Tuesday, October 10, 2017

Theory

Every die-hard fan of the scientific method knows that Karl Popper was a baller. While his achievements clearly extend far beyond analysis of the scientific method alone, he is arguably best known for his work on empirical falsification. In essence, the idea behind his argument is that a theory is only any good if there exists a direct and clear experimental/observational way to demonstrate that it is incorrect. In other words, it is more important to point out avenues in which your theory can be wrong than to flaunt all the possible ways it could be right.

Why am I writing about this? Mike and I just spent a week at 14,000ft on the Big Island directly searching for Planet Nine, and I’ve been thinking a lot about how Popper’s falsifiability criteria apply to the Planet Nine hypothesis… Obviously, if we search the entire sky at sufficient depth and don’t find Planet Nine, then we are plainly wrong. But I don’t think this is going to happen. Instead, I think we (or some other group) are going to detect Planet Nine on a timescale considerably shorter than a decade - maybe even this year if we/they get lucky. Which begs the question: if a planet beyond Neptune is found, how would we proceed to determine that the Planet Nine theory is actually right?


Figure 1. Mike and I at the telescope - where colors don't exist.

I’m sure this question sounds incredibly stupid, so let me back up a bit. The Batygin & Brown 2016 AJ paper is by no means the first to predict a trans-Neptunian planet with a semi-major axis of a few hundred astronomical units. That accolade goes to George Forbes, who in 1880 proposed a planet located at ~300AU, based upon an analysis of the clustering of the aphelion distances of periodic comets (sound familiar?). Since then, a trans-Neptunian planet has been re-proposed over and over again, which brings us to problem at hand: whose trans-Neptunian planet theory is right and whose is wrong?

In my view, there is a very clear and intelligible way to answer this question. Each proposition of a trans-Neptunian planet is uniquely defined by (i) the data it aims to explain and (ii) the dynamical mechanism that sculpts the observations. So in order to be deemed correct, the discovered planet must match both of these specifications of the theorized planet.


Figure 2. The current observational census of distant KBOs.


When it comes to the Planet Nine hypothesis, point (i) is well-established: Planet Nine is invoked to explain (1) physical clustering of distant Kuiper belt orbits, (2) the perihelion detachment of long-period KBOs such as Sedna and VP113, as well as (3) the origin of nearly-perpendicular orbits of centaurs in the solar system. Embarrassingly, until recently our understanding of the “machinery” behind how Planet Nine generates these observational signatures has been incomplete. That is, although we have plenty of numerical experiments to demonstrate that Planet Nine can nicely reproduce the observed solar system, the theory that underlies these simulations has remained largely elusive.

The good news is that this is no longer a problem. In a recently accepted paper that I co-authored with Alessandro “Morby” Morbidelli, the theory of Planet Nine is characterized from semi-analytical grounds. So, for the first time, we not only know what Planet Nine does to the distant Kuiper belt, but we understand how it does it.

The first lingering question that Morby and I tackled is that of stability: how do the distant Kuiper belt objects avoid being thrown out of the solar system by close encounters with Planet Nine, when their orbits intersect? Turns out, the answer lies in an orbital clockwork mechanism known as mean motion resonance (MMR). When a Kuiper belt object is locked into an MMR with Planet Nine, it completes an integer number of orbits per (some other) integer number of orbits of Planet Nine. This strict rationality of the orbital periods allows the bodies to exchange orbital energy in a coherent fashion, and ultimately avoid collisions.

But how do such configurations arise in nature? Remarkably, the answer in this case is “by chance.” When the Kuiper belt first formed, a staggering number (roughly 30 Earth masses worth) of small, icy asteroid-like bodies were thrown out into the distant realm of the solar system by Neptune (for the interested reader, see papers about the Nice model here and here). Most of these objects were not fortunate enough to accidentally land into mean motion resonances with Planet Nine and were ejected from the solar system. However, the few that were, survive in the distant Kuiper belt to this day, and comprise the anti-aligned cluster of orbits that we observe. As a demonstration of this point, check out the simulated orbital period distribution of surviving Kuiper belt objects in one of our idealized simulations, and note that all distant bodies have rational orbital periods with that of Planet Nine:



Figure 3. Orbital distribution of long-term stable KBOs in an idealized P9 simulation.

All of this said, the full picture is of course not as clear-cut. Within the context of our most realistic calculations of distant Kuiper belt evolution, the clustered KBOs chaotically hop between resonances, instead of staying put. Still, the qualitative framework provided by analysis of isolated resonances holds well, even in our most computationally expensive simulations.

Ok so this resolves the question of how Kuiper belt objects survive, but it leaves open the question of why their orbits are clustered together. Intriguingly, a qualitatively different dynamical mechanism - known as secular interactions (see here for a neat discussion) - is responsible for the orbital confinement that we see. Plainly speaking, over exceedingly long periods of time (e.g. hundreds of orbits), Planet Nine and the Kuiper belt objects it perturbs will see each-other in almost every possible configuration along their respective orbits. Thus, their long-term evolution behaves as if the mass of Planet Nine has been smeared over its orbital trajectory, and its gravitational field torques the elliptical orbit of the test particle. The plot below shows the eccentricity-longitude of perihelion portrait of this secular dynamic inside the 3:2 mean motion resonance, where the background color scale and contours have been computed analytically and the orange curve represents a trajectory drawn from a numerical simulation. 


Figure 4. Eccentricity-perihelion diagram showing the secular trajectories of stable KBOs trapped in a 3:2 MMR with P9.

Indeed, the fact that the semi-analytic theory predicts looped trajectories that cluster around a P9 longitude of perihelion offset of 180 degrees implies that the raising of perihelion distances (i.e. lowering of eccentricities) of long-period KBOs and anti-aligned orbital confinement are actually the same dynamical effect. In other words, the reason that objects such as Sedna and VP113 have orbits that are not attached to Neptune is because they are entrained in the peculiar anti-aligned secular dynamic with Planet Nine. 

Finally, there is the puzzle of the highly inclined orbits. Whenever one sees cycling of orbital inclination and eccentricity, there is a temptation to invoke the Kozai-Lidov mechanism as the answer.  In the case of Planet Nine, however, the high-inclination dynamics are keenly distinct from those facilitated by the Kozai-Lidov effect. Perhaps the most obvious reason why the dynamics we observe in numerical simulations is not the Kozai-Lidov effect is that in our calculations, highly inclined KBOs develop the highest eccentricities when their orbits are perpendicular to the plane of Planet Nine’s orbit, in direct contrast with perpendicular and circular orbits entailed by the Kozai-Lidov effect. 

So if it’s not the Kozai-Lidov resonance, then what is it? As it turns out, the high-inclination dynamics induced by Planet Nine is characterized by the bounded oscillation of a octupole-order secular angle which is equal to the difference between the longitude of perihelion of the KBO relative to that of Planet Nine and twice the KBO argument of perihelion. How could we have ever thought it was anything else?… The plot below shows the high-inclination secular resonant trajectories executed by test-particles in our simulation plotted in canonical action-angle coordinates, with the observed objects shown in orange. Examining this plot closely, one detail that I’m reminded of is the fact that the few high-inclination large semi-major axis centaurs that we know of are actually the “freaks” of the overall population, since they all have perihelia on the order of ~10AU. Certainly, detecting a sample of these objects with perihelia well beyond 30AU would be immensely useful to further constraining the parameters of the model.
Figure 5. High-inclination dynamics, depicted in action-angle variables.
With the above rambling in mind, I will admit that all I’ve mentioned here is an introductory account of the paper. As such, it represents a considerable simplification of our actual calculations, so if you want to better understand the full picture, I can only urge you to read the paper itself. Importantly, however, the work presented in this manuscript not only provides us with a better understanding of how the observed census of distant KBOs has been sculpted by Planet Nine, it finally places the P9 hypothesis within the framework of Popper’s demand for falsifiability, and sincerely allows for the confrontation of the Planet Nine theory with the observational search. The final step is now to find it.

36 comments:

  1. I'm curious; do you know what the albedo of Planet 9 must be in order for you to detect it? If for some reason it is very dark and reflects almost nothing, would it be possible to detect it by checking if any stars are eclipsed?

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    2. We typically assume that it has a similar albedo to that of Neptune. There is reason to believe that it is in fact much *more* reflective (as pointed out in this paper: https://arxiv.org/pdf/1604.07424.pdf) If it is really dark, we would have to go out to higher magnitude. Occultations of stars, however is unlikely to be a viable avenue for discovering P9.

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  2. Thank you for keeping us up to date! It's amazing to follow your search. I know it's a bit early to know, but how thin or thick do you think the atmosphere will be? Given that it's supposed to be less massive than Uranus and Neptune, how different could it be?

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    1. There is really no way to know for sure. But my bet would be on a ~10% H/He envelope fraction - consistent with typical extrasolar super-Earths.

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  3. Could a secular or other interaction between P9 and the 30 Earth masses of Neptune-scattered KBOs be responsible for raising P9's perihelion?

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    1. In principle, absolutely. This is not something we've looked into in detail yet.

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  4. Hi Mike and Konstantin. I think,...I also calculated,..found from simulations and other sources 15 years ago that there should be Planet,...on orbit with period cca 20000 years. You proposed, that that one planet should be cca 10Earth masses. What is but exact limit for that mass? Is it 8-20,..or 5-100 Earths masses,...? Did you invoved into sim. also influence of Alpha Centauri, for to find mass of P9,10....?

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    1. No - the influence of Alpha Centauri is not in our simulations...

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    2. If we consider that gravitational influence of Alpha Centauri stars on our planets is cca like 1/2Earth mass in distance of supposed P9,10,..(300AU),..it is not too much

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    3. little more precise with calculator,...gravit. influence of all 3 Alpha Centauri in our Solar system is equivalent to 1 Earth mass body placed to distance 346AU-2Ldays-cca distance of P9,10,... Pavel Smutny

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    4. I dont know if you think about motions of barycenter for our Sun...there should be amendment of each planet proportional to its mass and to distance fron Sun.....

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    5. Unknown: I think you've calculated the gravitational acceleration caused by Alpha Centauri (which is proportional to the inverse square of distance). But this effects all Solar System objects nearly equally. If you want to know how the Solar System orbits are influenced, you'd need the tidal acceleration (proportional to the inverse cube of distance).

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    6. Little more about motions of Sun,..barycenter,..Maximal distance of Sun from barycenter is 4,4 radiuses of Sun. If only Jupiter is there (in our solar system), so 1 radius of Sun should be there,,,Amendment of Saturn could be till 2/3 of 1 radius,...Uranus, Neptune,..could give cca 3/5 of 1 radius,... together cca 3 radiuses,..but where is the rest,.._? Amendment, shift of barycenter due to P9,10 could be visible ,..especially after longer time,...Pavel Smutny

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  5. In your simulations how much does the inclination of the orbit of the perturb in P9 planet change as a result of the interactions? In the course of altering the inclination (Momentum) of the solar system by about 6 degrees, P9's orbit must also be altered (?) What would the change look like over time and might you be over-estimating the present inclination because of the its history of a higher inclination? Would the scattering reflect more of the historical average inclination than on the contemporary one?
    -jdk
    Thanks for the ongoing communications.

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  6. You're absolutely right, the inclination of P9 (with respect to the ecliptic) changes, too (because the ecliptic changes). We fully track all of that for our current inclination predictions. Sadly, our current inclinations aren't as precise as I would like, so in the end it doesn't make all that much difference.

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  7. Do the objects move clockwise around the trajectory in Fig. 4?

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  8. I would like to see a more practical visualization of how the orbits of the high inclination objects evolves.

    Perhaps a plot of the location of the perihelion of one of the high inclination objects plotted in ecliptic longitude and latitude, with arrows showing the direction it travels in its orbit to indicate the orbit's orientation, possibly scaled by eccentricity would work.

    This could also provide predictions of where they should be observed, in the interest of falsification.

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    1. Better yet, check out the orbits in physical space on the second figure of the very first post: http://www.findplanetnine.com/2016/01/premonition.html

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    2. After reading in the paper that the longitude of perihelion is a dog-leg angle I recalled that in the first paper latitude and longitude was used in one of the figures instead. That and Fig. 10 left me wondering what sort of path the perihelion (and the orbital pole) follows. Would they follow circles if plotted on a globe or a more complex curve?

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  9. In the initial 2016 papers a range of a9=200-2000AU, e9=0.1-0.9, m9=0.1-30Me was explored with w=0,180. When looking at the most up to date trans-Neptunian orbits (those highly aligned for a>250AU), there seems to be a small void (little to no orbital overlap) for small a9 near w=0. More specifically, it is quite apparent for all angles within 0 < w < 90 and 270 < w < 360, only a small angular window near w=0 is cleared out along the ecliptic. Your simulations included w=0, a9=200AU, e9=0.9, m9=10Me, which would roughly appear to fit within this region. What are the secular versus numerical results of these calculations? In particular, what is revealed for plots of: 1) e vs. delta(w) for different values of a, and 2) e vs. delta(w) for different MMR?

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  10. Intriguingly, the secular dynamics inside MMRs mirrors the purely secular dynamics (where the orbital motion of P9 and the KBO is assumed to be uncorrelated) at the corresponding semi-major axes rather well. I note that formally, this purely secular dynamics is inapplicable when orbits cross because of emergent singularities in the gravitational potential. Remarkably, however, these singularities are integrable allowing one to essentially get the right answer while being formally out of bounds of where the theory is supposed to work.

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  11. As ever, thanks for the time and effort you and Mike expend on explaining what's going on to we blog-followers, Konstantin.

    You wrote:
    "So if it’s not the Kozai-Lidov resonance, then what is it? As it turns out, the high-inclination dynamics induced by Planet Nine is characterized by the bounded oscillation of a octupole-order secular angle which is equal to the difference between the longitude of perihelion of the KBO relative to that of Planet Nine and twice the KBO argument of perihelion. How could we have ever thought it was anything else?…" 😂

    You're in good company ;-)

    "My reflection, when I first made myself master of the central idea of the ‘Origin’ was, ‘How extremely stupid not to have thought of that!’"

    (Huxley, Thomas Henry. 1887. On the Reception of the ‘Origin of Species,’ in Darwin, Francis (ed.). 1887. The Life and Letters of Charles Darwin, Including an Autobiographical Chapter (London: John Murray), volume 2, pp. 179-204. Quote is on v2:p197. View original page here:
    http://www.ucl.ac.uk/sts/staff/cain/projects/huxley/how_extremely_stupid)

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  12. Is data analyses from your last Hawaii trip over or there is still possibility that it is already found? :)

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    1. Barely started. I give it 15-20% chance that we found it in this run.

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  13. Is the production of the high inclination objects connected to Planet Nine's inclination relative to the Solar System, its eccentricity, or a combination of the two?

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    1. Oh, wait. The secular angle wouldn't include the difference between the longitude of perihelion if Planet Nine didn't need to be on an eccentric orbit to produce perpendicular orbits.

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    2. It's a combination of both. You are correct that this effect simply does not exist for circular angles, since P9's longitude of perihelion is ill-defined.

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    3. Does the number of perpendicular objects vary with Planet Nine's inclination?

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  14. We know lot of planetoids on prolongated eliptic orbits.There is group like on:https://upload.wikimedia.org/wikipedia/commons/2/2a/Planet_nine-150au-etnos_now-close_Jan-2017.png which have main axis cca 150AU, what is also in good coincidence with Titus-Bode Law. Closer P9 should have some such orbit, orientation. P10-could be on orbit like Brown with Batygyn foretold too. The second, ore distant group like on:http://www.nature.com/news/planets-graphic-jpg-7.33255?article=1.19182 could be in other resonance ratio but with P10,... or are those orbits such only due to projection of relative motions of planetoids towards Sun,..known planets, due to much more massive P10,P9,... than oficially foretold,... Pavel Smutny

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  15. In the paper its stated that 38% of all stable objects experience at least one high inclination excursion.

    Looking at Fig. 10 it appears that the excursions last 1.5 - 2 Gyr. Would roughly 15% undergoing one of these at a particular time be accurate?

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