How the Planet Nine would affect the furthest asteroids

Hi there! You have heard of the hypothetical Planet Nine, which could be the explanation for an observed clustering of the pericentres of the furthest asteroids, known as eTNOS for extreme Trans-Neptunian Objects. I present you today a theoretical study investigating in-depth this mechanism, in being focused on the influence of the inclination of this Planet Nine. I present you Non-resonant secular dynamics of trans-Neptunian objects perturbed by a distant super-Earth by Melaine Saillenfest, Marc Fouchard, Giacomo Tommei and Giovanni B. Valsecchi. This study has recently been accepted for publication in Celestial Mechanics and Dynamical Astronomy.

Is there a Planet Nine?

An still undiscovered Solar System planet has always been dreamed, and sometimes even hinted. We called it Tyche, Thelisto, Planet X (“X” for mystery, unknown, but also for 10, Pluto having been the ninth planet until 2006). Since 2015, this quest has been renewed after the observation of clustering in the pericentres of extreme TNOS. Further investigations concluded that at least 5 observed dynamical features of the Solar System could be explained by an additional planet, now called Planet Nine:

  1. the clustering of the pericentres of the eTNOs,
  2. the significant presence of retrograde orbits among the TNOs,
  3. the 6° obliquity of the Sun,
  4. the presence of highly inclined Centaurs,
  5. the dynamical detachment of the pericentres of TNOs from Neptune.

The combination of all of these elements tends to rule out a random process. It appears that this Planet Nine would be pretty like Neptune, i.e. 10 times heavier than our Earth, that its pericentre would be at 200 AU (while Neptune is at 30 AU only!), and its apocentre between 500 AU and 1200 AU. This would indeed be a very distant object, which would orbit the Sun in several thousands of years!

Astronomers (Konstantin Batygin and Michael Brown) are currently trying to detect this Planet Nine, unsuccessfully up to now. You can follow their blog here, from which I took some inspiration. The study I present today investigates the secular dynamics that this Planet Nine would induce.

The secular dynamics of an asteroid

The secular dynamics is the one involving the pericentre and the ascending node of an object, without involving its longitude. To make things clear, you know that a planetary object orbiting the Sun wanders on an eccentric, inclined orbit, which is an ellipse. When you are interested in the secular dynamics, you care of the orientation of this ellipse, but not of where the object is on this ellipse. The clustering of pericentres of eTNOs is a feature of the secular dynamics.

This is a different aspect from the dynamics due to mean-motion resonances, in which you are interested in objects, which orbital periods around the Sun are commensurate with the one of the Planet Nine. Some studies address this issue, since many small objects are in mean-motion resonance with a planet. Not this study.

The Kozai-Lidov mechanism

A notable secular effect is the Kozai-Lidov resonance. Discovered in 1961 by Michael Lidov in USSR and Yoshihide Kozai in Japan, this mechanism says that there exists a dynamical equilibrium at high inclination (63°) for eccentric orbits, in the presence of a perturber. So, you have the central body (the Sun), a perturber (the planet), and your asteroid, which could have its inclination pushed by this effect. This induces a libration of the orientation of its orbit, i.e. the difference between its pericentre and its ascending node would librate around 90° or 270°.

This process is even more interesting when the perturber has a significant eccentricity, since the so-called eccentric Kozai-Lidov mechanism generates retrograde orbits, i.e. orbits with an inclination larger than 90°. At 117°, you have another equilibrium.

Now, when you observe a small body which dynamics suggests to be affected by Kozai-Lidov, this means you should have a perturber… you see what I mean?

Of course, this perturber can be Neptune, but only sometimes. Other times, the dynamics would rather be explained by an outer perturber… which could be the Planet Nine, or a passing star (who knows?)

Methodology

Before mentioning the results of this study I must briefly mention the methodology. The authors made what I would call a semi-analytical study, i.e. they manipulated equations, but with the assistance of a computer. They wrote down the Hamiltonian of the restricted 3-body problem, i.e. the expression of the whole energy of the problem with respect to the orbital elements of the perturber and the TNO. This energy should be constant, since no dissipation is involved, and the way this Hamiltonian is written has convenient mathematical properties, which allow to derive the whole dynamics. Then this Hamiltonian is averaged over the mean longitudes, since we are not interested in them, we want only the secular dynamics.

A common way to do this is to expand the Hamiltonian following small parameters, i.e. the eccentricity, the inclination… But not here! You cannot do this since the eccentricity of the Planet Nine (0.6) and its inclination are not supposed to be small. So, the authors average the Hamiltonian numerically. This permits them to keep the whole secular dynamics due to the eccentricity and the inclination.

Once they did this, they looked for equilibriums, which would be preferential dynamical states for the TNOs. They also detected chaotic zones in the phase space, i.e. ranges of orbital elements, for which the trajectory of the TNOs would be difficult to predict, and thus potentially unstable. They detected these zones in plotting so-called Poincaré sections, which give a picture of the trajectories in a two-dimensional plane that reduces the number of degrees-of-freedom.

Results

And the authors find that the two Kozai-Lidov mechanisms, i.e. the one due to Neptune, and the one due to the Planet Nine, conflict for a semimajor axis larger than 150 AU, where orbital flips become possible. The equilibriums due to Neptune would disappear beyond 200 AU, being submerged by chaos. However, other equilibriums appear.

For the future, I see two ways to better constrain the Planet Nine:

  1. observe it,
  2. discover more eTNOs, which would provide more accurate constraints.

Will Gaia be useful for that? Anyway, this is a very exciting quest. My advice: stay tuned!

To know more…

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