Answers

Orbital elements describe a conic (most commonly an ellipse) in inertial space. They also describe an object's state (equivalent to its Cartesian position and velocity) at a specific epoch. Typically orbital elements are used to express an object's osculating orbit (an orbit tangent to and approximating the actual orbit) at the specified epoch. On this site, we use both Keplerian elements and so-called comet elements.

Keplerian elements are eccentricity, semimajor axis, mean anomaly, inclination, longitude of the ascending node, and argument of perihelion. In some cases, longitude of perihelion is used instead of the argument of perihelion. Comet elements are eccentricity, perihelion distance, time of perihelion passage, inclination, longitude of the ascending node, and argument of perihelion.

Osculating orbital elements are often used in two-body propagation to estimate a body's state (position and velocity) at some time other than the epoch. It is important to realize that for some bodies, especially planetary satellites and comets, that such estimates may be grossly in error with respect to the actual orbit. In general, the farther away in time from the epoch, the greater the error.

A complete description of orbital elements and their use in celestial mechanics is beyond the scope of this web site. More information can be found in a number of texts. See the FAQ entry below for some relevant books.

Original reference: IAUC 7555, January 2001

• "Explanatory Supplement to the Astronomical Almanac", ed. P. K. Seidelmann, 1992, University Science Books.
• "Fundamentals of Astrodynamics", R.R. Bate, D.D. Mueller, J.E. White, 1971, Dover Publications, Inc.
• "Fundamentals of Celestial Mechanics", J.M.A. Danby, 1992, Willmann-Bell.
• "Methods of Orbit Determination for the Micro Computer", D. Boulet, 1991, Willmann-Bell.
• "Orbital Mechanics", J.E. Prussing, B.A. Conway, 1993, Oxford University Press.
• "Orbits for Amateurs with a Microcomputer", D. Tattersfield, 1984, Halsted Press.
• "Spherical Astronomy", R. M. Green, 1985, Cambridge University Press.
• "Vectorial Astrometry", C.A. Murray, 1983, Adam Hilger Ltd.

The astronomical unit (AU) may be defined as the distance from the Sun that a massless particle would require for a circular, Keplerian orbit with an exact period of 2*pi/0.01720209895. The denominator, denoted by "k", is the "gaussian constant", and its value, in radians/day, comes from a measurement of the earth's mean motion taken back in the 19th century when it was far easier to measure the orbital period of the earth than it was to measure its distance from the Sun.

In such an idealized (circular, Keplerian) orbit, the AU would be equal to the semi-major axis of the orbit and would also be equal to the mean distance of the particle from the Sun. However, neither is exactly the case. With an eccentric orbit, the mean distance is not equal to the semi-major axis, a; instead, it is equal to a*[1+(e^2/2)], where e is the eccentricity. Furthermore, the earth's motion is not Keplerian; only approximately so. Thus, the orbit is not periodic and there is no exact semi-major axis nor is there an exact mean distance. A simple analogy to this is given in the FAQ concerning exact values.

To have an exact value, a quantity must be either strictly constant, or else, exactly periodic.

The orbits of the planets are only approximately elliptical; their motions are only approximately periodic; not exactly. Therefore, it doesn't make much sense to ask questions about "exact" Keplerian (elliptical) elements.

A simple analogy would be to take a pencil and draw a free-hand circle on a piece of paper, going round-and-round a number of times. Then ask, "what is the EXACT radius of that circle?"

It is impossible to give an answer; the curve that you have drawn is not exactly a circle.

One may define an "osculating" radius, for example: the radius of curvature at any given point on the curve. However, this value is exact at that given point only. The value will change for a different place on the curve; or, if averaged over some portion of the curve; or, if averaged over some other portion of the curve.

Which result gives the "exact" answer? None; there is no "exact" radius for the curve.

It's a whole different situation with the JPL ephemerides. We do not use things such as periods, eccentricities, etc. Instead, we integrate the equations of motion in Cartesian coordinates (x,y,z), and we adjust the initial conditions in order to fit modern, highly accurate measurements of planetary positions. As a result, we are able to produce ephemerides which are far more accurate than those based upon elliptical elements.

In the analogy above, it could be possible to measure each point of the hand-drawn curve very accurately; however, one still could not give a unique value for the curve's radius.

People used to think that the orbits of Uranus and Neptune could not be properly fit to the observations (measurements of their positions). Therefore, it was assumed that there was an additional planet out in the farther reaches of the solar system which was perturbing those planets' motions: i.e., Planet X. It is now known, however, that the orbits of Uranus and Neptune can be adjusted to the accuracy of the data if done properly (as in the reference below). Thus, no need for Planet X.

Over the past decade or so, a number of bodies have been found out past the orbit of Pluto. In fact, one of them is even bigger than Pluto; so, in some sense, it should be considered to be a planet. Is it "The Planet X"? No; neither it nor Pluto is close in enough or massive enough to significantly affect the orbits of Uranus or Neptune.

Standish,E.M.: 1993, "Planet X: No Dynamical Evidence in the Optical Observations", Astronomical Journal, 105, no.5, 2000-2006.