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Spaceflight

The tube that leads out of a neck

Below a certain energy the forbidden region opens at a Lagrange point, and a trajectory may pass. Which ones do is decided by a surface with no width at all — and two tubes that meet give a transfer that costs nothing at the join.

Assumes Lagrange points and The three-body problem.

16 min read 8 figures Two bodies, and no moreSmall perturbations, large consequencesEverything is a conic

The zero-velocity curves of the restricted three-body problem divide the plane into a region a body of a given energy may occupy and a region it may not. Below the energy of the first Lagrange point the two lobes around the primaries are separate, and nothing can pass between them at all. Raise the energy a little and a neck opens at L1L_1; raise it further and a second opens at L2L_2.

An open neck sounds like a gate: pass through it or do not, depending on aim. It is not a gate. The trajectories that pass are organised into structures with a shape, and the shape is what the last thirty years of low-energy mission design have been about.

Two outcomes, and a boundary with no width. 26 trajectories launched from one point beyond L₂, all at the one speed the Jacobi constant C = 3.5124 permits there, differing only in the direction they set off in. The heavy curve is the zero-velocity boundary at that constant — the region no trajectory of this energy may enter — and it is open at L₂ by the neck the trajectories are aimed at. 11 of the 26 pass through into the secondary's realm and 15 turn back, and they are not interleaved — sweeping the launch direction through 65° finds one changeover and nothing in between. Bisecting the first of them pins it to 2.6e-12 radians, and the integrator runs out of digits before the boundary runs out of sharpness. That surface is the tube. It is the stable manifold of the periodic orbit about L₂, it separates transit from non-transit everywhere and not only in this fan, and a mission that wants to arrive for nothing has to be put inside it.
Fig. 1 Twenty-six trajectories launched from one point beyond L2L_2, all at the one speed the Jacobi constant permits there, differing only in the direction they set off in. The heavy curve is the zero-velocity boundary at that constant. Eleven of the twenty-six pass through the neck into the secondary’s realm and fifteen turn back, and they are not interleaved — sweeping the launch direction through 65° finds one changeover and nothing in between. Bisecting it pins the boundary to 2.6×10−122.6\times10^{-12} radians, and the integrator runs out of digits before the boundary runs out of sharpness.

What the boundary is

That surface is not an artefact of the fan. It is the stable manifold of a periodic orbit about the Lagrange point, and its existence is a theorem rather than a numerical observation.

The collinear points L1L_1, L2L_2 and L3L_3 are saddle points of the effective potential, and the linearised motion about them has one pair of eigenvalues that is real — one growth and one decay — and one pair that is imaginary. The imaginary pair gives a family of periodic orbits about the point: planar Lyapunov orbits, and out of the plane the halo orbits that spacecraft actually use. The real pair means each such periodic orbit is unstable, and an unstable periodic orbit has a stable manifold — the set of trajectories that approach it asymptotically as time runs forward — and an unstable manifold, the set that leave it.

In the planar problem those manifolds are two-dimensional surfaces in the three-dimensional energy level set, and a two-dimensional surface in a three-dimensional space separates it. Everything on one side transits; everything on the other does not; and there is nothing in between, because a surface has no thickness.

Three topologies of the allowed region, mass fraction 0.15. The boundary of the region a particle may occupy, at three values of the Jacobi constant. Above C(L₁) = 3.717 the two bodies have separate lobes and nothing can pass between them; below it the lobes merge at the inner point; below C(L₂) = 3.524 the merged region opens to the rest of the plane. Nothing has been integrated: the boundary is a level set of a conserved quantity.
Fig. 2 The three topologies the energy selects between. Above C(L1)C(L_1) the two lobes are separate and nothing can pass; below it they merge at the inner point; below C(L2)C(L_2) the merged region opens to the rest of the plane. Nothing has been integrated in this figure — the boundary is a level set of a conserved quantity. What the energy decides is whether a passage exists; what the manifold decides is which trajectories take it, and the two questions are quite separate.

Why a tube

In the spatial problem the picture acquires the name it is usually known by. The manifolds of a halo orbit are three-dimensional surfaces in a five-dimensional energy level set, and their intersection with a plane of section is a closed curve — a circle, roughly — so the manifold looks like a tube whose cross-section is that circle and whose axis runs from the neck out into one realm or the other.

A trajectory inside the tube transits. One outside it bounces off the neck region and returns. The tube’s interior is not a region of high probability; it is the whole of the transit set, and its boundary is exact.

What transferring mass does to the lobe it came from. The donor's Roche-lobe radius against how much of its mass it has left, for conservative transfer — total mass and total angular momentum both held fixed — starting at mass ratios 2.5 and 0.4. Both curves come from J = M₁M₂√(Ga/M) rearranged for a, times Eggleton's lobe radius at the running mass ratio; nothing is fitted. They go opposite ways, and that is the whole of the Algol paradox. When the donor is the heavier star the separation shrinks, the lobe shrinks with it, more mass is pushed through L₁, and the transfer accelerates until the ratio reverses. When the donor is the lighter star the separation grows, the lobe grows away from it, and the transfer is self-limiting. So a system found transferring mass is almost always found with the less massive star overflowing — which is why Algol's evolved secondary weighs less than its unevolved primary, and why that looked for fifty years like a star ageing faster than a heavier one.
Fig. 3 What the tubes are for. A trajectory that enters the neck on a stable manifold leaves it on an unstable one, and the two together connect regions of the state space at essentially no cost in velocity — so a transfer that would take a burn to arrange by patched conics can be arranged instead by arriving in the right tube. The saving is not free in time: these routes take months where a Hohmann transfer takes days, and the currency being traded is exactly that.

The free transfer

The consequence that turned this from dynamics into engineering is what happens when two tubes meet.

Take the unstable manifold of a halo orbit about the Earth–Moon L2L_2, and the stable manifold of a halo orbit about the Sun–Earth L2L_2. Each is a surface; in the four-dimensional space they both live in, two two-dimensional surfaces generically intersect in points, and a point of intersection is a single trajectory that lies on both. Such a trajectory leaves the first periodic orbit asymptotically and arrives at the second asymptotically, and the join costs nothing at all, because it is one trajectory rather than two joined by a burn.

That is the origin of the phrase “interplanetary transport network”, and it is not a metaphor. A route through it is a sequence of tube segments joined at intersections, and the manoeuvres are the small ones needed to get onto the first tube and off the last.

What was actually measured

The technique has a first flight and it was an accident.

Japan’s Hiten was launched in 1990 as an engineering test, and after its main mission it had a few tens of metres per second of propellant left — far too little for a lunar capture by any conventional route, which needs several hundred. Edward Belbruno and James Miller designed a trajectory that took the spacecraft out to about 1.5 million kilometres, into the region where the Sun’s perturbation on an Earth-orbiting spacecraft is comparable to the Earth’s own gradient, and back in to arrive at the Moon already captured, at essentially zero relative energy. Hiten entered lunar orbit in October 1991 on a manoeuvre of a few metres per second.

The transfer takes about five months against the three days of a direct Hohmann-class route, and it requires no capture burn at all. That is the trade, and it is not always worth taking: a crewed mission cannot spend five months in transit to the Moon, and a spacecraft’s own systems have to survive the extra time.

Since then the route has been flown deliberately. GRAIL used it in 2011, sending two spacecraft on three-and-a-half-month low-energy transfers so that they arrived at the Moon a day apart with the small relative velocity their gravity-mapping mission needed. ARTEMIS in 2011 moved two spacecraft from Earth orbit through Earth–Moon L1L_1 and L2L_2 into lunar orbit using manifold segments, and Genesis, WMAP, Herschel, Gaia and JWST have all used the manifolds of Sun–Earth halo orbits to reach or to hold their stations.

The weak stability boundary

Belbruno’s own route to this was not through manifold theory. He defined a region around the Moon — the weak stability boundary — by a numerical experiment: start a spacecraft at a given point with a given velocity and integrate, and ask whether it completes an orbit about the Moon before escaping. The set of initial conditions for which the answer is “just barely” is a fractal-looking region, and a trajectory arriving into it is captured without a burn.

The two descriptions turned out to be the same thing. The weak stability boundary is, to within the accuracy of the numerical definition, the intersection of the manifold tubes with the region round the smaller body — an equivalence established a decade after the Hiten flight. A trajectory was flown before anybody could say what it was.

The gap between the two descriptions is worth dwelling on, because it is the difference between a recipe and an explanation. Belbruno’s algorithm answers the question “is this initial condition captured?” and can be run on any point, giving a map of the region that is correct and offers no reason. The manifold description answers “why is the boundary where it is?”, and its answer — that the boundary is the stable manifold of a periodic orbit round a saddle point — also predicts things the numerical experiment does not, including that the boundary should be a smooth surface rather than the fractal it appeared to be, and that its structure should repeat at every energy for which the neck is open.

Both were needed. The numerical map made the technique flyable and the manifold theory made it designable, and the decade between them is a fair illustration of how mission design and dynamical systems theory have come to operate on each other.

What it costs, in the only unit that matters

A comparison in numbers, for the same journey — low Earth orbit to lunar orbit.

Route Total Δv Time
Hohmann transfer with a capture burn ~4.1 km/s 3 days
Bi-elliptic via a high apogee ~3.9 km/s 2 weeks
Low-energy transfer via the weak stability boundary ~3.7 km/s 3–5 months

The saving is a few hundred metres per second out of four thousand, which is between five and ten per cent. That sounds modest and is not, because the rocket equation is an exponential: at an exhaust velocity of 3 km/s, saving 400 m/s multiplies the delivered payload by e0.13=1.14e^{0.13} = 1.14, and on a mission where the payload is a tenth of the departure mass a fourteen per cent gain in payload is a different spacecraft.

The saving is entirely in the capture burn. The departure is much the same either way; what the low-energy route removes is the several hundred metres per second normally spent killing the arrival speed, because the arrival speed is already near zero. That is what “ballistic capture” means, and it is the only part of the manoeuvre budget the three-body dynamics can give away for free.

How a tube is actually found

Nothing about the tubes can be written down in closed form, and the procedure that produces them is worth setting out because it explains what kind of object they are.

Start with a periodic orbit around the equilibrium point — a closed loop that the equations return to exactly after one period. Such orbits exist in families around each collinear point, and finding one is a boundary-value problem solved by shooting: guess a starting state, integrate for one period, and correct until the endpoint matches the start.

Then linearise around it. Integrating the variational equations alongside the orbit for one full period gives the monodromy matrix, which says what happens to a small displacement after one circuit. Its eigenvalues come in reciprocal pairs; one of them is large, and its eigenvector points along the direction in which a nearby trajectory runs away.

The manifold is what that runaway sweeps out. Displace the state along the unstable eigenvector by a tiny amount — a hundred metres, say, on an orbit hundreds of thousands of kilometres across — and integrate forward. Do it from every point around the periodic orbit and the trajectories together form a surface: a tube, extending away from the neck.

The stable manifold is the same construction with the small eigenvalue, integrated backwards in time.

So a tube is a family of trajectories rather than a place, computed by a linear analysis at the start and a nonlinear integration afterwards, and its usefulness comes from the fact that the linear step identifies which direction to take. Anywhere else in the problem there is no such direction, and finding a trajectory means searching.

Everything above is drawn at a mass fraction of 0.15, which is far larger than any real pair. The structure survives the reduction.

Two outcomes, and a boundary with no width. 26 trajectories launched from one point beyond L₂, all at the one speed the Jacobi constant C = 3.2661 permits there, differing only in the direction they set off in. The heavy curve is the zero-velocity boundary at that constant — the region no trajectory of this energy may enter — and it is open at L₂ by the neck the trajectories are aimed at. 11 of the 26 pass through into the secondary's realm and 15 turn back, and they are not interleaved — sweeping the launch direction through 65° finds one changeover and nothing in between. Bisecting the first of them pins it to 2.6e-12 radians, and the integrator runs out of digits before the boundary runs out of sharpness. That surface is the tube. It is the stable manifold of the periodic orbit about L₂, it separates transit from non-transit everywhere and not only in this fan, and a mission that wants to arrive for nothing has to be put inside it.
Fig. 4 The same fan of trajectories at a mass fraction of 0.03 — roughly a very massive planet around a small star. The boundary between the two outcomes is still a curve of zero width and the trajectories still sort themselves cleanly across it. Nothing about the mechanism needs the secondary to be large.
Three topologies of the allowed region, mass fraction 0.03. The boundary of the region a particle may occupy, at three values of the Jacobi constant. Above C(L₁) = 3.318 the two bodies have separate lobes and nothing can pass between them; below it the lobes merge at the inner point; below C(L₂) = 3.278 the merged region opens to the rest of the plane. Nothing has been integrated: the boundary is a level set of a conserved quantity.
Fig. 5 The three topologies at the same reduced mass fraction. The necks are narrower and they open at almost the same values of the Jacobi constant, because what decides the topology is the shape of the effective potential near the collinear points rather than the depth of the secondary’s own well.

The same structure, delivering rocks

The tubes were found by people designing spacecraft trajectories, and the same geometry governs a transport problem nobody designed.

A meteorite arriving at the Earth was until recently an asteroid, and it got here because its orbit was chaotic. The mechanism is now well understood: fragments produced by collisions in the main belt drift slowly under radiation forces until they reach one of the resonances with Jupiter, and inside a resonance the eccentricity grows on a timescale of a million years until the orbit crosses the inner planets’.

What the manifold picture adds is the shape of that escape. A resonance’s boundary in phase space is not a wall; it is a surface woven from the invariant manifolds of the unstable periodic orbits inside it, and the same tube structure that lets a spacecraft leave a neck for free is what lets an asteroid leave a resonance. The routes are narrow, they are determined by the geometry rather than by the details of the body, and the transit times along them are set by the same linearised growth rate.

That explains a fact about meteorites that would otherwise be a coincidence: their cosmic-ray exposure ages — the time spent as small bodies in space, measurable from the isotopes their surfaces accumulated — cluster at a few million to a few tens of millions of years, which is the transit time along the routes rather than anything about the collisions that started them.

The same argument applies at the other end of the solar system, where the tubes associated with Jupiter’s neighbourhood mediate the passage of comets between the outer reservoirs and the inner system — the transitions that a Tisserand contour permits and that nothing else says the timing of.

Why the routes are narrow

One feature of the transport picture deserves separating out, because it is what makes the tubes an explanation rather than a description.

A tube is a surface of one dimension less than the space it sits in, so the set of initial conditions lying exactly on it has measure zero — nothing is ever exactly on a manifold. What matters is the region enclosed by the tube, because a trajectory inside it is committed: it will pass through the neck, and one outside will not.

That turns a question about a continuum of possible orbits into a question about which side of a surface a body is on, and it is why transport in these systems is so sharply selective. Two asteroids with almost identical orbits can have entirely different fates over a million years, and the difference is not sensitivity to initial conditions in the usual chaotic sense — it is that one of them is inside the tube and the other is not.

The tubes are therefore the geometry underneath the chaos, and a system whose long-term behaviour cannot be predicted by integration can still have its possible behaviours enumerated by finding them.

That is also the reason the structures were found by mission designers rather than by dynamicists studying the solar system. A designer asks which trajectories reach a target and is therefore already thinking about the boundary between those that do and those that do not; a dynamicist integrating orbits sees only that the outcomes are unpredictable. The tubes are the answer to the first question and are invisible to the second.

It is a good example of a pattern worth noticing: a structure that organises a system’s behaviour can be invisible to the standard way of studying that system, and become obvious as soon as somebody asks an engineering question about it.

The same has happened with the resonance transport above, which was described qualitatively for decades before the manifolds gave it a geometry, and the geometry then predicted the transit times that the meteorite exposure ages confirmed.

Which is a reasonable summary of the whole rung: a set of surfaces nobody can see, computed from a linear analysis around orbits nobody has flown, that between them decide which rocks arrive here and how long they took.

Where the model stops

The tubes are objects of the circular restricted three-body problem, and reality is neither circular nor restricted.

The Moon’s orbit has an eccentricity of 0.055 and the Sun perturbs the Earth–Moon system substantially, so the periodic orbits are not periodic and the manifolds are not invariant. What survives is a set of quasi-periodic structures that behave similarly over the timescale of a transfer, and every real mission design begins with the restricted problem and finishes in a full ephemeris model with numerical continuation between them.

The tubes are also defined at a single energy. A spacecraft that fires an engine changes its Jacobi constant and steps onto a different family; a route through several tubes at different energies requires a burn at each step, and the design problem is to make those burns as small as the geometry permits.

And the whole picture is planar in its simplest form. The out-of-plane structure is where the useful halo orbits live, and it is a genuinely three-dimensional problem with no closed form at all.

How big a Roche lobe is. The volume-equivalent radius of a star's Roche lobe, as a fraction of the orbital separation, against the mass ratio q = M/M_companion. The curve is Eggleton's 1983 fit; the marked points are the lobe's actual volume, counted on a three-dimensional grid of the same rotating potential the lobes themselves are drawn from — 0.267 against 0.268 at q = 0.25, 0.380 against 0.379 at q = 1, 0.504 against 0.501 at q = 4. Two features do all the work. The radius is 0.379 a for an equal pair and it approaches 0.49/0.6 = 0.817 a as the mass ratio grows without bound, so a star cannot avoid its lobe by being heavy — at q = 100 it is only 0.720 a. And the curve is shallow: a tenfold change in mass ratio moves the lobe by less than a factor of two, which means a star that fills its lobe and starts transferring mass does not escape by changing q — it changes the separation instead, and that is the runaway.
Fig. 6 And the size of the region a neck connects. The Roche lobe’s radius depends on the mass ratio through a fit that is accurate to a per cent over four decades, and the neck sits at its inner edge — so how much of the state space a tube can reach is fixed by one number. That is why the same construction describes a spacecraft leaving the Earth’s neighbourhood and a star overflowing onto its companion: the geometry is one function of the mass ratio, and everything else is scale.

Two more readings of the same effective potential, at the other end of the problem, where the question is not how a body escapes a lobe but how large the lobe was to begin with.

What transferring mass does to the lobe it came from. The donor's Roche-lobe radius against how much of its mass it has left, for conservative transfer — total mass and total angular momentum both held fixed — starting at mass ratios 1.2 and 0.8. Both curves come from J = M₁M₂√(Ga/M) rearranged for a, times Eggleton's lobe radius at the running mass ratio; nothing is fitted. They go opposite ways, and that is the whole of the Algol paradox. When the donor is the heavier star the separation shrinks, the lobe shrinks with it, more mass is pushed through L₁, and the transfer accelerates until the ratio reverses. When the donor is the lighter star the separation grows, the lobe grows away from it, and the transfer is self-limiting. So a system found transferring mass is almost always found with the less massive star overflowing — which is why Algol's evolved secondary weighs less than its unevolved primary, and why that looked for fifty years like a star ageing faster than a heavier one.
Fig. 7 What mass transfer does to the lobe it left, started from a much less extreme mass ratio. The donor’s lobe shrinks as it loses mass and the accretor’s grows, and for a ratio near unity the two effects nearly cancel — which is why transfer at comparable masses is slow and transfer from the heavier star is a runaway.
How big a Roche lobe is. The volume-equivalent radius of a star's Roche lobe, as a fraction of the orbital separation, against the mass ratio q = M/M_companion. The curve is Eggleton's 1983 fit; the marked points are the lobe's actual volume, counted on a three-dimensional grid of the same rotating potential the lobes themselves are drawn from — 0.167 against 0.168 at q = 0.05, 0.321 against 0.321 at q = 0.5, 0.442 against 0.440 at q = 2. Two features do all the work. The radius is 0.379 a for an equal pair and it approaches 0.49/0.6 = 0.817 a as the mass ratio grows without bound, so a star cannot avoid its lobe by being heavy — at q = 100 it is only 0.720 a. And the curve is shallow: a tenfold change in mass ratio moves the lobe by less than a factor of two, which means a star that fills its lobe and starts transferring mass does not escape by changing q — it changes the separation instead, and that is the runaway.
Fig. 8 The lobe radius over a wider range of mass ratio, checked at three points including one far into the low-mass regime. The fit is good to a fraction of a per cent everywhere, which is why a formula published in 1983 is still the one every binary-evolution code uses.

Where this ladder goes next

The rungs below this one establish the five points and the curves that bound the motion; this rung is about the structures that decide which trajectories go where. The rungs above are about using them.

The nearest is station-keeping at an unstable point, which is the price of the same instability: a spacecraft at Sun–Earth L2L_2 drifts off its halo orbit exponentially with an ee-folding time of about twenty days, and holding it there costs a few metres per second a year in manoeuvres timed to cancel the unstable component before it grows. That is an extraordinarily cheap place to sit for how useful it is.

Beyond it lies the question of what the network looks like as a whole — which points connect to which, at what energies, and whether a route exists between any two given places in the solar system at arbitrarily low cost. The answer appears to be yes in principle and centuries in practice, which is a characteristic result for this kind of dynamics: what is possible and what is useful are separated by the one quantity the method spends freely, which is time.

Where this goes next

The five Lagrange points at mass fraction 0.12. The five points at which a small body can keep station with two larger ones. The three on the line of centres are roots of a quintic and are unstable; the two forming equilateral triangles are stable for a sufficiently lopsided mass ratio. Five places that keep station, in a problem with no solution Three bodies under gravity cannot be solved. Restrict the problem slightly and five exact answers fall out anyway — three of them roots of a quintic, two of them perfect equilateral triangles. Zero-velocity curves, mass fraction 0.15. Level sets of the Jacobi constant in the frame that rotates with the two bodies. A particle with a given value of C is confined to the side of its own curve where the kinetic energy would be positive; as C falls the curves open, first at the inner point between the bodies, then behind the smaller one, and finally around the whole system. The curve that says where a body cannot go The restricted three-body problem has no solution and one conserved quantity. That quantity is enough to draw a boundary the body can never cross — without integrating anything, without knowing where it started, and for all time. Total Δv against the radius ratio. Total transfer Δv, in units of the starting circular speed, against the ratio of the two circular radii. The Hohmann transfer is cheapest at small ratios; the bi-elliptic transfers overtake it, and the limiting one — an intermediate apoapsis taken to infinity — crosses at a ratio of 11.94. Above about 15.6 every bi-elliptic transfer beats the Hohmann. Going too far in order to arrive cheaply The Hohmann transfer is the cheapest two-burn route between circular orbits. Past a radius ratio of 11.94 the cheapest route is three burns, and it goes far beyond the destination first.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Ballistic captureHalo orbitInvariant manifoldJacobi constantLow-energy transferMission designSeparatrixTransit orbitWeak stability boundaryZero-velocity curve