An orbit moved by heat

Every force in this collection so far has been gravity or a consequence of it. The ellipse comes from an inverse square, the wobbles come from other masses, and even the corrections that broke Newton are still corrections to a law about mass and distance.

There is a force on a small asteroid that has nothing to do with mass at all. It is a recoil, the recoil of thermal photons leaving a warm surface, and it is imperceptible beside anything the elements normally respond to: for a body a few hundred metres across it is a few parts in ten billion of the Sun’s pull. It changes the semi-major axis by a few hundred metres a year. And it is the largest single uncertainty in any prediction of where such a body will be a century from now.

Where the force comes from #

A body in sunlight absorbs and re-radiates. If it re-radiated instantly, every photon would leave along the line to the Sun and the recoil would be purely radial — an outward push, indistinguishable in its effect from a slightly weaker Sun. Radiation pressure does exactly that, and for a body larger than a few centimetres it does nothing interesting: a constant fraction subtracted from the solar attraction leaves the orbit a conic of the same shape, with a slightly different scale.

Real surfaces have heat capacity. A patch of ground absorbs sunlight through the morning, reaches its highest temperature some time after local noon, and is still warm at dusk. So the hottest part of a rotating body is not the sub-solar point but a patch rotated past it, and the thermal photons — which carry momentum whether or not anybody is watching them — leave preferentially from there.

The recoil is therefore tilted away from the sunward line by a thermal lag angle, and the component perpendicular to that line is along-track. That component is the whole story.

Why the size decides it #

The power a body absorbs is the solar flux times its cross-section, $\pi R^2$. The mass it has to push is $\tfrac{4}{3}\pi R^3\rho$. The acceleration is the ratio,

$$a_{\text{rad}} = \frac{3(1-A)F}{2\rho D c},$$

with $A$ the albedo, $F$ the solar flux at the body’s distance, $D$ the diameter and $c$ the speed of light. Everything about the geometry has cancelled except one over the size.

That single factor is why this force is a curiosity for planets and a controlling term for asteroids. Halve the diameter and the drift doubles. The Earth’s Yarkovsky drift, if the phrase meant anything for a body with an atmosphere and an ocean and a magnetic field, would be a few nanometres a year. A three-hundred-metre asteroid’s is a few hundred metres.

Converting the force into a change in the orbit needs one more step, and it is a step this collection already has: Gauss’s variational equations say which component of a force moves which element, and for the semi-major axis the answer is that only the along-track component matters,

$$\frac{da}{dt} = \frac{2a_T}{n\sqrt{1-e^2}}.$$

So the drift is the along-track fraction of the radiative acceleration, divided by the mean motion. A slow orbit gives the force longer to act per revolution — the period grows as the three-halves power of the size — which is why the same body drifts further per year the further out it is, even though the sunlight is weaker.

The sign is the spin #

Here is where this force stops resembling anything else in the collection.

The lag rotates the hot patch in the direction the body turns. For a prograde rotator — spin axis roughly parallel to the orbital angular momentum — the hot patch is on the trailing side as seen in the direction of motion, and the recoil pushes the body forward along its orbit. Forward means energy gained, which means the semi-major axis grows and the body spirals outward.

For a retrograde rotator the same argument runs backwards. The hot patch is on the leading side, the recoil pushes backwards, and the body spirals inward.

The dependence is a cosine of the obliquity, and it changes sign at ninety degrees. Nothing else here behaves that way. Gravity does not care which way a body turns; neither does drag, neither does radiation pressure, neither does a resonance. A body’s spin state — a property of its interior stresses and its collisional history, having nothing whatever to do with its orbit — decides whether its orbit grows or shrinks.

There is a second mechanism hiding in the same physics. The seasonal effect uses the orbit rather than the rotation as its clock: for a body whose spin axis lies in the orbital plane, one hemisphere faces the Sun for half a year and cools through the other half, and the thermal lag then acts over an orbital period rather than a rotational one. It is always negative, because the hemisphere that is radiating hardest is always the one moving away from where it was heated.

What was actually measured #

None of this would deserve an essay if the effect were only computed. It is measured, and the measurement is a beautiful piece of orbit determination.

The drift enters an orbit solution as a fitted parameter: instead of solving for the six elements that fix an orbit, the fit solves for seven, the seventh being a constant transverse acceleration. Whether that seventh parameter is significantly non-zero is then an ordinary question of statistics — and it becomes answerable only when the observations span long enough for the accumulated effect to exceed the noise — which is the same conditioning problem that decides whether a short arc determines an orbit at all.

The accumulation is the key, and it is quadratic. A constant change in the semi-major axis is a constant change in the period; a period error grows linearly into a phase error; a phase error is a position error along the orbit. So the displacement grows as the square of the elapsed time even though the force is constant.

The first detection was (6489) Golevka, in 2003, from three radar apparitions spread over twelve years. Radar ranging measures a distance directly rather than an angle, so a single apparition pins the orbit far better than a season of optical astrometry can, and three of them separated by years gave a baseline over which a few kilometres of accumulated displacement was unmistakable. The fitted drift was about ninety-six metres a year, inward.

The best-measured is (101955) Bennu, whose semi-major axis falls by 284.6 metres a year with a quoted uncertainty of two tenths. That number rests on twelve years of ground-based astrometry and on two years during which a spacecraft in orbit around the asteroid was itself being tracked from Earth, which turns the asteroid’s position into something known to a few metres rather than a few hundred kilometres.

The efficiency nobody can compute #

The model above has one number in it that no observation supplies: the fraction of the radiative acceleration that comes out along-track. Call it $\kappa$. It depends on the thermal conductivity of the surface, on the rotation period, on the size, and on the shape — and of those, the conductivity of a rubble-pile surface nobody has touched is essentially unknown. Estimates for the same body have differed by an order of magnitude.

So the model cannot predict any of these drifts. What it can do is be run backwards. Take each measured drift, divide by everything the model does know — the size, the density, the albedo, the distance, the obliquity — and see what $\kappa$ is left over.

For the four objects with measured drifts the answers are 0.084, 0.085, 0.089 and 0.129. Four bodies with unrelated sizes, orbits, densities and spin states, four independent measurements made by different techniques over thirty years, and the dimensionless number they imply agrees to within a factor of one and a half — for a quantity that could in principle have taken any value between zero and about a fifth.

That agreement is the evidence that the mechanism is understood. It is essentially the only evidence there is, and it is the same shape of argument as solving a satellite’s measured heat for its dissipation number: a theory with one unmeasurable parameter is tested by asking whether several independent systems agree about its value.

Why every measured object is retrograde #

All four sit past ninety degrees of obliquity, drifting inward. It would be a startling fact about asteroid spin states if it were one, and it is not. It is a selection effect with a mechanism.

An asteroid in the main belt becomes a near-Earth object by drifting until it reaches one of the resonances that clear the Kirkwood gaps, where its eccentricity is pumped until its perihelion falls inside the terrestrial planets. The resonances that do the delivering are mostly inside the main population, so a body drifting inward reaches one and a body drifting outward drifts away from them.

Retrograde rotators drift inward, reach a resonance, and become near-Earth objects with observable orbits. Prograde ones drift outward and stay in the belt where nobody measures their drift. The four objects are a sample of the ones the mechanism selected.

What it does to a prediction, and why anyone cares #

The practical consequence is an entry in an error budget. An impact prediction a century out is a statement about where a body will be to within a few Earth radii, and it is made by propagating an orbit determined from decades of astrometry.

That propagation has a shape this collection has already met. The uncertainty does not spread as a ball; it collapses onto a line of variations along the track, because a semi-major-axis error is a period error and a period error is a phase error. So an impact probability is a one-dimensional integral along a curve, and a gravitational keyhole is a short interval of that curve. An unmeasured thermal drift adds an uncertainty to the semi-major axis, and therefore a segment to the line of variations, and therefore an interval of possible arrival times. For Bennu the drift is measured well enough that it contributes a few kilometres of the total uncertainty at the 2135 approach. For a body discovered last year with no thermal model at all, the same term is thousands.

The inverse of the argument is a proposal that gets made regularly: if a small along-track force moves an asteroid this much over a century, then painting one, or parking a spacecraft near it, or letting it absorb sunlight differently, would do the same. The arithmetic is honest — the required change in velocity is millimetres a second if applied decades early — and the difficulty is entirely in the phrase decades early.

A pamphlet nobody kept #

The effect is named after a man who was not an astronomer, published it in a form that has not survived, and was remembered by one person.

Ivan Yarkovsky was a civil engineer working in Russia, and around the turn of the twentieth century he wrote a pamphlet — privately printed, in Russian — setting out the idea that a rotating body in sunlight would feel a force from its own thermal emission, and that over long times this would move small bodies about the solar system. The pamphlet was not published in any journal, was not abstracted, and no copy of it is known to exist.

It was read by Ernst Öpik, who came across it as a student some time before 1910. Forty years later, writing about the dynamics of meteoroids, Öpik recalled the argument and reproduced it from memory, crediting Yarkovsky and noting that he had long since lost the original. Everything the effect is known by descends from that recollection.

Two things about the history are worth more than their anecdotal value.

The first is that the idea was correct and was unusable for half a century. Nothing could be measured: asteroid orbits were known from optical astrometry over short arcs, the drift is a few hundred metres a year on an orbit hundreds of millions of kilometres round, and the accumulated displacement over the available baselines was far below the observational error. The effect became a measurement only when radar ranging supplied distances directly, which was in the 1990s.

The second is that in the interval it was doing real work as a hypothesis. The question of how meteorites reach the Earth from the main belt — and how the near-Earth population is resupplied, since its dynamical lifetime is far shorter than the age of the solar system — had no satisfactory answer without a mechanism that slowly moves bodies into resonances. Collisions were proposed and are not frequent enough. A steady drift is, and the numbers work out.

So the effect was accepted on the strength of a population argument long before any individual object was seen to drift, and the direct measurements when they came confirmed a mechanism that was already being relied on.

A force can be established by what a population requires rather than by what any member of it is seen to do, and this one was, from an argument recalled from a document nobody has read.

What the model leaves out #

Three things, and each is where the subject currently is.

The spin state does not stay put. The same asymmetric re-radiation that pushes the body also exerts a torque, and that torque changes both the rotation rate and the obliquity over a few million years. The mechanism is the same physics under a different acronym, and its consequence is that a body’s drift is not constant over the timescales the drift itself matters on. Spin-up has been measured directly on several small asteroids, and it is thought to be what makes contact binaries and what sheds the moonlets of binary asteroids.

The shape matters and is usually unknown. A perfect sphere with a uniform surface is the case the algebra is written for. A real asteroid is lumpy, and the along-track component of the recoil from a lumpy body depends on which face is where — which is a shape model, which needs radar or a spacecraft.

And the conductivity is inferred from the very effect it is meant to explain. Thermal-infrared observations do measure a surface’s thermal inertia, and where they have been made the value is consistent with the $\kappa$ recovered above. Where they have not, the argument is closed on itself: the drift is used to estimate the conductivity and the conductivity is used to predict the drift. Bennu is the one case where a spacecraft measured the surface directly and the loop was cut, and it is why the number in the first figure of this essay is quoted to four figures for one object and to one for the rest.

Both halves of the practical consequence are worth reading at other settings, since one is a material property and the other is a prediction horizon.

Where this ladder goes next #

This rung has taken a force that is not gravity and shown what it does to a conic: an along-track acceleration, invisible instantaneously, accumulating quadratically, and signed by a property of the body that has nothing to do with its orbit.

The rung above is the family of the same kind. Cometary non-gravitational forces are the same shape of problem with a far larger and far less predictable coefficient, because a comet’s outgassing turns on and off; solar radiation pressure on a body small enough matters directly; and a spacecraft’s own thermal emission is a term in its navigation that had to be modelled before its trajectory could be understood.

Beside it lies the torque rather than the force: what asymmetric re-radiation does to a spin state, and why the rotation periods of small asteroids pile up against a limit that has nothing to do with light.

And below it, the habit this rung is an instance of: a force too small to see instantaneously is measured by letting it integrate. Nobody has ever detected a thermal recoil on an asteroid. What is detected is a body arriving somewhere it should not have been, decades later, by an amount that grows as the square of the wait.