Six numbers that fix an orbit for all time, and the sixth is the awkward one

A body moving under gravity has three coordinates of position and three of velocity, and those six numbers determine everything it will ever do. That is not a claim about orbits; it is a claim about second-order differential equations, and it is true of a thrown stone.

What is peculiar to the two-body problem is that the six numbers can be traded for six others that barely change. Position and velocity vary continuously and say nothing at a glance. The orbital elements are a repackaging in which five of the six are constants of the motion — they describe a fixed curve fixed in space — and only the last one advances, uniformly, with time.

That repackaging is why an orbit can be written on a postcard and why a catalogue of thirty thousand pieces of orbiting debris is a table rather than a simulation.

Two for the shape, three for the attitude, one for the clock #

The six divide cleanly into three groups, and the division is more informative than the list.

The size and the shape. The semi-major axis $a$ fixes how big the orbit is and, through the harmonic law, how long it takes; the eccentricity $e$ fixes which conic it is and how elongated. Those two describe the curve completely as a shape, drawn on a flat sheet with no reference to anything outside it.

The orientation. Three angles are needed to place that sheet in space, and they are applied in a fixed order. The inclination $i$ is the tilt of the orbital plane against a reference plane — the ecliptic for the solar system, the Earth’s equator for a satellite. The longitude of the ascending node $\Omega$ says where, going round the reference plane, the orbit crosses it heading north; that crossing line is the line of nodes, and it is the intersection of two planes rather than anything on the orbit. The argument of periapsis $\omega$ then says how far round the orbit’s own plane, from that crossing, periapsis lies.

The clock. The sixth is a time, and it is the odd one out in every respect. It is usually given as the epoch of periapsis passage $T$ — the instant the body was last at its closest point — or equivalently as the mean anomaly at some stated moment. It is the only element that changes, and it changes at a perfectly constant rate.

The asymmetry is worth dwelling on. Five numbers describe a curve that, in an ideal two-body system, is eternal. The sixth is a clock reading, and a clock reading is only meaningful with a date attached — which is why every set of orbital elements ever published carries an epoch, and why elements without one are useless.

The order the rotations are applied in #

Three angles orienting a plane sounds like a job with an obvious answer, and it is not. Rotations do not commute, so the same three numbers applied in a different order give a different orbit, and the convention has to be stated.

The standard is a 3-1-3 sequence: rotate by $\omega$ about the orbit’s own normal, then by $i$ about the resulting node line, then by $\Omega$ about the reference pole. Undoing it in the reverse order recovers the orbital plane. The figures on this page are generated by exactly that composition, applied to a conic drawn in its own plane, which is why the drawn node line really is where the two planes meet.

The three orientation angles are not equally important in practice, and the reason is which of them a perturbation moves. An oblate primary — which is not a point mass — leaves the inclination and the semi-major axis alone to first order and makes $\Omega$ and $\omega$ drift steadily. Every Earth satellite therefore has an orbital plane that rotates, and the rate is entirely predictable — which is a nuisance for most missions and a design tool for a few.

The elements that vanish #

The set is not well behaved everywhere, and the failures are worth knowing because they are not rare cases.

If the orbit is exactly circular, there is no periapsis. The argument of periapsis is undefined, and so is the epoch of periapsis passage, because there is no distinguished point to pass. Two of the six elements have evaporated. The orbit is still perfectly well determined — it simply cannot be written in this parameterisation.

If the orbit is exactly in the reference plane, there is no node. The line of nodes is undefined, so $\Omega$ is undefined, and $\omega$ — measured from the node — goes with it.

Neither degeneracy is a physical event. Nothing happens to the body. What breaks is the coordinate system, in the same way that longitude fails at the poles while the pole itself is an ordinary place. And near-degeneracy is worse than degeneracy: an orbit at $e = 0.0001$ has a periapsis, technically, whose direction is set by a perturbation smaller than the measurement error, so $\omega$ swings wildly from one determination to the next while the orbit itself sits still.

The standard remedy is a change of variables to equinoctial elements, which replace the two problematic angles with combinations like $e\cos(\omega + \Omega)$ and $e\sin(\omega + \Omega)$. These go smoothly to zero as the eccentricity does, instead of becoming undefined, and every serious propagator uses them.

What was actually measured #

An element set is never observed. What is observed is a sequence of directions, and the conversion is a fit — which means that the six numbers in a catalogue entry are not six measurements but six parameters of a model, with a covariance between them.

The classical version of the problem is Gauss’s: three observations, six unknowns, and each observation supplying two numbers. It works, and it worked spectacularly in 1801 when Ceres was tracked for forty-one days, lost in the Sun’s glare, and recovered a year later within half a degree of where Gauss’s elements put it. The method is still the standard initial-orbit determination, refined afterwards by least squares over every subsequent observation.

The precision achieved varies enormously with which element is asked about, and the pattern is instructive.

For a tracked spacecraft the elements come from radio ranging and Doppler rather than from angles, and they are extraordinarily good: a deep-space probe’s position is known to hundreds of metres at a distance of billions of kilometres, because a two-way coherent Doppler measurement gives the line-of-sight velocity to a fraction of a millimetre per second and the integrated velocity is the position.

For an asteroid seen on three nights the semi-major axis may be uncertain by several percent and the mean anomaly by rather less, because the along-track position is what the observations most directly constrain and the distance is what they least constrain. This is why a newly discovered near-Earth object’s impact probability changes dramatically as more observations arrive: the orbit’s shape is being pinned down long after its phase was.

For a low Earth satellite the five geometric elements are stable and the sixth is the problem. Atmospheric drag removes energy continuously, at a rate depending on solar activity, so the mean anomaly of a satellite predicted a month ahead may be wrong by many kilometres along track while the orbital plane is right to metres. Conjunction assessment — deciding whether two objects will collide — is dominated by that single uncertainty, and it is why the published two-line element sets carry a drag term and an epoch and go stale within days. Nothing about the geometry has moved; only the clock reading is wrong.

Elements that were chosen rather than found #

Because five of the six are geometric constants, choosing them is how an orbit is designed — and several standard mission orbits are nothing more than a particular value of one element.

Geostationary is a statement about $a$ alone. Set the period to one sidereal day and the harmonic law returns 42,164 km from the Earth’s centre, with $e = 0$ and $i = 0$ to keep the satellite fixed rather than tracing a figure-of-eight.

Sun-synchronous is a statement about $i$, and it is the elegant one. The Earth’s equatorial bulge makes the node $\Omega$ drift, at a rate that depends on the inclination and reverses sign at 90°. Choosing an inclination of about 98° — slightly retrograde — makes the drift exactly one turn per year, so the orbital plane keeps a constant angle to the Sun and the satellite crosses the equator at the same local solar time every day. Every imaging satellite that needs consistent shadows is on one, and the whole arrangement is a perturbation being used deliberately.

Molniya is a statement about $i$ and $\omega$ together. The same bulge makes $\omega$ drift too, except at the two inclinations where the effect vanishes — 63.4° and 116.6°. A highly eccentric orbit at 63.4° keeps its apoapsis where it was put, and putting it over the northern hemisphere gives a satellite that spends most of a twelve-hour period loitering at high latitude where a geostationary satellite cannot be seen at all.

Frozen orbits go further and choose $e$ and $\omega$ so that the two perturbations cancel between them, leaving an orbit whose altitude over any given latitude does not change from one revolution to the next. Altimetry missions use them because a changing altitude would be indistinguishable from a changing sea surface.

Osculating, mean, and the elements in a catalogue #

There are two kinds of element set in circulation and they are not interchangeable, which is a routine source of error.

Osculating elements are the conic a body would follow if every perturbation were switched off at that instant. They are exact, they describe the true position and velocity precisely, and they change continuously — a low satellite’s osculating semi-major axis oscillates by a kilometre or so within every revolution as the equatorial bulge pushes it about.

Mean elements have those short-period oscillations averaged out. They do not describe the instantaneous position at all; they are the input to a particular analytic propagator, which puts the oscillations back when it computes a position.

The published two-line element sets for Earth satellites are mean elements of the second kind, and they are meaningful only when fed to the propagator they were fitted with. Using a TLE’s semi-major axis as though it were the real one gives an error of kilometres; using it in a general-purpose numerical integrator gives an orbit that drifts away from the truth within days. The elements and the theory that consumes them are a matched pair, and neither is portable without the other.

The generalisation: elements are a coordinate system, not a fact #

The deeper point about the elements is that they are a canonical transformation — a change of variables in which the equations of motion become as simple as possible — and once that is recognised the same trick appears everywhere in the subject.

In the two-body problem the transformation makes five coordinates constant and the sixth linear in time. That is the simplest form any dynamical system can be put into, and a system that admits it is called integrable.

Add a perturbation and the elements stop being constant, but they stay useful, and the method that exploits that is the whole of classical perturbation theory. Variation of parameters, due to Lagrange, keeps the same six elements and lets them be slow functions of time: the orbit is treated as an ellipse that is continuously changing into a slightly different ellipse. Neptune was found this way. So was the correction to the Moon’s motion that made lunar tables good enough for navigation.

The elements then split by how they respond, and the split is the practically important one. Some perturbations produce periodic changes, which oscillate and average to nothing. Others produce secular changes, which accumulate without bound. The node drift that Sun-synchronous orbits exploit is secular; the wobble in a satellite’s altitude over one revolution is periodic. Predicting an orbit years ahead means getting the secular terms right and being fairly relaxed about the periodic ones.

Which plane the angles are measured from #

Two of the six elements are meaningless until a reference plane is named, and the plane is chosen by convention rather than by physics. Four conventions are in general use and they are not interchangeable.

For a body orbiting the Sun the reference is the ecliptic — the plane of the Earth’s orbit — with the direction of the vernal equinox as the zero of longitude. That is a choice with no dynamical significance whatever: it is the plane the observers happen to sit in.

For a satellite of the Earth the reference is the equator, because the perturbation that dominates such an orbit is the Earth’s oblateness and the oblateness is symmetric about the equator. Here the choice is dynamical, and it is why the elements of an artificial satellite are the ones that behave simply.

For long-term dynamics of the solar system the reference is the invariable plane, perpendicular to the total angular momentum of the whole system. It is the only one of the four that does not move, and it is used precisely for calculations long enough that the movement of the others matters.

For stellar and galactic work the reference is the galactic plane, which is a different object again.

The practical consequence is that an inclination is not a number that can be compared without checking what it was measured against. A comet’s inclination to the ecliptic and a satellite’s to the equator differ by up to the obliquity of twenty-three and a half degrees for the same physical orientation, and quoting one where the other is expected is a routine error that survives every internal check because both numbers are individually correct.

There is a further wrinkle: the ecliptic and the equator both move, so a set of elements referred to either must also state an equinox — the epoch of the reference frame — as well as the epoch of the elements themselves. The two are often different, and a catalogue that omits either is incomplete.

Where the model stops #

Two bodies. Elements are exact only for an isolated pair. With a third body they become osculating elements — the ellipse the body would follow if every other force were switched off at that instant — which is a useful fiction and a fiction. With three bodies there is no closed orbit to osculate to for longer than an instant.

Constant masses, no thrust, no drag. A thrusting spacecraft has elements that change discontinuously at each burn, and a low satellite’s change continuously.

A point-mass primary. The oblateness terms discussed above are the leading correction and they are not small: for a low Earth orbit the node drifts by several degrees a day.

A stated reference frame. Inclination and node are meaningless without saying what they are measured against, and the reference frames themselves move — the Earth’s equator precesses, so a satellite’s inclination in equatorial coordinates and a comet’s in ecliptic coordinates are not comparable numbers.

The figures on this page share a limitation that is intrinsic rather than accidental. They are three-dimensional scenes projected onto a page, and the projection destroys exactly the information the figure exists to convey: a tilt seen nearly edge-on and a tilt seen nearly face-on look completely different, and the same orbit drawn from another viewpoint is unrecognisable. That is not a flaw in the drawing. It is the reason the elements exist — an orbit’s orientation cannot be read off a picture, so it is given as three numbers instead.

One of the six carries a sign convention that is not a convention at all, and it deserves drawing.

That is an elegant piece of bookkeeping and it is the reason inclination runs from 0 to 180 rather than 0 to 90. The alternative — an inclination in the first quadrant plus a direction bit — would carry the same information and would break every formula that treats the elements as continuous, because a body perturbed across the pole would have to jump discontinuously from one representation to another. Letting the angle run past a right angle keeps the parameterisation smooth through a configuration that physically has nothing special about it.

It also puts a real physical distinction into a coordinate that looks purely geometric. Almost every natural satellite in the solar system is prograde with respect to its planet’s spin, because it formed from the same disc; the retrograde ones were captured. So an inclination above 90° in a catalogue of moons is a statement about origin, and it is read directly off the fifth column of a table of elements without any further calculation at all. That is the best argument for the six elements as a choice of coordinates rather than as an arbitrary convention: each of them is the answer to a question somebody actually asks. The semi-major axis is the energy, the eccentricity is the shape, the inclination is the origin, the node and the argument are the orientation, and the epoch is where the body is now. Six questions, six numbers, and no redundancy anywhere in the set — which is what makes the parameterisation feel inevitable rather than chosen, and is also what makes its degenerate cases genuinely awkward rather than merely inconvenient.

The ladder from here #

Later rungs on this anchor: the state-vector-to-elements conversion, done explicitly. Equinoctial and Delaunay elements. Osculating against mean elements, and why the two-line element sets use the latter. The $J_2$ perturbation and the node and apsidal drift rates it produces. Sun-synchronous, Molniya and frozen orbits derived. Gauss’s method of initial orbit determination. Lambert’s problem in element form. Covariance of an element set, and how an impact probability is computed from it. And the proper elements of asteroid families, which are the parts of an element set that survive averaging over millions of years — the same averaging that makes a resonance visible as a gap and are used to identify collisional fragments of a common parent.

The two-line element format used for every catalogued Earth satellite fits an entire orbit into 138 characters, in a fixed-width layout designed for punched cards. It has not changed since, and the format’s inability to express an epoch beyond the year 2056 is a problem currently being deferred.