Q: How do planets and systems of planets form ?

Brief Description of the Project

The formation of planets and planetary systems of different types is an extremely complex physical phenomenon. Understanding the processes that govern planet formation requires the use of extensive, high-resolution computer simulations, which translates to building dedicated software and having immense computing power, not always available (or extremely costly to develop). This can be overcome with the use of general-purpose Graphics Processing Units (gp-GPUs, a.k.a. programmable graphics cards). New-generation GPUs can turn a normal PC to a small super-computer at a very low cost.

In this project, we use GPU-based simulations to study planet formation and the dynamical stability of planetary systems. We used our GPU-based platform at the Lab. Of Mechanics, Dept. of Physics, AUTh: 2 Quad-core PCs equipped with 4x and 2x (resp.) Nvidia Tesla-C1060 GPU boards – each with a ~1 Tflops performance for single-precision arithmetic) and developed the necessary software. This has been done with the help of our collaborators (see above).

In the following we present the technical/scientifc achievements of our project, namely:

(a) software that has been developed (link to the code and a short manual is included below),

(b) results from simulation of planet formation and dynamical evolution of systems, and

(c) relevant scientific and educational material (papers/presentations)

Results

Software development

We have build a series of codes, based on the well-established N-body code “SyMBA” (Lee et al. 2001). The new subroutines that we built were integrated in SyMBA and tested, both on our GPU platform as well as on the newer Tesla – K20 GPU platform at the Southwest Research Institute (SwRI, Boulder CO, USA). The main new routines are responsible for:

1. performing the necessary O(N²) force calculations at every step between N massive bodies on a GPU. This has been implemented in different ways:

   - using the SAPPORO library (i.e. using sub-routines that are similar to those needed for GRAPEs): symba_GPU_Sap

   - using the OPEN_ACC protocol : symba_openacc

   - using CUDA/Fortran : symba_cuda

2. performing a nearest-neighbor search (necessary for better resolving close encounters between massive bodies in SyMBA) on the GPU. Again this has been implemented both using CUDA/Fortran and OPEN_ACC

3. calculating the accelerations felt by a massive body (or a small, test-particle) due to a massive gas disk, present in the system. This includes tidal forces (leading to Type I/Type II migration for planets and massive embryos) and drag forces (Stokes and Epstein drag) for smaller particles (which are considered not to be interacting with each other, gravitationally). For the time-being this is included in standard SyMBA and in our Sapporo-based GPU version (symba_drag).

An additional version of symba_GPU_Sap has been developed, in which an additional (distant) stellar perturber on a fixed elliptical orbit is considered. This code was developed with the intention to be used for studying planet formation in well-separated binary stars. It goes without saying that this treatment is an approximation, since the binary orbit remains unaffected by the gravitational perturbations that the planetary embryos exert on the binary. The code has been tested by studying the evolution of a proto-satellite disk around a planet, which itself revolves on a fixed elliptical orbit around a star. An “extra” feature is that the inclination of the orbit can change in time, with respect to the planetary equator, so that we could study the formation and dynamical evolution of satellites (or planets, in the context of a binary star), while the spin axis of the central body is changing in time (e.g. formation of the satellites of Uranus, during the “event” that has tilted the planet's rotation axis by 98 deg, with respect to the orbit normal).

Some benchmarking of the codes is shown below. Note that the GPU-based codes can be ~ 20 times faster in execution that normal SyMBA (in one CPU), as shown in the graphs / tables.

The codes can be found in the following link, together a short user's manual.

Simulations

Here's a collection of some simulations performed using these codes, on different problems in planet formation and dynamical stability of planetary systems. These include, (I) vertical stability of two-planet systems, (II) evolution of a self-gravitating disk and the precession of embryo orbits therein, and (III) formation of Earth-like planets and Giant planet cores in a self-gravitating disk

I) First, let's start with the problem of dynamical stability of an initially nearly co-planar two-planet system and the possibility of forming a “3-D” system. It is known that extrasolar systems exist, containing more than two giant planets; in some of which a resonance relation between the orbital periods of the planets exists (e.g. the outer planet takes twice as long to go around the star as the inner one, i.e. they are in a 2/1 orbital resonance). Previous studies have shown that such a resonance can be established while the system is still forming, as the gas disk forces the planets to migrate towards the star at different rates. Then, when the period ratio becomes commensurate (e.g. 2/1), the planets can enter a gravitational resonance, which tries to maintain itself against migration. This leads to the increase of orbital eccentricities; a well-known phenomenon in planetary dynamics. The question we want to answer is whether it is also possible for very small (less than 0.1 deg) deviations from a co-plane configuration can be enhance by the resonance, so to produce a system that will have a significant mutual inclination. This would mean that the system of planets does not move on the same plane anymore but, rather becomes a '3-D' system. This was shown to happen, in previous works, for specific resonance combinations, but the explanation was not clear.

As was shown before by our collaborators, a stable two-planet system follows a resonant trajectory that belongs to one of the families of resonant periodic orbits, as shown below (these orbits can be found with accuracy, using specialized algorithms). The interesting feature is that some of these orbits can be shown to be vertically unstable, i.e. the system can no longer stay on the original mean plane but has to have a non-zero mutual inclination. What we need to understand is whether gas-driven migration can force the system to move along the family of resonant orbits, starting from a circular and co-planar configuration, cross a vertically critical orbit and then become '3-D'.

A series of simulations were performed (using “symba_drag” and a similar code in C++), where tidal (Type II) migration was included, in the form of a Stokes-type drag (see …), assuming different values of the migration parameters (i.e. migration rate and eccentricity damping rate) and planetary masses. The systems generally behave as shown in the results below

As seen in the graphs, the systems indeed cross a vertical critical orbit (depending on the mass ratio), if the eccentricity manages to grow beyond some value, during the resonance capture. This leads to the formation of 3-D systems! Such systems can exist in nature, if gas disks are not efficient in damping the eccentricities of the planets. The eccentricity of the critical orbit depends on the mass ratio and resonance considered, as shown in the respective graph.

These results have been presented in the “11^th Hellenic Astronomical Conference” in Athens, September 2013 (PDF presentation). A paper has been submitted for publication in the international journal “Celestial Mechanics and Dynamical Astronomy” (PDF).

II) A self-gravitating disk of gas and solid material is the environment in which planets can form. Similarly, a gas-poor disk of solids (of different sizes) is the outcome of planet formation. Such a system will subsequently evolve dynamically, under the effects (primarily) of self-gravity between its constituents (planets, embryos, small planetesimals). It is therefore important to be able to simulate effectively the gravitational interactions of at least tens-hundreds of thousands of bodies. Note that this is still an oversimplification (a “real” system will contain thousands of relatively big objects, but also billions of small bodies), but it's the best we can do so far.

It is known from analytical theories that the orbits of bodies in a massive self-gravitating disk

III)