Every orbit one force allows, and the number that picks between them

Apollonius of Perga wrote eight books on the conic sections around 200 BC. He was slicing a cone with a plane and cataloguing the curves that resulted: circle, ellipse, parabola, hyperbola. There is no physics in it anywhere, no force, no motion, and no suggestion that the curves were the shape of anything.

Eighteen hundred and eighty-seven years later, Newton showed that those four curves — all of them, and nothing else — are the complete set of paths a body can follow under an inverse-square attraction. The catalogue was finished long before there was anything to put in it.

One family, one parameter #

Every one of these curves obeys the same equation,

$$r(\nu) = \frac{p}{1 + e\cos\nu},$$

with $r$ the distance from the focus, $\nu$ the angle from the closest approach, $p$ a scale, and $e$ the eccentricity. That single formula produces all four, and the transitions between them are entirely in the denominator.

At $e = 0$ the cosine term vanishes and $r$ is constant: a circle. For $0 < e < 1$ the denominator stays positive at every angle, so $r$ is finite everywhere and the curve closes: an ellipse. At $e = 1$ the denominator reaches zero at $\nu = 180°$, so $r$ runs to infinity there and the curve never closes: a parabola. For $e > 1$ the denominator hits zero earlier, at $\cos\nu = -1/e$, and beyond that angle the formula returns nothing at all: a hyperbola, whose asymptotes are exactly those forbidden directions.

That last case is the one worth pausing on. The angles at which the equation has no solution are not a defect of the formula. They are the directions the body came in from and leaves along, and they are as much part of the trajectory as the curve is.

The apoapsis distance is $p/(1-e)$, which grows without bound as $e \to 1$. So the parabola is what an ellipse becomes when its far end is pushed to infinity, and the hyperbola is what happens when it is pushed past. Nothing discontinuous occurs anywhere in the family, which is a good deal less obvious than the four names suggest.

What the number really is #

The eccentricity looks geometric and is really an energy.

The total energy per unit mass of an orbiting body is kinetic plus potential:

$$\varepsilon = \frac{v^2}{2} - \frac{GM}{r},$$

and it is conserved. Its sign has an immediate meaning. Negative means the body is bound — it has less kinetic energy than it would need to climb out of the well, and it will come back. Positive means it is unbound and leaving for good. Zero is the boundary.

The eccentricity is related to it by

$$e = \sqrt{1 + \frac{2\varepsilon h^2}{(GM)^2}},$$

where $h$ is the specific angular momentum. Every feature of the family follows from the sign of $\varepsilon$ under the square root: negative energy gives $e < 1$, zero gives exactly 1, positive gives more than 1.

So the four shapes are three energy regimes and a boundary case, and the boundary is a set of measure zero. No real orbit is a parabola. A comet whose energy is within a part in $10^{12}$ of zero is on an extremely long ellipse or a barely hyperbolic path, and the difference is measurable; a parabola requires exactly zero, which nothing has. Parabolic orbits appear constantly in the literature as approximations for long-period comets, and every one of them is a convenience.

The vis-viva relation makes the energy visible. Written as $v^2 = GM(2/r - 1/a)$, the second term carries the energy: for a bound orbit $a$ is positive and the term is a subtraction, for a hyperbola $a$ is negative and the term becomes an addition, and for a parabola it is absent entirely, leaving $v^2 = 2GM/r$, which is the escape speed exactly.

A negative semi-major axis is a strange object and it is the right bookkeeping. A hyperbolic orbit has no far end, so “half the long axis” has nothing to measure, and the quantity that appears in the equations is the one that keeps the algebra working.

The visitors that settled it #

For two centuries every known object in the solar system was on a bound orbit. Comets appeared on paths so elongated that their eccentricities were indistinguishable from 1, and the question of whether any of them came from outside was unanswerable with the available precision.

Then in October 2017 a small object was found moving at 26 km/s relative to the Sun while still beyond the Earth’s orbit — far too fast to be bound. Its eccentricity came out at 1.20, which is not marginally above 1 and not a measurement error. ʻOumuamua came in from interstellar space, swung round the Sun on a hyperbola, and left. Borisov followed in 2019 at $e = 3.36$, which is emphatically unbound.

Those two objects are the observational proof that the upper half of the conic family is occupied. Before them it was a mathematical possibility with no known instance; the theory had predicted an entire branch of the curve family that nothing had ever been seen on.

Why exactly these curves #

The result that an inverse-square force gives conics is not obvious, and the reason is a hidden conserved quantity.

Any central force conserves energy and angular momentum. That is enough to confine the motion to a plane and to fix the range of distances, but not enough to make the orbit close: in general the direction of closest approach drifts round, and the path is a rosette that fills an annulus and never repeats.

The inverse square conserves one thing more — a vector that points from the focus toward periapsis and has a length equal to the eccentricity. It is called the Laplace–Runge–Lenz vector, and its constancy is exactly the statement that the direction of closest approach does not move. That is what makes the orbit close, and it is the extra symmetry that singles out the inverse square from every other exponent.

Bertrand’s theorem makes the uniqueness precise: of all central forces, exactly two produce closed orbits for every bound trajectory — the inverse square and the linear spring. Both were known to Newton, both are conic sections, and the difference between them is where the centre of attraction sits. For the spring the force pulls toward the centre of the ellipse; for gravity, toward a focus. Same curve, two different laws, distinguished by which point is occupied — which is the geometric content of Kepler’s first law restated as a claim about force.

The conserved vector also gives the cleanest test of whether gravity is exactly inverse-square. If it is not, the vector is not conserved, and the orbit precesses. Mercury’s orbit precesses by 574 arcseconds per century, of which 531 come from the other planets. The remaining 43 are not Newtonian, and they are among the most consequential leftovers in the history of physics.

What was actually measured #

An eccentricity is never observed. What is observed is a sequence of directions — a moving point of light against the background stars — and the conic is fitted to them. The fit is where the whole difficulty lives, and knowing how it works explains why “$e = 1.20$” is a strong claim and “$e = 1.0000$” would not be.

Six numbers specify an orbit. Each observation supplies two — a right ascension and a declination — so three observations are the minimum, and turning three directions into an orbit is a problem Gauss solved in 1801 when Ceres was lost after forty-one days of observation and reappeared where his calculation said it would be. His method is still the starting point: guess the distances at the three times, use the geometry to correct the guess, iterate.

The difficulty is that direction alone gives no distance, and a short arc of directions is compatible with a wide range of orbits. Over a few nights an object’s arc barely curves, and the fitted eccentricity has an uncertainty that easily spans the bound–unbound boundary. This is why newly discovered comets are announced with parabolic elements: not because anyone believes the orbit is parabolic, but because with $e$ fixed at exactly 1 the fit has one fewer parameter and converges on a short arc.

Against that background, ʻOumuamua’s numbers are worth stating properly. Its eccentricity was determined as $1.201 \pm 0.002$ from an arc of about eighty days, with observations from several telescopes and — critically — a measurable curvature to the path. The excess hyperbolic speed, $v_\infty$, came out at 26.3 km/s, which is far outside anything the solar system could impart. The claim is not that a number came out slightly above 1. It is that the energy came out positive by a hundred standard deviations.

The one real complication was non-gravitational. ʻOumuamua accelerated slightly as it left, by an amount consistent with outgassing, and any non-gravitational force makes the fitted conic a fiction to some degree. The acceleration was small enough not to threaten the unbound conclusion and large enough to have consumed several years of argument about what the object was made of.

The same curves, repelling #

The conic family belongs to the inverse-square force, and gravity is not the only inverse-square force there is. Switching the sign produces the identical mathematics with a single change, and the consequence was the discovery of the atomic nucleus.

Two charges of the same sign repel with a force going as $1/r^2$. The orbit equation is unchanged apart from the sign of the constant, and the solutions are conics — but the bound branch is gone. A repelled particle can never have negative energy, so only the hyperbolas survive, and the attracting focus becomes a focus the particle is pushed away from. The trajectory is the other branch of the same hyperbola.

Ernest Rutherford’s students fired alpha particles at gold foil in 1909 and found that about one in eight thousand came back toward the source. On the prevailing model of the atom — charge spread smoothly through its whole volume — the maximum possible deflection was a fraction of a degree, and a backward scattering was impossible by many orders of magnitude. Rutherford’s remark that it was as if a fifteen-inch shell had bounced off tissue paper is the standard quotation, and it is a statement about a probability, not about a surprise.

What he then did was compute. Assuming all the positive charge sat in a point, the trajectory is a hyperbola; the impact parameter fixes the eccentricity; the eccentricity fixes the turn angle. Integrating over impact parameters gives the fraction scattered past any given angle — the Rutherford cross-section, which goes as $\csc^4(\theta/2)$ — and the measured counts matched it across five orders of magnitude in rate. The nucleus was discovered by fitting a conic section.

The scaling in that formula also delivered the nucleus’s size. The agreement holds only while the alpha particle stays outside the nucleus; at high enough energies, or small enough impact parameters, the particles get close enough for the strong force to intervene and the counts depart from the curve. The energy at which the departure begins gives the radius. Apollonius’s curves, applied to a foil of gold, measure something $10^{-14}$ metres across.

The family used deliberately #

Interplanetary navigation lives in this family and moves between its branches on purpose. A spacecraft leaving Earth is on a hyperbola with respect to the Earth and an ellipse with respect to the Sun, simultaneously, and the two descriptions are stitched together at the edge of Earth’s gravitational influence. That patched-conic method — treat the trajectory as a sequence of two-body conics, each valid in its own region — is how every mission from Mariner to New Horizons was planned. It is an approximation with no rigorous justification and it works to within a course correction.

A gravity assist is a hyperbolic pass, and the turn angle it delivers is set entirely by the eccentricity of that hyperbola. The tighter the pass, the lower the eccentricity, the sharper the turn, and the more speed the manoeuvre extracts.

Moving between branches on purpose #

The family is not just a classification. A spacecraft changes which member of it is on, deliberately, several times per mission. Each burn moves the vehicle to a different conic through the same point, because a burn changes velocity and not position. Add enough energy at once and the conic becomes a hyperbola, which is what leaving means. The whole of trajectory design is a sequence of hops between members of this one family.

A flyby does the same thing without a burn. Passing a planet puts the spacecraft on a hyperbola with respect to that planet while leaving it on an ellipse with respect to the Sun — two conics at once, in two frames, both exact.

Drawn as a family, the hop across the boundary is less dramatic than the vocabulary suggests.

That is the practical content of the classification and it cuts both ways. On one hand the threshold is real: the difference between returning and not returning is a difference of kind, not of degree, and it is decided by an arbitrarily small change in speed at the right moment. On the other hand nothing local distinguishes the cases. A spacecraft on any of the three curves above, observed for an hour near periapsis, is on all three within the measurement error.

So the eccentricity of an orbit is never measured; it is inferred from an arc, and its uncertainty near 1 is always large because the curves are least distinguishable exactly where a body is easiest to observe. That is not a defect of any particular telescope. It is a property of a one-parameter family whose members agree to second order at the point they share, and it is the reason the interstellar-object question is settled in energy rather than in eccentricity. The same degeneracy is why a newly discovered comet’s orbit is published as a range of solutions rather than as a single conic, and why the range collapses only after the body has moved far enough along its path for the members of the family to have visibly separated. Waiting is the measurement, and the geometry says how long the wait has to be. For a comet at a few astronomical units that is months; for an object discovered on its way out, as the first interstellar visitor was, the arc available is short and the answer has to be extracted from astrometry precise enough to separate curves that differ in the fourth decimal place.

Where the model stops #

Two bodies. The conics are exact only for two point masses alone. With three, no closed-form solution exists at all, and the trajectory is not a conic and not any other named curve.

Point masses. Extended bodies deviate, and close binaries deviate a lot.

No drag, no thrust, no radiation pressure. A comet outgassing near perihelion is being thrust, and the resulting non-gravitational acceleration is large enough to have made ʻOumuamua’s orbit briefly controversial.

Newtonian gravity. Near a compact object, orbits precess and eventually decay by radiating gravitational waves — a possibility the conic family has no room for at all, since it conserves energy exactly.

And a limitation of the picture. All these figures are drawn in a plane and face-on, which no observer ever sees. A conic in space needs three more numbers to say how the plane is tilted and turned, and those three are where the practical difficulty of orbit determination actually lies. Knowing that a newly-found object is on a hyperbola is easy; knowing where it will be next month is not.

The member with no width #

There is a fifth case in the family that no figure on this page can draw, and it is the one that describes falling.

Set the angular momentum to zero. The orbit equation loses its meaning — the parameter $p$ is proportional to the square of the angular momentum and vanishes — and what is left is motion straight along a line through the centre. The body falls in, or is thrown out, with no transverse motion at all.

That is the radial trajectory, and it belongs to the family in a precise sense: it is the limit of an ellipse whose eccentricity approaches one at fixed semi-major axis, so that the width goes to zero while the length does not. The path degenerates to a line segment traversed out and back, and the period is still given by the harmonic law with the same semi-major axis — which is a useful and slightly startling fact, because it means the time for a body to fall from rest into a central mass is a quarter of the period of a very thin ellipse with the same starting distance.

The solution in time has no closed form in the elementary sense either, and for the same reason the elliptic case does not: the equation relating position to time is transcendental. What it does have is a parametric solution in which the position and the time are both simple functions of an auxiliary angle, and the curve traced by that pair is a cycloid.

The case matters more than its measure-zero status suggests, because it is the limiting behaviour every collapse approaches. A cloud with negligible rotation collapses along nearly radial paths, and the free-fall time computed from this degenerate orbit is the standard timescale of star formation. It is also the case in which the two-body approximation fails soonest, because a genuinely radial orbit ends in a collision rather than in a periapsis.

The ladder from here #

Later rungs: the derivation of the orbit equation from Newton’s laws. The Laplace–Runge–Lenz vector and the hidden symmetry behind it. Bertrand’s theorem. Hyperbolic anomaly, and Kepler’s equation for unbound orbits. The patched-conic method and spheres of influence. Interstellar objects, and what their arrival rate implies about how many there are. Radial trajectories, the degenerate case with zero angular momentum, which the family contains and no figure shows. And the orbits of general relativity, where the conics acquire a precession and, close in, stop closing at all.

Apollonius called the curves ellipse, parabola and hyperbola — deficient, exact and excessive — after how a certain area compared with a reference rectangle. The names describe an arithmetic comparison in a geometry problem, and they now describe the energy budget of everything that moves under gravity.