth-mass. A planet formed at L4 or L5 is known as a Trojan planet or Trojan Earth (if it has ~ 1 Earth mass). The word “Trojan” comes from the Trojan asteroids that exist around Jupiter’s L4 and L5 Lagrangian points.
Lyra et al. (2008) investigated the formation of Trojan planets by running simulations using a gas giant planet with the mass of Jupiter and solid particles with diameters of 1 cm, 10 cm, 30 cm and 1 m. The 10 cm and 30 cm solid particles readily underwent gravitational collapse at the Lagrangian points. For the 1 cm solid particles, they were so well coupled to the gas that gravity is not the main factor driving their collapse. Instead, the 1 cm solid particles collapse as the high gas density regions around L4 and L5 gradually shrunk. For the large 1 m solid particles, they remained unbound because they were too heavy to be captured by gas drag into the high gas density regions.
Each simulation by Lyra et al. (2008) was run using only one particle size. Gravitational collapse was more efficient for 10 cm solid particles than for 30 cm ones. The 10 cm solid particles collapsed to form a 0.6 Earth-mass planet at L4 and a 2.6 Earth-mass planet at L5. For the case involving 30 cm solid particles, a 0.1 Earth-mass planet was formed at L4, but the solid particles remained unbound at L5. These planet masses are only applicable for the particular set of assumptions used in the simulations by Lyra et al. (2008).
Figure 2: Artist’s conception of an Earth-size planet with vegetation covering most of the planet’s surface from pole-to-pole. Credit: Goran Licanin.
Figure 3: Artist’s conception of a habitable Earth-size planet with a large ice sheet on one of the planet’s poles.
Lagrangian trapping can produce a wide range of planetary masses depending on many conditions such as mass of gas giant planet, density of protoplanetary disk, size distribution of solid particles, etc. As a result, a huge diversity of Trojan planets is expected. Trojan planets formed at the Lagrangian points of a gas giant planet orbiting within the habitable zone of its parent star can potentially be Earth-like and habitable. Such “Trojan Earths” can appear very similar to the Earth. For a gas giant planet orbiting further from its parent star, Trojan planets formed at its Lagrangian points can acquire a significant fraction of icy material if temperatures are cool enough for ices to exist in the protoplanetary disk. Such a Trojan planet is expected to have a surface ice shell overlying a vast global ocean of liquid water.
If Trojan objects with masses in the regime of terrestrial planets can form, it is worth considering why Trojan planets are absent in the Solar System. One reason could be that the initial Trojan objects that formed at Jupiter’s L4 and L5 Lagrangian points were de-stabilized when Jupiter and Saturn crossed the 2:1 mean motion resonance. Following that, Jupiter acquired a new population of Trojan objects. However, without gas drag, the new population of Trojan objects could not assembly into planets and hence left behind what is now observed as the Trojan asteroids at Jupiter’s Lagrangian points. As a result, Trojan planets are probably more common in planetary systems with only one gas giant planet or in planetary systems where the gas giant planets did not undergo resonance crossing.
Lyra et al. (2008), “Standing on the shoulders of giants: Trojan Earths and vortex trapping in low mass self-gravitating protoplanetary disks of gas and solids”, arXiv:0810.3192 [astro-ph]