A drift rate that says what the surface is made of

An asteroid’s orbit is moved by heat: the body absorbs sunlight, warms, and re-radiates the energy from a hemisphere that has had time to rotate away from the Sun, so the recoil is not aimed back along the incoming beam. The force is a few parts in $10^{10}$ of gravity and it has been measured for several bodies to four significant figures.

The interesting thing about that force is not its size. It is that the size depends on a property of the surface that nothing else measures — how long the surface holds heat — and depends on it in a way that has a maximum. The dependence’s shape is what turns a drift rate into a statement about what the ground is made of.

Why there is a peak #

The diurnal Yarkovsky force exists because the afternoon side of a rotating body is warmer than the morning side. That statement conceals a competition.

If the surface has no thermal inertia at all, each patch is in instantaneous equilibrium with the sunlight falling on it. The temperature pattern is then symmetric about the sub-solar point — hottest at local noon, cooling symmetrically either side — and integrating the recoil over the body gives a force directly along the Sun line. There is no transverse component, so the orbit does not drift.

If the surface has infinite thermal inertia, heat diffuses instantly around the body and the whole surface sits at one temperature. There is no contrast at all, and again no transverse force.

Between those two the temperature pattern is both asymmetric and substantial, and the transverse component is largest where the surface’s thermal time constant is comparable to the rotation period. The dimensionless group that expresses this is the thermal parameter

$$\Theta = \frac{\Gamma\sqrt{\omega}}{\varepsilon\sigma T^{3}},$$

with $\Gamma = \sqrt{\rho c \kappa}$ the thermal inertia and $T$ the subsolar temperature, and the standard one-dimensional treatment gives a response proportional to $\Theta/(1+2\Theta+2\Theta^{2})$, which peaks at $\Theta = 1/\sqrt{2}$.

The rotation period enters through $\omega$ and the heliocentric distance through $T$, so the location of the optimum is a property of the orbit and the spin rather than of the material. The material enters only as $\Gamma$, and reading a $\Gamma$ off the curve requires the other two to be known.

There is a second, slower version of the same effect that behaves differently, and it is worth separating because the two are often conflated. The seasonal Yarkovsky term arises not from the daily rotation but from the annual one: a body with its spin axis in the orbital plane presents alternate hemispheres to the Sun over an orbit, and the thermal lag in that much longer cycle produces a force along the orbital velocity. It is always decelerating, so it always shrinks the orbit, and it depends on the thermal inertia through the same kind of response function evaluated at the orbital frequency instead of the rotational one.

Because the orbital frequency is thousands of times lower, the seasonal term peaks at a thermal inertia thousands of times higher. A surface can therefore be near-optimal for one term and far from optimal for the other, and the two terms have different signs for most obliquities. The observed drift is their sum, and separating them requires knowing the obliquity — which is why a pole solution is not optional.

What a thermal inertia is a proxy for #

$\Gamma = \sqrt{\rho c \kappa}$ combines a density, a specific heat and a thermal conductivity, and in a granular material the conductivity is the variable. Solid rock conducts through its bulk and has $\Gamma$ in the thousands. A powder conducts only through the small contact areas between grains and through whatever gas is in the pores — and in vacuum there is no gas — so its conductivity collapses. Lunar regolith has $\Gamma \approx 50$; a surface of centimetre pebbles is in the hundreds; bare rock is above a thousand.

The relation between the two is not linear and not universal, but it is monotonic and steep enough to be useful: conductivity in a vacuum-packed granular medium scales roughly with the contact area between grains, which scales with grain size, so a factor of ten in $\Gamma$ is a factor of a hundred or so in particle diameter. Calibrations come from laboratory measurements on meteorite powders under vacuum and from the lunar samples, where the thermal inertia was measured in situ and the grain size measured in a laboratory afterwards.

So a thermal inertia is a grain size. That is the payoff: a number measured from an orbital drift, on a body a hundred million kilometres away, says whether the surface is dust, gravel or stone.

And a grain size is a history. Fine regolith is produced by micrometeorite bombardment grinding a surface down, and it is retained only if the body’s gravity can hold it against the ejection speeds involved. A kilometre-sized asteroid has an escape velocity of about half a metre a second; an impact that produces dust launches most of it faster than that. So the absence of fine material on small bodies is not surprising in retrospect, and the presence of centimetre pebbles says that the grinding stopped at a size the body could keep. The thermal inertia is measuring where that threshold falls.

Solving for it #

A drift rate on its own is one number containing several unknowns: the size, the density, the obliquity, the spin period, the albedo and the thermal inertia. Measuring the thermal inertia means measuring everything else.

Size comes from radar, from a stellar occultation, or from thermal infrared photometry combined with an albedo.

Spin period and obliquity come from a light curve, or from several light curves at different viewing geometries inverted for a shape and a pole.

Density is the difficult one. For a body with a satellite it comes from the satellite’s orbit. For a body a spacecraft has orbited it comes from the tracking. For everything else it is assumed from the spectral class, and the assumption carries most of the final uncertainty.

Albedo comes from comparing the reflected and thermal fluxes.

With all of those, the drift rate leaves $\Gamma$ as the only unknown, and the curve above converts it. It is worth noticing how many independent observations that chain contains — radar, photometry, spectroscopy, astrometry over decades — and that the answer is a single number about a surface layer a few centimetres thick.

The astrometry deserves a word of its own, because it is the part that makes the whole thing possible and the part that is least like an experiment. What is measured is the position of a faint moving point among stars, on plates and images taken by many observers over decades, referred to a reference frame that has itself been revised several times. The Yarkovsky signal is a quadratic departure from a Keplerian ephemeris, amounting after fifty years to something like a hundred kilometres — a few hundredths of an arcsecond as seen from Earth. Extracting it means that every systematic in fifty years of positional astronomy has to be modelled, and the detections that are believed are the ones with radar ranging in them, because a radar range is a distance measured directly and is immune to almost everything that afflicts an angle.

The answers, and the photographs that followed #

The bodies with the best-measured drifts are near-Earth asteroids tracked by radar over several apparitions, and the thermal inertias that come out sit in the hundreds: 310 for Bennu, 225 for Ryugu, around 700 for Itokawa. All three are far above lunar regolith and far below solid rock.

The reading was that these surfaces are covered in pebbles and gravel rather than in fine dust — a conclusion drawn from an orbital drift and from thermal light curves, with no image involved. Spacecraft subsequently visited all three, and photographed exactly that: surfaces of centimetre-to-metre fragments, with strikingly little fine material anywhere.

That is worth dwelling on as a piece of method. The inference chain ran from an astrometric residual, through a force model, through a thermal model, to a statement about grain size, and it was later checked against a photograph. It was right. Very few inferences in this collection get that kind of confirmation, and it is a reasonable argument for trusting the same chain where no photograph is coming. Two of the four have been visited and returned samples, which turns the inference into something closer to a controlled experiment. In both cases the spacecraft’s own tracking gave a mass, hence a bulk density, hence an independent check on the number that had been assumed in the drift calculation. In both cases the assumed density was low by ten to twenty per cent — the bodies are more porous than the spectral class suggested — and the recovered thermal inertias moved accordingly. That is a small correction and an instructive one: it says which link in the chain was weakest, and it was not the physics.

The material property that is not a property of the material #

There is a complication in reading a thermal inertia as a grain size that is easy to miss and that undermines comparisons between bodies at different distances from the Sun.

In a granular medium in vacuum, heat crosses from grain to grain by two routes: conduction through the small solid contacts, and radiation across the gaps. The first is independent of temperature. The second is not — a radiative conductance goes as the cube of the temperature, because that is how the emission from one grain face to its neighbour scales.

So the effective conductivity of a regolith rises steeply with temperature, and the thermal inertia derived from it rises as roughly the temperature to the three-halves. A surface of a given grain size therefore has a different thermal inertia at one astronomical unit from the one it has at three, and the difference is not small: across that range the subsolar temperature falls by a factor of about 1.7, which is a factor of more than two in the radiative term.

Two consequences follow, and both bite on the arguments made above. Comparing near-Earth asteroids against main-belt asteroids without correcting for this compares surfaces at different temperatures, and the belt objects will appear to have finer grains than they do. And a single eccentric body has a thermal inertia that varies around its own orbit, so the drift integrated over a revolution is not the drift computed at the mean distance — an error that grows with eccentricity and is largest for exactly the objects whose orbits are best measured.

The correction is straightforward to apply once it is recognised, since the temperature dependence follows from the geometry of grains radiating at each other rather than from anything about the particular material. What it removes is the comfortable idea that a thermal inertia is a number belonging to a surface. It is a number belonging to a surface at a temperature, and quoting one without the other is quoting half a measurement.

Why the peak’s flatness is a limitation #

The response function falls as $\Gamma$ on one side of the peak and as $1/\Gamma$ on the other, so it is symmetric in the logarithm and gentle. A surface three times more conductive than optimal still drifts at a third of the best rate, and so does one three times more insulating.

Two consequences follow. The measurement is good at distinguishing dust from pebbles, because those differ by a factor of ten or more in $\Gamma$ and sit on opposite sides of the peak. It is poor at distinguishing pebbles from cobbles, which differ by a factor of two on the same side.

And the two-sided ambiguity is real: for a body whose drift is below the peak value, two thermal inertias fit, one on each side. Breaking that requires a second observable, and the usual one is the thermal light curve — how the infrared emission varies through a rotation, which is a direct measurement of how far the surface temperature lags the illumination. That is the same physics, measured photometrically rather than dynamically, and the two are combined.

Where it matters practically #

The drift rate is not merely diagnostic. It is the largest uncertainty in predicting where a small asteroid will be in a hundred years, and therefore in deciding whether one is going to hit the Earth.

An impact prediction works by propagating an orbit forward through a keyhole — a small region of a future close approach whose passage would set up a later collision — and the width of the predicted position distribution grows with time. Gravity is known exactly. The Yarkovsky drift is not, and after a century it dominates.

Reducing that uncertainty means measuring the thermal inertia, which is the argument this essay has been making from the other direction. For the objects of most concern, a thermal-infrared observation is a planetary defence measurement.

There is a further use that runs in the opposite direction, and it is the one that turns this from a curiosity into a tool for the whole belt. Bodies too faint for their drifts ever to be measured individually still drift, and the drift’s dependence on size — inversely proportional to diameter — sorts a population. A family of fragments produced by one collision spreads in semi-major axis over time, with the small ones spreading furthest, and the resulting distribution in the plane of size against semi-major axis is a V whose opening angle is the family’s age.

That is the measurement a collision dated by a scatter plot makes, and its calibration is exactly the thermal physics of this essay. An age from a V depends on the assumed thermal inertia of the fragments, and the ages published for asteroid families carry that assumption in them. Getting the thermal inertia of a handful of near-Earth objects right therefore propagates into the dating of collisions in the main belt hundreds of millions of years ago.

What the model leaves out #

The treatment above is one-dimensional: a rotating half-space, heated sinusoidally, with a single thermal inertia. Three things it omits are known to matter at the tens-of-per-cent level.

Shape. An irregular body has self-shadowing and self-heating between facets, and both change the emission pattern. Full thermophysical models with a shape from radar do the integral properly and typically shift the answer by twenty to forty per cent.

Roughness. Sub-resolution roughness beams the thermal emission back toward the Sun, which changes the effective albedo and the temperature contrast. It is parameterised rather than computed.

Layering. The thermal skin depth at a rotation period is a few centimetres and at an orbital period is a metre, so the diurnal and seasonal terms sample different depths. A body with fine material over coarse has two different thermal inertias, and quoting one is quoting an average over a depth that depends on which term is being discussed.

None of these changes the shape of the hero figure or the location of its peak. They change the number by factors that matter for a keyhole prediction and not for a grain size.

The peak’s position depends on the rotation and its sign on the obliquity, and both are worth reading at a second value.

One more horizon shows what the drift’s uncertainty costs a prediction two centuries out, which is the span over which a close-approach forecast has to be made if it is to be actionable at all.

Where the ladder goes #

The two earlier rungs of this anchor treated the effect as a force to be modelled: what it does to an orbit, and what its cometary cousin does to a period. This one has treated it as an instrument. The next step in that direction is the torque rather than the force — the same asymmetric re-radiation spins a body up or down, and the spin-up drives small asteroids into the barrier that says they are piles, sheds material from their equators, and appears to be the main way small binaries are made.

There is also a version of the argument for much larger bodies, where the thermal lag is not diurnal but seasonal and the drift is inward regardless of obliquity. That term dominates for slow rotators and for bodies whose thermal inertia is high, and it is what removes small fragments from the families the belt’s collisions leave behind — sorting them by size, because the drift goes as one over the diameter, into the V-shaped patterns that date the collisions.