Memories of past close encounters in extreme trans-Neptunian space: Finding unseen planets using pure random searches
AAstronomy & Astrophysics manuscript no. memories © ESO 2021February 5, 2021 L etter to the E ditor Memories of past close encounters in extreme trans-Neptunianspace: Finding unseen planets using pure random searches
C. de la Fuente Marcos and R. de la Fuente Marcos Universidad Complutense de Madrid, Ciudad Universitaria, E-28040 Madrid, Spain AEGORA Research Group, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Ciudad Universitaria, E-28040Madrid, SpainReceived 11 January 2021 / Accepted 3 February 2021
ABSTRACT
Context.
The paths followed by the known extreme trans-Neptunian objects (ETNOs) e ff ectively avoid direct gravitational perturba-tions from the four giant planets, yet their orbital eccentricities are in the range between 0.69–0.97. Solar system dynamics studiesshow that such high values of the eccentricity can be produced via close encounters or secular perturbations. In both cases, thepresence of yet-to-be-discovered trans-Plutonian planets is required. Recent observational evidence cannot exclude the existence, at600 AU from the Sun, of a planet of five Earth masses. Aims.
If the high eccentricities of the known ETNOs are the result of relatively recent close encounters with putative planets, themutual nodal distances of sizeable groups of ETNOs with their assumed perturber may still be small enough to be identifiablegeometrically. In order to confirm or reject this possibility, we used Monte Carlo random search techniques.
Methods.
Two arbitrary orbits may lead to close encounters when their mutual nodal distance is su ffi ciently small. We generatedbillions of random planetary orbits with parameters within the relevant ranges and computed the mutual nodal distances with a setof randomly generated orbits with parameters consistent with those of the known ETNOs and their uncertainties. We monitoredwhich planetary orbits had the maximum number of potential close encounters with synthetic ETNOs and we studied the resultingdistributions. Results.
We provide narrow ranges for the orbital parameters of putative planets that may have experienced orbit-changing encounterswith known ETNOs. Some sections of the available orbital parameter space are strongly disfavored by our analysis.
Conclusions.
Our calculations suggest that more than one perturber is required if scattering is the main source of orbital modificationfor the known ETNOs. Perturbers might not be located farther than 600 AU and they have to follow moderately eccentric and inclinedorbits to be capable of experiencing close encounters with multiple known ETNOs.
Key words. methods: data analysis – methods: numerical – celestial mechanics – planets and satellites: detection – minor planets,asteroids: general – Kuiper belt: general
1. Introduction
Extreme trans-Neptunian objects (ETNOs) serve as uniqueprobes into the gravity perturbations shaping the outer solar sys-tem beyond the classical trans-Neptunian or Kuiper belt (seee.g., Kaib et al. 2019). The trajectories followed by the knownETNOs e ff ectively avoid direct gravitational perturbations fromthe four giant planets, yet their orbital eccentricities are in therange 0.69–0.97. E ffi cient drivers for the eccentricity excita-tion of small bodies include close encounters with planets (seee.g., Carusi et al. 1990) and the von Zeipel-Lidov-Kozai mecha-nism (von Zeipel 1910; Kozai 1962; Lidov 1962; Ito & Ohtsuka2019). In both cases above and in the case of ETNOs, the pres-ence of yet-to-be-discovered trans-Plutonian planets is required.Fienga et al. (2020) used the INPOP19a planetaryephemerides that include Jupiter-updated positions by the Junomission and a reanalysis of Cassini observations to show thatthere is no clear evidence for the existence of the so-calledPlanet 9 predicted by Batygin & Brown (2016) as an explana-tion for the orbital architecture of the known ETNOs. However,Fienga et al. (2020) concluded that if Planet 9 exists, it cannot Send o ff print requests to : C. de la Fuente Marcos, e-mail: [email protected] be closer than 500 AU, if it has a mass of 5 M ⊕ , and no closerthan 650 AU, if it has a mass of 10 M ⊕ . The latest version ofthe Planet 9 hypothesis (Batygin et al. 2019) predicts the exis-tence of a planet with a mass in the range 5–10 M ⊕ , followingan orbit with a value of the semi-major axis in the range of 400–800 AU, eccentricity in the range of 0.2–0.5, and inclination inthe interval between (15 ◦ , 25 ◦ ). A number of exoplanets havealready been observed orbiting at hundreds of AUs from theirhost stars (see e.g., Bailey et al. 2014; Naud et al. 2014; Nguyenet al. 2021) and theoretical calculations have confirmed plausi-ble pathways for their formation (see e.g., Kenyon & Bromley2015, 2016).If the high eccentricities of known ETNOs are the result ofrelatively recent close encounters with putative planets, the mu-tual nodal distances of sizeable groups of ETNOs with their as-sumed perturber may still be small enough to be identifiable geo-metrically. Here, we use Monte Carlo random search techniquesto identify orbits that may lead to the maximum number of po-tential close encounters with synthetic ETNOs whose orbital pa-rameters are consistent with those of the real ETNOs and theiruncertainties. This letter is organized as follows. In Sect. 2, wereview our methodology and present the data used in our analy- Article number, page 1 of 12 a r X i v : . [ a s t r o - ph . E P ] F e b & A proofs: manuscript no. memories ses. In Sect. 3, we apply our methodology and discuss its results.Our conclusions are summarized in Sect. 4.
2. Methods and data description
This work explores a "what if" scenario in which the starting hy-pothesis states that a sizeable number of known ETNOs have ex-perienced relatively recent close encounters with putative plan-ets; the timescale comes constrained by their orbital periods thatare in the range 1867–50116 yr, so encounters may have takenplace during the last 10 –10 yr. In this work, therefore, we aretesting the hypothesis statistically. If this hypothesis is plausible,a statistically significant number of compatible planetary per-turber orbits should emerge from the analysis of a very largesample of orbits. If the distributions of one or more of the or-bital parameters of the candidate are flat, the starting hypothesismust be rejected as this would show that there is no favored or-bital solution for the perturber; conversely, if all the distributionsproduce consistent intervals that are statistically significant, theplausibility of the starting hypothesis can be considered as con-firmed. Plausibility concerns the likelihood of acceptance, notthe likelihood of being true or better than competing scenarios.The problem under investigation here is equivalent to a nondif-ferenciable optimization that is well-suited for a uniform MonteCarlo random search (Metropolis & Ulam 1949). Our method-ology brings together geometry and statistics in our attempt tofind the confocal ellipse that passes the closest to the maximumnumber of known orbits of a certain dynamical class; this ap-proach is fundamentally di ff erent from those involving N -bodycalculations and statistics (see e.g., de la Fuente Marcos et al.2016; de León et al. 2017; de la Fuente Marcos et al. 2017).Kalinicheva & Chernetenko (2020) recently applied a geometricapproach within the context of the Planet 9 hypothesis. Two arbitrary orbits may experience close encounters when theirmutual nodal distance is su ffi ciently small. Recurrent (or evensingle) encounters within 1 Hill radius (Hamilton & Burns 1992)of a massive body may change the orbit of a small body signifi-cantly. The mutual nodal distance between the orbits of a smallbody (an ETNO in our case) and an arbitrary planet can be com-puted as described in Appendix A. Orbits are defined by the val-ues of semi-major axis, a (that controls size), eccentricity, e (thatcontrols shape), and those of the angular elements — inclination, i , longitude of the ascending node, Ω , and argument of perihe-lion, ω — that control the orientation in space of the orbit; theperihelion distance or pericenter, q , is given by the expression q = a (1 − e ). The actual position of an object in its orbit is con-troled by a fourth angle, the mean anomaly, M . Our geometricapproach leaves this angle out of the analysis and, therefore, it isnot capable of predicting the current location of the perturber, ifit is, in fact, real.We generated 2 × random planetary orbits with uni-formly distributed relevant parameters: Ω p and ω p ∈ (0 ◦ , ◦ ), i p ∈ (0 ◦ , ◦ ), e p ∈ (0 , . , and q p ∈ ( x , − x ) AU, with x = , , , and 600 AU so a p = q p / (1 − e p ). For eachrandom planetary orbit, we computed the mutual nodal distancesbetween a set of synthetic ETNOs and the planet. Each set ofsynthetic ETNOs was generated using the mean values and stan-dard deviations of the orbit determinations of the known ETNOsas pointed out in Appendix B.For each combination of random planetary orbit and syn-thetic (but compatible with the observations) set of ETNOs, we have counted how many synthetic ETNOs had at least one mu-tual nodal distance with the planet under 5 AU (for x =
300 AU),7.5 AU (for x =
400 AU), and 10 AU (for x =
500 and 600 AU).We then proceeded to record the random planetary orbit if thecount was ≥
5, in order to maximize the number of potentialclose approaches between planet and set of ETNOs. We thenstudied the resulting distributions. In order to analyze the results,we produced histograms using the Matplotlib library (Hunter2007) with sets of bins computed using NumPy (van der Waltet al. 2011) by applying the Freedman and Diaconis rule (Freed-man & Diaconis 1981). Instead of using frequency-based his-tograms, we considered counts to form a probability density sothe area under the histogram will sum to one.The nodal distance separation criteria for selective countingare not arbitrary but motivated by the results in Fienga et al.(2020), a 2 M ⊕ has a Hill radius of 4.5 AU (if a p =
400 AU and e p = . M ⊕ has a Hill radius of 9.2 AU (if a p =
542 AUand e p = . Here, we work with publicly available data from Jet PropulsionLaboratory’s (JPL) Small-Body Database (SBDB) and HORI-ZONS on-line solar system data and ephemeris computationservice, both provided by the Solar System Dynamics Group(Giorgini 2015). Assuming the definition in Trujillo & Sheppard(2014), that the ETNOs have q >
30 AU and a >
150 AU,the sample of known ETNOs now includes 39 objects with re-liable orbits (see Appendix B) whose data have been retrievedfrom JPL’s SBDB and HORIZONS using tools provided by thePython package Astroquery (Ginsburg et al. 2019). In the fol-lowing section, we use barycentric elements because within thecontext of the ETNOs, barycentric orbit determinations betteraccount for their changing nature as Jupiter follows its 12 yr orbitaround the Sun; Appendix C shows results based on heliocentricorbits that are consistent with those of the barycentric ones.
3. Results and discussion
As pointed out above, the motivation behind this study is thebelief that a fossil record of planetary encounters might be pre-served in the distribution of the orbital elements of ETNOs.With this hypothesis in mind, we monitored which planetaryorbits had the maximum number of potential close encounterswith synthetic ETNOs and analyzed the resulting distributionsthat are shown in Fig. 1. The first (leftmost) column of panelsshows the results based on the assumption that q p >
300 AU,including 20304 orbits with a number of potential close ap-proaches in the range between 5–7; the next one, shows resultsfor q p >
400 AU also with a range of potential close approachesof 5–7 for 5671 orbits; the following column of panels displaysresults for q p >
500 AU with a range of 5–6 for 2635 orbits; theright column shows panels with results for q p >
600 AU and thenumber of potential close approaches between planet and set ofETNOs is just 5 for 56 orbits. From these values and the plots, itis increasingly di ffi cult to find consistent perturbers as we movefarther away from the Sun.Table 1 shows a summary of the central values and disper-sions of the orbital parameters of the sample of orbits that mayexperience close encounters with multiple known ETNOs. We https: // ssd.jpl.nasa.gov / sbdb.cgi https: // ssd.jpl.nasa.gov / ?horizonsArticle number, page 2 of 12. de la Fuente Marcos , R. de la Fuente Marcos: Close encounters in extreme trans-Neptunian space
400 600 800 1000Semi-major axis (AU)0.0000.0020.0040.0060.008 P r ob a b ilit y d e n s it y
400 600 800 1000Semi-major axis (AU)0.0000.0020.0040.0060.008 P r ob a b ilit y d e n s it y
600 800 1000Semi-major axis (AU)0.0000.0050.0100.015 P r ob a b ilit y d e n s it y
600 800 1000Semi-major axis (AU)0.0000.0010.0020.003 P r ob a b ilit y d e n s it y P r ob a b ilit y d e n s it y P r ob a b ilit y d e n s it y P r ob a b ilit y d e n s it y P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y
20 40 60 80Inclination ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y
100 200Longitude of the ascending node ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y
100 200 300Argument of perihelion ( o ) P r ob a b ilit y d e n s it y Fig. 1.
Barycentric orbital elements of putative perturber planets. Distributions of barycentric orbital elements of planetary orbits that may resultin close encounters (under 5 AU for q p >
300 AU, under 7.5 AU for q p >
400 AU, and under 10 AU for q p >
500 AU and q p >
600 AU) withfive or more present-day extreme trans-Neptunian objects (ETNOs). Each column of panels shows the cumulative results of 2 × experiments.From left to right, the panels show the results of imposing q p >
300 AU (left column, 20304 orbits), q p >
400 AU (5671 orbits), q p >
500 AU(2635 orbits), and q p >
600 AU (right column, 56 orbits). Median values are shown as vertical blue lines. Based on the available data on knownETNOs, the presence of massive perturbers well beyond 600 AU is strongly excluded within the context of the hypotheses centered in this study. consider that results for q p >
300 AU and q p >
400 AU are sta-tistically consistent and are compatible with a p ∈ (331 , e p < . i p ∈ (10 ◦ , ◦ ), Ω p ∼ ◦ , and ω p ∈ ( − ◦ , ◦ ). Thevalue of a p is probably ∼
400 AU but the uncertainty is signifi-cant, the value of the eccentricity is well constrained and it hasto be low, the inclination is perhaps the best constrained valueand it is ∼ ◦ , the value of Ω p is very likely ∼ ◦ , but the valueof ω p is poorly constrained, perhaps ∼ ◦ . If scattering is themain source of orbital modification for the group of ETNOs thatmainly move at (300, 400) AU from the Sun, the orbit of the pu-tative perturber is relatively well-constrained and according toFienga et al. (2020), it must have a mass < M ⊕ .Table 1 shows that our approach is far less conclusive inthe cases of q p >
500 AU and q p >
600 AU as these valuesproduce distributions of the orbital parameters that may not bestatistically compatible. In any case, we must emphasize that >
50% of the known ETNOs cannot interact directly with aperturber with q p >
500 AU because their aphelion distances, Q = a (1 + e ), are below 500 AU: 9 ETNOs have Q <
300 AU,18 have Q <
400 AU, and 23 have Q <
500 AU. Therefore,the distribution in Q and our calculations suggest that more thanone perturber is required if scattering is the main source of or-bital modification for the known ETNOs. Perturbers might notbe located farther than 600 AU and they have to follow moder-ately eccentric and inclined orbits to be capable of experiencingpresent-day close encounters with multiple known ETNOs.At this point, it can be argued that the results in Fig. 1 maybe a ff ected by a statistical artifact. In order to test the statisticalsignificance of our results, we repeated the Monte Carlo randomsearch on an input sample of scrambled data as explained in Ap-pendix D. By randomly reassigning the values of the orbital ele-ments of the ETNOs, we preserve the original distributions of the Article number, page 3 of 12 & A proofs: manuscript no. memories
Table 1.
Summary of central values and dispersions of optimal barycentric orbits.
Orbital parameter q p >
300 AU q p >
400 AU q p >
500 AU q p >
600 AUSemi-major axis, a p (AU) 375 + − (376) 489 + − (420) 569 + − (581) 762 + − (680)Eccentricity, e p + . − . (0.01) 0.10 + . − . (0.01) 0.05 + . − . (0.02) 0.16 + . − . (0.07)Inclination, i p ( ◦ ) 20 + − (14) 24 + − (14) 14 + − (12) 21 + − (24)Longitude of the ascending node, Ω p ( ◦ ) 167 + − (180) 157 + − (161) 149 + − (148) 137 + − (156)Argument of perihelion, ω p ( ◦ ) 135 + − (51) 124 + − (12) 234 + − (275) 164 + − (56) Notes.
Median values and 16th and 84th percentiles (absolute maximum in parentheses) from the Monte Carlo random searches whose distributionsare shown in Fig. 1. parameters, but destroy any possible correlations that may havebeen induced by close encounters with massive perturbers (or thee ff ects of hypothetical mean-motion or secular resonances). Theresults of this significance test are shown in Fig. D.1: the dis-tribution of i p becomes flat and almost the same happens to thedistribution of Ω p . In other words, for the scrambled data, it isnot possible to find a statistically significant orbital solution thatis compatible with the starting hypothesis and we conclude thatthe results in Fig. 1 are unlikely to be due to statistical artifacts.
4. Summary and conclusions
When considering the well-studied case of Neptune and the reg-ular trans-Neptunian objects, we observe that these objects arenot part of a single population, but they are organized into sev-eral dynamical classes. Some objects never experience close en-counters with Neptune due to the existence of protection mecha-nisms such as mean-motion or secular resonances, as in the caseof Neptune’s Trojans or Pluto (see e.g., Milani et al. 1989; Wanet al. 2001), others undergo close encounters that may send themtowards the inner solar system (centaurs) or outwards to becomescattered objects (see the recent review by Saillenfest 2020). Iftrans-Plutonian planets exist, their perturbations may shape theextreme trans-Neptunian space in a similar fashion and the cur-rent sample of ETNOs may include the signatures of their pres-ence (see e.g. Saillenfest et al. 2017).Our results show that it is possible to find suitable orbits forwhich the mutual nodal distances of sizeable ( ≥
5, comprising atleast 13% of the sample) groups of ETNOs with their assumedperturber could be small enough for a close encounter to occur,at least in theory (assuming that no protection mechanisms, suchas mean-motion or secular resonances, are in place to avoid theflyby). This was our original aim. In addition, the results pre-sented in Sect. 3 clearly indicate that the most probable planetaryorbit consistent with the starting hypothesis is the one obtainedfor the experiment with q p >
300 AU. The number of consis-tent orbits in this case is 3.6 times higher than the one found for q p >
400 AU. Our results seem to be incompatible with thoseattributed to a statistical artifact (see Appendix D).Prior to the announcement of the Planet 9 hypothesis (Baty-gin & Brown 2016), Trujillo & Sheppard (2014) had already ar-gued for the existence of a planet at 250 AU within the contextof the ETNOs — driving von Zeipel-Lidov-Kozai librations on2012 VP — and de la Fuente Marcos & de la Fuente Marcos(2014) suggested that the limited data available at the time weremore compatible with the presence of two massive perturbers,one of them close to 300 AU. These massive perturbers wereinitially proposed based on data corresponding to a sample of 13objects, whereas the current sample has tripled this number. Ifwe repeat the experiment discussed in Sect. 3 for q p >
200 AU(see Appendix E), we find that the number of consistent orbits, although larger than the one generated in the experiment for q p >
400 AU, is still lower than that of the most statisticallysignificant case, namely, 8234 versus 20304 for q p >
300 AU.The most likely orbit is still similar, in terms of shape and ori-entation in space, to the most probable one in the q p >
300 AUcase.Our results are consistent with the conclusions of the studypresented in de la Fuente Marcos & de la Fuente Marcos (2017).Although our approach has not been able to single out a statisti-cally significant, unique planetary orbit that may be responsiblefor the orbital architecture observed in extreme trans-Neptunianspace, we provide narrow ranges for the orbital parameters ofputative planets that may have experienced orbit-changing en-counters with known ETNOs. Some sections of the available or-bital parameter space are strongly excluded by the findings ofour analysis.
Acknowledgements.
We thank the anonymous referee for a constructive andtimely report, S. J. Aarseth, J. de León, J. Licandro, A. Cabrera-Lavers, J.-M.Petit, M. T. Bannister, D. P. Whitmire, G. Carraro, E. Costa, D. Fabrycky, A. V.Tutukov, S. Mashchenko, S. Deen and J. Higley for comments on ETNOs andA. I. Gómez de Castro for providing access to computing facilities. This workwas partially supported by the Spanish ‘Ministerio de Economía y Competitivi-dad’ (MINECO) under grant ESP2017-87813-R. In preparation of this letter, wemade use of the NASA Astrophysics Data System and the MPC data server.
References
Bailey, V., Meshkat, T., Reiter, M., et al. 2014, ApJ, 780, L4Batygin, K. & Brown, M. E. 2016, AJ, 151, 22Batygin, K., Adams, F. C., Brown, M. E., et al. 2019, Phys. Rep., 805, 1Carusi, A., Valsecchi, G. B., & Greenberg, R. 1990, Celestial Mechanics andDynamical Astronomy, 49, 111de la Fuente Marcos, C. & de la Fuente Marcos, R. 2014, MNRAS, 443, L59de la Fuente Marcos, C. & de la Fuente Marcos, R. 2017, MNRAS, 471, L61de la Fuente Marcos, C., de la Fuente Marcos, R., & Aarseth, S. J. 2016, MN-RAS, 460, L123de la Fuente Marcos, C., de la Fuente Marcos, R., & Aarseth, S. J. 2017, Ap&SS,362, 198de León, J., de la Fuente Marcos, C., & de la Fuente Marcos, R. 2017, MNRAS,467, L66Fienga, A., Di Ruscio, A., Bernus, L., et al. 2020, A&A, 640, A6Freedman, D. & Diaconis, P. 1981, Probability Theory and Related Fields, 57,453Ginsburg, A., Sip˝ocz, B. M., Brasseur, C. E., et al. 2019, AJ, 157, 98Giorgini, J. D. 2015, IAUGA, 22, 2256293Hamilton, D. P. & Burns, J. A. 1992, Icarus, 96, 43Hunter, J. D. 2007, Computing in Science and Engineering, 9, 90Ito, T. & Ohtsuka, K. 2019, Monographs on Environment, Earth and Planets, 7,1Kaib, N. A., Pike, R., Lawler, S., et al. 2019, AJ, 158, 43Kalinicheva, O. V. & Chernetenko, Y. A. 2020, Astrophysical Bulletin, 75, 459Kenyon, S. J. & Bromley, B. C. 2015, ApJ, 806, 42Kenyon, S. J. & Bromley, B. C. 2016, ApJ, 825, 33Kozai, Y. 1962, AJ, 67, 591Lidov, M. L. 1962, Planet. Space Sci., 9, 719Metropolis, N. & Ulam, S. 1949, J. Am. Stat. Assoc., 44, 335Milani, A., Nobili, A. M., & Carpino, M. 1989, Icarus, 82, 200
Article number, page 4 of 12. de la Fuente Marcos , R. de la Fuente Marcos: Close encounters in extreme trans-Neptunian space
Naud, M.-E., Artigau, É., Malo, L., et al. 2014, ApJ, 787, 5Nguyen, M. M., De Rosa, R. J., & Kalas, P. 2021, AJ, 161, 22Saillenfest, M., Fouchard, M., Tommei, G., et al. 2017, Celestial Mechanics andDynamical Astronomy, 129, 329Saillenfest, M. 2020, Celestial Mechanics and Dynamical Astronomy, 132, 12Trujillo, C. A. & Sheppard, S. S. 2014, Nature, 507, 471van der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, Computing in Scienceand Engineering, 13, 22von Zeipel, H. 1910, Astronomische Nachrichten, 183, 345Wan, X.-S., Huang, T.-Y., & Innanen, K. A. 2001, AJ, 121, 1155
Article number, page 5 of 12 & A proofs: manuscript no. memories
Appendix A: Mutual nodal distance: formulae
Our statistical analyses are based on studying the distribution ofnodal distances between two Keplerian trajectories (one ETNOand one hypothetical planet) with a common focus; therefore,the core of our approach is purely geometrical and gravitationalinteractions are not directly taken into account. The mutual nodaldistance between the orbits of a small body (an ETNO in ourcase) and an arbitrary planet can be written as (see Eqs. 16 and17 in Saillenfest et al. 2017): ∆ d pn = a (1 − e )1 ± e cos (cid:36) − a p (1 − e )1 ± e p cos (cid:36) p , (A.1)where cos (cid:36) = cos ω (sin i cos i p − cos i sin i p cos ∆Ω p ) + sin ω sin i p sin ∆Ω p √ − (cos i cos i p + sin i sin i p cos ∆Ω p ) , (A.2) cos (cid:36) p = − cos ω p (sin i p cos i − cos i p sin i cos ∆Ω p ) + sin ω p sin i sin ∆Ω p √ − (cos i cos i p + sin i sin i p cos ∆Ω p ) , (A.3) ∆Ω p = Ω − Ω p , and a , e , i , Ω and ω are the orbital elements ofthe small body, and a p , e p , i p , Ω p , and ω p those of the arbitraryplanet. For each random search experiment (each analysis con-sists of 2 × ), we compute the orbital elements of the putativeperturber using the expressions: q p = x + (1000 − x ) r e p = . r a p = q p / (1 − e p ) i p = r Ω p = r ω p = r , (A.4)where x = , , , and 600 AU (see Sect. 2, or 200 AUfor Appendix E), and r j with j = ,
5, are random numbers inthe interval (0, 1) with a uniform distribution.
Appendix B: Extreme trans-Neptunian objects: Data
The barycentric orbit determinations used as input data in theuniform Monte Carlo random search discussed in Sect. 3 areshown in Table B.1. The data are referred to epoch 2459000.5Barycentric Dynamical Time (TDB) and they have been re-trieved (as of 11-January-2021) from JPL’s SBDB and HORI-ZONS using tools provided by the Python package Astroquery(Ginsburg et al. 2019).
Article number, page 6 of 12. de la Fuente Marcos , R. de la Fuente Marcos: Close encounters in extreme trans-Neptunian space T a b l e B . . B a r y ce n t r i c o r b it a l e l e m e n t s a nd1 σ un ce r t a i n ti e s o f kno w n ET NO s w it h r obu s t o r b it d e t e r m i n a ti on s . O b j ec t a b σ a e b σ e i b σ i Ω b σ Ω ω b σ ω ( AU )( AU )( ◦ )( ◦ )( ◦ )( ◦ )( ◦ )( ◦ ) ( FP ) . . . . × − . . × − . . × − . . S e dn a . . . . × − . . × − . . . . ( CR ) . . . . . . . . . . ( V Z ) . . . . × − . . × − . . . . ( VN ) . . . . . . . . . . ( G P ) . . . . . . . . . . ( U T ) . . . . . . . . . . ( H B ) . . . . . . . . . . ( S O ) . . . . × − . . × − . . × − . . ( T G ) . . . . × − . . × − . . . . ( V J ) . . . . × − . . × − . . × − . . L e l ea kuhonu a . . . . . . . . . . G B . . . . . . . . . . SS . . . . . . . . . .
173 2005 R H . . . . . . . . . . G B . . . . . . . . . . V P . . . . . . . . . . FS . . . . . . . . . . F T . . . . . . . . . . R F . . . . . . . . . . R A . . . . . . . . . . S Y . . . . . . . . . . S L . . . . . . × − . . . . UH . . . . . . . . . . F E . . . . . . . . . . S R . . . . . . . . . . W B . . . . . . . . . . B P . . . . . . × − . . . . G T . . . . . . . . . . KG . . . . . . . . . . KH . . . . . . . . . . R X . . . . . . . . . . R Y . . . . . . . . . . GA . . . . . . . . . . QU . . . . . . . . . . QV . . . . . . . . . . S G . . . . . . × − . . . . T P . . . . . . . . . . V M . . . . . . . . . . N o t e s . T h e o r b it d e t e r m i n a ti on s h a v e b ee n c o m pu t e d a t e po c h J D . t h a t c o rr e s pond s t o00 : : . T D B on2020 M a y31 ( J . ec li p ti ca nd e qu i nox ) . I npu t d a t a s ou r ce : J P L ’ s S B D B . Article number, page 7 of 12 & A proofs: manuscript no. memories
The orbits of the synthetic ETNOs are obtained using theexpressions: a s = a b + σ a R e s = e b + σ e R i s = i b + σ i R Ω s = Ω b + σ Ω R ω s = ω b + σ ω R , (B.1)where R j with j = ,
5, are univariate Gaussian random numbers.
Appendix C: Heliocentric orbit determinations:results
If we repeat the calculations discussed in Sect. 3 using helio-centric orbit determinations instead of barycentric ones as inputdata, we obtain the distributions in Fig. C.1 with central valuesand dispersions summarized in Table C.1.
Appendix D: Statistical significance
In order to verify that our results do not come from statisticalartifacts, we randomly scramble the orbit parameters data usedas input and repeat the uniform Monte Carlo random searchesdiscussed in Sect. 3 and Appendix C. In these experiments, theset of synthetic ETNOs is such that, for example, the first ob-ject may have the value of a of object e of i of Ω of ω of i p and Ω p are flattened and no statistically significant perturbing orbitsare produced. Appendix E: Results at 200 AU
Fienga et al. (2020) focused on testing for the presence of pos-sible planets at 400, 500, 600, 650, 700, 750, and 800 AU withmasses of 5 or 10 M ⊕ . The existence of 5 M ⊕ planets at 400 or500 AU is strongly disfavored by their results (see their Fig. 5,top panels). However, a hypothetical Earth-like planet at 200–400 AU from the Sun may still induce significant gravitationale ff ects if close encounters are possible, due to its relatively largevalue for the Hill radius (e.g., 2.2 AU if a p =
200 AU, e p = . M ⊕ ). We repeated the analysis, imposing q p >
200 AU,and we obtained 8234 orbits with a number of potential closeapproaches in the range between 5–7. Here, we count howmany synthetic ETNOs had at least one mutual nodal distancewith the planet under 2 AU. The median values and 16th and84th percentiles (absolute maximum in parentheses) from theMonte Carlo random searches whose distributions are shown inFig. E.1 are: a p = + − AU (258 AU), e p = . + . − . (0.03), i p = ◦ + ◦ − ◦ (10 ◦ ), Ω p = ◦ + ◦ − ◦ (196 ◦ ), and ω p = ◦ + ◦ − ◦ (350 ◦ ). Article number, page 8 of 12. de la Fuente Marcos , R. de la Fuente Marcos: Close encounters in extreme trans-Neptunian space
400 600 800 1000Semi-major axis (AU)0.0000.0020.0040.0060.008 P r ob a b ilit y d e n s it y
400 600 800 1000Semi-major axis (AU)0.0000.0020.0040.0060.008 P r ob a b ilit y d e n s it y
600 800 1000Semi-major axis (AU)0.0000.0050.010 P r ob a b ilit y d e n s it y
600 800 1000Semi-major axis (AU)0.0000.0020.0040.006 P r ob a b ilit y d e n s it y P r ob a b ilit y d e n s it y P r ob a b ilit y d e n s it y P r ob a b ilit y d e n s it y P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y
20 40 60 80Inclination ( o ) P r ob a b ilit y d e n s it y
20 40 60 80Inclination ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y
50 100 150 200 250Longitude of the ascending node ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y Fig. C.1.
Heliocentric orbital elements of putative perturber planets. Distributions of heliocentric orbital elements of planetary orbits that mayresult in close encounters (under 5 AU for q p >
300 AU, under 7.5 AU for q p >
400 AU, and under 10 AU for q p >
500 AU and q p >
600 AU) withfive or more present-day extreme trans-Neptunian objects (ETNOs). Each column of panels shows the cumulative results of 2 × experiments.From left to right, the panels show the results of imposing q p >
300 AU (left column, 18369 orbits), q p >
400 AU (6594 orbits), q p >
500 AU(3567 orbits), and q p >
600 AU (right column, 105 orbits). Median values are shown as vertical blue lines. Based on the available data on knownETNOs, the presence of massive perturbers well beyond 600 AU is strongly excluded within the context of the hypotheses centered in this study.
Table C.1.
Summary of central values and dispersions of optimal heliocentric orbits.
Orbital parameter q p >
300 AU q p >
400 AU q p >
500 AU q p >
600 AUSemi-major axis, a p (AU) 379 + − (378) 486 + − (420) 576 + − (584) 684 + − (645)Eccentricity, e p + . − . (0.01) 0.09 + . − . (0.01) 0.05 + . − . (0.04) 0.09 + . − . (0.05)Inclination, i p ( ◦ ) 19 + − (13) 25 + − (13) 14 + − (12) 17 + − (16)Longitude of the ascending node, Ω p ( ◦ ) 170 + − (181) 153 + − (148) 150 + − (151) 157 + − (160)Argument of perihelion, ω p ( ◦ ) 127 + − (7) 136 + − (35) 212 + − (305) 62 + − (22) Notes.
Median values and 16th and 84th percentiles (absolute maximum in parentheses) from the Monte Carlo random searches whose distributionsare shown in Fig. C.1. Article number, page 9 of 12 & A proofs: manuscript no. memories
400 600 800 1000Semi-major axis (AU)0.0000.0020.0040.0060.008 P r ob a b ilit y d e n s it y
400 600 800 1000Semi-major axis (AU)0.0000.0020.0040.006 P r ob a b ilit y d e n s it y
600 800 1000Semi-major axis (AU)0.0000.0020.0040.0060.008 P r ob a b ilit y d e n s it y
600 800 1000Semi-major axis (AU)0.0000.0020.004 P r ob a b ilit y d e n s it y P r ob a b ilit y d e n s it y P r ob a b ilit y d e n s it y P r ob a b ilit y d e n s it y P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y Fig. D.1.
Barycentric orbital elements of putative perturber planets. As Fig. 1 but using scrambled data as input.Article number, page 10 of 12. de la Fuente Marcos , R. de la Fuente Marcos: Close encounters in extreme trans-Neptunian space
400 600 800 1000Semi-major axis (AU)0.0000.0020.0040.006 P r ob a b ilit y d e n s it y
400 600 800 1000Semi-major axis (AU)0.0000.0020.004 P r ob a b ilit y d e n s it y
600 800 1000Semi-major axis (AU)0.0000.0020.0040.0060.008 P r ob a b ilit y d e n s it y
600 800 1000Semi-major axis (AU)0.0000.0010.0020.0030.004 P r ob a b ilit y d e n s it y P r ob a b ilit y d e n s it y P r ob a b ilit y d e n s it y P r ob a b ilit y d e n s it y P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y Fig. D.2.
Heliocentric orbital elements of putative perturber planets. As Fig. C.1 but using scrambled data as input. Article number, page 11 of 12 & A proofs: manuscript no. memories
200 400 600 800 1000Semi-major axis (AU)0.00000.00250.00500.00750.0100 P r ob a b ilit y d e n s it y P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y ( o ) P r ob a b ilit y d e n s it y Fig. E.1.
Barycentric orbital elements of putative perturber planets. AsFig. 1 but for q p >>