text:
Talk:Apsis

Missing definitions
I miss the definitions of the two kinds of Perihelion times, namely: 1. The perihelion of the earth itself, and 2. The perihelion of the centre of gravity of the earth-moon system. The difference between these two times ("moments")can be as much as 26 hours, depending on the moonphase around January 4.

Alex Vermeulen, Zoetermeer, The Netherlands (Jordaan12 (talk) 15:24, 4 November 2009 (UTC))

Is that really correct? The original Latin term is apsis, plur. apsides. From apsis derives Eng. apse which can hardly have apsis as plur.

Sebastjan 11 March 2004 10:45 (CET)

Isn't there something slightly odd in the term "furthest approach"? Evercat 01:40, 15 Mar 2004 (UTC)

Apogee
There is a link on this page to apogee which simply links back to this page. This page says nothing about the term apogee though.

Perijove and apojove
Why should "perijove" and "apijove" be avoided? Sez who? They are probably plainer.

Rlquall 19:00, 19 Aug 2004 (UTC)

 * It is considered very bad form to mix Latin roots with Greek, which is what "perijove" does. Jupiter is Zeus in Greek, and the corresponding Greek root is Zeno, hence perizene/apozene.

 * Urhixidur 15:41, 2004 Aug 20 (UTC)

 * "This is the kind of impertinence up with which I shall not put!"--Churchill Seriously, to say it's "very bad form" is extreme pedantry, along the lines of "never split an infinitive". Wikipedia doesn't seem to have a List of words combining Latin and Greek roots, but 'automobile' is the canonical example. Perijove and apojove are far more commonly used than perizene and apozene. NASA uses them; why shouldn't we?
 * —wwoods 19:37, 4 Oct 2004 (UTC)

 * If NASA uses them then that should at least deserve a mention in the footnote. Wikipedia should be descriptive rather than prescriptive, IMO; we should use the most widely-used term even if it's "wrong", though its "wrongness" should still be pointed out. Bryan 23:27, 4 Oct 2004 (UTC)

 * And the "wrongness" is of the "Thou shalt not wear a cloth woven of wool and linen" (Deut. 22:11, I ♥ Wikipedia) variety. The usual terms are better, since someone who hasn't seen them before stands a good chance of understanding -jove = -Jupiter, while nobody is going to make the leap -zene = -Zeno = -Jupiter. Without context, I'd have guessed they were geologic periods.
 * —wwoods 16:12, 7 Oct 2004 (UTC)

Further to this, I have just toned down the 'it is considered more correct to use the greek form for the body'. It's not specified who considers it correct, and as these are all scientific terms, I think it's best to go with what the scientists actually use, regardless of any linguistic concerns. Otherwise, you might as well put a note in the Television article saying that 'proculvision' or 'teleopsis' are considered more correct. Also, I couldn't find any use in the literature of the '-cytherion' or '-areion' terms, so I moved mention of those from occasionally used to never used. Worldtraveller 08:43, 22 Apr 2005 (UTC)

 * Absence of evidence is not evidence of absence, as Einstein used to say. The problem with trying to find quotes in the litterature through the Web is that there are still huge gaps. You wouldn't be able to find the first use of -saturnium mentioned in the article because it only exists as scanned-in GIFs on the web (I stumbled across it serendipitously). A lot of the litterature of the mid-20th century has yet to be made available either. The term -cytherion, apparently, was introduced by Carl Sagan in the 1960s...But just try and prove it! Anyway, I think the article is reaching a pretty decent equilibrium point by now.
 * Urhixidur 17:32, 2005 Apr 22 (UTC)

Indeed, very true, but if we can't cite our sources, then the information is not verifiable, and I think that's very important for an encyclopaedia. I'd say anyway that the gaps in online coverage of astronomy literature are not that huge. The Astrophysics Data System contains scanned versions of a huge number of journals right back to issue 1 in most cases, and even allows you to search OCR'd text of scanned journal articles. Checking now uncovers a total of 87 refs to -saturnium including the one you chanced across, of which 8 are in the last 50 years. Still no sign of perizenes, -hermions or others though. I still question, I'm afraid, the statement that some scholars object to these combined forms. Who are these scholars? If you can show me an example of a scholar actually saying that perijove should not be used, then that is worthy of including, but seeing as perizene is never used, it seems odd to try and give it credence by including it here in the table with perijove. We don't do the same with other hybrid words such as television, although they do have an article about them at hybrid words. I've now commented out the forms which do not appear anywhere in the available searchable astronomical literature. If they are actually used in journal articles anywhere, then they should be included, and sources should be cited, but otherwise they're just neologisms. Worldtraveller 19:18, 22 Apr 2005 (UTC)

Ten years on, the objections to perizene and apozene are even more relevant. They still do not appear in the full text of the ADS database of astronomy and physics papers, or on Google Ngram. Likewise for peri- and apocytherion, -uranion, -poseidion or -hadion. These words seem to have no scientific currency. Peri- and apohermion do appear in ADS, but are outnumbered by peri- and apoherm. So the list here doesn't reflect scientific vocabulary, and hasn't managed to significantly penetrate scientific usage during the last decade. I'm amazed they're still here. Should Wikipedia really be trying to perpetuate words of its own devising? There's a summary of the current situation with regard to apsidal names at https://oikofuge.com/perihelion/

JimmieBlue (talk) 19:41, 12 January 2017 (UTC)

Excellent explanaiton
That would explain with regard to most of the planets, then, the more obscure (to most of us) Greek vs. Latin form, and why "Perilune/Apilune" would be totally incorrect for "Periselene/Apiselene". Excellent.

Rlquall 15:54, 20 Aug 2004 (UTC)

Aphelion's Pronunciation
It's not clear to me what the proper pronunciation is for "aphelion", from a linguistic point of view. My copy of the American Heritage Dictionary (2nd college edition) gives the options of "a-feel-yon" and "a-feel-e-on". Indeed, almost every (perhaps every) Greek work that's come into English turns the greek letter Φ (phi) into a FFF sound.

However, it is the case that, rather than being a mere hypercorrection, 'ap-helion' would indeed be the ancient pronounciation, if we were able to bring some ancient Greek speaker to this time and show him or her the written word. (Source: Vox Graeca, by Sidney Allen, the standard book on the prounciation of Ancient Greek)  I guess I'm uncomfortable hearing it called a Hypercorrection, since it does, in fact, reflect the classical pronunciation, which is being taught increasingly these days to students of Ancient Greek. (When I was studying Ancient Greek twenty-five years ago, the pronounciation we were taught was the Erasman Pronounciation, which pronounces Φ (phi) as FEE. The Modern Greek pronounciation of the letter is also FEE, and that's how I hear it pronounced when I attend a Greek Orthodox church service.)

The comment made in the article -- about "Greek από, from, which becomes απ before a vowel, and αφ before rough breathing)" -- is absolutely true. However, the sound change only makes sense when you understand that the ancient Greeks really did pronounce Φ (phi) as an aspirated P sound, like the P that starts the English word "pin", where we release a puff of air, a sort of soft H, after the P.  Try it yourself, then compare with the way you say the P in "stop", where you don't release the puff of air.  Again, this change makes sense only if you understand that "apo helion", by the ancient sound transformation rules, naturally turns into ap' helion, which turns into "aphelion".   And the only way that change is natural is if it's pronounced in the classical way.  So go figure.

Saugart 06:04, 27 August 2006 (UTC)

Missing Definitions
The article needs to define what mu and 'e' are. I know e is called the eccentricity. I have no idea what mu is. --69.5.156.155 14:11, 14 Dec 2004 (UTC)

"MU" is gravitational parametr for central body, i.e. product of its mass and gravitational constant.

217.11.255.138 3 July 2005 19:06 (UTC) (A.Vitek from CS.wiki)

earth's apogee/perigee
When are Earth's apogee and perigee? (Presumably, it's at around the same time each year.) I know it's not the same as the solstice, so when are they? --Carl 01:59, 1 Jan 2005 (UTC)

 * Apogee around July, perigee around January, I think. However, unlike Mars, the Earth's orbit has low eccentricity and thus the distance difference between perigee and apogee is more or less negligible and doesn't affect the weather noticeably. -- Curps 07:46, 1 Jan 2005 (UTC)

 * Shame on you Curps for not spotting the mistake. Earth has no apogee or perigee because it does not revolve around itself! For Epoch J2000, Earth's perihelion occurred on January 3, 1999, aphelion on July 5,1999. The following years, perihelion occurred on January 4, 2000, January 3, 2001-2003, January 4, 2004. Aphelion was on July 4, 2000-2001, July 5, 2002-2003, July 4, 2004. Next occurrences are January 3, 2005 and July 5, 2005.
 * Urhixidur 15:25, 2005 Jan 1 (UTC)

Pronunciation of aphelion
I'm sure that many books stating that the correct pronunciation of aphelion is ap-helion can be found. However, the Greek word suggest the exact opposite. The classical Greek word for sun is ἥλιος (hēlios). The preceding preposition is ἀπό (apo). When this is prefixed to another word that begins with a vowel it looses its final ο, and if that word begins with the spiritus asper, the π is aspirated into φ. Thus the compound word is ἀφήλιον (aphēlion), not ἀπήλιον (apēlion). If you feel that the ph in photograph, pheromone and physics should be pronounced with an /f/, then the same holds true for aphelion. Gareth Hughes 01:31, 6 Mar 2005 (UTC)

 * Presumably your typical classical Greek would have a lot more trouble with the meaning of the word than with its pronunciation. :-) How would a modern Greek pronounce it? Anyway, whatever its roots, it's an English word, and not bound by its ancestry.
 * —wwoods 19:00, 22 Apr 2005 (UTC)

 * All right, let's start looking at sources. For now I'll put just this one because that's all I've got here. More to follow:

 * Webster's Ninth New Collegiate Dictionary (1989) : afelyon (also attests -cynthion, -gee, -lune; note that -astron and -apsis are absent)
 * Urhixidur 19:16, 2005 Apr 22 (UTC)

Fcw July 4, 2005 00:34 (UTC)
 * The OED concurs with the /af-/ pronunciation; they cite that Kepler coined the term around 1596, and also that there was an alternative form, aphelium, for a time.

Redirection from Pericenter
I was redirected to this page from the term 'pericenter', as used in article S2_(star), but the page does not contain the word, let alone the definition. --128.9.216.23 17:36, 17 August 2005 (UTC)

mnemonics
Are thre any useful mnemonics to help remember which one (peri- or ap-) refers to the closest point and which refers to the far point? I can never remember without checking this page. SigmaEpsilon → ΣΕ 05:14, 9 December 2005 (UTC)

Use the word "away" (as in away from the earth) to remember aphelion. Russ

Question
I wanted to find out what a perihelion is, but this article is completely incomprehensible. Could there be a friendlier explanation? I dont think it's really that complicated. —The preceding unsigned comment was added by 194.80.49.125 (talk • contribs) 15:43, 9 August 2006.

 * I'm sorry that you find the article difficult. Please explain what could be done to make it more accessible/relevant. A perihelion is a point in space at which an object (usually in orbit around the sun) is closest to the sun. The Earth's perihelion is the moment when it is closest to the sun. THe furthest point from the sun is called the aphelion. — Gareth Hughes 14:58, 9 August 2006 (UTC)

The diagram doesn't show any apsis
203.144.143.7 15:19, 4 September 2006 (UTC)Chris

Perigee but no apogee article
Is there a reason that there is an article for perigee, but not apogee? Perhaps these should all be merged? Or should the apogee article be created? Lunokhod 21:35, 16 January 2007 (UTC)

Advance of perihelion
Obviously the perihelion of the Earth varies a few days correponding to a few degrees or some thousend seconds of arc every year. That's much more than the relativistic advance of 43“ per century – not year – as predicted by general relativity for the planet Mercury. 84.169.216.201 09:38, 21 March 2007 (UTC)

 * As clearly seen form the table of data, the advance of perihelion is fluctuating by some days (one day is about 3,600“) from year to year. There is not a continuous advance every year or century. Therefore, it just nonsense to say Earth or Mercury have a certain advance of perihelion per century caused by some effect due to general relativity, since the fluctuations every year are larger than this effect in one century. 84.59.54.27 18:21, 2 April 2007 (UTC)

 * The variations are caused by the moon and to a lesser extent the other planets in the solar system. This is not explained in the article (it should be). -- B.D.Mills  (T, C) 06:42, 28 May 2010 (UTC)
 * If what you say were true Milankovitch theory would be impossible as the orbits would be too noisy to extrapolate them with the necessary accuracy to a million years.
 * Just as annual climate is made meaningful by averaging out the daily and seasonal temperature fluctuations, so can an orbit of a million years be made meaningful by averaging out the fluctuations. Even more so than climate because the orbital fluctuations are more predictable than the temperature fluctuations since the orbits of the perturbing bodies are predictable.
 * Apropos of the influence of the Moon, NASA JPL's orbital elements for Earth are preferably given for the Earth-Moon barycenter so as to eliminate that influence. The remaining two big influences are then Jupiter and to a less extent Venus, which appear as pulses along an otherwise almost perfectly elliptical orbit coinciding with when those planets are respectively in opposition and inferior conjunction, essentially their points of closest approach to Earth.
 * I would however think that giving distances to the Sun to the nearest km only makes sense for a best-fit Keplerian (elliptical) orbit defined by a choice of orbital elements, and even then 1,000 km precision would be more appropriate. Giving the actual distance to the nearest km at any specified instant conveys nothing for the reason you give, namely those perturbations.   Vaughan Pratt (talk) 23:44, 19 December 2013 (UTC)

Confused
Could anybody please explain the difference between, say, the Perihelion of a planet and its semi-minor axis? --Overpet 02:14, 10 June 2007 (UTC)
 * A semi-minor axis is half of the shortest possible diametric section of an ellipse. It is measured from the centre of an ellipse to the ellipse itself. In an elliptical orbit, the gravitational centre is not at the centre of the ellipse (unless it happens to be a circle, it has zero orbital eccentricity), but at one of the two foci of the ellipse (a circle is an ellipse with both foci at the centre). As you can see from the diagram, the place where the ellipse is closest to F1 is A, and closest to F2 is B, whereas the semi-minor axis can only be measured to C or D. The place of farthest approach is at the opposite end of the major-axis from the closest. Perhaps it's good to think of an extremely eccentric orbit, like that of a comet orbiting the sun: it disappears off out of the solar system fo long periods, and so the sun is clearly not at the centre of the ellipse, but far away at one of its foci. — Gareth Hughes 11:01, 10 June 2007 (UTC)
 * Wow, I didn't expect such a good response, thanks very much. If you wouldn't mind clarifying one thing for me. If, for example, the sun was at the centre of an ellipse, which I assume from your above comment is not possible, as it would be at one of the two foci, but for arguements sake, if it were at the centre, then would the Perihelion of an orbiting planet and its semi-minor axis be the same? Thanks very much. --Overpet 12:25, 10 June 2007 (UTC)
 * It is possible for the gravitational centre of attraction of an orbit to be at the geometrical centre of the orbit, but then it creates an ellipse with zero eccentricity: a circle. A circle is an ellipse in which the axes are equal and the two foci merge on the geometric centre. Of course, there is no notion of closest approach with a circular orbit: the orbit remains equidistant at all points. If an object were at the centre of a true ellipse (one that is not a circle or a hyperbola), it would not be at a focus and thus would not be the gravitational centre of attraction. But, in such a case, the orbit would pass closest to it at the two points where the minor axis intersects the orbit. — Gareth Hughes 16:40, 10 June 2007 (UTC)

There is no real definition for the terms listed on this page, or if I am mistaken, the article needs some clean-up. I was looking up perihelion in this article and fail to understand it.. WinterSpw 05:46, 12 June 2007 (UTC)

Ceres and Eris
I don't know if the other two dwarf planets besides Pluto have special names for their apses. Wouldn't they be "-demetrion" for Ceres and "-eridion" for Eris? Or am I wrong? —Preceding unsigned comment added by 151.203.56.155 (talk) 01:21, 23 December 2007 (UTC)

Simplify Intro, at least
This article needs to start off simply and answer the most common questions first, such as when and why the earth is closest to and furthest from the sun. Going to "perihelion" results in a redirect to this article, so someone looking for information on perihelion wants something at the top. The article is too technical too quickly and will leave the average non-expert baffled from the start. It could use some editing to make it more user-friendly.Tmangray (talk) 00:54, 6 January 2008 (UTC)

time of periastron
what is it? a field named like this apears on the exoplanet infoboxes, like in 70 Virginis b, XO-1b, WASP-1b and others. can anyone explain it? thanks. 190.21.218.236 (talk) 02:40, 25 April 2008 (UTC)

 * It is a moment of time when (exo)planet passes periastron. — Chesnok (talk) 17:28, 27 April 2008 (UTC)

 * I don't quite catch it. On 70 Virginis b says "Time of periastron (T0)= 2,447,239.82 ± 0.21 JD" . That means that the planet passes it's apsis each 2,447,239.82 days? or that passes it 2,447,239.82 days after j2000? thanks again, by advance 190.21.217.7 (talk) 04:07, 28 April 2008 (UTC)

 * JD is Julian day (number of days since noon Universal Time (UT) Monday, January 1, 4713 BC).
 * JD 2,447,239.82 ± 0.21 is a moment of time when planet passes periastron. — Chesnok (talk) 08:23, 28 April 2008 (UTC)
 * Now I understand. 190.21.250.10 (talk) 20:44, 28 April 2008 (UTC)

What caused Image:Inner Planet Orbits.jpg to go up for deletion per CSD
I noticed the remaining text still refers to two images, though there is only one image. I was going to correct the text, but upon viewing Image:Inner Planet Orbits.jpg I couldn't figure out why it is deleted. I found it very informative, and its copyright annotations seemed all in order. —Preceding unsigned comment added by 99.226.205.27 (talk) 15:42, 28 November 2008 (UTC)
 * I see no reason why it was removed and so have re-inserted it. AstroMark (talk) 22:35, 14 December 2008 (UTC)

aphelia?
Does anyone know why the word aphelia (Greek for brotherly love) redirects here? Jeffrey Pierce Henderson (talk) 07:22, 11 December 2008 (UTC)
 * Aphelia is the point of an orbit furthest from the Sun (Sun being helios in Greek. Are you perhaps getting confused with philia? Also see other Greek words for love. AstroMark (talk) 13:05, 11 December 2008 (UTC)
 * Thank you. Jeffrey Pierce Henderson (talk) 16:41, 11 December 2008 (UTC)

Lunar perigee
Hi. This article still doesn't say anything about perigees of the moon, which is what I was really looking for. Today, the moon is the closest to Earth in all of 2008. Apparently in 1999 the moon was the closest for a century. Either there should be an article on lunar apses, or this article should contain the specific information. Or, maybe this article could link to such articles. Either way, please add the information. Thanks. ~ A H  1 (TCU) 03:07, 13 December 2008 (UTC)

Tiny tweak to date
In case anyone wonders why I'd interfere in the article just to remove 'AD' from "2000 AD", the reasoning is basically like this: "Hmmm, someone's put 'AD' after the year number when, a popular British comic title aside, it should really be before it ("the Year of Our Lord 2000"). But then, that's going to seem a bit religiously fussy.  Let's switch it to 'CE'.  On the other hand, 'CE' isn't really any better, despite common practice, since it's still just the Gregorian calendar based on the traditional birth year of Jesus and therefore still carries specifically Christian overtones.  And let's face it, most of the time we don't bother with either: you rarely hear someone say that the Second World War started in AD1939, or that it ended in 1945CE, so let's just leave it out entirely and avoid the whole issue."

Sorry. I have far too much time on my hands. - Laterensis (talk) 08:29, 4 August 2009 (UTC)

Opening paragraph: Center vs. focus
The opening paragraph of this article is    "An apsis, plural apsides (pronounced /ˈæpsɨdiːz/) is the point of greatest or least distance of a body from the center of its elliptical or eccentric orbit. In modern celestial mechanics the center is also the center of attraction, which is usually the center of mass of the system. Historically, in geocentric systems, apsides were measured from the center of the Earth." The use of "center" in sentence 1 is misleading because, without a qualifying adjective before it, the reader's default interpretation is bound to be the "geometric center", which is not correct. Also, the word "focus" never appears in the opening paragraph, although the diagram makes it clear that the "center" referred to is really one of the foci.

I propose to replace the opening paragraph with the following:

"An apsis, plural apsides (pronounced /ˈæpsɨdiːz/) is the point of greatest or least distance of a body from one of the foci of its elliptical orbit. In modern celestial mechanics this focus is also the center of attraction, which is usually the center of mass of the system. Historically, in geocentric systems, apsides were measured from the center of the Earth."

Would anyone object to this? Duoduoduo (talk) 23:14, 22 June 2010 (UTC)

Dubious time period for Milankovitch cycles

 * The dates of perihelion and aphelion progres through the seasons, making one complete cycle in 22,000 to 26,000 years. This is a mechanism behind one of the many Milankovitch cycles, but represents an insufficient period to cause the ice ages which occur on a period exceeding 100 million years. (emphasis mine)

The highlighted time period is dubious because so far as I remember Milankovitch cycles operate in periods of 100,000 years and not 100 million years. Ice ages are about 100,000 to 200,000 years apart so 100 million years is probably incorrect here. -- B.D.Mills  (T, C) 07:08, 6 July 2010 (UTC)
 * You're right. The article on Milankovitch cycles says ice ages happen on a 100,000 year cycle. I don't know where the 100 million year number came from but it's wrong. I'm going to correct it. Dr. Morbius (talk) 19:11, 6 July 2010 (UTC)

Apoasis

 * The point of farthest excursion is called the apoapsis (ἀπό, apó, "from", which becomes ἀπ-, ap- or ἀφ-, aph- before an unaspirated or aspirated vowel, respectively), apocentre or apapsis (the latter term, although etymologically more correct, is much less used).

I'm confused. Some articles use apoasis (61_Cygni, Season, Earth's_orbit and Seasons1). Should we add this form to the article, or should we correct the others? Sharkb (talk) 10:43, 25 July 2010 (UTC)
 * The others are wrong. They are misspelled. Dr. Morbius (talk) 20:26, 25 July 2010 (UTC)

Stop redirect, create two less technical pages. (Apogee/Aphelion & Perigee/Perihelion)
I think this article could gain a lot by avoiding a redirect from the commonly-used words of Apogee and Perigee. Apsis is an overarching concept, and to understand the basics of "Apogee" one need not first grasp the overarching term "Apsis". It's confusing to the general encyclopedia reader, who may be seeking to answer the question of "What is an Apogee?" or "What is a Perihelion?"

I suggest that Perigee/Perihelion and Apogee/Aphelion are given their own, less technical pages (each having a link to the general concept of Apsis). -PhysGrad —Preceding unsigned comment added by 128.146.34.151 (talk) 20:49, 19 October 2010 (UTC)

 * Sounds like duplicating information to me. I think the 3rd para in the intro partially remedies for the complications you're adressing, although improvements could well be considered. Weird items such as perimelasma for black holes could redirect to Apsis. Rursus dixit. ( m bork3 !) 20:00, 22 January 2011 (UTC)

 * The trouble is, the 3rd paragraph doesn't explain what these words mean. Currently, one has to wade through a lot of irrelevant notes about greek grammar to try and extract the basic meanings of the two apsis forms, and then figure out how to apply this to the derived terms. I think a simple paragraph near the beginning, perhaps under a separate heading, defining in simple terms what perigee/apogee and perihelion/aphelion mean would be useful.

 * Also, although we are told what the genitive and plural forms are and given some (irrelevant?) info about aspirated/unaspirated vowels, it doesn't actually say what the greek word ἁψίς means. If anyone thinks the apo/apho stuff is relevant, or even interesting, then maybe it should be moved to a footnote - or maybe add a separate etymology section. Thirteenangrymen (talk) 11:51, 4 April 2011 (UTC)

Not useful
This is another of those annoying pages that is only useful if you already know what is being explained. It's of no value to anyone who is trying to understand either the definition or mechanics of perihelions. It's just one of those pages where the preacher is speaking to the converted. Of what use is that to Wikipedia? Anyone who wanted this sort of technical information would not rely upon Wiki, which is notoriously inaccurate anyway, but a more technical source. Very disappointed. NaySay (talk) 19:37, 8 April 2011 (UTC)

 * I agree, if I want to get an idea of what an apogee is, I either have to infer it from the jumble of words in the opening section, or just look somewhere else where the writers aren't so impressed with themselves. The Wikipedia isn't a technical journal. — Preceding unsigned comment added by 24.11.145.12 (talk) 01:15, 15 October 2012 (UTC)

Apsis distances vary
The tables or article should make it clear. As an example, Solex 11.0gives, from 1900-2100, Earth perihelions ranging from 147.083345 to 147.112547 Gm, with a mean of 147.098078 Gm. Saros136 (talk) 10:26, 10 April 2011 (UTC)

Dates of Perihelion and Aphelion shifting (long term)
"On a very long time scale, the dates of the perihelion and of the aphelion progress through the seasons, and they make one complete cycle in 22,000 to 26,000 years"

Could it please be included in the article which way the dates are currently tending. For example is the July apsis currently tending towards june or august!? I think this would be a very good informative addition to this article. Thanks in advance! —Preceding unsigned comment added by 119.18.19.65 (talk) 02:03, 24 April 2011 (UTC)

International Units Astronomical Measures
The standard system for Astronomical measures is the decimal system. Distances should always be expresed as Kilometers and if available the milage in parenthesyst and not the other way arround. —Preceding unsigned comment added by 108.38.23.35 (talk) 06:22, 12 May 2011 (UTC)

 * Agreed, except that Unsigned does not mean the decimal system. The standard system of measurements used in 99% of countries in civil life and in 100% for almost all scientific work is the SI. for which the unit of length is the metre; and, for Solar System distances, the kilometre is a popular multiple. 94.30.84.71 (talk) 10:51, 23 September 2011 (UTC)

First Diagram
In the first diagram, the periapsis is not the nearest point on the ellipse to the focus. It should be. The diagram should be re-drawn in true proportion. The diagram in "Confused" above looks correct in this respect, and could be copied and modified. 94.30.84.71 (talk) —Preceding undated comment added 10:55, 23 September 2011 (UTC).

Terminology problem.
So the article about the International Space station says that it has a perigee of 380 km and an apogee of 410 km ( or numbers like this ). So according to the stated formula, the major axis of its elliptical orbit is 790 km  and the length of its semi-major axis is 395 km ? Which would put its orbit where ! Umm no. And repetitive and pedantic waffle about the distinction between the centre of the earth  and the barycenter of the Earth-ISS system does not solve this problem. It is, what, about 3 centimeters in the centre of the earth ?

The "altitude" of satellites and other man-made objects in earth orbit is often described in this way, which people can clearly understand. Yet the actual major axis length of the ISS orbit is more like 14,000 km (whatever).

Is it actually correct to state the apogee and perigee of the space station in this manner ? I think this issue needs to be clearly and plainly discussed at some part of this article.Eregli bob (talk) 00:56, 1 July 2012 (UTC)

 * According to policy and style guidelines, it is correct for Wikipedia to reflect real-world usage. NASA has traditionally used the altitude definition since the beginning of the space age. I would assume this is because the public is more interested in how high a spacecraft is above the ground, than in its distance from the Earth's center of mass. The problem is, this article seems to have been written for and by physicists and astronomers, rather than engineers and people interested in real-world spaceflight. I absolutely agree, this article should mention the common usage in the altitude sense. JustinTime55 (talk) 17:33, 25 July 2012 (UTC)

Geometric mean of two positive values cannot be an imaginary number
The Apsis/Mathematical formulae section states:
 * The geometric mean of the two limiting speeds is $$\sqrt{-2\epsilon}$$

First of all, the geometric mean should not be an imaginary number; second of all, using the speeds given the g.m. appears to be simply $$\sqrt{\mu/{a}}$$ hgilbert (talk) 11:32, 3 December 2012 (UTC)

 * Specific mechanical energy of an orbit is $$\epsilon=-\frac{\mu}{2a}$$, so the radicand is actually positive. Your second point is correct though. Jaxcp3 (talk) 15:26, 7 August 2013 (UTC)

Pericynthion and apocynthion
I spent a lot of time watching Walter Cronkite and the Apollo program on TV. I never heard pericynthion and apocynthion. I heard perilune and apolune several times.  Randall Bart    Talk   22:14, 3 December 2012 (UTC)
 * Pericynthion and apocynthion were the terms NASA used. I remember tuning in during the early hours of the Apollo 13 accident and hearing the astronauts discussing a post-pericynthion abort with mission control. The networks may have preferred the shorter terms.--agr (talk) 00:42, 3 January 2013 (UTC)

Apogee and Perigee
No useful information about apogee and perigee. This is especially bad because those definitions are complex -- there are a number of different standards used by different countries depending on what Earth radius they used (equatorial radius, mean radius, height above geoid). This leads to a lot of confusion when people get historical orbit info from US, British or Soviet sources, because they all used different definitions. DonPMitchell (talk) 20:24, 11 March 2013 (UTC)

What does the distance refer to?
Being as the distances being given are precise to a distance smaller than the Earth's diameter, are the measurements mean-surface to mean-surface, or center to center? That is, does the distance between the objects include portions of the sun and Earth, or does it only refer from the surface of the Earth to the surface of the sun? GBC (talk) 02:24, 5 July 2013 (UTC)

Alternate pronunciation for aphelion
(This topic has already been mentioned on this page, but that was all the way back in 2005) I added the alternate pronunciation of aphelion to the lede. As far as I know, both pronunciations are used(I've only ever heard it as "ap-helion"), regardless of what the "correct" version is Jaxcp3 (talk) 16:12, 6 August 2013 (UTC)

Earth-Moon barycenter is ~74% of the way from Earth's center to its surface
I just tweaked the following so as to be more accurate.... "The center of mass of the Earth-Moon system or Earth-Moon barycenter, as the common focus of both the Moon's and Earth's orbits about each other, is about 74% of the way from Earth's center to its surface." - Benjamin Franklin 65.34.181.75 (talk) 14:24, 21 June 2014 (UTC)

Periapsis and apoapsis
Can anyone explain these terms to me? If an apsis is the point at which a body is nearest or furthest from the body it is orbiting, then what is the periapsis close to? Itself?

I see these terms have spread around the web, but they aren't in my Collins English Dictionary which is usually pretty good on technical terms. Even in the OED, the first occurrence of periapsis is 1959, whereas the first occurrence of perigee is 1595, quite a difference. Neither apoapsis or apapsis are in the OED at all (but pericentre and apocentre are there). It seems a rather misguided generalisation. Do we need them in the article? Chris55 (talk) 11:45, 10 August 2014 (UTC)

 * I think you have a point. To quote Meeus (Astronomical algorithms, 2nd ed., p. 411, note): “… ‘periapse’ would really mean the point closest to the apse; but this is rediculous, because what is meant is the apse itself!”. — Leen 94.212.249.212 (talk) 22:45, 10 December 2020 (UTC)

Silly 9 and 10 digit precisions in "Planetary perihelion and aphelion"
Okay, the numbers are taken from a NASA page, but clearly someone went overboard at NASA and mindlessly wrote down what their pocket calculator displayed. They give the Earth radius as 6,371.00 km and a Saturn Perihelion of 1,349,823,615 km. What is the *actual* precision here? --62.156.155.14 (talk) 10:06, 6 July 2015 (UTC)
 * I don't know when that calculation was done, but today we know them within a couple kilometers,. So assuming that value is recent, it's probably to the correct number of significant figures. Just add a + or - 5 onto the end as the error.

 * Earth radius is a little bit harder, because Earth isn't a sphere. Hard to give a radius of an object that has multiple radii to give.— Gopher65talk 22:01, 6 July 2015 (UTC)

 * The normal practice is for sources presenting calculations to give it to way too many digits after the decimal place, regardless of the uncertainty or significance. This is nothing new either. The problem is that is that it comes from JPL, people take it very seriously and assume they are all valid. The numbers come from a 250 year linear best fit. A 6000 year one (by the same source) gave different figures.

 * The precision for the Earth's radius is very high. The mean radius is the radius of a sphere with the same volume as the body. For the reference ellipsoid of the Earth, very high. Such as the World Geodesy Survey. Not nearly as good for Saturn. Saros136 (talk) 22:37, 7 October 2015 (UTC)
 * The geoid. In Encyclopædia Britannica.
 * Figure of the Earth
 * (talk) 22:47, 7 October 2015 (UTC)
 * Here's a table comparing the perihelion distances between the two fits from JPL:

"Best fit" meaning, JPL tweeked the orientations and sizes of the orbits around until it was a mathematical best fit to our current best observed data. This doesn't mean that we know Neptune's periapse is 4,459,753,056 km to the exact kilometer. It is indeed misleading to present it that way. It's like saying, "my friend lives halfway to the next town, which is about 5 miles away. Therefore my friend lives 2.5000000000 miles away." Probably not exactly, but that's the best fit given the data we have. We do actually know the orbits of the inner planets (Mercury through Mars) about that well due to direct radar and laser ranging measurements — the outer planets, no. Beside all of that, planet orbits aren't fixed things. Due to the action of all of the other planets, any particular orbit is constantly changing. If you give a number that exact, you have to specify exactly what it is and what time period it represents. Is it a mean value? — across what range of time? Is it an instantaneous value? — if so, at what instant? Tfr000 (talk) 17:13, 16 January 2016 (UTC)

explanation of sentence removal, regarding effect of Moon on dates.
''The dates and times of the perihelions and aphelions vary much more than those of the equinoxes and solstices due to the presence of the Moon. Because perihelion and aphelion are defined by the distance between the center of the Sun and the center of the Earth, the Earth's position in its monthly motion around the Earth–Moon barycenter greatly affects the time when the Earth is at its shortest or longest distance from the Sun.''

I removed the sentence. The anomalistic year is greater that the calendar year, so perihelia and aphelia advance through the dates in the calendar, eventually happening on each. The design of the calendar year keeps the equinoxes and solstices from varying much. In any case the the perihelion dates in this century for Earth are never more than two days different from those of the EMB. Saros136 (talk) 11:37, 4 October 2015 (UTC)

All orbits are elliptical
In the article there is a statement "...For elliptic orbits about a larger body...".

All orbits are, in fact, elliptical. I understand the implication behind the statement is probably intended to denote greater or lesser eccentricity about an orbital focus. While merely a pedantic point, a novice may pass over that subtlety without a more factual understanding.

As there are no non-elliptical orbits for the IRL mechanics of this universe, I would edit the statement thus: s//elliptic — Preceding unsigned comment added by 2607:FCC8:BD15:BD00:5CF1:6EE6:EBC2:530E (talk) 17:25, 22 December 2015 (UTC)

 * Only orbits in a two-body system are guaranteed to be elliptical. cherkash (talk) 23:22, 20 January 2018 (UTC)
 * I was about to write that. Thanks for getting it first. Gah4 (talk) 06:10, 29 September 2020 (UTC)
 * I was about to write that. Thanks for getting it first. Gah4 (talk) 06:10, 29 September 2020 (UTC)

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Diagram with (1) farthest (3) focus (2) nearest
I found the first (upper right) diagram deceptive with these labels, since the focus (3) doesn't coincide with the larger body, but it also makes a smaller eclipse around the center of mass. So I added this diagram. The original diagram might be considered accurate if the primary body is considered stationary, but that would be a noninertial reference frame. I'm not sure how to best explain the difference, but my new diagram could show a third image with that, and then the center of mass could be drawn as a red ellipse perhaps? Tom Ruen (talk) 10:29, 21 October 2016 (UTC)
 * Would it be possible to redraw this diagram to label the apoapsis and periapsis as being the maximum and minimum distances from the orbiting body to the center of mass, rather than to the other body, to make it consistent with the text? Geoffrey.landis (talk) 18:56, 29 September 2017 (UTC)

Proposed merger January 2018
On 10 January 2018, has proposed to merge Perihelion and aphelion into Apsis (this article); and he posted: The reason is that the "Apsis" article has much more information about the subject, including detailed info about perihelion/aphelion of Earth and other planets.

 * neutral — I'm not sure if a merger makes the subject simpler, as there are already too many terms to cover under the title "Apsis", but I have no strong opinion. I do, however, think that any proposal for a merger should be accompanied with a post on the corresponding talk page in order to allow others to easily join a conversation.   R fassbind  – talk  23:06, 19 January 2018 (UTC)
 * Strong support – These two articles basically cover the same exact subject. The material in the "Perihelion and aphelion" article is either already stated in the Apsis article or original content that can easily fit in as a section on the "Apsis" article. The "Perihelion and aphelion" article is too small to justify its own existence as a separate page, in my opinion. – PhilipTerryGraham (talk · contribs · [//xtools.wmflabs.org/ec/?username=PhilipTerryGraham&project=en.wikipedia.org count]) 13:01, 9 February 2018 (UTC)
 * Strong support – I am with PhilipTerryGraham on this one. For me Perihelion and aphelion just barely passes beyond a stub and the information present therein is not particularly distinct or detailed. If it is the case that too many terms are or become included in the Apsis article then I would advocate the creation of an Apsis (disambiguation) page as opposed to the continuation of the status quo. Fritz1776 (talk) 18:47, 9 February 2018 (UTC)
 * Support – Subject matter and explanations are essentially the same irrespective of the central body under consideration. We would still need a dedicated section in the apsis article, because heliocentrism applies to many familiar celestial objects. Then the "aphelion" and "perihelion" titles should redirect to that section. — JFG talk 22:01, 19 February 2018 (UTC)

 * Strong oppose – Wikipedia is not a scientific text book. People come here more for general knowlege than they do scinetific in dept details, although I use it myself for that sometimes too, with a proper degree of skepticism. People curious about space flight and trajectories of satellites and the space station come here looking for definitions for apogee and perigee. In fact the title page called Perihelion and aphelion should be "apogee and perigee" because that is mostly likely what the search term was. Then explain the entomology of perihelion and aphelion, instead of the other way around. To make things worse under the page called "Perihelion and aphelion" under "entomology" has links under "apogee" and "perigee" that don't link to definitions of those words but instead linked to the "Apsis" page. Trying to get a proper definition of those two words is like chasing your tail. I don't think the proposed merger helps that at all. Why direct a search on a common term to one most people have never heard of? I think this is another example of editors here getting carried away with letting complex definitions get in the way of the reason people come to Wikipedia and why they come to a particular page or enter a certain word into a search. I don't know for sure, but I'd guess there's some way to see what the most common words are that somebody puts in a search box that takes you to a page. I bet real money that "Apsis" doesn't rank is high as "apogee" or "perigee". I seriously doubt that most people arrive on that page after searching the term "heliocentrism". For example, I was simply writing about ISS visible passes and did up Google search on apogee to make sure that I didn't have the meaning of the two words flipped. It doesn't matter that I first learned the words during the Gemini missions of project Apollo. That was a long time ago and I haven't used the word in a long time. To me that's one of the main things Google is for. So I clicked on the Wikipedia page because I'm a user and an editor and it takes me to a page called "Apis". I was unfamiliar with the word. I start reading and the first thing I read is the entomology of the word I was looking for instead of the word. People need to have the reasonable expectation that they're on the right page when they've come here. Merging these two pages just makes that problem worse. Like I said I'll bet somebody knows how to search how often a word is typed in a search. Jackhammer111 (talk) 06:06, 23 March 2018 (UTC)
 * About the most likely words to access this page, redirects do the job fine, and "Perigee" would appear in the search box in bold. In addition, each redirect could point to the appropriate section where this particular case is discussed, i.e. would redirect to the "Geocentric orbit" section, and  would redirect to the "Solar orbit" section. That would indeed be an improvement for readers. — JFG talk 13:57, 24 March 2018 (UTC)
 * On the origins of the words, surely you mean etymology, not entomology, unless we are discussing orbital insects. — JFG talk 13:57, 24 March 2018 (UTC)
 * Jackhammer, the point is that there's very little to say about the apsides of orbits around the Sun that's different from those of orbits around Earth, the Moon, or another body. Rather than duplicate all the same material about what apsides are, the etymology of the terms, and so forth in a half dozen different articles, it would be more helpful to write one good article about apsides with sections about the apsides or particularly important kinds of orbits. As JFG says, redirects would bring all of those terms to this article. -Bryanrutherford0 (talk) 20:03, 6 November 2018 (UTC)

 * Strong support: There's nothing special about the Sun. All of the content in Perihelion and aphelion belongs in this article, and the combined article wouldn't be too long by any means. -Bryanrutherford0 (talk) 20:03, 6 November 2018 (UTC)
 * Comment, there is something special about the Sun, Earth orbits it. This has profound impact on astronomy both now and historically. Fotaun (talk) 16:52, 21 November 2018 (UTC)
 * Oppose (aka Keep) This is really a distinction worth keeping as separate article. The Sun, and its related terminology have a unique history. Its the sun and apsis is already a Christmas tree article which is loaded with similar terms. Fotaun (talk) 16:52, 21 November 2018 (UTC)
 * Yet the minimal amount of original content in Perihelion and aphelion is not enough to justify an entire separate article from the modestly-sized Apsis. – PhilipTerryGraham (talk · articles · reviews) 23:36, 8 December 2018 (UTC)

Terminology and loops
As per this page here, some of the terms in this article were made up by one of the contributors to this article. If we strip each of those terms out (as we should, until and unless they can be cited to sources older than the point at which they were first added to the article), the table will be left with some blank spaces. What should we do about those? DS (talk) 20:28, 5 February 2019 (UTC)
 * Bah. Not only was the issue raised two years ago, citing the exact same source I just raised, the issue was argued about in 2005. "Should we do this", it went. No. No we shouldn't. I have removed the unsourced neologisms, from here and from Wiktionary both. Please do not restore them unless you have a source older than 2005. DS (talk) 20:47, 5 February 2019 (UTC)

uncertainty
What are the uncertainties for aphelion and perihelion? Some values come from Kepler's laws, but not them, so they aren't so easy to measure. It looks like the article takes a value in A.U., and converts to km and mi with more digits. But how many should it have?

Terminology section
This section kind of repeats itself and duplicates it's information several times just structured differently. It clearly needs restructuring. Question is in which way the information could be most clearly presented? I would prefer to remove the point list, expand the normal text (under the etymology heading) slightly and keep the table of different terms. If no one objects, I will try to get back and do it. In the meantime suggestions are welcome and by all means feel free to edit it (who knows if I ever get around to do it). Amphioxi (talk) 12:53, 30 April 2021 (UTC)

table of aphelion and perihelion dates
due to the presence of leap years, it is difficult to guess the length of the years. Is it possible to add this datum? I suppose that should be easy for someone that manages well the software. thanks. pietro 151.29.59.56 (talk) 23:40, 27 May 2022 (UTC)

title: Talk:Apsis


text:
Pisot–Vijayaraghavan number

In mathematics, a Pisot–Vijayaraghavan number, also called simply a Pisot number or a PV number, is a real algebraic integer greater than 1, all of whose Galois conjugates are less than 1 in absolute value. These numbers were discovered by Axel Thue in 1912 and rediscovered by G. H. Hardy in 1919 within the context of Diophantine approximation. They became widely known after the publication of Charles Pisot's dissertation in 1938. They also occur in the uniqueness problem for Fourier series. Tirukkannapuram Vijayaraghavan and Raphael Salem continued their study in the 1940s. Salem numbers are a closely related set of numbers.

A characteristic property of PV numbers is that their powers approach integers at an exponential rate. Pisot proved a remarkable converse: if α > 1 is a real number such that the sequence

 * $$\|\alpha^n\|$$

measuring the distance from its consecutive powers to the nearest integer is square-summable, or ℓ 2, then α is a Pisot number (and, in particular, algebraic). Building on this characterization of PV numbers, Salem showed that the set S of all PV numbers is closed. Its minimal element is a cubic irrationality known as the plastic ratio. Much is known about the accumulation points of S. The smallest of them is the golden ratio.

Definition and properties
An algebraic integer of degree n is a root α of an irreducible monic polynomial P(x) of degree n with integer coefficients, its minimal polynomial. The other roots of P(x) are called the conjugates of α. If α > 1 but all other roots of P(x) are real or complex numbers of absolute value less than 1, so that they lie strictly inside the unit circle in the complex plane, then α is called a Pisot number, Pisot–Vijayaraghavan number, or simply PV number. For example, the golden ratio, φ ≈ 1.618, is a real quadratic integer that is greater than 1, while the absolute value of its conjugate, −φ−1 ≈ −0.618, is less than 1. Therefore, φ is a Pisot number. Its minimal polynomial is x2 − x − 1.

Elementary properties

 * Every integer greater than 1 is a PV number. Conversely, every rational PV number is an integer greater than 1.
 * If α is an irrational PV number whose minimal polynomial ends in k then α is greater than |k |.
 * If α is a PV number then so are its powers αk, for all positive integer exponents k.
 * Every real algebraic number field K of degree n contains a PV number of degree n. This number is a field generator. The set of all PV numbers of degree n in K is closed under multiplication.
 * Given an upper bound M and degree n, there are only finitely many of PV numbers of degree n that are less than M.
 * Every PV number is a Perron number (a real algebraic number greater than one all of whose conjugates have smaller absolute value).

Diophantine properties
The main interest in PV numbers is due to the fact that their powers have a very "biased" distribution (mod 1). If α is a PV number and λ is any algebraic integer in the field $$\mathbb{Q}(\alpha)$$ then the sequence
 * $$\|\lambda\alpha^n\|,$$

where ||x|| denotes the distance from the real number x to the nearest integer, approaches 0 at an exponential rate. In particular, it is a square-summable sequence and its terms converge to 0.

Two converse statements are known: they characterize PV numbers among all real numbers and among the algebraic numbers (but under a weaker Diophantine assumption).

 * Suppose α is a real number greater than 1 and λ is a non-zero real number such that

 * $$ \sum_{n=1}^\infty \|\lambda\alpha^n\|^2 < \infty. $$

 * Then α is a Pisot number and λ is an algebraic number in the field $$\mathbb{Q}(\alpha)$$ (Pisot's theorem).

 * Suppose α is an algebraic number greater than 1 and λ is a non-zero real number such that

 * $$ \|\lambda\alpha^n\| \to 0, \quad n\to\infty. $$

 * Then α is a Pisot number and λ is an algebraic number in the field $$\mathbb{Q}(\alpha)$$.

A longstanding Pisot–Vijayaraghavan problem asks whether the assumption that α is algebraic can be dropped from the last statement. If the answer is affirmative, Pisot's numbers would be characterized among all real numbers by the simple convergence of ||λαn|| to 0 for some auxiliary real λ. It is known that there are only countably many numbers α with this property. The problem is to decide whether any of them is transcendental.

Topological properties
The set of all Pisot numbers is denoted S. Since Pisot numbers are algebraic, the set S is countable. Raphael Salem proved that this set is closed: it contains all its limit points. His proof uses a constructive version of the main diophantine property of Pisot numbers: given a Pisot number α, a real number λ can be chosen so that 0 < λ ≤ α and

 * $$\sum_{n=1}^\infty \|\lambda\alpha^n\|^2 \leq 9.$$

Thus the ℓ 2 norm of the sequence ||λαn|| can be bounded by a uniform constant independent of α. In the last step of the proof, Pisot's characterization is invoked to conclude that the limit of a sequence of Pisot numbers is itself a Pisot number.

Closedness of S implies that it has a minimal element. Carl Siegel proved that it is the positive root of the equation x3 − x − 1 = 0 (plastic constant) and is isolated in S. He constructed two sequences of Pisot numbers converging to the golden ratio φ from below and asked whether φ is the smallest limit point of S. This was later proved by Dufresnoy and Pisot, who also determined all elements of S that are less than φ; not all of them belong to Siegel's two sequences. Vijayaraghavan proved that S has infinitely many limit points; in fact, the sequence of derived sets

 * $$S, S', S'', \ldots$$

does not terminate. On the other hand, the intersection $$S^{(\omega)}$$ of these sets is empty, meaning that the Cantor–Bendixson rank of S is ω. Even more accurately, the order type of S has been determined.

The set of Salem numbers, denoted by T, is intimately related with S. It has been proved that S is contained in the set T' of the limit points of T. It has been conjectured that the union of S and T is closed.

Quadratic irrationals
If $$\alpha\,$$ is a quadratic irrational there is only one other conjugate, $$\alpha'$$, obtained by changing the sign of the square root in $$\alpha$$ from

 * $$\alpha = a + \sqrt D \text{ to } \alpha' = a - \sqrt D\, $$

or from

 * $$\alpha = \frac{a + \sqrt D}{2}\text{ to }\alpha' = \frac{a -  \sqrt D}{2}.\,$$

Here a and D are integers and in the second case a is odd and D is congruent to 1 modulo 4.

The required conditions are α > 1 and −1 < α' < 1. These are satisfied in the first case exactly when a > 0 and either $$(a-1)^2 < D < a^2$$ or $$a^2 < D < (a+1)^2$$, and are satisfied in the second case exactly when $$a>0$$ and either $$(a-2)^2 < D < a^2$$ or $$a^2 < D < (a+2)^2$$.

Thus, the first few quadratic irrationals that are PV numbers are:

Powers of PV-numbers
Pisot–Vijayaraghavan numbers can be used to generate almost integers: the nth power of a Pisot number approaches integers as n grows. For example,

 * $$(3+\sqrt{10})^6=27379+8658\sqrt{10}=54757.9999817\dots \approx 54758-\frac{1}{54758}.$$

Since $$ 27379 \, $$ and $$ 8658\sqrt{10} \, $$ differ by only $$ 0.0000182\dots,\, $$

 * $$\frac{27379}{8658}=3.162277662\dots$$

is extremely close to

 * $$\sqrt{10}=3.162277660\dots .$$

Indeed

 * $$\left( \frac{27379}{8658}\right)^2=10+\frac{1}{8658^2}.$$

Higher powers give correspondingly better rational approximations.

This property stems from the fact that for each n, the sum of nth powers of an algebraic integer x and its conjugates is exactly an integer; this follows from an application of Newton's identities. When x is a Pisot number, the nth powers of the other conjugates tend to 0 as n tends to infinity. Since the sum is an integer, the distance from xn to the nearest integer tends to 0 at an exponential rate.

Small Pisot numbers
All Pisot numbers that do not exceed the golden ratio φ have been determined by Dufresnoy and Pisot. The table below lists ten smallest Pisot numbers in increasing order.

Since these PV numbers are less than 2, they are all units: their minimal polynomials end in 1 or −1.

The polynomials in this table, with the exception of

 * $$x^6 - 2x^5 + x^4 - x^2 + x - 1,$$

are factors of either

 * $$x^n(x^2 - x - 1) + 1$$

or
 * $$x^n(x^2 - x - 1) + (x^2 - 1).$$

The first polynomial is divisible by x2 − 1 when n is odd and by x − 1 when n is even. It has one other real zero, which is a PV number. Dividing either polynomial by xn gives expressions that approach x2 − x − 1 as n grows very large and have zeros that converge to φ. A complementary pair of polynomials,

 * $$x^n(x^2 - x - 1) - 1$$

and
 * $$ x^n(x^2-x-1) - (x^2-1)\,$$

yields Pisot numbers that approach φ from above.

Two-dimensional turbulence modeling using logarithmic spiral chains with self-similarity defined by a constant scaling factor can be reproduced with some small Pisot numbers.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Pisot–Vijayaraghavan number


text:
Talk:2003 New Democratic Party leadership election

Untitled
It may be helpfull to indicate with what country we are dealing here....:-)

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Name change
This article has been moved without any prior discussion from "New Democratic Party leadership election, 2003" to "New Democratic Party leadership convention, 2003". I disagree with this move, and not just because moving an article should be discussed first.

There was a good reason for using "election" in the title instead of "convention", which is used for the other NDP leadership articles: it was an election -- it was a fundamentally different way of selecting a leader than the previous delegated conventions. As the article says:

 * "The leadership race was the first to be conducted under the NDP's new partial one member, one vote system, in which the popular vote of the members is weighted for 75% of the result. The rest are votes cast by delegates for affiliated organizations (mainly labour unions). It was also the first Canadian leadership convention to allow Internet voting; delegates who chose to vote electronically were given a password to a secure website to register their votes.
 * "The race was heated, with the leaders campaigning to NDP audiences across Canada. One of the most notable events of the campaign occurred at the convention in Toronto, the day before the election, when candidate Pierre Ducasse made a stirring speech. [1] Ducasse's speech attracted widespread praise, [2] [3] although its late delivery was unable to sway the postal and internet votes which had already been cast."

I propose to move the article back to its previous name, and ask for comments from other editors. Ground Zero | t 00:18, 21 November 2009 (UTC)
 * It's called a convention by the party, and that is what it is. As well, other Canadian parties use the name of the party plus convention, plus year.  That seems to be standard.--Abebenjoe (talk) 04:19, 2 January 2010 (UTC)

Move discussion in progress
There is a move discussion in progress on Talk:British Columbia Liberal Party leadership contest, 2011 which affects this page. Please participate on that page and not in this talk page section. Thank you. 117Avenue (talk) 02:37, 7 December 2010 (UTC)


text:
Michael Tenzer

Michael Tenzer (born 1957) is a composer, performer, and music educator and scholar.

Career
Tenzer was born in New York City and studied music at Yale University (BA. 1978) and University of California, Berkeley (Ph.D. 1986). After teaching at Yale from 1986 to 1996, he moved to University of British Columbia where he teaches ethnomusicology, composition, music theory and gamelan performance, co-directs the doctoral program in ethnomusicology.

Tenzer's compositions for chamber, solo and orchestral media have been performed in North America, Europe, and Asia, featuring performers such as Pandit Swapan Chaudhuri (tabla), Alex Klein (oboe) and Evan Ziporyn (clarinet). His publications have been recognized with the Society for Ethnomusicology's Alan P. Merriam Prize (best book of 2000) and the 34th annual ASCAP-Deems Taylor award, and his research has been supported with grants from the National Endowment for the Humanities and Fulbright. Among his composition prizes are a Library of Congress/Koussevitzky commission for a chamber work, Sources of Current. After its premiere the New York Times called it "deft, sophisticated and inventive." He received the Charles Ives Center award for his percussion quartet (1981), the DiLorenzo prize for the octet Daya (1985) for string quartet and clarinets, and the Morse Fellowship to complete his Symphony for Strings (1988). Tenzer's music is available on New World, Cantaloupe and Bali Stereo labels.

Since 1977, Tenzer has been deeply involved with the gamelan music of Bali, Indonesia. He carried out several years of research and writing about it on a series of fellowships, among them a Fulbright (1982), a grant from the Asian Cultural Council (1987) the Morse Fellowship (1989), and a National Endowment for the Humanities University Teacher's Fellowship (1994). An experienced performer and teacher of gamelan, Tenzer is the author of two books on the subject: Balinese Music (Periplus: 1991; 2nd ed. 1998) and Gamelan Gong Kebyar: The Art of 20th Century Balinese Music (University of Chicago Press 2000). More recently he published Analytical Studies in World Music (Oxford 2006). He was the first Western composer invited to compose for Balinese ensembles in Bali and has completed a series of works of an increasingly experimental character for the gamelan since 1982, among them Sinar Jegog (1985), Situ Banda ("Bridge of Monkeys"; 1989), Banyuari ("Tributary"; 1992), Talakalam for gamelan with tabla (1999), "Puser Belah" for 2 simultaneous gamelan ("Unstable Center"; 2003), and "Buk Katah" for gamelan with a nonet of brass, winds and piano ("Underleaf"; 2006). These works have been cited by Balinese critics as "an important and unique contribution to our cultural heritage". The last three compositions cited plus others are featured on the 2009 CD Let Others Name You on New World records. In 1979, Tenzer co-founded the Sekar Jaya gamelan ensemble in Berkeley, California, an organization of Americans dedicated to the performance of Balinese arts that is now internationally known. Since 1996 he has directed Gamelan Gita Asmara in Vancouver.

title: Michael Tenzer


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Action libérale nationale

The Action libérale nationale (ALN)('National Liberal Action') was a short-lived provincial political party in Quebec, Canada. It was founded during the Great Depression and led by Paul Gouin. The ALN played an important role in the foundation of the Union Nationale.

Origin and beliefs
The party was created in 1934 by dissidents from the Liberal Party of Quebec. It soon received the support of French Canadian nationalists as federal Liberal Member of Parliament Édouard Lacroix, Liberal Members of the Legislature Oscar Drouin and Philippe Hamel, and Québec City mayor Joseph-Ernest Grégoire.

The ALN promoted social justice, nationalism and was not affiliated to any federal party. Its platform included the following proposals:

 * Farm credit;
 * Voluntary migration from cities to rural areas;
 * Rural electrification;
 * Nationalization of electricity;
 * Corporatism;
 * Electoral reform;

Its ideology was influenced by the social doctrine of the Catholic Church promoted by the priest Lionel Groulx, and the economical ideas of Esdras Minville.

Electoral breakthrough
In order to unite the vote against the Liberal government of Alexandre Taschereau, the Action libérale nationale (ALN) and the Conservative Party of Quebec decided to run only one candidate of either party in each district for the 1935 Quebec election. With 29% of the vote, the ALN elected 26 out of 57 candidates; the Conservatives received 19% of the vote and won 16 seat out of 33 in which they ran a candidate.

Decline
Less than a year later, Conservative Leader Maurice Duplessis, a rising star in provincial politics, tried to pressure ALN Leader Paul Gouin into merging both parties. While Gouin cuts ties with Duplessis, 22 of his Members of the Legislative Assembly (MLA) joined the new party, known as Union Nationale, which won the 1936 Quebec election.

Gouin did not run for re-election to the legislature in 1936. ALN MNAs Vital Cliche, Wilfrid-Eldège Lauriault and Frederick Arthur Monk ran for re-election as Independents. None of them were elected. Cliche ran as an Action libérale nationale candidate in a by-election held on March 17, 1937, in his home district of Beauce. He was defeated.

The ALN opposed conscription and contested the 1939 Quebec election. Other than Gouin who took the leadership, the party was unable to attract any of its former candidates from the 1935 election. Instead René Chaloult and Oscar Drouin, who had grown disillusioned with Duplessis after they joined the Union Nationale, ran as Liberal candidates. The party won only 4.5% of the vote and none of its 56 candidates were elected. It disappeared soon after.

Legacy
Even though the ALN did not survive the 1930s political realignment in Quebec politics, many of its policies were eventually implemented by the provincial governments of Maurice Duplessis, Adélard Godbout and Jean Lesage.

title: Action libérale nationale


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Lujo Brentano

Lujo Brentano (18 December 1844 – 9 September 1931) was an eminent German economist and social reformer.

Biography
Lujo Brentano, born in Aschaffenburg into a distinguished German Catholic intellectual family (originally of Italian descent), attended school in Augsburg and Aschaffenburg. He studied in Dublin (Trinity College), Münster, Munich, Heidelberg (doctorate in law), Würzburg, Göttingen (doctorate in economics), and Berlin (habilitation in economics, 1871).

He was a professor of economics and state sciences at the universities of Breslau, Strasbourg, Vienna, Leipzig, and most importantly, Munich (1891–1914). With Ernst Engel, the statistician, he made an investigation of the English trade unions.

In 1872, he became involved in an extended dispute with Karl Marx and Friedrich Engels. Brentano accused Marx of falsifying a quotation from an 1863 speech by William Gladstone.

In 1914, he signed the Manifesto of the Ninety-Three. After the revolution of November 1918, he served in minister-president Kurt Eisner's government of the People's State of Bavaria as People's Commissar (Minister) for Trade, but only for some days in December 1918. Brentano died in Munich in 1931, aged 86.

Legacy
Brentano was a Kathedersozialist, a professor advocating social reform, and a founding member of the Verein für Socialpolitik. His influence on the social market economy, and on many Germans who would be leaders just after the end of World War II, can hardly be overrated. He also influenced later economists, such as his doctoral student Arthur Salz.

title: Lujo Brentano


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Sabino Arana

Sabino Policarpo Arana Goiri (in Spanish), Sabin Polikarpo Arana Goiri (in Basque), or Arana ta Goiri'taŕ Sabin (self-styled) (26 January 1865 – 25 November 1903), was a Spanish writer and the founder of the Basque Nationalist Party (PNV). Arana is considered the father of Basque nationalism.

He died in Sukarrieta at the age of 38 after falling ill with Addison's disease during time spent in prison. He had been charged with treason for attempting to send a telegram to President Theodore Roosevelt, in which he praised the United States for helping Cuba gain independence from Spain.

Background
One of the consequences of the First Carlist War was the substitution of the Ancien Régime Basque home rule (fueros) by a limited still relevant autonomy. A majority in Navarre and the rest of the Basque districts supported the pretender to the Spanish crown Carlos V for his support to their institutions and laws (characterized for being more liberal than elsewhere in Spain). However, they were defeated in 1839, and Navarre, Biscay, Álava and Gipuzkoa were integrated into the Spanish customs system. Basque industrialists profited from privatization of exploitations and the Spanish captive market with the iron ore and the Bessemer converter, and Biscay became "the iron California". Workers from all of Spain were attracted to the area as labourers for the burgeoning industry.

Arana was born in a jauntxo ("petty noble") family from Abando, a neighbourhood that had been recently incorporated into the city of Bilbao as the new extension for the growth of the industrial era. Abando was a Basque speaking town, but following the attitudes of the elites in the area of Bilbao during this period, Basque was not transmitted to Arana's siblings within the family. Abando and its port were at the centre of the Zamacolada uprising against attempts by the Spanish premier Manuel Godoy to recruit Basques for the Spanish army (1804), a contrafuero or breach of basic Basque legislation.

In the aftermath of the Third Carlist War (1876), Arana attended the Jesuit School of Orduña along with his brother Luis (1876–1881). Orduña became a hotspot and meeting point for the pro-fueros, primeval Basque nationalists concerned with the loss of the Basque native institutions. Arana claimed that he had a quasi-religious revelation at Easter 1882, one that he communicated to his brother Luis Arana. From then he devoted himself to the nationalist cause of Biscay, later extended to the Basque Country.

Basque language proponent
He was an early proponent of the use of the Basque language in all areas of society to avoid its increasing marginalization in the face of Spanish language penetration, which was imposed as mandatory in schooling, administration and even cultural events such as theater.

Arana learned the language as a young man, and competed for a position as a Basque language professor at the Instituto de Bilbao, contending against Miguel de Unamuno and the winner, Resurrección María de Azkue who also became a scholar of the language. Arana attempted to establish a codified orthography and grammar for the Basque language, and proposed several neologisms to replace loanwords of Spanish origin.

Some of these innovations, like the characters ĺ and ŕ, were ultimately rejected in Basque language standardization efforts beginning in 1968 that led to the establishment of Standard Basque—the Euskara Batua.

Ideology
His first published work was Bizkaya por su independencia ("Biscay for its independence"), where he calls for the independence of the Biscay district from Castile-Spain ("as it was before 1200"), echoing like proposals put forward by Gipuzkoa's governmental representatives to the National Convention officials Pinet and Cavaignac in Getaria during the War of the Pyrenees (1793–1795).

The document is a collection of historical events, mythical stories and sometimes inaccurate accounts of earlier battles of the ancient people of Biscay. Just as others nationalist ideologies were doing during the period, e.g. Spanish nationalism, Arana's historic accounts distorted and magnified events from Basque history.

Distancing himself from the pro-fueros advocates, Arana refused to demand a reversal of the fueros suppressed in May 1876, instead putting an emphasis on the full restoration of home rule suppressed in 1839. He considered the 1839 Spanish law 'upholding' home rule as the act putting an end to the Basque own sources of authority and 'secular Basque independence', as well as a violation of international law.

In 1894, he founded the first center for the new nationalist party (Partido Nacionalista Vasco - Eusko Alderdi Jeltzalea), the second-oldest political party in Spain, to provide a place for gathering and proselytizing. Sabino Arana, like many Europeans of his time, believed that the essence of a country was defined by its blood or ethnic composition. In Spain, the supremacy of the Spanish race and its "civilizing" pursuit over peoples held to be inferior was defended by the main political figures and parties, while a number of intellectuals Spanish and even Basque, including the Socialists, advocated for the extinction of the Basque language — ever more marginalized to family and informal environments.

He is considered by many Basques to be a gadfly that sparked the movement for the cultural revival of the Basques, and for the freedom of his people. In that respect, Arana defended the Constitutional foundations of the abolished Basque institutional and legal framework (the fueros). The PNV, the party in power in the Basque Autonomous Community from the end of Francoism (except during 2009–2012), developed along more nuanced and pragmatic lines in respect of religion and views on race, moving away from his most controversial ideas but not from his political persona.

Sabino Arana created am ideology centered on the purity of the Basque race and its alleged moral supremacy over others (a derivation of the system of limpieza de sangre of Modern-Age Spain), anti-Liberal Catholic integrism, and deep opposition to the migration of Spaniards to the Basque Country. By contrast, unlike many contemporary conservative politicians in Spain, he was against slavery in Spanish colonies such as Cuba.

He was disturbed by the migration into Biscay of many workers from western and central Spain during the industrial revolution, into a small territory whose native political institutions had recently been suppressed (1876), believing that their influence would result in the disappearance of the 'pure' Basque race. He presented the Basque as opposed to the maketo (people from the rest of Spain): ""It is necessary to isolate ourselves from the maketos. Otherwise, in this land we walk on, it is not possible to work toward the Glory of God." Bizkaitarra, No. 19"

He was a prolific writer, with over 600 journalism articles, most of them with a propaganda purpose. He liked to shock and provoke, in order to get attention from a society that he deemed unaware of its fate. There are two key aspects of Sabino Arana's political persona:

 * He was not a conventional conservative; he strongly opposed slavery (legal in Spanish-held Cuba until ten years before its independence) and defended the right of South African Zulus to their land.
 * He was an indefatigable worker, taking action in many areas; he learned the Basque language as an adult, undertook a number of activities to promote the Basque language and culture, created a political movement, and inventing the symbols (flag, anthem, country name) used to this day not only by Basque nationalists, but other political parties and representatives also, especially in the Basque Autonomous Community.

Another essential part of his ideology was devout Catholicism. He considered this to be an essential part of the Basque identity that contrasted with the secularism imported from other parts of Spain and abroad along with new means of production and labour, often unprecedented internal migration. However, his Basque nationalism kept him away from Carlism that was the dominant ultra-Catholic and conservative movement in the area and the ideology of his father.

Arana and his Basque nationalist movement were persecuted for their ideas against Spanish imperialism, for which he was convicted when he submitted a telegram to Theodore Roosevelt to congratulate him for his assistance in 'liberating' Cuba in 1902, and put to harsh time in prison, which ruined his health and would die soon after. Ahead of his demise, a baffling manifesto attributed to him was released by which he relinquished the core of his ideas to everyone's surprise. The nature of this document establishing the Liga de Vascos Españolistas ('League of the Spanish-minded Basques') is still subject to debate as to whether he had sincerely changed his views or whether he was instead merely trying to improve the conditions of his imprisonment. Arana's death left the question unanswered and neither his brother Luis nor the party followed through with his proposal.

Legacy
Sabino Arana's ideas are considered to have spawned the Basque nationalist movement.

In 2015, a made-for-television movie called Sabin was released. In February 2017 the People's Party of the Basque Country called for a street named after Arana to be renamed because he “made possible the existence of phenomena such as the ETA terrorist group”. The Partido Nacionalista Vasco has kept only the more moderate part of his message. On the other hand, some Basques still revere him as the father of the Basque nationalist movement, who managed to start the turnaround of the decay of the Basque language and culture. Many Basque cities have streets named after him.

The estate of his Abando home is now Sabin-Etxea ("Sabino-House"), the EAJ-PNV headquarters.

Jon Juaristi has remarked that perhaps the most influential part of his heritage is the neologistic list of Basque versions of names in his Deun-Ixendegi Euzkotarra ("Basque saint-name collection", published in 1910). Instead of the traditional adaptations of Romance names, he proposed others he made up and that in his opinion were truer to the originals and adapted to the Basque phonology.

For example, his brother Luis became Koldobika, from Frankish Hlodwig. The traditional Peru, Pello or Piarres ("Peter") became Kepa from Aramaic כיפא (Kepha). He believed that the suffix -[n]e was inherently feminine, and new names like Nekane ("pain"+ne, "Dolores") or Garbine ("clean"+ne, "Immaculate [Conception]") are frequent among Basque females. Even the name of the brother-in-law of the king of Spain is Iñaki Urdangarin, Iñaki being Arana's alternative for Ignatius instead of the Basque traditional Inazio.

title: Sabino Arana


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Mourne Mountains

The Mourne Mountains (Beanna Boirche), also called the Mournes or the Mountains of Mourne, are a granite mountain range in County Down in the south-east of Northern Ireland. They include the highest mountains in all of Ulster, the highest of which is Slieve Donard at 850 m. The Mournes are designated an Area of Outstanding Natural Beauty and it has been proposed to make the area Northern Ireland's first national park. The area is partly owned by the National Trust and sees many visitors every year. The Mourne Wall crosses fifteen of the summits and was built to enclose the catchment basin of the Silent Valley and Ben Crom reservoirs. The wall, and the area inside it, are owned by Northern Ireland Water.

Name
The name 'Mourne' is derived from the name of a Gaelic clan or sept called the Múghdhorna. The older name of this mountainous territory was Bairrche, which is likely a collective noun derived from the Irish barr, meaning 'top, peak'. This survives in the Irish name for the mountains, na Beanna Boirche, literally "the peaks of the peak district". It was historically anglicized as 'Bennyborfy'. The name Bairche or Boirche was also personified as the mythical shepherd of the mountains.

Many of the mountains have names beginning "Slieve", from the Irish sliabh, meaning "mountain".

Mountains
On clear days, the Mourne Mountains can be seen from the Isle of Man and Dublin.

Other features
The below sub-headings detail other features and visitor attractions found in the Mourne Mountains.

The Mourne Wall
The Mourne Wall is a dry stone wall measuring 31.4 km in length that crosses fifteen summits and was constructed to define and protect the 36 km2 catchment area purchased by Belfast Water Commissioners in the late 19th century. This followed a number of Acts of Parliament allowing the sale, and the establishment of a water supply from the Mournes to the growing industrial city of Belfast. Construction of the Mourne Wall was started in 1904 and was completed in 1922.

The Mourne Wall has been a listed building since 1996, and 600 repairs were completed in 2018 by Geda Construction.

Forests
Tollymore Forest Park is at Bryansford, near the town of Newcastle in the Mourne and Slieve Croob Area of Outstanding Natural Beauty. It covers an area of 630 ha at the foot of the Mourne Mountains and has views of the surrounding mountains and the sea at nearby Newcastle. The Shimna River flows through the park where it is crossed by 16 bridges, the earliest dating to 1726. The river is a spawning ground for salmon and trout and is an Area of Special Scientific Interest for its geology, flora and fauna. The forest has four walking trails signposted by different coloured arrows, the longest being the 8 mi "long haul trail". The Forest Park has been managed by the Forest Service since they purchased it from the Roden Estate in 1941.

Donard Forest is near Newcastle, County Down. It borders Donard Park at the foot of the Mourne Mountains. The Glen River flows through the forest, crossed by three stone bridges.

Rostrevor Forest is near the village of Rostrevor, County Down, between the Mourne Mountains and Carlingford Lough, in the Mourne Area of Outstanding Natural Beauty. The first trees, primarily sitka spruce, douglas fir and pine, were planted in 1931.

Reservoirs
All water reservoirs are owned and maintained by Northern Ireland Water (NIW).

Silent Valley Reservoir is a reservoir in the Mourne Mountains near Kilkeel, County Down. It supplies most of the water for County Down, surrounding counties and most of Belfast via the Aquarius pipeline. The reservoir was built between 1923 and 1933 by a workforce of over one thousand men, nine of whom died during construction. The catchment area is 9,000-acres (3,600 ha / 36 km2).

Ben Crom Reservoir is upstream of Silent Valley in the Kilkeel River valley, and supplies the same areas. It was constructed between 1953 and 1957.

Spelga Reservoir is in the townland of Spelga (Irish: Speilgeach), close to Hilltown, in the North West of the Mourne Mountains. It was formed by construction of Spelga Dam and sits at over 1,200 ft (370 m) above sea level. The dam was constructed between 1953 and 1957, and has a volume of 2,700,000 cubic metres and a catchment area of 1,340-acres (542 ha / 5.423 km2).

Fofanny Dam Reservoir is approximately 2 km north-east of Spelga Dam and is much smaller.

Flora and fauna
Aside from grasses, the most common plants found in the Mournes are heathers and gorse. Of the former, three species are found: cross-leaved heath, bell heather, and common heather. Of the latter, two species are found: common gorse and western gorse. Other plants which grow in the area are: common cottongrass, roseroot, harebell, marsh St John's-wort, wild thyme, wood sorrel and heath spotted orchid.

Sheep graze high into the mountains, and the range is also home to birds, including the raven, peregrine falcon, wren, buzzard, meadow pipit, grey wagtail, stonechat and snipe. The golden eagle, a former inhabitant, has not been seen in the Mournes since 1836.

Recreation
"Discover Northern Ireland", a website operated by Tourism NI, promotes the Mourne Mountains as a popular destination for hiking and taking in views of the surrounding landscape, including local forests and the coastline. The Mournes offer a range of activities for visitors, including hiking, forest and beach walks, cycling and rock climbing, with nearly three-quarters of visitors choosing the Mournes as a place to go walking and hiking.

The Mournes are a popular destination for Duke of Edinburgh's Award expeditions. However, there are also a number of walking challenges which take place in the Mournes. The Mourne Wall challenge, which is also referred to as the 7-peak challenge because it takes into account 7 of the 10 highest Mourne mountains, is advertised by WalkNI. The Mourne six peak challenge is advertised by DiscoverNI and takes hikers up Slieve Donard, Commedagh, Bearnagh, Slieve Binnian, Slieve Meelmore and Slieve Meelbeg across three days of hiking.

Information and statistics on tourism to the Mournes were gathered by TourismNI in 2014. In a survey of leisure visitors, 79% were found to come from Northern Ireland, 15% from elsewhere in the British Isles or Republic of Ireland, and 5–6% were international visitors. Two-thirds of all visitors made a single-day trip rather staying overnight, and party sizes averaged between 3 and 4 people.

There are many granite cliffs, in the form of outcrops and tors, scattered throughout the range, making the Mournes one of Northern Ireland's major rock-climbing areas since the first recorded ascents in the 1930s. The rockforms are generally quite rounded, thus often requiring cams for protection, but with good friction. The 1998 guidebook lists 26 separate crags, with a total of about 900 routes of all grades.

Conservation
Following a fundraising drive in 1993, the National Trust purchased nearly 1300 acre of land in the Mournes, which included a part of Slieve Donard (at 850 m) and nearby Slieve Commedagh (at 767 m), the second-highest mountain in the area.

It has been proposed that the Mourne Mountains be made Northern Ireland's first national park. The plan has been subject to controversy because of the area's status as private property, with over 1,000 farmers based in the proposed park, and also because of fears over the impact on local communities, bureaucracy and house prices.

Gorse burning
Historically, gorse had many uses in the rural economy and hill farmers often cleared gorse by hand. There is also a tradition in the Mournes of controlled gorse burning to improve grazing for sheep. Today, however, many of the fires are unmanaged and some become out-of-control wildfires. In the 1950s, Emyr Estyn Evans had written that some shepherds in the Mournes tended to burn gorse and heather recklessly. He said that such over-burning "results in widespread destruction" and, along with other mismanagement, had "greatly impoverished the mountain environment". In the 21st century there have been hundreds of heather and gorse wildfires in the Mournes each year, the vast majority started deliberately, with "farmers and vandals" often blamed. It is claimed many of the wildfires are caused by hill farmers and landowners carrying out unapproved burning to clear gorse/heather and thus maximize the subsidy payments they receive for the amount of grazing-land they have. Some are also caused by careless visitors. In April 2021, more than a hundred firefighters tackled a major gorse wildfire in the eastern Mournes, which blazed for three days and devastated habitat in the area. The over-clearing of gorse, heather and trees also heightens the risk of landslides.

Wind farm proposal
In 2015, German-owned company ABO Wind applied to build a wind farm at Gruggandoo in the western Mourne Mountains. Its first two applications were turned down, and its revised application is to build eight turbines standing 142 m high, along with a network of access tracks, substations and a control building. The turbines would be among the tallest structures in Ireland. The company claims they could power 37% of homes in the district. There is opposition, as the wind farm would be in an Area of Outstanding Natural Beauty and would impact wildlife and habitats. Local protest group, Mourne AONB Against Windfarms, warn it would open the door for further wind farms to be built in the Mournes and other protected areas. Planning officers for Newry, Mourne and Down District Council deem the wind farm "unacceptable" and recommended the council reject it. In 2020, councillors instead voted to ask for a public inquiry.

Popular culture
The mountains are immortalised in a song written by Percy French in 1896, "The Mountains of Mourne". The song has been recorded by many artists, including Don McLean, and was quoted in Irish group Thin Lizzy's 1979 song "Roisin Dubh (Black Rose): A Rock Legend". Frank Baker's famous Ulster-based comic novel Miss Hargreaves refers to it: "I'm not going to tell you much about the holiday except to say it was a grand month and we enjoyed every bit of it even though it rained much of the time. We went miles in the car, swam in the river, messed about in an old tub of a boat belonging to a farmer; and we spent a good many evenings in the hotel at Dungannon, drinking Irish whiskey and flirting with a cheeky girl Henry rather fell for. We climbed the Mourne Mountains and sang the right song on the top, though we couldn't remember the words".

"The Mountains of Mourne" are also mentioned in John Lennon's song "The Luck of the Irish" on the album Some Time in New York City.

The scenery of the Mourne Mountains have also provided the backdrop for a number of productions, most famously HBO's Game of Thrones.

Many local painters have depicted the mountains, including the same Percy French, who immortalized them in song.

The Mourne Mountains also influenced C. S. Lewis to create the mythical world featured in his The Chronicles of Narnia series.

Helicopter crash
On 23 October 2010 an AgustaWestland AW109 (tail number: N2NR) was operating a VFR flight from Enniskillen Airport to Caernarfon Airport, Wales. While en route the helicopter crashed into the western side of Shanlieve, killing all three passengers and crew on board. The cause of the accident was determined to be pilot error in heavy fog.

title: Mourne Mountains