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Whether the Moon (Earth's natural satellite) is a planet is an old debate in planetary science.
Though I do not agree that Moon is a planet, I found something difficult to clearly understand.

According to the 2006 IAU definition of planet, a planet must satisfy Clearing the neighbourhood condition. And some criteria for the condition are introduced in the Wikipedia article about Clearing the neighbourhood.

  • Stern–Levison's $\Lambda$

$$\Lambda = \frac {m^2}{a^{\frac 3 2}} k$$

  • Margot's $\Pi$

$$\Pi=\frac {m}{m^{\frac 5 2}a^{\frac 9 8}} k $$

In the above equations, $m$ is the mass of the body, $a$ is the body's semi-major axis, and other factors are described in the Wikipedia article. Both criteria are said to be satisfied if the calculated value is greater than 1.

My calculations are simple. Let $m_{Moon} = 0.0123 \times m_{Earth}$ and Moon's other factors (such as $a$, $k$ and $M$) have the same values as Earth's. Then I can get...

$$\Lambda_{Moon} = 0.0123^2 \times \Lambda_{Earth} = 23.147$$

$$\Pi_{Moon} = 0.0123 \times \Pi_{Earth} = 9.963$$

Both $\Lambda_{Moon}$ and $\Pi_{Moon}$ thus satisfy the Clearing the neighbourhood condition. Though I do not calculate the Soter's $\mu$, it will not change my results.

What is wrong with my calculations and results? Do I miss something about the condition and the criteria? Or is there something wrong with the Clearing the neighbourhood condition?


Edit in response to comments:

This question can be related to the idea that the Clearing the neighborhood condition is inadequate and ambiguous for the planet definition. More deeply, this question can be related to the idea that 2006 IAU definition of planet is logically ambiguous and clumsy.
For example, the 2006 definition of planet defines the Moon as a satellite and not a planet. If one celestial body can be defined as a planet or not without rules and conditions, why other conditions are needed? All we need is just a planet list, not other planet conditions.

And...

Frankly speaking, the Clearing the neighborhood condition is like comedy. The condition says "Planet must sweep out other bodies" ... Then says "Except satellites" ... Then again says "Except bodies having no significant size" ... Then again "Except bodies having resonant orbit" ... Then again "It may take billions of years" ... Then again "Sweep-out process and time are influenced by other planets and bodies, actually" .......

It is obvious that sometimes the Clearing the neighborhood theory is useful. But it is inappropriate for the rule and definition requiring strictness.

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    $\begingroup$ The 2006 IAU definition was designed to deal with non-satellites, such as Pluto and other large Kuiper Belt objects (and Ceres as well). Complaining that the "clearing the neighborhood" condition is superfluous for the Moon is missing the point. $\endgroup$ – Peter Erwin Nov 23 '18 at 10:28
  • $\begingroup$ @PeterErwin : Your point is already mentioned and commented many times by others. And I also know the meaning of 2006 IAU definition. Do you think astronomers who claim that Moon is a planet are missing the point? $\endgroup$ – pdh0710 Nov 23 '18 at 11:19
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    $\begingroup$ @uhoh : Thank you for your advice. I also think the Edit section has some problems. But the original purpose of the Edit section is to prevent repeated comments and queries. As you can see, this question has been involved with more complex and broader issues than I expected. So I hesitate to delete the section. $\endgroup$ – pdh0710 Nov 25 '18 at 3:42
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    $\begingroup$ @uhoh : I edited the paragraph. My intention for the paragraph is "Why it is like comedy? Because.......". Would you please let me know whether my intention is expressed well? (I'm not good at English) $\endgroup$ – pdh0710 Nov 25 '18 at 4:15
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    $\begingroup$ @uhoh : Oh, thank you. It looks very good and what I wanted to express are contained well. The only problem is... It looks too nice. It does not look like a comedy at all. $\endgroup$ – pdh0710 Nov 25 '18 at 5:49
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According to the 2006 IAU definition of a planet, a celestial body must meet all 3 criteria to be considered a planet. The Moon fails the first criteria "1. Is in orbit around the Sun". The Moon orbits Earth, not the Sun, and thus it doesn't have a legitimate "neighbourhood" (solar orbit) to clear in that context.

Also to note, according to the Clearing the neighbourhood article, this criteria is contentious as it is vaguely written and Earth, Mars, Jupiter and Neptune have not "cleared the neighbourhood" of their respective solar orbits.

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    $\begingroup$ You think this problem as too simple one. First of all, this question can be related to the idea that the "Clearing the neighbourhood" condition is inadequate and ambiguous for planet definition. More deeply, this question can be related to the idea that 2006 IAU definition of planet is logically ambiguous and clumsy. 2006 IAU definition of planet begins with Moon is already set as a satellite, not a planet. If you can set a celestial body whether planet or not a planet, why other conditions are needed? All you need is just a list of planetary bodies, not other conditions. $\endgroup$ – pdh0710 Nov 22 '18 at 3:04
  • $\begingroup$ Perhaps I misunderstood your question. It just seems to me that your pretense to isolate the "clearing the neighbourhood" criteria from the others is to ignore the reality of the moon. You seem to be asking if the moon has the mass to "clear the neighbourhood" as if it was orbiting the sun, perhaps in a hypothetical sense? Is that correct? $\endgroup$ – Blaelph Nov 22 '18 at 3:42
  • $\begingroup$ Yes, it's similar. And I'd also considered the idea that Earth-Moon system is a double planet system. Though I do not agree with the idea, I think the idea is worth considering. If Earth-Moon system is a double planet system, Earth is orbiting around the center of mass of Earth-Moon. Then the center of mass of Earth-Moon is orbiting around Sun. $\endgroup$ – pdh0710 Nov 22 '18 at 3:57
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    $\begingroup$ Downvoted due to the second paragraph. "Clearing the neighborhood" does not mean capturing or ejecting every last spec of matter. The question itself alludes to this, using two of the three contending mathematical formulations for what qualifies a body as having the potential to have cleared its neighborhood. $\endgroup$ – David Hammen Nov 22 '18 at 8:37
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    $\begingroup$ @Duckisaduckisaduck : Although the position of barycenter is often said to be a criterion for satellite, it can not be an appropriate criterion. Currently Earth-Moon barycenter lies inside Earth. However, as Moon is drifting away from Earth, eventually it will lie outside Earth. Another notable case is that Sun-Jupiter barycenter lies outside Sun. And though Pluto-Charon barycenter lies outside Pluto, Charon is a satellite of Pluto. $\endgroup$ – pdh0710 Nov 24 '18 at 21:15
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What you have shown is that the Moon is large enough so that it would be a planet by the IAU definition if the Earth were not present. But, because the Moon does orbit the Earth, it is not a planet by the IAU definition. If the "clearing the neighborhood" criterion were sufficient by itself to define a planet, then the other criteria would not be part of the definition.

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  • $\begingroup$ What I have shown is that Moon satisfies the qualification of the planet, according to the "Clearing the neighbourhood" conditions, Of course, satisfying qualification can not mean that Moon is a planet. However, the condition causes a contradictory situation, as I mentioned : Moon is classified as a satellite, not a planet, without rules and conditions. But Moon satisfies the qualification of the planet....... 2006 IAU definition of planet induces this contradictory situation, and instigates the old debate whether Earth-Moon system is a double planet system, instead settles the old debate. $\endgroup$ – pdh0710 Nov 23 '18 at 1:54
  • $\begingroup$ Additionally... Moon orbits around Earth? In Pluto–Charon system, it is not clear whether Pluto and/or Charon orbit around Sun, or orbit around their barycenter, or Charon orbits around Pluto. In Earth-Moon system, the situation is not much different. $\endgroup$ – pdh0710 Nov 23 '18 at 2:12
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    $\begingroup$ This is a good example showing what a disaster the 2006 definition was and the process leading to it was every bit as ugly as any gerrymandered decision. And is the Moon really in orbit about the Earth? The gravitational force between the Sun-Moon is greater than the force between the Earth-Moon, so it is very easy to argue that the Moon is in orbit about the Sun with the Earth providing a periodic perturbation to the orbit. $\endgroup$ – Dave Nov 23 '18 at 16:42
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(Please excuse my English)

Above question arose during my investigation on the double planet/dwarf planet system. As we know, the definition of satellite and the definition of double planet/dwarf planet system are closely related. And currently there is no clear definition of the satellite, and so is the double planet/dwarf planet system.
I thought a simple and obvious criterion is better. However, the simple size criterion is not sufficient for complex combinations of celestial bodies. In other hand, the position of barycenter is often said to be a criterion for satellite, but it can not be an appropriate criterion as described here.
What I thought as a criterion is : If two celestial bodies bound by gravity are classified as same class, we should classify the two bodies as a double body system. For example, if two stars are bound by gravity, we classify it as double star system. And if two black holes are bound, we classify it as a double black hole system.
So if two planet/dwarf planet bodies are bound, it is natural to classify it as a double planet/dwarf planet system.

For test, I applied the criterion to Pluto-Charon system. Pluto is classified as a dwarf planet by 2006 IAU definition. If we consider Charon as a seperated body, Charon also satisfies the conditions for dwarf planet. So we can classify Pluto-Charon system as a double dwarf planet system.

Next, I applied the criterion to Earth-Moon system. As I mentioned in above question, I do not agree that Moon is a planet and Earth-Moon system is a double planet system. I thought Earth and Moon should be classified to different classes, because there are too many differences between two bodies.
However, Moon as a seperated body satisfies IAU planet conditions(especially Clearing the neighborhood condition) as I described in above question. I was embarrassed. So I asked my calculation is correct.

I have thought for a long time that there are many problems with the Clearing the neighborhood condition for the planet definition. Above situation increased my distrust of the Clearing the neighborhood condition. Thus, as I replied the comments and answers, I became over-expressing my opinion on the condition.

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  • $\begingroup$ I forgot the fact that the two definitions do not necessarily have to be mutually exclusive. Here, the two definitions are the definition of satellite and the definition of double planet/dwarf planet system. So, in a double planet system, the smaller planet can be a satellite. Whether the two definitions are mutually exclusive or not seems to be a matter of preference. $\endgroup$ – pdh0710 Nov 25 '18 at 19:14
  • $\begingroup$ As we can see in the comments and answers here, 2006 IAU definition of planet has two serious problems ....... 1) The definition of satellite is missing. 2) The definition of double planet/dwarf planet system is impossible ....... To prevent misunderstandings, I agree that the 2006 IAU definition re-classified Pluto as a non-planet body. Although many people are sorry about it, the re-classification of Pluto is necessary and inevitable. However, the way of re-classification does not look good. $\endgroup$ – pdh0710 Nov 25 '18 at 19:52
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At present, the Moon is defined as a satellite of the Earth, but over time the Earth-Moon barycenter will drift outwards (see tidal acceleration) and could eventually become situated outside of both bodies.25 This development would then upgrade the Moon to planetary status at that time, according to the definition. The time taken for this to occur, however, would be billions of years, long after many astronomers expect the Sun to expand into a red giant and destroy both Earth and Moon.26

25 Robert Roy Britt (2006). Moon Mechanics: What Really Makes Our World Go 'Round'

26 Robert Roy Britt (2006). Earth's moon could become a planet

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$Wikipedia : Criticism of the IAU definition of 'planet'

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  • $\begingroup$ Yes, this is already mentioned in the comments and answers here. $\endgroup$ – pdh0710 Nov 26 '18 at 21:22

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