Astronomy & Celestial Mechanics
the sky, computed
The collection

Every essay — page 25

One idea per essay, ordered so that the earlier ones set up the later ones — but nothing here depends on being read in sequence. Essays 481–500 of 534.

Exoplanets

Planets nobody has seen, weighed and measured from a dip, a wobble and a delay.

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.

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 · Detection bias 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.

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 · Detection bias

Galaxies

Where the unknown stops being a number and becomes a profile — and most of it is not light.

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.

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 · Luminosity function 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.

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 · Luminosity function 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.

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 · Luminosity function

The observed sky

Coordinates, seasons, phases and shadows — geometry seen from inside it.

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°.

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 · Seasons 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 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 · Seasons 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 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 · Seasons

Spaceflight

Celestial mechanics used forwards — where to burn, and what it costs.

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.

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 · Low-thrust transfer 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.

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 · Low-thrust transfer 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.

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 · Low-thrust transfer

Starlight

Distance, brightness and colour: what can be measured when nothing can be visited.

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.

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 · Stellar colour 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.

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 · Stellar colour 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.

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 · Stellar colour

Orbits

Kepler's three laws, and the family of curves a single force allows.

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.

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 · Circularisation 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.

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 · Circularisation

Cosmology

One object, seen once, from inside — and every number in it the output of a model.

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.

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 · Sunyaev zeldovich 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.

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 · Sunyaev zeldovich 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.

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 · Sunyaev zeldovich

Exoplanets

Planets nobody has seen, weighed and measured from a dip, a wobble and a delay.

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.

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 · Habitable zone

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