Orbital Stability of Closely-Spaced Four-planet Systems

Abstract

We investigate the orbital dynamics of four-planet systems consisting of Earth-mass planets on initially-circular, coplanar orbits around a star of one solar mass. In our simulations, the innermost planet's semimajor axis is set at 1 AU, with subsequent semimajor axes spaced equally in terms of planets' mutual Hill radii. Several sets of initial planetary longitudes are investigated, with integrations continuing for up to $10^{10}$ orbits of the innermost planet, stopping if a pair of planets pass within 0.01 AU of each other or if a planet is ejected from the system. We find that the simulated lifetimes of four-planet systems follow the general trend of increasing exponentially with planetary spacing, as seen by previous studies of closely-spaced planets. Four-planet system lifetimes are intermediate between those of three- and five-planet systems and more similar to the latter. Moreover, as with five-planet systems, but in marked contrast to the three-planet case, no initial semimajor axes spacings are found to yield systems that survive several orders of magnitude longer than other similar spacings. First- and second-order mean-motion resonances (MMRs) between planets correlate with reductions in system lifetimes. Additionally, we find that third-order MMRs between planets on neighboring orbits also have a substantial, though smaller, destabilizing effect on systems very near resonance that otherwise would be very long-lived. Local extrema of system lifetimes as a function of planetary spacing occur at slightly smaller initial orbital separation for systems with planets initially at conjunction relative to those in which the planets begin on widely-separated longitudes. This shift is produced by the asymmetric mutual planetary perturbations as the planets separate in longitude from the initial aligned configuration that cause orbits to spread out in semimajor axis.

Figures (9)

• Figure 1: Initial angular locations for four longitude prescriptions used at $\beta$ = 7.000, corresponding to a period ratio of $P_{i+1} / P_{i} \approx 1.142$. The five-pointed star at the center represents a one solar mass star, the rings trace the initially-circular planetary orbits, and the colored shapes show the initial planetary positions for the four initial sets of planetary longitudes listed in the legend on the right. All planets orbit in the counterclockwise direction.
• Figure 2: The top panel shows simulated lifetimes of four-planet systems with SL09 (grey) and Aligned (blue) initial longitudes with MMRs overlain. The bottom panel displays the simulated lifetimes for the Random (purple), $\Delta_{10}$ (red), and Hexagonal (light blue) systems. Here and elsewhere in this work, the order of the overlaid plots corresponds to the order in the legend, as can be seen here with the grey SL09 points on top of the blue Aligned points. Period ratios between neighboring planets are marked above each panel. All results are plotted with a resolution of 0.001 $R_H$ (mutual Hill radii, see Equation \ref{['eq:beta_def']}) for systems with $\beta \leq 8.5$, above which system lifetimes are plotted at a resolution of 0.01 up to $\beta = 10$ for SL09 longitudes and $\beta = 8.7$ for Aligned longitudes. Likewise, the resolution was 0.001 $R_H$ for Random longitudes up to $\beta = 8.5$, while $\Delta_{10}$ and Hexagonal were completed up to $\beta = 8.2$. Upward pointing triangles represent systems which remained stable at $10^{10}$ years, when the integrations were terminated. The vertical lines correspond to the regions of first-order (solid lines), second-order (dashed lines), and third-order (dotted lines) MMRs of neighboring planets (blue), pairs with a single planet orbiting between (purple), and the innermost planet with the outermost planet (black). Note that some resonances near the two-planet stability limit do not have callout boxes. No clear pattern of reduction was observed in the lifetimes of systems near second-order resonances between the innermost and outermost planets nor third-order resonances between neighboring planets for systems more tightly-spaced than $\beta \approx 8.52$, so these resonances are not displayed for small values of $\beta$. Third-order resonances between neighbors are labeled beginning at $26/23$, with the first clear effect observed at $20/17$.
• Figure 3: Overlain plots of the lifetimes of three- (red dots), four- (gray dots), and five-planet (blue dots) systems with SL09 (Top) and Random (bottom) initial longitudes at a resolution of 0.001 in mutual Hill radii. Red points are plotted first, followed by grey points, and finally blue points. The three-planet lifetimes are from lissauer_gavino_2021. Additionally, five-planet Random lifetimes were retrieved from Obertas. Initial orbital period ratios of neighboring planets are listed on the top axes. The vertical lines correspond to the regions of first-order (solid lines), second-order (dashed lines) and third-order (dotted lines) MMRs of neighboring planets (blue), pairs with one intermediate planet orbiting between (purple), pairs with two intermediate planets orbiting between (black), and pairs with three intermediate planets orbiting between (pink) for the five-planet systems.
• Figure 4: Lifetimes of three-, four-, and five-planet systems with SL09 initial longitudes and orbital separations at a resolution of $10^{-6}$ mutual Hill radii are superimposed in the region corresponding to a 3/2 first-order resonance between the first and fifth planets in five-planet systems. Within this small window, typical lifetimes of the three- and four-planet systems increase roughly linearly with orbital separation. However, there is a significant decrease in system lifetimes in the five-planet systems close to and somewhat narrow of the 3/2 first-order resonance between the first and fifth planets, a resonance not present for systems with fewer planets.
• Figure 5: Lifetimes of three-, four-, and five-planet systems with SL09 longitudes from lissauer_gavino_2021 and this work, respectively are compared against the theoretical system lifetimes of petit2020path. Four- and five-planet systems are plotted at differing resolutions past a mutual Hill radius of 8.5, denoted by a $+$, to demonstrate when 10 gigayears is reached. For the five-planet extension, utilizing random longitudes, the results from Obertas are used.