version 2.X

I finished the last post by stating the obvious: the Planet Nine hypothesis - as imagined by us - is NOT the first (or even the latest) proposal of a trans-Neptunian planet. So what distinguishes all these hypotheses? Are all putative trans-Neptunian planets the same Planet? Clearly not. Put simply, each theory is characterized by 1) the anomalous data it seeks to explain, and 2) the dynamics through which the putative planet explains the data. One of the key goals of the new paper was to place P9 hypothesis within this framework by delineating a series of purely analytical models, as well as a suite of large-scale numerical simulations. As a result, the discussion below will inevitably touch on some technical points. For those of you only interested in the final answer, however, I will take a shortcut and summarize our conclusions:

Planet Nine is a factor of ~2 smaller in all quantities compared to what we reported in the original paper. The new estimate of the semi-major axis is a~400-500AU (could potentially be even smaller, but only marginally so). P9’s orbital eccentricity is about e~0.15-0.3. The inclination is around i~20 deg. Last but not least, the mass is about m~5 Earth masses. Planet Nine is probably not a relative of Neptune — it’s a Super-Earth. Now let’s dig into the details a bit.

As always, the best starting point is the observational data. Compared with three years ago, the dataset has expanded by a factor of a couple, and now contains 14 objects with semi-major axis beyond 250 AU and inclinations lower than 40 degrees. The diagram below shows the orbits in physical space, as viewed from the north ecliptic pole, while the inset presents a polar plot of the orbital inclinations and longitudes of ascending node (which dictate the orientations of the orbital planes):

Generally, the distant KBOs span a broad array of perihelion distances, ranging from ~35AU to ~80AU. As a result, some KBOs interact with Neptune (which lives at 30AU) much more strongly than others, yielding a wide spread in dynamical lifetimes. For the purposes of the P9 hypothesis, it is useful to sub-divide the data into three categories: stable, metastable, and unstable orbits, which are shown on the above plot as purple, gray, and green ellipses respectively. Although the orbital confinement among the plotted long-period KBOs is easy to see by eye, it is also evident that the degree of clustering is far more striking among dynamically stable (purple) and metastable (gray) orbits than their unstable (green) counterparts.

Intuitively, the weaker clustering of unstable orbits is easy to understand — KBOs that interact with Neptune most strongly experience relatively rapid dynamical chaos. In turn, these stochastic perturbations erase any innate orbital structure of the distant belt. While all KBOs tell some story within the framework of the Planet Nine hypothesis, some tell a deeper story than others. The most cautious/conservative thing to do then, is to focus exclusively on the (meta)stable objects, which are not contaminated by strong interactions with Neptune.

Ok, so these orbits are clustered together — why is that a big deal (i.e., couldn’t they just have formed that way)? In essence, this dynamical structure is puzzling because if left to their own devices, the orbits would disperse on a timescale far shorter than the age of the solar system (due to precession induced upon them by Jupiter, Saturn, Uranus, and Neptune). To give you an example, we can plug in numbers into the above equations, and obtain that while Sedna’s orbit precesses at about 0.15 degrees per million years, 2014 SR349 precesses at 0.8 deg/Myr. Give it a few hundred million years, and the orbits will disband. Therefore, some kind of an external gravitational pull is actively keeping them confined.

As discussed in numerous published papers (see here for a list (papers that cite BBa)), the above orbital anomalies collectively point to the existence of P9. Nevertheless, to date, the parameters of P9 have remained relatively poorly determined. In the new paper, we went to some lengths to remedy this problem. As an example of the type of calculations we carried out, consider one constraint that comes from matching the critical semi-major axis that corresponds to the transition point between randomized and clusters orbital distributions. In other words, if a planet is responsible for the structure that we see, how does it keep all the orbits aligned, and why does this alignment suddenly “turn on” at ~250AU? The simplest way to understand this analytically is to restrict the orbits to a common plane and to examine the functional form of their gravitational coupling. But to do so without resorting to simulations (which is important if we want to understand the dynamics rather than just model them), we have to rely on the so-called orbit-averaging procedure.

Back in the late 1700’s, Joseph-Louis Lagrange and Pierre-Simon Laplace realized that over very long periods of time, gravitational interactions between planets can be approximated by smearing the planets along their respective orbits and computing the gravitational torques that the massive wires exert upon one another. This brilliant insight was better formalized in the mid-1800’s by Gauss, and forms the qualitative basis of secular perturbation theory. Employing this secular approximation, we can write down the total gravitational potential experienced by a KBO, under the influence of the known giant planets as well as Planet Nine, and it looks like this:

Importantly, this expression is the Hamiltonian for the problem at hand, and acts like a stream-function (or a topographic map) for secular evolution of the KBO’s orbit. This means that the long-term changes in the KBO’s orbit will simply follow the contours of the above equation. Neat right? Now, mapping the level curves of this Hamiltonian on a eccentricity vs. longitude of perihelion (relative to P9) plane at different KBO semi-major axes reveals the emergence of a stable equilibrium at ∆w=π for a>250AU. Qualitatively, this is why only distant orbits are clustered — perihelion anti-aligned orbits are only stable beyond a critical semi-major axis, and this value depends on the assumed orbital parameters of P9.

The reason behind why the orbits all cluster to a common orbital plane can also be understood in this manner, but excitation of KBO orbits onto highly inclined/retrograde orbits is considerably more complicated. If you’re interested in learning more, check out section 4 of the manuscript.

Analytical Intuition is important, but a thorough comparison between P9-sculpted synthetic Kuiper belt and real data requires a more detailed description of dynamics. Realistically, such a description can only come from N-body simulations. In this review, we did thousands of them, varying P9's mass and orbital parameters. Typical results from N-body simulations look like the fig below. Notice how in agreement with the above analytical formula, beyond a critical semi-major axis a~250AU, long-term stable KBOs - shown in this plot as blue points - cluster in longitude of perihelion:

In the simulations, the specific choices of P9 parameters directly translate to the characteristics of the distant Kuiper belt sculpted by P9’s gravity. For instance, the critical semi-major axis at which the transition from randomized to clustered orbits ensues, the actual degree of clustering, the mean tilt of the orbits, etc. all depend sensitively on P9 parameters. Although a little abstract, the most transparent approach to visualizing the relationship between simulation and data is in Poincare action-angle variables, and the figure below gives you a taste of the simulations success criteria that we used, as well as a representation of one example simulations

In section 5 of the paper, we carried out a quantitative comparison between statistical properties of the synthetic data and the real objects for each one of our N-body sims. Intriguingly, it was this analysis that revealed that a comparatively low mass Planet Nine fits the data better than our original estimates for P9 in every respect. If you don’t believe me, feel free to examine the full ensemble of 5 Mearth simulations for yourself in the plot below.

In principle, there is so much more that I would like to say, but at this point I think it’s becoming progressively clearer that my coffee supply ran out a couple paragraphs ago, and in an effort to prevent further degradation of the text, I will get straight to the final point: if Planet Nine is smaller, does that mean it's harder to find with a telescope? Counterintuitively, it's the opposite. The smaller distance from the sun more than makes up for the diminished surface area. Indeed, if we make naive baseline assumptions about P9’s albedo and adopt the interpolated exoplanet mass-radius relation to estimate P9’s size, Planet Nine turns out to be about one magnitude brighter than we previously thought. Annoyingly, though, the aphelion is very close to (in?) the galactic plane, where confusion due to background stars can readily impede detection. Still, unless we are unlucky and P9 is unexpectedly small and/or dark, it should be within the reach of LSST and comparable telescopes like Subaru. The good news is that in the case of Planet Nine hypothesis, time truly will tell.

Stories

The history of Le Verrier’s mathematical discovery of Neptune is my favorite story, period. It’s literally got everything you’d want in a good novella - differential equations, integrals, telescopes, intrigue, you name it. Rather than try to rehash it here without doing it justice, I’ll point the interested reader to an excellent 2016 article by Davor Krajnović called “The contrivance of Neptune.” Here, I only want to call attention to what Le Verrier (and Adams) got right and what they got less right.

It’s widely known that Neptune was discovered “with the tip of a pen.” Indeed, Le Verrier was able to derive Neptune’s location on the sky from orbital anomalies of Uranus with exquisite accuracy, such that Galle and D’Arrest’s observational campaign to discover this elusive planet took less than a single night. What is somewhat less well known is that Le Verrier and Adams’ calculations of Neptune’s orbit and mass were not as precise. The figure below shows the true orbits and locations of Uranus (gray) and Neptune (black) between 1830 and 1860, as well as the predicted orbits of Neptune (in purple).

Notice that the inferred semi-major axis of Neptune was about 40 (rather than 30) AU and the derived mass (36 and 50 Earth masses for Le Verrier and Adams, respectively) also significantly exceeded that of Neptune. In light of the fact that the discovery of Neptune represents the only successful mathematical prediction of a planet to date, this level of uncertainty sets the gold standard for dynamically motivated planetary predictions. In other words, if we get Planet Nine to a similar level of precision, I’ll be satisfied. It is also useful to point out that the most significant quantity in perturbing the orbit of Uranus was the anomalous acceleration in the radial direction produced by the new body - GM/r^2 - a ratio that was calculated to higher accuracy than the individual values of mass and semi-major axis. As I will highlight later, the general framework of the Planet Nine hypothesis is characterized by comparable degeneracies between P9’s mass and orbital parameters.

Following Le Verrier’s mathematical discovery of Neptune, the planet prediction business didn’t simply get a boost - it exploded. Jacques Babinet (1848), David P. Todd (1877), George Forbes (1880), Camille Flammarion (1884), William Pickering (1909-1932) all took turns predicting trans-Neptunian planets that later turned out to not be there. But no planetary prediction is quite as emblematic as Percival Lowell’s hypothesized “Planet X.” Briefly, the story goes as follows: despite the addition of Neptune to the solar system’s ledger of planets, small apparent discrepancies in the orbits of the giant planets remained, and pointed to the existence of a ~7 Earth mass planet beyond Neptune. The search continued well past Lowell’s death, and in 1930, a bright moving object was discovered by Clyde Tombaugh in the approximate location on the sky where Planet X was envisioned to be. Because Planet X was the object of the original search, the newly found body was initially considered to be the long-sought-after Planet X.

Immediately, however, there was a problem. Planet X was supposed to be like Neptune, but Tombaugh’s new planet appeared dim and point-like, and therefore much much smaller. It soon became clear that the new member of the solar system could not be THE Planet X. The object was subsequently named Pluto, and its estimated mass steadily declined for the next 5 decades.

Although Planet X - as originally formulated by Lowell - does not exist, the discovery of Pluto turned out to be the tip of an extraordinary trans-Neptunian iceberg called the Kuiper belt. The mapping and subsequent characterization of the Kuiper belt in the ‘90s and the ‘00s, generated a new wave of planetary proposals — check out Brunini & Melita (2002), Gladman & Chan (2006), Gomes et al. (2006), Lykawka and Mukai (2008), Trujillo and Sheppard (2014), Volk and Malhotra (2017) and many others that are referenced therein. All of these hypothetical planets were invoked to explain different observational puzzles, and attempt to do so through individual dynamical mechanisms. Stepping away from specific predictions, however, it is worthwhile to examine the question of where still-undetected planets can hide in the solar system, from a completely model-independent perspective. As it turns out, the combination of ephemerides, orbital stability, and definition of a planet alone leave only a limited parameter space where additional solar system planets can hide (shown as the shaded region on the plot below):

Remarkably, Planet Nine falls right in the center of that region.

Progress

The scientific process is, at its core, iterative. There is no such thing as a truly final/definitive answer - there are only solutions that are good enough for now. In a day-to-day sense, what this really means is that when you’ve been chugging away at some problem and you finally arrive at the result, you can be damn sure that you probably haven’t understood the Full Picture (even if in the moment it feels like you have). “Maybe there is a more elegant way to arrive at the solution? Maybe some approximation has introduced a hidden uncertainty? Maybe there’s a better way to look at the data?” Every step in the right direction is inevitably haunted by questions that fall along these lines. 

The Planet Nine story is no exception to this rule. Back when Mike and I published our first P9 paper three years ago, we didn’t worry that there might be a lot more work to be done on this problem - we were certain of it. Instead, what we worried about was that there exists a simpler, or perhaps more natural resolution to the anomalies we were seeing in the data, and that the Planet Nine hypothesis will be rendered irrelevant shortly after publication. That didn’t happen.

To our joint relief (and to some extent surprise), thus far, the P9 hypothesis has fared the test of time rather well. Inevitably, questions have come up regarding the role of observational biases in shaping the orbital clustering we see in the distant Kuiper belt, but these concerns have been largely put to rest. Alternative theories, on the other hand, require the existence of a hidden, coherent, and massive belt of icy planetesimals at hundreds of AU - a scenario that suffers from a number of astrophysical drawbacks. The P9 story thus continues to be in pretty good shape. Nevertheless, we have always felt the need to drive our understanding of the Planet Nine hypothesis a little bit further (and then a bit further after that). So, in collaboration with Juliette Becker and Fred Adams from University of Michigan (as well as Elizabeth Bailey here at Caltech and Alessandro Morbidelli from Nice observatory on earlier works), we spent the last couple years characterizing P9-induced dynamics from analytical grounds and trying to constrain the mass and orbit of Planet Nine to better precision.

The results of these endeavors are compiled in our new review article entitled “The Planet Nine Hypothesis,” published in Physics Reports today. Admittedly, in writing this manuscript, we ended up erring on the side of completeness over completion, so the paper is not exactly short. As a result, with an eye towards providing an “executive summary” of the results, in the next couple posts, I will highlight some of the main take-away points of the article, beginning with brief historical account of planetary predictions based on dynamical evidence.