GPU accelerated stability maps for extrassolar planetary systems: the Kepler-46 system

Abstract

We develop a code to compute different chaos indicators for the planetary N body problem. The code uses a second order symplectic integrator to advance the Hamiltonian flow of the system in heliocentric canonical coordinates, and the corresponding variational equations. The chaos indicators are computed over a dense grid of initial conditions, allowing to map the stability of the phase space in detail. The code computes the MEGNO indicator and the time variance of the orbital elements, among others. We take advantage of the GPU parallelization with CUDA to simulate the nodes of the grid simultaneously. We apply this code to analyze the stability of the Kepler-46 system of exoplanets. This system is constituted by a Jupiter and a Saturn size planets in tightly packed orbits. The presence of a third, super-Earth size, planet has been also hypothesized. Our analysis helps to reveal the main resonant structure of the system, which shows many similarities with our own solar system. All the planets in the system lie in regions of regular motion, even considering their orbital uncertainties. Our results also put constraints to the possible radial migration of the planets in the system, indicating that the pair of Jovian planets could have never got closer than the location of the 2:3 resonance.

Appendix A

The Hamiltonian Eq. (1) has a drawback when dealing with a few body problem. In a system with only two bodies ($n=1$), it reduces to

$$\begin{aligned} H&=\frac{\left| {\textbf {p}}_1\right| ^2}{2m_1} -\frac{Gm_0 m_1}{\left| {\textbf {r}}_1\right| } + \frac{1}{2m_0}\left| {\textbf {p}}_1\right| ^2 \end{aligned}$$

(A1)

so a two body system is not represented by its corresponding Keplerian Hamiltonian. The symplectic integrator treats the last term in Eq. (A1) as a perturbation of the Keplerian motion, and this introduces a spurious precession of the periastron of the orbit. In a many body system, this effect is negligible compared to the mutual gravitational perturbations (Eq. 3), but for a few body system it may become relevant. A possibility to overcome this consists to incorporate the pure quadratic terms of Eq. (4) into the Keplerian part Eq. (2). The Hamiltonian then takes the form:

$$\begin{aligned} H&=\sum _{i=1}^{n}\nu _i\left( \frac{\left| {\textbf {p}}_i \right| ^2}{2m_i}-\frac{GM_i m_i}{\left| {\textbf {r}}_i \right| }\right) \nonumber \\&\quad -\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}\left( \frac{Gm_i m_j}{\left| {\textbf {r}}_i -{\textbf {r}}_j\right| } -\frac{{\textbf {p}}_i \cdot {\textbf {p}}_j}{m_0}\right) \end{aligned}$$

(A2)

where $\nu _i=(m_0+m_i)/m_0$ and $M_i=m_0/\nu _i$. The second order symplectic integrator in this case reduces to the same steps as before, except that steps 3 and 6 become

$$\begin{aligned} {\textbf {r}}_i \leftarrow {\textbf {r}}_i+\displaystyle \frac{h}{2}\sum _{j\ne i}\frac{{\textbf {p}}_j}{m_0}, \qquad \delta {\textbf {r}}_i \leftarrow \delta {\textbf {r}}_i +\displaystyle \frac{h}{2}\sum _{j\ne i}\frac{\delta {\textbf {p}}_j}{m_0} \end{aligned}$$

and the Keplerian solution in steps 4 and 5 must be advanced over the re-scaled time step $h^\prime =\nu _ih$.

Alternatively, we may introduce the reduced masses $\mu _i=m_0m_i/(m_0+m_i)$ to write the Keplerian part as [28, e.g.]:

$$\begin{aligned} H_{\textrm{Kep}} =\sum _{i=1}^{n}\frac{\left| {\textbf {p}}_i\right| ^2}{2\mu _i}-\frac{G(m_0+m_i) \mu _i}{\left| {\textbf {r}}_i\right| } \end{aligned}$$

where ${\textbf {p}}_i=\mu _i{\textbf {v}}_i^\prime$ and ${\textbf {v}}_i^\prime = \nu _i{\textbf {v}}_i$.