Dynamical Characterization of v1298 b-e planetary pair

The planetary system v1298 Tauri can be considered a landmark in our current theories of planetary formation: It harbors four near-resonant super-earths orbiting its host star, whose age is estimated to be 20Myr. Recent analysis of  Transiting Time Variations (TTV) data allowed to precisely characterize the dynamical architecture of the system and to pinpoint it in the early stages of it's formation and evolution, shortly after the dispersal of it's protoplanetary gas disk and after the break-up of the resonant chain 6:4:2:1.

The efforts to characterize the dynamical architecture of v1298 Tauri system allowed for tight constrains on the orbital elements of the three inner planets (c, d and b), but planet e's eccentricities e_e are not as sharply determined as those of the inner planets. Additionaly, its longitude of periapsis ϖ_e display an essentially uniform distribution, whereas it was clearly defined for the other planets of the system.

In this context, we propose to better understand how the different possible apsidal configurations of planet e could impact the dynamical stability and the domains of different regimes of motion in v1298 Tauri planetary system.

For this, we mapped the distinct regimes of motion in the (e_e,ϖ_e) plane, which allowed us to observe apsidal corotation for a range of longitudes and eccentricities. After understanding which ϖ_e give rise to apsidal corotation, we dynamically probed stability of the phase space of planets b and e in the (P,e) plane for selected apsidal configurations by making use of the Spectral Number, which calculates through a Fast Fourier Transform (FFT) how many dominant frequencies dominate the motion of the planets. We also mapped the regions in which the secular and the resonant angles (Δϖ = ϖ_b-ϖ_e and σ_be=2λ_b-1λ_e-ϖ_b,e, respectively) were librating for the b-e planet pair. The stability maps are shown in Figure 1.

Figure 1: Stability maps of the phase space of planet e for an anti-aligned apsidal orientation (left panel) and aligned (right panel). Darker blue regions are considered dynamically stable due to being subject to few identifiable frequencies, whereas yellow ones are considered highly chaotic, with motions composed of a high number of noise-like frequencies in the FFT spectra. Greener regions are related to dynamical structures related to the  separatrix of the Mean Motion Resonance. The red dot represents the mean value of P_e, e_e and its errorbars.

We observed that systems that are initially in apsidal alignment (ϖ_e=ϖ_b) allow for persistent apsidal corotation for larger values of e_e, while shrinking the instability region associated with the 2:1 Mean Motion Resonance (MMR) between planets b and e. On the other hand, in the case of anti-aligned apsis (ϖ_e=ϖ_b-π), the instability strip related to the MMR increases around the MMR location, giving rise to regions where the resonant angle λ¹_be librates rather than circulates. We also observed an extended region in low eccentricity regime where λ¹_be librates for different orientations of ϖ_e, hinting for a low-eccentricity resonant regime.

We further investigated sections of the phase space for fixed values of e_e, in which we applied the Frequency Map Analysis, so as to understand the behaviour of the independent frequencies associated with the MMR and with the secular angle, which allowed us to better understand in detail the dynamical mechanisms acting in this system.

Considering how young this planetary system is, better characterizing the possible regimes of motion it can be subject to is fundamental in pointing possible formation and evolutionary pathways.