Which direction moves which element

The rung below established that the six numbers describing an orbit do not stay constant once a third body is present, and split the ways they move into the periodic and the secular.

It left the mechanism alone. Knowing that the elements drift is a long way from knowing which element a given force moves, and where in the orbit it moves it. That is what the variation-of-parameters formulation supplies, and in the form Gauss gave it the answer is unusually blunt: resolve the disturbing acceleration into three directions and each element responds to a specific combination of them, with a specific dependence on position.

The three directions, and the equations #

Take the instantaneous orbit — the conic the body would follow if the perturbation stopped now — and resolve the disturbing acceleration into

• $R$, radially outwards from the primary;
• $S$, perpendicular to $R$ in the orbital plane, positive in the direction of motion;
• $W$, perpendicular to the orbital plane.

Then, with $n$ the mean motion, $p$ the semi-latus rectum, $r$ the current radius and $f$ the true anomaly,

$$\frac{da}{dt} = \frac{2}{n\sqrt{1-e^2}}\left[e\sin f\,R + \frac{p}{r}\,S\right],$$

$$\frac{de}{dt} = \frac{\sqrt{1-e^2}}{na}\left[\sin f\,R + (\cos f + \cos E)\,S\right],$$

$$\frac{di}{dt} = \frac{r\cos(\omega+f)}{na^2\sqrt{1-e^2}}\,W, \qquad \frac{d\Omega}{dt} = \frac{r\sin(\omega+f)}{na^2\sqrt{1-e^2}\sin i}\,W.$$

These are not approximations. They are an exact change of variables from position and velocity to the six elements, valid for any disturbing acceleration whatever — a third body, a drag, an engine, an oblate primary, radiation pressure. The physics enters entirely through what $R$, $S$ and $W$ are.

The trick that produces them is worth naming, because it is not obvious that such a change of variables should exist at all. The elements are defined as the constants of the unperturbed problem, so under a perturbation they are no longer constants — but they are still six functions of position and velocity, and differentiating those functions along the perturbed trajectory gives six rates. What makes the result usable is that the unperturbed part of the motion contributes nothing: by construction the elements are exactly the combinations that a Keplerian flow leaves alone, so every term on the right-hand side is proportional to the disturbing acceleration. The method turns a problem about a trajectory into a problem about six slowly varying numbers, and it works precisely to the extent that they do vary slowly.

The first exact statement: out of plane does no work #

$W$ appears in the equations for the inclination and the node. It appears in neither $da/dt$ nor $de/dt$, and the absence is exact rather than approximate.

The reason is one line of mechanics. The rate of change of orbital energy is $\dot{E} = \mathbf{v}\cdot\mathbf{a}$, and $W$ is perpendicular to the orbital plane, in which $\mathbf{v}$ lies. A force perpendicular to the velocity does no work, so it cannot change the energy, and since $a$ is a function of the energy alone it cannot change $a$ either. That single fact is the reason plane changes are the most expensive routine manoeuvre in spaceflight. Every other burn buys something: raising an orbit stores energy that can be traded back later. A plane change buys a rotation and nothing else, and the propellant it consumes has bought no energy at all.

The second exact statement: perigee is worth more #

Look again at the along-track term in $da/dt$. It carries the factor $p/r$, and since $p/r = 1 + e\cos f$, its value at perigee is $1+e$ and at apogee $1-e$. The ratio is

$$\frac{(da/dt)_{\text{peri}}}{(da/dt)_{\text{apo}}} = \frac{1+e}{1-e},$$

which for $e = 0.35$ is 2.08, for $e = 0.7$ is 5.7, and for a highly eccentric transfer orbit can be a factor of thirty.

This is the Oberth effect, arriving as a term in a differential equation rather than as an argument about kinetic energy. The two derivations agree, as they must, but the differential-equation version says something the energy argument does not: the advantage is a property of where the body is on the orbit, and it is available to a continuous thrust as much as to an impulse. The radial term deserves a look too. It carries $e\sin f$, which is antisymmetric about the line of apsides: outward force helps on the way out and hinders on the way in, and over a full revolution its contribution to $a$ integrates to exactly zero. A purely radial thrust changes the shape and orientation of an orbit and cannot change its size, which is why nobody uses one to raise an orbit and why a radial burn in a rendezvous returns the vehicle to where it started.

A number, to fix the scale #

The equations are worth once being made numerical, because the coefficients are not intuitive.

Take a circular orbit at the height of the space station: $a = 6{,}791$ km, and $n = \sqrt{\mu/a^3} = 1.128\times10^{-3}$ rad/s. With $e = 0$ the along-track term reduces to $da/dt = 2S/n$, so an impulse $\Delta v$ applied along the track gives

$$\Delta a = \frac{2\,\Delta v}{n} = 1{,}773\ \text{km per km/s}.$$

One metre per second buys 1.77 kilometres of altitude. The station loses about a hundred metres a day to drag at solar minimum and considerably more at maximum, so a reboost of a few metres per second every few weeks is the whole of the arithmetic — and the figure explains why the manoeuvre is described in altitude and executed in velocity.

The same number read the other way is the phasing rule. The period changes by $\Delta P/P = \tfrac{3}{2}\,\Delta a/a$, which for that one metre per second is $3.9\times10^{-4}$, or 2.2 seconds per revolution. A vehicle that wants to close a lead of a few kilometres on the station therefore burns for a fraction of a second and waits, and the waiting rather than the burning is what costs the schedule.

Where each element is most responsive #

The equations also say when to apply a force, and the answers are not all the same.

For the semi-major axis, apply along-track thrust at perigee.

For the eccentricity, the along-track term carries $\cos f + \cos E$, largest at the apsides and of opposite sign at the two of them: thrust at perigee raises $e$, thrust at apogee lowers it. Circularising an eccentric orbit therefore means burning at apogee, which is exactly what the second burn of a Hohmann transfer does. For the inclination and the node, the responses are in quadrature. Inclination responds as $\cos(\omega+f)$ — maximum at the nodes — and the node as $\sin(\omega+f)$, maximum a quarter turn from them.

Running the equations backwards #

So far the equations have been read forwards: given a force, find the element rates. Read backwards they are a measuring instrument, and this is where the subject earns its place in a collection about what was actually observed.

A satellite’s node is watched. Its rate is measured. The equation for $d\Omega/dt$ then says what $W$ must have been — and if the only plausible source of an out-of-plane force is the primary’s oblateness, the measured rate is a measurement of $J_2$.

The same inversion is how drag coefficients, radiation-pressure areas and thruster performance are all determined in flight. Nobody measures the force. What is measured is an element rate, and the equations above are what converts one into the other.

Two formulations, and why this one #

There are two classical ways to write variation of parameters, and the difference between them is not cosmetic.

Lagrange’s planetary equations express the element rates in terms of the partial derivatives of a disturbing function $\mathcal{R}$ — a scalar potential whose gradient is the perturbing acceleration. They are elegant, they preserve the Hamiltonian structure, and they are what secular theory is built on. They also require the perturbation to have a potential.

Gauss’s form — the one above — takes the acceleration itself, resolved into three directions. It is uglier and it does not care whether the force is conservative.

That distinction decides which one gets used. Drag has no potential. Neither does thrust, nor a thermal recoil, nor a spacecraft venting propellant. Every non-conservative force in the subject is inaccessible to Lagrange’s form and routine in Gauss’s, which is why the astrodynamics literature uses Gauss’s almost exclusively and the celestial-mechanics literature uses Lagrange’s almost exclusively. They are the same change of variables; they differ in what is allowed to appear on the right.

The equations as a control law #

Read as a design tool rather than as an analysis, these equations answer a question that has no other closed-form answer: given a thruster that can point in any direction, where should it point?

For each element there is a direction of maximum response, obtainable by reading the coefficients as a vector and normalising it. Steering the thrust along that direction moves the chosen element as fast as the engine allows. Steering it along a weighted sum of several such directions moves several elements at once, and the weights are a control law. The same reading explains a rule of thumb that otherwise looks arbitrary. To change the inclination of a low orbit with a low-thrust engine, thrust out of plane near the nodes and coast elsewhere — because $di/dt \propto \cos(\omega+f)$ means the thrust is doing almost nothing at the quadratures, and a duty cycle that skips the useless part converts directly into propellant saved.

What the formulation cannot do #

Three limits are worth stating, because the equations are so clean that it is easy to over-trust them.

They are singular at zero eccentricity and zero inclination. The expression for $d\Omega/dt$ has $\sin i$ in the denominator and the one for $d\omega/dt$ has $e$; a circular equatorial orbit has neither a node nor a perigee, so the elements that locate them stop existing. The cure is a different set of elements, not a different physics.

They are exact but not integrable. Writing $da/dt$ down does not solve for $a(t)$; the right-hand side depends on the elements, which are changing. Everything after this point is either numerical integration or an averaging scheme that removes the fast angle deliberately.

And they describe the osculating elements, which oscillate over a revolution by far more than they drift over a year. A quoted element from a satellite catalogue is usually a mean element belonging to a specific propagator, and the difference between the two is kilometres.

The same three readings at other settings show which of the conclusions are about the equations and which are about the orbit they were read at.

The same equations with nobody steering #

The equations are introduced here as a way of deciding where to point a thruster, and they are just as useful read the other way: a natural perturbation has a direction, and the direction says which elements it moves.

Drag acts opposite to the velocity, so it is a purely along-track force. By the essay’s second exact statement it is most effective at perigee, where the speed is highest and — for an atmosphere — the density is greatest too. The result is that drag lowers the apogee preferentially, circularising the orbit while shrinking it, which is why a decaying elliptical orbit becomes nearly circular before it re-enters.

The Earth’s equatorial bulge produces a force with an out-of-plane component that reverses sign twice per orbit. Out-of-plane forces do no work, so the semi-major axis and the eccentricity are untouched, and what moves is the orientation: the node regresses and the argument of perigee precesses, at rates depending on the inclination.

Solar radiation pressure acts in the anti-solar direction, which is fixed in inertial space while the orbit turns beneath it — so it is along-track on one side of the orbit and against-track on the other. The along-track components largely cancel over a revolution and the residual drives the eccentricity vector around an annual circle.

A third body pulls from a direction that changes slowly, and its averaged effect is out of plane for an inclined perturber — which is why a distant companion moves inclinations and orientations rather than energies.

So each perturbation has a fingerprint in the element set, and a satellite’s orbit determination identifies what is acting on it by which elements are drifting rather than by measuring any force. That is the standard diagnostic when a spacecraft’s orbit does not behave: not “how large is the anomalous force” but “which elements are moving”, which the equations turn into a direction.

Averaging over one revolution #

There is a step between these equations and their practical use that is worth naming, because it changes what they describe.

As written, the equations give the instantaneous rate of change of each element, and those rates oscillate strongly around an orbit — the semi-major axis rises and falls within a single revolution under a perturbation that does not change it at all over the whole.

The remedy is to average. Integrating each rate over one revolution, holding the elements fixed on the right-hand side, gives the secular rates: what remains after the periodic terms have cancelled.

Two things then become visible that the instantaneous equations obscure. Some perturbations average to nothing, and the fact that a force moves an element within an orbit is not evidence that it moves it at all in the long run. And the ones that survive averaging are the ones that matter over years, so a long-term propagation can be done with the averaged equations at enormously less cost — hours of integration replaced by a rate multiplied by a time.

The distinction between a periodic and a secular effect is therefore the whole content of the averaging, and nearly every long-term result in this subject — the node’s regression, the drift of a resonant argument, the decay of an orbit — is a statement about what is left when one revolution has been integrated over.

There is one caution that goes with the averaged form. Averaging assumes the elements change little over one revolution, which fails precisely where the perturbation is strong — during a close encounter, inside a resonance, or under a thrust large enough to alter the orbit within a single pass. Those are the cases the averaged equations describe worst and the ones a designer most often cares about, so the practical arrangement is to average where the motion is slow and to integrate directly where it is not, with the boundary between the two chosen by comparing the answers.

The comparison is also the only honest way to set the boundary, since the averaged equations give no warning when they stop being valid — they return a smooth answer whether or not it means anything, which is a failure mode worth watching for in any method built on an average.

The same caution applies to the fingerprints of the previous section. Attributing a drift to one perturbation assumes the others are modelled correctly, and the residual that identifies an unmodelled force is the difference between what was observed and what the model predicted — so the identification is only as good as everything already in the model, which is why an anomalous acceleration is always the last conclusion rather than the first.

The history of the subject contains several accelerations that were announced and then absorbed into a better model of something ordinary, and no anomalous force has yet survived that process.

And two more readings of what the equations do when nothing is thrusting.

Where this ladder goes next #

This rung has established the mechanism: which component moves which element, and where in the orbit each is most effective.

The rung above removes the fast angle. Averaging the right-hand sides over a revolution kills the periodic terms and leaves the secular ones, which is what makes long-term integration tractable and is the first step towards the eigenvalue problem the whole planetary system obeys.

Beside it lies low-thrust trajectory design, which is these equations used as a control problem: given a thruster that can point anywhere, choose the direction at every instant that moves the elements towards their targets fastest. The answer is a feedback law written directly in terms of the partial derivatives above.

And below it, the habit: a perturbation is not a nuisance to be estimated but a channel with a known transfer function. Knowing which direction moves which element turns every small force into either a design tool or an instrument, depending on which end of the equation is unknown.