Talk:Orbital elements

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 However, its earliest sketch was offered yet by Newton in his unpublished Portsmouth papers and was briefly mentioned in his "Principia."

For more details on the orbital elements, and for historical references see: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

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

[edit] 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

[edit] ============================================================

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

[edit] 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)