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Three years (and 2 fortnights) ago @plutokiller and I published our first #PlanetNine paper. But science is an iterative process, and since then we've made more progress. The result: an updated P9 Hypothesis (w/ Adams, Becker): arxiv.org/pdf/1902.10103…. Highlights in thread below. 
First, an exec summary of the paper: we did thousands of new P9 simulations, and realized that P9 is smaller (m~5Mearth), closer (a~400-500AU), more circular (e~0.2) and visually brighter (V~23) than we originally thought. Pic: @jtuttlekeane Now, let’s dig into the details a bit.
For starters, some history and an acknowledgement: the Planet Nine hypothesis is by no means the first inference of a new solar system member that is based upon anomalous orbital behavior of known objects. But this class of scientific proposals has a long and uneven record.
Despite countless attempts, to date, Neptune represents the only successful planetary discovery motivated by dynamical evidence. The figure shows the predictions of Le Verrier & Adams. Location was spot-on, but not orbit or mass. If we get P9 orbit to similar accuracy, A+
Following Le Verrier’s mathematical discovery of Neptune, the planet prediction business exploded - Babinet, Todd, Flammarion, Forbes, Pickering all spun the wheel. But no hypothesis was more famous than Lowell's Planet X. It was while searching for PX that Tombaugh found #Pluto.
Immediately there was a problem. PlanetX was supposed to be like Neptune, but Tombaugh’s new planet appeared dim and point-like. It soon became clear that the new member of the solar system was not THE PlanetX, and Pluto’s estimated mass steadily declined for the next 5 decades.
Since the discovery of the Kuiper belt, there has been a new wave of planet proposals: Brunini & Melita (2002), Gladman & Chan (2006), Gomes et al. (2006), Lykawka and Mukai (2008), Trujillo and Sheppard (2014), Volk and Malhotra (2017) just to name a few.
So what distinguishes all these hypotheses? 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. So let's place the P9 hypothesis within this framework.
Data first. The fig. shows long-period KBOs with i < 40 deg in physical space (inset plot shows the orbital poles). You can see clustering by eye, but it is far more striking among dynamically stable (purple) and metastable (gray) orbits than their unstable (green) counterparts.
Even in absence of P9, integrations of observed objects reveal how some KBOs experience much stronger chaotic interactions with Neptune. These perturbations erase any innate orbital structure of the distant belt - all KBOs are important but some are more important than others.
So suppose we only consider the (meta)stable KBOs. Why is orbital clustering among them a big deal? This is because if left to their own devices, the orbits would disperse on a timescale far shorter than the age of the solar system. SOMETHING (hint: P9) IS KEEPING THEM CONFINED.
Intriguingly, there is more. The distant solar system also contains a host of highly inclined/retrograde objects. KBOs with inclinations larger than ~30 deg are not a natural result of the solar system formation. An external gravitational influence is required to produce them.
Let's pump the breaks for a sec. How do we know all this clustering is not just by chance, or due to observational bias? This question is the subject of a new paper that @plutokiller led: arxiv.org/pdf/1901.07115…. To make a long story short, the false-alarm probability is ~0.2%.
Let's move on to theory. My grad-school advisor (Dave Stevenson) used to tell me that if you can't explain something on the board, then you don't really understand it. Section 4 of the paper aims to do just that for the P9 hypothesis (caution: it's gonna get wonkish).
1st step: alignment of orbits. By restricting the orbits to the plane, we can compute the orbit-averaged gravitational coupling between KBOs and P9 (+other planets). This yields a function (secular Hamiltonian) that approximately describes the long-term evolution of KBOs.
In fact, this Hamiltonian is time-independent (autonomous) and only depends on one angle and one action (so it is integrable). Therefore, it acts like a stream-function in phase-space, meaning that motion of KBOs just follows its contours.
Mapping the Hammy 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. This is why only distant orbits are clustered, and why P9 orbit must be anti-aligned.
A very similar exercise can be done for the degree of freedom of KBO dynamics related to inclination. The trick here is to assume that the KBO inclinations are not too high, which allows us to approximate forcing due to P9 with its quadrupole component of phase-averaged potential
Mapping this Hamiltonian in inclination-relative ascending node space, we see the bending of the Laplace plane: orbit poles of distant KBOs precess around the ang-momentum vector of P9, while a<250AU orbits circulate around Neptune. This is why the distant Kuiper belt is warped!
The case of retrograde KBO orbits is more complicated, and I’m not gonna try to explain it in a tweet. The key takeaway message though is that high-inclination dynamics ensues due to a well-defined secular resonance, which is driven by the *retrograde* longitude of perihelion.
Analytical Intuition is great, but comparison of P9-sculpted synthetic Kuiper belt with real data requires a more detailed description of dynamics, which can only come from N-body simulations. So we did thousands of them, varying P9's mass and orbital parameters.
Typical results from N-body simulations look like the fig below. Beyond a critical semi-major axis a~250AU, long-term stable KBOs cluster in longitude of perihelion, but this transition point, degree of clustering, etc. depend sensitively on P9 parameters.
As an example, the transition point from randomized to clustered orbital distribution in the simulations - a_crit - exhibits a very particular dependence on P9 eccentricity and semi-major axis, but depends much more weakly on P9’s mass and inclination.
Other statistical properties of the simulated distant belts are considerably more subtle, and to make matters worse (well, actually better), we carry out our analysis in terms of Poincare action-angle variables.
At the end of the day, we can compare the whole ensemble of simulations against the data in a relatively compact way. It is at this point that we find that m~5 Earth mass simulations work the best (this is clearly evident from the figure below).
