# Description of Orbits and Ephemerides

## Background

Since objects in the solar system are constantly moving with respect to (and gravitationally pulling on) one another, a reference frame and system of time-keeping must first be adopted to describe and organize all this motion.

We adopt the International Celestial Reference Frame (ICRF), a three-dimensional cartesian (x,y,z) system whose three basis axes are defined relative to over 4500+ remote radio sources, mostly quasars. These objects are so far across the universe from Earth, they have no apparent angular motion themselves, so serve as fixed beacons in the sky.

For time-keeping, we adopt the Barycentric Dynamical timescale (TDB), a uniform free-flowing coordinate timescale in the solar system barycentric frame consistent with General Relativity (GR). This timescale is the independent variable in the numerically integrated second-order relativistic differential equations of motion. All other coordinate systems and timescales used here are derived as conversions from these fundamental position (ICRF) and time (TDB) systems.

## Physics

A characteristic of physical objects that have mass is that they move toward each other. In General Relativity, this can be viewed as mass deforming four-dimensional spacetime such that objects are “rolling down-hill” toward each other in four dimensions, producing an effect we see in three dimensions and call “gravity”. Picture a bowling ball and baseball rolling on a rubber sheet suspended in space, each deforming the sheet, but with the sheet having four dimensions instead of three. (Good luck with that!)

If the objects have enough existing momentum in a direction not toward the other object, they frictionlessly roll around each other instead of colliding, “orbiting” each other. If one object is much more massive than the other (like the Sun), it is the small object that does most of the moving and is said to be “orbiting” the massive object, though both are moving relative to the center of mass of the system.

Unlike a rubber-sheet, the geometry of spacetime does not have friction, so energy and momentum is conserved, not dissipated, and the motion continues undiminished unless acted on by an external force.

When there are many such masses moving around the solar system, they each deform four-dimensional spacetime to produce a combined, time-varying gravitational slope in 3-D space that does not exactly repeat.

For this “n-body” perturbed situation of the real world, it is generally not accurate to refer to an object’s “orbit” as if it is a fixed thing, a noun, repeating over and over like a train on a track. While an object can be “orbiting” as a process (a verb), it actually has a “trajectory”, an open-ended path that does not exactly repeat, because of all the other masses constantly changing the gravity gradient.

## Orbital Elements

It is often convenient to discuss an object’s osculating orbital elements as if they define “the” orbit, but this is a classical concept from the 16th century, prior to perturbation physics. Such orbital elements are now primarily used as a way to encode the object’s position and velocity at a single instant in a geometrically useful way, in terms of instantaneous inclination, eccentricity, or orbit period, among other parameters. These things are actually changing at the ~5 significant figure level all the time but, given the exact instant they are valid (the “epoch”), can serve as the starting point for a fully perturbed trajectory propagation by numerical integration.

## How it is Done

Estimating trajectories in the solar system to find where an object has been, is, and will be, with a quantified level of confidence, is done through a statistical process which works like this:

The best known physics is used to predict what future measurements will be. Actual measurements are then considered. The difference between actual measurements and predicted measurements is used in a mathematical process to improve the physical model and the location and motion of the object in three-dimensional space.

The process is then repeated: the new physical model of forces acting on the object is used to generate new predicted measurements, then new measurements are obtained, and again the differences are used to improve the physical force model.

Measurements can be of many types, but mostly include the position of an object relative to background stars (RA and Dec angles), radar measurements (time-delay and Doppler-shift), VLBI, spacecraft radiometric tracking data, occultations, or meridian transit timings, among others.

Over time, this method iteratively converges on a physical force model and trajectory solution that best describes all the measurements, assuming continuity of the underlying physics, but also quantifies the uncertainties in those best-fit solutions.

Since large, well-observed objects in the solar system like the planets have been through many thousands of rounds of such interative improvement, and use comparisons against measurements spanning many centuries as well as modern high-precision spacecraft tracking, the trajectory of the planets can be accurately predicted over many thousands of years, even corresponding with records in ancient clay tablets.

The situation is different from the notorious problem of predicting the weather, since solar system objects are fairly isolated from each other, with well-defined gravitational physics driving the motion, and having limited correlations to other physical processes even as the necessary measurements are possible to obtain.

## Ephemeris Development

A representation of an object’s position and velocity over time, or coordinates derived from that, is called an “ephemeris”. Historically, the term “ephemeris” referred to a printed table of position coordinates at discrete instants, but is now extended to modern computational concepts like representations in time-continuous polynomial data-files.

Given these underlying trajectory solutions, many ephemeris products can be derived that express the basic position and velocity data in different ways convenient to specific purposes.

At JPL, ephemeris development is grouped into three interrelated categories:

#### Planetary Systems

The planetary ephemeris (PE) development provides simultaneous solution and perturbed trajectory files for the large planetary-system barycenter masses, Earth’s Moon, and the Sun. It also defines the reference frame and timescale used for all other products, with the reference axes being as closely aligned with the ICRF as possible.

#### Natural Satellites

Natural satellite ephemerides for the nearly 200 moons of other planets, are developed using separate software and a planetary ephemeris. This process also produces the baryentric shift vector that gives the time-varying motion of the planetary mass offset relative to the system barycenter of the planetary ephemeris.

Natural satellite trajectories can be usefully determined over decades and centuries, but not over as long a time-span as the planetary system barycenters. This is due to weaker observational datasets and more correlated dynamics (the satellites being close to a large primary mass).

#### Small Bodies

Small-body ephemeris development is handled by another separate process that estimates their trajectories from measurement data, given the planetary ephemeris and gravitational perturbations from major asteroid perturbers. Radiation pressure and outgassing dynamics are considered for some of the objects if the astrometric measurements warrant it.

Because of the large number of small bodies, their orbit solutions are stored compactly in a database as osculating orbital elements at one epoch, the geometrically meaningful form of position and velocity at a single instant. For those few cases requiring a non-standard dynamical model, information on that force model is also stored. When a full trajectory is needed, those initial conditions are then propagated to other times by numerically integrating those initial conditions using the appropriate dynamical model of forces acting on the object.

Larger, well-observed “small-bodies” in the Main Belt can be usefully predicted over thousands of years like the planets. However, a large percentage (30-40%) of the objects currently have sparse measurement histories and can only be predicted for weeks or months around their discovery time. Predictability for most small-bodies lies somewhere in between those extremes, and we provide the tools and information here to quantify the uncertainties for each small-body