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 However, its earliest sketch was offered yet by Newton in his unpublished Portsmouth papers and was briefly mentioned in his "Principia."

For more details on the orbital elements, and for historical references see: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

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)

Perhaps that would improve the article. If you have a source, please do add such a discussion. N2e (talk) 14:44, 28 December 2014 (UTC)

File not found Ice.Queen (talk) 04:33, 6 February 2017 (UTC)

Please clarify the role of mass among the orbital elements

According to this article, the Keplerian elements include neither the masses nor the gravitational parameters of the bodies involved in the orbit. Are these properties assumed to be known? If so, the article should mention that, because it seems very important to an understanding of the orbital elements.

Some of the paragraphs in the "Alternative parameterizations" section make special note that the gravitational parameter is required to derive some orbital parameters. For instance: "It is common to specify the period instead of the semi-major axis in Keplerian element sets, as each can be computed from the other provided the standard gravitational parameter, GM, is given for the central body." If there is a presumption that the masses are known, then isn't this qualification redundant? On the other hand, if there is no presumption that the masses are known, then mustn't similar caveats be added elsewhere? For instance, instead of saying "Keplerian elements can be obtained from orbital state vectors", mustn't we say instead that "Keplerian elements can be obtained from orbital state vectors, provided the gravitational parameters are known"?

On a related note, the section "Orbit prediction" seems like a great place to clarify whether or not a set of orbital elements contains the information required to make orbital predictions. (Also, nitpicking, mu is used as a symbol for the gravitational parameter in the formula for mean motion in that section without having been defined elsewhere in article.)

184.187.180.7 (talk) 06:52, 18 June 2015 (UTC)

You raise some good points. And you are right: the mass (or, more directly, gravitational parameter) of the central body (or of the bodies for the more general two-body problem) is required in order to make orbit predictions. I feel (but I don't work in this field) that it is not traditionally mentioned because 1) it is so crucial in calculating anything related to orbital motion that it's generally assumed to be a known quantity, and 2) it is the same for all bodies moving in a given central field, so including it as part of the orbital elements of every body would be redundant. Without it, the orbital elements only specify the motion up to "time scaling" (you can predict any past or future position of the body in question in time units where t = 1 is the orbital period, whose value is directly tied to the gravitational parameter).

So yes, I agree: perhaps the article should include an explicit mention that all discussion about orbits usually assumes the gravitational parameter of the attracting body is known.

Regarding the section "Alternative parametrizations", I don't think it's redundant: indicating that the set of elements is not unique and describing some of the (fully equivalent) choices that are typically chosen is in my opinion relevant. Also, this section does mention twice that the mass is needed for some of these orbital elements conventions. A note, though: obtaining the Keplerian elements does not require knowing the gravitational parameter of the central body; the state vectors alone at a given point in time are enough. But as you may infer, without the gravitational parameter the state vectors are not enough to make predictions.

Finally, good catch in noticing that mu is not properly defined in the "Orbit prediction" section; I think this should be clarified. And this would also be a good place to include an explicit that without this quantity, mu, the elements alone are not sufficient to make predictions for arbitrary times.

Meithan (talk) 15:37, 19 June 2015 (UTC)

In Meithan's reply above and in the discussion here, I believe it has been established that the mass of the central body (or equivalent information) is required to characterize an orbit; that this information is not represented among the Keplerian elements; and that this information is generally assumed to be known (thus justifying the claim that the Keplerian elements are sufficient to define an orbit). Based on this understanding, I recommend these changes:

1) First, make it perfectly clear, both here and in the "Orbit" article, that an orbit defines a path through space and time. These articles are currently written in a way that makes it unclear whether an orbit is a purely geometric path, or a path that describes motion.

2) Edit this article to say that exactly seven parameters are required to unambiguously define an orbit, but that, by convention, the information required to establish the pace of the orbit is assumed to be given, and is therefore not included among the orbital elements, reducing their number to six.

3) In the "Alternative parameterizations" section, explain that, just as there is redundancy among the orbital parameters defining the shape, orientation, and initial condition of the orbit, there is also redundancy among the orbital parameters defining the pace of the orbit. Thus, the pace of the orbit may be established by any one of the following commonly-used parameters: Gravitational parameter of the central body, mass of the central body, period, mean anomaly, and mean motion

My knowledge in this area is still shaky, so I won't make any of these changes until I get confirmation that they are appropriate.

184.187.180.7 (talk) 16:03, 19 June 2015 (UTC)

I like your suggestions. I agree that an orbit should be regarded as the description of a time-dependent trajectory, not just of the geometric shape of such trajectory, and the article should perhaps more clearly state this fact.

Regarding points 2 and 3, I think that the "seventh" element should generally be considered as the gravitational parameter of the central body (and not really considered as part of the orbital elements, simply an extra "global" piece of information -- as I pointed out before, this "parameter" is the same for all bodies in the same central field), for it usually appears in most of the orbit equations (while other equivalent parameters, such as period or mean motion, should be considered "derived" from the it).

If you agree with these points (and if nobody else has an objetion) I'll go ahead and introduce these ideas in the article.

Meithan (talk) — Preceding undated comment added 20:17, 20 June 2015 (UTC)

Meithan, let's discuss a little more. Here are some things to think about:

This article is about orbital elements. Before I take responsibility for any change, I want to be sure I know what orbital elements are. The definitional sentence in the article says "Orbital elements are the parameters required to uniquely identify a specific orbit". Is that definition correct? Or are orbital elements any set of parameters containing equivalent information to the six traditional Keplerian elements? Given the changes we are discussing, the distinction matters.

I believe an orbit describes motion, and you seem to agree. However, I don't feel I've received strong confirmation of this. Note that earlier on this talk page, Michael Efroimsky says "The long-established convention... has it that the notion of orbit embraces both the geometric locus and the initial condition". He says nothing about motion. I don't want to read too much into that, but I do want to be cautious. I'm particularly concerned that perhaps 'orbit' is ambiguous, and in the context of orbital elements, it does not describe motion -- in which case our proposed changes are inappropriate. (See the 'Orbit' talk page for my attempt to clear this up.)

If we are right about an orbit describing motion, then seven parameters are required to characterize an orbit. Following on, it would seem appropriate to explain, in the article, the historical and semantic reasons why there are only six orbital elements. I don't know those historical and semantic reasons. If I understand you correctly, you have suggested that one reason may be that in the astronomy culture, the gravitational parameter is "generally assumed to be a known quantity" and is considered "an extra 'global' piece of information". I have no cause to doubt this reason. However, I think it would be a mistake to add it to the article without a citation or direct personal knowledge. My instinct is to focus on explaining the orbital elements correctly, and to avoid explaining why they are what they are.

184.187.180.7 (talk) 02:14, 22 June 2015 (UTC)

True/mean anomaly in initial enumeration

The initial enumeration of orbital parameters lists the "true anomaly" as the parameter to determine where on the orbit the object is. However, the very next sentence explains the "mean anomaly" as if it was a term that was previously used. The section "alternative parameterization" goes on saying "instead of the mean anomaly, ...". All of this makes it sound like someone wanted to write "mean anomaly" in the initial enumeration of parameters. — Preceding unsigned comment added by 46.19.152.35 (talk) 13:09, 16 April 2017 (UTC)

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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)