# Talk:Trojan (astronomy)

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## Comment

What is the mass limit for a trojan, as a fraction of its parent's mass? After a point, the orbit should become unstable. kwami (talk) 16:45, 10 June 2008 (UTC)

About 1/24.95. It is fairly accurate to say that the sum of the two smaller bodies must not exceed that. 94.30.84.71 (talk) 13:09, 17 June 2012 (UTC)
That's the limit for the second "large" body. Are you saying that two moons of equal mass, each 1/50 of their primary, could be a stable trojan pair? —Tamfang (talk) 17:44, 17 June 2012 (UTC)
You are all thinking too much in terms of absolute, deterministic problems. This is a probabilistic problem. As for s Trojan asteroid, the answer for the mass is "The smaller the better." If you wish for a number, the answer of "smaller than 1/100 of the mass of the planet" is a good one. Do not think of all of this in deterministic terms. As for the asteroid, the situation is that the larger its mass is in relation to its planet, the higher the probability that it was escape from the L4 or L5 position permanently, and the sooner.
Given one primary object and two moons of equal mass, each having 1/50 of the mass of the primary, this is an unstable situation, and it would not last - no matter what their initial separation (in degrees) in their common orbit. Within a short period of time (in terms of astronomy), one of three things will happen: 1) The two satellites will collide and merge into one. 2) One of the satellites will collide with the primary and "stick" (merge into it). 3) One of the satellites will be expelled from the system permanently. This is the classic case of the Three body problem. The Lagrange case with a small body at the L4 or L5 point is a very special case of the Three Body Problem, and that is the whole reason for even thinking about it.
98.81.14.72 (talk) 15:47, 6 September 2012 (UTC)
.72, you were not invited to alter the signed writings of the three other contributors to this "Comments" section. —Tamfang (talk) 19:12, 6 September 2012 (UTC)

SINCE THREE SIGNED CONTRIBUTIONS ABOVE HAVE (Tamfang says) BEEN CORRUPTED BY 98.81.14.72, THE WHOLE OF THIS SECTION ABOVE SHOULD BE IGNORED. 98.81.14.72 WRITES NONSENSE. I WILL NEXT RE-INTRODUCE THE QUESTION, OR AT LEAST ITS ANSWER. 94.30.84.71 (talk) 10:25, 17 October 2012 (UTC)

.72's changes were trivial (24.95→25, trojan→Trojan) and I already undid them. —Tamfang (talk) 18:54, 17 October 2012 (UTC)

In a three-body system with circular orbits, it is well-established that the equilateral configuration is stable if 27(m1m2 + m2m3 + m3m1) < (m1 + m2 + m3)2 (http://www.lesia.obspm.fr/perso/bruno-sicardy/biblio/biblio/Sicardy_Gascheau_CelMec10.pdf refers).

Thank you, I came to this page to ask whether there was such a formula, and to suggest that if there is, it go on the main page. Of course, there should also be words saying that this stability applies only if the Newtonian Universe is otherwise empty, but that the larger the ratio by which it is satisfied, the greater the robustness to external perturbation. The special cases are of interest: m3 being a mote of dust imposes a lower bound on m1÷m2 of ½(1–√(23/27)) ≈ 25.9599. (Do give the surds: some of the readers of this page are geeks, and geeks like surds.) The other limit should also be mentioned: if m1÷m2 ≈ +∞, then it is stable for any m2÷m3 ratios. JDAWiseman (talk) 13:28, 17 November 2013 (UTC)

If m3 is negligible, the ratio of m1 and m2 must exceed 24.9599, say 24.96.

I was not saying that m1=1, m2=m3=0.02 would be stable, because I wrote "It is fairly accurate to say that the sum of the two smaller bodies must not exceed that" and not "It is exact ...". In fact, to be stable, the two equal bodies must be slightly smaller than 0.5/24.96; they must each have mass less than 0.019819 times the mass of m1.

In writing about Trojans, there are two cases, both of which need to be considered and distinguished.

(1) There is the theoretical case of three bodies in an otherwise empty Euclidean Newtonian non-quantum universe. Any three masses, if perfectly initialised, will follow perpetually an exact equilateral configuration in general conic-section paths - linear, elliptical, circular, parabolic, hyperbolic. If imperfectly initialised to circular orbits, the bodies will remain permanently in near-equilateral configuration if the above expression is satisfied. For non-circular orbits, there must be a related expression involving the eccentricity (I've never seen one; it may remain to be discovered).

(2) There is the practical case of three bodies in, for example, our Solar System. There will be perturbations arising from other bodies (e.g. the Sun perturbs bodies at the Earth-Moon Lagrange points, Jupiter perturbs bodies at the Sun-Earth points). These perturbations will, almost invariably, diminish stability; and can easily eliminate it.

94.30.84.71 (talk) 10:25, 17 October 2012 (UTC)

Amendment : re-reading cited Sicardy - the above expression may apply independently of eccentricity. 94.30.84.71 (talk) 11:10, 17 October 2012 (UTC)

... or eccentricity may initially increase stability. 94.30.84.71 (talk) 11:39, 17 October 2012 (UTC)

The same stability condition expression holds for non-circular orbits, but with a constant depending on the eccentricity (Danby, May 1964). 94.30.84.71 (talk) 22:04, 28 October 2012 (UTC)

The condition “stable if 27(m1m2 + m2m3 + m3m1) < (m1 + m2 + m3)2” behaves in a way that seems wrong. Of course, it could be the seeming. The condition is satisfied if m1/(m2+m3) > ½(25+18√2) ≈ 25.2279. Our sun is 333000× the mass of the Earth, massively massively more than 25.2279. So if there had been a planetary-mass object in an L4 or L5 point of the early Earth, that would have been very stable. This seems surprising. Am I wrong to be surprised, or is there a deeper misunderstanding? Perhaps a “deeper misunderstanding” that could be explained in the article? JDAWiseman (talk) 13:30, 25 September 2015 (UTC)

According to that, m2=m3 should be stable for Earth, which is wrong, because Theia would then have stayed where it was. --JorisvS (talk) 16:35, 25 September 2015 (UTC)
Exactly the source of my reasoning. What have I (we?) misunderstood? JDAWiseman (talk) 18:16, 25 September 2015 (UTC)
It's just plugging in numbers. Maybe it's only in the absence of any perturbations? If so, it would mean that this formula is practically useless. At any rate, the entire section is completely unsourced. --JorisvS (talk) 11:54, 26 September 2015 (UTC)
I will investigate the link earlier in this comment section, cite it if appropriate, or amend/delete. JDAWiseman (talk) 13:06, 26 September 2015 (UTC)

## Capital Trojan or small trojan?

The usage needs regularising. I'm inclined to think there's no need for caps as it is a generic term. So, trojan moons, Jupiter trojans...? As opposed to the Trojans of Troy, of course. Rothorpe (talk) 00:00, 29 July 2010 (UTC)

[Neptune trojan] Neptune trojan is stupid, no matter what some stupid sources might say. It must be [Neptunian trojan]. You can just count on so-called "science writers" getting it wrong. Just count on it. Write [Martian trojan], [Trojan of Jupiter], [Jovian trojan], [Saturnian trojan], [Neptunian trojan], [Trojan of Neptune]. I also prefer to see "Trojan" capitalized, just like "English", "French", "German", etc.
Would you believe that some writers in the Internet are so dim that they write stuff like these? {british, dutch, finnish, hungarian, mexican, polish, and russian}. They also do not know that "Internet" is capitalized because it is a proper noun. Watch out for those writers, especially Eastern Europeans, who do not understand that proper adjectives and proper nouns in English are always capitalized. A salient thing is that you can count on people whose native language is Chinese, Japanese, Spanish, Swedish, or Portuguese getting it RIGHT. Why can't the others pay us the same courtesy?
98.81.14.72 (talk) 16:12, 6 September 2012 (UTC)

The word "trojan" refers to a class of objects, just like "planet", "asteroid", and "moon"/"natural satellite", and hence should not be capitalized. --JorisvS (talk) 16:22, 6 September 2012 (UTC)

## Trojan asteroid

Shouldn't Trojan asteroid be a separate article from this? Trojan moon and Trojan planet are separate articles. 184.144.167.193 (talk) 07:12, 11 December 2010 (UTC)

There's no substantial content to split - splitting a stub just makes 2 stubs! In fact, it would be better to merge the other two articles here - the concept is the same, and the particulars are quite minor. - BilCat (talk) 09:32, 11 December 2010 (UTC)

## Image

Perhaps File:Minor Planets - Martian L5.svg is better than File:InnerSolarSystem-en.png for illustrating general trojan asteroid zones. 184.144.167.193 (talk) 07:46, 11 December 2010 (UTC)

Unfortunately, it is not labeled. Ruslik_Zero 17:51, 11 December 2010 (UTC)

## 2010 TK7

Should we create a 2010 TK7 article ? --Stone (talk) 18:08, 27 July 2011 (UTC)

## Historical Error?

Article has ; "In 1772 the French mathematician and astronomer Joseph-Louis Lagrange predicted the existence and location of two groups of small bodies located near a pair of gravitationally stable points along Jupiter’s orbit."

I don't think he did. 1772 is the date of his "three-body problem" paper, Chapters 1 & 2 (of 4) are the only relevant ones, and they contain no reference to Jupiter.

Lagrange did establish properties of solutions of the three-body problem, including the constant-pattern property; and L4 and L5 are then a trivial deduction; but Lagrange should not be asserted to have made that deduction public without trustworthy evidence. Moreover, while Lagrange probably would have considered L1 L2 L3 likely to be unstable in principle, to predict actual groups of bodies at Jupiter's L4 and L5 points requires not only a belief that L4 and L5 are stable in the Sun-Jupiter system alone, but that they remain stable in the presence of perturbations from other planets such as Saturn.

Before writing in an encyclopaedia about a person such as Lagrange, one should read the relevant portion of his own works.

Lagrange's entire "Oeuvres" are on the web, at Gallica.
94.30.84.71 (talk) 19:44, 5 August 2011 (UTC)

Yes, I agree. Lagrange's calculation of the special case of the Three Body Problem, with the LARGE body, the smaller body, and the tiny body at the L4 or L5 position didn't have anything to do with Jupiter. His was the general case. Then, it just so happens that the Sun-Jupiter-Trojan system fits the general case, as do the cases with Sun-Saturn, Sun-Neptune, and Earth-Moon-(L4 or L5). Collections of dust particles have been observed at the latter L4 and L5 positions.
98.81.14.72 (talk) 15:55, 6 September 2012 (UTC)
Lagrange did NOT consider, in regard to mass, ANY special cases in his 'Essai' (apparently his only relevant work). Before writing about Lagrange's work, please read Lagrange's work. The original is at http://www.ltas-vis.ulg.ac.be/cmsms/uploads/File/Lagrange_essai_3corps.pdf and a translation is at http://www.merlyn.demon.co.uk/essai-3c.htm. 94.30.84.71 (talk) 09:25, 17 October 2012 (UTC)
...but such observations are difficult and controversial, I gather. —Tamfang (talk) 19:18, 6 September 2012 (UTC)

## Inappropriate limitation?

Article has : "Trojan asteroids are small Solar System bodies that reside in Trojan points.". I see no need to restrict the term to the Solar System. It is certainly not so restricted in fiction (Pournelle, the Mote system). 94.30.84.71 (talk) 19:52, 5 August 2011 (UTC)

Makes sense to me. The sentence loses nothing by deletion of the words Solar System. —Tamfang (talk) 05:33, 29 August 2011 (UTC)
"Small Solar System Body" is a term introduced by the IAU in 2006, alongside "dwarf planet" and the definition of planet. It is an unfortunate term, as it simply cannot be used outside the Solar System for obvious reasons (contrary to the term "dwarf planet", which extends naturally). I agree that the sentence here loses nothing by deleting Solar System; though, on a theoretical level, wouldn't it be dynamically possible to have an asteroid–dwarf planet in a Trojan point? --JorisvS (talk) 21:39, 29 August 2011 (UTC)
Sure, though a sufficiently massive body would be unstable over astronomical timescales. — kwami (talk) 06:04, 30 August 2011 (UTC)
How massive would it have to be to become unstable over, say, 5–6 Ga? --JorisvS (talk) 08:20, 30 August 2011 (UTC)

## Trojans vs. non-trojans

What exactly makes a trojan (asteroid) a trojan, as opposed to the non-trojan co-orbital bodies? --JorisvS (talk) 19:22, 15 November 2011 (UTC)

Being essentially "tied" to L4 or L5; having the same average period about the primary as the secondary, and never crossing the line L3 - primary - L1 - secondary - L2. IMHO. 94.30.84.71 (talk) 13:07, 17 June 2012 (UTC)

## Dates of Discovery, etc.

The years when the first few Trojans were discovered should be given, and an indication of the subsequent rate of discoveries. 94.30.84.71 (talk) 13:12, 17 June 2012 (UTC)

98.81.14.72 (talk) 15:59, 6 September 2012 (UTC)
Which associated article? It would have been better to copy the link. 94.30.84.71 (talk) 09:27, 17 October 2012 (UTC)
Here is a once-checked draft sortable table covering, I think, the first 50 years, with data from the Wikipedia pages of the asteroids. Perhaps someone would re-check it, and then it can be put into the Article. Sorting is stable.
=== Sun-Jupiter Trojans === 94.30.84.71 (talk) 15:13, 19 October 2012 (UTC)
Asteroid
Number
Name Nation Camp Discovery Discoverer
588 Achilles Greek L4 1906-02-22 Max Wolf
624 Hektor Trojan! L4 1907-02-10 August Kopff
659 Nestor Greek L4 1908-03-23 Max Wolf
911 Agamemnon Greek L4 1919-03-19 K W Reinmuth
1143 Odysseus Greek L4 1930-01-28 K W Reinmuth
1404 Ajax Greek L4 1936-08-18 K W Reinmuth
1437 Diomedes Greek L4 1937-08-03 K W Reinmuth
617 Patroclus + Greek! L5 1906-10-17 August Kopff
884 Priamus Trojan L5 1917-09-22 Max Wolf
1172 Äneas Trojan L5 1930-10-17 K W Reinmuth
1173 Anchises Trojan L5 1930-10-17 K W Reinmuth
1208 Troilus Trojan L5 1931-12-31 K W Reinmuth
+ Now known to be double, (617) Patroclus | Menoetius
===Trojan Discovery Sequence=== 94.30.84.71 (talk) 09:32, 24 October 2012 (UTC)
Sequence Name Nation Camp Discovery
1 588 Achilles Greek L4 1906-02-22
10 1208 Troilus Trojan L5 1931-12-31
100  ?  ?  ?  ?
1000  ?  ?  ?  ?
10000  ?  ?  ?  ?
100000  ?  ?  ?  ?
Partially filled in - comment?
=== First Trojan Found in Region === 94.30.84.71 (talk) 15:20, 23 October 2012 (UTC)
Primary Secondary Side Name Date Discoverer
Sun Earth L4 2010 TK7 2010-10 NEOWISE
" " L5 2010 SO16 2010 WISE
" Mars L4 1999 UJ7 1999-10-30 LINEAR
" " L5 5261 Eureka 1990-06-20 Palomar
" Jupiter L4 588 Achilles 1906-02-22 Max Wolf
" " L5 617 Patroclus + 1906-10-17 August Kopff
" Neptune L4 2001 QR322 2001-08-21 Deep Ecliptic Survey
" " L5 2008 LC18 2008-06-07 Subaru
Saturn Tethys L4 Telesto 1980-04-08
" " L5 Calypso 1980-03-13
" Dione L4 Helene 1980-03-01 Pic du Midi
" " L5 Polydeuces 2004-10-24 Cassini

94.30.84.71 (talk) 15:07, 19 October 2012 (UTC)

## Introduction

The present introduction seems amateurish, and is in part wrong. It is :

In astronomy, a trojan is a minor planet or natural satellite (moon) that shares an orbit with a planet or larger moon, but does not collide with it because it orbits around one of the two Lagrangian points of stability (trojan points), L4 and L5, which lie approximately 60° ahead of and behind the smaller body, respectively. It is also sometimes called a Lagrangian object. Trojan objects are one type of co-orbital object. In this arrangement, the massive star and the smaller planet orbit about their common barycenter—a location in space where the forces of their mutual gravitational attraction balance each other out. A much smaller mass located at one of the Lagrangian points is subject to a combined gravitational force that acts through this barycenter. Hence the object can orbit around the barycenter with the same orbital period as the planet, and the arrangement can remain stable over time.

Comment on that above : An equilateral Lagrange Point does not exactly share the orbit of the secondary, unless the secondary is massless. The "does not collide" part is superfluous at best. There is an exact angle of 60°, but not quite "ahead" or "behind or 'in the path of'. The first sentence allows for a planet/moon system to have a trojan, but the third sentence uses "star" and "planet" - it would be better to use 'primary' and 'secondary' throughout. "... orbit about their common barycenter" - not quite, when a smaller mass is introduced. "... barycenter—a location in space where the forces of their mutual gravitational attraction balance each other out" - that is just plain false, and gravitational balance actually occurs moderately near, but not at, L1. "Hence" is inappropriate, because it is not sufficient to have a force in the right direction; it must also be of the right magnitude.

It might be better to start with a loose description, then to state the equilateral constant-pattern solution to the general three-body problem (as found by Lagrange), then to make the mass of one body negligible. Rough initial draft :-

In astronomy, a trojan is a lesser body which orbits in, approximately, a constant equilateral pattern with a more massive secondary body and a much more massive primary body. It will remain near to the path of the secondary and about 60° away.

Consideration of the "General Three-Body Problem" can easily show that any three gravitating bodies, given exact suitable initial velocities, will remain in an equilateral configuration, with conic-section paths about their mutual barycentre. If one body is of negligible mass, its possible positions are termed Lagrange Points L4 and L5, honouring Lagrange's publication in 1772 of the two types of constant-pattern configuration. If the primary body is sufficiently heavier (for circular orbits, more than about 25 times) than the secondary body, those configurations are stable against perturbations, and small bodies can remain near those points indefinitely. Such bodies form one class of co-orbital objects.

94.30.84.71 (talk) 17:38, 18 October 2012 (UTC), polished 94.30.84.71 (talk) 10:18, 25 October 2012 (UTC), 94.30.84.71 (talk) 22:25, 25 October 2012 (UTC) 94.30.84.71 (talk) 10:23, 27 October 2012 (UTC)

I have not checked in detail lately; but outside Wikipedia it seems more common to use "Lagrange Point" rather than "Lagrangian Point", and "L4" rather than "L4". 94.30.84.71 (talk) 17:38, 18 October 2012 (UTC)

Thank you for writing this; I was independently reaching the conclusion that L4 and L5 cannot possibly be on the exact same orbit as the secondary and was slowly going barking mad to see them nevertheless drawn onto the same circle. This really should be fixed and pointed out on the main page!

How about update Introduction with more recent discoveries about Uranus trojans, Neptune trojans, and some temporary Venus trojans?--Bobbylon (talk) 12:15, 2 May 2017 (UTC)

## First Figure

In the first figure, which (except for the positions of L1 & L2) is drawn for large primary:secondary mass ratio, L3 should be on the dotted circle. While things are being changed, the -x axis is a dash longer than need be, and L2 should be slightly further from the secondary than L1 is. I don't do SVG. 94.30.84.71 (talk) 21:51, 23 October 2012 (UTC)

## Trojans of Uranus

The last sentence in the introduction seems to imply that Uranus cannot/does not have Trojan asteroids, which contradicts the following source: http://www.space.com/22590-uranus-trojan-asteroid-discovery.html 12.250.50.174 (talk) 19:46, 30 August 2013 (UTC)

It's probably related to the word 'primordial' in there. The article already mentions that Uranus has Trojans. According to the ref on the sentence you refer to, it means the Trojans developed independently of the formation of Uranus, so it got them some point later. Chris857 (talk) 19:55, 30 August 2013 (UTC)
How about update Introduction with more recent discoveries about Uranus trojans, Neptune trojans, and some temporary Venus trojans?--Bobbylon (talk) 12:15, 2 May 2017 (UTC)

It may be a rather trivial question, since one can assume that like the trojan horse, the trojan moons/planets are a hidden threat to their host, due to possibility of destabilisation of their orbit and collision with the host, but it's still noteworthy, who named these objects and why. — Preceding unsigned comment added by 217.12.97.50 (talk) 03:06, 30 October 2013 (UTC)

I think it's in the article. AFAIK, it was coincidence that several of these bodies were named for characters in the Trojan War, and it became a tradition to name them that way. — kwami (talk) 10:27, 17 November 2013 (UTC)

## Trojan Planet?

There is a problem with the use of this term. 1. There is no corresponding IAU definition. 2. It could not be a "planet" per se, any more than a "dwarf planet" is, the scientific terms would be a "co-orbital planetary mass object", "planetary companion" (like a moon) or "trojan of planetary mass". Not only are such objects hypothetical, but Google returns only a couple of thousand results, and all but a few refer to different things, for example, what is classically referred to as a trojan, or objects that orbit with a star.--EvenGreenerFish (talk) 08:10, 24 September 2015 (UTC)

Trojans can be planets, because an orbital resonance is a special case of clearing the orbit. Resonant objects are specifically excluded from the parameter μ (point (5) in [1]). --JorisvS (talk) 13:01, 24 September 2015 (UTC)

## Etymology

Where does this name come from? I can't see any relation to the ancient city of Troy.--80.141.5.35 (talk) 11:21, 1 January 2016 (UTC)

## Planets

On Category:Trojan minor planets, I saw categories for trojans of all eight major planets (MercuryNeptune) except for two; namely Mercury and Saturn. Have any Saturn trojans been discovered? How about Mercury trojans? (The latter might not exist; due to Mercury's highly elliptical orbit and its closeness to the Sun.) (There seem to be no Vulcanoids [asteroids closer to the Sun than Mercury]; except maybe very small ones.)--Solomonfromfinland (talk) 01:37, 3 July 2016 (UTC)

It's simply because no Saturn and Mercury trojans are known, regardless of whether they exist. One Uranus and one Venus trojan are known. The one Uranus trojan is not stable, but I find this article lacking in information about how stable Venus, Earth, and Mars trojans are. --JorisvS (talk) 10:48, 3 July 2016 (UTC)
I believe it’s more than that. Jupiter’s presence destablises any potential Saturn Trojans: there are no stable orbits for Saturnian Trojans to occupy. So it’s not just that we haven’t seen any, there can’t be any. JDAWiseman (talk) 21:05, 3 July 2016 (UTC)
Your first part is correct: There are no possibilities for stable Saturn trojans. The same is true for Uranus. But there is a Uranus trojan known...it's in an unstable, temporary orbit. The same can happen for Saturn: minor planets temporarily in a trojan configuration. --JorisvS (talk) 21:09, 3 July 2016 (UTC)
Oooh, 2011 QF99 — and not very unstable. JDAWiseman (talk) 21:24, 3 July 2016 (UTC)

How about update Introduction with more recent discoveries about Uranus trojans, Neptune trojans, and some temporary Venus trojans?--Bobbylon (talk) 12:15, 2 May 2017 (UTC)

## Trojan asteroid

I've just spotted we had two pages for this concept - as the other one had much less information than is found here, I've redirected it. Andrew Gray (talk) 12:57, 20 May 2017 (UTC)