The curve that says where a body cannot go

The three-body problem has no general solution, and the sentence is usually delivered as an ending. It is closer to a beginning, because a problem can be enormously constrained without being solved, and the restricted three-body problem is the standing example of how much can be extracted from a single conserved quantity.

One integral, in a problem with none #

Set two massive bodies on circular orbits about their common centre, and let a third body of negligible mass move in their field without affecting them. That is the circular restricted three-body problem: it has three degrees of freedom and, famously, no complete set of integrals.

Now write the equations in a frame that rotates with the two primaries, so that both of them sit still. Two fictitious forces appear — centrifugal, which is the gradient of a potential, and Coriolis, which is not. The Coriolis force is perpendicular to the velocity, so it does no work; the centrifugal term folds into the potential; and the result is that a single combination is constant along every trajectory:

$$C_J = 2\Omega(x,y,z) - v^2, \qquad \Omega = \frac{1-\mu}{r_1} + \frac{\mu}{r_2} + \tfrac12(x^2+y^2).$$

This is the Jacobi constant, published by Carl Jacobi in 1836. It is not the energy: energy is not conserved in the rotating frame, because the frame does work on the particle. It is not the angular momentum either. It is the one combination that survives, and it survives because of the Coriolis force’s one virtue.

Why a constant becomes a boundary #

The step from a conserved quantity to a wall is one line, and it is the same step the effective potential of the two-body problem takes.

Rearranged, $v^2 = 2\Omega - C_J$. Speed squared cannot be negative, so the particle can only be where $2\Omega \ge C_J$. The surface where equality holds is the zero-velocity surface: the place the particle would arrive with no speed left. It cannot cross, because crossing would require an imaginary velocity, and it cannot be argued around, because $C_J$ does not change.

What makes this powerful rather than merely true is that it holds without integrating the trajectory. The three-body problem’s difficulty is that a trajectory cannot be written down; the Jacobi constant sidesteps the trajectory entirely and reports where it can and cannot be, for all time, from the initial conditions alone.

The shape changes, and the changes have names #

The allowed region depends on $C_J$, and as the constant falls the boundary changes its topology at four values. Each of those changes is a named phenomenon.

The closed region around each body above the first threshold is its Roche lobe. It is the region within which material belongs to that body rather than to the other, and its size is a computed quantity rather than a definition.

This has a consequence in stellar astronomy that runs the entire theory of interacting binaries. A star that expands — as every star does when it leaves the main sequence — eventually fills its own lobe, and the material at the L1 point is then free to move to the companion. That is Roche-lobe overflow — the mechanism by which the pairs of stars whose masses can be measured directly stop being ordinary pairs of stars, and it is why the majority of exotic objects in the sky are in binaries: novae, X-ray binaries, most type Ia supernova progenitors, and the accretion discs that make an ordinary white dwarf outshine its own galaxy for a fortnight. The threshold is not a matter of degree. Below it nothing crosses; above it a stream does.

Two things named after Édouard Roche appear in this collection and they are different. The Roche lobe is the region defined by the neck at L1, and crossing it is a statement about where material is allowed to be. The Roche limit is the distance at which a satellite’s own tidal stretching exceeds its self-gravity, and crossing it is a statement about whether a body holds together.

The Hill sphere, and a theorem about the Moon #

If the boundary at $C(L_1)$ closes a region around the smaller body, then anything inside that region with a sufficiently large Jacobi constant is trapped. That region’s characteristic size is the Hill radius,

$$r_H \approx a\left(\frac{m}{3M}\right)^{1/3},$$

which is 1.5 million kilometres for the Earth against the Sun, 0.06 astronomical units for Jupiter, and about 60,000 kilometres for the Earth against the Moon.

George William Hill turned this into a genuine theorem in 1878. Taking the Moon’s actual Jacobi constant in the Sun–Earth problem, he showed that the Moon’s zero-velocity surface is closed around the Earth. The Moon is therefore inside a region it cannot leave, whatever it does, for as long as the approximations hold — and this is a rigorous statement of permanent capture in a problem that has no solution. It remains one of the few results of that kind in celestial mechanics.

The Hill radius is also a design constraint. Every artificial satellite of every planet must be inside it. Every irregular moon — the captured, inclined, often retrograde outer satellites of the giant planets — sits inside its planet’s Hill sphere and typically within about half its radius, because orbits further out are unstable even though the boundary formally allows them. That gap between allowed and stable is the most important caveat in this essay, and it is the subject of the last section.

What was actually measured: a comet identified across an encounter #

The Jacobi constant’s most-used consequence is an identification, not a boundary.

A comet that passes close to Jupiter has its orbit changed drastically — that is what a close encounter is — so its semi-major axis, eccentricity and inclination before and after bear no simple relation to each other. This was a serious practical problem in the nineteenth century: a comet seen in one apparition, perturbed by Jupiter, and seen again decades later might be the same object under an unrecognisable set of elements.

Tisserand’s answer, in 1889, was to express the Jacobi constant in the comet’s heliocentric elements rather than in rotating-frame coordinates. The result is the Tisserand parameter,

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

which is approximately conserved through the encounter because $C_J$ is exactly conserved and the two differ only by terms of order the comet’s distance from Jupiter during the encounter. Two sets of elements that look nothing alike, sharing a $T_J$ to two decimals, are the same object.

That is the measurement, and it is still the working definition of the categories the small bodies are sorted into. Objects with $T_J > 3$ cannot cross Jupiter’s orbit in the relevant sense and are asteroids — the same population whose semi-major axes carry the gaps a resonance clears; those with $2 < T_J < 3$ are Jupiter-family comets; below 2 are the nearly-isotropic comets from further out. The boundary at 3 is not a convention chosen for tidiness — it is the value of $T_J$ for a body on Jupiter’s own orbit, which is to say a critical value of a conserved quantity. Comet 2P/Encke sits at 3.03 and 1P/Halley at −0.61, and those two numbers say more about their histories than their orbital elements do.

Two different radii for “where the planet wins” #

The Hill radius is not the sphere of influence, and the collection now contains both.

Patched conics divides a trajectory at the sphere of influence, whose radius is $a(m/M)^{2/5}$ — 924,000 kilometres for the Earth. The Hill radius is $a(m/3M)^{1/3}$, which is 1,496,000. They differ by 60 per cent and they answer different questions. The sphere of influence is where the perturbation from one body stops being the smaller of the two, and it is a statement about which two-body problem to use next; the Hill radius is where the zero-velocity surface closes, and it is a statement about what can escape. One is an accuracy criterion and the other is a theorem, and a spacecraft is routinely outside the first and inside the second.

The Jacobi constant also puts a hard limit on what a flyby can do. A gravity assist changes a spacecraft’s heliocentric energy enormously — that is its whole purpose — and leaves $C_J$ untouched, because $C_J$ is conserved through the encounter like everything else. So the Tisserand parameter of a spacecraft is the same before and after every unpowered flyby of the same planet, and mission designers use exactly that: a Tisserand graph plots the reachable orbits as contours of the conserved quantity, and a tour of the Jovian moons is planned by moving along one such contour and hopping to another only where a second body allows it. Cassini’s seven-year tour of Saturn is a walk on that diagram.

The last region to close #

Three of the four critical values have been used and the fourth has not, so it is worth saying what happens there.

As the Jacobi constant falls, the forbidden region shrinks. Past the value at the outer collinear point the particle can leave the system; past the value at the third collinear point, on the far side of the primary, the last connection round the outside opens. What remains forbidden is then two small islands, one around each triangular point.

Those islands shrink as the constant falls further, and at the value of the potential at the triangular points they vanish. Below that there is no forbidden region at all: the particle may in principle be anywhere.

That is worth stating because it bounds what the whole construction can do. The Jacobi constant is a powerful constraint only when its value is high — when the particle is slow-moving, deep in one of the wells, or far out and barely bound. A fast particle has a low constant, no boundary, and the conserved quantity tells nothing about where it goes.

Most of the objects the technique is applied to are in the useful regime. A satellite of a planet, a comet on a low-energy orbit, a spacecraft in a transfer between Lagrange points — all have constants near the critical values, which is exactly why the topology of the boundary is the interesting thing about them.

The particles with low constants are the ones that arrive from elsewhere at speed, and for those the Jacobi constant is still conserved and still useful, in the form the Tisserand parameter takes: not as a boundary in space but as a constraint on which orbits are reachable from which.

The same surfaces, applied to two stars #

The construction has been described for a massless third body, and its most-used application in stellar astronomy has no third body at all.

Two stars in a close orbit are each surrounded by their own equipotential surfaces in the rotating frame, and the classification of close binaries is a classification of which surfaces the stars fill.

A detached pair has both stars comfortably inside their own lobes. Nothing is transferred, and the two evolve as though they were single, except that tides keep their rotation synchronised and their orbits circular.

A semi-detached pair has one star filling its lobe exactly. Material at the inner Lagrange point is on the boundary between the two regions and flows across, so such a system is transferring mass — and because the transfer is driven by the donor’s own expansion, it continues for as long as the donor keeps evolving.

A contact pair has both stars overflowing, so the system is enclosed by a single surface and the two share an envelope. Such systems are common among low-mass stars and their light curves are continuously variable rather than showing distinct eclipses, because the shared envelope has no flat parts.

The classification is observational — it is read off the shape of an eclipsing light curve — and it is a direct measurement of the topology this essay has been drawing. A light curve with sharp, well-separated eclipses is a detached system; one with a smoothly varying baseline between eclipses is a distorted star nearly filling its lobe; one with continuous variation and equal minima is a contact binary.

A boundary derived from a conserved quantity in a problem with no solution is, in this application, a category a light curve is sorted into.

What the picture cannot show #

Necessary, not sufficient. The forbidden region is genuinely forbidden. The allowed region is not therefore visited. A particle inside a closed lobe with an enormous Jacobi constant might still be on a trajectory that hits the surface of the primary within a week. The boundary constrains and does not predict, and a great many orbits inside the Hill sphere are unstable on timescales of a few orbits despite being permanently bounded in principle.

The primaries’ orbit is circular, and real ones are not. Every element of a real orbit drifts, and the two primaries’ separation is no exception. With an eccentric binary the rotating frame is no longer uniformly rotating, $\Omega$ acquires an explicit time dependence, and the Jacobi constant is not constant. Everything above becomes approximate, and the approximation degrades with eccentricity. The Sun–Jupiter system is at $e = 0.048$ and the treatment is excellent; a binary star at $e = 0.6$ is a different problem.

The third body has no mass. The whole construction assumes the particle does not affect the primaries. For a comet or a spacecraft this is exact for every practical purpose; for a third star it is worthless. The mass fraction is the one parameter the whole picture depends on, and it is worth drawing the two readings of the curves at values either side of the one used above.

The same trick, twice #

It is worth putting the two side by side, because this collection now contains both and they are easy to run together.

In the two-body problem the ignorable coordinate is the angle, the conserved quantity is the angular momentum, and eliminating one with the other produces a curve in $r$ whose level sets are the turning points of an orbit. In the restricted three-body problem the ignorable coordinate is the time in a rotating frame, the conserved quantity is the Jacobi constant, and eliminating one with the other produces a surface in $(x,y,z)$ whose level sets bound the motion.

Both are instances of a habit worth naming: find what does not change, and let it draw the boundary. It is the only technique in this subject that survives the loss of an exact solution intact.

One more consequence is worth stating because it is the reason the whole construction is not merely elegant. A boundary that holds for all time is rare. Almost every result in celestial mechanics beyond two bodies is either a numerical integration, valid until its errors accumulate, or a perturbation series, valid until its denominators go small. The zero-velocity surface is neither: it is an exact consequence of an exact integral, and a body inside a closed lobe stays there whatever chaos its trajectory undergoes in between. Hill’s theorem about the Moon is two hundred years old and has not needed revision, which is not a sentence that can be written about many statements in this subject.

And the third dimension. Every figure here is a slice through the plane of the two primaries, and the real object is a zero-velocity surface. Out of the plane the potential has no centrifugal term along the rotation axis, so the surface closes over the top and bottom sooner than the in-plane curves suggest — which matters for exactly the case a reader is most likely to apply this to, a satellite on an inclined orbit inside a Hill sphere.

And the topologies at the mass fraction the essay’s own figures use, drawn over a wider span so the outer boundary is visible.

Where the ladder goes next #

The next rung is the dynamical structure inside the neck. The zero-velocity surface says a particle may pass through the L1 gateway; what governs whether it does, and where it goes afterwards, is the family of periodic orbits around the collinear points and the tubes of trajectories that wind onto them. Those tubes are the low-energy transfers that carried Genesis and SMART-1, and they are a rung of this ladder that begins exactly where this essay’s boundary stops being informative.