The orbit that has no period

Every path an inverse-square force allows is a conic section, and the eccentricity alone decides which one. Above $e = 1$ the curve stops closing, and more goes with it than the closure. There is no period, no aphelion, no mean distance, no epoch at which the body was last here: almost every quantity ordinarily used to describe an orbit is defined by the return.

What is left is two numbers, and they are enough. A closest approach and an eccentricity fix the impact parameter, the direction the body arrived from, the angle it turned through and the speed it leaves with. The bookkeeping asks for one concession first: a semi-major axis of $-1.273$ AU.

That is the sort of formal move a reader is right to distrust, and the figure below draws the sign rather than asserting it. The asymptotes cross 1.529 AU from the Sun on the same side as periapsis, where an ellipse’s centre lies on the far side. The negative sign is not a device for keeping an equation tidy; it says which side of the focus the centre of the conic sits on.

The sign that carries the energy #

The specific energy of a two-body orbit, kinetic plus potential, evaluates on any conic to

$$\varepsilon = \frac{v^2}{2} - \frac{\mu}{r} = -\frac{\mu}{2a}.$$

That is the whole of the argument. Energy and semi-major axis are one quantity written two ways, so a positive energy — the condition for never coming back — requires a negative $a$, and there is no freedom to define it otherwise.

The geometry agrees without being asked. Any conic has $q = a(1-e)$, so $a = q/(1-e)$: positive below $e = 1$, infinite at $e = 1$, negative above. The figures use $|a| = q/(e-1)$, the same number with its sign carried separately.

Then the vis-viva relation survives the change untouched:

$$v^2 = \mu\left(\frac{2}{r} - \frac{1}{a}\right) = \mu\left(\frac{2}{r} + \frac{1}{|a|}\right).$$

No case has been distinguished; the second term changes sign with $a$, and thereby changes what happens at large $r$. On an ellipse the two cancel at $r = 2a$ and the speed runs out; on a parabola the second is absent and the speed falls to zero; on a hyperbola the first vanishes and the second does not.

What is left over is the whole subject of this essay. As $r \to \infty$,

$$v_\infty = \sqrt{\frac{\mu}{|a|}},$$

and for ʻOumuamua’s $|a| = 1.273$ AU that is one division: the circular speed at 1 AU is 29.79 km/s, and $29.79/\sqrt{1.273} = 26.40$ km/s. Squared, the same quantity is the characteristic energy $C_3$ that launch vehicles are advertised against — a first hint that an interstellar visitor’s orbit and a spacecraft’s departure are one piece of arithmetic.

The angle a single number fixes #

The conic equation $r = p/(1 + e\cos\nu)$ has no solution once its denominator turns negative, which above $e = 1$ happens at a finite angle. Setting $\cos\nu_\infty = -1/e$ gives the asymptote’s direction, $146.37°$ at $e = 1.201$. The body sweeps that on the way in and again on the way out — $292.74°$ against the $360°$ a closed orbit turns through — and the deficit is the deflection:

$$\delta = 2\nu_\infty - 180° = 112.74°.$$

A little trigonometry collapses that into the identity the figure hangs on, $\sin(\delta/2) = 1/e$, which at $e = 1.201$ reads $0.8326$. The eccentricity is the reciprocal of the sine of half the turn, and nothing else enters — not the Sun’s mass, not the periapsis distance, not the speed.

The impact parameter, $b = |a|\sqrt{e^2-1} = 0.847$ AU, is where the body would have passed had gravity been switched off. It is also the angular momentum in disguise, since $h = b\,v_\infty$ exactly, so both conserved quantities are lengths on one drawing: energy is where the asymptotes cross, angular momentum is how far off centre they run.

The eccentricity is a poor label because it is a shape, measured against a periapsis distance that has nothing to do with where the body came from.

Time, on a path that never repeats #

Position against time on an ellipse has no closed form and is computed anyway, by solving $M = E - e\sin E$. The open case is the same construction with hyperbolic functions:

$$M = e\sinh H - H, \qquad r = |a|(e\cosh H - 1),$$

with $M = nt$ and mean motion $n = \sqrt{\mu/|a|^3}$. It is transcendental in the same way, yields to the same Newton iteration, and degenerates near $e = 1$, where Barker’s equation — a cubic, solvable exactly — takes over. What has changed is the meaning of $n$. On an ellipse it is $2\pi/T$ and every value of $M$ recurs; here $M$ runs once from $-\infty$ to $+\infty$, $n$ is a rate with no cycle attached, and $H$ is not an angle at all.

Everything ever known about ʻOumuamua was gathered inside the narrowest part of that curve.

The speed that is left over #

Because $v_\infty$ depends on $|a|$ and nothing else, it separates objects the eccentricity files together.

A long-period comet with perihelion at the Earth’s distance, nudged by Jupiter to $e = 1.0001$, has $|a| = 10^4$ AU and leaves at $29.79/100 = 0.30$ km/s. The Sun’s own motion with respect to the neighbouring stars is sixty times larger, so the comet goes nowhere in any galactic sense.

C/1980 E1 is the measured version. Jupiter caught it in 1980 and left it at $e = 1.057$ with $q = 3.36$ AU, the largest eccentricity of any comet known to have originated here — and $|a| = 3.36/0.057 = 59$ AU gives $v_\infty = 3.9$ km/s, a seventh of ʻOumuamua’s speed.

The classification lives in the gap between 0.30 and 26.40 km/s, while the eccentricities differ by two parts in ten. An object at $e = 1.0001$ and one at $e = 1.201$ are the same conic class and unrelated things.

What was actually measured #

Nothing above was observed. A telescope produces two angles and no distance — a right ascension and a declination measured against catalogue stars on the same frame — and every quantity here belongs to a conic fitted to a sequence of those. ʻOumuamua was found by Pan-STARRS on 19 October 2017 and followed for about eighty days; its eccentricity came out as $1.201 \pm 0.002$.

Propagating that uncertainty into $v_\infty$ is where the essay’s claim could fail. Since $|a| = q/(e-1)$, a fractional error in $|a|$ is $\Delta e/(e-1)$, which at $e = 1.201$ multiplies the eccentricity error by five: two parts in a thousand on $e$ becomes one per cent on $|a|$ and, since $v_\infty \propto |a|^{-1/2}$, half a per cent on the speed. So 26.40 km/s carries an uncertainty near 0.13 km/s, and the positive energy stands at about a hundred times its own error bar. At $e = 1.0001 \pm 0.0002$ the amplification is two thousand and the sign of the energy is not determined at all, which is why marginal comets are catalogued with parabolic elements.

The direction is the same fitted asymptote. ʻOumuamua arrived from within a few degrees of the solar apex in Lyra — the direction the Sun itself travels with respect to the neighbouring stars — so subtracting the Sun’s own 18 km/s from the 26.4 km/s of arrival leaves roughly 10 km/s, which is a slow star rather than a fast one. What was measured is a speed relative to the Sun; the statement it supports is about motion relative to the local stars, and needed the Sun’s own. No spectrum contributed: the Doppler shift that reads a speed off a line is useless on a twentieth-magnitude object reflecting sunlight, so the whole velocity came out of astrometry and the two-body problem.

The comets that were hyperbolic on paper #

Long before 2017, catalogues listed comets with eccentricities above 1, and none was interstellar. Published elements are osculating elements: the conic a body would follow if every other force were switched off at that instant. With Jupiter placed conveniently, an ordinary comet’s osculating eccentricity can exceed 1 near perihelion while the orbit it will actually follow is bound.

Jan Oort’s 1950 paper turned that nuisance into a measurement by working not in $e$ but in $1/a$ — proportional to the energy, positive when bound, negative when not, smooth through the boundary — computed in barycentric coordinates before the planets had touched it. The distribution of original $1/a$ for long-period comets has a sharp spike within about $10^{-4}\ \text{AU}^{-1}$ of zero, with a scatter consistent with measurement error rather than with any population beyond.

That spike is the Oort cloud, discovered as a histogram of a reciprocal semi-major axis — and its other half is the negative result this essay depends on: no comet’s original energy was convincingly positive. Two centuries of astrometry had produced hyperbolic elements by the hundred and interstellar objects by the none, and it took a change of variable to see that.

The two cases are separated by about three orders of magnitude in $v_\infty$, and nothing in an eccentricity above 1 distinguishes them. That is the whole content of Oort’s change of variable: $e$ crosses its threshold on a scale set by the perihelion distance, which is an accident of where the comet happens to be, while $1/a$ crosses zero on a scale set by the energy, which is the quantity the question is about. A catalogue that sorts on the first will keep producing false interstellar objects; one that sorts on the second produced none in two centuries and then two in six years.

The same identity, performed on purpose #

The turn angle has a separate life in mission design. A gravity assist is a hyperbolic pass of a planet: the spacecraft arrives with some $v_\infty$ relative to it, is turned through $\delta$, and departs with the same $v_\infty$, nothing having been added in the planet’s frame. The turn obeys $\sin(\delta/2) = 1/(1 + r_p v_\infty^2/\mu)$, which is the identity above with the eccentricity written out in terms of the approach speed — so the figure below is drawn at ʻOumuamua’s own deflection, by a generator written for spacecraft that needed no change to accept it. The surprising part is that the Sun performed this manoeuvre on ʻOumuamua, and that it counts. In the Sun’s frame nothing was gained: the object arrived at 26.40 km/s and left at 26.40 km/s, turned by $112.74°$. But the Sun is itself moving at 18 km/s with respect to the local stars, and adding that motion back to a velocity rotated by $112.74°$ does not return the velocity it had before. ʻOumuamua’s galactic velocity was changed by the encounter, by the same arithmetic that changes a probe’s heliocentric velocity at Jupiter. The only difference between a slingshot and an interstellar flyby is which frame the third velocity is added in — and whether anybody chose the geometry.

The direction, and why it does not name a star #

The asymptote is a direction as well as a speed, and the obvious next question is which star the object left. It is a question the measurement cannot answer, and the reasons are worth setting out because they are about the Galaxy rather than about the fit.

Start with what is available. The incoming asymptote gives a direction — the radiant — and $v_\infty$ gives a speed with respect to the Sun. Adding the Sun’s own motion converts that into a velocity with respect to the local stars, which for ʻOumuamua came out near 10 km/s and, remarkably, close to the mean motion of the stars in the solar neighbourhood. An object at rest in the local standard of rest is what a body that has been drifting for a long time looks like: the velocity dispersion of stars grows with age, so a slow visitor is more likely to be an old one than a young one from nearby.

Running the trajectory backwards is where it fails. The Galaxy’s gravitational field is not smooth on the scale of an orbit through the disc — the object passes molecular clouds, spiral arms and the disc’s own vertical potential, each perturbing it — so an integration backwards diverges from the truth on a timescale of tens of millions of years. Over that interval the object covers a few hundred parsecs, and the number of candidate stars within the error volume grows into the thousands.

The travel time is almost certainly far longer than the integration is good for. At 10 km/s a body crosses a parsec in about a hundred thousand years, and the plausible ages of these objects — set by how long a planetesimal ejected during planet formation has been drifting — run to hundreds of millions or billions of years. A trajectory reliable for ten million years and a journey lasting a thousand times that do not meet.

Attempts have been made anyway, matching ʻOumuamua’s incoming velocity against the catalogued motions of nearby stars, and they return a handful of candidates each of which is a coincidence at the level the statistics permit. The honest conclusion is that the object’s origin is not recoverable, and that this is a property of the Galaxy rather than a limitation of the astrometry.

What the direction does support is a statistical statement. A population of interstellar bodies with a velocity distribution matching the local stars would arrive from directions weighted towards the solar apex, because the Sun is running into them — the same reason more meteors are seen after midnight. Both of the first two objects arrived from broadly that hemisphere, which is consistent and, with two objects, proves nothing.

The sample has since grown. A third such object was found in 2025, on a considerably more extreme orbit than either of the first two — an eccentricity of about six and a speed at infinity near 58 km/s, which by the arithmetic of this essay means a very small $|a|$ and a body that is moving fast with respect to the local stars rather than drifting with them. Three objects with excess speeds of 26, 32 and 58 km/s are the beginning of a velocity distribution, and a velocity distribution is the quantity that says something about where such bodies come from — which the individual trajectories, for the reasons above, cannot.

An orbit with no period is still an orbit, and everything the two-body problem says about it holds exactly — what is lost is only the convenience of a quantity that repeats, and with it every method that was built on one.

What the picture cannot show #

The plane. Every drawing here is face-on, and the tilt is what turns a speed into a place in the sky. ʻOumuamua’s inclination is $122.7°$ — it went round the wrong way — and no in-plane figure carries that, or the two further angles fixing the asymptote among the constellations. Those three are most of what “where it came from” means.

Which way the body went. A conic is a curve; a trajectory is a curve traversed. The hero figure would be identical for an object arriving along either asymptote, and only the anomaly figure, which has time on an axis, tells them apart.

Whether the orbit is real at more than one epoch. ʻOumuamua showed a small non-gravitational acceleration on the way out, and any such force makes $a$ a function of time. Then $v_\infty$ belongs to the fit’s epoch rather than to the object, and the clean hyperbola above is one instant’s orbit extended both ways.

The margin. Every curve here is drawn exactly, and a real determination draws a band — one whose width, near $e = 1$, spans the boundary. No figure on this page can show a curve that might be either.

Where the open conic stops being the orbit #

The heliocentric hyperbola is a two-body solution and the Sun has eight planets; what saves it is volume rather than principle. Those bubbles are also the whole mechanism of the middle case: C/1980 E1 was ejected because it went inside one, where a different hyperbola applies and where Jupiter can change a heliocentric energy by more than the entire energy of a marginal orbit. One figure explains both why the interstellar visitors are clean single conics and why solar-system comets on unbound orbits exist at all.

At the other end the model fails differently. The infinity in $v_\infty$ is a limit taken on a two-body problem, and the Sun’s dominance ends around $10^5$ AU, where the galactic tide takes over. At 26.40 km/s the limit is reached long before that matters; for the comet at 0.30 km/s it is not, and its asymptotic speed belongs to a region it never crosses undisturbed.

The family is also Newtonian and point-mass, and neither correction is worth a term at a periapsis of 0.2559 AU. This model does not fail for want of relativity; it fails, when it fails, because a third body exists. The two known interstellar objects also make a statement about the ones that were not seen. A detection rate of two in six years from surveys of a known depth implies a number density of such objects in the solar neighbourhood, and the number that comes out is of order one per hundred cubic astronomical units — far more mass in unbound debris than any straightforward account of planet formation ejects. Either the ejection efficiency is much higher than modelled, or the population is dominated by objects from a source nobody has identified, or both detections were lucky. Each of those is testable by the next decade of wide-field surveys, and the quantity they will test is a number extracted entirely from an eccentricity above one. A number that large, resting on two objects, is the kind of estimate that moves by an order of magnitude on the third detection.

Where the ladder goes next #

Later rungs on this anchor: the hyperbolic Kepler equation solved numerically, and the universal-variable formulation that handles all three conics without branching on the eccentricity. The $B$-plane, the impact parameter turned into a targeting coordinate, which is how a flyby is aimed. Original and future barycentric $1/a$, computed properly for a real comet. The capture problem, in which a visitor loses just enough energy to a planet to stay. And the arrival-rate inference — what two objects in two years implies about the number density of interstellar bodies, which is an argument about a survey’s sensitivity rather than about either object.

Apollonius named the hyperbola for an excess — a comparison of areas, in a geometry problem with no motion in it. The excess turned out to be an energy, and the number it delivers is the only thing an interstellar object says about the star it left.