Talk:Orbital elements

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 However, its earliest sketch was offered yet by Newton in his unpublished Portsmouth papers and was briefly mentioned in his "Principia."

For more details on the orbital elements, and for historical references see: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

Louis strous

Reply

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

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Satellites

I was curious about orbital elements of satellites, doesn't look like anything on Wikipedia. I found this Satellite Orbital Parameters which seems similar but adds some "Rate of change" of perigee, ascending node among others. If there's a standard, it would be good to add somewhere here. Tom Ruen (talk) 01:01, 8 September 2017 (UTC)

Euler Angle triplet --> Rotator Matrix R math only works correctly for positive inclinations < 180 degrees

I wanted to test the math shown claiming that the generalized Rotator R can be composited by using 3 Euler angle (single-axis) transforms.

Just to be clear about assumptions: I'm assuming that the orbit path begins abstractly as an elliptical path in 2D. Periapsis is located on the +x-axis, the focus closest to it at the origin, and the apoapsis and more distant focus on the negative x-axis. Also, I'm assuming the circulation direction is standardized in the abstract 2D coordinates -- with circulation of the orbiting body always circulating CCW wrt the +z-axis.

In testing the formulation given (to convert inclination angle, longitude of ascending node angle, and argument of periapsis angle into a 3 x 3 rotational matrix R), my assumptions held up for the case of positive inclination angles < 180, but failed for negative inclination angles.

The article is vague and incomplete about the precise meanings of the 3 angles. Each of these Keplerian angles should have a number range given, and indicate how the sign of the angular measure is defined. For instance, let's take the "argument of periapsis". Clearly, a value of 0 degrees indicates that periapsis coincides with the ascension node. We might also infer that, for prograde orbits, the argument of periapsis increases in the direction of travel of the orbiting body, or CCW in the reference x-y axis system. In this case, both ways of defining positive angle increase agree. However, what if the orbit is retrograde (CW circulation in the reference x-y frame)? Does "argument of periapsis" increase positively in the direction of object travel? Or, is the sign of the angle based on angular measure in the x-y frame? These two definitions conflict. So, the article is incomplete to fail to mention, for each angular element, how positivity is determined for measuring the angle.

It's difficult to critique this article's math in words only. When with complex 3D geometries, graphics are very helpful.

I'm wondering if people agree that the math in the article should be tested with software, and proven to "work" correctly with ALL possible cases of orbits? My assertion is that, since an authoritative article will be taken as a software math spec, then it should address all possible cases of input (every possible elliptical orbit path). — Preceding unsigned comment added by Pbierre (talk • contribs) 22:44, 11 December 2017 (UTC)

Continuing my software testing of the "Euler angle --> R" math published in the article, my primary contribution to clarifying the topic is to point out that, the Keplerian Elements (3-angle) spec for the orbit's orientation DOES NOT convey circulation direction. Every unique 3D heliocentric ellipse represents two possible orbit paths, representing the two opposite circulation paths. I'm assuming that circulation direction is NOT part of the Keplerian orbit elements because almost all orbits found by solar system astronomers were prograde (CCW circulation wrt solar North direction vector). However, any modern treatment must accomodate satellite orbits, which may be chosen to run in retrograde circulation, with a full 180 range of inclinations between the polar orbits.

The use of a 3x3 Rotator matrix is able to encode both the unique 3D orientation of the ellipse, but also the circulation direction, and therefore, posits a better standard means of specifying the "3D spatial orientation" parts of an orbit. To give better intuitive meaning, the 3x3 Rotator can be simplified by specifying 2 of its 3 axis directions:

   Plane of orbit normal (unit vector), chosen so that body circulation runs CCW to this vector.  (This is the z-axis of R)
   Direction of periapsis (unit vector). (This is the x-axis of R)

The elegance of this style of orbit specification is that it provides a 1:1 representation (there is only one numerical representation for one distinct orbit), it is a complete system (all orbits are representable), and it offers continuity (orbits very close to one another have numerics very close to one another).

I believe the practical utility of this article could be improved by stating explicitly that the Keplerian angles are not adequate to differentiate between prograde and retrograde orbit paths sharing the same ellipse geometry. The restriction of inclination angle to 0..180 degrees would make this point clear, and steer software developers away from attempting to extend inclination angle to 180..360 (or negative values (0..-180), as a means of encoding circulation direction. As the heading of my post states, the Keplerian system of specification doesn't work toward this purpose. Pbierre (talk) 17:44, 12 December 2017 (UTC)pbierrePbierre (talk) 17:44, 12 December 2017 (UTC)

Rotational elements?

Is anyone planning on creating an article like this one but about rotational elements? ➧datumizer  ☎  20:39, 20 September 2018 (UTC)

Stellar systems

This article, Orbital_elements#Keplerian_elements and Orbital plane (astronomy), don't describe their definition in regards to stars like the stat table for Sirius. As best I can tell those coordinates are defined as if looking down Ecliptic south axis, so inclination 0 is orthogonal ccw, and inclination 90 is perpendicular to view, and 180 is orthogonal cw orbits. Tom Ruen (talk) 02:40, 18 February 2019 (UTC)

Greek letters

Ω is commonly used for longitude of the ascending node, ν or θ is commonly used for true anomaly, and ω is commonly used for argument of periapsis.

My question is, how common is it for ι (iota) to be used for inclination?

There is evidence for this at Greek letters used in mathematics, science, and engineering#Ιι (iota). Maybe it should be mentioned in the Orbital Elements article. 214.6.80.22 (talk) 22:27, 29 October 2019 (UTC)

Feedback needed on my rework

I just published my rework of this article. The main thing I changed was including many more elements, along with sections dedicated to describing the mathematical relations between them. I wanted to not give preference to any particular set of elements, and instead describe all parameters as equally valid ways of describing an orbit.

The first thing I want feedback on is my change to how orbital elements are defined. Previously, the article stated only six elements are needed (or seven if epoch is included), but I changed this to eight elements to include all parameters needed to find the shape of an orbit and the position of an object at a given time along that orbit, even if some of these values (such as the gravitational parameter and epoch time) are traditionally not considered as being orbital elements. I feel this is the best way of doing this, as it means all values used to describe an orbital trajectory can be described equally, and it makes it clear that elements like μ are equally valid ways of describing orbital motion.

My first question is this: Is this broader definition of orbital elements okay, or should the article describe the more typical six element view?

The second thing I want feedback on is my grouping of orbital elements. I state that "When describing an orbit with orbital elements, typically two are needed to describe the size and shape of the trajectory, three are needed describe the rotation of the orbit, one is needed to describe the speed of motion, and two elements are needed to describe the position of the body around its orbit along with the epoch time at which this occurs." and these 4 groupings are referred to throughout the article. These groupings of "2 size and shape, 3 orientation, 1 motion, and 2 (or 1) epoch" are not something I personally have seen described in any great detail, and they are mostly my personal analysis; however, I felt it was the best way to explore and explain the topic in the most generalized way.

My second question is this: Are these groupings okay to include in the article, even if they are my own personal analysis, or should they be revised?

I tried my best with this, but I may have gotten things wrong! So if you have any additional concerns, please do tell. Have a good day :3

Fusion (talk) 23:18, 22 February 2025 (UTC)

orbital element systems missing

There are many different ways to display the 6 parameters. Some of them mostly used in the FreeFlyer software, which is still relevant since its used by e.g NASA. I would love a section that explains for example Nonsingular Keplerian that removes singularities in the edgecases of eccentricity=0. SGP4 also uses a slightly different representation of a orbital vector. A possible good surce for this would be Fundamentals of Astrodynamics and Applications, fifth edition, David A. Vallado 83.229.93.117 (talk) 08:01, 20 October 2025 (UTC)

Removal of the 'Euler-angle transformations' section

On 27 December 2025 this subsection, containing transformations between the orbit-frame and the reference frame, as well as explanations of how the main orbit angles are indeed Euler angles was removed because "its content is not relevant to this article". I would like to argue for its re-inclusion. Explicit constructions of normal vectors to a given orbit are profoundly useful, not (to my knowledge) present elsewhere on Wikipedia, and are a natural desire to have when working with orbital elements. I have referred several students to this section in the past for easy reference and return to it regularly. Moreover, such a concise and detailed discussion isn't present in Murray & Dermott, making its inclusion here even more useful (the best you get is Eq 2.119-2.121 which will get you there but lack the explicit descriptions present here that make this section good reference material). I really do think that explicit relations to how the orbital elements relate to orientations in Cartesian coordinates is relevant to the article and deserves to be reinstated here, just like how conversions to Cartesian coordinates are relevant to the article on spherical coordinates. KuriGohan and Kamehameha (talk) 22:17, 22 January 2026 (UTC)

Details of this topic are over my head but a quick look at the section before it was removed makes me think that retaining it would require a short paragraph explaining its purpose for Orbital elements, preferably with a reliable source. The original is available at permalink. Johnuniq (talk) 00:42, 23 January 2026 (UTC)

In my opinion, the purpose of this article is to explain what Orbital Elements are, which ones exist, and give a basic overview of how they are used. This article has a fairly broad scope, and as such I believe it should not be overly technical and bloated with complex mathematics. More detailed information about formulae used in Celestial Mechanics should be delegated to other articles. For this case in particular I would suggest the content in The 'Euler-angle transformations' section be moved to Kepler Orbit, a much more technical article more focused on the mathematical details. (Actually, the content was originally from Kepler Orbit, before an editor moved it to this article back in 2008! [Original content in Kepler orbit]).

I do apologize for removing it outright instead of attempting to discuss moving it elsewhere beforehand, but I still don't think the section has much of a place in this article. I based my decision to remove it both on my own opinion of its relevance to this article, and on this ancient post on this article's talk page from an editor also concerned with the section's inclusion.

I do certainly believe that the section has utility though! And actually solving for the normal direction of an orbit from its elements is actually a problem I've been looking for a solution to! Although, I did not even fully understand that was what this section did until you just described it. But the fact that I did not even realize the function of the section despite having looked it over several times is likely just further evidence that it is too technical for this article.

I'll look into starting a discussion on Talk:Kepler orbit over potentially adding the 'Euler-angle transformations' back over there. Fusion (talk) 02:51, 24 January 2026 (UTC)

I feel like this information doesn't really belong on Kepler orbit. That article is about the fundamental physical object of the orbit. You'll note that beside parameters fundamental to the shape of the orbit like eccentricity, the other orbital elements only very, very rarely appear in that article. The conversions and vectors we're talking about are entirely products of one's choice of coordinates to represent this object. The most common are the Keplerian elements, so it makes sense to have a section describing how the Keplerian elements orient orbits in Cartesian space.

Moreover, this section actually provides a mathematical formula for the Inclination, Longitude of Ascending Node, and Argument of Periapsis down at the bottom, so the contents of the section is a two-way conversion between Cartesian and Keplerian. It feels very strange to have in our hands a formula for calculating the orbital elements the Orbital Elements article is about and choose not to include them.

Perhaps renaming the section to something better conveying the conversion that's taking place would make the relevance more clear? That would more better emphasize the easily missed value of the section. KuriGohan and Kamehameha (talk) 20:00, 27 January 2026 (UTC)

After thinking it over a bit more I think I have a proper solution here. I do agree that Kepler Orbit might not be the best place to put this. Still, it does not seem that Orbital Elements is the correct place either. The section about deriving the orbital elements is especially beyond this article's scope, since that topic is covered extensively in both Orbit Determination, and the articles for the specific elements (i,Ω,ω), which are wiki-linked in this article such that readers may find that information there.

However, I do agree that topics about specific applications for Orbital Elements and how to derive Orbital Elements do need to be covered in this article. Although, I believe this should take the form of sections that give general overviews (without formulae) of the subject, with links to other articles that cover the subject in greater detail and with specific formulae. Specifically, the article Perifocal coordinate system does contain a section that, while not exactly the same, could be referenced to fulfill the duty that the 'Euler-angle transformation' section did (the section in that article could also be fully replaced with the 'Euler-angle transformation' section if needed). This approach should make this article more approachable and understandable to the average reader, while also allowing more expert readers like yourself to access the finer details.

I think this approach could be a fair compromise, but if you have any further concerns, please do share. Fusion (talk) 20:38, 28 January 2026 (UTC)