Three rotations that put an orbit in space, and they do not commute

Two of an orbit’s six numbers describe the ellipse itself, and one says where the body has got to along it. The remaining three do something quite different: they take a curve that exists in a flat plane and put that plane into space.

Three angles for that job is exactly the right count — three is the dimension of the rotation group — but it is also the source of the one property of orbital elements that catches people out. Rotations do not commute. Applying an inclination and then a node rotation gives a different result from applying the node rotation and then the inclination, and the difference is not small.

So a set of orbital elements is not six numbers. It is six numbers and a convention, and the convention is the part that gets lost when the numbers are copied out of one document into another.

The three angles, and what each one is measured against #

The construction starts from a reference plane and a reference direction in it. For a heliocentric orbit the plane is the ecliptic and the direction is the vernal equinox; for an Earth orbit the plane is the equator and the direction is again the equinox; for a binary star it is the plane of the sky and the direction is north, which is how a spectroscopic orbit is reported.

Given those, the three angles are defined in this order:

The argument of periapsis $\omega$ is measured within the orbital plane, from the ascending node to the periapsis, in the direction of motion. It says where in the ellipse the body comes closest — which end of the ellipse points toward the node.

The inclination $i$ is the angle between the orbital plane and the reference plane, measured at the ascending node. It runs from 0° (prograde and coplanar) through 90° (polar) to 180° (retrograde and coplanar).

The longitude of the ascending node $\Omega$ is measured in the reference plane, from the reference direction to the point where the orbit crosses it going north. It says which way the tilted plane is facing.

Each is measured in a different plane from the others, and that is precisely why the order matters: the second rotation moves the plane the third one is measured in.

Why the order cannot be swapped #

The standard convention is the 3-1-3 sequence: a rotation about the third axis by $\omega$, then about the first axis by $i$, then about the third axis by $\Omega$. Written as matrices acting on a point in the orbital plane,

$$\mathbf{r} = R_z(\Omega)\,R_x(i)\,R_z(\omega)\,\mathbf{r}_{\text{plane}}.$$

Matrix multiplication is associative but not commutative, and $R_z(\Omega)R_x(i) \neq R_x(i)R_z(\Omega)$ for any non-zero pair of angles. The failure is easy to feel with a book: lay it flat, tip it 90° toward yourself, then spin it 90° clockwise about the vertical; start again and do the spin first. The book ends in two different attitudes.

For the orbit the consequence is that the last panel of the figure above is not a mistake or a rounding artefact. It is a genuinely different orbit, reachable from the same three numbers by reading them in the wrong order, and at the angles drawn its plane differs from the intended one by more than fifty degrees.

The degenerate cases, where an element stops existing #

The three angles are a coordinate system on the rotation group, and like every coordinate system on a curved space it has places where it breaks down. There are two, and they are both common rather than exotic.

Zero inclination. If the orbit lies in the reference plane there is no ascending node, because the orbit never crosses the plane — it is in it. $\Omega$ is then undefined, and so is $\omega$, which is measured from a node that does not exist. What remains well defined is their sum, $\varpi = \Omega + \omega$, called the longitude of periapsis, and it is the quantity that survives the degeneracy. Almost all the planets have small inclinations to the ecliptic, so planetary tables quote $\varpi$ rather than $\omega$ as a matter of routine.

Zero eccentricity. A circular orbit has no periapsis, so $\omega$ is undefined again, and with it the true anomaly measured from periapsis. The surviving combination this time is the argument of latitude $u = \omega + \nu$, the angle from the ascending node to the body itself.

An orbit that is both circular and equatorial — which is roughly what a geostationary satellite is trying to be — loses four of its six elements at once, and is described instead by the true longitude $l = \Omega + \omega + \nu$: one angle that is the sum of three undefined ones. This is not a failure of the physics. The orbit is perfectly well determined; it is the parameterisation that has run out of meaning. The practical form of this is not that the elements become infinite but that they become ill-conditioned. Near zero inclination, a tiny change in the observed orbit produces an enormous change in $\Omega$ and a compensating one in $\omega$, while their sum stays put. An orbit determination that reports $\Omega$ to four decimal places for a body at $i = 0.02°$ is reporting noise, and a covariance matrix that has not been transformed into non-singular elements will say so if anyone looks.

Which reference frame, and at what date #

The three angles are meaningless without the frame they are measured in, and this is where real orbital data goes wrong more often than anywhere else.

The vernal equinox — the reference direction for both the ecliptic and the equatorial frames — is the intersection of two planes that are both moving. The equinox precesses westward along the ecliptic at about 50.3 arcseconds per year, which is 1.4° per century. So an inclination or a node quoted “in equatorial coordinates” is quoted with respect to an equator that has since moved, and the numbers are only usable with a date attached.

The conventions in use are a short list and they are not interchangeable:

• J2000.0 — the mean equator and equinox of 2000 January 1.5 TT, and the frame most catalogues are on.
• Of date — the true equator and equinox at the instant of the observation, which is what a telescope actually points in.
• ICRF — the International Celestial Reference Frame, defined by the positions of several hundred extragalactic radio sources and not tied to any plane of the solar system at all. It is within about 20 milliarcseconds of J2000 and is the modern standard precisely because it does not move.

The difference between J2000 and of-date elements for a current observation is around 0.3°, which is thirty times the accuracy of a routine determination. Mixing them is not a rounding error; it is a wrong answer that looks plausible.

An inclination chosen to make the node drift on purpose #

The three orientation angles are usually treated as things to be measured. One of them is routinely chosen, and the choice is the neatest application of the whole apparatus.

The Earth is not a sphere. Its equatorial bulge adds a term to the potential that a point mass does not have, and the leading effect of that term on a satellite orbit is to make the line of nodes rotate steadily. To first order the rate is

$$\dot{\Omega} = -\frac{3}{2}\,J_2\left(\frac{R_\oplus}{a(1-e^2)}\right)^{2} n \cos i,$$

with $J_2 = 1.0826 \times 10^{-3}$ and $n$ the mean motion. The $\cos i$ is what matters: the drift is westward for a prograde orbit, eastward for a retrograde one, and zero for a polar one.

Now demand that the node drift eastward at exactly the rate the Sun appears to move along the ecliptic — 360° in a year, or 0.9856° per day. The orbital plane then keeps a fixed angle to the Sun, and the satellite crosses every latitude at the same local solar time on every pass, for years. That is a sun-synchronous orbit, and for a typical altitude of 700 km it requires an inclination of about 98.2°.

Ninety-eight degrees is retrograde. A sun-synchronous satellite goes around the Earth the wrong way, by eight degrees, and it does so for no reason connected to what it is looking at — it does so because $\cos i$ has to be slightly negative for the drift to have the right sign. Almost every Earth-observation satellite ever flown is in one of these orbits, which is why satellite imagery of a given place is always taken at the same time of morning and the shadows in it always fall the same way.

What was actually measured #

None of these three angles is observed. Every one of them is inferred from positions, and the inference is the subject of orbit determination.

The classical result is that three complete observations determine an orbit. Each observation of a solar-system body gives two numbers — a right ascension and a declination, a direction and nothing more — so three observations give six numbers for six unknown elements. That is Laplace’s and Gauss’s method, and the historical demonstration of it is famous: Piazzi observed Ceres on 24 nights in early 1801 over an arc of only 3°, lost it in the Sun’s glare, and Gauss computed elements from that arc which let von Zach and Olbers recover it a year later within half a degree of the prediction.

What the observations actually constrain is uneven, and the unevenness is the honest part of the story. A short arc constrains the direction of the orbital plane well — the node and inclination follow almost directly from the great circle the body traces on the sky — and constrains the size of the orbit hardly at all, because the same apparent motion is produced by a slow near object or a fast distant one. That is why a newly discovered asteroid has an inclination good to a hundredth of a degree and a semi-major axis good to perhaps ten per cent, and why a single further observation months later collapses the uncertainty.

For Earth satellites, the measurement is radar or laser ranging and the elements are fitted to thousands of observations rather than three. The published two-line element sets are the visible output: a fixed-column format devised for punched cards, carrying inclination, right ascension of the ascending node, eccentricity, argument of perigee, mean anomaly and mean motion, each to a fixed number of digits. They are quoted with respect to a specific analytic theory — SGP4 — and using them with any other propagator gives systematically wrong positions, because the “elements” in them are not osculating elements at all but parameters fitted to make that particular theory reproduce the observations.

That is a general and useful caution. An element set is a set of numbers that reproduces the observations when fed to the model it was fitted with, and swapping the model without refitting is the same error as swapping the reference frame.

The generalisation: three angles are never enough, everywhere #

The structure here is not specific to orbits. Any three-parameter description of an orientation has the same two properties — non-commutativity and a coordinate singularity — and the same two consequences.

Aircraft and spacecraft attitude uses Euler angles in a 3-2-1 sequence (yaw, pitch, roll), and hits its singularity at a pitch of 90°, where yaw and roll become the same rotation. The failure has a name, gimbal lock, and it is a real mechanical problem in a physical gimbal set as well as a numerical one: the Apollo guidance platform had a “no-go” region marked on the display for exactly this reason, and Michael Collins famously asked for a fourth gimbal to be added.

The mathematical statement is that the rotation group is a three-dimensional manifold with the topology of real projective 3-space, and no three-parameter chart can cover it without a singularity. The way out is not a cleverer set of angles but a redundant parameterisation: four numbers with one constraint — a quaternion — which has no singularity anywhere and no ordering ambiguity, at the cost of not being readable by a human.

Orbital mechanics has its own version of the same escape, the equinoctial elements, which replace $(e, i, \Omega, \omega)$ with four combinations chosen so that no denominator vanishes at zero eccentricity or zero inclination. Every modern orbit-determination code integrates in something like them — the same instinct that puts the vis-viva relation in terms of $GM$ and never splits it and converts to classical elements only for printing.

What is used instead of angles #

Three angles are the classical description and they have two defects that matter in practice, and the working representations avoid both.

The first defect is the singularity. When the second rotation reaches zero or a straight angle, the first and third turn about the same axis and only their sum is determined — a gimbal lock, in which one degree of freedom has been lost from the description while the body retains all three. Anything computed by differentiating the angles blows up there, which for an attitude-control system is a failure at exactly the orientation the angles were chosen to make simple.

The second is cost. Composing two rotations expressed as angle triples means converting both to matrices, multiplying, and extracting angles again, with trigonometric functions at every step.

The representations that replace them are the rotation matrix — nine numbers with six constraints, singularity-free, composed by multiplication — and the quaternion, four numbers with one constraint, composed by a product of sixteen multiplications and twelve additions with no trigonometry at all.

The quaternion is what flies. It is compact, it never loses a degree of freedom, its constraint is restored by dividing by its own length, and interpolating smoothly between two orientations is a well-defined operation on it in a way it is not on angle triples.

What survives of the angles is their readability. Nobody looks at four quaternion components and pictures an orientation, so the angles remain the interface: the computation is done in quaternions and the result is converted for the display, which is the same division of labour as between spherical trigonometry and vectors elsewhere in this collection.

Where the model stops #

The elements are osculating. They describe the ellipse the body would follow if all other forces stopped now. Under perturbations all three orientation angles drift — $\Omega$ and $\omega$ of a low Earth satellite move several degrees per day from the Earth’s oblateness — so an element set without an epoch is not merely imprecise, it is meaningless.

Three angles assume a rigid plane. For a body in a strong three-body interaction there is no orbital plane to orient, and the elements oscillate through large ranges without describing anything.

The figures cannot show the reference frame moving. Every panel here draws the reference plane as fixed, and it is not — it precesses, nutates and is defined differently by different conventions. That motion is a thousand times smaller than anything visible at this scale and is the source of most of the errors this essay is about, which is an uncomfortable combination.

The ladder from here #

Later rungs on this anchor: the rotation matrices written out and composed. Equinoctial and modified equinoctial elements, and why every propagator uses them. Quaternions, and the topology that forces the fourth number. Orbit determination from three observations, worked through Gauss’s method. The two-line element format and the theory it is welded to. Frame transformations between ICRF, J2000 and of-date, and the arcsecond-level corrections that separate them.

The order of the rotations is a convention, and like most conventions it is older than any argument for it. The 3-1-3 sequence comes from Euler’s 1775 treatment of rigid-body rotation, where the three angles are the precession, the nutation and the proper rotation of a spinning top — and an orbit’s node, inclination and argument of periapsis are those same three angles, applied to a plane instead of a body.