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Unit 3 · Lesson 3-1
Frames of Reference
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Frames of Reference

Inertial and rotating frames, ECI, ECEF, perifocal and topocentric coordinates

23 September 1999, 09:00:46 UTC
The Mars Climate Orbiter fires its engine to slow into orbit around Mars. It should settle at about 226 km above the surface. Telemetry stops as the spacecraft passes behind the planet, exactly as planned. It never comes back. The review board found it had entered the atmosphere at roughly 57 km — deep inside a region no orbiter could survive. The cause was not a broken thruster or a bad line of code. Lockheed Martin's ground software reported the small course-correction impulses in pound-force seconds. NASA's navigation software consumed them as newton seconds. A factor of 4.45, applied a few hundred times over nine months of cruise. Nobody's arithmetic was wrong. Every number was correct. The numbers simply did not mean what the other team thought they meant. This is the lesson where we stop being casual. "The satellite is at [8000,2000,0][8000, 2000, 0]" is not a fact. It is not even a claim. It is an incomplete sentence until you finish it: relative to what origin, along which axes, at what instant, in what units. Space has no up, no north, no fixed marker of any kind. Whatever reference you use, you put it there — and so did the other team.
Watch first · presented by Shoaib Aziz
The numbers in this film are the labs' own: at 12:00 UTC on 1 January 2024 Greenwich has turned 280.6° past the equinox; a point held still in ECI keeps latitude 14.7° while its longitude falls 30° every two hours; from Kourou a geostationary satellite sits 29.5° up, almost due east, 38,658 km away; and 82 wrong-unit pushes over nine months drift the Mars Climate Orbiter about 240 km.
A reference frame is three things, and all three have to be pinned down before a triple of numbers means anything:
An origin — where [0,0,0][0,0,0] sits. Three axes — the directions, mutually perpendicular and right-handed. A time — because most useful frames are moving, so "which axes" depends on when you ask.
A map is not the territory it represents, but, if correct, it has a similar structure to the territory, which accounts for its usefulness.
– Alfred Korzybski, Science and Sanity, 1933
💡Inertial versus rotating frames
An inertial frame is one in which Newton's laws hold as written. No fictitious forces, no Coriolis, no centrifugal terms. F⃗=ma⃗\vec{F} = m\vec{a} and nothing else. Every equation you have derived in this course — the two-body equation, conservation of h⃗\vec{h}, conservation of ε\varepsilon — is true only in an inertial frame. Write r⃗¨=−μr⃗/r3\ddot{\vec{r}} = -\mu\vec{r}/r^3 in a rotating frame and it is simply false; you have to bolt on Coriolis and centrifugal terms, and the tidy conservation laws stop being tidy. A rotating frame is one that spins with something — usually the Earth. It is enormously convenient (it is where cities are) and dynamically treacherous (it is where the equations break). Every frame in this lesson is one or the other, and the whole practical craft of astrodynamics is knowing which frame you are in at any moment and how to get out of it.
ECI — Earth-Centred Inertial. Origin at Earth's centre. Axes fixed relative to the distant stars, so they do not rotate with the planet. This is where physics happens, and where every propagation in this course runs.
Figure 3.1
ECI — inertialaxes locked to the starsx̂ → ♈ẑω⊕satelliteECEF — Earth-fixedaxes turn with the crustx̂ → Greenwichẑω⊕satelliteone satellite, one instant — the numbers differ only because the axes do
Same instant, same satellite, two frames. ECI axes stay locked to the stars while Earth spins inside them; ECEF axes are bolted to the crust and turn with it. The satellite's ECI position changes smoothly; its ECEF position also picks up the 15°/hour of Earth's rotation, which is the entire reason ground tracks drift west.
The z^\hat{z} axis is easy — Earth's rotation axis, pointing north. But which way is x^\hat{x}? There is nothing out there to point at.
The answer is one of the more elegant tricks in the subject. Earth's equatorial plane and Earth's orbital plane around the Sun (the ecliptic) are tilted 23.44° apart, so they intersect along a line. Point x^\hat{x} along that line, in the direction the Sun appears to cross going north — the vernal equinox, the moment spring begins in the northern hemisphere.
We have manufactured a direction out of the intersection of two planes, using nothing but geometry.
Figure 3.2
equatorecliptic♈23.44°Two planes that are notparallel meet in a line.That line is a direction,not a place — and it ishow x̂ is defined.
The vernal equinox direction, ♈, is where the ecliptic crosses the equator going north. It is a direction, not a place — an arrow with no origin, valid anywhere in the solar system. Its symbol is the ram, because two thousand years ago it pointed into Aries. Precession has since dragged it into Pisces, which is why frames carry a date.
⚠What "fixed stars" means
Earth's axis wobbles like a slow top, completing one circuit every 25,772 years. That is precession of the equinoxes, and it means the vernal equinox direction moves about 50 arcseconds per year — around 1.4° per century. Layered on top is nutation, an 18.6-year nod of about 9 arcseconds driven by the Moon. So "ECI" is not one frame. It is a family, and each member carries a date: J2000 / EME2000 — axes frozen at 12:00 TT on 1 January 2000. The classical workhorse. GCRF — the modern replacement, tied to quasars via the International Celestial Reference Frame rather than to Earth's axis at all. Agrees with J2000 to about 20 milliarcseconds. TEME — True Equator Mean Equinox, an awkward intermediate frame nobody would choose, except that it is what TLEs and SGP4 output, so everyone has to handle it. Mix two of these up and you get errors of a few kilometres. Not mission-ending, usually. Merely wrong in a way that takes weeks to find.
ECEF — Earth-Centred, Earth-Fixed. Same origin, but the axes are riveted to the crust: x^\hat{x} through the Greenwich meridian at the equator, z^\hat{z} north, y^\hat{y} completing the set. Longitude and latitude live here. GPS coordinates are ECEF.
ECEF rotates with Earth at ω⊕=7.2921159×10−5\omega_\oplus = 7.2921159\times10^{-5} rad/s, so converting between the two is a single rotation about z^\hat{z} through the Greenwich Mean Sidereal Time angle θg(t)\theta_g(t):
r⃗ECEF=R3(θg) r⃗ECI=[cos⁡θgsin⁡θg0−sin⁡θgcos⁡θg0001]r⃗ECI\vec{r}_{ECEF} = R_{3}(\theta_g)\,\vec{r}_{ECI} = \begin{bmatrix}\cos\theta_g & \sin\theta_g & 0\\ -\sin\theta_g & \cos\theta_g & 0\\ 0 & 0 & 1\end{bmatrix}\vec{r}_{ECI}

The whole ECI→ECEF conversion, ignoring polar motion and precession. θ_g is how far Greenwich has turned past the vernal equinox at this instant.

💡A day is not 24 hours
Earth turns 360° in 23 h 56 m 4.1 s — the sidereal day, measured against the stars. The 24-hour solar day is longer because Earth has also moved about 1° along its orbit, so it must turn an extra 1° to bring the Sun back overhead. That four-minute difference is not a footnote. Use 24 hours where you should have used 23 h 56 m and your ground track is wrong by 1° of longitude per day, drifting steadily. Over a year it accumulates to a full lap. It is also why geostationary altitude corresponds to a sidereal day: a GEO satellite must match Earth's rotation against the stars, not against the Sun.
Two more frames, each earning its place by making one specific job trivial.
Perifocal (PQW). Origin at the focus, P^\hat{P} pointing at perigee — that is, along e⃗\vec{e} — and W^\hat{W} along h⃗\vec{h}. In this frame the orbit is flat, zz is always zero, and the position is just the orbit equation written in Cartesian form:
r⃗PQW=p1+ecos⁡ν[cos⁡νsin⁡ν0],v⃗PQW=μp[−sin⁡νe+cos⁡ν0]\vec{r}_{PQW} = \frac{p}{1+e\cos\nu}\begin{bmatrix}\cos\nu\\ \sin\nu\\ 0\end{bmatrix}, \qquad \vec{v}_{PQW} = \sqrt{\frac{\mu}{p}}\begin{bmatrix}-\sin\nu\\ e+\cos\nu\\ 0\end{bmatrix}

All of Unit 2, in the frame that makes it look easy. Everything is two-dimensional because the frame was chosen to make it so.

Topocentric horizon (SEZ). Origin at an observer on the ground, S^\hat{S} south, E^\hat{E} east, Z^\hat{Z} straight up. This is the frame of a person with an antenna, and it answers the only question a ground station actually asks: which way do I point, and how far away is it?
el=arcsin⁡ ⁣(ρZρ),az=atan2⁡ ⁣(ρE, −ρS)\text{el} = \arcsin\!\left(\frac{\rho_Z}{\rho}\right), \qquad \text{az} = \operatorname{atan2}\!\left(\rho_E,\, -\rho_S\right)

Elevation and azimuth from the SEZ components of the range vector ρ. Elevation below zero means the satellite is under the horizon and the pass is over.

Figure 3.3
R₃(θg)R₃(−Ω)R₁(−i)R₃(−ω)translate + rotate to siteECIorigin: Earth centrex̂ → vernal equinox, ẑ → polewhere the physics is trueECEForigin: Earth centrex̂ → Greenwich, ẑ → polewhere latitude and longitude livePerifocalorigin: the focusP̂ → perigee, Ŵ → hwhere the orbit is flatSEZorigin: the ground stationŜ south, Ê east, Ẑ upwhere the antenna points
The four frames side by side, with the transformation that links each pair. ECI is where the physics runs, ECEF is where the ground is, perifocal is where the geometry is simple, and SEZ is where the antenna lives. Real operations move between all four constantly.
🛰A timing error: the 1991 Patriot failure
25 February 1991, Dhahran. A Patriot missile battery failed to intercept an incoming Scud, which struck a barracks and killed 28 American soldiers. The cause was time, not space, but it is the same failure. The system tracked time as an integer count of tenths of a second, then multiplied by a 24-bit truncated binary representation of 0.1 — a number that does not terminate in binary. The error was 0.000000095 seconds per tick. The battery had been running for 100 hours, so the accumulated clock drift was 0.34 seconds. A Scud travels 1.7 km in 0.34 seconds. The radar looked in the wrong place and found nothing. A frame error and a clock error are the same species of mistake: the numbers were fine, and the meaning was not. In this subject, the units, the frame and the epoch are the physics.
↓But there is a problem
Fix a frame, and now a satellite can be described unambiguously. The obvious way is the state vector: three numbers of position, three of velocity, six in total. But a state vector is a terrible way to describe an orbit. Every one of its six numbers changes continuously. Read it out and you know where the satellite is this second and essentially nothing about what it is doing — you cannot see whether the orbit is circular, which way it is tilted, or how long it takes to come round. There is a second way to package the same six numbers, chosen so that five of them do not change at all and only one advances with time. Those are the classical orbital elements, and they are how every satellite on Earth is actually catalogued. Six in, six out. And the reason it is exactly six — not five, not seven — is the next thing to understand.