Talk:Trojan (celestial body)

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).

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.

Read http://www.merlyn.demon.co.uk/gravity4.htm and associated pages.

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)

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)

Yes, I agree that at least a few of these should be listed HERE in this article. However, do look at the list at the associated article, whose wikilink is here in this article -- BUT this article does not explain anything about what information about what could be found there, not even two sentences worth. This article should say, "Look THERE for such-and-such information....."
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)

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)