MNRAS 494, 2045–2052 (2020) doi:10.1093/mnras/staa790 Advance Access publication 2020 April 14 An attempt to constrain Planet Nine’s orbit and position via resonant confinement of distant TNOs Brynna G. Downey 1‹ and Alessandro Morbidelli2 Downloaded from https://academic.oup.com/mnras/article/494/2/2045/5819972 by Royal Library Copenhagen University user on 14 September 2020 1Department of Earth and Planetary Sciences, University of California Santa Cruz, 1156 High Street, Santa Cruz 95064, USA 2Laboratoire Lagrange, Universite Cote d’Azur, Observatoire de la Cote d’Azur, CNRS, CS 34229, F-06304 Nice, France Accepted 2020 March 17. Received 2020 February 13; in original form 2019 August 2 ABSTRACT We considered four TNOs on elongated orbits with small semimajor axis uncertainties: Sedna, 2004 VN112, 2012 VP113, and 2000 CR105. We found two sets of simultaneous near commensurabilities for these objects with a putative Planet Nine that are compatible with the current uncertainties in the objects’ orbital periods. We conducted a large number of numerical simulations of quasi-coplanar simulations (i.e. inclinations of Planet Nine and TNOs set to zero but not the giant planets) to find which values of Planet Nine’s mean anomaly and longitude of perihelion could put these objects in stable mean motion resonance (MMR) librations. We found no cases of simultaneous stable librations for multiple TNOs for more than 800 My, with most librations lasting much shorter than this time-scale. The objects 2004 VN112 and 2000 CR105 are the most unstable. Being in an MMR is not a strict requirement for long-term survival in 3D simulations, so our result cannot be used to refute Planet Nine’s existence. Nevertheless, it casts doubt and shows that theoretical attempts to constrain the position of the planet on the sky are not possible. Key words: celestial mechanics – Kuiper belt: general – planets and satellites: dynamical evolution and stability. studies have tried to refine the predictions of Planet Nine’s orbital 1 INTRODUCTION parameters by analysing how the planet could maintain the TNOs Starting with the discovery of 2012 VP113 by Trujillo & Sheppard in orbital confinement. (2014), an alleged pattern arose in the known distant Trans- The quasi-2D numerical simulations of Batygin & Brown (2016) Neptunian Objects (TNOs) with a > 150 au, in which their (i.e. particles starting with inclination i = 0◦, coplanar with Planet arguments of perihelion (ω) seemed to cluster around ∼0◦.There Nine and only the known giant planets having their real inclinations) are no obvious orbital biases favouring the discovery of objects with and the finite inclination simulations of Batygin et al. (2019) found ω ∼0◦ versus ω ∼ 180◦, so this observation called for a dynamical evidence for long-lasting TNOs on elongated orbits being trapped explanation, which the authors proposed could be the presence of for a long time in mean-motion resonances (MMRs). Using 2D a distant planet. Batygin & Brown (2016) noticed that for orbits simulations, where all objects are coplanar, Batygin & Morbidelli with semimajor axis a > 250 au, the orbital clustering concerns not (2017) showed that all bodies surviving for the age of the Solar only the argument of perihelion but also the longitude of the node system are in MMRs, although some can hop from one resonance () and hence the longitude of perihelion ( ). In other words, the to another during a short phase of instability. In full 3D simulations, orbits are almost aligned with each other in physical space. Using with TNOs at their observed inclinations and Planet Nine at an ◦ ◦ simplified analytic models and numerical simulations, Batygin & inclination ip9 = 20 –30 , the importance of MMRs for long-term Brown (2016) showed that an eccentric ∼10M⊕ perturber at a survival is reduced because close encounters with Planet Nine are distance a > 500 au could generate an orbital clustering of distant less frequent (Becker et al. 2017; Khain et al. 2018). Nevertheless, TNOs over the lifetime of the Solar system and maintain it over several objects spend significant time in resonances, and the final time. The observed orbital clustering would be anti-aligned with semimajor axis distribution of surviving objects clearly shows ◦ the orbit of the planet itself, i.e. − p9 ∼ 180 ,where p9 preference for resonant configurations (Becker et al. 2017). denotes the longitude of perihelion of this putative ninth planet of Malhotra, Volk & Wang (2016) noticed that a Planet Nine with the Solar system. Since Batygin & Brown (2016), several further orbital semimajor axis ap9 ∼ 665 au could put the six extreme TNOs known at the time (with a > 150 au and q > 40 au) near MMRs. They also put constraints on the planet’s eccentricity E-mail: [email protected] to make these resonances stable. Millholland & Laughlin (2017) C 2020 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society 2046 B. G. Downey and A. Morbidelli Table 1. Periods (yrs) and semimajor axes (au) with 1σ uncertainties of considered TNOs from JPL Small-Body Database and semimajor axes from Malhotra et al. (2016), labelled M16, and Millholland & Laughlin (2017), labelled ML17. Object P (JPL) P 1σ (JPL) a (JPL) a 1σ (JPL) a (M16) a 1σ (M16) a (ML17) Downloaded from https://academic.oup.com/mnras/article/494/2/2045/5819972 by Royal Library Copenhagen University user on 14 September 2020 2000 CR105 3307.22 14.44 221.98 0.64626 221.59 0.16 226.1 2012 VP113 4242.42 32.43 262.07 1.3355 265.8 3.3 260.8 2004 VN112 5910.63 47.54 326.91 1.7528 319.6 6.0 317.7 2010 GB174 6586.69 715.78 351.38 25.462 350.7 4.7 369.7 Sedna 11430.27 21.10 507.42 0.62442 506.84 0.51 499.4 Table 2. TNO-Planet Nine MMRs for this work, Malhotra et al. (2016) and Millholland & Laughlin (2017). The first two of the MMR solutions, MMR1 and MMR2 end up being more stable and so are followed more in-depth. There are two 2000 CR105 solutions in MMR3, which adds no complexity since 2000 CR105 was too unstable to give meaningful constraints in our simulations. TNO MMR1 MMR2 MMR3 MMR4 MMR5 MMR6 M16 ML17 a 2000 CR105 9:2 9:1 7:3 or 9:4 3:1 9:2 3:2 5:1 5:1 2012 VP113 7:2 7:1 9:5 7:3 7:2 7:6 4:1 4:1 2004 VN112 5:2 5:1 9:7 5:3 5:2 5:6 3:1 3:1 Sedna 4:3 8:3 2:3 6:7 9:7 3:7 3:2 3:2 ap9 615 au 976 au 387 au 458 au 600 au 288 au 665 au 654 au a Malhotra et al. (2016) deemed 2000 CR105 too far outside the error bounds to really be in the 5:1 MMR resonance with Planet Nine. similarly searched for low-order Planet Nine-TNO resonances and Four TNOs were used with a > 200 au: Sedna (90377), 2004 derived a semimajor axis probability distribution for Planet Nine, as VN112, 2012 VP113, and 2000 CR105 (148209). We did not consider well as probability distributions for the other orbital elements. The 2010 GB174 because its semimajor axis uncertainty of 25 au was most prominent peak of the probability distribution corresponded too high to make it a useful constraint. ◦ ◦ to ap9 ∼ 654 au, ep9 ∼ 0.4–0.5, p9 ∼ 50 , ωp9 ∼ 150 ,andip9 ∼ Following a suggestion by S. Chesley, we used the barycentric 30–40◦. orbital elements of the distant TNOs. The barycentric elements are This paper builds off of the ideas of Malhotra et al. (2016)and more constant over time, while the heliocentric elements vary due Millholland & Laughlin (2017) but with notable differences. Most to the rotation of the Sun relative to the Solar system’s barycenter, importantly, we look for orbital configurations that minimize the mostly associated to the orbital motion of Jupiter. Each TNO’s libration amplitude of the resonant angles, unlike previous works barycentric orbital elements came from the JPL HORIZONS Web- that checked the libration of these angles only a posteriori. Because Interface1 and the element uncertainties from the JPL Small-Body the mean longitudes and longitudes of perihelion of the TNOs are Database Browser2. The semimajor axes and periods are listed in well constrained by observations, whereas their semimajor axes Table 1 along with the barycentric semimajor axes used in Malhotra (and hence orbital periods) have large errors, checking the actual et al. (2016) from their own orbit-fitting software. Millholland & libration of the resonant angles is a much more severe test for Laughlin (2017) used the heliocentric elements from the JPL Small- possible simultaneous resonant configurations than simply looking Body Database, which differed by 2–9 au in semimajor axis from at orbital period ratios. the barycentric elements. The difference in using heliocentric versus This paper is structured as follows. In Section 2, we look at barycentric elements alone can explain the difference in resonances orbital period ratios among the TNOs on elongated orbits to see found between Millholland & Laughlin (2017) and this work. The which resonances with Planet Nine are potentially real. In Section 3, comparison between the JPL Small-Body Database elements and we describe the simulations that we have conducted and how we those from orbit-fitting software highlights the fact that elements for analysed them to find values of the planet’s mean anomaly and long-period bodies in our Solar system vary greatly from data base longitude of perihelion that can put all the TNOs in long-term MMR to data base, with one potentially being outside of the 1σ bounds libration.