Planet Nine -- Andrew Lowe's Prediction of Orbit Elements

Where is Planet Nine? An Analysis of the Elements of the Hypothetical Planet Nine

Let's be honest. The discovery of a new planet in our Solar System would be huge. So why don't we have some fun, join in the discussion, and offer some insights into where Planet Nine might be hiding


Introduction

I've been recently asked about elements I provide in a couple of datasets (Centaurs and TNOs (Skymap format) and Centaurs and TNOs (Celestia format)) on my website for the orbit of the hypothetical Planet Nine. For the last few months, I've used numbers culled from various technical papers, but on June 3, 2018, I published my first orbital solution derived from my own research. My Lowe20180622 numbers are the most current. On this page, I'll describe how the elements were computed.

In the first section, I'll discuss the size and shape of Planet Nine's orbit. These parameters are described by the semi-major axis (a) and the eccentricity (e). In the second section I'll address the orientation of the orbit, which is defined by the argument of perihelion, the angle of the ascending node, and the inclination. In the third section, the position of Planet Nine within its orbit is determined by the value of the mean anomaly at the current standard epoch.

I'll assume you are familiar with the background to the Planet Nine story. A good review is available here, at Wikipedia.


Part 1 -- The Size and Shape of Planet Nine's Orbit

Trujillo and Sheppard (2014) suggested a massive outer Solar System perturber to account for a clustering in the value of the argument of perihelion for objects beyond the gravitational influence of Neptune (defined here as extreme trans-Neptunian objects with a > 250 A.U., or eTNOs).

Brown and Batygin (2016) conducted extensive simulations to account for the distribution of eTNOs. They recognized ranges of allowable orbital elements, but used a = 700 A.U. and e = 0.6 for some of their auxilliary computations. In particular, they identified a relationship between a and e; for a ten-earth-mass object, e = 0.75 - (450 A.U./a)^8.

Millholland and Laughlin (2017) investigated the possibility of mean-motion resonances (MMRs) between Planet Nine and eTNOs, following a suggestion by Malhotra et al. MMRs have the potential for greater orbital stability for eTNOs since close approaches to Planet Nine are avoided, similar to the situation between Neptune and Pluto. The authors concluded that a = 654 A.U. and e = 0.45.

Since the publication of the last paper, several new eTNOs have been discovered. These new objects have confirmed and refined the notion that Planet Nine is in resonance motion. The table below summarizes the fractional period relationships to Planet Nine.
For eTNOs (a > 250 A.U.) in the first block of data, then selected TNOs (a > 150 A.U.), I show barycentric values of a from JPL's HORIZONS system. Derived data for Planet Nine is on the last two lines.

Columns from left to right show the individual object, its reference orbit at the Minor Planet Center, the epoch of the elements, semi-major axis (a), period (P), P/P9 (ratio of periods of the individual object and Planet Nine (assumed to be 17017.62 years), and predicted period ratio, based on M and N values. M represents the number of revolutions that the object would complete during the time that Planet Nine completes N revolutions. Note that Neptune and Pluto interact in a similar manner with M = 3 and N = 2, where M refers to Neptune and N refers to Pluto.

The key information from this table is the comparison of P/P9 with M/N; that is, do many of the TNOs have a period ratio which can be expressed as the ratio of two small integers. It appears in many cases that this is true.

Planet Nine's period was based on the 3/2 period ratio for (90377) Sedna and 2007 TG422, giving an average of 17017.62 years. This value gave credible period ratios for the other eTNOs which could be expressed with small integer values of M and N. Note that because the eTNOs will tend to librate around the point of maximum stability, it is expected that there will be a small deviation from an exact fit to small integer values of M and N. Assuming that the period is 17017.62 years, I estimate that Planet Nine's barycentric orbit has a = 661.893 A.U. The Find_Orb displays below show the conversion from barycentric to heliocentric elements.
A comparison of Find_Orb orbital elements for Planet Nine, for barycentric and heliocentric solutions.

Barycentric values for the semi-major axis of Planet Nine were combined with heliocentric values for the orbit orientation.

A heliocentric value of a = 656.11 A.U. results in e = 0.701 from Brown and Batygin (2016)'s relationship for a and e.


Part 2 -- The Orientation of Planet Nine's Orbit

Brown and Batygin (2016) sought to determine orientation angles from their simulations. In this and subsequent publications, they have suggested that the angle of ascending node = 94, longitude of perihelion = 235, and inclination = 30. Since the perihelion direction of TNOs affected by Planet Nine is a key factor, a quick review of its computation follows.

It may be of some interest to show how Brown derived his longitude of perihelion for Planet Nine. The following table provides heliocentric orbital details (courtesy of the Minor Planet Center) of some distant objects that are relevant to this discussion.
This table shows anti-aligned eTNOs (perihelion direction opposite to that of Planet Nine), aligned eTNOs (perihelion direction aligned with Planet Nine), and TNOs with orbits that approach closely to the orbit of Planet Nine. In addition to previously defined parameters, the MOID (minimum orbit intersection distance) to Planet Nine is shown in the last column.

(469750) 2005 PU21 is included in this list because of its low MOID value. It is, however, capable of interactions with Neptune over the long term because its Neptune MOID value is only 0.62 A.U. according to the Minor Planet Center. Its perihelion direction is used in Part 2 in the estimation of the orientation of the orbit of Planet Nine, but it is neglected in Part 3 to constrain the value of the mean anomaly.

I'll examine three independent approaches.

Consider first the anti-aligned eTNOs. Since their orbits at aphelion are well beyond the perihelion distance of Planet Nine, there is a large angle of libration around the point of maximum stability (180 separation of longitudes of perihelion). The simplest approach is to take the largest and smallest longitudes of perihelion values for the anti-aligned objects and assume they represent the limits of libration (in fact the degree of libration is dependent upon a, but let's keep things simple at this stage). Ignoring 2015 GT50, which has an apsidally circulating orbit in contrast to the other apsidally aligned objects, the two candidates are 2015 BP519 with 128.33 and 2014 SR349 with 16.70. The midpoint is 72.52. Add 180 to obtain the perihelion direction of Planet Nine of 252.52.

Secondly, the two aligned eTNOs have perihelion directions of 256.81 and 250.33 (not only are these values in excellent agreement, but the angle of the ascending node and the inclination are very similar, suggesting that a common perturber is acting on these two bodies). With two data points, the best solution is to take the average, or 253.57.

Thirdly, the three TNOs with orbits that closely approach the orbit of Planet Nine have a very limited angle of libration, since their orbits at aphelion are not as distant as the eTNOs, and they tend to move more slowly as they cross Planet Nine's orbit, which allows for a greater degree of interaction. Assume that (148209) 2000 CR105 at 87.55 and (469750) 2005 PU21 at 60.22 represent the limits of libration, then the average is 73.89. Since these three objects are anti-aligned, add 180 to obtain the perihelion direction of Planet Nine of 253.89.

The agreement of these three independent values is truly remarkable, again suggesting that some external perturber is responsible for the arrangement of their orbits. It is entirely reasonable to assume at this point that Planet Nine's longitude of perihelion is 253.

The final refinement comes from examining Figure 4 from Brown and Batygin (2016). Note that in addition to a cluster of anti-aligned test objects at 180 from Planet Nine's perihelion direction, there is a string of test objects nearly aligned with Planet Nine, but offset by +18. It is not entirely clear why the simulation has created this misalignment (inclinations, for example, were not considered), but if it is accepted as truth, then the previously-derived perihelion direction of Planet Nine should be reduced by 18 to obtain 235, Brown's currently preferred value.

Incidentally, Figure 4's range of perihelion direction for the anti-aligned test objects is about 105, which compares favorably with the difference between 2015 BP519 and 2014 SR349 of 111.63.
The uncertainty in the inclination of Planet Nine's orbit is relatively large, with a reasonable range extending from 25 to 35. In this SkyMap display, I show the sky tracks for the nominal inclination of 30, with the 25 orbit to the north and the 35 orbit to the south. In order to maintain the perihelion direction at 235, the 25 orbit has argument of perihelion = 138.22; that of the 35 orbit is 135.33. The mean anomaly (M) for each track is labelled every 10.

From Part 3, potential sky locations at M = 116, M = 159, and M = 234 are shown along the 30 inclination line.


Part 3 -- Current Position on the Orbit

While eTNOs are the best objects to constrain the size, shape, and orientation of Planet Nine's orbit, it turns out that TNOs with a > 150 A.U. with small MOID to the orbit of Planet Nine are ideal for setting limits on the location of Planet Nine within its orbit. They have periods from one-fifth to one-ninth that of Planet Nine, and the fact that they can approach closely under certain circumstances limits the position of Planet Nine in its orbit.

If Planet Nine was at its nearest point to the Sun, it would be bright enough to have been found in archival surveys, and would have a measurable influence on the known major planets, so I've excluded any analysis for its mean anomaly (M) within 60 of perihelion. For values of M from 60 to 300 at 1 increments, I used Aldo Vitagliano's Solex 11.09 numerical integrator to compute the interaction of Planet Nine with all the objects listed in the table in Part 2. The results are plotted below referenced to M for the current epoch of 2018 March 23. For this display, the mass of Planet Nine was ignored. Encounters during the previous perihelion passage are shown in red, while those for the upcoming perihelion passage are shown in blue.
Any value of M that results in a close approach with a TNO would be excluded from the range of acceptable numbers. I have assumed that any near approaches greater than 20 A.U. can be neglected. Examining the graph, it appears that a default assumption that the object is at aphelion is not allowed, as there are a number of close approaches at M = 180. In fact, there are only three points in the orbit with any appreciable width at which close approaches are avoided: M = 116, M = 159, and M = 234. Of these three options, M = 116 has the narrowest width in allowable range of M measured at the 20 A.U. cutoff and is probably the least likely of the three choices to be correct. If M = 159, then the geocentric coordinates for June 22, 2018 are (J2000.0) RA = 3h32.75m Dec = -0326.7', in the constellation of Eridanus; for M = 234, the geocentric coordinates are (J2000.0) RA = 4h34.83m Dec = +0745.2', in Taurus.

If I had to choose between these two options, I like the fact that for M = 159, the close approaches of 2012 VP113 at M = 155 for the last and next perihelion passage of Planet Nine are mirror images of each other (note the precise overlap of the close approaches curves, when 2012 VP113 completes exactly four revolutions for one revolution of Planet Nine, even though the period of Planet Nine was fixed by two other eTNOs). This may suggest that 2012 VP113 is locked into a MMR situation, not allowing a closer approach than 22.44 A.U., and unable to stray into adjacent M space where it can approach to the MOID value of 8.72 A.U. and be perturbed by Planet Nine.

Consequently, my latest set of elements for Planet Nine are as follows:

Epoch 2018 Mar. 23.0 TT = JDT 2458200.5     
M = 159.00                    
a = 656.11 A.U.
e = 0.701
Peri. = 136.92   
Node  = 94.00    
Incl. = 30.00
  
Based on these numbers, here is a Celestia orbit display of Planet Nine and Extreme Trans-Neptunian Objects. Here is a Skymap sky display of Planet Nine and some Extreme Trans-Neptunian Objects.

These elements will be updated in the future as further eTNOs are discovered... or Planet Nine itself is found.


Acknowledgements

It's a pleasure to thank all the Planet Nine researchers whose work forms the basis of my computations. The Minor Planet Center and JPL's HORIZONS system provided heliocentric and barycentric elements of associated distant objects. I also thank Aldo Vitagliano (Solex 11.09 numerical integrator), Bill Gray (Find_Orb), and Chris Marriot (SkyMap) for their computational and visualization tools.