# Talk:Invariable plane

## Orbital axes ?

I don't understand your phrase "the rotational axes are not parallel to the orbital axes" (3rd para.). It seems to make no sense. Do you perhaps mean "orbital planes" ? Kiwi137 12:12, 23 April 2007 (UTC)

The rotational axis of a planet is its spin axis, around which it rotates. Earth's rotational period is 24 hours. The orbital axis of a planet is a line perpendicular to its orbital plane, passing through the center of mass of the solar system, around which it revolves. Earth's orbital period is 365¼ days. Earth's rotational axis is tilted 23.5° to its orbital axis, which causes the seasons. Thus Earth's rotational axis is not parallel to its orbital axis. — Joe Kress 20:14, 23 April 2007 (UTC)
Thanks Joe, that makes it much clearer. I can follow the reasoning in the article now.--Kiwi137 21:45, 24 April 2007 (UTC)

## Couple of questions

1) What is the inclination of the invariable plane with respect to a) the galactic plane; and b) the ecliptic? 2) The article says that the barycenter moves over time. Does the plane's inclination change as well? (The article only says that the orbital planes for the planets fluctuate.) SharkD (talk) 02:11, 15 February 2008 (UTC)

The plane of Earth's orbit is the plane of the ecliptic so the inclination of the invariable plane to it is already in the article. I've corrected the article to state that the Sun moves, not the barycenter. The invariable plane does not move so its inclination relative to inertial space does not change. The orbital planes of all the planets move relative to the invariable plane and to inertial space.— Joe Kress (talk) 01:44, 14 November 2008 (UTC)

## Question

Is the sun's angular momentum included in these calculations? --Stepheng3 (talk) 17:17, 1 April 2009 (UTC)

Yes—both the Sun's axial angular momentum and its orbital angular momemtum around the barycenter of the solar system, which is usually outside the Sun, is included within the 2% non-jovian residual as the article states: "The orbital angular momenta of the Sun and all non-jovian planets, moons, and minor solar system bodies, as well as the axial rotation momenta of all bodies, including the Sun, total only about 2%." — Joe Kress (talk) 19:26, 1 April 2009 (UTC)

## Questions about inclination of orbital planes

The article says: "All planetary orbital planes wobble around the invariable plane, meaning that they rotate around its axis while their inclinations to it vary, both of which are caused by the gravitational perturbation of the other planets."

Q1: Does this mean that in a hypothetical solar system consisting of only one sun and one planet, the orbital plane would necessarily have an inclination of 0 degrees relative to the invariable plane?

Observation: There no orbital planes with really "big" inclinations (e.g., 70 degrees). In fact, the maximum inclination seems to belong to Mercury, which tops the charts at a whopping 6.34 degrees (pardon the sarcasm).

Q2: So why don't the inclinations of the orbital planes vary more than they do? (It seems obvious that SOME force is pulling the planets' orbital planes towards the invariable plane, but I'd like to understand this mechanism a bit more.)

Q3: Is there a reason that the planets closest to the sun also have orbits with greatest inclination? Shawn Fitzpatrick (talk) 21:01, 23 August 2009 (UTC)

Shawn,
1) Inclination has little meaning in a two-body system. The planet's orbital angular momentum would dominate the system, so the invariable plane would basically be the same as the orbital plane. Maybe the spin angular momentum of the sun would make a difference, but it would be very small.
2) The sun and planets all formed from the same proto-planetary disk, so all of them have angular momenta roughly similar to that of the initial system.
3) If you think in terms of vertical excursion from the invariable plane, z = a sin(I), three of the four giant planets (not Jupiter) stray much farther from the invariable plane than Mercury does (and Mars' vertical excursion is slightly larger than Mercury's). But because Mercury's distance from the Sun is shorter, the angle is larger. Of course, the reason Jupiter does not stray far from the invariable plane is because the latter is largely defined by Jupiter's orbit.
--BlueMoonlet (t/c) 18:21, 27 August 2009 (UTC)

## Invariable vs Laplacian vs Laplace plane

I reverted recent edits by BlueMoonlet which are contrary to the Explanatory Supplement to the Astronomical Almanac, which says that the invariable plane is the Laplacian plane. It says this on page 11: "[The] invariable plane or Laplacian plane [is] the plane that is normal to the axis of angular momentum of a system and passes through its center." and on page 327: "The Laplacian plane is also known as the invariable plane through the planet." Four sentences earlier on page 327 is the definition, "The Laplacian plane is the reference plane about whose axis the satellite's orbit precesses ..." which is similar to that in Laplace plane in Eric Weisstein's World of Physics, "The plane normal to the mean pole about which the pole of a satellite's orbit precesses on a nearly circular path for satellites which are not too close or too far from the attracting body." or that in Laplace plane, "the plane normal to the orbital precession pole of the satellite." Hence all three planes (invariable, Laplacian, and Laplace) are one and the same. Thus even the article immediately prior to BlueMoonlet's edits must be appropriately changed. — Joe Kress (talk) 07:17, 27 August 2009 (UTC)

Firstly, I think we need to dispense with the idea that there is a difference between the "Laplace plane" and the "Laplacian plane". Nomenclature is never this precise, and it was to correct this confusing distinction that I was drawn to edit on this topic. The sources cited here by JK show that the two terms are interchangeable. But how do they relate to the invariable plane? Weisstein's definition of the Laplace plane is correct; unfortunately, he does not define the invaraible plane. Seidelmann, I'm afraid I have to say, seems to be a little sloppy. As JK has cited, on page 11 he says that both the invariable plane and the Laplace plane are defined as the plane normal to the total angular momentum, while on page 327 he says that both the invariable plane and the Laplace plane are defined as the plane about which an orbital plane precesses. The problem is that these two things are not always equivalent. For example, Saturn's inner satellites precess about a plane that is roughly perpendicular to Saturn's angular momentum; for them, the Laplace plane is roughly equivalent to Saturn's equator. On the other hand, Saturn's outer (irregular) satellites precess about a plane that is roughly perpendicular to the angular momentum of the entire solar system; for them, the Laplace plane is roughly equivalent to the invariable plane. You might argue that Saturn's equator can be thought of as a kind of invariable plane for just the Saturn system, but then you have Iapetus, which is far enough from Saturn that the Sun's perturbations are very important but not so far that they dominate; Iapetus' Laplace plane is midway between Saturn's equator and the invariable plane.[1]
My understanding has always been, and this comes verbatim from my PhD advisor among other sources, that the invariable plane is the same everywhere (it's just the global average), while the Laplace plane is the plane about which an orbit at a particular location precesses. The two are equivalent only if you are not too close to a particular perturber (such as Saturn). Unfortunately, I am not finding this distinction being clearly made in my textbooks. But JK has just shown that there are two definitions, one of which is usually associated with one term and the other with the other. I think I have just shown that the two definitions are not equivalent. One reference that focuses on Iapetus' orbital plane is below. --BlueMoonlet (t/c) 15:18, 27 August 2009 (UTC)
[From Terry0051] I hope this might help. I think you'll find, if you go further into the sources, that there are two planes, or kinds of plane, both of them going by the name of Laplace. One, as noted, is the invariable plane of the solar system, and as the discussion has already pointed out, it is defined by the vector sum total of all the orbital angular momenta in the solar system. The other Laplacean plane is a kind of plane associated with the orbit of a planetary satellite, and it is a plane relative to which the current orbital plane of the satellite undergoes precession _at a constant angle_, so the pole of this plane is defined by the track of the moving pole of the satellite's orbital plane -- I can't say how accurately, offhand. That doesn't appear necessarily to be a plane defined by a sum of angular momentum vectors. I'm away from sources at the moment but will try and find some if it's still a problem later.
Best wishes. 86.14.225.140 (talk) 18:51, 27 August 2009 (UTC)
Terry, you are correct that the second kind of plane (the subject of another article) is not necessarily a sum of angular momentum vectors; rather, it has to do with a balance among various perturbations whose strength decreases as some power of the distance from the various perturbers. What I'm not convinced of is that the first kind of plane (the subject of this article) is properly called a Laplace plane. Of the sources JK cited, Weisstein doesn't really address the issue, and Seidelmann (the Explanatory Supplement) uses both terms to apply to both kinds of plane, when we seem to agree that they are not really the same. --BlueMoonlet (t/c) 00:15, 28 August 2009 (UTC)

What we need is some sense of how the various terms are used in the literature, then we can determine how best to word both articles, which may involve some disambiguation at the head of both articles. Seidelmann is only the editor of the Explanatory Supplement—each chapter has individual authors. The chapter containing page 11 was written by Seidelmann and Wilkins whereas the chapter containing page 327 was written by Rohde and Sinclar, so they may be using these terms with different meanings. — Joe Kress (talk) 02:39, 28 August 2009 (UTC)

[From Terry0051]

• Usage of 'Laplace' name for solar system invariable plane:

Besides the ESAA 1992 reference (at page 11) already noticed by Joe K, here are several examples:

(1) Archie E. Roy, "Orbital motion", 4th edition (CRC Press) 2005, at page 104, referring to the "invariable plane of Laplace" (the context makes it clear that the topic is the invariable plane of the solar system).

(2) M Bursa, "Tidal contribution of planets to removing angular momentum from the sun", Astronomical Institutes of Czechoslovakia, vol.37 (1986), pp.142-146, refers in its abstract to the "Laplace invariable plane of the solar system".

(3) T J J See, "On the Degree of Accuracy Attainable in Determining the Position of Laplace's Invariable Plane of the Planetary System", Astronomische Nachrichten, vol.164 (1904), p.161.

(4) And then see p.147 in "Planetary Astronomy from the Renaissance the the rise of astrophysics: B, the eighteenth and nineteenth centuries" (eds. R Taton & C Wilson), in ch.22 'Laplace' by Bruno Morando: "Laplace defined _his_ invariable plane" (emphasis added), the context being clear as before.

• Disambiguation:

Here's a couple of proposals for a disambiguation statement for this article. When flipped around it would also pretty much fit the other I suppose.

"The invariable plane of the solar system, sometimes called Laplace's invariable plane, is to be distinguished from the invariable or Laplacean plane of a planetary satellite (which relates to the satellite's orbit around its primary planet)."

"The invariable plane of the solar system, sometimes called the invariable plane of Laplace (determined by the sum of the orbital angular momenta of all the masses in the solar system), is to be distinguished from the invariable or Laplacean plane of a planetary satellite (intermediate between the plane of the equator of the satellite's oblate primary planet, and the orbital plane of the primary planet around the Sun)."

Terry0051 (talk) 17:52, 28 August 2009 (UTC)

Terry, thank you for supplying these references. I would point out that in each of the cases you cited, the word "invariable" invariably appears, whether or not the name "Laplace" is also used, when discussing the plane perpendicular to the total angular momentum. Clearly Laplace deserves credit for originating the concept, but the defining word is really "invariable". On the other hand, for the plane about which orbits precess (the Laplace plane), the word "invariable" is never used unless the Laplace plane does in fact happen to be equivalent to the invariable plane for the orbit under consideration.
This is important: The invariable plane is defined by the angular momentum of the entire system. It gets its name because it is the same everywhere in the system. The Laplace plane is the "mean plane" of a precessing orbit; because it changes depending on where in the system the object is orbiting, the word "invariable" should not be applied to it. If you are talking about the precession of a particular orbit, you should speak of its Laplace plane, not its invariable plane. Your second proposed statement above is problematic because it defines the Laplace plane based on where it is, not on what it is. As this article already states, the Laplace plane "is the plane about which orbital planes precess". It's as simple as that.
Also, I have only heard the term "invariable plane" applied to the total angular momentum of the entire solar system. I have never heard it applied to a planetary system, as you seem to be doing. Do you have a reference for that? --BlueMoonlet (t/c) 03:30, 30 August 2009 (UTC)

1. The citations answer the question originally posed, and confirm the existence of a usage or practice 'out there', of (sometimes) including Laplace's name when naming the invariable plane of the solar system. The fact itself is not altered by analyzing the content of the expressions used to see which parts convey most of the descriptive meaning. I guess we just have to include a disambiguating explanation.
2. You wrote "On the other hand, for the plane about which orbits precess (the Laplace plane), the word "invariable" is never used unless the Laplace plane does in fact happen to be equivalent to the invariable plane for the orbit under consideration.", but here I think you inadvertently overlooked Joe K's comment (above). He already cited and quoted the very source you ask for (or seem to doubt the existence of), it is page 327 in the 1992 Explanatory Supplement ('ESAA'). It clearly confirms that the expression 'invariable plane' is sometimes used in the satellite-orbit connection and I'm sure that if you looked you could find further confirmatory examples. (The discussion on this talk page throws no doubt on the credibility of ESAA as a reliable source. Its pages 11 and 327 are not in conflict with each other, nor with facts otherwise established, and the most that can fairly be said against them is that they omit to disambiguate two real-world usages that are clearly similar enough on very different matters -- unfortunately -- that there is a likelihood of creating confusion.) I sympathize to some extent with your prescriptive statements about how the word 'invariable' should and should not be used. But it's not our role here on WP to legislate and make fresh rules. We have to take the world as we find it, and explain as needed. (I also suggest it would be unhelpful to the cause of explaining and disambiguating to discuss in the article the completely coincidental and probably rare case that "the Laplace plane" (the plane related to the satellite orbit??) "does in fact happen to be equivalent to the invariable plane" (of the solar system??) "for the orbit under consideration.")
3. 'Planetary system': the solar system itself is an example of a planetary system -- see the title of reference citation no.(3) previously given above, which both explains the usage and amounts to a citation in support.

Perhaps more importantly than all of this discussion, the main article needs a cleanup, at least to disambiguate and make sure the right planes are mentioned in the right places! With good wishes Terry0051 (talk) 19:34, 30 August 2009 (UTC)

Terry, I just edited the article to show how I think the disambiguation should look. To your #2, I maintain that the ESAA is rather confused, as I described above. It clearly seems to think that the invariable plane and the Laplace plane are not actually different, when it seems well established that they are. I don't think we should cite it for nomenclature on this particular topic where we know it is not quite right on the physics. To your #3, I have only ever heard of "the" invariable plane, referring to the mean plane of the solar system. I was asking whether you have a reference for a broader usage of the term. I don't deny that such broader usage might be logical; I just am curious to see an example. --BlueMoonlet (t/c) 00:22, 31 August 2009 (UTC)
Two different Laplacian planes were described by Laplace in Mécanique Céleste. The first plane, concerning angular momentum, he called the "plane of maximum areas" in vol. I book I § 21. Laplace described it in a rather cryptic form:
The sum of the areas, traced by the projections of the radius vector of each of the bodies, multiplied by its mass, is a maximum
In The system of the world (1830) Laplace named it an "invariable plane". The second plane, concerning precessing orbits, he called the "intermediate plane" in vol. IV book VIII § 35. Laplace described it in almost modern form:
The inclination $\begin{smallmatrix}\varpi\end{smallmatrix}$ of the orbit of the outer satellite, upon the intermediate plane between the orbit and the equator of Saturn, remains always the same, notwithstanding the variations of this plane. ... Saturn's equator carries with it, in its motion, the intermediate plane and the orbit of the satellite, which preserves always, upon this plane, the same mean inclination; with a retrograde and nearly uniform motion, varying however a little on account of the variations of the respective inclinations of the equator and the orbit of Saturn.
Quoting several sources which call the invariable plane of the Solar System a Laplacian plane:
• AIAA Astrodynamics Standard: Propagation Specifications, Test Cases, and Recommended Practices:
Invariable Plane or Laplacian Plane: the plane that is normal to the axis of angular momentum of a system and passes through its center.
• Planetary Geometry Handbook:
Invariable (or Laplacian) Plane, defined as the plane normal to the total angular momentum vector of the solar system.
• Modeling the 3-D secular planetary three-body problem:
The direction of the constant total angular momentum defines a plane orthogonal to $\begin{smallmatrix}\overrightarrow{C}\end{smallmatrix}$, which is an invariant of the problem (this plane is known as the Laplace invariable plane). The choice of the Laplace plane as the reference plane of coordinates ...
• Satellite dynamics of the Laplace surface:
More generally, the Laplace plane is usually defined as the plane normal to the axis about which the plane of a satellite's orbit precesses. Note: Unfortunately, the term is sometimes also applied to the invariable plane, the plane perpendicular to the total angular momentum of an N-body system.
• Evolution of the Solar System (NASA)
the orbital inclination of the planets i refers to the orbital plane of the Earth (the ecliptic plane). It would be more appropriate to reference it to the invariant plane of the solar system, the so-called Laplacian plane.
Conversely, quoting several sources which call the Laplacian plane for precessing orbits an invariable plane:

[From Terry0051] Hallo Joe K, thanks for posting your list of references, it certainly gives a strong showing -- and contains material of wider interest too (separate msg on your talk page). With good wishes Terry0051 (talk) 15:43, 1 September 2009 (UTC)

All right, you've convinced me that there is considerable confusion in the literature as to which concept should have which name. I think we need to keep these articles separate, as they really are separate concepts, but mention in both the variety of names that can be applied. I've made an attempt at that. --BlueMoonlet (t/c) 14:48, 11 September 2009 (UTC)

## Angular momentum not conserved?

The second paragraph of section "Description" seems wrong to me. It says that the transfer of axial angular momentum to orbital angular momentum can change the invariable plane. But the invariable plane is defined by the total angular momentum, which does not change by a transfer between its constituents, neither in its magnitude nor in its direction. A valid statement would be: If the invariable plane was defined only by the orbital angular momentum, it would not be truly invariable. As such a statement seems useless, I suggest to delete the paragraph. Comments please. --Peter Buch (talk) 09:26, 26 September 2009 (UTC)

[From Terry0051] Interesting point. It seems similar to one raised by Louis Poinsot in about 1830. This looks like another of those cases where an astronomical quantity was initially thought to be invariable, but turns out later not to be perfectly constant after all. The article surely needs improving, but I suggest that deletion is not the best answer. (If imperfection alone was a reason for deletion, there would be nothing left.)
There is an element of history in the article, and it's important not to introduce anachronism, let alone try to re-write history. When Laplace identified his solar system invariable plane he did not use the language of angular momentum, which in his time was yet to be developed: he discussed planetary integrals of areas (and weighted them by the respective planetary masses for their combination). It is clear that he considered in effect only the sum of the orbital angular momenta.
Louis Poinsot later objected that the rotation of the planets (and orbits of the satellites) should be taken into account too. This seems to have had little practical impact. Also, I've seen nothing to indicate that Poinsot, let alone Laplace, envisaged any mechanism for exchange between orbital and spin angular momenta.
The numerical data for practical evaluation of the invariable plane are not well determined. There are not only remaining uncertainties in the planetary masses, there is an appreciable amount of solar system mass and momentum unaccounted for, in the main belt of minor planets, and in the Kuiper belt as well.
Current estimates of rotational (spin) angular momentum suffer from uncertainty too, but such as they are, indications have been that the rotational angular momentum is several orders of magnitude smaller than the orbital angular momentum, and in literature it seems often specially labelled as negligible in its contribution to the overall parameter, where it is not disregarded altogether.
While there is the theoretical possibility of spin-orbit interaction, with the exception of effects on the Moon it does not seem to be taken into account in current equations of motion used to derive solar-system motions (Newhall, Standish & Williams 1983 seem to be still regularly cited for the form of those equations of motion, at least I haven't seen any update).
So it looks as if, within current limits of measurement uncertainty, there is no current practical difference between the 'invariable plane' with and without the spin contributions, and no distinct consistency of usage of the term -- even though the concepts can of course be theoretically distinguished, and made relevant to very-long-term solar-system evolutionary considerations.
As there is a clear historical basis for identifying the 'invariable plane' in a practical sense with the sum of the orbital angular momenta, this should continue to be acknowledged. The fact, that such a sum is not theoretically invariable after all, does not seem to be a good reason to try to erase or re-write the historical record, but it surely needs explaining. It's also worth pointing out that, in principle, exchange with the spin momenta could change this apparently 'invariable plane', but also it seems worth adding that in practice the changes are insignificantly or immeasurably small. I don't have time to track sources at present, but I suggest that article improvement is indicated, rather than the suggested deletion. Terry0051 (talk) 14:09, 26 September 2009 (UTC)

## 23 degrees?

I'm pretty sure the inclination of the Earth is about 23 degrees, and that of Uranus is about 90 degrees. I'm not sure if someone is just screwing with the numbers in the tables or what, but they appear to be way, way, off. Br77rino (talk) 13:36, 2 March 2010 (UTC)

Hi Br77rino. You are speaking of the obliquity of the planets, which is the angle between the spin axis and the orbit plane. The article is speaking of the inclination, which is the angle between the orbit plane and a reference plane (in this case, the invariable plane). --BlueMoonlet (t/c) 15:13, 2 March 2010 (UTC)

## Above or Below?

Simple question: which planets are "above" the plane and which are "below"? (I assume you define "above" as the side pointed to by the Earth's north pole.) It might be computable from the numbers presented, but it's not readily apparent.

It might make a nice addition to the chart. 208.127.93.29 (talk) 00:36, 6 June 2010 (UTC)

I suppose, on second thought, that it depends which part of the orbit you're talking about. But that just makes it more interesting - do all the planets' orbits all reach their "upper" most point at roughly the same spot? 208.127.93.29 (talk) 00:38, 6 June 2010 (UTC)

Hi, 208.127.93.29. You've already given the right answer to your first question. Your second question is answered by the parameter called the longitude of the ascending node, which is one of the six parameters called orbital elements that describe any orbit's characteristics. I'm not finding a table of the planets' OEs that's available online, but suffice to say that the planets' nodes are more or less randomly distributed. --BlueMoonlet (t/c) 13:20, 7 June 2010 (UTC)
The orbital elements of all the planets valid for −4000 to +8000 are in J.L. Simon et al., Numerical expressions for precession formulae and mean elements for the Moon and the planets. All of the elements on pages 677–680 are referred to the mean dynamical ecliptic and equinox of date, a plane that always coincides with the plane of the ecliptic no matter how it moves, so Earth's orbit has no inclination i or ascending node Ω of its own. Surprisingly, the ascending nodes of the planets are not randomly distributed at the moment—all occupy less than one quarter of the ecliptic, 48°–132°. On pages 674–677 they are referred to the mean dynamical ecliptic and equinox J2000, a fixed inertial plane, so both exist for Earth's orbit. I do not know of a similar list relative to the invariable plane. — Joe Kress (talk) 06:14, 9 June 2010 (UTC)
The giant planets have inclinations to the invariable plane that are small compared to Earth's, so you would expect their nodes on Earth's orbital plane to be similar to Earth's node on the invariable plane. For the three non-Earth terrestrial planets, the odds of having all three in the same quadrant by random chance are no worse than 1/4^3 = 1/64. That's my quick-and-dirty analysis. --BlueMoonlet (t/c) 09:02, 9 June 2010 (UTC)

## Inclination to the galactic plane

What is the inclination of the invariable plane to the galactic plane? --JorisvS (talk) 11:33, 17 July 2012 (UTC)