Astronomy & Celestial Mechanics
the sky, computed
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Non gravitational forces — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Every model curve has slope −1, and four measurements agree on κ to 1.5×. Semi-major-axis drift against body diameter, for a thermal recoil in which a fraction κ = 0.085 of the absorbed sunlight comes back out along-track. The three curves are the same expression at 1, 1.6, 2.5 astronomical units, and each has a slope of exactly −1: the acceleration is the absorbed power divided by the mass, which is a cross-section over a volume, so it falls as one over the size and nothing else on this axis changes it. A kilometre-wide body drifts a few metres a year; a ten-metre one drifts hundreds. The four filled marks are the bodies whose drift has actually been measured as a fitted parameter in an orbit solution, and they do not lie on any single curve because each carries its own density, distance and obliquity. What they agree about is the number beside each: solve every measured drift for the efficiency that would produce it and the four answers are 0.084, 0.085, 0.089, 0.129 — a factor of 1.5 apart, for a quantity that could in principle have been anything from zero to a fifth. That agreement is the evidence that the mechanism is understood, and it is the only evidence there is, because the thermal conductivity that sets κ has never been measured for any of them.

    An orbit moved by heat

    A rotating body re-radiates absorbed sunlight from the hemisphere that has had time to warm, so the recoil is not aimed at the Sun. The resulting force is a few parts in ten billion of gravity, it is the only orbital force whose sign depends on which way the body spins, and it has been measured to four figures.

    part 1 · orbits
  2. An impulse delivered inside 0.5 AU, and a comet 2050 hours early. Above: Marsden's outgassing law, the factor g(r) that scales a comet's non-gravitational acceleration, against distance from the Sun over one orbit of a comet with perihelion at 0.336 AU and aphelion at 4.09. It is close to an inverse square inside the water snow line and then falls off a cliff, because water ice that is not being heated does not sublimate. Half the whole revolution's impulse is delivered inside 0.55 AU — a few weeks out of a 3.3-year orbit — so the force is effectively a kick at perihelion rather than a perturbation spread around the path. Below: what a kick of that kind does to the timekeeping. A transverse component changes the semi-major axis and so the period, by 2.5 hours per revolution here, and a constant change in the period accumulates as the square of the number of revolutions rather than in proportion to it. After 40 returns the comet arrives 2050 hours — more than 85.4 days — before an orbit fitted without the term predicts, and doubling the number of returns multiplies the discrepancy by 3.90. That is why the effect was found in the eighteen-twenties from nothing but arrival times, and a century and a half before anyone photographed a jet.

    A comet that arrives a day early

    Encke's comet returned two and a half hours ahead of prediction, every revolution, for decades before anybody could say what was pushing it. The force is a rocket — a few tonnes a second of vapour leaving the sunward side of a rotating nucleus, delivered almost entirely in the few weeks around perihelion, and it accumulates in the arrival time as the square of the number of returns.

    part 2 · orbits
  3. Drift against thermal inertia: a peak at Γ = 93, in the same place for every size. How fast an asteroid's orbit drifts under its own re-radiated heat, against the thermal inertia of its surface, at a rotation period of 4.3 hours and 1.13 astronomical units. Both axes are logarithmic, and the curves are four diameters. The non-monotonic shape is the content. A surface that conducts nothing re-radiates its heat the instant it receives it: the emission is then symmetric about the sub-solar point and the transverse push cancels exactly. A surface that conducts perfectly is isothermal, has no temperature contrast at all, and again pushes nowhere. The force lives between those two nothings, and peaks where the surface's thermal time constant is comparable to the rotation period — here at Γ = 93 in SI units, and at the same place on every curve, because the size scales the drift without moving the optimum. That separation is what makes the effect a measurement. A drift rate on its own is a single number with several unknowns in it; a drift rate together with a size from radar, a spin from a light curve and a density from a flyby leaves the thermal inertia as the only thing not measured, and solving for it says what the surface is made of. Fine dust sits near 50, bare rock in the thousands, and the values measured for the bodies spacecraft have visited — Bennu at 310, Ryugu at 225, Itokawa at 700 — straddle the peak, with the two rubble piles a factor of two or three above it and the Moon's dust well below. Being past the optimum is not a small effect but it is a gentle one: the curve falls as one over the thermal inertia on that side, so a surface three times more conductive than optimal still drifts at a third of the best rate, while one three times more insulating drifts at a third as well. The shape is symmetric in the logarithm, which is why the measurement is a good one for telling dust from pebbles and a poor one for telling pebbles from boulders. The curve is one-dimensional linear theory for a rotating half-space: it has the right limits and the right peak, and it omits the body's shape, which for an irregular asteroid changes the answer by tens of per cent.

    A drift rate that says what the surface is made of

    The thermal recoil that moves an asteroid's orbit depends on how long its surface holds heat, and the dependence is not monotonic — a perfect insulator and a perfect conductor both push nothing. The peak in between means a measured drift is a measurement of thermal inertia, which is a measurement of grain size.

    part 3 · orbits
  4. Four known pieces of hardware, and the anomaly is the sum of them. The reported anomalous acceleration of a deep-space probe, in units of 10⁻¹⁰ metres per second squared, built up from the heat the spacecraft was known to be radiating. The generators put out about two and a half kilowatts of waste heat and sit on booms beside a large dish that reflects a share of it backwards, which is 62 per cent of the total on its own; the instrument compartment radiates through louvres on one face; the radio transmitter beams eight watts at the Earth, which is a torch pointing the wrong way. Sunlight is negligible this far out and is drawn to show that it is. The four sum to 8.65 against a measured 8.74 ± 1.33, and the agreement is the answer. What makes the episode worth keeping is that none of these numbers was discovered later: every one was in the spacecraft's own thermal documentation from before launch, and the model that produced the anomaly was a model of a point mass.

    An acceleration that was the spacecraft's own heat

    Two probes leaving the solar system were tracked for thirty years and both drifted from their predicted paths by a tenth of a nanometre per second squared. The residual was real, it was constant, and it was the same on both. It was also the waste heat of the reactors that powered them, radiating slightly more one way than the other.

    part 4 · orbits

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