# Are ECI and ECEF both frames and/or coordinate systems? Is there a difference?

I don't know really know the difference between a frame and a coordinate system. I'd proposed ECEF and ECI tags, do we need them? Frames seems to be the standard in meta but there is concern so any changes are on hold.

I'd always thought that ECI and ECEF were the same thing except one rotated and the other didn't. They were two cartesian coordinate systems centered on the geocenter with one inertial and one rotating with the planet.

But maybe it's much more complicated than that, it usually is!

Questions:

1. Are ECI and ECEF both frames?
2. Are ECI and ECEF both coordinate systems?
3. In spaceflight lingo, there a difference between a frame and a coordinate system?

Going backwards, and starting with 3. In my experience the two are used interchangeably. The Wikipedia article on frame of reference implies that coordinates are the orthogonal directions, while a frame of reference has a defined origin. The AGI help reference for STK treats the two terms as interchangeable.

So the answer to 1 and 2 yes, they are both frames and coordinate systems, at least in broad colloquial use.

By the way the difference between ECI and ECEF is more than just earth's rotation. It also takes into account precession, nutation, and polar motion.

• My trepidation comes from the use of "frame" in relativity, where we also have time to consider, but then again perhaps time is a coordinate? Also, aren't precession, nutation, and polar motion simply behaviors that "coordinate systems centered on the geocenter... rotating with the planet" would exhibit and not really separate things from them? – uhoh Nov 7 '20 at 0:58
• Most orbit mechanics problems don't worry about special relativity :D. For ECEF precession, nutation, and define the axis of rotation of the frame (coordinate system?). Polar motion defines the alignment of the physical body, and hence the coordinate frame, relative to the axis of rotation. And the UT1 offset defines the variability of the rotation rate. – Carlos N Nov 7 '20 at 1:03
• Certainly in the engineering world of shuttle ops "frame" = "coordinate system" so my experience agrees with yours. – Organic Marble Nov 7 '20 at 1:15
• The SPICE frame tutorial and Navipedia have useful definitions of frames and coordinates systems, particularly slide 4 of the SPICE tutorial. I also recommend the SOFA cookbook on Earth Attitude for precession, nutation and earth orientation. The modern way is to think of these as a matrix, changing with time, which describes where Earth's pole is right now – astrosnapper Nov 7 '20 at 1:56
• Not sure, the use of ECI and ECEF seems to be much more prevalent in the near-Earth/satellite communities - in astronomy, I've never really come across them being used, only ITRF/ITRS. But as far as tags go, I would consider eci and ecef as examples of frames. Maybe frame, frame: eci and frame: ecef tags ? – astrosnapper Nov 7 '20 at 23:54

It really depends on how pedantic you want to be.

In the most extreme phrasings, there is a frame centered on the earth which is inertial, sometimes marked as "Earth inertial" and there is a frame centered on the earth which is fixed to the geoid of the Earth, sometimes marked as "Earth fixed."

The key attribute of frames is that, within a frame, a vector always has a single given direction and magnitude. However, when one converts between frames, that vector may change. For example, the acceleration vector of an object fundamentally changes when going between the Earth fixed frame and the Earth inertial frame, accounting for the rotational effects of the Earth.

A coordinate system is a way of measuring these vectors as a string of numbers. ECI and ECEF are technically coordinate systems (or a class of coordinate systems, more on that later). To do this, one chooses a set of basis vectors. Those basis vectors may change over time in a frame, but typically we choose vectors that remain fixed in the frame for sanity reasons.

We treat these as roughly interchangeable because you need a coordinate system to write the vector in component form, and we work in component form so much that it's hard to imagine a vector without them. Vectors do exist without them. But in this day and age, with computers doing as much as they do, we almost always have vectors described in a coordinate system.

We can see this issue by looking at the class of ECI coordinate systems. There are actually many such coordinate systems. If its an inertial coordinate system, and its centered on the earth, its ECI. The most famous, of course, is ICRS, based on directions to many distant celestial bodies, but you can create an ECI coordinate system at any moment in time just by picking your 3 axes in an inertial frame and leaving it at that. As a practical example, I have worked with simulations where ECI coordinate system (at least, the one recognized by the sim) is defined to be aligned with ECEF at t=0, an arbitrary moment in time where the simulation clock happened to read 0. It was simply a matter of convenience.

Thus I could have two vectors (1, 0, 0) in ECI coordinate system A, and (0, 1, 0) in ECI coordinate system B, and have them actually be the same vector. They're just broken into components differently. A simply had a different ordering of basis than B did.

However, if I take a vector like (1, 0, 0) in ECI coordinate system A and transform it into ECEF, I have to recognize that ECI was in the inertial frame, and ECEF is in the fixed frame, so the vectors can fundamentally change in direction and magnitude. This is different from a coordinate system change, where the numeric change in components is little more than a shell game.

(ECEF could be thought of as a class, just like ECI, except that ECEF is defined with respect to the international reference pole and the international reference meridian -- aka the North pole and the prime meridian. So it truly is a coordinate system)

So in the most technical of settings, ECEF is a coordinate system, ICRS is a coordinate system, and ECI is a class of coordinate systems. None of them are frames -- Earth Fixed is a frame and Earth Inertial is a frame. However, in the practical world, they are treated interchangeably as is is rare that we find a chance to leverage the mathematical beauty of vectors without components tying them down. Indeed, it is not unusual to hear ECI and ECEF be referred to as frames. Know your audience, and use the terms in the most effective way to convey your meaning.

In the language of the International Earth Rotation Service's Technical Note 36, a "reference system" is a mathematical definition of the relationships between the coordinates, and a "reference frame" is a physical realization of a theoretical system. For example, the ITRS (International Terrestrial Reference System) is just a set of equations, while the corresponding ITRF (International Terrestrial Reference Frame) is a set of values to plug into those formulas based on a particular set of measurements of extragalactic radio sources.

In the language of pure mathematics, I would say that a coordinate system is a collection of charts (the local homeomorphisms to $$\mathbb{R}^n$$) for a differentiable manifold $$M$$, and a frame is a section of the tangent vector fiber bundle over $$M$$. The terminology used in General Relativity --- frame field and coordinates --- is a specialization of this to the case of the Lorentz group, SO(3,1).

Some work with satellites definitely does require relativity, both special and general, but I agree many applications don't. As for whether precession and nutation are part of rotation, I would have to say that while the phenomena nearly always happen together, in this terminological context they are a specific choice of coordinate system that decomposes the general motion of the frame into simpler components. Does that help?