Monday, October 25, 2021

The hunt is on!

 Back in August, we published our detailed analysis which gave predictions of where we should expect to find Planet Nine and how big and bright it might be when we do find it. One of the very interesting parts of the analysis was the result that Planet Nine should be on the closer and brighter end of our initial range of expectations. Planet Nine might be close enough and bright enough, that it might even already have been imaged in one of the many ongoing sky surveys taking place on moderately small telescopes all around the world. 

In many ways, the idea that Planet Nine might have already been imaged should not be a surprise. All of the planets discovered had been seen before they were recognized. Galileo saw Neptune -- without realizing it was a planet -- in 1612, more than 200 years before Neptune was found. Flamsteed cataloged Uranus as a star in 1690 --  a century before its discovery -- and it is even possible that Uranus was observed and cataloged by eye in the second century BCE. Eris, which I first spotted in 2005, showed up on photographic plates from the 1950s (from the same Palomar 48-inch Schmidt telescope used to discover it in 2005!). Images of Pluto have been found from 16 years before its discovery.

The problem, of course, is that just getting an image of an object in the solar system is not enough. You need to recognize it as something interesting. In all of the historical cases, the solar system body was simply cataloged as a star. The only way to know that it is not a star is to see it move.

Can't be too hard, right? Herschel saw Uranus one night and came back the next and saw that it had moved. We found Eris by taking three images of the right place in the sky over three hours and noticing one little bright star slowly sliding east.

Sadly, Planet Nine will be a little harder, but only because the data are not designed for searching for Planet Nine. Most of the big surveys out there cover large fractions of the sky, but only ever couple of days or weeks or months. If you look one night and see something where nothing has ever been before (a "transient" in astronomical language) it could be an asteroid, a supernova, a variable star, a satellite glint, a defect in the camera. Who knows? And when you see the same area of sky a month later there might be dozens of other transients nearby. Is one of them the same object, just moved? Who knows? 

In our new paper out today, we search for Plane Nine by combing through the transients from the Zwicky Transient Facility (ZTF; which, once again, uses the Palomar 48-inch Schmidt!), of which there are ~13 million over a three year period. Finding one Planet Nine in that big of a haystack is daunting.

The good news is we figured out a cool trick to make the searching moderately efficient. It relies on a simple yet delightfully clever insight from Matt Holman at the Harvard-Smithsonian Center for Astrophysics. While it is true that, when viewed from the Earth, asteroids and planets have complicated looping paths across the sky that are difficult to link, if viewed from the sun they simply travel on great circles. Since we're unlikely to move all of our telescopes to the sun, we have to fake it, by transforming everything we see in the sky into what it would look like if we were on the sun. It's more complicated than this, as we don't know how far away solar system things that we haven't discovered are, so we have to search over a range of potential distances and do more stuff too, but in the end the algorithm is able to smoothly link single detections of Planet Nine over many years. Check out the paper for the gory details.

Given the 13 million transients in the sky, sometimes they will accidentally line up and look like something real. We find that requiring 7 detections over 3 years allows us to avoid all of the coincidental detections and only find the real Planet Nine.... which we don't. It's not in the ZTF data. Insert sad face emoji.

Searching for and not finding Planet Nine is only useful if we can somehow figure out what it means, and here is the second part of this paper about which I am excited. We take the results from our paper two months ago and make what we call the Planet Nine Reference Population -- a collection of 100,000 statistically sampled realizations of our Planet Nine prediction, we inject these into the ZTF data, and we see which we would detect. Astonishingly, this one survey would detect 56% of the reference population, or, said differently, this survey covers 56% of the Planet Nine parameter space. 

Knowing this 56% is great, of course, but even better is that we know where (and how bright) the remaining 44% of the reference population is. Basically we took the treasure map from the last paper and we've removed 56% of the search area, giving us an updated treasure map (the updated treasure map is noisier looking than the original because it is a little more sparsely sampled with only 100,000 members of the reference population, but you can see the important points there).

 In this map you can see not only where Planet Nine is predicted to be, but how far away and how bright at all of the points across across the sky. And you can see what the ZTF analysis has done. ZTF would have found most Planet Nines brighter than ~20.75th magnitude, as long as they weren't too far south or in the southern galactic plane. 

My final favorite part of this paper is that now the reference population and treasure map are easily accessible for anyone who either wants to look for Planet Nine, wants to think about how to look for Planet Nine, or wants to ponder whether or not they have a telescope/survey/whatever that could possibly detect Planet Nine. Just go to the permanent archive of the reference population , download the reference population, and start making plots like the above. Software to help is even included, conveniently written in IDL for those born before 1970 (ahem). 

The reference population includes a flag of everything that would have been detected in ZTF, so you know where not to look. We're happy to include similar flags for any other survey that would have detected any part of the reference population. We've got a couple of new surveys that we are working on including, but get in touch if you'd like yours there. 

Planet Nine is out there. Except in the 56% part of parameter space we have now ruled out.

Monday, August 23, 2021

The orbit of Planet Nine

 Five and a half years after our proposal of the existence of Planet Nine, we have finally accomplished what is perhaps the most important task in aiding the search: we now know where to look.  We've had a pretty good general idea for years now, but we couldn't really give a full assessment of the range of uncertainties for where in the sky Planet Nine might be, how massive it might be, and how bright it might be. Now we can. You can read all of the details here.

Figuring all of this out sounds somewhat straight forward. Our task is simply to take the observations of all of the distant Kuiper belt objects (KBOs) whose orbits are affected by Planet Nine and use those to determine everything we possibly can. It only took five years. 

You can read the full paper for the details, but the five years consisted of a couple of critical steps:

  • Understand the physics. Until we had explored all of the various ways that P9 affects objects in the outer solar system, we didn't know which objects were most critical for constraining properties of P9. Many objects are more influenced by Neptune than by P9; if we had included those the signal from P9 would basically be washed out.
  • Understand the observations. There has been a lot of talk of observational bias in the population of objects in the outer solar system. Why? Because there is a lot of bias. Without fully characterizing the bias, we would have a difficult time disentangling what we are seeing from just the simple bias.
  • Understand how changes to parameters of Planet Nine change the outer solar system. We accomplished this through a huge number of numerical simulations where we placed a bunch of objects in the outer solar system, added the 4 giant planets, and then added different incarnations of Planet Nine, and watched the outer solar system evolve. Every parameters of Planet Nine (almost) has huge impacts on the outer solar system. Which is good, since that means we have leverage to learn about Planet Nine from the outer solar system.
  • Figure out a statistical treatment to compare the observations and the simulations. In practice this meant creating a maximum likelihood model combining the simulations, the bias, and the observations of each relevant object. 
  • Put it all together! Using the maximum likelihood model and what appears to be everyone's favorite statistic technique (MCMC), we now have probability distributions of all of the Planet Nine parameters.

Sorry it took so long.

Here are some fun numbers from the analysis. Planet Nine has a mass of  6.2/+2.2/-1.3 Earth masses, a semimajor axis of 380/+140/-80 AU, and inclination of 16+/-5 degrees,  and perihelion of 300/+85/-60 AU. We can turn all of that into a map of where to look in the sky and of how bright and far away Planet Nine would be at any position in the sky (brightness depends on some assumption; you might want to read the detail in the paper).

That plot on top if a full-sky view going from 360 to 0 in Right Ascension. The equator, ecliptic, and +/-15 degrees of the galactic plane are shown, if that helps to orient you. The colors show the probability of find P9 at any point in the sky. The highest probability is near aphelion, around ~60 degrees in RA, pretty close to the galactic plane. Not surprisingly, this is where we have been concentrating with our Subaru search. The bottom two panes show the distance and brightness. The median distance at aphelion is around 500 AU. The median brightness is around 22nd magnitude. There is a lot of room for variation however! At its brightest predicted magnitude, P9 could be found in many ongoing all-sky surveys. At its faintest, it will require dedicated searches with large telescopes.


That plot above is truly the point of the whole paper, but there are two more plots of which I am a big fan so I want to share these also. 

As mentioned earlier, there had been a lot of discussion of bias and its effect on what we see in the outer solar system. I am here to tell you bias is real. Also I am here to show you that it doesn't cause the clustering that we see.


Here is a plot that shows a lot of things all at the same time. First, lets concentrate on the green points. Those are 11 real objects in the sky and they show those objects semimajor axis ("a") versus their longitude of perihelion aka the direction their orbits point in space aka "delta varpi". The fact that 10 of those 11 are aligned is the whole reason we are here. 

But wait! There is bias. Maybe those are all lined up like that simply because that's where people have looked? Now look at the blue bars. These show, for each object, the probability of the object having been found at all values of longitude of perihelion if the true longitudes of perihelion were distributed uniformly in space. There are two major regions of bias. One is at longitude of perihelion of ~90, and a weaker one is at ~300. These are caused by the DES and OSSOS surveys which both looked in very specific directions and found many objects. But that's not where the cluster is! 

Now look at the red in the background. This shows how the green points should be distributed if Planet Nine were out there messing things up. Looks pretty good, no?

OK, one more:

This is the same general idea as above, but now we are plotting the pole position of the orbit of each of the distant objects. A point right in the middle has an orbital plane exactly the same as the planets, a point displaced in any direction from the center is tilted, and the direction of displacement tells the direction of the tilt. As you can see from the green points, the real objects have, on average, about a 15 degree tilt towards zero. Red again shows the Planet Nine prediction. Blue shows the biases. The biases here look weird and spider webby, again mostly because of DES and OSSOS. There is a lot of bias, and the observations generally fal along the lines of bias. But the bias clearly cannot account for the fact that the orbits are tilted and that they are tilted in one direction.

Put these two plots together and you get a 99.6% chance that the objects are clustered, rather than uniform. That sounds pretty good to me.


The paper has even more inside (at 26 pages believe me, there is a lot more), but these are my favorite highlights. I think of that first plot as the Planet Nine Treasure Map. Time to start hunting, my fellow pirates.

Tuesday, April 13, 2021

The Inner Oort Cloud Connection

The census of Kuiper belt objects spanning the distant solar system has sure come a long way. Even if we strictly limit our object sample to those with semi-major axes above 250 AU, or equivalently, an orbital period greater than about 4,000 years, the number of TNOs clocks in at nineteen — a staggering improvement from the six we had back in 2016. 

Some of these TNOs cluster together, while others do not. More importantly, a simple glance at the orbital structure of the outer solar system reveals an intriguing pattern: the degree of clustering is intimately connected to their present-day dynamical stability. In other words, orbits strongly perturbed by Neptune and those on their way out of the solar system (rendered in green) are all over the place. Meanwhile, long-term stable Sedna-like orbits (rendered in purple) appear as though a cosmic florist has carefully arranged them into a bunched bundle of trajectories. Meta-stable orbits (shown in gray) experiencing slow orbital diffusion are caught somewhere in the middle.

The degree to which the overall orbital clustering is a physical effect, as opposed to being a figment of observational bias or a statistical chance, has been a subject of discussion ever since the Planet Nine Hypothesis saw the light of day (see here, here, here, here and here, as well as Mike’s post below). To this end, individual surveys having only searched limited areas of the sky cannot sufficiently overcome their own observational biases to determine the absence or presence of orbital alignment rigorously. However, even after taking observational biases into account, a combined observability analysis of all available data shows that distant KBOs jointly cluster in eccentricity and angular momentum vectors at the ~99.8% significance level.

Dynamical sorting of KBOs is another matter: although there are distinct biases pertinent to high-eccentricity orbits documented in the literature, no conceivable observational bias can differentiate KBOs based upon their dynamical stability, thereby causing long-term stable orbits to cluster more strongly than their unstable counterparts. Put simply, when astronomers discover TNOs, pretty much the only well-known information about them is RA/DEC, heliocentric radius, and Vmag for approximately one year preceding follow-up. For this reason, it is impossible for surveys to bias themselves based on the Lyapunov exponent, or the rapidity of exponential divergence of cloned initial conditions, of an orbit. On the contrary, within the framework of the Planet Nine hypothesis, there is a close link between the degree of orbital clustering and dynamical stability. Thus, if we interpret the picture insinuated by the data at face value, the observed “width” of the stable-orbit cluster tells us something about Planet Nine’s mass and orbit.

A couple of years ago, we published a review article detailing thousands of simulations suggesting that P9 is less eccentric and less massive than we initially thought (i.e., m~5 Earth masses and e~0.3 instead of m~10 Earth masses and e~0.6). Even more refined models having more particles and matching the data more precisely are possible and are worthwhile to pursue. However, the question we asked ourselves during the height of the pandemic is a different one: are essential physics missing from our simulations? Through our continued and incessant probing of the model, we have discovered that the answer to this question is “yes.”

All our P9 calculations thus far have treated the solar system as an isolated entity, with the sun at the center of the simulation and all objects that attain a heliocentric distance over 10,000 AU as being forever lost to the abyss. Although this is a viable approximation scheme for understanding the present-day solar system, it is an unrealistic full-scale model considering the sun’s birth environment. After all, enrichment of meteorites in short-lived radiogenic isotopes suggests that the sun formed in a massive cluster of stars, somewhat akin to the Orion Nebular Cluster. This fact immediately renders the formation of a perihelion-detached quasi-toroidal cloud of icy debris at a few thousand AU all but inevitable. As our newly published P9 simulations show, this cloud of debris does not remain dormant throughout the lifetime of the solar system. Instead, Planet Nine’s gravity acts to inject these “inner Oort Cloud” objects into the trans-Neptunian region of the solar system, mixing them in with the distant scattered disk. A necessary consequence of the process is that newly-injected debris tends to broaden the cluster, requiring a more eccentric P9 to match the data.

Let us discuss this process in greater detail. The formation of giant planets, particularly Jupiter and Saturn, happens while the sun is encircled by its primordial disk of gas and dust during the first three million years of its lifetime. As the giant planets gain mass, they necessarily scatter icy debris outwards. If the solar system were to evolve in isolation, the debris would eventually attain positive orbital energy and become unbound from the sun. In a star cluster, however, the situation is more delicate. As the planetesimal obits become sufficiently elongated, they become susceptible to perturbations from passing stars that can detach them from the gravitational bashing of newly-born Jupiter and Saturn, thereby “freezing” them in place. Distinct from the classical Oort Cloud that likely formed later, we refer to this hypothetical population of icy asteroids envisioned to orbit the sun at thousands of astronomical units as the inner Oort Cloud.

We have simulated the formation of the inner Oort Cloud, modeling the sun’s birth cluster as a Plummer sphere, with parameters chosen to mimic the Orion Nebular Cluster and solar residence time chosen such that it does not violate cold classical Kuiper belt constraints. Shown in terms of Keplerian orbital elements, a typical inner Oort Cloud that forms in our simulations looks as follows:

Now, the conventional story is that the inner Oort Cloud formation stops the moment the sun leaves its birth cluster, and this frigid cloud of icy debris sits there “forever.” Planet Nine, however, alters this picture on a qualitative level. Due to the long-term gravitational pull of P9’s orbit, inner Oort Cloud objects evolve on billion-year timescales, slowly getting re-injected into the outer solar system. So what happens to them? We have simulated this process, accounting for perturbations from the canonical giant planets, P9, passing stars, as well as the galactic tide, and have found that these re-injected inner Oort Cloud objects can readily mix in with the census of distant Kuiper belt objects, and even exhibit orbital clustering. However, the degree of clustering they show is prominently weaker, suggesting that a more eccentric P9 is required to explain the data. Another intriguing aspect of these calculations is that they produce a more radially extended distribution of long-period TNOs, providing an intriguing potential explanation for the abundance of highly long-period orbits, such as that of the Goblin.

All in all, as scientists, our primary role is to continue hammering away at the hypothesis, and our new simulations indicate that there exists an intriguing additional mode of P9-induced dynamics affecting our efforts to pin down the orbit of Planet Nine. With each theoretical breakthrough, the observational search comes into sharper focus.

Tuesday, February 16, 2021

Is Planet Nine finally dead?

 Konstantin and I have a recurring conversation. One of us says "Wait! I think we're wrong about the evidence for Planet Nine. Here's why..." and proceeds to spin a hypothesis about how we could have been misled by the data or that we are predicting something that is not happening, and then we will carefully go through the entire Planet Nine analysis over again before convincing ourselves that, no, all is normal in the outer solar system (where by normal we mean that Planet Nine is happily orbiting the sun waiting to be seen). So far, we have not found a reason to discard the Planet Nine hypothesis. Will one some day? Perhaps. If the evidence for the existence of Planet Nine were to unravel we would be sad, but we would have to give up the idea. We have been prepared for that moment since the day we first proposed Planet Nine.

Has the moment now come? A new paper by Napier et al. appeared on the archive the other day with the fairly conclusive title of No Evidence for Orbital Clustering in the Extreme Trans-Neptunian Objects, which, well, sounds pretty bad for Planet Nine. There was a quick flurry of requests for comments on the paper, but it seemed like it was better to carefully read and go over the analysis than to simply glibly comment. It's a substantial paper with a lot buried in it, so it took a while, but I think I have now fully digested the paper and can comment on what is going on.

The central idea of the paper is pretty simple: do a careful analysis of where three separate surveys pointed their telescopes and use that analysis to examine whether or not their is any clear sign of orbital clustering in the outer solar system. The specific method is different, but the paper follows the general prescription of our paper from 2019 in trying to determine if the orbits of distant KBOs are aligned in the same general direction and if the orbits are tilted in the same direction. These effects are precisely what is predicted by the Planet Nine hypothesis. In our 2019 paper, we analyzed the 14 objects known as of that time that have semimajor axis (average distance from the sun) beyond 230 AU. We found that we could rule out a non-clustered outer solar system at the 99.8% confidence level. Pretty good right?

Our 2019 paper performed one other critical analysis (this will matter below!). We showed that the 4 objects found by the OSSOS survey were too few to be able to detect the clustering. If you think of the measure of clustering as a single number (which it isn't; it's a 4-dimensional point, but we'll pretend), you could say that we measured a clustering of 4+/-1, for example. But the OSSOS survey could not detect clustering because they measured 0+/6. The uncertainty on their measurement was larger that the expected effect. All was happy in Planet Nine land.

OK, so back to the Napier et al. analysis. They look at  14 objects that have been discovered since our original 2015 paper. After much work they conclude that they can rule out a non-clustered outer solar system at only the 83% confidence level. Bad news for Planet Nine! Taken at face value it seems that the evidence for clustering has weakened or gone away entirely. 

 Statistically, it is weird that a 99.8% significant result would fade so quickly with just a little new data. Interestingly, when Napier et al. looked only at the objects which are identical between their analysis and our analysis, they see clustering at the 99.5% significance level in excellent agreement with our original analysis, so we all agree that those 2019 results appear on firm ground.  So: what is up with the 7 new objects in their analysis that suddenly drops the significance? Let's find out who they are and what surveys they came from.

First, let's look at the measure of clustering from our 2019 paper:


What you are looking at are measures of two parameters for each of the 14 distant objects in our analysis. x,y shows, basically, the direction that the orbit points (with some additional complications). p,q shows the direction that the orbit tilts (with some additional complications). As you can see, the orbits of 11 of the 14 objects point towards one quadrant and, on average, they are tilted in a common direction. Napier et al. followed our prescription here, so you can see their versions of these same plots in their Figure 4. The big red dots are the average position of the black dots and show the strength of the clustering. (To complete the analysis we must determine if, when observational biases are included, the red dots are far enough away from zero [unclustered] to be significant; they are, at the 99.8% confidence level). 

Let's add the new Napier et al. objects to these plots:


 The green points are from the spatially concentrated (thus highly biased) DES survey, the purple from the much broader survey of Sheppard and Trujillo (there is one additional object that Napier et al. include that was never reported to the Minor Planet Center; we restrict our analysis to objects whose detection history we can track; that one unreported object is down in the lower left of the lefthand plot). According to Napier et al., the inclusion of these points which are clearly consistent with the previous measurement of clustering into the analysis makes the evidence for clustering inconclusive. WHAT?

It took my a long time to understand this strange-looking behavior (again, instant commentary is not always the best commentary). But I think I get it. I believe it is all about the measurement uncertainty that we talked about above. Our paper says that the clustering statistic is 4+/-1. Their paper says the clustering statistic is consistent with zero. But I believe that their clustering statistics is something more like 4+/-6. Thus consistent with zero, as they correctly claim, but not good enough to detect the clustering that exists. Do I know this for sure? No, because they did not publish their uncertainties (an interesting question would be how did whoever refereed the paper let them get away without publishing their uncertainties!). 

OK, but wait. They have about the same number of objects in their analysis as we did in our 2019 analysis, so why would our uncertainties be smaller? I think that the answer is primarily due to the fact that DES, which only looked in one direction in the sky, happened to look right in the direction of the clustering. Why? Dumb luck. Their survey was designed long before there was an inkling of Planet Nine. And the Planet Nine clustering is independent of any DES data.

Why is that a problem, though? Imagine a situation where I look around for a few nights and notice that the sun is setting in the west the modest number of times I happened to notice the sunset. You then decide to studiously look west. You see sunsets, but only in the west. You have performed a very biased survey, so when you do your statistically correctly you state that you cannot confirm that the sun sets exclusively in the west because your large survey would only see western sunsets thus the direction of the setting sun is statistically conclusive with being in all directions. It seems weird, but adding in highly biased data that is biased in the precise direction as the signal for which you are looking makes it harder to confirm the signal in the first place. But there is a solution. That solution? Publish your uncertainties.   

Napier et al. get it right, in the end:

Ah ha! Sadly, they don't check for consistency with the previously measured clustering, or they would see that, indeed, the ETNOs were already known to be clustered precisely where DES has looked. 

I think this solves the mystery of how adding in objects which appear quite clustered makes the significance of your clustering appear to go away.

I think that the right conclusion is that the highly-biased DES data is consistent with the previous measurements of clustering, but that the bias from DES is strong enough that we should probably not be surprised by this. In the end, the previously measured clustering from our 2019 paper is still valid (and has actual uncertainties published), and the conclusions of that paper remain. The clustering of distant Kuiper belt objects is highly significant. It's hard to imagine a process other than Planet Nine that could make these patterns. The search continues.

Tuesday, February 26, 2019

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.


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.


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.