Finding Planet Nine: apsidal anti-alignment Monte Carlo results Free

Abstract

The distribution of the orbital elements of the known extreme trans-Neptunian objects or ETNOs has been found to be statistically incompatible with that of an unperturbed asteroid population following heliocentric or, better, barycentric orbits. Such trends, if confirmed by future discoveries of ETNOs, strongly suggest that one or more massive perturbers could be located well beyond Pluto. Within the trans-Plutonian planets paradigm, the Planet Nine hypothesis has received much attention as a robust scenario to explain the observed clustering in physical space of the perihelia of seven ETNOs which also exhibit clustering in orbital pole position. Here, we revisit the subject of clustering in perihelia and poles of the known ETNOs using barycentric orbits, and study the visibility of the latest incarnation of the orbit of Planet Nine applying Monte Carlo techniques and focusing on the effects of the apsidal anti-alignment constraint. We provide visibility maps indicating the most likely location of this putative planet if it is near aphelion. We also show that the available data suggest that at least two massive perturbers are present beyond Pluto.

1 INTRODUCTION

The distribution of the orbital parameters of the known extreme trans-Neptunian objects or ETNOs is statistically incompatible with that of an unperturbed asteroid population following Keplerian orbits (de la Fuente Marcos & de la Fuente Marcos 2014, 2016b; Trujillo & Sheppard 2014; de la Fuente Marcos, de la Fuente Marcos & Aarseth 2015, 2016; Gomes, Soares & Brasser 2015; Batygin & Brown 2016; Brown & Batygin 2016; Malhotra, Volk & Wang 2016). A number of plausible explanations have been suggested. These include the possible existence of one (Trujillo & Sheppard 2014; Gomes et al. 2015; Batygin & Brown 2016; Brown & Batygin 2016; Malhotra et al. 2016) or more (de la Fuente Marcos & de la Fuente Marcos 2014, 2016b; de la Fuente Marcos et al. 2015, 2016) trans-Plutonian planets, capture of ETNOs within the Sun's natal open cluster (Jílková et al. 2015), stellar encounters (Brasser & Schwamb 2015; Feng & Bailer-Jones 2015), being a by-product of Neptune's migration (Brown & Firth 2016) or the result of the inclination instability (Madigan & McCourt 2016), and having been induced by Milgromian dynamics (Paučo & Klačka 2016).

At present, most if not all of the unexpected orbital patterns found for the known ETNOs seem to be compatible with the trans-Plutonian planets paradigm that predicts the presence of one or more planetary bodies well beyond Pluto. Within this paradigm, the best studied theoretical framework is that of the so-called Planet Nine hypothesis, originally suggested by Batygin & Brown (2016) and further developed in Brown & Batygin (2016). The goal of this analytically and numerically supported conjecture is not only to explain the observed clustering in physical space of the perihelia and the positions of the orbital poles of seven ETNOs (see Appendix A for further discussion), but also to account for other, previously puzzling, pieces of observational evidence like the existence of low perihelion objects moving in nearly perpendicular orbits. The Planet Nine hypothesis is compatible with existing data (Cowan, Holder & Kaib 2016; Fienga et al. 2016; Fortney et al. 2016; Ginzburg, Sari & Loeb 2016; Linder & Mordasini 2016) but, if Planet Nine exists, it cannot be too massive or bright to have escaped detection during the last two decades of surveys and astrometric studies (Luhman 2014; Cowan et al. 2016; Fienga et al. 2016; Fortney et al. 2016; Ginzburg et al. 2016; Linder & Mordasini 2016). A super-Earth in the sub-Neptunian mass range is most likely and such planet may have been scattered out of the region of the Jovian planets early in the history of the Solar system (Bromley & Kenyon 2016) or even captured from another planetary system (Li & Adams 2016; Mustill, Raymond & Davies 2016); super-Earths may also form at 125–750 au from the Sun (Kenyon & Bromley 2015, 2016).

The analysis of the visibility of Planet Nine presented in de la Fuente Marcos & de la Fuente Marcos (2016a) revealed probable locations of this putative planet based on data provided in Batygin & Brown (2016) and Fienga et al. (2016); the original data have been significantly updated in Brown & Batygin (2016). In addition, independent calculations (de la Fuente Marcos et al. 2016) show that the apsidal anti-alignment constraint originally discussed in Batygin & Brown (2016) plays a fundamental role on the dynamical impact of a putative Planet Nine on the orbital evolution of the known ETNOs. Here, we improve the results presented in de la Fuente Marcos & de la Fuente Marcos (2016a) focusing on the effects of the apsidal anti-alignment constraint. This paper is organized as follows. Section 2 presents an analysis of clustering in barycentric elements, pericentre and orbital pole positions, which is subsequently discussed. An updated evaluation of the visibility of Planet Nine virtual orbits at aphelion is given in Section 3. Conclusions are summarized in Section 4.

2 CLUSTERING IN BARYCENTRIC PARAMETERS

The six (Batygin & Brown 2016) or seven (Brown & Batygin 2016) ETNOs singled out within the Planet Nine hypothesis (see Appendix A for details) have a > 226 au (heliocentric) and they exhibit clustering in perihelion location in absolute terms and also in orbital pole position. In order to better understand the context of these clusterings we study the line of apsides of the known ETNOs (see Table 1 and Fig. 1) and the projection of their orbital poles on to the plane of the sky (see Table 1 and Fig. 2). Here, we consider barycentric orbits as it can be argued (see the discussion in Malhotra et al. 2016) that the use of barycentric orbital elements instead of the usual heliocentric ones is more correct in this case.

In Trujillo & Sheppard (2014), the ETNOs are defined as asteroids with heliocentric semimajor axis greater than 150 au and perihelion greater than 30 au; at present, there are 16 known ETNOs. Because of the nature of their orbits, the ETNOs cannot experience a close approach to the known planets, including Neptune. Nevertheless, the orbits of the ETNOs are affected by indirect perturbations that induce variations in perihelion location. The perihelion distance of an ETNO depends on the value of its semimajor axis and eccentricity, e, as it is given by q = a(1 − e). However, its absolute position on the sky is only function of the inclination, i, the longitude of the ascending node, Ω, and the argument of perihelion, ω. In heliocentric ecliptic coordinates, the longitude and latitude of an object at perihelion, (l_q, b_q), are given by the expressions: tan (l_q − Ω) = tan ω cos i and sin b_q = sin ω sin i (see e.g. Murray & Dermott 1999). For an orbit with zero inclination, l_q = Ω + ω and |$b_q=0^\circ$|⁠; therefore, if |$i=0^\circ$|⁠, the value of l_q coincides with that of the longitude of perihelion parameter, ϖ = Ω + ω. In Brown & Batygin (2016), l_q is called ‘perihelion longitude’ and b_q is the ‘perihelion latitude’. Considering barycentric orbits, we denote the barycentric ecliptic coordinates of an ETNO at pericentre as (L_q, B_q). In Table 1 we show the values of q, L_q and B_q computed using the barycentric data in Table 2. The input values are from Jet Propulsion Laboratory's Small-Body Database¹ (SBDB) and Horizons On-Line Ephemeris System (Giorgini et al. 1996).

Fig. 1 shows the position in the sky at pericentre as a function of the pericentre distance for the known ETNOs. The objects in Brown & Batygin (2016) are plotted in red; the average angular separation at pericentre for this group is 44|$^\circ$|±30|$^\circ$|⁠, but the pair 2012 VP₁₁₃–2013 RF₉₈ has a separation of 2| $_{.}^{\circ}$|5, 2004 VN₁₁₂–2007 TG₄₂₂ has 5| $_{.}^{\circ}$|6, and 2004 VN₁₁₂–2013 RF₉₈ has 9| $_{.}^{\circ}$|8. In addition to this clustering, other obvious groupings are also visible. The dispersion of these additional groupings in ecliptic coordinates is similar to that of the set singled out by Brown & Batygin (2016), but their dispersion in q is considerably lower. ETNOs (82158) 2001 FP₁₈₅, 2002 GB₃₂ and 2003 HB₅₇ have a nearly common line of apsides; the same can be said about (445473) 2010 VZ₉₈, 2007 VJ₃₀₅ and 2015 SO₂₀. The case for the first grouping is particularly strong. Out of 16 known ETNOs, only five reach pericentre north from the ecliptic, i.e. |$B_q>0^\circ$|⁠; these objects are 82158, 2002 GB₃₂, 2003 HB₅₇, 2005 RH₅₂ and 2013 GP₁₃₆. This represents a 2σ departure from an isotropic distribution in B_q, where |$\sigma =\sqrt{n}/2$| is the standard deviation for binomial statistics. This marginally significant result suggests that an unknown massive perturber has aligned the apsidal orientation of these objects, but the Planet Nine hypothesis cannot explain this pattern (Batygin & Brown 2016); another perturber is needed as the line of apsides is scarcely affected by indirect perturbations from the known planets. A nearly common line of apsides is also expected in a set of objects resulting from the break-up of a single object at pericentre. Such scenario might be linked to some of the groupings observed.

In Table 1 we show the current values of the position in the sky of the poles of the orbits of the known ETNOs computed using the data in Table 2; the ecliptic coordinates of the pole are |$(L_{\rm p}, B_{\rm p}) = (\Omega -90^\circ , 90^\circ -i)$|⁠. Fig. 2 shows the poles of the known ETNOs and also those of the known planets of the Solar system and various nominal orbits of Planet Nine (epoch 2457600.5 JD). The objects singled out in Brown & Batygin (2016) and Planet Nine appear to exhibit a relative arrangement in terms of position of their orbital poles similar to the one existing between Neptune and Pluto. For this group, the average polar separation is 16|$^\circ$|±8|$^\circ$|⁠, but the pair 148209–2010 GB₁₇₄ has 1| $_{.}^{\circ}$|5, 2004 VN₁₁₂–2013 RF₉₈ has 4| $_{.}^{\circ}$|1, 148209–2007 TG₄₂₂ has 6| $_{.}^{\circ}$|8, 2007 TG₄₂₂–2010 GB₁₇₄ has 6| $_{.}^{\circ}$|8, and 2007 TG₄₂₂–2012 VP₁₁₃ has 9| $_{.}^{\circ}$|6. On the other hand, the ETNOs 2002 GB₃₂ and 2003 HB₅₇ not only exhibit apsidal alignment but their orbital poles are also very close.

As for the overall clustering in orbital parameter space, it has been claimed that the ETNOs exhibit clustering in the values of their ω (Trujillo & Sheppard 2014), e and i (de la Fuente Marcos & de la Fuente Marcos 2014), and Ω (Brown & Firth 2016). Table 2 presents the descriptive statistics of the known ETNOs; in this table, unphysical values are shown for completeness. The bottom block of statistics in Table 2 (see also Appendix A) focuses on the seven ETNOs singled out in Brown & Batygin (2016). The overall statistics is only slightly different from that of the heliocentric orbital elements. The mean value of the barycentric e of the known ETNOs amounts to 0.81±0.06, the barycentric i is 20|$^\circ$|±8|$^\circ$|⁠, the barycentric Ω is 134|$^\circ$|±72|$^\circ$|⁠, and the barycentric ω is −26|$^\circ$|±49|$^\circ$|(see ω* in Table 2). Clustering in e may be due to selection effects, but the others cannot be explained as resulting from observational biases, they must have a dynamical origin (de la Fuente Marcos & de la Fuente Marcos 2014). For the ETNO sample in Brown & Batygin (2016), the average values of the barycentric orbital parameters are e = 0.84 ± 0.07, |$i=22^\circ \pm 6^\circ$|⁠, |$\Omega =106^\circ \pm 31^\circ$| and |$\omega =314^\circ \pm 20^\circ$|⁠.

Regarding the issue of statistical outliers and assuming that outliers are observations that fall below Q₁ − 1.5 IQR or above Q₃ + 1.5 IQR, where Q₁ is the first quartile, Q₃ is the third quartile, and IQR is the interquartile range, we observe a small but relevant number of outliers. For the entire sample, 2003 SS₄₂₂ is an outlier in terms of ω* (see Table 2), Sedna and 2012 VP₁₁₃ are outliers in terms of q, and Sedna and 2007 TG₄₂₂ are outliers in terms of orbital period. In terms of size (not shown in the table), Sedna is a very significant outlier with H = 1.6 mag when the lower and upper limits for outliers are 4.7 mag and 8.7 mag, respectively. As for the sample of ETNOs in Brown & Batygin (2016), Sedna is an statistical outlier in terms of i.

3 VISIBILITY ANALYSIS

Here, we apply the Monte Carlo approach (Metropolis & Ulam 1949) described in de la Fuente Marcos & de la Fuente Marcos (2014) to construct visibility maps indicating the most probable location of this putative planet if it is near aphelion. Each Monte Carlo experiment consists of 10⁷ test orbits uniformly distributed in parameter space. The analyses in e.g. Cowan et al. (2016), Fienga et al. (2016), Fortney et al. (2016), Ginzburg et al. (2016) and Linder & Mordasini (2016) strongly disfavour a present-day Planet Nine located at perihelion and they do not discard the aphelion which is also favoured in Brown & Batygin (2016).

3.1 Batygin & Brown (2016)

The resonant coupling mechanism described in Batygin & Brown (2016) emphasizes the existence of simultaneous apsidal anti-alignment and nodal alignment, i.e. Δϖ librates about 180|$^\circ$|and ΔΩ librates about 0|$^\circ$|⁠. The relative values of ϖ and Ω of the ETNO with respect to those of the perturber must oscillate in order to maintain orbital confinement but see Beust (2016) for a detailed analysis. In Batygin & Brown (2016), the value of ω of the putative Planet Nine is 138|$^\circ$|±21|$^\circ$|⁠. It is also indicated that the average value of Ω for their six ETNOs is 113|$^\circ$|±13|$^\circ$|and that of ω is 318|$^\circ$|±8|$^\circ$|⁠. Applying the conditions for stability and using the other values discussed in their work, we generate a synthetic population of Planet Nines with a ∈ (400, 1500) au, e ∈ (0.5, 0.8), i ∈ (15, 45)|$^\circ$|⁠, Ω ∈ (100, 126)|$^\circ$|and ω ∈ (117, 159)|$^\circ$|⁠. Fig. 3, left-hand panels, shows the distribution in equatorial coordinates of the set of studied orbits. In this figure, the value of the parameter in the appropriate units is colour coded following the scale printed on the associated colour box. The location of the Galactic disc appears in panel D (inclination). The background stellar density is the highest towards this region. The distribution of Q, a and e is rather uniform as expected because the location in the sky of Planet Nine does not depend on these parameters (see above). The distribution in declination depends on i and ω; those orbits with higher values of i reach aphelion at lower declinations, the same behaviour is observed for the ones with lower values of ω. The location in right ascension mainly depends on the value of Ω, both increasing in direct proportion. Fig. 4, left-hand panels, shows that the frequency distribution in right ascension and declination is far from uniform; the probability of finding an orbit reaching aphelion is highest in the region limited by α ∈ (4.5, 5.5)^h and |$\delta \in (6, 9)^\circ$|⁠. However, the mean values of Ω and ω for the six ETNOs of interest differ from those in Table A1.

3.2 Brown & Batygin (2016)

The Planet Nine hypothesis has been further developed in Brown & Batygin (2016). In this new work, the volume of orbital parameter space linked to the putative planet for an assumed mass of 10 M_⊕ is enclosed by a ∈ (500, 800) au, e ∈ (0.32, 0.74), i ∈ (22, 40)|$^\circ$|⁠, Ω ∈ (72.2, 121.2)|$^\circ$|and ω ∈ (120, 160)|$^\circ$|⁠. Figs 3 and 4, second to left-hand panels, show similar trends to the previous ones but now the distribution in equatorial coordinates of the aphelia is wider. The probability is highest in the region limited by α ∈ (3, 5)^h and |$\delta \in (-1, 10)^\circ$|⁠. This enlarges the optimal search area significantly. However, they use a value of l_q of 241|$^\circ$|±15|$^\circ$|that is based on a value of 61|$^\circ$| for the seven ETNOs so Δl_q – not Δϖ – librates about 180|$^\circ$|⁠; the average value of l_q for these ETNOs from Table 1 is 62|$^\circ$|±38|$^\circ$|⁠, the average value of ϖ is 60|$^\circ$|±39|$^\circ$|⁠.

3.3 Enforcing apsidal anti-alignment

Here, we enforce apsidal anti-alignment and nodal alignment, using the data in Table 2, bottom section, and considering the Planet Nine orbital parameter domain defined by a ∈ (600, 800) au, e ∈ (0.5, 0.7), i ∈ (22, 40)|$^\circ$|⁠, Ω ∈ (74.4, 137.3)|$^\circ$|and ω ∈ (113.5, 154.4)|$^\circ$|⁠. Figs 3 and 4, second to right-hand panels, show that in this scenario, Planet Nine is most likely to be moving within α ∈ (3.0, 5.5)^h and |$\delta \in (-1, 6)^\circ$|⁠, if it is near aphelion. If the data in Table A1 are used instead, then Ω ∈ (87, 117)|$^\circ$|and ω ∈ (118.5, 148.8)|$^\circ$|and we obtain Figs 3 and 4, right-hand panels. In this case, the putative planet is most likely moving within α ∈ (3.5, 4.5)^h and |$\delta \in (-1, 2)^\circ$|⁠, i.e. projected towards the separation between the constellations of Taurus and Eridanus. In terms of probability, now the most likely location of Planet Nine is at α ∼ 4^h and δ ∼ 0| $_{.}^{\circ}$|5, in Taurus. This area is compatible with Orbit A in de la Fuente Marcos et al. (2016).

4 CONCLUSIONS

In this paper, we have re-analysed the various clusterings in ETNO orbital parameter space claimed in the literature and explored the visibility of Planet Nine within the context of improved constraints. Our results confirm the findings in Batygin & Brown (2016) and Brown & Batygin (2016) regarding clustering but using barycentric orbits. However, the observed overall level of clustering may not be maintained by a putative Planet Nine alone, other perturbers should exist. Summarizing:

• We confirm the existence of apsidal alignment and similar projected pole orientations among the currently known ETNOs. These patterns are consistent with the presence of perturbers beyond Pluto and/or, less likely, break-up of large asteroids at perihelion.

• If Planet Nine is at aphelion, it is most likely moving within α ∈ (3.0, 5.5)^h and |$\delta \in (-1, 6)^\circ$| if Δϖ ∼180|$^\circ$|and ΔΩ ∼0|$^\circ$|⁠.

We thank two anonymous referees for their constructive reports, and S. J. Aarseth, D. P. Whitmire, G. Carraro, D. Fabrycky, A. V. Tutukov, S. Mashchenko, S. Deen and J. Higley for comments on ETNOs and trans-Plutonian planets. This work was partially supported by the Spanish ‘Comunidad de Madrid’ under grant CAM S2009/ESP-1496. In preparation of this paper, we made use of the NASA Astrophysics Data System, the ASTRO-PH e-print server and the MPC data server.

REFERENCES

APPENDIX A: Further details on the comparison with the results of Brown & Batygin (2016)

According to the heliocentric orbital solutions publicly available from the SBDB, the six (not seven) objects singled out in Batygin & Brown (2016) all have values of the heliocentric semimajor axis, a, larger than 257 au; Brown & Batygin (2016) indicate that their semimajor axes are larger than 227 au. The six original objects in Batygin & Brown (2016) – namely, (90377) Sedna (a = 499 au), 2004 VN₁₁₂ (a = 318 au), 2007 TG₄₂₂ (a = 483 au), 2010 GB₁₇₄ (a = 370 au), 2012 VP₁₁₃ (a = 258 au) and 2013 RF₉₈ (a = 307 au) – do not have the largest values of the perihelion distance, only four of them – Sedna (q = 76 au), 2004 VN₁₁₂ (q = 47 au), 2010 GB₁₇₄ (q = 49 au) and 2012 VP₁₁₃ (q = 80 au) – are in this situation. In Brown & Batygin (2016), it is said that the seven objects with a > 227 au are singled out. However, fig. 1(a) in Brown & Batygin (2016) highlights in red only six objects (those singled out in Batygin & Brown 2016), not seven; (148209) 2000 CR₁₀₅ (a = 226 au, q = 44 au), which has the eighth largest value of the heliocentric semimajor axis and the fifth largest perihelion distance, appears in green in that figure. The ETNO with the seventh largest value of the heliocentric semimajor axis is (82158) 2001 FP₁₈₅ (a = 227 au, q = 34 au), not 148209 as erroneously stated in Brown & Batygin (2016). Our analysis in Figs 1 and 2 shows that 82158 is unlikely to be dynamically connected with the original six ETNOs singled out in Batygin & Brown (2016), but 148209 very probably is. Therefore, the seven objects of interest within the context of the Planet Nine hypothesis are: Sedna, 148209, 2004VN₁₁₂, 2007 TG₄₂₂, 2010 GB₁₇₄, 2012 VP₁₁₃ and 2013 RF₉₈.

In terms of barycentric (not heliocentric) orbits, see Table 2, 148209 has the seventh largest value of the semimajor axis and 82158 has the eighth largest.

Table A1 shows the statistics for the six original objects in Batygin & Brown (2016); the average values of the orbital parameters are e = 0.85 ± 0.08, |$i=22^\circ \pm 6^\circ$|⁠, |$\Omega =102^\circ \pm 33^\circ$| and |$\omega =314^\circ \pm 22^\circ$| (these values have been used in the discussion in Section 3.1). The average angular separation at pericentre for this group is 45|$^\circ$|±34|$^\circ$|and the average polar separation is 16|$^\circ$|±8|$^\circ$|⁠. For the sample of ETNOs in Batygin & Brown (2016), both 2007 TG₄₂₂ and 2012 VP₁₁₃ are statistical outliers in terms of e (see Tables 2 and A1).

The expression for the eccentricity of the putative Planet Nine in Brown & Batygin (2016) produces negative values for masses under ∼6.38 M_⊕ and the lower limit of the value of the semimajor axis. Production of unphysical values is the main reason why it has not been applied to select the range in eccentricities used in Section 3.3.