# Calculation of Osculating Elements from State Vectors in Horizons

I figured out that the osculating semi-major axis of the planet orbits in Horizons is calculated from the observed/computed state vectors by assuming the kinetic energy as given by the observed velocity vector (i,e. by the disturbed orbit), but the potential energy as given by the gravitational field of the sun only i.e. the undisturbed orbit. The osculating semi-major axis is then calculated from the total energy (kinetic + potential energy) via the classical relationship for the gravitational 2-body problem.

Does anybody know what the philosophy behind this is? Would it not be more meaningful and consistent to calculate the potential energy including the effect of the other bodies in the solar system as well?

Of course, one can in principle 'encode' the observations through any scheme one likes, but this may then have no more value than using Ptolemy's theory of epicycles to represent the orbits of the planets. The point is that the Keplerian elements are still nowadays frequently used in quantitative astronomy, especially when discussing secular (long-tern) changes of orbits. And even when averaging over long times, the gravitational potential in the solar system will be different from that of the sun alone, so the orbits won't be given by the classical 2-body equations anymore.

To further clarify my point:

My question is about the way the osculating elements (and the mean orbital elements derived from this) are calculated in Horizons, namely, as I found, by considering only the main mass (e.g. the sun) when computing the gravitational potential (from the measured x,y.x positions) but ignoring the gravitational potential of the other masses (e.g. other planets).

Including instead also the gravitational potential of the disturbing masses would (negatively) increase the overall gravitational potential energy of the system and therefore also the total energy. This in turn would reduce the semi-major axis and orbital period by a significant amount. Considering that the semi-major axis is identical to the average distance between the masses in the two-body problem, it should, like the orbital period, be however an objective and unique quantity, and there should therefore be only one unique way of calculating these from the state vectors.

So in my view it is more than just a question of being useful or not.

• – uhoh
Aug 22 '20 at 22:00
• It's possible that answers to those may shed some light on this, but I think a thorough new answer about "the philosophy behind this" would be great!
– uhoh
Aug 22 '20 at 22:38
• What you are proposing in the edit does not make sense, which is why no one does this,. Aug 23 '20 at 17:50
• @DavidHammen I would be interested to know why you think it does not make sense to include the disturbances by the other planets both for the calculation of the potential and kinetic energy rather than only for the latter Aug 24 '20 at 7:30

Does anybody know what the philosophy behind this is?

Osculating elements are easily calculated from position and velocity, and are sometimes useful. At some times and under some circumstances they are very useful. And that is it.

On the other hand, given an Earth-centered inertial position and velocity of an object in low Earth orbit, a solar system ephemeris can be used to convert those Earth-centered inertial position and velocity vectors to Neptune-centered inertial coordinates. The very same equations that are used to compute Earth-centered osculating orbital elements could then be applied to the Neptune-centered position and velocity to yield Neptune-centered osculating orbital elements. The resulting elements would be absolutely meaningless.

Would it not be more meaningful and consistent to calculate the potential energy including the effect of the other bodies in the solar system as well?

Yes and no. Alternatives to the (somewhat) easily calculated osculating orbital elements do exist. A very widely used example is the two line element set used to describe an object orbiting the Earth. Despite having names that match up very nicely with those of classical (i.e., osculating) orbital elements, the elements in a two line element set are not osculating orbital elements.

Two line elements are one kind of mean orbital elements. The intent is to account for features that are not considered in calculating osculating orbital elements such as atmospheric drag, a non-spherical Earth, and perturbing effects from the Moon and the Sun.

Over the past few centuries, several concepts of mean orbital elements (as opposed to osculating orbital elements) have been developed to describe objects that orbit the Sun. Mean orbital elements for objects orbiting the Earth are much more recent. The rationale is that these mean orbital elements, whether used to describe objects orbiting the Sun or orbiting the Earth, are somewhat easy to use for predicting where a body will be at some point in the future and are much more closer to truth than would be the predicted state from using osculating elements.

The downside is that, like osculating elements, the usefulness of mean orbital elements is time-limited. Over time, the deviations between the predictions from an orbital element set (osculating or mean) and reality will eventually render the predicted state meaningless.

• Thanks for your reply. I am aware that the osculating elements only have a time-limited meaning (in fact, strictly speaking they refer to that particular point in time only to which the corresponding state vectors refer). Aug 23 '20 at 16:00
• continued from above.....However, my question is not about the applicability of the osculating elements but the way they are computed. Please see my edited original post for more on this. Aug 23 '20 at 16:12