Why isn't Eris considered a planet despite being the body of dominant mass?

Question

The 2006 definition of a planet states that a planet is a celestial body that

(a) is in orbit around the Sun

(b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium shape

(c) has cleared the neighbourhood around its orbit.

Point c is why Pluto allegedly is no longer a planet. However some "proponents" of the definition, namely those who use logic, figured out that according to this vague definition, if we take it literal and set a arbitrary border on "clearing the orbit", then either Pluto is still a planet or we have no planets at all. Therefore these guys actually set up an own definition, thereby replacing point c by the requirement that a planet must be the dominant mass in its orbit. This is not what the 2006 definition is telling, so these anti-Plutoers aren't actually proponents of the 2006 definition. They have their own definition according to which really only Pluto is no planet because Neptune dominates its orbit.

However their definition still doesn't exclude Eris from planethood. Its orbit is far from any of the eight recognized planets. From a vertical point of view, Eris' orbit intersects with that of Pluto. However Eris' orbit is highly tilted to the ecliptic and the two bodies don't really come close enough that their gravities would influence each other strongly. Eris is more massive than any Kuiper belt object (including Pluto). Eris itself is not really a KBO because it only intersects the Kuiper belt while most of its year being outside it. So there is no reason to ban Eris from planethood if point c states (according to some anti-Plutoers) that a planet must be the dominant mass throughout its orbit. So why don't these guys consider Eris a ninth planet?

Answer 1

You are correct that the IAU definition of "clearing the orbit" has the problem of being not explicitly quantified. And a complete clearing was obviously never the intention behind the definition. I like this statement by Steven Soter:

The IAU definition of a planet as a heliocentric body that "has cleared the neighborhood around its orbit” is problematic. Clearance is never complete because the asteroid and comet reservoirs are leaky, and resonant planet-crossing orbits can be stable. A more accurate criterion for planetary status is "dynamical dominance". An object is dynamically dominant if it sweeps up or scatters other objects from its orbital zone in a time much less than the age of the system (theoretical criterion), and/or if its mass is much greater than the total mass of all other objects in its orbital zone (observational criterion), where "orbital zone” can also be simply quantified.

However, there really are substantial difference in between Eris/Pluto and the regular planets concerning their dominance in their respective orbits. The Wikipedia article on "Clearing the neighbourhood" has a neat overview over a few of the proposed ways to make this difference more quantifiable.

Probably the most straight forward definition is the fraction $\mu = M/m$ between the mass of the object in question $M$ and the combined mass of other objects in its orbits zone $m$. But there are other options as well:

$$ \begin{array} { r |r |r |r |r |r |r |r } \mathrm{ Name} & \mathrm{Margot's \,\,} \Pi & \mathrm{Soter's \,\, } \mu & \mathrm{Stern–Levison\, parameter \,\, } \Lambda & \mathrm{Mass} (kg) \\ \hline Jupiter & 4.0 \times 10^4 & 6.25 \times 10^5 & 1.30 \times 10^9 & 1.8986 \times 10^{27} \\ Saturn & 6.1 \times 10^3 & 1.9 \times 10^5 & 4.68 \times 10^7 & 5.6846 \times 10^{26} \\ Venus & 9.5 \times 10^2 & 1.3 \times 10^6 & 1.66 \times 10^5 & 4.8685 \times 10^{24} \\ Earth & 8.1 \times 10^2 & 1.7 \times 10^6 & 1.53 \times 10^5 & 5.9736 \times 10^{24} \\ Uranus & 4.2 \times 10^2 & 2.9 \times 10^4 & 3.84 \times 10^5 & 8.6832 \times 10^{25} \\ Neptun & 3.0 \times 10^2 & 2.4 \times 10^4 & 2.73 \times 10^5 & 1.0243 \times 10^{26} \\ Mercury & 1.3 \times 10^2 & 9.1 \times 10^4 & 1.95 \times 10^3 & 3.3022 \times 10^{23} \\ Mars & 5.4 \times 10^1 & 5.1 \times 10^3 & 9.42 \times 10^2 & 6.4185 \times 10^{23} \\ Ceres & 4.0 \times 10^{−2} & 0.33 & 8.32 \times 10^{−4} & 9.43 \times 10^{20} \\ Pluto & 2.8 \times 10^{−2} & 0.08 & 2.95 \times 10^{−3} & 1.29 \times 10^{22} \\ Eris & 2.0 \times 10^{−2} & 0.10 & 2.15 \times 10^{−3} & 1.67 \times 10^{22} \\ Haumea & 7.8 \times 10^{−3} & 0.02 & 2.41 \times 10^{−4} & 4.0 \times 10^{21} \\ Makemake & 7.3 \times 10^{−3} & 0.02 & 2.22 \times 10^{−4} & 4.0 \times 10^{21} \\ \end{array} $$

In this paper by Sober he also made some nice plots showing the categorical difference between the two types of objects depending on their M, m and their semi-major axis a:

So, you see, while the IAU definition of "clearing an orbit" really is ambiguous in its interpretation, following its intent, there are quantifiable ways to show the existing difference between planets and dwarf-planets.

Answer 2

Because the "orbital neighbourhood" of Eris (and Ceres and Pluto) is defined as the entire Kuiper belt or in Ceres' case the entire Main belt in the planetary dominance discriminants user SpaceBread is showing. If the IAU defined the orbital dominance region as the body's hill sphere all three bodies would have to be classified as planets. Except if they took a strict approach that no body must even cross their hill sphere, in such case there wouldn't exist any planets at all. And they'd perhaps take the strict approach since, as you say, Eris actually isn't part of the Kuiper belt but just crosses it. Therefore, the IAU took the approach to call the entire belts their "orbital region" and from a 2-dimensional point of view rather than a 3-dimensional one (despite Pluto's and Eris' orbits being highly inclined to the ecliptic) in order to fit their definition of 2006.

According to the argument by some you provide, that another body dominates more than the three bodies, you're right that Neptune dominates Pluto's orbit but one might also conclude that Jupiter dominates Ceres' and farther on that Neptune dominates the entire Kuiper belt and bodies crossing it (like Eris). Still, Eris is the most massive known object beyond Neptune.