User talk:Rich Farmbrough/temp331

Rename article

If things like "synchronous rotation" are to be kept, then it makes sense to rename this article to "The Moon (motion)" from the current "The Moon (orbit)". mdf 15:03, 11 July 2006 (UTC)

distance

I had an argument with a friend who said that the distance between the earth and the moon was increasing. From what I know about physics, this is impossible. Orbital energy must be lost due to tides etc. However, he argued so convincingly that I came here to know for sure. If he is correct, I think it deserves a section in the article, if he is wrong, the decay rate of the orbit would be an interesting factoid to add to this article.

the Moon is indeed receding from the earth. "Measurements show that the Moon is receding from Earth at a rate of about 3.8 centimeters per year" http://sunearth.gsfc.nasa.gov/eclipse/SEhelp/ApolloLaser.html. But the is also a corrsponding change in the rotation which keep the engergy conserved. Roguebfl 11:01, 25 August 2006 (UTC)

Predictions suggest that the range will increase until the Earth and Moon become double synchronised, that is, both are tidally locked to one another. (So the Earth's day length would match the Moon's future orbital period of about 47 days, and the Earth-Moon distance would be about 550000km, compared to today's figure of 400000km). This won't occur for something like 50 billion years, by which point the Sun will be a white dwarf and will have passed through a red giant stage, which may result in the destruction of the Earth. [1]

 :Roguebfl 11:05, 25 August 2006 (UTC)

Query re Axial tilt

In the heading Inclination of the rotation axis, this article explains the axial tilt as 6.69° to ecliptic (my emphasis).

However, in the table lower down Other properties of the Moon's orbit Mean inclination of lunar equator to ecliptic is listed at 1° 32'Roo60 12:50, 15 July 2006 (UTC)this is really confusing for some people

Merge results

The following discussion is an archived debate of the proposal. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

Proposal: Merge Earth and Moon with either The Moon's orbit or Tidal acceleration.

The result of the debate was merge with tidal acceleration. — Lunokhod 21:12, 4 December 2006 (UTC)

Suggested Merge

It has been suggested that Earth and Moon be merged with either The Moon's orbit or Tidal acceleration.

FOR: I am for the move for the following reasons

1. The title of the Earth and Moon papge is really bizarre.
2. The Earth and Moon topic is really short and I don't see how it will ever be expanded.
3. The orbital evolution of the Moon is clearly under the domain of The Moon's orbit.

I think that the some portion of this material could be placed in The Moon's orbit, but that Tidal acceleration is where the bulk of it should go. Lunokhod 18:46, 22 November 2006 (UTC)

ALTERNATIVE: The Earth and Moon page mainly deals with tidal evolution of the orbit. That topic has been treated at length on a separate page: Tidal acceleration. So I propose to merge Earth and Moon with that page instead. Do change the link on this page though. Tom Peters 11:47, 23 November 2006 (UTC)

I agree, tidal acceleration is a more appropriate place. Lunokhod 23:14, 29 November 2006 (UTC)

double planet?

Recently the statement that Earth+Moon form a double planet has been reversed. That apparently has been done on the grounds that the COM lies within the Earth. That is only one possible criterium. IMNSHO it also is a poor one: if the Moon were twice as small but four times more distant, the COM would lie outside of the Earth, and the smaller Moon would be part of a double planet anyway? Asimov's proposal, based on the fact that the Moon orbits the Sun rather than the Earth (also looking at the actual shape of its orbit in space) makes more sense. Anyway, with even the concept of "planet" in confusion, I don't believe we have a solid base to securely classify E&M as a double planet or not. Tom Peters 10:45, 4 December 2006 (UTC)

I agree this is a bad definition. The Moon is evolving outwards as a result of tidal interactions and will someday become a double planet. I'll find the official IAU definition and reference for this later today. Nevertheless, I find all proposals at defining what a double planet are to be arbitrary at best. The Moon only appears to orbit the Sun from a Sun fixed perspective; If the Sun disappeared, the Moon would still continue to orbit about the Earth as if nothing happened. The fact that the trajectory of the Moon looks like a closed loop (instead of sometimes going backwards) about the Sun is (in my opinion) an illusion based on the fact that its orbital velocity (about the Earth) is small compared to the Earths orbital velocity about the Sun. I've been thinking of coming up with a definition based on angular momentum, but this would suffer some of the same problems as with the barcycenter definition, even though (again, imho) it would adress the manner in which planets and moons form.Lunokhod 11:14, 4 December 2006 (UTC)

After looking into this, it appears that the barycenter definition is only an informal one. It was considered by the IAU at the last general assembly, but was dropped. See 2006_redefinition_of_planet. Lunokhod 19:14, 4 December 2006 (UTC)

The Moon orbits the Sun? No, it doesn't. Not even close. You can see my comments on this notion at http://en.wikipedia.org/wiki/Talk:Moon/Archive_3. mdf 16:13, 30 March 2007 (UTC)

Asimov's proposal of considering the Earth-Moon system a double-planet system is based, yes, on Moon's orbit around the Sun, Moon's size and mass in comparison to Earth's (only the Pluto-Charon system come close in proportions - that one fully recognised as a double-planet) and also (I believe this is the most important point) the "tug of war" (Asimov's name): Sun's gravitational pull on Earth is stronger than Earth's. This is true only for Moon among all other big satellites in Solar System (some of the outer, tyniest moons of Jupiter and Saturn also have this property), false even for Charon. It also makes me doubt that, if Sun would dissappear suddenly, Moon's orbit around Earth would continue as if nothing had happened. Since Sun's gravitational pull on the Moon is stronger than Earth's, I'd expect Moon decaying into a lower orbit in case the Sun dissappears. Can anyone support the claim that "nothing would happen" with some calculations, or providing an animation using "Gravity" or any other simulation program? Thanks —Preceding unsigned comment added by 148.244.69.177 (talk) 22:54, 16 October 2007 (UTC)

Interesting thoughts, but I'm sure "nothing would happen" to the moon's orbit if the sun disappeared. A simple two-body conservation of energy disallows the orbit to change its semimajor axis length. Tom Ruen 21:04, 17 October 2007 (UTC)

Proposed rename results

The following discussion is an archived debate of the proposal. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

Proposal: Rename The Moon's orbit to Orbit of the Moon.

The result of the debate was move. — Lunokhod 10:50, 7 December 2006 (UTC)

Proposed rename

FOR I am for the proposed rename/move because (1) it is not wikipedia policy to have pages started with the word "the", and (2) when listing this page in an alphabetic list of see alsos, this topic falls under "the" (hence the reason for the naming policy). Lunokhod 21:26, 4 December 2006 (UTC)

Makes sense. I agree. Tom Peters 23:23, 4 December 2006 (UTC)

You know, I just came over here to propose that idea myself. 100% agree. The Moon's Geology, The Moon's Exploration etc. would look just as bad. Fine if we're writing in Swedish (Indiens flagga for Flag of India for example), but not English. Mithridates 02:10, 7 December 2006 (UTC)

Perigee distance should be fixed

The perigee listed in the article may be good for an "average" month but the moon has been known to come as close as 356,300 km. Someone should find a better source. Sagittarian Milky Way 05:00, 23 March 2007 (UTC)

Yes, but the question is, where does it end? We could always find a closer perigee. The distance you cited was very, very rare. The closest perigee within the range of A.D. 1500 to A.D. 2500 is 356371 km; the Moon approaches within 356425 km 14 times within that range. This comes from Meeus' Astronomical Algorithms. He used the ELP-2000/82 lunar theory. Solex90, an excellent numerical intergration program, could only find one as close as 356,313 km going bace to -15,000. (15001 B.C.) Although the predictions get pretty uncertain that far back and one could have easily been less than 356,000 km.

It is a good question in general-should a record or typical perigee be used. Saros136 07:01, 23 March 2007 (UTC)

Right, I suppose 364,000 km is the elliptical perigee, which is perturbed to as much as ~356,000 / ~370,000 km by the other two orbs. The Moon's orbit is amazing, like an elastic thing continously being stretched and played with. Sagittarian Milky Way 01:54, 24 March 2007 (UTC)

My bad, [2] states the extremes for 1750-2125 as 356,375 / 406,720, I remembered it wrong, thought it was 356,325, and rounded it down just to make sure I wasn't overstating accuracy!

When were the top ten closest of -3000 to 3000 AD? Sagittarian Milky Way 02:22, 24 March 2007 (UTC)

A quick scan of the DE-406 follows. The ten closest are:

356337.064 -2338 Nov 09 03:22:22
356349.827 -2683 Nov 13 02:33:02
356352.945 -1055 Nov 13 21:40:37
356354.171 -2665 Nov 23 13:31:54
356356.491 -0851 Dec 08 02:26:27
356356.657 -2356 Oct 28 16:22:36
356360.917 -1400 Nov 17 21:11:04
356365.136 -2320 Nov 19 14:21:41
356365.621   796 Dec 19 05:44:44
356366.204 -2869 Oct 30 08:25:51

The ten furthest are:

370389.858 -0367 Nov 19 13:58:27
370390.249 -2915 Nov 20 20:59:18
370391.003 -2588 Nov 05 10:27:44
370392.976 -0256 Dec 06 21:55:54
370393.790 -2699 Oct 18 19:22:03
370394.088    90 Dec 03 02:19:14
370395.483 -2933 Nov 08 21:26:07
370397.683 -1212 Nov 16 20:04:09
370404.210 -2088 Nov 11 23:55:44
370407.525 -1650 Nov 15 01:16:01

When you plot the ranked data (79532 points) one obtains a roughly straight line, so the distribution is about as uniform. At least on a 6000 year time-scale, where dynamics are being smeared. On a month-to-month basis, things are more correlated. Graphics can be uploaded etc on request. mdf 16:51, 30 March 2007 (UTC)

Right, if you draw it out for that long it'll look like a line. What's neat is that there are cycles upon cycles upon cycles. Where apogee and perigee repeat like a sine wave every month, which then has a ~yearly sine wave superimposed on it, and then higher order cycles on top of that and so on.

A series of graphs of distance vs time showing this would be nice. (you might have to show the max-apo, min-apo, max-peri, and min-peri top to bottom on different scales, otherwise a few kilometers out of 50,000 wouldn't be seen)

The article could also use a illustration showing the movement of the nodes and apsides. Sagittarian Milky Way 08:09, 1 April 2007 (UTC)

CCW?

I see from the article that the ellipse of the lunar orbit rotates counterclockwise, and that the precession of its orbital plane is clockwise. How about the orbit itself? The animation of the Moon as it cycles through its phases allows one to infer that the orbit is counterclockwise (same as the rotation of the Earth, and the Earth's orbit around the Sun) but this really should be stated explicitly both here and at Moon. --Wfaxon 22:01, 28 July 2007 (UTC)

The physical Moon itself moves counterclockwise. (directions are always measured from above the north pole). Sagittarian Milky Way 21:18, 29 August 2007 (UTC)

Secular acceleration of the Moon

Someone looking for info on the "secular acceleration of the Moon" in Wikipedia will search long and hard. Perhaps it is here but searching on this familiar term yields nothing.Cutler 09:51, 24 August 2007 (UTC)

That is already discussed in this article under Orbit of the Moon#Tidal evolution of the lunar orbit as well as the main article Tidal acceleration already listed under that heading. But I've added your term, secular acceleration of the Moon, as a redirect to tidal acceleration. — Joe Kress 23:58, 24 August 2007 (UTC)

Image, The Earth and Moon's path

The equations which are said to be the basis of the image are not clear. In the first place the given equations of the earth's orbit would describe a circle, not an ellipse? This is easily corrected and should be shown as precisely as possible in an encyclopedia article. In the second place, the origin of the symbol " p:(synodic months+1)=14 " is not clear. If "p" is a constant(14), then how can it be an integer? Should it be more precisely the number of synodic months in a sidereal year ~12.368? And why plus one? Will the contributor of this image and these equations please clear this up? PSpace —Preceding unsigned comment added by Alexselkirk1704 (talk • contribs)

There is no reference point!

I've searched the internet, and can only find recorded dates for one of the lunar cycles, the synodic. This is a problem due to the importance of the draconic (nodical) cycle in predicting eclipses. Someone should do some research. 68.144.80.168 (talk) 08:00, 30 March 2008 (UTC)

Spin orbit resonance

According to this article, cited by a source I can't access, the Earth and the Moon will achieve spin orbit resonance in 2 billion years. This seems a remarkably short time. I've heard estimates as high as 50 billion years. Serendi^podous 16:08, 8 June 2008 (UTC)

Concave or Convex?

There are two changes under discussion with both being in the Orbit of the Moon#Path of Earth and Moon around Sun section.

1. The illustration caption was modified
   • < The Earth and Moon's path around the Sun is always concave to the Sun (far left down)
   • > The Earth and Moon's path around the Sun is always convex to the Sun (far left down)
2. The body text was modified
   • < Unlike most other moons in the solar system, the annual trajectory of the Moon is very similar to the one of the Earth.
     It is always concave towards the Sun, and is nowhere convex or looped.
   • > Unlike most other moons in the solar system, the annual trajectory of the Moon is very similar to the one of the Earth.
     It is always convex towards the Sun, and is nowhere concave or looped.

There is a misunderstanding about the usage/meaning of the orbit being 'convex' or 'concave to the sun' here. Both of the currently-cited references are agreed that the moon's orbit around the sun is like a slightly flattened circle or ellipse but it does not change sign of its curvature (no loops or dents). Reference #5 specifically calls this 'convex outward'. Reference #4 calls it 'convex' but without saying in which direction. The older references used to call this 'concave to the sun', but they clearly meant the same thing. See "The Moon's Orbit Around the Sun, Turner, A. B. Journal of the Royal Astronomical Society of Canada, Vol. 6, p.117, 1912JRASC...6..117T"; and

"H Godfray, Elementary Treatise on the Lunar Theory". I believe if you look at these as well "the existing reference by H L Vacher" you will see that 'convex to the sun' is clearly wrong. Please may I suggest that you reconsider and amend? Terry0051 (talk) 08:12, 25 March 2009 (UTC)

So apparently there are some sources that describe the Moon's orbit as convex, and some that describe it as concave. People have a tendency to mix up the terms convex and concave. Damn whatever idiot decided to use such similar words for such an obscure concept. Rather than attempt to determine which sources have more authority, I suggest we look at the actual definitions of the words. All the sources agree that the Moon's orbit has no loops. And the Wikipedia article Convex and concave polygons seems to imply that a circle without loops would be convex, rather than concave.

Some of the sources describe it as convex/concave inwards vs. convex/concave outwards. I think that distinction is nonsense. Something is either convex or concave. Convexity and concavity have no distinction with respect to inwards or outwards. When they say that the Moon's orbit is convex "with respect to the Sun" I think they just mean that as opposed to convex with respect to the Earth.

So I still think I'm right on this, and I'm not inclined to change my edit. Except maybe to link to the article on the definition of convex, to avoid this misunderstanding in the future. But now that I've expanded on my reasoning, if you still think it contains a mistake, feel free to let me know why. - Shaheenjim (talk) 08:38, 25 March 2009 (UTC)

The article now contains a mistake because it says "always convex towards the Sun" which is the opposite of what the citations say, and also the opposite of what the diagram shows. May I suggest you read the citations which were provided as links, there is no disagreement amongs them. The distinction between convex and concave is very easy: if you are looking at a convex curved boundary, the centre of curvature is on the other side from you; if you are looking at a concave boundary, the centre of curvature is on the same side as you. There is no 'nonsense' in recognizing that the convexity and concavity are each directed to one of the sides of the boundary. Terry0051 (talk) 11:33, 25 March 2009 (UTC)

I'm not sure what your position is. To clarify it, answer these four yes or no questions:

1. Do you think the Moon's orbit is concave inward?

2. Do you think the Moon's orbit is concave outward?

3. Do you think the Moon's orbit is convex inward?

4. Do you think the Moon's orbit is convex outward?

- Shaheenjim (talk) 18:31, 25 March 2009 (UTC)

Terry0051, I'd want to recheck that math. If you look at File:Moon_trajectory1.svg then you can see that the moon's path relative to the sun has "bumps" and "dents" with the peak to trough height being ~770,000km. The Earth/moon system orbit is fast enough that there are no loops and is also fast enough that those bumps and dents are very smooth with the peak to peak being spaced at roughly 27 degree intervals around the sun. FWIW, the Earth's orbit around the sun also has bumps and dents and it's only the barycenter that has a smooth orbit. I think I'll leap out of my chair and move the barycenter momentarily. :-) --Marc Kupper|talk 09:54, 25 March 2009 (UTC)

Thanks for your comment Marc: you said "I'd want to recheck that math". and I have indeed checked the moon-orbit file/diagram you mention, and it doesn't show dents, it shows no more than slight flattenings not sufficient to change the sign of the curvature. May I suggest that you in turn look at the 4 references, they all show -- and indeed prove -- the same thing: There is no change in the sign of curvature of the orbit, and it therefore remains always convex outwards and concave inwards towards the sun. The article -- after Shaheenjim's edits -- now states the opposite "convex towards the sun", whereas it was correct before. I think you might have been using the word 'dents' in a different sense. My message used it to denote what is shown in the citations. It would be appreciated if you would reply on my talk page. Terry0051 (talk) 11:35, 25 March 2009 (UTC)

For The illustration caption: The illustration caption is easier to deal with as the picture is of an open line that could be described by a concave function. I personally prefer the former wording as the current wording is only correct if one places oneself in the upper right corner and views the moon's path from outside the entire Sun/Earth/Moon system. With both versions the caption text leads the reader to the lower left corner and seems to ask that the reader view the moon's path from the point of view of the Sun when in fact to the Sun means to position yourself outside and to look to the sun. Thus while the new wording is syntactically correct it's only so if you read it carefully. Rather that just changing the article and possibly edit warring I'd rather discuss the wording here and am proposing that the illustration caption be changed to “From the point of view of the Sun (far left down), the Earth and Moon's path around the Sun is always concave.” --Marc Kupper|talk 20:23, 25 March 2009 (UTC)

For The body text: I wrote up a really detailed reply but need to proofread and think more before posting it. The issue is that within a single year, and much ignoring of details, the path described is a convex set and is thus always concave from the point of view of the Sun but convex if you are describing the overall shape. However, because of precession the path actually describes a complex convex shape as the moon's path is not exactly the same from year to year. I believe it repeats every 61,000 years or so but need to get the cites lined up. There's also the matter of the entire solar system and galaxies movement meaning we really have a spiral... --Marc Kupper|talk 20:23, 25 March 2009 (UTC)

Marc, I direct my questions from my post at 18:31, 25 March 2009 (UTC) above to you as well. - Shaheenjim (talk) 20:45, 25 March 2009 (UTC)

Shaheenjim, thank you for consolidating the threads here. As for the questions; let me back up a moment which to explain how I remember convex vs. concave which is I see the cave in "concave" and visualize myself standing in front of the mouth of a cave and looking at it. Or I can visualize myself as being inside the cave. In either case the center is further away and the edges are closer to me. If I'm the colon in this picture, ":)", then that parentheses is concave. If I'm faced with ":(" then it's easy for me to see I'm looking the opposite of concave and that the parentheses is convex with respect to me. I have no idea how I came up with this mnemonic device and if others use it.

I believe the average person is faced with the concept of convex vs. concave shapes (either polygon or non-polygon) so rarely that if we employ this usage in the article then it needs to be explained in the article along with a main article link to convex set to help people visualize it. I would not use the convex and concave polygons article as the orbit is not a polygon.

On to the questions. The first thing I see is that they ask about "Moon's orbit" which is normally thought of as the Moon's orbit around Earth but that this discussion had started with edits to the article's "Path of Earth and Moon around Sun" meaning I'm not sure if you are asking about "the Moon's orbit around the Earth" or "the path of the Moon around the Sun as the Moon participates in its own relationship with Earth." To clarify the questions I modified all four to replace the word "orbit" with "path around the Sun."

I then see that the questions include the terms "concave inward", "concave outward", "convex inward", and "convex outward". None of those mean anything to me and so I will modify all four questions by replacing "inward" with "when looking inward at the entire Moon/Earth/Sun system" and "outward" with "when looking outward from the POV of the Sun."

1. Do you think the Moon's path around the Sun is concave when looking inward at the entire Moon/Earth/Sun system? A: No.
2. Do you think the Moon's path around the Sun is concave when looking outward from the POV of the Sun? A: Yes.
3. Do you think the Moon's path around the Sun is convex when looking inward at the entire Moon/Earth/Sun system? A: Yes if you you view part of the orbit and treat it as an open curve) and "sort of" if you look at the entire system as the correct term is that we what the orbit defines is a convex set and not just "convex."
4. Do you think the Moon's path around the Sun is convex when looking outward from the POV of the Sun? No. --Marc Kupper|talk 22:38, 25 March 2009 (UTC)

For the record, I'm officially fine with the changes you made to the wording of my questions. And thanks for your answers, I think I get what you're saying now. But until now, I've never heard anyone make the distinction between convex/concave when looking outward from the inside, and convex/concave when looking inward at the entire system. Wikipedia's article on Convex and concave polygons doesn't make that distinction. It says a regular pentagon is convex. End of sentence. No "when looking inward at the entire system." Just "convex." The article on Convex set doesn't seem to make that distinction either, unless I missed it. When I learned about convexity and concavity in middle school, they didn't make that distinction. Where makes you think you should make that distinction? And don't you think it's strange that the Wikipedia articles on convexity and concavity don't make the distinction? - Shaheenjim (talk) 00:05, 26 March 2009 (UTC)

(Unindent otherwise the lists below get really to read.) The distinction is made on both the convex and concave polygons and convex set articles in that they note that the observer's position is inside the shape when taking the measurements that determine if it's convex or concave.

I have two questions:

• Is this line, ), concave or convex?
• Is this line, (, concave or convex?

While pondering that; Here are definitions from my dictionary starting with convex. The italics are my comments.

• curved out, like the outside of a circle or sphere. They place the observer on the outside of the object.
• The lens of an automobile headlight is convex on the outside. Again, they place the observer on the outside of the object.
• The crystal of a watch is slightly convex. Mildly ambiguous. The assumption made is that you are looking at the watch from the outside as you would to tell the time.
• The back of a spoon is a convex mirror. Again, they carefully note the relationship of the observer to the spoon and its orientation.
• Derived from Latin concavis < com- (intensive) + cavis hollow.

That's it for the convex definitions. Here is concave

• Hollow and curved like the inside of a circle or sphere. Observer's position is defined.
• curving in. Ambiguous. This was part of the previous definition but separated from it by a semicolon.
• The palm of one's hand is slightly concave. Mildly ambiguous but most people will interpret this as looking at the surface of own palm's meaning it's from the outside. Another interpretation is that the surface and area bounded by that surface are the "palm" and that, yes, it's concave.
• Obsolete. Hollow.
• Derived from Latin convexus vaulted, arched, probably < com- around + an unrecorded root vac- to bend, related to vacillare totter, sway.

The point is that when words like convex and concave are used care is taken to note the observer's position, and if needed, the orientation of the subject described.

This is why I commented on the illustration caption. The caption itself fails to orient the observer and then in parentheses leads the reader, and presumed observer, to a spot where the subject would be interpreted as concave per any of the dictionary definitions. This is why I suggested that the illustration caption be changed to “From the point of view of the Sun (far left down), the Earth and Moon's path around the Sun is always concave.”

An alternative wording: "The area inside the Moon's orbit around the Sun forms a convex set. The sun is to the far left and down." Or, "Detail of the Moon plus Earth system as it orbits the Sun which would be to the far left and down. The orbit is convex when viewed from the outside (upper right corner)." Either attempt to use convex correctly seems more convoluted than identifying where both the Sun and observer are and calling the Moon's path concave.

I'll see if I can make a better image. I found the web site it was stolen from a bit ago but would need to Google that one up again. --Marc Kupper&;;;;;;;;#124;talk 01:58, 26 March 2009 (UTC)

What's your source for the distinction between convex/concave when looking outward from the inside, and convex/concave when looking inward at the entire system? Is it those definitions from your dictionary? Because I really don't think those definitions supported your argument. You seemed to be really stretching in your comments in order to make them fit. The first one said convex is the outside of a circle, but it didn't say that the inside would be concave. And where in the Convex and concave polygons article did it say the observer's position is inside the shape? I think you're reading things into this that aren't there. It said convexity relates to the inside of a shape. But it didn't say that it differed depending on whether or not the observer is inside the shape.

Maybe POV is relevant when dealing with lines like ) or (. But I'm still not convinced that it's relevant when dealing with closed systems, like polygons, or the Moon's path around the Sun. - Shaheenjim (talk) 12:17, 26 March 2009 (UTC)

[From Terry0051:] I agree with all of Marc's answers to the modified questions quoted as follows (with clarifications as offered below):-

"Q1: Do you think the Moon's path around the Sun is concave when looking inward at the entire Moon/Earth/Sun system? A: No.

"Q2: Do you think the Moon's path around the Sun is concave when looking outward from the POV of the Sun? A: Yes.

"Q3: Do you think the Moon's path around the Sun is convex when looking inward at the entire Moon/Earth/Sun system? A: Yes if you you view part of the orbit and treat it as an open curve) and "sort of" if you look at the entire system as the correct term is that we what the orbit defines is a convex set and not just "convex."

"Q4: Do you think the Moon's path around the Sun is convex when looking outward from the POV of the Sun? No.

The clarifications offered are:

A: We are talking about the Moon's curved path, relative to a non-rotating solar-system-barycentric reference frame - or heliocentric reference frame. (For the present purpose it does not matter which.)

B: All of the above answers have been given from the physical POV of an observer located either near the Sun, or else at a great distance away and from somewhere on or near the solar system's invariable plane.

C: None of the above answers have been given from the physical POV of an observer located on the Moon's path itself, and in relation to the small element of that path which is close to the observer (which seems to match the 'open curve' referred to in Marc's answer 3). (A physical POV located on the path-element is the one effectively used in the online astronomical references provided.)

D: Every element of the Moon's orbital path has its own curvature and center of curvature -- regular sources on calculus and orbital dynamics provide. From the physical POV of an observer located on this curved path-element, it is concave towards its center of curvature, and convex away from its center of curvature. This corresponds with the usage of concave and convex that occurs in the astronomical references already supplied.

E: There is no reason to believe the Moon's path is precisely periodic, many reasons to believe it is not, and it seems unnecessary for present purposes to go into the long-term behavior. The local curvature of the path-element that the Moon is traversing right now is defined by the dynamics of the Moon's position right now. Also, the motion is not, in physical fact, confined precisely to a single plane. These astronomical facts, and the probable absence of any closed curve for the orbital path, also make the language of convex sets seem somewhat alien from the physical situation under discussion.

As I read the original text, its task was to convey, in brief and less-technical language, the physical fact that the center of curvature of every element of the Moon's curved orbital path (in the non-rotating reference frame of the solar-system barycenter) is always on the inward side, towards the Sun, and there is no inwards-outwards alternation as the Moon makes its progress along its path.

The purpose of making this whole point seems to have been to convey, that while intuitive thinking might indicate that the center of curvature of the Moon's path at new-moon is away from the Sun and towards the Earth, that is not in fact so.

The less-technical language omitted a number of details, such as, that the center of curvature is not precisely in the Sun-Earth-Moon plane, nor in any other single plane. That simplification seems reasonable in a brief description, because the out-of-plane deviations remain small enough to neglect for this purpose. That is, the component, in the line from Moon to Sun, of the direction vector from the Moon towards the center of curvature of the path-element that the Moon is currently traversing, always points M->S, never the other way.

But it looks as if the recent edits have uncovered a point in the simplified language used in the original article, which was unclear to the general reader: i.e. the usage and physical POV of 'convex' and 'concave' -- even though these were clear to the authors of the cited mathematical-astronomy references, and to others familiar with the calculus of orbital dynamics.

I still think it is clear that the original text, before the recent edits, was correct -- given prevailing usages of the words employed in the mathematical-astronomy field -- and is now in error. But I agree that it would be helpful if the text on this point is given brief and correct clarification. If the words chosen differ from those that are usually encountered in the astronomical refernces, it would also be helpful to let the reader know in some way what to expect as the usage in the outside world if s/he is interested enough to look up the cited astronomical references.

I suggest to Shaheenjim that he looks at the Godfray and Turner references which are online and only need one click of the mouse to reach. Terry0051 (talk) 15:58, 26 March 2009 (UTC)

It may be true that the Moon's annual path isn't precisely periodic, and isn't precisely a closed curve, and that its center of survature isn't precisely in the Sun-Earth-Moon plane. But it's close enough. So let's not get distracted by things that aren't important to the broader discussion.

I acknowledge, again, that the Godfray and Turner references say the Moon's annual path is concave. But there are other references that describe it as convex. I see no reason to believe that the Godfray and Turner references are any more reliable on this point than the other references.

Terry0051, I'm going to ask you the same questions that I asked Marc Kupper about what your source is for the distinction between convex/concave when looking outward from the inside, and convex/concave when looking inward at the entire system. But before I ask you all those questions, I'll let Marc respond to the questions I asked him, since he might resolve some of them. - Shaheenjim (talk) 16:38, 26 March 2009 (UTC)

[From Terry0051] Shaheenjim, You've just said "I acknowledge, again, that the Godfray and Turner references say the Moon's annual path is concave. But there are other references that describe it as convex. I see no reason to believe that the Godfray and Turner references are any more reliable on this point than the other references."

It has already been pointed out that the substance of the references is in agreement (look at the diagrams and other details of their proofs), but you are making it out that they are in conflict, seemingly merely because they use different terms or different physical POVs to describe the same thing.

What is your source for making it out that the references are in conflict? Terry0051 (talk) 17:38, 26 March 2009 (UTC)

The Godfray and Turner references say that the Moon's annual path around the Sun is concave. Other reference like this one say that it's convex. That seems like a conflict to me. Now, maybe you'll claim that the Godfray and Turner references are looking outward from the POV of the Sun, and the other references are looking inward at the entire system, and that explains the difference. But I still don't acknowledge a distinction between convex/concave when looking outward from the inside, and convex/concave when looking inward at the entire system. Hopefully Marc will clear up some of that when he responds to my question above. - Shaheenjim (talk) 18:17, 26 March 2009 (UTC)

My source for the distinction is the World Book Dictionary which I quoted above. Nearly all of the examples and usages required that the observer and subject have a specific orientation. Look at the inside of a spoon the bowl is concave. Look at the outside of a spoon and the bowl is convex. It surely can't be both unless a distinction is made.

This morning I woke up and realized when rocket scientists talk to one another and say that "an orbit is convex" that they are using a mathematical definition, and not standard English definition of convex, and also that they are using "convex" as a shorthand. They mean that the orbit forms convex curves and that the area inside the orbit is a convex set. Convex curve is not defined in my English dictionaries but here's a definition from the McGraw-Hill Dictionary of Scientific and Technical Terms: “Convex curve [Math] A plane curve for which any straight line that crosses the curve crosses it at just two points.” Note that here too they are careful to note the measurement method and by implication, its orientation with respect to the subject. The fact that the straight line must cross the curve itself is the distinction.

As the intended audience for Wikipedia articles is lay people, and not the rocket scientists, any use of convex should explain that it's in a mathematical sense and to explain how they define and use the word. With that in mind, the word concave should not be used in the article other than if someone wants to say that the Moon and Earth related curves or orbits are not concave, again using the mathematical or rocket science definition and usage, and not how it's often used in standard English. --Marc Kupper|talk 20:02, 26 March 2009 (UTC)

[From Terry0051]: Marc, there is a more informative definition in "Wiktionary - convex" "curved or bowed outward". This shows clearly that the current wording in the article "convex towards the Sun" is clearly wrong, it would mean 'curved or bowed outward towards the Sun'.

The way Shaheenjim has put it, "a distinction between convex/concave when looking outward from the inside, and convex/concave when looking inward" is both a confusing formulation in itself, and not the way in which the point has been formulated, whether in the references, in the article, nor by the other participants in the discussion. It would be more helpful to the discussion to use one of the clearer formulations on record.

The ostensible 'conflict' based on the use of the words 'convex' and 'concave' has already been explained as not a conflict, based on the differing physical POV relative to which the words were applied. Please look again at the Vacher reference, page 5, Fig. 2B, and the description "everywhere convex outward", and then compare that with Turner, page 119, where it is made clear in connexion with an idealized case in which the moon moves around the sun in a circle, that is an example of a path that is 'always concave to the sun'. I am sure that you can see very easily how those two statements can be made about the same curve without being in conflict.

The Turner and Godfray references are both in the mainstream astronomical literature, and it would be well if the general reader is alerted, at least in a footnote, to expect their usage, for the reason already offered. The orbit of the moon is not a closed curve and better descriptions of its curvature would probably be given in terms that include center of curvature and curvature. Terry0051 (talk) 20:35, 26 March 2009 (UTC)

Now that Marc has seen the light, it's time for me to ask you, Terry0051. What is your source for the allegation that whether something is convex or concave depends on the perspective of the observer (or whether it's outward or inward, or however you want to phrase it)? - Shaheenjim (talk) 20:40, 26 March 2009 (UTC)

Note that I edited the article and hopefully that'll address your concerns. - Shaheenjim (talk) 20:43, 26 March 2009 (UTC)

[From Terry0051] Thank you Shaheenjim for your further edit, and I agree with it as far as it goes, but it leaves the ambiguity in 'convex' unaddressed. I was about to do something almost the same, retaining the word 'convex', along with a clarifying phrase based on Marc's answer #3, plus a footnote that advises of the alternative usage in respect of the same geometry. The clarifying phrase, and the explanatory footnote, I've now put in. I believe this is securely sourced in the cited references and hope and believe it otherwise reflects common ground between parties to the current discussion. Terry0051 (talk) 21:14, 26 March 2009 (UTC)

I'm still waiting on you to cite your source for the allegation that whether something is convex or concave depends on the perspective of the observer (or whether it's outward or inward, or however you want to phrase it). My position is that there is no ambiguity in describing it as simply "convex." I don't think the Turner and Godfray articles are correct, and just use a different phrasing. I'm alleging that they're just outright wrong. - Shaheenjim (talk) 21:21, 26 March 2009 (UTC)

[From Terry0051] I've already referred you to a figure and description in Vacher and to description in Turner. Godfray clearly matches Turner in the pages already specifically cited. I didn't put the point in the way that you are putting it, "the allegation that whether something is convex or concave depends on the perspective of the observer", but insofar as that is clear, it is supported by the sources just mentioned and by their clearly different usages in respect of the clearly corresponding geometry as already pointed out specifically. The statement in the footnote is also very clearly supported by the citations -- even if your suggestion is correct that they are somehow wrong. If you are alleging that Turner and Godfray are just outright wrong then please would you cite some specific support for that allegation. Terry0051 (talk) 21:41, 26 March 2009 (UTC)

Sun-Earth-Moon resonance?

As the Moon orbit becomes larger, eventually the year will become exactly 12 synodic (or 13 sideric) months. Will this cause any resonance effect, like the orbit getting more excentric, or is the orbit too irregular for such resonance effects? Ambi Valent (talk) 23:52, 30 April 2009 (UTC)

The Sun will become a red giant long before that happens. In about a billion years the Sun's increasing luminosity will vaporize the oceans, removing virtually all tidal acceleration from the Moon, stopping its regression from Earth and preventing any further change in the length of the month. In 7.6 billion years both the Earth and Moon will be destroyed as the Sun increases in size and engulfs their mutual orbit. Furthermore, all calculations that I have seen presume that the present tidal acceleration will continue unchanged, producing double Earth/Moon spin-orbit resonance in 50 billion years, whereas we now know that the present acceleration is at least twice the long term (620 Ma) average deduced from tidal rhythmites, doubling all calculated times. — Joe Kress (talk) 06:18, 1 May 2009 (UTC)

Clarification of diagram

The diagram in the section "Path of Earth and Moon around Sun" only shows a small part of the orbit of the Moon relative to the Sun, in order to illustrate the "convexity" (or "concavity") point. The diagram is not apt to illustrate the direction of movement of the Moon along its orbit (and in any case, insufficient information is given to define this direction, because the diagram might equally be considered as seen from the north, or from the south). But to clarify the "convexity" point, it is important to say what is the location of the Sun relative to the portion of orbit shown in the diagram: the Sun is located below and to the left. Terry0051 (talk) 12:33, 2 May 2009 (UTC)

Cusps on cycloidal moon orbit

The Earth-Moon system as a binary planet

Joe and Terry: The diagram of lunar orbit 'convexity/concavity' you prefer has been around for at least a hundred years, since Young's 1902 Manual of Astronomy. Considering the voluminous discussion above about whether your reverted diagram makes clear orbital direction and whether viewed from up, down, inside or out I felt that the diagram I submitted made those key points absolutely clear to the casual reader of Wikipedia. Perhaps you would care to clarify your objections to describing the orbit as a modified cycloid and why the diagram that shows this is unacceptable?Geologician (talk) 10:16, 26 June 2009 (UTC)

[From Terry0051] The edit summary reverting your edit said -- "there's no support in RS for the erroneous image and description which also contradict the existing RS, see Turner and Godfray citations". Have you looked at the Turner and Godfray citations in the main article? -- They have been provided in a form where you can easily see the full text (pdf) just by clicking on the links. If you want to maintain that your contradictory diagram really is correct, please would you provide a reliable source for it (which you have not yet done), and clarify how you want to differ (if at all) from the facts set out in the existing reliable sources, the Turner and Godfray citations? Specifically, what they set out makes it clear among other things that the lunar orbit with respect to the Sun does not have cusps, whereas your diagram shows cusps, so that is one feature which alone makes it clearly incorrect. If you think about it, cusps are clearly quite unphysical here, because they would (if true) directly imply that the velocity of the Moon is subject to frequent discontinuities, and in turn that would imply intermittent sudden momentary forces of indefinitely large, even infinite size, on the Moon, for which there is clearly no basis in physics or in reliable sources. (In saying this I do not suggest that the current text could not be improved. Compromises arose to try to settle previous misunderstandings, and that alone has made the text a little clumsier in places than it was before, and clumsier than it needs to be, but the route to improvement clearly must lie in setting out correct facts, reliably sourced.) Terry0051 (talk) 11:34, 26 June 2009 (UTC)

A cycloid is the mathematical curve formed by a point on the perimeter of a circle rolling along a straight line. This is similar to the curve formed by a point on the rim of a bicycle wheel rolling along a road. Geologician drew an epicycloid, a point on a circle rolling along the outside perimeter of a fixed circle. In both cases, the speed of the point is comparable to the speed of its circle's center (the axle of a bicycle wheel)—there are no sudden changes in speed relative to the axle. If the Moon's orbit was an epicycloid, its orbital speed relative to Earth would be the same as Earth's orbital speed around the Sun. When the Moon passes between Earth and Sun, the Moon is moving opposite to Earth, so the two speeds cancel, apparently stopping the Moon momentarily (the cusp) relative to the Sun, but not relative to Earth.

But the Moon's orbital speed is only 1/30 of Earth's, so the desired curve is an epitrochoid, formed by a point on the radius of the moving circle much closer to its center than to its perimeter. Epitrochoids are the curves formed by the planets in the Ptolemaic system of deferents and epicycles. The Moon's orbit around Earth would be an epicycle and Earth's orbit around the Sun would be a deferent. Because there are 12.37 lunations in a solar year, the radius of the moving circle would be 1/12.37 radius of the fixed circle. To match the relative size of the Moon's orbit to Earth's orbit, the point on the radius of the moving circle would be 384,000/150,000,000 or 0.00256 of the combined radii of the two circles from the center of the moving circle, which was correctly shown in the original image. An epitrochoid is a much smoother curve than an epicycloid. The very small separation between the Moon and Earth relative to the separation between Earth and Sun is critical to show the convexity outward of the Moon's orbit relative to the Sun. An epicycloid fails to show any convexity at the cusps. Even an epitrochoid with a large undulation fails to show convexity at many points. — Joe Kress (talk) 20:05, 26 June 2009 (UTC)

[From Terry0051] The cusps drawn by Geologician are in a circumsolar orbit. That is rather near to an inertial reference frame. It's in this frame that the forces would rise to physically quite unfeasible levels if there were cusps. In the frame of reference relative to the moving earth, the fact that there is at the same time no cusp in the relative velocity of Moon and Earth is not enough by itself to define the forces on the Moon in an inertial reference frame: the Earth's local frame is a long way from that. Otherwise I agree with Joe Kress, and the only minor qualification I'd offer is that the epitrochoid is an approximation, though a good one for this context, because of the lunar orbital eccentricity and perturbations. But the slight differences of shape away from circular that these bring are not such as to change the main point here at all, which is that the proposed diagram and associated alleged facts are untrue to the physical reality. Terry0051 (talk) 21:22, 26 June 2009 (UTC)

The cusps of an epicycloid would produce no additional force on the Moon whatsoever. Every point on the rim of a bicycle wheel traces a cycloid relative to the near inertial frame of the road without any physical harm. Ditto for every point on the tread of a motorcycle wheel. If that motorcycle cycled around the inside of a spherical cage of death at a county fair, every point on the tread of its wheel would describe a hypocycloid with cusps, also without physical harm (at least to the motorcycle wheel).

More to the point, Europa, the second Galilean moon of Jupiter, has an epicycloid orbit with numerous cusps relative to the Sun without any adverse effect. Its orbital speed around Jupiter (13.74 km/s) is almost the same as Jupiter's orbital speed around the Sun (13.07 km/s), so when Europa passes between Jupiter and the Sun, it almost stops relative to the inertial frame of the Sun. This is a true epicycloid relative to the dimensions of its orbit as well, with about 1200 Europa months per Jupiter year and an orbital radius from Jupiter that is about 1/1200 of Jupiter's orbital radius from the Sun. — Joe Kress (talk) 01:57, 27 June 2009 (UTC)

I checked Young's 1902 Manual of Astronomy cited by Geologician and find that it explicitly states on page 174 that describing the lunar orbit with near cusps is an "erroneous representation of the Moon's path". — Joe Kress (talk) 03:54, 27 June 2009 (UTC)

[From Terry0051] The model of a single point of a rolling wheel returning to rest momentarily when it touches a static surface has little or no bearing on the motion of the Moon. Any suggestion that cusps arise in the Moon's motion by its regularly losing all of its kinetic energy, and returning to rest, would not really be an improvement on the extreme unlikelihood that it suffers unfeasibly large impulses of no possible origin, otherwise needing to be invoked to produce the alleged cusps. It would not help to replace one implausible thing by another. (Btw, cusps in motion would not 'produce', but would need to be produced by, some arrangement of forces.) All this illustrates how reliable sources do not support Geologician's alleged facts. The Turner reference of 1912 as well as the Young (1902) reference were written partly to correct similar misconceptions that were already in circulation a century ago. This is getting old. Terry0051 (talk) 14:51, 27 June 2009 (UTC)

[From Geologician] At Terry's suggestion, I have read both Godfrey's 1859 paper and Turner's 1912 paper, which concludes with the sentence "So that at full moon our satellite is travelling in an orbit having a curvature approximately that of the earth, while at new moon she is travelling in an orbit more than one and a half times as great as the earth's." Presumably Terry's citation implies that at New Moon our satellite, at 1.5 AU, is close to the orbit of Mars? I think we can at least agree that this possibility has been disproved long since.

Young's diagram extended to show cusps

Joe Kress misrepresents Young's comment. His figs 67 and 68 show waves and loops that he points out are erroneous. He does not say that cusps are erroneous as his diagram in fig 66 can be extended for a synodic month to show cusps, as shown here.

My diagram is merely an exaggerated version of this diagram to show the full year, as I explained in the caption.

As Young writes "the resulting path in space is one which deviates very slightly from the orbit of the earth and is always concave towards the sun.

Thus the moon orbit describes a cycloid curve [[3]] with the Earth-Moon centre of gravity at its axis. —Preceding unsigned comment added by Geologician (talk • contribs) 21:33, 27 June 2009 (UTC)

[From Terry0051] Geologician has read a phrase in Turner (1912) out of context, the subject is the curvature, not the size of the orbit, and to suggest that Turner is saying nonsensically that the moon's orbit is at any point the size of Mars' orbit is a clear and surely obvious misreading. The point labelled 'cusp' by Geologician in the page extracted from Young clearly has no resemblance to a cusp at all, and clearly Young's figure also needs to be read in conjunction with Young's statement at the top of page 174 that "the moon's orbit is always concave towards the sun". Whatever the semantic trouble that can be and has been created over 'convex' or 'concave', one thing at least is clear, 'always concave' must mean the sign or direction of the curvature does not change, and this alone clearly excludes the alleged cusps. 'Cycloid', 'epitrochoid', 'hypocycloid' etc cover a wide range of curves with or without cusps or loops. So while one of the heavily-smoothed-out versions of these curves might perhaps become an approximation for the moon's path if bent round enough into the orbit so that the remaining reverse curvature entirely disappears (as it must, in view of the proofs that the sign or direction of curvature does not change), that's not an opportunity to insinuate the unrealistic cusps back into the argument under cover of a name. There is clearly a complete lack of RS support for these alleged facts. Terry0051 (talk) 00:01, 28 June 2009 (UTC)

[From Geologician] Terry, I cannot understand how a direct quotation of the last sentence in Turner's paper can be considered out of context and how your citations from 1859 and 1912 must be considered reliable sources, whilst Young's 1902 Manual of Astronomy is not comparably RS. Your assertion, without supporting evidence, that the sign or direction of curvature cannot change is obviously at variance with any reasonable extrapolation of Young's Fig 66 that includes more than one synodic month.The true orbit of the moon is not a simple mathematical construct such as a circle, ellipse, cycloid or epicylcoid but is an infinitely complex 3D path that changes continuously over time. Young's point about concave towards the sun refers to individual synodic month segments, not to the shape during the entire lunar year, as that would result in a heliocentric spiral. If you submit a complete annual lunar path that demonstrates lunar phases using continuous concavity then we shall have a clearer picture of your ideas to discuss further. If that is not possible, then please desist from removing mine from the article.

The point of a diagram in Wikipedia is not to capture the subtleties of astrophysics in 2 mega-pixels but to convey the essence of a complex subject so that it can be understood by the interested amateur. The existing pair of diagrams confuse the matter rather than elucidate it, which is why I was obliged to substitute them with a diagram that captures the complete solar circumnavigation of the moon and its earth companion using the time-honoured device in astronomical texts of judicious exaggeration.

Geologician (talk) 16:58, 28 June 2009 (UTC)

The Moon's or Earth's orbit if the other were to disappear suddenly is completely concave to the Sun at all times, as are all closed orbits. All regular orbits are concave to the primary, otherwise it would just keep going without turning and never get around. What the text must be talking about is that even at new moon, when we would expect a loop, or spiky curve, or at least some undulation or convexity, all that is within the power of the Moon's feeble orbital speed of it's own to do is shallow up the radius of curvature of it's still concaveness to that of a 1.5 AU orbit's (temporarily of course, no spiral) instead of the nominal 1 AU radius. It would have to go all the way out to an infinity radius of curvature to be neither concave nor convex, and beyond that to be convex to the Sun. Why doesn't anybody arguing just make a scale model and see what it looks like? Sagittarian Milky Way (talk) 20:03, 28 June 2009 (UTC)

[From Terry0051]Hallo Sagittarian Milky Way:-- Yes, I believe you're completely correct in your reading of the text, fwiw I agree with you that's what the author meant.

Hallo again Geologician:

-- A direct quote of an isolated sentence is out of context if it doesn't take account of the overall message, what the text as a whole is talking about. You didn't take the overall message into account, your isolated sentence was out of context.

-- (Also, there's something like assuming good faith, giving the author reasonable credit, assuming that he probably had some sense and probably wasn't trying to say a crazy thing that nobody ever believed, and choosing a sensible reading of the whole if one is available. If a reading doesn't do that, it may end up looking like a perverse reading.)

-- I agree that Young (in its original form) is a 'reliable source' under the usual conditions: but that doesn't apply to any altered version.

-- You misquote me more than once, (a) I didn't deny Young as a reiable source and (b) I didn't say that the sign of the curvature 'cannot' change, only that there are proofs here that it does not change: those proofs are in the cited articles where they show that the orbit is always concave. It doesn't appear reasonable for you to assert that there is 'no supporting evidence'. But your suggestion that Young's statements apply only to 'an individual synodic month segment' does appears to be an invention of something not present in Young or anywhere else at all.

-- Where is your reliable source for 'judicious exaggeration'? I would say that whatever may be 'judicious' exaggeration, changing the shape is not, and the onus is on an editor who wants to 'exaggerate' to find RS in support.

-- You haven't shown any reliable source for the cusps, and your diagram does not resemble any of the diagrams in the reliable sources -- please read Sagittarian Milky Way's very good description of the real characteristics or the orbit by somebody who does 'get the point': [Quote: "all that is within the power of the Moon's feeble orbital speed of it's own to do is shallow up the radius of curvature of it's still concaveness to that of a 1.5 AU orbit's (temporarily of course, no spiral) instead of the nominal 1 AU radius."] Terry0051 (talk) 23:56, 28 June 2009 (UTC)

"Exaggerating" the Moon's path as Geologician does by increasing its orbital radius while maintaining the number of synodic months per year at 12.37 fundamentally changes the character of the Moon's path. The Moon's orbital radius is now b=a/389.2 using Turner's notation, producing a curve at new moon that has a slight bulge outward with a negative radius of curvature, that is, a radius of curvature on the inside of the closed curve. Because the curve at full moon also has a slight bulge outward, the radius of curvature of the Moon's path is always negative, hence always convex outward (concave inward). If the Moon's orbital radius were increased to about twice its present value or b=a/169.4, the curve at new moon (but nowhere else) would become a straight line, which has an undefined radius of curvature, although it could be viewed as simultaneously infinitely negative and infinitely positive. When the Moon's orbital radius is further increased, the curve at new moon will have a bulge inward which has a positive radius of curvature, that is, on the outside of the closed curve. The positive radius of curvature gradually decreases from near infinite to a very small value. Such a curve is called a curtate epitrochoid. When the Moon's orbital radius becomes b=a/12.37 an epicycloid is formed, drawn by Geologician, producing a cusp at new moon which has an undefined radius of curvature, although it could be viewed as having an infinitesimally small positive radius of curvature. A still further increase in the Moon's orbital radius produces a prolate epitrochoid with a curve at new moon that has a small loop with a small positive radius of curvature. See Handbook and atlas of curves by E. V. Shikin for examples of both a curtate epitrochoid (fig 191) and a prolate epitrochoid (fig 193). Thus Geologician's image does not have the same character as the present Moon's path, showing a cusp where no cusp exists. The only correct way to "exaggerate" the Moon's path is to magnify it as the image in the article now does, which unavoidably means that a full year cannot be shown if the undulations in the Moon's path are to be larger than the width of the line representing the Earth's orbit. — Joe Kress (talk) 00:56, 29 June 2009 (UTC)

"Straight line" above should be critical point. At larger oribital radii inflection points are present, separating the regions of positive from negative radii of curvature. Although increasing the size of the Moon's orbit is not physically possible for the Earth/Moon system, all curves described are physically possible somewhere. Specifically, the Galilean moons of Jupiter exhibit epitrochoid orbits. Because the innermost moon Io has a much higher orbital speed than Jupiter, it clearly has a prolate epitrochoid orbit with large loops at new moon, so it moves in the opposite direction of Jupiter at new moon as viewed from the Sun. (The Moon is so slow that it moves in the same direction as Earth at a slightly slower speed at new moon.) The orbit of the next moon, Europa is close to an epicycloid with cusps at new moon, although it could be argued that it has very small loops. Because both outer moons, Ganymede and Callisto, have much smaller orbital speads than Jupiter, they clearly have curtate epitrochoid orbits with positive radii of curvature at new moon. — Joe Kress (talk) 19:09, 1 July 2009 (UTC)

[From Geologician] Joe Kress says "The only correct way to "exaggerate" the Moon's path is to magnify it as the image in the article now does," That image is described by its author AM de Campos as "Moon trajectory in proper scale". This is misleading for the following reasons.

Firstly it is not in proper scale as the curvature of the Earth's orbit shown implies that the ratio between the Earth's orbital radius and the moon's orbital radius is about 150, whereas it is really about 389.

Secondly the upper left segment is shown as parallel to the Earth;'s orbit rather than converging with it as is shown correctly in the lower right segment.

Thirdly the sector illustrating one synodic month occupies approximately 21 degrees of the Earth's Orbit rather than 29.05 degrees.

Moon's trajectory showing cusp at new moon

The orbital ratios shown in de Campos' diagram make it impossible to discern what happens at new moon, which is the whole point of this discussion. I don't think it productive to discuss what characteristics greater lunar orbits might exhibit but accept your point that an epicyloid having a cusp of infinitesimally small positive radius of curvature is an appropriate model to proceed with. Your earlier analogy with what happens at the perimeter of a bicycle wheel is very apt. Both the Earth and the moon rotate around their mutual centre of mass, as I showed in my diagram. As the moon always has the same face towards the Earth, the farthest point of the new moon is directly analogous to the extreme perimeter of your bicycle wheel with its axle at the centre of mass. So the moon's orbit can be visualized as "rolling" on an imaginary surface one moon orbit radius inside the earth's orbit. Thus the cusps are real and it only remains to agree on how they should be depicted on a diagram.Geologician (talk) 18:02, 29 June 2009 (UTC)

Regardless of any problems with de Campos' diagram, it is more accurate than yours. Turner clearly shows that the radius of curvature at new moon (θ=0°) is negative for case 3, which approximates the Earth/Moon/Sun system. A more accurate value using n=13.37 and b=a/389 is ρ=−1.725a. This means that the radius of a circle which has the same curvature as the Moon's path points toward the origin (the Sun), that is, the Moon's path bulges outward at new moon, just as it does at full moon, although it has a less pronounced bulge at new moon than at it has at full moon. The outward bulge of Earth's orbit has a radius of curvature of ρ=−a, midway between that at new moon, ρ=−1.725a, and that at full moon, ρ=−0.733a (obtained using θ=14.551°). Absolutely critical for any image is that the Moon's path near new moon must be in the circular segment between a straight line or chord between last and first quarter moons (ρ=−∞) and the circular arc of Earth's orbit between those phases. If the Moon's path at new moon had a positive radius of curvature, the radius of curvature would point away from the Sun and the path would bulge inward at new moon (be closer to the Sun than a chord). It is definitely not a cusp which you show as a corner, which has an undefined radius of curvature because the denominator of the radius of curvature equation is zero at a cusp and division by zero is undefined. — Joe Kress (talk) 19:09, 1 July 2009 (UTC)

There are cusps in a pseudocartesian x=heliocentric longitude y=heliocentric distance; coordinate system. Sagittarian Milky Way (talk) 09:18, 2 July 2009 (UTC)

Wait, even the barycenter's radial velocity relative to the Sun can often be greater than the Moon's sine-adjusted orbital speed. Elliptic pseudocartesian? Sagittarian Milky Way (talk) 09:57, 2 July 2009 (UTC)

[From Geologician] Joe Kress says"Turner clearly shows that the radius of curvature at new moon (θ=0°) is negative for case 3"

It is a trivial matter to set up Turner's equations on an Excel spreadsheet to test his ideas.

shows the curve resulting from a similar case to his Case (2) with b=20 and n=13. The graph shows just 90 degrees but is easily extended to 360. It shows that the resulting curve is cycloidal and marginally cuspate.

shows the curve resulting from a case with b=5 and n=13. The graph shows that the 'cusps' have become convex towards the Sun at new moon.

shows the curve resulting from a case with b=1 and n=13. The graph shows that the 'cusps' are reduced further and are absorbed into the thickness of the line at this scale.

Using Turner's model the cusps develop into loops above b=30 and the cusps become convex below b=20.

Thus it is clear that you cannot support Turner's model and also maintain that the orbit is always concave towards the sun.

Since Young is clear that the moon's orbit is always concave towards the sun, I maintain that Turner's analysis is mistaken and that the orbit is actally a modified cycloid. —Preceding unsigned comment added by Geologician (talk • contribs) 19:33, 2 July 2009

None of your cases represent the Moon's path. Only n=13.37 and b=a/389 do (which Turner approximates in case 3 as n=13 and b=a/400). Turner's cases 1 and 2 do not represent the Moon's orbit. I have already stated many times that other moons can have any conceivable orbital shape relative to the Sun, including hypothetical and unrealistic exaggerations of the Moon's present orbit—those that maintain 12.37 synodic months per year regardless of size even though those orbits violate the universal theory of gravity. In my "exaggerating" discussion above I stated that if the Moon's orbital radius is increased beyond b=a/169.4 (while maintaining n=13.37) it developes a bulge inward, which means it ceases to be everywhere convex outward. This only shows that Turner's model is correct because it can generate all possible orbits, including the Moon's orbit whose convex outward shape is only possible with parameters near those of the Moon's actual orbit. — Joe Kress (talk) 03:52, 3 July 2009 (UTC)

400 in the PDFs isn't even the same distance in both axes. Sagittarian Milky Way (talk) 09:53, 3 July 2009 (UTC)

[From Terry0051] I believe it's relevant to the improvement of the main article to give cross-references here to the following two matters (especially because of the amount of editorial efforts that have been put into this section here, and because of the length and character of the discussion):

(a) a message on 'Geologician' 's talk page asking him to adhere to WP guidelines, especially in regard to disruptive editing; and

(b) an application for a 'sockpuppet investigation' to check whether 'Geologician' is in effect an 'alter-ego' of 'Shaheenjim' (a now-blocked user who had also contributed in a seemingly similar way to the 'convex or concave' issue further up this talk page before being blocked for activities elsewhere). Terry0051 (talk) 19:27, 4 July 2009 (UTC)

Moon's orbital speed: 'correction' was mis-correction

Recent change was said to be a 'correction' from 1.023 'km/s' to 'm/s'. Mean distance from the Earth is about 385000 km (not m), mean sidereal period is 27.321661 days of 86400 secs, (385000 x 2 π )/(27.321661 x 86400) comes to about 1.024 in km/s not m/s. After a chance for anybody to point out mistakes in this , the 'correction' should be reverted. Terry0051 (talk) 22:47, 1 December 2009 (UTC)

Unlike all other moons ?

In Path of Earth and Moon around Sun the article says "Unlike all other moons in the solar system, the trajectory of the Moon is very similar to that of the Earth". In context, I assume this refers to the fact that the motion of the Moon as viewed from the Sun is never retrograde. This is a consequence of the fact that the Moon's orbital speed relative to the Earth is smaller than the Earth's orbital speed relative to the Sun. But the same is true of both satellites of Mars - the orbital speed of Mars relative to the Sun is about 24 km/s, whereas the orbital speeds of Deimos and Phobos relative to Mars are about 1.4 km/s and about 2 km/s respectively. It must also be true for the outer moons of both Jupiter and Saturn - by my calculation, any moon of Jupiter beyond Europa and any moon of Saturn beyond Rhea has this property. So how is the phrase "Unlike all other moons in the solar system" justified ? Gandalf61 (talk) 14:13, 12 March 2010 (UTC)

Not only is the Moon's path never retrograde, it is never concave outward; can other (known) bodies say the same? —Tamfang (talk) 08:04, 14 March 2010 (UTC)

If "concave outward" is what this sentence means, then I agree that is a stronger requirement that never retrograde. So let's see if the Moon is unique in that respect. This reference says that in a simplified model with circular orbits in the same plane, there is no retrograde motion when p < d and a convex path (which I assume is what you mean by "concave outwards") when p² < d where p is the ratio of primary's period to satellite's period, and d is the ratio of primary's orbital radius to satellite's orbital radius. We have:

Primary	Satellite	p	d	p < d	p2 < d
Earth	Moon	13	389	Yes	Yes
Mars	Phobos	2,160	24,308	Yes	No
Mars	Deimos	544	9,716	Yes	No
Jupiter	Megaclite (outermost moon of Pasiphaë group)	5.4	31.5	Yes	Yes
Saturn	Fornjot (Outermost known moon)	7.9	60.7	Yes	No
Uranus	Ferdinand (Outermost known moon)	10.7	137.7	Yes	Yes

So it seems possible that some of the outermost moons of Jupiter and Uranus may also have "concave outwards" orbits. A more detailed calculation would be required to take account of non-circular orbits and inclination to the ecliptic. But I certainly think the "Unlike all other moons ..." claim needs to be either sourced or watered down - "Unlike most other moons .." would be more accurate. Gandalf61 (talk) 12:39, 14 March 2010 (UTC)

No response, so I have been bold and changed "all other moons" to "most other moons". Gandalf61 (talk) 13:58, 18 March 2010 (UTC)

A plane orbit is convex if the planet's pull on the satellite does not exceed the Sun's. (Surprisingly, the Sun's pull on the Moon is 2.5 times Earth's.) That criterion can be applied to inclined orbits too, so that's one way to generalize. —Tamfang (talk) 01:03, 20 March 2010 (UTC)

Another is to ask whether the dot-product of the momentary acceleration vector and the position vector (relative to the barycenter) ever be positive. —Tamfang (talk) 22:09, 12 September 2010 (UTC)

May not have always been receding ?

In contrast to all texts that I have ever read, this maths site www.mathpages.com/home/kmath273.htm plausibly proposes that the moons orbit has not always been receding from Earth. I think that mention should be made of this in the article. I will likely be unable to pursue this matter as I rarely have appropriate internet access. 93.187.145.247 (talk) 20:12, 11 September 2010 (UTC)

Angular momentum considerations

Both the earth and the moon carry stored Kinetic energy related to their orbital path around the sun. They also carry stored kinetic energy related to their orbital path around their center of gravity and rotation. However these values interact in such a manner as to allow the earth-moon system to perturb the sun orbit parameters of particularly the moon by adding to and subtracting kinetic energy (and angular momentum) from the moon's orbit around the sun during certain phases of its orbital path around the sun. This is presumably an important factor related to the ballistics of the moon's orbit around the sun, and explains why the moon speeds up and rises in orbit over the Earth's orbit radius during half the moons orbit, and then slows back down and passes back under the earth's orbit radius during the other half. Is there any discussion of this angular momentum transfer subject matter available to be read?WFPM (talk) 02:31, 14 February 2011 (UTC)

The Moon's orbital speed is not larger when above (north) of the ecliptic (Earth's orbit) and smaller below. The Moon's speed is largest at perigee and smallest at apogee, opposite apsides. This line of apsides takes 8.85 years to precess once in the direction of the Moon's motion. Conversely, the Moon's orbit intersects Earth's orbit at a line of nodes, the ascending and descending nodes. The line of nodes takes 18.6 years to precess once opposite to the Moon's motion. Thus a fast Moon is equally likely to be above the ecliptic as to be below it.

However, there is a semiannual variation in the Moon's inclination to the ecliptic. The Moon's average inclination is 5.14° but it varies from 4.99° to 5.30° or about ±0.15°. It is a maximum when the line of nodes is aligned with the Sun and a minimum when they are perpendicular to it. The moon's inclination gradually rises over about three months, gradually falls over three months, then repeats. See Jeen Meeus, Mathematical astronomy morsels 11–12, who mentions neither kinetic energy nor angular momentum. — Joe Kress (talk) 06:06, 14 February 2011 (UTC)

Does that mean that that you think that there is no transfer of angular momentum to the moon and and then back to the earth during the period of the moons orbit around the earth? That's the idea that I'm trying to promote.WFPM (talk) 15:00, 14 February 2011 (UTC) Then we can generate the idea that although the orbit of the moon is always around the sun, it includes 1 incidence where it advances over the top of the earth's orbit and another incidence where it slides back under the earth's orbit. And I'm thinking about the stability of such a situation.WFPM (talk) 15:08, 14 February 2011 (UTC)

The orbit of the Moon is NOT around the Sun, it is around the Earth. Simply because the path of the Moon is always concave towards the Sun does not mean that it orbits the Sun. That is simply a artifact of the relative size of the Moon's orbit around the Earth to the size of Earth's orbit around the Sun. If the Moon did orbit the Sun, it would no longer orbit the Earth. However, the Moon is far enough from Earth that the Sun perturbs the Moon's orbit significantly, causing the Earth-Moon-Sun to be the classic three-body problem which does not have a closed-form solution. — Joe Kress (talk) 07:07, 15 February 2011 (UTC)

Interesting difference in concept!! So the Sun is in orbit around the sun. And the moon is on a point behind the earth along the path of the earth's orbit around the sun. And then the moon rises up and over the earth's orbit around the sun (during a 15 day period) and then is on a point ahead of the earth on the earth's orbital path around the sun. And then the moon drops down and back to a point between the earth and the sun, and then moves up and back to a position behind the earth in its path around the sun, (also within the next 15 day period) And the question is how it managed to do that. And I thought that that was because the gravitational force of the earth on the moon was able to speed up the velocity of the moon during a 15 day period so that so that it had sufficient angular momentum in its path such as to raise its radius of motion around the sun to be greater than that of the earth. Then during the next 15 day period it fell back behind the earth in orbit due to having lost back to the moon the change in angular momentum around the sun. I'll leave it there for now for any additional comments that you might want to make. Cordially, WFPM.WFPM (talk) 21:19, 15 February 2011 (UTC)