Two-Dimensional Numerical Simulations

Chapter VI: Feasibility of a resonance-based planet nine search

6.2 Two-Dimensional Numerical Simulations

By now, a considerable number of studies, employing variable levels of approxi- mation aimed at simulating the dynamical evolution induced by Planet Nine, have been published in the literature (Batygin and Brown 2016a, 2016b; Brown and Batygin 2016; Bailey, Batygin, and Brown 2016; Fuente Marcos, Fuente Marcos, and Aarseth 2016; Shankman et al. 2016; Millholland and Laughlin 2017; Lawler et al. 2017; Becker et al. 2017; Batygin and Morbidelli 2017; Hadden et al. 2018).

It has been found, in𝑛-body simulations accounting for the observed inclinations of distant KBOs and the∼20−30 degree inclination of Planet Nine, that objects tend to chaotically skip among commensurabilities. The primary aim of this work is to

0 2 4 0

180 360

φ = 4λ - 3λ₉ - φω φ = 5λ - 17λ₉ + 3ω + 9ω₉

φ = 2λ - 3λ₉ - ω + 2ω₉ φ = 9λ - 10λ₉ - 8ω + 9ω₉

φ

Time [Gyr]

Figure 6.1: Four examples of resonant angles 𝜑 = 𝑗₁𝜆+𝐹 𝜆₉+ 𝑗₃𝜛+ 𝑗₄𝜛₉, for a variety of resonances.

characterize a prior distribution of mean motion resonances in order to ascertain the feasibility of resonance-based constraints on Planet Nine, and such transitional behavior obfuscates the classification of specific MMRs. Thus, we employ a sim- plified, two-dimensional model of the solar system to understand the degree of resonance-based constraints that can be made. If significant resonance-based Planet Nine constraints can be obtained from present observational data, this capability should be best reflected in this highly idealized two-dimensional model.

Within the framework of this two-dimensional model, we confine all objects to the plane and average over the Keplerian motion of the known giant planets. Ac- cordingly, the solar system interior to 30 au is treated as a central mass with a 𝐽₂ gravitational moment having magnitude equivalent to the mean-field contribution of the canonical giant planets to the secular evolution of exterior bodies (Burns 1976; Batygin and Brown 2016a). Hence, Planet Nine is the only massive perturber in these simulations. This model omits various realistic details. Notably, modu- lations in the eccentricity and inclination of KBOs due to close-range interactions with Neptune, as well as dynamics induced by the mutual inclination of the KBOs

green:

perihelion distance q9 1/2 1/1 3/2 2/1

dark blue trajectories: confinement in Δω e9 = 0.1e9 = 0.2e9 = 0.3e9 = 0.4e9 = 0.5e9 = 0.6e9 = 0.7e9 = 0

Semi-major axis of test particles (AU)

0 200 400 600 800 1000

Δω [degrees]

180 360

0

circulation in Δω

Figure 6.2: Trajectories in semimajor axis and longitude of perihelion offset Δ𝜛 for all bod- ies surviving the entire 4-Gyr duration of simu- lations including a 10𝑀 Planet Nine with 𝑎₉ = 600 au. The anti-aligned population (dark blue) is distinguished from other bodies (light blue) by li- bration in Δ𝜛. Further- more, the approximate ra- dius below which con- finement does not occur is typically lower than the perihelion distance𝑞₉ of Planet Nine (green).

Each plot corresponds to the result for a specific eccentricity 𝑒₉ of Planet Nine. Among simula- tions having an eccen- tric Planet Nine, sev- eral low-order resonances are preferentially occu- pied, including the 1/2, 1/1, 3/2, and 2/1 reso- nances. However, pre- dominantly occupied are a variety of high-order resonances.

200

150

100

50

0100 200 300 400 500 600 700 800 900 1000 1100

Semimajor axis [au]

Maximum apsidal libration width

e₉ = 0.1 0.2 0.3 0.4 0.5 0.6 0.7

2/1 5/3

3/2 4/3

Period Ratio P/P₉ = 1/2 3/5 1/1

Tighter confinementLooser confinement

Figure 6.3: The range of semimajor axis and maximum apsidal libration width exhibited in simulations by apsidally confined objects in specific resonances, across simulations featuring a range of Planet Nine eccentricities𝑒₉.

with Planet Nine, are absent from our calculations. Crucially, however, due to the lack of repeated transitions between resonances induced among surviving objects, these simplified simulations allow rigorous identification of the resonances in which objects reside, and their capacity to reveal Planet Nine’s parameters.

We implemented direct𝑛-body simulations using the mercury6 integration package (Chambers 1999), employing the built-in Hybrid symplectic/Bulirsch-Stoer integra- tor (Wisdom and Holman 1992; Press et al. 1992), with time step chosen to be 1/8 the orbital period of Neptune. In the simulations, we evolve an initially axisym- metric disk of eccentric test particles having uniformly random angular distribution and perihelion distance and semi-major axis randomly drawn from the𝑞 ∈ [30,50]

au and 𝑎 ∈ [50,1000] au range, respectively. While the initial distribution of test particles does not reflect the complete evolution of KBOs into resonance with Planet Nine, it serves as a probe of relative strengths of resonances. For each of the eight values of𝑒₉tested, 6000 such test particles were randomly initialized and simulated.

Particles attaining radial distances𝑟 <10,000 au or𝑟 <30 au were removed.

In principle, the relative strengths of resonances are not expected to vary signifi- cantly with𝑎₉, as the relative strengths of individual terms associated with specific resonances in the usual expansion of the disturbing function only depend on the semi- major axis ratio of the interacting bodies (Murray and Dermott 1999b). Therefore, Planet Nine was assigned a single, nominal semimajor axis,𝑎₉=600 au. However, we note that different semimajor axes of𝑎₉would, in reality, subject the innermost

resonances to variable levels of secular coupling with the canonical giant planets, altering the resonant widths slightly. Still, 𝑎₉ =600 au is roughly in keeping with the semimajor axis predictions of Millholland and Laughlin 2017; Malhotra, Volk, and Wang 2016; thus, we choose this value of𝑎₉. Moreover, eccentricities𝑒₉were tested ranging from 0 to 0.7 in increments of 0.1. The simulations in this work span 4 Gyr in approximate accordance with the solar system’s lifetime.

Because this work addresses the distribution of closely-spaced, high-order mean motion resonances, which have finite width in semimajor axis, the period ratio alone is insufficient to confirm a specific resonance. Instead, we confirmed specific mean- motion resonances with Planet Nine among the surviving objects by searching for a librating resonant argument (Figure 6.1). The general form of such a resonant argument can be stated as 𝜑 = 𝑗₁𝜆+ 𝑗₂𝜆₉+ 𝑗₃𝜛+ 𝑗₄𝜛₉, where the d’Alembert relation, following from rotational symmetry, restricts the integer coefficients 𝑗 to satisfyÍ4

𝑖=1 𝑗_𝑖 =0 (Murray and Dermott 1999b). (For the 2-dimensional case, we adopt the standard convention that longitude of ascending nodeΩ =0, thus𝜛=𝜔.) Based on the values found for coefficients 𝑗₁ and 𝑗₂, identification of a critical argument informs the individual resonance in which a particle resides. Specifically, for an object in 𝑝/𝑞 resonance with Planet Nine, the resonant argument takes the form𝜑 =𝑞𝜆− 𝑝𝜆₉+ 𝑗₃𝜛+ 𝑗₄𝜛₉.