Mean Motion Resonances and the Search for Planet Nine

Greetings! My name is Elizabeth Bailey, and I am a graduate student here at Caltech. As part of my work so far, I have addressed the ongoing search for Planet Nine, in particular the use of mean-motion resonances to infer its present-day location on the sky. 

A mean-motion resonance occurs when two bodies orbiting a central body have orbital periods related by an integer ratio. A great example is Pluto and Neptune. Pluto’s orbit is not a perfect circle, but rather a little elongated (e ~ 0.25). It actually crosses Neptune’s orbit, which might lead one to ask if they are on a collision course with each other. Fortunately, the answer is a confident “no.” Neptune and Pluto will never collide, because they are in a 3:2 (pronounced “three-to-two”) mean motion resonance with each other. Meaning, for every three trips Neptune completes around the sun, Pluto completes exactly two. It’s as if they’re dancing with each other. Every three times Neptune steps into the intersection of their orbits, Pluto steps twice somewhere else, and they don’t step on each other. 

So what does this have to do with Planet Nine? If Planet Nine exists, the distant KBOs it shepherds may very well experience resonant interactions with it. In fact, this was already pointed out in Konstantin & Mike's original Planet Nine paper, and is at this point relatively well understood. As a result, we can reasonably expect that at least some of the observed KBOs are currently locked into resonances with Planet Nine, and if we can understand the machinery of these interactions, perhaps we can infer the location of P9.

In a sense, the distant solar system is a lot like a giant cosmic nightclub. In this analogy, we are scanning the dance floor for Planet Nine, but it's hanging out in a dark corner somewhere in the back, while everyone is doing a P9-themed dance. So rather than looking for P9 itself, we are instead trying to figure out where it is by studying the KBO mosh-pit. This brings us to the key problem at hand: is this feasible in practice? We address this question in our recent work, published in the Astronomical Journal. 

The short answer is no - using resonances does not appear to be a feasible approach to find Planet Nine. Here's a figure from the paper comprised of seven histograms, corresponding to simulations with seven different eccentricities of Planet Nine (e_9 = 0.1, ..., 0.7) showing the count of objects occupying individual resonances. (The 2:1 resonance is located at "2" on the horizontal axis, and the 3:2 resonance is located at "1.5," and so forth.)

The takeaway point from this figure is that although you do find a lot of KBOs at the big-name resonances like 3:2 or 1:1, there are many objects occupying other resonances with larger integers in their names, like 14:5 or 2:7. There is a disturbing consequence of the mathematical nature of the planetary disturbing function (yes, it is actually called "The Disturbing Function" in celestial mechanics literature) which, upsettingly, suggests that these so-called high-order resonances become increasingly important when dealing with eccentric planets like Planet Nine, and the results of computer simulations presented in this work confirm this. In summary, because Planet Nine is eccentric, it carries out very complicated dance moves with the KBOs. 

It's worth mentioning that the simulations used to make this figure were simplified in comparison to reality. The canonical giant planets Jupiter through Neptune were treated as a static ring of mass (this is often referred to as the “secular” approximation), and the solar system is treated as a flat 2-dimensional object even though Planet Nine is, in reality, inclined. Think of it as a best-case scenario of sorts: in this physical setup, Planet Nine is the only active perturber of the KBOs. In the real solar system, Neptune is also on the dance floor, behaving in a very disruptive fashion. When KBOs get too close to Neptune, it flings them around. Sometimes those KBOs resume dancing with Planet Nine, but other times they just head out the door into interstellar space. 

Suppose, despite these complications, you could determine which individual KBOs are indeed in mean motion resonances with Planet Nine at this time. Then, if this information were to be of any use, you would then need to know the specific resonance of each KBO. In 3-D simulations, there is no obvious concentration of objects at particular resonances (see figure below). Hence, no matter how long we wait for more KBOs to be observed, we have virtually no hope of using resonances to predict Planet Nine's current location along its orbit. 

Although based on the results of this work it does not appear feasible to predict the present-day location of Planet Nine along its orbit, this does not by any means imply that Planet Nine is invisible to telescopes. There is still a well-defined swath of sky in which the search for Planet Nine continues. We have merely shown that mean-motion resonances with KBOs are not a useful tool for deciding where point the telescope, so we're back to systematically scanning the sky. Turns out that even in astronomy, the easy way is the hard way.

Planet Nine makes some KBOs go wild

Hi, everyone! I’m Tali, an undergrad at the University of Michigan. Last summer, I worked on a Planet Nine project with Konstantin and Mike, and although we didn’t find Planet Nine (yet!), we did look further into the stability of objects in the presence of Planet Nine. Turns out, not everything is stable!

In his last post, Konstantin explained that the main cluster of anti-aligned objects is able to remain stable due to mean motion resonances with Planet Nine. Their orbits always cross Planet Nine’s orbit, but such resonances allow the objects to avoid collisions. Here’s an example of what the dynamics looks like: the green orbit is Planet Nine, and pink orbit is an anti-aligned Kuiper belt object. The little blue circle is Neptune’s orbit, and the star is the Sun (not to scale).

What we see here is that the anti-aligned object experiences librations (=bounded oscillations) in the direction its orbit points (the longitude of perihelion). Meanwhile, Planet Nine’s orbit slowly precesses and changes direction as well.

BUT, it turns out that being in resonance with Planet Nine is not enough for stability. That’s because Neptune is still in the picture. Let’s look back at the animation above. Notice that as the pink orbits wags back and forth, its perihelion distance (=the shortest distance from the orbit to the Sun) changes. The pink orbit stretches (and hugs Neptune’s orbit) and then circularizes (and detaches from Neptune). The wider the “wagging the tail” oscillations are, the more pronounced the in and out behavior becomes. If the object librates with too large of an amplitude, it comes suuuuuper close to Neptune. And when that happens, it either gets ejected from the solar system or its dynamics changes entirely, and its behavior is no longer relevant to Planet Nine.

SO, the stability of the anti-aligned objects can be summarized by the two gifs below. When the longitude of perihelion libration (tail-wagging) is mild, our object experiences small changes in perihelion distance, and thus remains at a safe distance from the inner solar system. But, if the librations become too wide (and too wild), the object goes unstable, thanks to Neptune.

STABLE LIBRATIONS

UNSTABLE BEHAVIOUR

Now, the anti-aligned population is not the only one we looked at. Planet Nine carves out an aligned cluster of objects as well, which experience librations in longitude of perihelion, but this time, inside the orbit of Planet Nine:

As you can see this object is in the perfect stable location - it stays far away from Neptune (blue) AND doesn’t cross Planet Nine’s orbit, and just quietly librates in longitude of perihelion. This object is all set for life. Nothing will make it budge from this configuration.

BUT, there are objects that seem to be aligned at first, but suffer because their libration amplitudes are too large. Here’s an example of such an object:

In the animation above, the orbit spins too far and crosses Planet Nine’s orbit. This is not good for two reasons: (i) Planet Nine starts having collisions with this object and knocking it about, and, (ii) UNLESS the object is in a resonance with Planet Nine, it gets swept by Planet Nine into the Neptune scattering region. If you look at the animation carefully after the pink orbit crosses the green orbit, you’ll see that the perihelion distance of the object is slowwwly decreasing. When it gets small enough - when the object starts interacting strongly with Neptune - we get the same output as for the unstable anti-aligned objects (i.e. instability and a crazy jumping dog.)

So, what’s the bottom line? Not all anti-aligned objects are stable! And not all aligned objects are stable. And it all depends on their perihelion distance, which is closely tied with their librations in longitude of perihelion.

Moreover, it turns out that what kind of objects we find surviving through the end of our simulations depends on the initial conditions we put in. What do I mean by initial conditions? Well, for example, we expect that different scenarios of Planet Nine’s formation would have affected the initial configuration of the Kuiper belt in different ways. So, suppose we start with two different initial conditions: a “narrow” Kuiper belt (objects initially within a narrow interval of perihelion distances) and a more widely spread “broad” Kuiper belt.  And now we integrate these populations forward, in the presence of Planet Nine, in two separate simulations. Do these populations end up creating the same Kuiper belt?

In our recently accepted paper, we find that they don’t! In fact, the stable aligned population discussed above is completely missing from the “narrow” Kuiper belt. So, as the astronomy community continues to find more and more of these distant Kuiper belt objects, we might be able to start to tell which initial Kuiper belt we started with, and maybe how Planet Nine formed…

Read our paper here to find out more about Planet Nine, initial conditions, and stability!