An invariant that is only almost one

One combination of a comet’s orbital elements survives a close encounter with Jupiter unchanged, and that fact does an enormous amount of work. It identifies the same comet before and after an encounter that changed everything else about its orbit; it sorts small bodies into dynamical families; it bounds what a gravity assist can achieve. The first rung of this anchor established it, and the second turned it into a map of which transfers are free.

Both rungs were careful to say that the invariant belongs to the circular restricted three-body problem. This rung asks what that qualification costs, by integrating the same encounter twice.

Where the invariant comes from #

The Jacobi constant is the energy of the restricted three-body problem in the rotating frame, and it exists because that frame rotates uniformly. A uniformly rotating frame has a time-independent effective potential, so the Hamiltonian in it is conserved even though the energy in the inertial frame is not.

Writing the Jacobi constant in terms of the small body’s heliocentric osculating elements, and expanding to first order in the planet’s mass, gives

$$T = \frac{a_J}{a} + 2\sqrt{\frac{a\,(1-e^2)}{a_J}}\cos i,$$

which is the Tisserand parameter. Nothing in the derivation requires the encounter to be weak: the Jacobi constant is exact for all time in that problem, so the parameter is the same before and after however violently the elements change. That is unusual and it is worth savouring — the three-body problem has no closed-form solution, and here is an exact statement about it anyway.

What an eccentric planet breaks #

Jupiter’s orbit is an ellipse, so a frame corotating with Jupiter does not rotate uniformly: the angular rate varies over the orbit by about ten per cent. The effective potential in such a frame is time-dependent, the Hamiltonian is not conserved, and there is no Jacobi constant.

The question is not whether the parameter drifts — it must — but by how much per encounter, and whether that is large or small compared with the boundaries drawn in it.

The drift per encounter comes out at a few parts in a thousand for a deep passage. That is small in absolute terms and it has to be compared against something.

The boundaries it is compared against #

The parameter is used to classify comets, and the classification uses boundaries at exactly 2 and exactly 3.

A body with a parameter above 3 cannot cross Jupiter’s orbit at all — the contours with that value do not reach the planet’s semi-major axis — so it is dynamically decoupled from Jupiter. Between 2 and 3 it can encounter Jupiter and is called a Jupiter-family comet. Below 2 it arrives from the outer solar system on a nearly parabolic orbit and is called a Halley-type or long-period comet.

Those boundaries are physical rather than conventional: 3 is the value at which the zero-velocity surfaces close and the body cannot reach the planet, and 2 is where the encounter speed exceeds Jupiter’s own orbital speed.

A drift of a few parts in a thousand per encounter is therefore two orders of magnitude below the width of the classes, so a single encounter never reclassifies a body. But a Jupiter-family comet has hundreds of encounters over its dynamical lifetime, and the drifts accumulate as a random walk rather than cancelling — so after a hundred encounters the accumulated drift is a few per cent, which is a tenth of the way across a class.

That is the honest statement: the parameter is an excellent identity over one encounter, a good classifier over a lifetime, and not a conserved quantity over the age of the solar system.

The other reason it drifts #

Jupiter’s eccentricity is not the only thing wrong with the circular restricted problem. Saturn exists.

A comet’s orbit is perturbed secularly by every planet, and those perturbations change the eccentricity and inclination between encounters without changing the semi-major axis at all. The parameter is defined with respect to Jupiter alone, so anything Saturn does to the comet moves it.

Separating the two is possible in principle — one acts during encounters and the other between them — and in practice the integrations that would settle the balance are the same integrations whose accuracy the hero figure is testing.

What a drift of a part in a thousand looks like in practice #

It is worth converting the number into something an observer would notice, because a change in a dimensionless combination is hard to feel.

At the comet’s starting orbit the parameter is close to 2.9. A drift of 0.008 is a change of about three parts in a thousand. Holding the inclination fixed and asking what change in semi-major axis alone would produce it, the answer is a shift of about a fiftieth of an astronomical unit — which is a change in orbital period of a few weeks on an orbit of eleven years.

That is not detectable in a single apparition, because the encounter itself changed the period by years. It is detectable in the statistics of a population, because a systematic drift in one direction accumulates and a random one spreads.

Which of those it is turns out to depend on where in Jupiter’s orbit the encounter happens. An encounter at Jupiter’s perihelion, where the planet is moving fastest, drifts the parameter one way; one at aphelion drifts it the other. A comet encountering Jupiter repeatedly at random phases therefore random-walks; one locked in a resonance that puts its encounters at a preferred phase drifts systematically, and the resonant comets are exactly the long-lived ones.

Why the circular case has to be checked too #

The lower trace in the hero figure is not decoration. Without it the upper trace would prove nothing, because an integrator that could not conserve the parameter in the circular case would produce a drift in the elliptic case for reasons having nothing to do with Jupiter’s eccentricity.

This is the site’s general habit applied to a numerical experiment: the control is a case where the answer is known exactly, and the measurement is the difference between the control and the case of interest. The symplectic integrator used here conserves the parameter in the circular case to a part in a hundred thousand across a deep encounter, which is a thousand times finer than the effect being measured.

The spike, and why it is not a failure #

Both traces in the hero figure have a spike at the moment of closest approach, and it is worth explaining rather than cropping.

The parameter is computed from the comet’s heliocentric osculating elements: the semi-major axis and eccentricity of the Keplerian orbit it would follow if every other body vanished at that instant. During a deep encounter with Jupiter, that orbit is not a useful description of anything. The comet is momentarily bound more tightly to Jupiter than to the Sun, so its heliocentric energy swings violently, and the elements — and any function of them — swing with it.

Nothing has gone wrong. The osculating elements are recovering a two-body description of a moment that is not a two-body problem, and the parameter’s constancy is a statement about the asymptotic states before and after, not about every instant in between.

Any measurement made from a figure like this therefore has to be made on the flat parts, well before and well after, which is what the comparison of the two endpoints does.

What was actually measured #

The measurement here is a numerical experiment, so the observational content is in the inputs.

Jupiter’s eccentricity is known to eight figures from the same planetary ephemerides that predict eclipses. Its mass is known from the Galilean satellites and from spacecraft tracking. The comet’s initial elements are chosen rather than observed, because the experiment is about the mechanism rather than about a particular object.

What connects it to observation is the comet population itself. The distribution of Jupiter-family comets in this parameter has a shape near the boundaries that a strictly conserved quantity would not produce: there is a tail above 3, of objects that should not be able to encounter Jupiter and demonstrably have, and a spread below 2 in objects whose orbits are otherwise identical. Both are consistent with a drift of the size the hero figure measures. There is a corollary about how the classification should be quoted. A boundary drawn at exactly 3 implies a precision the quantity does not have over a comet’s lifetime, and a body reported at 3.02 is not thereby proved to be decoupled from Jupiter — it is proved to be within the drift of the boundary, which is a weaker and more useful statement.

The same defect, viewed as a feature #

There is a way to read the drift that makes it useful rather than annoying.

A quantity that is exactly conserved carries no information about the history of the system: knowing it says what the body could have done and nothing about what it did. A quantity that drifts by a computable amount per encounter is a counter. If the drift per encounter can be characterised, then the spread in the parameter across a population that started together is a measure of how many encounters its members have had — which is a dynamical age.

That is the same logic that makes a family of asteroid fragments datable from the spread the Yarkovsky drift has given them: a slow, size-dependent leak out of an initially sharp distribution is a clock, provided the leak rate is known.

For comets the leak rate is known only roughly, because it depends on the distribution of encounter geometries, and the population is small. But the principle stands, and it is the reason the size of the effect measured in the hero figure is worth pinning down rather than merely bounding.

One consequence of that is worth stating because it is a trap the figure was built to avoid. A code that samples the parameter at fixed time intervals and reports its extreme values will report the spike, and a code that samples at fixed intervals of true anomaly will miss it. Neither is measuring what the invariance claim is about. The claim is about the limit as the comet leaves Jupiter’s neighbourhood, and the only honest way to evaluate it is to run far enough out in both directions that the trace is flat, which is why the window drawn here is nine radians of Jupiter’s orbit wide on each side rather than the tenth of one the encounter occupies.

The wider point this rung makes is about how an exact result should be used once its assumptions are known to be false. There are two bad options and one good one. The bad options are to keep quoting the invariant as exact, which propagates an error nobody has bounded, and to abandon it because its derivation does not apply, which throws away a constraint that is right to three decimal places. The good option is the one taken here: measure the departure in the case of interest, in the units the result is used in, and carry it as a number. That is what turns a broken assumption into an error bar. It also identifies where the number matters and where it does not — negligible for one encounter, negligible for a mission, and decisive only for the long-term statistics of a population — which is the kind of statement that lets a result go on being used rather than being either overtrusted or discarded.

What the parameter cannot see #

Two limitations are worth separating from the drift, because they are not errors in the quantity — they are consequences of what it is a function of.

The first is that it is defined with respect to one planet. A body’s parameter with respect to Jupiter and with respect to Saturn are different numbers computed from the same elements, and they classify differently. That is not a defect: an object dynamically controlled by Saturn should be classified by its relation to Saturn. But it means the bare phrase “the Tisserand parameter” is incomplete, and the Centaurs — the population on unstable orbits between Jupiter and Neptune — are exactly the objects for which the choice of planet matters, since they encounter several.

The second is that it is a function of three elements out of six. The semi-major axis, the eccentricity and the inclination enter; the argument of pericentre, the longitude of the ascending node and the position along the orbit do not. Two bodies with identical parameters can therefore have completely different futures, because whether an encounter actually happens depends on where the orbits cross and whether the two bodies arrive there together.

The parameter constrains the possibility of an encounter and says nothing about its timing. A body whose contour reaches Jupiter’s orbit can encounter Jupiter; whether it does so next century or in ten million years is decided by the three elements the parameter discards.

That division is the reason the quantity is used the way it is. It is excellent for asking what a body could have been and what it might become — questions about reachable regions — and useless for asking what it will do next, which is what an integration is for.

The same quantity with no comet in it #

A mission designer uses this parameter constantly, under a different name and for a reason that is the mirror image of the cometary one.

A spacecraft doing a tour of a planet’s moons, or of the inner planets, gains what it gains from flybys, and a flyby cannot change the encounter speed relative to the body it flies past. The Tisserand parameter is that speed written in heliocentric elements, so the whole set of orbits reachable by any number of unpowered flybys of one body is a single contour — which is exactly the constraint the second rung of this anchor drew as a map.

That makes the parameter a design tool rather than a classification: a proposed sequence of encounters is checked by seeing whether each successive orbit lies on the same contour, and a sequence that requires leaving it requires propellant.

There is a technique for leaving it cheaply, and it is instructive because it works by exploiting exactly the assumption this essay has been testing. A small burn applied far from the planet, at the right point of the orbit, changes the period slightly; the changed period alters the phasing of the next encounter, so the spacecraft arrives at a different point and receives a different deflection. The propellant buys a change in geometry rather than a change in speed, and the leverage — the change in encounter conditions per unit of propellant — can be an order of magnitude better than a direct burn.

So the invariant that bounds what a flyby can do is also the thing a manoeuvre is designed against. A constraint that is exact is a constraint that can be planned around exactly, which is why the circular approximation’s small error is irrelevant for mission design and decisive for the population statistics the previous sections were about.

One more starting orbit shows how the invariant’s small violation depends on where the encounter happens.

Where the ladder goes #

The next rung is the generalisation. There is a quantity conserved in the elliptic restricted problem too — not a constant, but an adiabatic invariant when the planet’s eccentricity is small — and constructing it turns the drift measured here into a calculable correction rather than a nuisance.

The other direction is practical. Mission designers use the parameter to decide which flyby sequences are reachable, and the drift measured here is far smaller than the manoeuvre budgets involved, so for that purpose the circular approximation is exact enough. The one place it is not is in the long-term statistics of small-body populations, where a few parts in a thousand per encounter, accumulated over a billion years, decides which reservoir a body ends up in.