Astronomy & Celestial Mechanics
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Essays arrive in groups rather than one at a time. The most recent group is below in full, and every earlier one after it, newest first.

Essays arrive in groups rather than one at a time, and a group usually opens up a subject not covered before. Between one group and the next nothing changes, so a reader who has seen the most recent group has seen everything.

16 September 2026

38 essays on cosmology, exoplanets, galaxies, the observed sky, spaceflight, starlight and orbits

The darkness, itemised. The extragalactic background light as a spectrum: νIᵥ against wavelength, in nanowatts per square metre per steradian, with wavelength logarithmic. The total under the two humps is not read off a plot — it is the integral (c/4π)∫ε(z)(1+z)⁻¹|dt/dz|dz of the Madau–Dickinson star formation history, taking a continuously star-forming population to radiate 10¹⁰ solar luminosities per solar mass per year, and it comes to 37.5 nW m⁻² sr⁻¹. Half of it was emitted beyond z = 0.91, which is the figure's real content: the dark sky is dominated by light released when the universe was 45 per cent of its present age. The split between the two humps is an assumption rather than a result — 50 per cent of the starlight is taken to be absorbed by dust and re-radiated near 140 µm, and the shapes are lognormals of about the observed widths — but the AREA under each is the computed quantity. Measured, the two come to about 24 and 26: the calculation falls 25 per cent short, and whether the missing light is real or a residual foreground is unsettled. Cosmology

The darkness has a number in it

The night sky is not black. It carries about sixty nanowatts per square metre per steradian, and that number is the sum of every photon every star has ever emitted, redshifted and added up over thirteen billion years.

7 figures The sky's darkness, measured as an opacity. Optical depth to electron–positron pair production against gamma-ray energy, for sources at redshifts 0.03, 0.1, 0.3, 1, both axes logarithmic. A gamma ray of energy E is absorbed most readily by background photons near twice the square of the electron rest energy divided by E: at 1 TeV that is 2.37 µm and at 100 GeV it is 0.24 µm, so the energy axis is a wavelength axis for the background light, running backwards — and the background it is evaluated against is the same two-component spectrum the star formation history produced, not a flat number. Above the marked τ = 1 the universe is opaque. The depth is computed in the delta-function approximation, the cross-section replaced by 0.2 of the Thomson value over a bandwidth of order the energy, with the background's comoving density evolving as (1+z)^1.2. That is a factor-of-two calculation and the shape is what it gets right: the horizon closes from z = 0.59 at 100 GeV to z = 0.16 at 1 TeV. The measurement runs the other way. A blazar's spectrum is observed, the absorbed part is the difference between it and the spectrum the source is believed to have emitted, and that difference gives the background — in the near infrared, where no direct measurement can subtract the zodiacal light well enough to compete. Cosmology

A background weighed by what it stops

The faintest light in the universe cannot be photographed from inside the Solar System, because the zodiacal foreground is a hundred times brighter. It can be weighed instead, by the bite it takes out of a blazar at a trillion electronvolts.

7 figures Three messengers, and the three walls they end on. The redshift of the last opaque surface, for the three things that cross the universe, on one logarithmic axis spanning thirty-one decades. Olbers' argument compares how far a sight line runs before it ends on something against how far anything has had time to come — and for starlight the first is 1.7·10¹⁸ Mpc against a horizon of 1.4·10⁴ Mpc, which is why the optical sky is dark. That comparison is not what decides the other two. A relic neutrino's mean free path against ordinary matter works out at 7.7·10³⁷ Mpc — 4·10¹⁹ times starlight's, so on the mean-free-path argument alone the neutrino sky should be darker still. It is not, because the quantity that terminates a sight line is the WALL, and the walls are at z = 1,090, z ≈ 6×10⁹ and nowhere at all. Every neutrino sight line ends on a surface from one second after the beginning; every gravitational-wave sight line runs to the beginning, or to a binary. Both of those skies are saturated — which is what Olbers' argument predicted, and what the optical sky refuses to do. The three are also wildly unequal in brightness: the microwave sky is 996 nW m⁻² sr⁻¹, and the gravitational-wave background at Ω = 10⁻⁹ is 0.018 — a saturated sky five decades fainter than a dark one. Cosmology

Two skies where the paradox comes out right

Olbers argued that every sight line should end on a source and the sky should blaze. In neutrinos and in gravitational waves it does — the walls are at one second and at no time at all — and both of those skies have now been detected.

7 figures What comes back is a ramp, not a threshold. Detection efficiency against signal-to-noise: the fraction of synthetic transits injected into real photometry that the pipeline afterwards finds. The measured curve is a gamma cumulative distribution of shape 4.65 and scale 0.98 beginning at 4.1, which is the form a survey's own injection tests are fitted with; the dashed line is the step at 7.1 that a threshold calculation assumes instead. Half the injections are recovered at 8.33, 1.2 units above the nominal threshold — the ramp is a property of the search and the cut is a separate decision, so the two need not meet anywhere in particular. The rest of the disagreement is the area between the curves. The pipeline does not reach 99 per cent efficiency until 14.9, four units above the threshold, and it recovers 45 per cent one unit above it. Over a population whose signal-to-noise falls as s^-2 — which is what a planet population looks like, because there are far more small planets than large ones — the step function counts 1.21 times as many detections as the ramp does. That factor is not an error bar. It multiplies every occurrence rate computed without it, and it is larger for the small planets than for the large ones, because the small ones live where the ramp is. Exoplanets

The threshold that is not a threshold

A survey's detection limit is quoted as a number — seven point one — and a pipeline does not behave that way. Half the injected signals come back at the threshold, and full efficiency arrives four units above it.

7 figures Three biases against eccentricity, and they do not agree. Four quantities against orbital eccentricity, each relative to a circular orbit of the same semi-major axis, averaged over the argument of periastron. The transit probability rises as (1 − e²)⁻¹, because an eccentric planet spends part of its orbit inside its own semi-major axis: at e = 0.5 a transit is 1.33 times as likely. The transit duration falls as √(1 − e²), so the event carries less signal-to-noise, and the two together — probability times the square root of the time in transit — come to 1.24 at the same eccentricity. They very nearly cancel, and that is the surprise: a transit survey has almost no eccentricity bias at all. The radial-velocity curve is the one that does. A Keplerian of eccentricity e puts less of its variance in the fundamental and more into harmonics no sinusoidal search is looking at — 68 per cent remains at e = 0.6 and 47 per cent at e = 0.8 — so a velocity survey loses amplitude exactly where a transit survey does not. What no figure here can show is which of these the measured eccentricity distribution is made of, because the correction depends on a detection pipeline rather than on geometry, and the two surveys have to be corrected separately before their answers can be compared. Exoplanets

Every method prefers a circle, and not for the same reason

A transit is more likely on an eccentric orbit and shorter when it happens, and the two very nearly cancel. A velocity curve loses amplitude to harmonics no sinusoidal search is looking at, and that one does not cancel at all.

7 figures How many planets a star has is the hardest thing a catalogue measures. The multiplicity distribution a transit catalogue would contain, for systems that all truly hold 5 planets, at four mutual inclination dispersions. 40,000 systems are drawn per dispersion with an isotropic viewing direction and Rayleigh-distributed inclinations about a common plane, at semi-major axes of 12, 16, 21, 27, 34 stellar radii; the bars are conditioned on at least one planet transiting, which is what makes a system appear in a catalogue at all. At 0.5° of dispersion 33 per cent of the detected systems show all 5 planets and the mean apparent multiplicity is 3.13; at 10° it is 1.39, with 68 per cent of them showing exactly one. Every one of those systems has 5 planets. The entire difference between a catalogue of singles and a catalogue of compact multiples is one number that nothing in the light curve measures. And the two effects run in opposite directions: the fraction of stars showing any planet RISES with the dispersion — 8%, 9%, 12%, 19% across the four — because scattering the orbits gives more of them a chance to cross the line of sight, while the number seen per detected star falls by a factor of 2.2. A survey that scatters its systems finds more stars with planets and fewer planets per star, and neither number on its own says which has happened. What no figure here can show is the true dispersion, because the observable is the ratio of those two and a system with fewer planets and a tighter plane reproduces it exactly. Exoplanets

How many planets a star has is not a measurement

Draw five thousand identical five-planet systems, scatter their orbital planes by half a degree, and a third of the detections show all five. Scatter them by ten degrees and two thirds show exactly one. Every system has five.

7 figures The count theory predicts, and the inference it costs. The galaxy stellar mass function: galaxies per cubic megaparsec per dex of stellar mass, both axes logarithmic. Two Schechter components share a characteristic mass of 10^10.66 M☉ — one of slope -0.35 carrying the quenched galaxies at the knee, one of slope -1.47 carrying the star-forming ones below it — and the dashed line is the single component a luminosity function is usually fitted with. Integrated over the range drawn it gives 0.0487 galaxies per cubic megaparsec holding 2.22·10⁸ solar masses of stars, of which 51 per cent sits above the knee. This function is not measured. What is measured is a luminosity function; turning one into the other needs a mass-to-light ratio for every galaxy in the sample, and that ratio is not a constant — it runs by a factor of about five from the bluest galaxies to the reddest, so the conversion moves the red end of the distribution further than the blue end and changes the SHAPE rather than the units. A stellar mass function is a luminosity function plus a stellar population model, and the second half is where its disagreements live. Galaxies

The count theory predicts, and the inference it costs

A luminosity function is measured. A stellar mass function is inferred, one galaxy at a time, through a ratio that runs by a factor of six from the bluest galaxies to the reddest — so the conversion changes the shape and not merely the units.

7 figures Galaxy formation is inefficient nearly everywhere. Star formation efficiency — the stellar mass a halo has made, divided by the 0.157 of its mass that is baryons — against halo mass, by abundance matching. The n-th most numerous halo is assigned the n-th most numerous galaxy and nothing else is assumed: the halo count is a Sheth–Tormen mass function integrated from a linear power spectrum, the galaxy count is a measured double Schechter, and the matching is a monotone map between two cumulative counts. The curve peaks at 22 per cent, at a halo mass of 10^11.89 M☉, and falls to 0.6 per cent at the bottom of the range and 0.3 per cent at the top. A halo of the Milky Way's mass, 1.3·10¹² M☉, sits near the peak at 21 per cent — and near the peak means near the best any halo has ever managed. The two sides are two different problems and the figure cannot tell them apart: below the peak the shallow potential lets supernovae drive gas out, above it the gas falling in is shock-heated and cannot cool fast enough. What the abundance-matching assumption cannot show is scatter — it assigns one galaxy mass per halo mass by construction, and the real relation has about 0.15 dex of spread that this method is blind to by definition. Galaxies

Two counts that are not the same shape

Dark matter halos are counted by a calculation that knows nothing about stars. Galaxies are counted by a survey. Laid on the same axes the two curves disagree at both ends and agree nowhere, and the knee is where the two disagreements hand over.

7 figures The same count, taken in two places. The ratio of a cluster's luminosity function to the field's, per galaxy at the knee, against absolute magnitude. Both are Schechter functions — the field at a faint-end slope of -1.25 and a characteristic magnitude of -20.9, the cluster at -1.05 and -21.4 — and they are normalised to agree at -21 so that what is drawn is a difference of SHAPE rather than of density, a cluster being some 240 times denser than the field by construction. Two things differ. The cluster's faint end is shallower: at -15 it holds 0.22 of the field's dwarfs per bright galaxy. And its knee is 0.5 magnitudes brighter, which is a factor of 1.6 in luminosity. Neither difference can be read as a cause. A cluster's galaxies are also redder, and the same photometry measures both — so a shallower faint end could mean that dwarfs were destroyed, or that they were never made, or that they are still there and have faded below the survey's limit because their star formation was stopped. The count says the populations differ; it does not say which of a galaxy's life stages the difference happened in. Galaxies

The same census, taken in two places

Fit a Schechter function to a rich cluster and to the field around it and the two come back with different slopes and different knees. Both differences are real, and neither can be read as a cause — a cluster's galaxies are also redder, and the same photometry measures both.

6 figures Five zones, one angle. The Sun's highest and lowest noon altitude against latitude, for an obliquity of 23.4393°. The upper curve is noon on the summer solstice and the lower is noon on the winter one; they are the same function of latitude displaced by 23.4393° in each direction, which is why one angle fixes both boundaries. Where the upper curve reaches 90° is the tropic, at 23.44° — the Sun is overhead at noon there on exactly one day, and somewhere inside it on every other day of the year. Where the lower curve reaches 0° is the polar circle, at 66.56° — the Sun fails to clear the horizon on the winter solstice, and on more days the further poleward one goes. They are the same inequality: |φ| ≤ ε for the first and |φ| ≥ 90° − ε for the second, and an obliquity of zero would collapse the tropics to the equator and push the polar circles to the poles, leaving one zone. The areas are the part that is not intuitive. The fraction of a sphere between two latitudes is the difference of their sines, so the tropics — a band a quarter of the way to the pole — hold 39.8 per cent of the Earth's surface, the temperate zones 52.0 per cent, and the polar caps only 8.3. The zone where the Sun can be overhead is 4.8 times the area of the zone where it can fail to rise, and both boundaries are the same 23.44°. The observed sky

Five zones, and one angle

The tropics are where the Sun can stand overhead; the polar circles are where it can fail to rise. Both boundaries are 23.44° measured from opposite ends, and the zone the Sun can reach is nearly five times the area of the zone it can miss.

6 figures The sunniest place on Earth is the summer pole. Daily-mean insolation against latitude, on the June solstice, an equinox, the December solstice, computed from Q = (S₀/π)(ā/r)²[H₀ sin φ sin δ + cos φ cos δ sin H₀] with the half-day angle clipped to a whole day inside the polar circle. On the June solstice the north pole receives 524 W/m² and the equator receives 385 — a third more, at a place where the Sun never climbs above 23.4°. Both halves of the product move: the Sun is low, so each square metre catches little; and it never sets, so the catching goes on for twenty-four hours. The second wins. That is why the daily total and the noon altitude disagree about where summer is strongest, and the daily total is the one an ice sheet answers to. The two hemispheres are not equal either. The Earth is nearest the Sun in early January, so the December solstice delivers 559 W/m² to the south pole against the north pole's 524 in June — 6.7 per cent more. What this cannot show is any temperature. Antarctica is the coldest place on Earth and receives the most sunlight of anywhere on any day; between the insolation and the temperature sit an albedo, an altitude and an ocean. The observed sky

The sunniest place is the summer pole

On the June solstice the north pole receives 524 watts per square metre averaged over the day, and the equator 385. The Sun there never climbs above 23.4° and never sets, and the second wins.

6 figures The hottest month is not the sunniest. The annual cycle of insolation at latitude 45°, and the temperature three surfaces of different heat capacity answer it with — each curve scaled to its own peak, because what is being compared is timing and not degrees. A surface losing 2 W m⁻² per kelvin above equilibrium obeys C dT/dt = Q − λT, and for a sinusoidal forcing that is one line of algebra: the response lags by arctan(ωC/λ) and is reduced by (1 + (ωC/λ)²)^(−½). a continental interior, 2.4 m of water, lags by 45.6 days and swings by 71 per cent of what a massless surface would; a shallow shelf sea, 10 m of water, lags by 77.6 days and swings by 23 per cent of what a massless surface would; a deep ocean mixed layer, 50 m of water, lags by 88.5 days and swings by 4.8 per cent of what a massless surface would. The lag can never reach a quarter of a cycle — three months — because an arctangent cannot reach 90°, which is why no inhabited place has its warmest month in December in the north, and why the sea comes closest. And the two numbers are one number: the same ωC/λ that carries the phase towards the quarter cycle divides the amplitude, so a long lag is bought with a small swing and cannot be had any other way. The solstice falls on day 170 and the peak insolation with it; the warmest day at this latitude follows between 46 and 89 days later depending on what is underneath. What no curve here can show is the atmosphere's own transport, which carries heat sideways between the surfaces drawn and makes each one's effective capacity partly its neighbour's. The observed sky

The hottest month is not the sunniest

A surface with a heat capacity answers a sinusoid late and small, and the two are the same number. The lag can never reach a quarter of a cycle — three months for a year, six hours for a day — because an arctangent cannot reach ninety degrees.

5 figures One square root that raises the orbit and turns it. The velocity budget from a 300 km circular orbit to geostationary, against the plane change carried out along the way. Edelbaum's closed form — √(v₁² + v₂² − 2v₁v₂cos(½πΔi)) with Δi in radians — puts the whole continuous manoeuvre in one square root, and at Δi = 0 it collapses to |v₁ − v₂| = 4.651 km/s, which is the spiral's cost with no plane change in it. The two-impulse curve puts its rotation into the circularisation burn at apoapsis, where the vehicle is moving at 1.608 km/s and a rotation is cheap. At 28.5° the continuous transfer costs 5.951 km/s against the impulsive 4.256 — the plane change adds 1.300 to one and 0.363 to the other. That is the opposite of the usual claim that low thrust turns for free. It turns continuously, which is not the same thing: the gain is that the propellant is not the budget, and the Δv is worse here as it is everywhere else. Spaceflight

One square root that raises the orbit and turns it

Edelbaum put the plane change inside the same radical as the raise, and the half-pi in its cosine is the whole result — a continuous turn costs π/2 times an impulsive one below 140° and less above it.

5 figures There is a best exhaust speed, and the calendar picks it. Payload fraction against exhaust speed for a 11 km/s mission, at three thrusting durations, with a power plant of 0.025 kilograms per watt of jet power. The vehicle is payload plus power plant plus propellant and the arithmetic is one line: λ = e^(−Δv/c) − (αc²/2t)(1 − e^(−Δv/c)), the first term the rocket equation and the second the mass of the machine that makes the jet. They pull opposite ways. A slow exhaust burns propellant; a fast one needs power, and the power per newton rises in proportion to c, so the plant's mass rises as c². Neither end is where anybody builds, and the optimum is 3,211 s over 200 days, 6,523 s over 700 days, 11,424 s over 2000 days — the same mission, the same Δv, and the best engine for it changes by a factor of 3.6 depending only on how long there is to do it. A gridded ion engine at 3,100 seconds sits at 30.4 km/s, which suits the shortest of these and is slow for the longest. What the curve cannot show is that α is not a constant either: a solar array's mass per watt falls as the fourth power of the distance from the Sun, so the same vehicle is a different point on this plot at Mars and at Jupiter. Spaceflight

The engine is chosen by the calendar

A chemical stage's exhaust speed is fixed by chemistry. An electric one's is a dial, and turning it up costs power — so there is a best setting, and it is decided by how long the mission has rather than by how far it is going.

5 figures A sail has to be tilted, and tilting it throws most of it away. The thrust on an ideal flat sail, resolved into the orbit frame, against the angle between the sail's normal and the sunline. The force is along the normal and goes as cos²α — one cosine for the area the sail presents to the light, one for the momentum the reflection returns along the normal — so the radial component goes as cos³α and the transverse one as cos²α sin α. A sun-facing sail has no transverse push at all. Its thrust is purely outward and falls as 1/r² exactly as solar gravity does, so it merely replaces μ with μ(1 − β): the orbit stays the same conic with a smaller central mass, and the vehicle raises nothing. Every manoeuvre a sail makes it makes by tilting, and the transverse push peaks at 35.26° — arctan(1/√2), differentiated rather than tabulated — where it is 0.385 of the face-on force, or 2/(3√3). Two thirds of the thrust is the price of pointing any of it somewhere useful. The lightness number β is the sail's whole specification, radiation pressure and gravity both falling as 1/r² so their ratio is a constant: IKAROS, 2010 at 1607 g/m² gives β = 9.5e-4; LightSail 2, 2019 at 156 g/m² gives β = 9.8e-3; a 5 µm film with no structure at 7 g/m² gives β = 0.219, against the 1.53 g/m² at which the Sun would push as hard as it pulls. What no figure here can show is the thing a sail actually has instead of a rocket equation, which is nothing: the exponential that limits every other vehicle is absent, and what limits this one is a structure that has to hold a square kilometre of film flat. Spaceflight

A drive with no rocket equation

Radiation pressure and solar gravity both fall as the inverse square, so their ratio is a constant of the vehicle. A sun-facing sail therefore only rescales the central mass — it has to be tilted to do anything, and the best tilt throws away sixty-two per cent of the thrust.

5 figures The classical law gives every star the same colour. Planck's law and the Rayleigh–Jeans law at 3,000 K, 5,772 K, 10,000 K, both normalised to the 5,772 K Planck peak, on logarithmic axes. The classical law comes from counting standing waves in a cavity — 8πλ⁻⁴ of them per unit volume per unit wavelength — and giving each the kT that equipartition allows. It agrees with Planck's where the modes are crowded and each holds much less than kT, and it runs away where they are not: at 80 nm it exceeds the real spectrum by a factor of 1.1·10¹² while agreeing to within 55.9 per cent at 3000 nm, and the integral under it does not converge at all. That is the ultraviolet catastrophe, and it is the half everybody knows. The quieter half is that in 2ckT/λ⁴ the temperature is an overall factor, so the ratio of the law at two wavelengths is independent of it: the B − V index of a classical star comes out identical at 3,000 K, 5,772 K, 10,000 K — the same -0.968 magnitudes, to the last digit the quadrature carries — while Planck's law spreads the same three stars over 1.47 magnitudes. A classical universe has stars of every brightness and one colour. Colour is a thermometer only because the exponential in the denominator does not cancel, and the quantum of energy that put it there was fitted to this shape before anybody knew what it meant. Starlight

The classical law gives every star one colour

The ultraviolet catastrophe is the famous half. The quieter half is that in 2ckT/λ⁴ the temperature is an overall factor, so the ratio of the law at two wavelengths has no temperature in it — and a classical universe has stars of every brightness and one colour.

5 figures Two laws that are the peak and the area of one curve. Planck curves at 3,000 K, 5,772 K, 9,600 K on logarithmic axes, with each peak marked. Normalised by its own peak, Planck's law is a universal function of x = hc/λkT, and three exponents follow from that alone and are fitted here off the drawn curves rather than quoted: the peak wavelength goes as T^-1.000, which is Wien's displacement law with a constant of 2.897772 mm K obtained by solving 5(1 − e^(−x)) = x for x = 4.965114; the peak HEIGHT goes as T^5.000; and the area goes as T^4.000. The third is the first two multiplied. A peak five powers high on a curve one power narrow encloses four powers of area, so Stefan–Boltzmann is not an independent fact about radiation — it is Wien's law and the height of the peak, taken together. That is also why the two are worth having at once. A colour gives the temperature and a flux gives the luminosity, and L = 4πR²σT⁴ then gives a radius: for the Sun at 5,772 K receiving 1361 W/m² at 1.000 AU, the arithmetic returns 6.957·10⁸ m against a measured 6.957·10⁸. A thermometer alone cannot do that, because a colour is a ratio and a ratio has no size in it; the radius comes from the one law that is an absolute quantity rather than a shape. Starlight

Two laws that are one curve read twice

Wien's displacement and Stefan–Boltzmann are the peak and the integral of the same function. The peak is five powers high and one power narrow, so the area is four — and the fourth power that is taught as a separate law is the first two multiplied.

5 figures Three shifts larger than the error bar, and two that cancel. The blackbody B − V index against temperature, with three systematic shifts marked at 5,772 K. All three are computed from the same Planck integrals the relation itself is: a 3,800 K companion contributing 25 per cent of the V light reddens the index by 0.093 magnitudes, because the companion is relatively brighter in the redder band; 0.35 magnitudes of visual extinction at a total-to-selective ratio of 3.1 adds 0.113 directly, since the colour excess is the extinction divided by that ratio; and a metallicity of -1 dex subtracts 0.200, because the metal lines that eat the B band are the ones a metal-poor star is short of. Read as temperatures, the same star comes back at 5331 K, 5243 K and 7021 K against a true 5,772. Two of the three have opposite signs, and that is the difficulty rather than the relief. All three together shift the index by 0.005 magnitudes against 0.405 of combined magnitude — very nearly nothing, because the opposing pair removes almost all of it — and return 5744 K, which is 28 K from the truth by cancellation and not by accuracy. A reddened metal-poor star and an unreddened solar-metallicity one are the same point on this curve, and no amount of photometry in two bands separates them. What does separate them is a third band, or a spectrum — which is where the cheapest measurement in astronomy stops being cheap. Starlight

Three shifts larger than the error bar, and two that cancel

An unresolved companion, a reddening and a metallicity each move a colour index by more than any modern photometer's precision. Two of them move it in opposite directions, so the three together can return the right temperature by cancellation rather than by accuracy.

6 figures A wall measures a ratio, and a ratio is a line. The tidal quality factor against the system's age, with each line the locus of one measured circularisation boundary. For a planet the eccentricity damping time τₑ goes as Q′P^(13/3), so the cut-off period goes as (age/Q′)^(3/13) and only the ratio of the two appears. A measured wall is therefore a straight line of slope exactly 1 in this plane and never a point on it: a 5-day boundary is consistent with Q′ = 2·10⁵ in a one-billion-year-old system and with Q′ = 2·10⁶ in a ten-billion-year-old one, and nothing in the light curve chooses. The stellar version of this measurement escapes because the cluster supplies the age, from a main-sequence turn-off that owes the tide nothing — which is why a cut-off period read off four clusters is a dissipation measurement and the same wall in the hot-Jupiter plane is not. The shaded band is what a field star's age is actually worth: known to a factor of 3, it leaves Q′ known to a factor of 3 and no better, against a quantity whose published values for giant planets span 10⁴ to 10⁹. What would break the degeneracy is a second measurement with a different power of P in it — an orbital decay rate, which goes as Q′⁻¹ with no age in it at all, and which has now been measured for one planet. Orbits

A wall measures a ratio, and a ratio is a line

The same circularisation boundary drawn for planets probes the dissipation inside the planet rather than inside the star. But the boundary depends only on age over Q′, so with no cluster to date the system the measurement is a line in a plane and never a point on it.

6 figures Two damping times, one crossing, and the slope that separates them. Circularisation timescale against orbital period for the two tidal mechanisms, both logarithmic, normalised to 1.217 Gyr at 10 days. On logarithmic axes a power law is a straight line and the index is its slope, so the figure's content is that the two lines have different slopes and one crossing. The equilibrium tide gives 5.33, the bulge raised on a convective envelope being dragged ahead by a viscosity that is turbulent convection itself; the dynamical tide gives 7, gravity waves launched at the convective boundary carrying angular momentum to wherever they break. The horizontal lines are the ages of populations a boundary can be read in: where each curve crosses one is the wall that population shows — at 0.125 Gyr, 6.5 days against 7.2; at 0.625 Gyr, 8.8 days against 9.1; at 4 Gyr, 12.5 days against 11.9; at 10 Gyr, 14.8 days against 13.5. One cluster measures one number and both theories have a free normalisation, so one cluster cannot choose. The separation across every age available is a factor of 1.11. The discriminator that does not depend on the normalisation is mass: the equilibrium tide needs a convective envelope, and above about 1.3 solar masses there is not one, so the two predict different behaviour on either side of a boundary the theory names in advance. That is a measurement about where the wall stops behaving, not about where it is — and it is the reason the samples had to grow from tens of binaries per cluster to hundreds. Orbits

Two damping times, one crossing, and the slope that separates them

The equilibrium tide gives a damping time going as the sixteen-thirds power of the period and the dynamical tide as the seventh. One cluster measures one number and both theories have a free normalisation, so one cluster cannot choose.

5 figures A cluster moving at +500 km/s, and the frequency where only the motion is left. The two distortions one cluster imprints on the microwave background, against observing frequency, scaled to the largest excursion drawn. The thermal effect is from the random motion of electrons at 8 keV, with a central Compton parameter of 10⁻⁴; the kinematic effect is from the bulk motion of the same gas at 500 km/s along the line of sight, positive meaning receding, through an optical depth of 0.00639, which is the Compton parameter divided by kT/mₑc². A bulk velocity shifts every scattered photon by a common Doppler factor, and a blackbody shifted by a common factor is a blackbody at another temperature — so the kinematic distortion has exactly the shape of a temperature change, ΔT/T = −τv/c, which here is −29.0 µK at every frequency. In intensity that shape is the derivative of the Planck spectrum, and its largest value falls at 217.5 GHz, the same frequency at which the thermal distortion crosses zero, 217.5 GHz: both conditions reduce to x coth(x/2) = 4. At 150 GHz the thermal decrement is −260 µK, so the motion is 11.2 per cent of it there and all of the signal at the null. What the kinematic spectrum cannot be told apart from is the primary microwave background itself, which is also a temperature change with this shape — so the frequency that isolates the velocity from the gas is no help at all against the sky behind it. Cosmology

A velocity that has the colour of the sky

A cluster moving through the microwave background shifts the light it scatters by a common Doppler factor, which leaves a spectrum shaped exactly like a change of temperature. That shape is loudest precisely where the hot gas falls silent — and it is the one shape the background itself already has.

6 figures At 15 keV the decrement is 9 per cent shallower and the null has moved to 224.4 GHz. The thermal Sunyaev–Zel'dovich distortion at one fixed Compton parameter, 10⁻⁴, computed with the relativistic kinetic equation expanded to second order in kTₑ/mₑc² for gas at 5, 10, 15 keV, against the non-relativistic shape that is the same for every temperature. All are scaled to the non-relativistic curve's largest excursion. Heating the gas at fixed y does two things to the spectrum. The decrement becomes shallower — by 9.3 per cent at its deepest point for 15 keV — and the increment becomes lower and broader, by 17.3 per cent at its peak, because fast electrons scatter photons over a wider spread of frequencies than slow ones and some of the boost is carried to frequencies above the drawn range. The crossing moves up, from 217.5 GHz to 224.4 GHz. A cluster's temperature is therefore written into the shape of its distortion and not only its amplitude, which is what a thermometer needs; and a Compton parameter read off one frequency with the non-relativistic shape is biased low by the drawn amount, which is what a mass estimate does not need. The expansion is good to well under a per cent below 15 keV; above 20 it has to be replaced by the exact integral. Cosmology

A null that moves with the temperature

The frequency at which a cluster's hot gas vanishes from the microwave sky was derived for slow electrons. The electrons in a massive cluster move at a quarter of the speed of light, the null drifts half a gigahertz per keV, and what is left at the old frequency reads as a velocity as large as the ones being sought.

5 figures 1020 clusters or 372, and the survey cannot say which parameter moved. The number of clusters per unit redshift a survey finds above an integrated Compton signal of 8·10⁻⁵ arcmin² over 6 per cent of the sky, computed from the Sheth–Tormen halo count grown by the linear growth factor, the comoving volume in each redshift slice, and the calibrated relation between signal and mass. Each curve is one pair of assumptions: the amplitude of structure, σ₈, and the hydrostatic mass bias, 1 − b, which says how far the X-ray masses the relation was calibrated on fall below the true masses. With σ₈ = 0.811 and 1 − b = 0.8 the survey finds 1020; σ₈ = 0.811 with 1 − b = 0.6 gives 372; σ₈ = 0.75 with 1 − b = 0.8 gives 548. The counts fall at low redshift because there is little volume, and at high redshift because massive halos have not yet formed, and the peak sits near z = 0.24. Lowering 1 − b pushes the threshold onto more massive and rarer halos; lowering σ₈ makes every halo rarer. The two lower curves differ in total by 47 per cent and in normalised shape by at most 2 per cent of the peak. A survey that assumed 1 − b = 0.8 would read the 372 clusters of the curve with 1 − b = 0.6 as σ₈ = 0.716, with a redshift distribution that differs from it by at most 12 per cent of the peak — which is the only handle the survey has on the difference, and it is smaller than the counting noise in any redshift bin holding fewer than about 67 clusters. A total count cannot distinguish a universe with less structure from a survey that has misjudged its masses, and it is that degeneracy, not the counting, that has been argued about since the first large catalogue. Cosmology

Too few clusters, or a scale that reads light

A catalogue selected on the microwave shadow is, past redshift one half, very nearly a catalogue of everything above a fixed mass — so its count by redshift is the growth of structure read almost directly. Almost, because the mass behind the threshold comes from a calibration, and a scale that reads twenty per cent light is indistinguishable from a universe with less in it.

5 figures A thermostat that passes 40 per cent of the change it is meant to cancel. Surface temperature against the flux a planet absorbs, in units of the Earth's, for a planet round a 1 M☉ star, with and without the carbonate–silicate cycle. With carbon dioxide held at 280 µbar the temperature follows the flux directly. With weathering allowed to adjust — rock dissolves faster when it is warm and when there is more CO₂, and in the steady state it must remove exactly what volcanoes supply at 1 times today's rate — a colder planet accumulates CO₂ until the balance is restored. The feedback is real and it is not a set point. Near S = 1 it passes 40 per cent of a flux change through to the surface: the loop gain is k s / β = 1.49, with weathering rising one e-fold for every 9.7 K, a greenhouse of 4.33 K per e-folding of CO₂ and a CO₂ exponent of 0.3. The required CO₂ would reach 8.8 bar — the point at which more of it scatters sunlight faster than it traps heat, and the controller has nothing left to add — at S = 0.249. The steady state reaches 273 K at S = 0.531, before the CO₂ has run out — the outer edge of this planet's habitable zone is where the thermostat saturates or freezes, whichever comes first. The climate law is logarithmic in CO₂, which is right near today's values and only a calibration at several bar; ice-albedo feedback, which makes a cooling planet able to jump to a frozen state, is left out. Exoplanets

A thermostat that only halves the error

The carbonate–silicate cycle is credited with keeping a planet's water liquid across the whole width of its habitable zone. Written down with its own measured exponents, it is a proportional controller that cancels about three-fifths of a change in sunlight, takes half a million years to do it — and reaches the published outer edge only on a planet with an order of magnitude more volcanism than the Earth.

5 figures An eccentricity of 0.4 swings the surface by 126 K or by 0.1, depending on the length of the year. The peak-to-trough swing in surface temperature over one orbit, against orbital period, for a planet with eccentricity 0.4 receiving on average the flux the Earth does, for surface layers of 1, 10, 50 metres of water. The dashed line is the 126 K the surface would swing through with no heat capacity. Every curve rises from near zero at short periods, where the orbit is over before the layer can respond and the planet feels only the average flux, towards the full swing at long periods, where every part of the orbit lasts long enough to be felt in full. The crossover is where the orbital period is comparable with the layer's thermal time. The vertical marks are the orbital periods of the Earth-flux orbit round stars of 0.1 M☉ (7 days), 0.5 M☉ (0.2 years), 1 M☉ (0.9 years). At a fixed eccentricity a planet in the habitable zone of a small star is thermally averaging almost regardless of how much water it has, and one round a Sun-like star is not unless it has an ocean — the ordering is set by the star through the period, which is the one quantity the flux-averaged habitable zone discards. Exoplanets

A year too short to feel its own eccentricity

A planet on an eccentric orbit can have a comfortable average and murderous extremes, and the habitable zone is drawn from the average. Whether the surface lives on the average or on the extremes is not decided by the flux at all — it is the ratio of how long the surface takes to change temperature to how long the year lasts, and the star sets the year.

6 figures A planet in the middle of a 0.08 M☉ star's zone spends 94 Myr too hot to keep an ocean. How long a planet spends receiving more flux than the runaway-greenhouse limit while its star contracts onto the main sequence, against stellar mass, for planets at the inner edge, in the middle, at the outer edge of the zone the star will have once it settles. The luminosity is the contraction law of a fully convective star, falling as t^(−2/3) from an age of 1 Myr until arrival; the limit is the runaway flux for the star's temperature. A planet at the inner edge is too hot for 313 Myr round a 0.08 M☉ star and 17.8 Myr round a 0.6 M☉ one; a planet in the middle is too hot for 94 Myr round a 0.08 M☉ star and 5.5 Myr round a 0.6 M☉ one; a planet at the outer edge is too hot for 39 Myr round a 0.08 M☉ star and 2.0 Myr round a 0.6 M☉ one. A runaway greenhouse is not a hot climate; it is a state in which the ocean is entirely in the atmosphere as steam, where ultraviolet light splits it and hydrogen escapes. A planet in the eventual zone of the commonest stars in the galaxy begins its life in that state for tens to hundreds of millions of years — a span comparable with the whole assembly of the Earth. The durations are measured from 1 Myr; a rocky planet may take tens of millions of years to finish forming, and one that formed later misses the start of its exposure, while the smallest stars' arrival times are somewhat short in this model, which lengthens the end of it. Exoplanets

Steam before the zone existed

The smallest stars take hundreds of millions of years to contract onto the main sequence, shining at many times the luminosity they will settle at. A planet in the habitable zone such a star will eventually have spends that time with its ocean in the air as steam, while starlight splits the water and the hydrogen leaves — so the zone of the commonest star in the galaxy is a place that had to survive being too hot first.

5 figures A slope of 0.57 at the low-mass end, against the 0.59 and 0.24 two winds give there. Gas-phase oxygen abundance against stellar mass for star-forming galaxies. The solid measured curve is the relation found from electron-temperature abundances in stacked spectra, which rises as a power of mass below a turnover near 10^8.9 solar masses and saturates above it at 12 + log(O/H) = 8.798. The two model curves are a galaxy in equilibrium with its gas supply, whose metallicity is the yield divided by one plus the mass it ejects per unit mass of stars and one plus the dilution by the gas it must keep accreting. Only the ejection changes with mass. A wind driven by the energy of supernovae must lift gas out of a potential whose depth goes as v², so its loading goes as v⁻² and, with v ∝ M^(1/3), as M^(−2/3); a wind driven by momentum goes as v⁻¹ and M^(−1/3). Those set the limiting slopes far below the turnover, 0.67 and 0.33; at log M = 7.5, where dilution still matters, the drawn curves have slopes of 0.59 and 0.24. The measured slope there is 0.57. The dashed measured curve uses a strong-line calibration of the same galaxies' spectra; it sits higher, flattens sooner and has a slope of only 0.40 at log M = 9 — so which wind the data prefer is decided as much by the choice of abundance calibration as by the galaxies. Above the turnover every curve saturates, because a galaxy that ejects almost nothing keeps what it makes and its abundance is the yield, diluted. Galaxies

The metals a galaxy keeps measure what it threw away

Small star-forming galaxies are metal-poor and large ones are not, in a relation tight enough to be a law. Read as an equilibrium between inflow, star formation and wind, its slope says how the wind is driven, its turnover says where galaxies stop losing what they make — and its redshift evolution says galaxies were poorer because they were still being filled.

5 figures A disc built from the inside out, with a gradient of −0.066 dex per kiloparsec at 12 Gyr. The metallicity of the gas, in solar units on a logarithmic scale, against galactocentric radius, at ages of 2, 6, 12 Gyr, for a disc in which every ring is its own box with infall: pristine gas arrives on a timescale that grows with radius, from 1 Gyr in the centre to 7 Gyr at 8 kpc, turns into stars on the depletion time of a Kennicutt law, which is shorter where the gas is denser and much longer below a threshold of 7 M☉ pc⁻², and keeps everything it makes. The slopes fitted between 4 and 14 kpc: −0.259 dex/kpc at 2 Gyr, −0.124 dex/kpc at 6 Gyr, −0.066 dex/kpc at 12 Gyr. The inner disc has had its gas early and turned it over many times, so it is near the yield; the outer disc is still accreting and forming stars slowly, so its gas is diluted and young in the chemical sense. At 8 kpc the model's present abundance is 1.16 of the yield. Every ring is independent here: no gas flows between them and no star moves, which are the two processes that real discs add and which both act to flatten what is drawn. Galaxies

A gradient the old stars have walked away from

The gas in a disc galaxy is richer in metals near the centre than at the edge, by about six-hundredths of a dex per kiloparsec in the Milky Way. Two ingredients of disc growth make that slope, a disc that grows from the inside out makes it flatten with time — and the old stars that should carry the steeper history have moved several kiloparsecs from where they were born.

6 figures A power law has no timescale: half the Type Ia supernovae by 740 Myr and a tail to 13.7 Gyr. The fraction of all the Type Ia supernovae a single burst of star formation will ever produce that have exploded by a given delay, on a logarithmic time axis, for t⁻¹ from 40 Myr; t⁻¹·⁴ from 40 Myr; single delay of 1 Gyr; Gaussian, 3 ± 1 Gyr. The power law is what rates measured against host-galaxy ages and against the cosmic star-formation history both favour, and its cumulative fraction rises as the logarithm of the delay — equal numbers per decade of time. Half have exploded by the geometric mean of its limits, 740 Myr, 55 per cent by 1 Gyr, and the last are still exploding after a Hubble time. A single delay turns the whole population on at once; a Gaussian concentrates it at a characteristic age. The power law's shape has a physical reading: if white dwarfs explode when a pair of them merges by emitting gravitational waves, the merger time goes as the fourth power of their separation, and a broad distribution of separations becomes a distribution of delays with no preferred scale. What the drawing cannot say is which progenitors are involved — the measured rates constrain the shape and the normalisation, about one Ia per thousand solar masses of stars formed, and not the mechanism. Galaxies

The iron clock has no single delay

The α-element knee is drawn as though Type Ia supernovae switched on a billion years after the stars that made them. Measured rates say otherwise — the delays are spread evenly over every decade from forty million years to a Hubble time, as a power law with no timescale in it — and a clock with no timescale bends where a clock with one would break.

6 figures A right angle short by 0.147°, and a ratio of 389 hanging on it. The ratio of the Sun's distance to the Moon's implied by the angle between them at the moment the Moon is exactly half lit, on a logarithmic scale, for angles from 80° to 89.95°. At that moment the angle at the Moon between the directions to the Sun and the Earth is a right angle, so the ratio is the secant of the observed angle — a construction with no distance in it, the same right triangle that gives an inferior planet's orbit from its greatest elongation. Aristarchus measured 87°, which gives 19.1. The mean distances give 389, which corresponds to 89.853°. The curve's steepness is the whole story: across a tenth of a degree centred on each marked angle the ratio changes by 3.4 per cent at 87°, 10.5 per cent at 89° and 103 per cent at the true angle. The method was exact and it asked for a right angle measured to a hundredth of a degree, at an instant the eye can judge only to within hours, on a terminator that is never quite straight. The observed sky

A right angle short by a seventh of a degree

When the Moon is exactly half lit, the angle at the Moon between the Sun and the Earth is a right angle, so the angle seen from the Earth gives the Sun's distance in units of the Moon's. Aristarchus measured 87° and concluded the Sun was nineteen times further away. The construction was exact; the angle he needed was 89.85°, and at that angle a tenth of a degree is the whole answer.

5 figures Jupiter and Saturn meet every 19.86 years, tracing a three-cornered figure that turns 8.5° each round. The heliocentric longitudes at which Jupiter and Saturn are in conjunction — the same longitude seen from the Sun — for 21 successive conjunctions from 1800 to 2200, computed from Keplerian elements and dotted in three colours for the first, middle and last thirds of the span. The mean interval is 19.857 years, the synodic period the two mean motions give. Each conjunction falls 242.8° further round the orbit of Saturn than the one before, so 3 of them come back within 8.5° of where they started: the conjunctions sit near the corners of a 3-sided figure, and the figure itself rotates by 8.5° every 59.6 years. At that rate it returns to its starting orientation — a figure with 3 identical corners only needs to turn by a third of a turn — after about 838 years. The drawn corners are not exactly repeated because the orbits are ellipses: the planets move faster near perihelion, and the conjunction longitudes cluster where both are slow. That near-return is not a coincidence of dates. It is the statement that 3 synodic periods are close to a whole number of each planet's years, which is a near-commensurability of the two mean motions — and near-commensurabilities are where planets perturb one another most. The elements are a fit valid between 1800 and 2050; outside those years they are carried as fixed ellipses turning at their mean rates, which is right for the pattern and not for any individual date. The observed sky

A triangle of meetings that turns in eight centuries

Jupiter and Saturn meet every twenty years, and each meeting falls about two-thirds of the way round the sky from the last, so the meetings trace a triangle. The triangle turns a third of a turn in 838 years because five of Jupiter's years almost equal two of Saturn's — and that same near-fit is the largest perturbation in the solar system, the one that made Saturn appear to be slowing down.

6 figures Venus is brightest 51 days from inferior conjunction, 38 per cent lit. The brightness of Venus through one synodic period of 584 days, in magnitudes below its brightest, against days from superior conjunction — inferior conjunction at the two ends of the axis. The planet is treated as a matte, Lambert-scattering sphere on a circular orbit of 0.723 AU, so its flux is its phase function divided by the square of its distance from the Earth. The two factors fight: near inferior conjunction the planet is closest but shows only a thin crescent, and near superior conjunction it is fully lit but 1.72 AU away. For this orbit the contest has an interior winner. The brightest moment is 51 days either side of inferior conjunction, at an elongation of 44.6° from the Sun, with 38 per cent of the disc lit and the planet 0.539 AU away; greatest elongation, at 46.3°, comes 71 days from inferior conjunction, after the brightness peak on the way out from inferior conjunction. At superior conjunction the planet is 0.82 magnitudes fainter than its best. The marked point is the observed greatest brilliancy, about 36 days from inferior conjunction at an elongation near 39° — closer to conjunction and to a thinner crescent than any matte sphere on this orbit can be brightest at. A surface that sends more light forward, towards large phase angles, would move the peak exactly that way. Very near either conjunction the planet is lost in the Sun's glare, and the curve there describes light nobody sees. The observed sky

Brightest as a crescent, and not as a disc

Venus is fully lit when it is furthest away and nearest when it is barely lit, and it is brightest in between, as a crescent weeks from inferior conjunction. A matte planet only has such a peak if its orbit is wider than about 0.46 of the Earth's — Mercury's is not, and Mercury is brightest full — and Venus's real peak sits closer to conjunction than any matte sphere allows, which is its clouds throwing light forward.

6 figures At the Sun–Earth L₂ the cheapest correction is every 23 days, and it costs e σ per e-folding. The annual station-keeping cost at the Sun–Earth L₂ point, against the interval between corrections, on logarithmic axes, for velocity errors of 0.5 cm/s, 2.0 cm/s, 5.0 cm/s along the unstable direction at each correction. Correcting often costs a lot because every correction carries its own error σ; correcting rarely costs a lot because the error has grown by e^(T/τ) in between, with an e-folding time τ = 23.4 days set by the point's growth rate of 2.484 times the orbital mean motion. The product (365.25/T) σ e^(T/τ) has its minimum at exactly T = τ, where the annual cost is 365.25 e σ/τ: 0.21 m/s a year for σ = 0.5 cm/s, 0.85 m/s a year for σ = 2.0 cm/s, 2.12 m/s a year for σ = 5.0 cm/s. The minimum is broad, so an operator can correct at a convenient interval near the e-folding time for little penalty, and the cost scales linearly with how well the spacecraft's velocity is known and executed. This is a one-dimensional caricature: a real halo orbit's correction also removes a stable component it need not, and solar radiation pressure on a large sunshield is a steady error source of its own. The figure's claim is the structure — an unstable equilibrium is cheap to hold if the instability is caught while it is still small, and its cost is a navigation budget rather than a force budget. Spaceflight

An unstable point that costs less to hold than a stable orbit

A spacecraft at the Sun–Earth L₂ point sits on an equilibrium that throws it away, doubling any error every sixteen days. It holds station for a few metres per second a year — a twentieth of what a geostationary satellite pays to stay on an orbit that is stable. The difference is what is being paid for — an instability caught small costs a navigation budget, and a steady torque costs a force budget.

6 figures A bright star wants a wide aperture and a faint one wants 0.68 of the seeing. Signal-to-noise of simple aperture photometry against the aperture radius, in units of the seeing's full width at half maximum (1″), each divided by what optimal pixel weighting achieves for the same star, for stars of V = 12, 17, 20, 23 observed for 60 s through a 1 m telescope under a sky of 21 mag/arcsec². A small aperture loses starlight; a large one admits sky, and the balance depends on which dominates. For a bright star its own photons are most of the noise, so a wider aperture keeps gaining light almost for free and the best radius is large — 1.63 FWHM at V = 12, reaching 100.0 per cent of the optimum. For a star fainter than its sky the best radius shrinks to 0.680 FWHM and the best aperture reaches only 90.5 per cent of what weighting each pixel by its share of starlight divided by its variance achieves. That residual is exact in the background-limited limit: the best aperture captures 71.5 per cent of the light and 0.902 of the optimal signal-to-noise, so optimal weighting is worth 11 per cent in signal-to-noise, or 23 per cent in exposure time, and no more. The image is taken to be Gaussian; a real point-spread function has broader wings, which makes a fixed aperture a little worse and the optimal weights harder to know. Starlight

The best aperture throws away a tenth

Aperture photometry counts every pixel inside a circle equally and every pixel outside it not at all. For a faint star against its sky the best circle is two-thirds of the seeing wide, catches 71.5 per cent of the light, and reaches 90.2 per cent of the signal-to-noise that weighting each pixel by what it is worth achieves — a loss of 23 per cent in exposure time that no algorithm can beat by more.

6 figures Ten comparison stars as bright as the target cost 5 per cent in precision; ten 2 magnitudes fainter cost 23. The precision of a V = 12 target measured relative to an ensemble of comparison stars on the same 60-second frames, against the number of comparison stars, on a logarithmic precision axis, for comparisons 1 mag brighter, as bright as the target, 1 mag fainter, 2 mag fainter. A change in the atmosphere's transparency of 2.0 per cent — which would put the target's raw brightness out by 20.0 mmag — multiplies every star by the same factor and vanishes from the ratio. What is left is the target's own noise, 0.89 mmag, plus the ensemble's, which falls as the inverse square root of the number of stars in it. With comparisons as bright as the target the result is σ√(1 + 1/N): one comparison costs 41 per cent, ten cost 5. Fainter comparisons are noisier and need many more to reach the same point; brighter ones help, but the target's own noise is a floor the ensemble can only approach. Scintillation is treated as independent from star to star, which is right for stars more than a few arcseconds apart on a large telescope and makes it part of the noise that does not cancel. The figure also cannot show the defining weakness: every comparison star is assumed constant, and a variable among them injects its variability into every measurement made against the ensemble. Starlight

The comparison stars are part of the measurement

Measuring a star against others on the same frame cancels everything the atmosphere and the instrument do to all of them at once — a two per cent change in transparency vanishes completely. What does not vanish is the comparison stars' own noise, which the target inherits, and the variability of any comparison that is not constant, which the target reports as its own.

6 figures A wall at 2.33 hours that bends into a slope below 598 metres. The fastest rotation period a body of bulk density 2 g/cm³ can hold against its own spin, against its diameter, on logarithmic axes, for cohesive strengths of 0, 10, 100, 1000 pascals. With no cohesion the limit is the density-only barrier √(3π/Gρ) = 2.33 hours at every size. A cohesion C adds a stress that does not depend on size to a gravitational stress that goes as the square of the radius, so small bodies are held mostly by cohesion and can spin faster in proportion to their smallness: below the corner the limiting period goes as the diameter. The corner is where the two stresses are equal, at a diameter of 189 m for 10 Pa, 598 m for 100 Pa, 1.9 km for 1000 Pa. The scaling is the strength-regime form with a single coefficient of order one; detailed limits depend on the angle of friction and the shape. What the figure makes plain is why the observed spin barrier is sharp for kilometre-sized asteroids and fades below a few hundred metres, and why a handful of fast rotators a few hundred metres across can be rubble piles with a few tens of pascals of cohesion — a strength far below that of any rock — rather than monoliths. Orbits

A spin barrier with a corner in it

A rubble pile cannot spin faster than a period set by its density alone — 2.3 hours for most asteroids — at any size. Give it a cohesion of a few tens of pascals, weaker than any rock, and the barrier bends into a slope below a corner at a few hundred metres. The thermal torque that drives bodies to the barrier doubles their spin in a time that grows as the square of their size, so the bodies it pushes hardest are the ones that can go past.

6 figures A rubble pile that splits below a mass ratio of 0.204 can lose its piece; above it, the piece stays. The total energy of two spherical components of equal density in contact, spinning together at the rate at which their mutual gravity just holds them against the spin, against the mass ratio of the smaller to the larger, in units of G m₁²/R₁. The energy is the kinetic energy of the rotating pair minus their mutual gravitational binding. When the smaller piece is a small fraction of the whole, the spin carries more energy than the binding and the total is positive: a body spun to breakup that sheds a fragment of that size has enough energy for the fragment to escape entirely, becoming a separate asteroid on a nearly identical orbit. The total changes sign at q = 0.204. Above that ratio the pair cannot separate without an energy source; it stays as a binary, orbiting and eventually synchronising, or re-accretes. As q goes to zero the energy tends to 0.2 G m₁²/R₁, the rotational energy of the primary alone at its breakup rate. Nothing in the threshold depends on the size or the density of the body — it is a pure number from the geometry of two touching spheres — and asteroid pairs sharing an orbit have been found overwhelmingly with estimated mass ratios below it. Orbits

A split that decides whether the piece can leave

A rubble pile spun past its limit splits in two, and whether the smaller piece escapes or stays in orbit is not a matter of luck. Two touching spheres spinning at their shared limit have positive total energy only when the smaller is less than 0.204 of the larger's mass — a number with no size and no density in it. Below it the pieces can become a pair of asteroids on nearly identical orbits; above it, a binary. And the larger the piece that leaves, the slower the body left behind.

6 figures A gauge good to 7 per cent with a tenth of the load left, and to 23 per cent with three hundredths. The uncertainty in the propellant remaining in a spacecraft tank, as a percentage of what remains, against the fraction of the 450-kilogram load still in the tank, on a logarithmic uncertainty axis with the tank emptying to the right. Bookkeeping — summing every thruster firing through a flow-rate model — carries an error common to all burns of 2 per cent of the mass used, plus an independent 5 per cent per burn that averages down over 2000 firings; its absolute error grows with the mass used. Gauging by pressure and temperature infers the empty volume of the tank from the gas law applied to a known mass of pressurant, with a combined 0.66 per cent uncertainty in n R T / P and a 0.2 per cent uncertainty in the tank's volume; its absolute error grows as the gas fills the tank. The two methods are independent and are combined by inverse variance. With a tenth of the load left the combined estimate is uncertain by 2.9 kg, 7 per cent of what remains; with three per cent left, by 3.1 kg, 23 per cent. Near empty the absolute error barely changes, so halving what is left doubles the relative error — the gauge is at its worst exactly when the last manoeuvre has to be planned from it. Spaceflight

A fuel gauge that is worst when it is needed

A spacecraft's tank has no float and no dial. The propellant left is estimated by adding up every burn or by reading the pressure and temperature of the gas above the liquid, and both methods' errors grow with the propellant used. Relative to what remains, the error doubles every time what remains halves — so a geostationary satellite has to hold back months of station-keeping as a margin against a gauge that cannot see the last few kilograms.

6 figures

Before that

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