Talk:Orbital elements/Archive 1

Archive 1

From Harp: New figure

I made another figure for this article: commons:image:orbit.svg. If you have any suggestion, please write me to the talkpage of the figure. -- Harp 14:17, 26 May 2006 (UTC)

==New Figure Reproduces an Error from the original orbit.png by Urhixidur==

See discussion at Orbit.svg

In its current form, the illustration Image:Orbit.svg, composed by Harp, labels 'T' (True Anomaly) as if it were 'M' (Mean Anomaly); The mislabeling was carried forward from an original image Image:Orbit.svg composed by User:Urhixidur. The articles True Anomaly and Mean Anomaly and the accompanying diagram (Image:Kepler's-equation-scheme.png) illustrate the correct relationships.

While Urhixidur has posted a caveat in the caption, the illustration is still labeling one concept as another concept, so the misleading condition has not been truely eliminated. For this reason, I'm posting a Template:Confusing tag in the main article to properly alert readers. This is a short-term fix; I feel the cleaner solution is to have a correct drawing in the first place.

At User Talk:Urhixidur, xgarciaf proposes Image:Orbital elements.svg to address the issue; While the proposed illustration is correct insofar as it goes, it simply does not include a reference to mean anomaly. In truth, the visualization of M is problematical since it plots a position that doesn't correspond to the orbiting body, its position vector revolves around the geometric center of the ellipse, not the focus, and the head of the vector plots a position on the auxiliary circle, not in the orbital path. Introducing these new components into the diagram will make for a more complicated drawing. Perhaps the better way to go is to use the proposed drawing, and note the missing element in the caption. Still, the original attempt of Urhixidur to get all of the elements in one drawing is laudable; it would be nice to pull it off. — Gosgood 13:12, 7 August 2006 (UTC)

Further Comment: on reflection, maybe the easiest way to fix this drawing would be to simply correct the drawing, changing 'M Mean Anomaly' to 'T True Anomaly'. As the article points out, true anomaly is an alternate expression of 'mean anomaly'. — Gosgood 14:01, 7 August 2006 (UTC)

Patched the drawing, per reflection. — Gosgood 00:05, 8 August 2006 (UTC)

Shouldn't this be the other way around?

"We see that the first three orbital elements are simply the Eulerian angles defining the orientation of the orbit relative to some fiducial coordinate system. The next two establish the shape of the orbit, while the last establishes the location of the orbiting body at a particular time."

Shouldn't that be the first two and the next three? Eccentricity defines the shape of the orbit, not its orientation. --YFB ¿ 16:01, 6 February 2007 (UTC)

• Right. Marklark thought to rearrange the list of Keplerian elements to be in the same order as the NORAD two line elements, but didn't synchronize the paragraph you flagged, which is still written to the old ordering of the list, see the November 09 version. I have no love for the new ordering, preferring the original Euler rotations - shaping elements as distinct groups in the list. Gosgood 04:55, 7 February 2007 (UTC)

• Thanks for the insight. Feel free to revert my re-ordering. I should (and will in the future) have looked for its side-effects. Marklark 01:38, 25 February 2007 (UTC).

• Addressed with version 111056891; I restored the original ordering. Thank you all for catching this or commenting. — Gosgood 11:39, 26 February 2007 (UTC)

Lunarise, Lunaset, and transit

Can anyone tell me if the moon always rises in the southeast? I assume this varies depending on the time of year as it does with the sun? If so, does the moon always rise in the same place as the sunrise? I live near 45 03'20.71" N 77 47'33.05" W

66.30.240.85 17:41, 5 May 2007 (UTC) Linda Farley

Depends where on the planet you are, but in all locations, the azimuth of the Moon's rise and set points varies seasonally, in a manner similar to, but not quite like the sun. Nor will it's rise and setting points exactly coincide with the Sun. It can vary by a number of degrees azimuth, for reasons below.

In your locale, at one extreme, the Moon may rise at an azimuth of 47 degrees (zero degrees is north, ninety degrees is east, 180 degrees is south, and two hundred, seventy degrees is west). This puts the moonrise in the northeast. At the other extreme, the Moon may rise at an azimuth of 133 degrees, about due southeast. I gather this from the Rise/Set/Transit Tables courtesy of the US Naval Observatory. At this page, you can generate a table of specific times the Moon will rise, cross the southern azimuth, and set, with the azimuth of the rising and setting points, for your locale.

Upon what is this variation based? Recall the Earth is tilted with respect to the ecliptic by 23 degrees, while the orbit of the Moon is inclined 5 degrees with respect to the ecliptic. At one extreme, a northern latitude observer of the Moon's transit of the southern sky may be at a longitude that has just rotated to where the Earth's tilt into the plane of the ecliptic is at its greatest, 23 degrees, and it just so happens that the Moon is at the extreme of it's ascent above the plane of the ecliptic, placing the Moon about 28 degrees above the equator. The northern hemisphere observer will see the Moon rise in the northeast, climb very high in the southern sky, and set in the northwest. Conversely, the observer's longitude may have rotated to just be where the Earth is tilting away from the ecliptic by 23 degrees, just when the Moon has descended below the ecliptic by five degrees: the observer would see the Moon rise in the southeast, remain low in the sky even at the southern transit, and set in the southwest.

If I'm reading the USNO data for your latitude and longitude correctly, you will see this particular extreme on June 01, with the Moon one day past full. it will rise at 10:12 PM, EDT at an azimuth of 133 degrees (in the southeast). It will be only sixteen degrees above the horizon a quarter after midnight, and set at an azimuth of 229 degrees (southwest) at 5:15 AM. It will be at the other extreme even earlier, at May 18, but the Moon will be new and hard to observe, rising at an azimuth of 48 degrees, (northeast) at 6:33 AM (already past sunrise) set at an azimuth of 312 degrees (northwest) at 11:23 PM, past sunset. You will have to wait for the Northern Hemisphere's winter to see a full Moon rise in the northeast, go high in the southern sky, and set in the northwest: the full Moon for December 23 will do just that. Go to the US Naval Observatory website to generate a table for your locale. Take care. — Gosgood 01:47, 6 May 2007 (UTC)

Thank you so much

)

Transformations - 9 coordinates?

I think the transformations section needs some serious explanation. What are x1,x2,x3,y1,y2,y3,z1,z2,z3? Are these coordinates in 3 different systems? A citation in this section would be very helpful, since it looks like that section was copied and pasted from some FORTRAN code's help file. --Keflavich (talk) 16:49, 30 November 2008 (UTC)

${\hat {x}},{\hat {y}},{\hat {z}}$ represent mutually perpendicular vectors forming a rectangular orbital reference frame. This orbital reference frame is positioned so that its origin coincides with the focal point, the plane containing ${\hat {x}},{\hat {y}}$ vectors coincide with the orbital plane, ${\hat {x}}$ coincides with the semimajor axis, such that when an orbiting body passes through periapsis it passes through the ${\hat {x}}$ axis, and ${\hat {z}}$ is normal to the orbital plane at the origin/focal point. This is a suitable reference frame for an orbiting body.

We, may wish to plot the orbiting body with respect to a universal 'rest frame'. Let us suppose this rest frame consists of mutually perpendicular vectors $x,y,$ and $z$ and is positioned so that its origin coincides with the focal point, its $x-y$ plane corresponds to the reference plane and its $x$ axis coincides with the reference line directed toward the vernal point. As with the orbital reference frame, $z$ is normal to the $x-y$ plane at the focal point. This is a suitable reference frame for the solar system itself; it is not local to any particular orbiting body.

Now, if we treat:

• 1. the orbit's longitude of the ascending node,
  2. its argument of periapsis, and
  3. its orbital inclination

as three rotational twists — Euler angles — we can then find the transformation matrix to 'carry' points from the orbital reference to the universal rest frame.

If you toddle off to the article on Euler angles and look about midway down, you'll find a transformation matrix, $\mathbf {R}$, that relate rectangular coordinates from a rest frame to a second, re-oriented frame, one that has arisen from 'three twists' — the euler angles. The nine equations in this article, taken in groups of three, are none other than the column vectors from that transformation matrix, notated differently to confound the innocent. These nine equations, then, carry points written in terms local to the orbiting body over to the universal rest frame. Notation: $x_{1},x_{2},x_{3}$ are the unit component vectors of axis $x$.

As for the utility of this section — ? I think I can do without it. The mathematical manipulations transpiring in this section do not offer any new material for the discourse of Keplerian elements; its just a space transform, a particular application of Euler's angles. I think this article would be a good deal more terse and to the point if it just notes that the three rotational orbital elements can be regarded as Euler angles and, as such, can be harnessed to produce transformation matrices from orbit-local to solar-system local reference frames (both ways, really). A wiki link to Euler angles will serve those interested in the particular mechanics. That, after all, is what wiki linking is for — to direct people to various canonical references, so that material isn't needlessly replicated across the wiki, where it generates a tiresome problem of falling out of sync.

This section was introduced by Beland with this edit who copied them, pretty much unchanged, from a closely related, and relatively new article called Kepler orbit. I personally think the introduction is an unfortunate choice. The editors of Kepler orbit prefer a terse mathematical exposition. While it may not be especially obvious, this article attempts to fulfil a niche for the reader not trained in mathematics. The material introduced from Kepler orbit retains the terseness characteristic of that article, putting it kindly, and does not serve the reader this article is hoping to reach. That editor also undertook a number of copyedits from August 8 to 13, 2008 that, to a degree, improve the readability of this article. However, he commits the same gaffe that Marklark committed some time ago when he rearranged the order of the orbital element list in Keplerian elements. After rearranging elements, neither worthy deigned to look a mere one paragraph further to synchronize the follow-on discussion. So one looks up at the list from the follow-on discussion to read (1) eccentricity, (2) Semimajor axis, and (3) Inclination. The first two having nothing whatsoever to do with angles, the reader becomes understandably confused. This must bemuse more than a few of the one hundred or so people who have visited this page daily since August, who probably conclude that when you have a free encyclopaedia you get what you pay for. I shall be addressing that issue later, if time permits.

My inclination is to revert the transform section; but it would be civil to have Beland comment here first if he cares to; he may be implementing an approach that just isn't clear to me yet. This may be a bric-a-brac from the Talk:Kepler orbit discussion, subject: what is an apt balance between mathematical terseness and non-mathematical technical expository? A future version of this article, with improved prose, could function as a non-mathematical adjunct to portions of that article. Take care. Gosgood (talk) 03:54, 1 December 2008 (UTC)

Thanks for the note. I agree that this article (and all articles really, as explained by Wikipedia:Make technical articles accessible) should be accessible to non-mathematicians. My apologies for the confusing changes; I was trying to synchronize various articles but didn't have time to do a thorough job of it. I have just completed a thorough re-working of this article to remove the confusing discontinuities and redundancies in enumerating the Keplerian elements.

As for the "transformations" section, if you think it should be deleted from the encyclopedia entirely, that's fine with me. If you want to keep it, it seems to me that this article is the best place for it, considering that it relates to orbital elements, and the "Orbital elements" section of Kepler orbit should be a summary of this article which omits such details (following Wikipedia:Summary style). This difference between this article and this other one should be level of detail, not mathy vs. not mathy. I suspect in the long run, the two articles will be merged, but Kepler orbit is currently a mess. -- Beland (talk) 17:36, 1 December 2008 (UTC)

On the whole, a nice bit of work, Beland. Consider yourself commended. This article was the work of several authors with different styles, and writing at different times; you've done a nice job consolidating and giving the article a single 'voice.' I can (and probably will) quibble with some of your editorial choices, but that's what they are: quibbles. With me, fixing quibbles has a low priority. Tribbles, however, present another case. ;)

I do have issues with the transformation section. The prose is nonsensical. It declares that the accompanying math presents a 'transformation from the euler angles $\Omega ,i,\omega$ to ${\hat {x}},{\hat {y}},{\hat {z}}$'. Further down it claims the presence of the reverse transform. $\Omega ,i,\omega$ and ${\hat {x}},{\hat {y}},{\hat {z}}$ are not like things; the first three symbols reference rotations about axes, the second three symbols reference spatial measures along axes. In the parlance of software design, these are 'dissimilar types'. Rotations are not translations. I may as well ask what two and a half radians are in meters (please). To further muddle matters, it never discusses what the notation $x_{1},x_{2},x_{3}$, $y_{1},y_{2},y_{3}$ and $z_{1},z_{2},z_{3}$ represent. Keflavich initial complaint.

While it is nonsensical to write of translating from/to Euler angles, the three rotational orbital elements $\Omega ,i,\omega$ do have a bearing on spatial transformations; they record the orientation of the orbital reference frame with respect to some fixed frame of reference, say the International Celestial Reference Frame. As such, they may be regarded as Euler angles and harnessed to find a matrix $\mathbf {R}$ to translate position vectors in the fixed frame of reference $v$ to a representation ${\hat {v}}$ in the orbital reference frame:

${\hat {v}}=v\mathbf {R}$

The section Matrix rotation in Euler angles illustrates how to compose $\mathbf {R}$. To put the notation of the two articles on a common ground, let $\Omega =\alpha$, $i=\beta$ and $\omega =\gamma$. Let ${\hat {x}},{\hat {y}},{\hat {z}}$ represent the axes of the orbital reference frame and $x,y,z$ the axes of the fixed reference frame. Suppose they initially coincide.

1. To any position vector in the fixed reference frame, written as a row vector, post multiply with a rotation matrix representing a revolution of the orbital reference frame by an angle $\Omega$ around the ${\hat {z}}$ axis. By the definition of the longitude of the ascending node ($\Omega$), this aligns ${\hat {x}}$ with the line of nodes.
2. To the product developed so far, post multiply a second rotation matrix representing a revolution through angle $i$ around the ${\hat {x}}$ axis. This establishes the inclination of the orbit.
3. Finally, to the product developed so far, post multiply a third rotation matrix representing a revolution through angle $\omega$ around the ${\hat {z}}$ axis. This sets the argument of periapsis, orienting the orbit's periapsis with respect to the line of nodes.

This give us a product of one row vector and three rotation matrices, similar to the configuration illustrated in Matrix rotation. Multiplying the three rotation matrices gives us $\mathbf {R}$. and encodes the aggregate rotation of the three rotational orbital elements. Since these are all orthogonal matrices their transposed form quickly gives us the inverse transform, so, finding the reverse transform from the orbital reference frame to the fixed reference frame is trivial.

If we were to write out $\mathbf {R} ^{T}$ as a system of equations, we would get the mathematical part of the transformation section, as can be seen by reading down the columns of $\mathbf {R}$, depicted at the bottom of Matrix rotation. To (finally!) answer Keflavich question, $x_{1},x_{2},x_{3}$ are the vector components of the x axis of the orbital reference frame, ${\hat {x}}$ as represented in the fixed reference frame. Similarly, $y_{1},y_{2},y_{3}$ are the fixed reference frame components of the ${\hat {y}}$ axis, and $z_{1},z_{2},z_{3}$ are the components of the ${\hat {z}}$ axis.

I'll see if I have time to turn a (far more) terse version of these remarks into gloss notes for the transformation section. Gosgood (talk) 02:14, 3 December 2008 (UTC)

Applicability to exoplanets?

I've asked for some clarification of how the concepts described here apply to exoplanets on the argument of periapsis talk page. — Aldaron • T/C 03:29, 24 July 2009 (UTC)

Euler Angles are reversed

In the section on the Euler angles, I think large and small omega are consistently swapped. To see this most clearly, look at the z values. Obviously small omega cannot have any effect on the axis of rotation, but in the equation presented, it does. By my own calculations (from first principles -- not depending on the Euler Angles article), this error is consistent -- large and small omega are reversed in all nine equations. I think this comes of using the intrinsic rather than the extrinsic formulation of the Euler angles, although it's hard to square that with everything that's said in the Euler Angles article. —Preceding unsigned comment added by 98.237.244.126 (talk) 03:55, 15 October 2009 (UTC)

There is nothing reversed. The three last equations depend on large Omega, not on small omega. Everything is correct! Guaranteed! Stamcose (talk) 15:08, 13 March 2011 (UTC)

Argument of periapsis

The article says "Argument of periapsis defines the orientation of the ellipse (by the direction of the minor axis) in the orbital plane, as an angle measured from the ascending node to the semiminor axis. (violet angle \omega\,\! in diagram)". As far as I can understand this is wrong. Isn't it the angle to the semimajor axis on which the periapsis (for instance the perihelion) lies. --Episcophagus (talk) 16:50, 27 February 2013 (UTC)

Orbit prediction

"Under ideal conditions of a perfectly spherical central body, and zero perturbations, all orbital elements, with the exception of the Mean anomaly are constants, and Mean anomaly changes linearly with time, scaled by the Mean motion"

This seems at odds with Kepler's second law unless the orbit is circular. --IanOfNorwich (talk) 13:43, 13 May 2014 (UTC)

I think that is taken care of because it refers to *mean* anomaly, not *true* anomaly. Martijn Meijering (talk) 14:02, 13 May 2014 (UTC)

The page on Mean Anomaly seems to confirm that this works for elliptical orbits. TomatoCo (talk) 15:09, 4 September 2014 (UTC)

I agree that the Mean anomaly, based on the page and a quick google search, seems to change linearly with time. The equation "M=nt" on the mean anomaly page (where n is constant with time) couldn't be clearer. While I don't have access to the sources cited there, I don't think it's terribly improper to copy core facts from a main article to a location where it's referenced... removing dubious tag; if the matter is still disputed it's probably more appropriate to flag and discuss it there. Belovedeagle (talk) 08:41, 1 January 2015 (UTC)