Is Planet X Needed? – Further Comments on Trujillo & Sheppard and Batygin & Brown

In my last post, Does (Brown and Batygin’s) Planet 9 (or Planet X) Exist?, I ignored the media squall which accompanied the publication on 20th January 2016 of a paper in The Astronomical Journal, Evidence for a Distant Giant Planet in the Solar System, by Konstantin Batygin and Michael E Brown, and discussed the coverage of the issue in New Scientist (here [paywall] and here) and in Scientific American (here [paywall]).

The idea that there may be a Planet X is not original to the Batygin and Brown paper.  It was also proposed in particular by Chadwick A. Trujillo and Scott S. Sheppard in a Nature paper A Sedna-like body with a perihelion of 80 astronomical units dated 27th March 2014.  The New Scientist and Scientific American feature articles were not informed by Batygin and Brown.  Scientific American explicitly referenced Trujillo and Sheppard (as well as a paper by C and R de la Fuente Marcos).

A key part of the evidence for a “Planet X” is that for the orbits of a number of trans-Neptunian objects (TNOs) – objects outside the orbit of Neptune – including the minor planet Sedna, the argument of perihelion is near 0˚.  That is, they cross the plane of the planets near when they are closest to the Sun. The suggestion is that this is not coincidental and can only be explained by the action of an undiscovered planet, perhaps 10 times the mass of the Earth, lurking out there way beyond Neptune. An old idea, the “Kozai mechanism”, is invoked to explain how Planet X could be controlling the TNOs, as noted, for example, by C and R de la Fuente Marcos in their paper Extreme trans-Neptunian objects and the Kozai mechanism: signalling the presence of trans-Plutonian planets.

I proposed a simpler explanation for the key finding.  My argument is based on the fact that the mass of the inner Solar System is dispersed from its centre of gravity, in particular because of the existence of the planets. Consequently, the gravitational force acting on the distant minor planets can be resolved into a component towards the centre of gravity of the Solar System, which keeps them in orbit, and, when averaged over time and because their orbits are inclined to the plane of the Solar System, another component at 90˚ to the first, towards the plane of the orbits of the eight major planets:

My suggestion is that this second component tend will gradually reduce the inclination of the minor planets’ orbits. Furthermore, the force towards the plane of the Solar System will be strongest when the minor planets are at perihelion on their eccentric orbits, not just in absolute terms, but also when averaged over time, taking into account varying orbital velocity as described by Kepler. This should eventually create orbits with an argument of perihelion near 0˚, as observed.

Has such an effect been taken into account by those proposing a Planet X?  The purpose of this second post on the topic is to look a little more closely at how the two main papers, Batygin & Brown and Trujillo & Sheppard tested for this possibility.

Batygin & Brown

The paper by Batygin and Brown does not document any original research that would have shown AOPs tending towards 0˚ without a Planet X by the mechanism I suggest.  Here’s what they say:

“To motivate the plausibility of an unseen body as a means of explaining the data, consider the following analytic calculation. In accord with the selection procedure outlined in the preceding section, envisage a test particle that resides on an orbit whose perihelion lies well outside Neptune’s orbit, such that close encounters between the bodies do not occur. Additionally, assume that the test particle’s orbital period is not commensurate (in any meaningful low-order sense—e.g., Nesvorný & Roig 2001) with the Keplerian motion of the giant planets.

The long-term dynamical behavior of such an object can be described within the framework of secular perturbation theory (Kaula 1964). Employing Gauss’s averaging method (see Ch. 7 of Murray & Dermott 1999; Touma et al. 2009), we can replace the orbits of the giant planets with massive wires and consider long-term evolution of the test particle under the associated torques. To quadrupole order in planet–particle semimajor axis ratio, the Hamiltonian that governs the planar dynamics of the test particle is [as close as I can get the symbols to the original]:

Η=-¼(GM/a) (1-e²)^-3/2 Σ⁴_i=1(m_ia_i²)/Ma²

In the above expression, G is the gravitational constant, M is the mass of the Sun, m_i and a_i are the masses and semimajor axes of the giant planets, while a and e are the test particle’s semimajor axis and eccentricity, respectively.

Equation (1) is independent of the orbital angles, and thus implies (by application of Hamilton’s equations) apsidal precession at constant eccentricity… in absence of additional effects, the observed alignment of the perihelia could not persist indefinitely, owing to differential apsidal precession.” [my stress].

After staring at this for a bit I noticed that the equation does not include the inclination of the orbit of test particle, just its semimajor axis (i.e. mean distance from the Sun) and eccentricity.  Then I saw that the text also only refers to the “planar dynamics of the test particle”, i.e. its behaviour in two, not three dimensions.

Later in the paper Batygin and Brown note (in relation to their modelling in general, not just what I shall call the “null case” of no Planet X) that:

“…an adequate account for the data requires the reproduction of grouping in not only the degree of freedom related to the eccentricity and the longitude of perihelion, but also that related to the inclination and the longitude of ascending node. Ultimately, in order to determine if such a confinement is achievable within the framework of the proposed perturbation model, numerical simulations akin to those reported above must be carried out, abandoning the assumption of coplanarity.”

I can’t say I found Batygin & Brown very easy to follow, but it’s fairly clear that they haven’t modeled the Solar System in a fully 3-dimensional manner.

Trujillo & Sheppard

If we have to discount Batygin & Brown, then the only true test of the null case is that in Trujillo & Sheppard.  Last time I quoted the relevant sentence:

“By numerically simulating the known mass in the solar system on the inner Oort cloud objects, we confirmed that [they] should have random ω [i.e. AOP]… This suggests that a massive outer Solar System perturber may exist and [sic, meaning “which”, perhaps] restricts ω for the inner Oort cloud objects.”

I didn’t mention that they then referred to the Methods section at the end of their paper.  Here’s what they say there (and I’m having to type this in because I only have a paper copy! – so much for scientific and technological progress!):

“Dynamical simulation. We used the Mercury integrator to simulate the long-term behaviour of ω for the Inner Oort cloud objects and objects with semi-major axes greater than 150AU and perihelia greater than Neptune.  The goal of this simulation was to attempt to explain the ω clustering.  The simulation shows that for the currently known mass in the Solar System, ω for all objects circulates on short and differing timescales dependent on the semi-major acis and perihelion (for example, 1,300 Myr, 500 Myr, 100 Myr and 650 Myr for Sedna, 2012 VP113, 2000 CR105 and 2010 GB17, respectively).”

In other words their model reproduced the “apsidal precession” proposed in Batygin & Brown, but since Trujillo & Sheppard refer to ω, the implication is that their simulation was in 3 dimensions and not “planar”.

However, could the model used by Trujillo and Sheppard have somehow not correctly captured the interaction between the TNOs and the inner planets?  The possibilities range from apsidal precession being programmed in to the Mercury package (stranger things have happened!) to something more subtle, resulting from the simplifications necessary for Mercury to model Solar System dynamics.

Maybe I’d better pluck up courage and ask Trujillo and Sheppard my stupid question!  Of course, the effect I propose would have to dominate apsidal precession, but that’s definitely possible when apsidal precession is on a timescale of 100s of millions of years, as found by Trujillo and Sheppard.