# Talk:Orbital elements

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## Letter from an astronomer

Orbital elements are parameters defining a particular Keplerian orbit ( = Keplerian conic = unperturbed orbit = reduced-two-body orbit), i.e., an orbit described by a point mass about a nailed-down gravitating centre. Such orbit is either an ellipse or a hyperbola or a parabola. To define a particular Keplerian orbit, six orbital elements are sufficient (the seventh quantity, time, being the parameter describing motion along the orbit). Normally, an orbit is parameterised through the following six elements: - the longitude of the ascending node; - the argument of the pericentre; - the inclination; - the semimajor axis; - the eccentricity; - the mean anomaly at epoch (i.e., at some fiducial time). The first three elements fix the orientation of the Keplerian orbit, the other two define its shape, and the last one is the initial condition of motion. Sometimes an equivalent set is used, set with the mean anomaly at epoch substituted with the time of crossing the pericentre. Sometimes another equivalent set of parameters is used; in that set the mean anomaly at epoch is substituted with the current value of the mean anomaly (the latter being a combination of the time and the mean anomaly at epoch).

The relevance of the orbital elements lies in the fact that any realistic, perturbed, orbit may be represented as a sequence of points each of which belongs to some instantaneous Keplerian orbit. These instantaneous orbits share one of their foci. In case each such instantaneous unperturbed orbit is tangent to the physical orbit at the point of intersection, these instantaneous orbits are called osculating (and the orbital elements, wherewith these orbits are parametrised, are called osculating elements).

In the framework of representation of a perturbed orbit by a sequence of instantaneous unperturbed orbits, the motion along the perturbed orbit may be considered as an infinite series of infinetesimally small transitions from one instantaeous unperturbed orbit to another. In this sense, one may consider the orbital elements as functions of time. This treatment is called variation of parameters. It was introduced and explored by Euler and Lagrange. However, its earliest sketch was offered yet by Newton in his unpublished Portsmouth papers and was briefly mentioned in his "Principia."

For more details on the orbital elements, and for historical references see:

M. Efroimsky and P. Goldreich:
"Gauge Freedom in the N-body Problem of Celestial Mechanics."
Astronomy and Astrophysics, Vol. 415, pp. 1187 - 1199 (2004)


This paper also explains why the terms "orbital elements" and "osculating elements" are not always synonims.

For exact definitions of an orbit and an osculating orbit see the Glossary of the Astronomical Almanac published by the US Naval Observatory and HM Nautical Almanac Office.

Michael Efroimsky, Astronomer, US Naval Observatory, Washington DC 20392

## Overlap

This is overlapping Orbit#Orbital_parameters Kwantus 21:59, 2 Sep 2004 (UTC)

## how many orbital elements?

The article currently states that "The elements of an orbit are the parameters needed to specify that orbit uniquely" and "all sets of orbital elements have seven parameters", but the combination of these two statements seems wrong or at least insufficiently clear to me (even apart from the fact that one can trivially produce "sets of orbital elements" with fewer elements, merely by removing some).

• One needs only five parameters to specify the size, shape, and orientation of the orbit, i.e., "to specify the orbit uniquely": for example, the length of the semimajor axis, the eccentricity, the inclination, the length of the ascending node, and the argument of the periapsis. Given these five parameters for a particular orbit, one can tell whether this is (for example) the orbit of Mars. So, if seven orbital elements are insisted on, then the definition of "orbit" or of "orbital element" is currently insufficiently clear.
• With a sixth parameter (for example, the mean anomaly), one can also specify the position of the body in the orbit, but that is a property of the body and not of the orbit. If such a sixth parameter is to be included in the definition of "orbital element", then the current definition is insufficiently clear, and should be changed to something like "whatever parameters are necessary to specify the orbit and the body's position in it uniquely".
• The epoch (currently listed as one of the orbital elements) is not a property of the orbit or of the body. The purposes of the epoch are:
• to allow translation between the calendar date and the location of the body (in both directions).
• to identify for which time the semi-constant orbital elements (those that would be constant in the classical two-body problem) provide a good description of the osculating ("instantaneous") orbit of a particular celestial body. This highlights the difference between "orbit" as some fixed orbit in space that might for a moment happen to be occupied (in an osculating sense) by some particular planet and that is fully specified by five orbital elements, and "orbit" as whatever trajectory a particular planet follows through space, which requires far more than five parameters to specify, because of the perturbations compared to a fixed keplerian orbit.
• If one wants to designate the epoch as an orbital element, then "orbital element" must be defined something like "whatever parameters are necessary in order to be able to predict the position of a celestial body for any arbitrary calendar date, assuming that the orbit does not change with time".
• In that case, one should include the orbital size (e.g., the length of the semimajor axis) as an orbital element (in addition to the currently mentioned orbital period), because the orbital period depends not just on the orbital size but also on the masses of the central and orbiting objects, so the orbital size and orbital period represent different degrees of freedom. For example, a given orbital period does not correspond to the same orbital size for an orbit around the Sun and an similarly shaped orbit around Jupiter. Even for orbits around a single central object one cannot fully predict the orbital period from the orbital size, because the orbital period of any object is affected by the gravity of the other objects that also orbit around the same central object.

So, it seems to me that we have the following alternative definitions for "orbital elements":

• the five parameters needed to specify a fixed keplerian orbit uniquely.
• the six parameters needed to specify a fixed keplerian orbit uniquely and also a position of an object in that orbit.
• the seven parameters needed to specify a fixed keplerian orbit uniquely and to be able to predict the position of an object in that orbit for any arbitrary date and time, assuming a particular fixed relationship between orbital size and orbital period (which implies assuming a particular central object).
• the eight parameters needed to specify a fixed keplerian orbit uniquely and to be able to predict the position of an object in that orbit for any arbitrary date and time.
• any number of sets of orbital parameters (of whatever definition) for different times, from which the position of the object can be predicted more accurately than from just a single set.

I believe that in practice "orbital elements" is used in all of these senses, and that an update of the definition in the article is in order. Any objections?

To: Mr Louis Strous

Dear Mr Strous,

Thank you for your comment. It is indeed true that an orbit would be fully determined by only five elements, were it defined as a locus of points through which the point mass peregrinates. The long-established convention, however, has it that the notion of orbit embraces both the geometric locus and the initial condition. The origin of this convention comes from the fact that a Keplerian orbit is a particular solution to the Newton gravity law written in some inertial frame. By choosing a nonrotating Cartesian coordinate system fixed within this frame, we can express this law with its three projections, i.e., with three differential equations of the second order. A generic solution to such a system always depends upon time and exactly *six* adjustable constants. The role of these constants may be played by the afore mentioned six Keplerian orbital elements (or by some six algebraic combinations thereof, like the so-called Delaunay elements or the so-called Poincare elements). A further mathematical investigation shows that the six Keplerian orbital elements obey a closed system of six differential equations of the first order, the so-called planetary equations in the form of Lagrange. This is another justification for the said convention of keeping the amount of elements exactly six. (See the afore cited paper from the "Astronomy and Astrophysics" journal.)

An intriguing detail about this machinery is that it is possible to choose the set of six adjustable constants so that it includes the epoch (i.e., the initial instant of time), instead of the mean anomaly at epoch. This way, for a perturbed orbit, the epoch becomes a variable "constant." In this role, it enters the (accordingly transformed) system of planetary equations. At the first glance, this trick looks very counterintuitive. However, it is often employed. When it is used, they traditionally choose the epoch to be the instant of perihelion crossing. I think this is what Lagrange did in his "Mécanique analytique."

Michael Efroimsky,

Astronomer,

US Naval Observatory

## 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.
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. AldaronT/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)

## Inspiration

A discussion on the inspiration behind the selection of the orbital elements must be mentioned Iyer.arvind.sundaram (talk) 04:21, 11 June 2010 (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)