Does Soter take into account Pluto's and Orcus' masses when determining Neptune's planetary discriminant µ?

Does Steven Soter's planetary discriminant µ1 for Neptune take into account the masses of Pluto, Orcus and other Kuiper belt objects crossing or coming very close to Neptune's orbit? If so, Neptune would probably have the smallest planet mass versus orbit mass ratio, but the lowest µ among the eight recognized planets is that of Mars. Since Mars is clearly outside the main belt, the belt cannot be applied to Mars unlike Ceres.

While Pluto, Orcus and Neptune never get close enough to each other to influence themselves, Soter obviously takes all the mass approximately around the body's orbit into account, as is the case with Ceres: for Ceres' µ Soter took the mass of the entire main belt into account, resulting in a µ of 0.33 (as Ceres has 9.39e+20 kg and the entire belt a mass of about 3e+21 kg). It is unreasonable to do that because Ceres' perihelion is 2.56 AU and its aphelion 2.98 AU, while the entire main belt stretches from 2 AU to 3.5 AU, therefore Ceres' µ is actually higher.

Soter seemingly also took the mass of the entire Kuiper belt into account when calculating Pluto's and Eris' µ, despite Pluto leaving the Kuiper belt when going towards perihelion and Eris leaving the belt even for most of its orbit when going towards aphelion.

Consequently however, Soter should have taken Pluto's and Orcus' mass into account when determining Neptune's µ, and Neptune's mass in turn when determining the µ of Orcus or Pluto. Neptune's µ is 2.4e+4 and Pluto's µ is 0.077. Do these values include the masses of the other body of those? If no, that would be inconsistent with the calculations for Ceres, obviously, as well as for Eris.

1What is a planet? Steven Soter (2006) and Planetary discriminant

Does Steven Soter's planetary discriminant µ for Neptune take into account the masses of Pluto, Orcus and other Kuiper belt objects crossing or coming very close to Neptune's orbit?

No, it does not. It also excludes the mass of Neptune in calculating the planetary discriminant values for Kuiper Belt Objects that are in orbital resonance with Neptune. This treatment is very consistent with how a planet's moons are excluded from consideration in the planetary discriminant µ for that planet. The value for the Earth would be 81.3 if the Moon's mass was not excluded from the calculation of the Earth's µ.

There are several possible outcomes with regard to interactions between a planet-sized object and a lesser object:

1. The two objects can collide. This may result in the planet-sized object becomes even bigger, and if there is a partial rebound, some of the residual can be captured into orbit.
2. A close non-collisional encounter can result in the lesser object being ejected from the solar system or captured into orbit.
3. Repeated interactions can result in the lesser object being forced into a non-collisional, long-lived orbital resonance.

These are all examples of mechanisms by which a planet "clears its neighborhood". Forcing objects to be long-lived moons or in long-lived orbital resonances count as clearing the neighborhood just as much as do collisions and ejections.

Consistent with this line of thought, Soter excludes the Moon's mass when calculating the Earth's µ, and he similarly excludes the mass of the Plutinos when calculating Neptune's µ. By extension, he also excludes the mass of Neptune when calculating a Plutino's µ.

• So if I understand it correctly, in Soter's µ only bodies of similar size and mass that didn't affect each other significantly as yet can be included? Isn't it just a question of time then when two similarly-sized bodies come too close to each other, e.g. Pluto and Eris (who cannot come close enough to affect each other btw)?
– John
Jun 16 '21 at 17:13
• @John No, that's not correct. Soter makes an exception for Neptune and the Plutinos because those resonances are long-lived. Soter's calculation was fairly simple. Mercury, Venus and the Earth can eventually collide with any of the Near Earth objects; Mars with any of the Mars-Crossing Asteroids; Ceres with any of the Main Belt Asteroids; Jupiter, Saturn, and Uranus with any of the Centaurs; Neptune with any of the Centaurs and the non-resonant Kuiper Belt Objects; and Pluto with any of the Kuiper Belt Objects (but not Neptune). Eris, btw, is not a KBO. Jun 16 '21 at 17:57
• @David Hammen: The proper term for Eris is Scattered Disk Object? Jun 16 '21 at 18:50
• @JohnHunt Correct. The scattered disk (mostly) lies outside of the Kuiper Belt. Jun 16 '21 at 18:57
• @DavidHammen Right, Eris is outside the Kuiper belt for most of its orbit as written in my question description. Since there is little Eris can collide with, and being the most massive object beyond Neptune, why isn't it recognized as a planet then?
– John
Jun 17 '21 at 5:01

Pluto and Orcus are both Plutinos, and are locked into the 2:3 orbital resonance with Neptune. As such, Neptune's gravitational influence controls their orbital periods, and their miniscule relative masses to Neptune are excluded from determinations of the planetary discriminant.

• Neptune doesn't affect them significantly in the 2:3 orbital resonance. If Neptune suddenly disappeared, their orbits would remain the same.
– John
Jun 16 '21 at 12:21
• @John You are incorrect in that supposition. Neptune's influence )cause and maintains a libration)[en.wikipedia.org/wiki/Pluto#Other_factors} in Pluto's argument of perihelion of about 90°, over a period of about 10,000 years. Without Neptune, that doesn't happen. Jun 16 '21 at 13:04
• So which masses are included in determining Neptune's µ? Every mass around Neptune's orbit is "miniscule" in comparison to Neptune.
– John
Jun 16 '21 at 13:42
• @John: Soter estimated that over a long astronomical period of time, Neptune might collide with any of the Centaurs but only 5/783 of the Kuiper Belt Objects, resulting in a µ of $2.4\times10^4$ for Neptune. But even if he didn't use the orbital resonance exclusion rule, Neptune would still have a µ of 560. (And by extension, Pluto's µ would shrink to 0.00013. Pluto's µ excludes Neptune's mass.) Jun 16 '21 at 17:14