Astronomy & Celestial Mechanics
the sky, computed
Field

Starlight

Distance, brightness and colour: what can be measured when nothing can be visited.
Parallax for a star at 1.3 parsecs. The same star observed from two ends of a baseline. The two sight lines are 1.54″ apart, so the parallax — half of that, the shift seen from a one-astronomical-unit baseline — is 0.769″ for a star 1.3 parsecs away. The definition of the parsec is the distance at which it would be exactly one.

The triangle that reaches the stars, and stops

Parallax is the only distance measurement in astronomy that assumes nothing. It is also the only one with a hard ceiling, and everything beyond that ceiling rests on it.

 The magnitude scale, plotted. The logarithm of the received light against magnitude from -27 to 32, with the vertical scale left unlabelled because only its slope matters. The relation is a straight line of slope −0.4, which is what makes five magnitudes exactly a hundredfold — and the numbers run backwards, so brighter is smaller. 9 landmarks are marked, from the Sun to the deepest exposures, spanning a factor of 1.2·10²³ in received light.

A scale that runs backwards, multiplies, and works

Brighter stars have smaller magnitudes, and five steps is a factor of a hundred. A scale invented by eye in the second century BC turned out to be logarithmic, because eyes are.

 Blackbody curves at 3000, 5800, 10000 K. Thermal emission against wavelength, each curve scaled to its own peak so the shift can be seen on one plot. The peak moves to shorter wavelengths as the temperature rises, which is why colour is a thermometer.

Colour is a thermometer, and it reads across the galaxy

A star's colour gives its surface temperature, from two brightness measurements and no other information. It is the cheapest useful measurement in astronomy.

 The same spectrum, at rest and at 900 km/s. A set of absorption lines at rest and shifted by a radial velocity of 900 km/s. The displacement is proportional to wavelength, so the reddest line here moves 2.06 nm and the bluest 1.18 nm — which is why the measured quantity is the ratio Δλ/λ and not a distance.

A shift in a line is a speedometer, and it works at any distance

A spectral line has a wavelength fixed by physics. Measuring where it actually arrives gives the source's speed toward or away — and the measurement does not degrade with distance.

 An absorption spectrum at 5772 K. A blackbody continuum at 5772 K with absorption lines cut out of it, each line's depth computed from how much of the gas is in a state that can absorb it. 8 of the 8 lines are strong enough to see at this temperature, which is the whole reason the spectral sequence is a temperature sequence.

Composition, read from what is missing

The dark lines in a stellar spectrum are wavelengths that never arrived. Which ones are absent names the elements present — and the strength of a line says more about temperature than about abundance.

 The fractional distance error, rung by rung. Cumulative fractional uncertainty in a measured distance against the distance itself, both on logarithmic axes. Each rung of the ladder is calibrated against the one below it, so its scatter adds in quadrature to everything already inherited, and the total can only rise: radar to the planets 10⁻⁵%, parallax 1.0%, main-sequence fitting 5.1%, Cepheids 6.5%, Type Ia supernovae 8.2%.

Every distance is measured with the last one

No single method reaches from a planet to a distant galaxy. The ladder is built rung by rung, each calibrated on the one below, and the errors multiply all the way up.

 The distance modulus. The difference between apparent and absolute magnitude against distance, on a logarithmic distance axis. It is a straight line of slope five per decade, passing through zero at ten parsecs — the definition of the absolute magnitude. Reading a distance off it requires the absolute magnitude, which is never measured and always inferred.

A brightness is a distance only if something is known

The inverse square law turns a brightness into a distance in one line. The line contains a quantity that has never been measured for any object outside the solar system.

 Opacity against temperature, and the three things that supply it. The Rosseland mean opacity of a gas of composition X = 0.7, Y = 0.28, Z = 0.02, on logarithmic axes, at 10⁻⁷ g/cm³ and 10⁻⁶ g/cm³. The three faint curves are the separate processes at the first density — electron scattering, the Kramers bound-free and free-free term, and the negative hydrogen ion — and the solid curve is their sum. Every one of them is multiplied by the fraction of hydrogen the Saha equation says is ionised at that temperature and density, or by one minus it for H⁻, which is the only reason the low-temperature end is a picture of a star rather than of a formula outside its range: ungated, Kramers alone gives 10,343 cm²/g at 5,800 K, against the 0.40 drawn here. The peak sits at 15,400 K, where hydrogen is 78% ionised — that bump is not a detail, it is the engine of a Cepheid — and the flat floor at high temperature is electron scattering, which is the one term with no temperature in it at all.

The surface that is a depth

A star has no surface. What looks like one is the level at which the optical depth reaches about two-thirds — and the edge is sharp only because the opacity climbs so steeply that the transition takes a ten-thousandth of the radius.

 The extinction law, for three kinds of dust. How much of a star's light dust removes, against inverse wavelength, normalised to one at the V band. Blue light is extinguished more than red, which is why reddening and extinction are the same measurement — and the hump at 4.6 inverse microns is a feature of the grains themselves, present on almost every sight line and still without an agreed carrier. Larger grains give a flatter law and a larger R_V.

Dust makes everything look further away

A star behind dust is fainter, so a distance taken from its brightness comes out too large. The same dust also makes it redder, and the reddening is measurable where the dimming is not — which is the only reason the correction can be applied at all.

 The curve of growth. The equivalent width of an absorption line against the number of absorbers along the sight line, both logarithmic, computed by integrating a Voigt profile with damping parameter 0.005. Three regimes: the width grows in proportion to the abundance while the line is weak, then almost not at all for two decades once the core saturates, then as the square root once the damping wings dominate. A line measured in the middle stretch carries almost no information about the abundance, and most strong lines in a stellar spectrum are there.

How much of a line is not in its depth

An absorption line stops getting deeper long before it stops getting stronger. What keeps growing is its area, and the way the area depends on the number of absorbers has three distinct regimes over five decades — one of which carries almost no information at all.

 B − V against temperature, computed and measured. B − V against effective temperature. The curve is the colour index of a blackbody, obtained by integrating Planck's law against B and V response functions and subtracting a constant so that the index is exactly zero at 9600 K — the convention that an A0V star has every colour index zero, which is a choice and not a measurement. The 15 points are the main sequence as it is actually measured, and they do not lie on the curve: at the Sun's temperature the blackbody gives 0.446 where the sky gives 0.653, and at M0V 0.995 against 1.40. The model is too blue almost everywhere, and least wrong near 9600 K — which is why the zero point is put where it is. A colour index is a difference of two magnitudes, so it is a difference of two integrals, and changing either filter changes the number.

A magnitude has to say which light

The same star is a different magnitude in every filter, and a colour index is a difference of two conventions rather than a property of the star. Both are integrals of a spectrum against a piece of glass, and the zero point is a choice somebody made in 1953.

 Fringe visibility for a 47 mas disc at 575 nm. Fringe visibility against the separation of the two apertures, for a disc 47 milliarcseconds across seen at 575 nm. The solid curve is a uniform disc, |2J₁(x)/x| with x = πθB/λ; it is exactly one at zero baseline, where both apertures see the same wavefront, and falls to zero at 3.08 m — read off the drawn samples, and equal to 1.21967 λ/θ to better than one part in a million. That is the measurement: not a brightness, a baseline. The dashed curve is the same disc with linear limb darkening u = 0.4, whose null is 4.9% further out at 3.23 m — so the same observed null implies 47 mas as a uniform disc and 49.3 mas limb-darkened, and a diameter quoted without its model is a number without a unit. At the 2.54 m aperture of the telescope this was done on, the visibility is still 0.17: one mirror cannot reach the null, which is the same statement as saying it cannot resolve the star.

An angle of five hundredths of an arcsecond

No telescope has ever resolved a star other than the Sun, and stellar diameters are measured anyway — by finding the separation of two apertures at which the star's interference fringes vanish. What that returns is an angle; the radius arrives only when a distance is brought in, and the distance is the worse-known half.

 Three profiles of equal equivalent width, 28.6 mÅ. Three absorption profiles with the same equivalent width — 28.6 milliångström, 1.72 km/s at 500 nm, matched to better than 0.1% by root-finding over the quadrature — differing in nothing but shape. Left: the cores, on a common velocity axis. Right: the same three normalised to their own half widths, on a logarithmic depth scale. The thermal profile is a Gaussian set by Fe's mass at 6000 K, 1.34 km/s; the collisional one a Lorentzian of γ = 2.28e-3 nm; the rotational one the classical kernel of a disc turning at v sin i = 1.73 km/s, which is exactly zero beyond 1.28 half widths and is the only one of the three with an edge. Matching the areas does not match the widths: the half widths are 1.11 km/s (thermal), 1.35 km/s (rotational), 0.68 km/s (collisional), a factor of 1.98 between the widest and the narrowest. At three half widths the collisional wing is 49 times the thermal one and at five it is 1.27·10⁶ times — six decades, which is why a line's shape stays diagnostic long after its width has stopped being so. 11% of the Lorentzian's own equivalent width lies beyond the right-hand panel's edge and is not drawn anywhere.

The same width for three different reasons

Thermal motion, rotation and collisions each widen an absorption line, and they can be tuned to areas that agree to a part in a thousand. What separates them is the shape, and the shape carries a rotation speed from one profile and a surface gravity from another.

 The Balmer maximum is at 9,870 K, and it is a maximum in temperature. What two absorption lines actually count, each normalised to its own maximum. The first curve is the fraction of all hydrogen sitting in n = 2 — the only hydrogen a Balmer line can absorb — which is a Boltzmann factor climbing with temperature multiplied by the neutral fraction falling with it. The product peaks at 9,870 K, where 35.2% of the hydrogen is still neutral and only 8.73·10⁻⁶ of all of it is in n = 2 at all. Below the peak there is plenty of hydrogen and almost none of it excited; above it there is plenty excited and almost none of it neutral. The second curve is the fraction of calcium that is singly ionised, which is what the Ca II K line counts, and it peaks at 6,335 K — cooler, because calcium gives up its first electron at 6.113 eV. At 6,000 K the Balmer curve is at 0.1% of its own maximum while Ca II is near its peak, and calcium is 4.5·10⁵ times rarer than hydrogen in the same gas. A spectrum in which Ca II K is the strongest line is not a spectrum of a calcium star. Reading it as one is precisely the error that had the Sun made of iron until 1925.

Why hydrogen's lines are strongest where hydrogen is not

A Balmer line counts the hydrogen atoms sitting in one particular excited state, and that population peaks near 10,000 K — where a third of the hydrogen has already been ionised away. The strength of a line is a thermometer, and reading it as an abundance put hydrogen at one per cent of the Sun until 1925.

 The limb is 40% as bright as the centre, and that is a temperature gradient. Left: a stellar disc shaded by the grey-atmosphere law I(μ)/I(1) = (2 + 3μ)/5, in 26 steps, with μ = cos θ read from the centre outwards. Right: that law against μ, with linear laws at the measured solar coefficients from 400 to 1600 nm. The grey law's coefficient is exactly 3/5 and its very limb is exactly 2/5 of the central brightness — both read off the drawn curve rather than quoted — because the Eddington–Barbier relation makes the emergent intensity at angle μ the source function at optical depth τ = μ, and in radiative equilibrium that source function is linear in τ. The limb is not a cooler part of the star. A sight line entering at the edge reaches unit optical depth higher up, where the gas is cooler, so what the darkening measures is the run of temperature with depth; a star with an isothermal atmosphere would show a uniform disc, and one with a steeper gradient a darker limb. The measured coefficients fall from 0.9 at 400 nm to 0.35 at 1600 — the same gradient seen through a less steep Planck function — which is why a radius measured from a transit or a fringe null has to say which colour it was measured in.

The light that is missing from the edge

The Sun's limb is forty per cent as bright as its centre, and the reason is not that the edge is cooler. A sight line entering at the edge stops higher up, so what the darkening measures is the temperature gradient — and it is worth seventeen per cent on Betelgeuse's radius.

 A strong line at three gravities, 6000 K. Left, one strong absorption line of Fe computed at three surface gravities and drawn on the same wavelength scale — no normalisation to each profile's own width, which is exactly what the comparison is about. The thermal core is identical in all three, because the Doppler width is 0.00223 nm at 6000 K whatever the star's size. The wings are not: collisional damping is proportional to the density of perturbers, that density is proportional to the gas pressure, and in a grey atmosphere the pressure at the photosphere is proportional to g — so log g = 4.4 carries a damping parameter 794 times that of log g = 1.5. Right, the width at a tenth of the core depth against gravity, measured off those curves: a slope of 0.499, against the half a Lorentzian wing forces. That is the second dimension of a spectral classification. The first is the temperature, read from which lines are present; this one is the pressure, read from how wide they are, and it is the whole reason a spectrum can be turned into an absolute magnitude and then into a distance.

The second thing a spectrum says

Two stars of the same colour can differ in luminosity by ten magnitudes, and the difference shows in the widths of their lines rather than in which lines are present. That width is a pressure, the pressure is a gravity, and the gravity is a distance.

 The Bouguer line, and the intercept nobody observed. Instrumental magnitude against airmass for one star of magnitude 10 outside the atmosphere, observed at 5 airmasses in 5 bands. Each slope is that band's extinction coefficient, computed from Rayleigh scattering, an aerosol term and ozone rather than assumed: U 0.493, B 0.246, V 0.126, R 0.061, I 0.027 magnitudes per airmass. Every line is fitted through its points and extended to X = 0, and the intercept there is the published magnitude — a measurement made at an airmass no observation is ever taken at, because the smallest airmass available is 1 and that is already a whole atmosphere. Two consequences follow and neither is a detail. The slope has to be re-measured every night, because the aerosol term changes with the weather and is not a property of the site. And the U-band line is 3.9 times steeper than the V-band one, so the extrapolation is 3.9 times longer in exactly the band where photons are scarcest — which is why ultraviolet photometry from the ground was always the least trustworthy part of a magnitude system.

A magnitude measured where nothing was measured

Every published brightness is an extrapolation off the end of a graph. A star is observed through one atmosphere at least, a line is fitted against airmass, and the number quoted is its intercept at zero — a place no observation is ever taken from.

 The transform plane an array of 9 actually samples. Every point in this plane is a spatial frequency the array has measured, in thousands of wavelengths. The 9 antennas make 36 pairs, each pair measures one point at any instant, and turning the Earth sweeps each of them along an ellipse — so eight hours of tracking turns 36 measurements into the arcs drawn here. Two properties are structural rather than chosen. The plane is Hermitian: a real sky forces V(−u,−v) = V(u,v), so half the points are free and the coverage is symmetric through the origin. And every ellipse has axis ratio exactly sin δ = 0.707* at this declination, measured off the longest track as 0.707 — an array is squashed in one direction by where the source is in the sky, and at the equator the tracks collapse to lines whatever the array. What the figure cannot show is the hole in the middle: no baseline is shorter than an antenna is wide, so the largest structures on the sky are simply not measured, and no processing recovers them.

Resolution without a mirror

Two telescopes a kilometre apart do not make a kilometre-wide telescope. They measure one number — the Fourier component of the sky at the spatial frequency their separation sets — and an image is what you get by collecting enough of those.

 What the error bar is made of, on a 1 m in 60 s. The four contributions to a photometric error, against the brightness of the star, for a 1-metre aperture, a 60-second exposure and a sky of 21 magnitudes per square arcsecond. The star's own photons give a line of slope exactly 0.2 — σ ∝ N^−1/2 and N ∝ 10^−0.4m, so a magnitude of extra faintness costs a fifth of a magnitude of precision, and no instrument changes that. The sky and the read noise are fixed counts, so their lines have slope 0.4, twice as steep, and they overtake the star at V = 18.25 — that crossing is the faint limit of the night, and it moves when the Moon rises rather than when the telescope changes. Scintillation is flat, because the atmosphere modulates a bright star and a faint one by the same fraction: at 4.09e-4 relative it is 0.44 millimagnitudes here and it is what caps the bright end, up to about V = 10.8. Below all of them is the systematic floor at 0.3 millimagnitudes, which is flat-fielding and colour terms and does not integrate down at all.

The error bar that comes from counting

A brightness is a number of photons, so the precision of the measurement is fixed before any instrument is chosen. What follows is a slope of exactly 0.2 magnitudes of error per magnitude of star, a slope of 0.4 once the sky wins, and a floor that neither of them explains.

 A peak at 0.55 µm, and therefore a grain size. Interstellar polarisation against wavelength — the Serkowski law, p(λ) = p_max exp[−K ln²(λ_max/λ)] with K = 1.66 λ_max. The heavy curve peaks at 0.550 µm, measured off the drawing rather than read back from the parameter, and falls to half its peak at 1.314 µm on the red side, against the closed form λ_max exp√(ln2/K) = 1.315. That peak wavelength is the measurement. It is set by the size of the grains doing the aligning — bigger grains, longer λ_max — and it is tied to the shape of the extinction curve along the same sight line by R_V ≈ 5.5 λ_max, which gives 3.03 here against the diffuse-medium value of 3.1. The two faint curves are populations peaking at 0.35 µm and 0.75 µm: the same amount of polarisation, distributed differently, and a different dust. Nothing in a photometric measurement of the same star distinguishes them.

The direction a photon count throws away

A photometer records how many photons arrived. It discards a two-component quantity that survives every attenuation on the way, and that quantity carries a magnetic field direction, a grain size, and the shape of an exploding star nobody can resolve.

 A colour term of -0.059 magnitudes per magnitude, and one star that will not obey it. Synthetic photometry of blackbodies from 3,000 to 42,000 K through two V bands: the standard one, and a natural system whose effective wavelength is 9 nm longer and whose width is 1.12 times as large. The vertical axis is the difference between the two magnitudes for the same star — not a constant, because a wider redder band collects a different fraction of a hot spectrum than of a cool one. Fitting a straight line against B − V gives a colour term of -0.0585 magnitudes per magnitude and leaves a residual of 3.1 millimagnitudes, which is why a linear transformation is the standard reduction and why it works. The mark off the line is a cool star with molecular absorption bands in the red, drawn from the same blackbody with three synthetic bites taken out of it: it sits 27 millimagnitudes from the fit, 9 times the blackbody scatter. A colour term knows one number about a star and a spectrum has a shape, and that gap is the reason all-sky photometry stops at a per cent while differential photometry on one field reaches a millimagnitude.

The same star through two telescopes

A magnitude is defined by a response curve, and no two telescopes have the same one. The difference between two observatories' measurements of one star is not a constant to be subtracted but a function of the star's colour — and for a star whose spectrum has structure, not even that.

 A 28.7 km/s correction, and a 12.5 m/s planet underneath it. Two years of radial velocities of a star at ecliptic latitude 12°, orbited by a companion whose reflex semi-amplitude is 12.5 m/s — the Sun's own, from Jupiter. The upper panel is what the spectrograph measures: the Earth's motion about the barycentre of the solar system, amplitude 28.72 km/s, which is V⊕ cos β to a fraction of a per cent. The planet is in that curve and is 2,297 times smaller than it, which is a line thinner than the stroke it is drawn with. The lower panel is the same data after the correction, and the correction is not a fit: it is computed from an ephemeris, the observatory's position on a rotating deformable Earth, and the star's own coordinates and proper motion. To leave a centimetre a second it has to be right to one part in 2.9·10⁶ — the light-travel time across the Earth's orbit, the relativistic terms, and the fact that the star moves are all inside that budget. What remains is the planet, at 4333 days, and a scatter of 1.2 m/s that is the star rather than the instrument.

The metre per second that is not the star

A line shift is a speedometer, and the speed it reads is mostly the observer's. Getting to a metre a second means removing thirty kilometres of the Earth's own motion to a part in three million, and then confronting a floor that is the star's own surface rather than the instrument.

 20 closure phases, unmoved by an atmosphere that ruins every baseline. Closure phase measured against closure phase true, for all 20 triangles of a 6-antenna array observing a binary 3.2 mas apart with a flux ratio of 0.35. Each antenna has been given an independent atmospheric phase of 65° rms, which corrupts the individual baseline phases by 102° rms — several times the 31.2° the source itself produces, so no single visibility phase in this simulation carries usable information. Every point here nonetheless sits exactly on the diagonal: the per-antenna terms cancel identically round any triangle, and the largest departure over all 20 is 2.5e-14 degrees, which is round-off. The price is in the counting. 6 antennas give 15 baseline phases of which 5 are consumed by the unknowns, so of the 20 triangles only 10 closures are independent — a fraction (N−2)/N = 0.667 of the phase information. For two antennas that fraction is zero and there is no closure at all; the Event Horizon Telescope's image rests on quantities of this kind and on no absolute phase whatever.

A phase that survives what corrupts it

An atmosphere over each antenna adds an unknown to the phase of every baseline that antenna takes part in. Sum the phases round a triangle and every one of those unknowns cancels identically — which is the reason an image can be made across ten thousand kilometres, and the reason it has no position on the sky.

 A true peak at 0.3103 and a false one at 0.6897, from the same data. The Lomb–Scargle periodogram of 162 simulated observations taken over 419 nights from one site, of a star carrying a 4.2-unit sinusoid at 0.3103 cycles per day under 3 units of Gaussian noise per point. The injected signal is recovered at 0.3103 cycles per day, within the 0.0024 resolution element the baseline allows. The second peak, at 0.6899, is 87 per cent as tall and corresponds to nothing: it is 1 − f, the signal reflected in the one-cycle-per-day spike of the sampling. Neither peak is more real than the other in this picture — deciding between them needs a second site at a different longitude, or a run long enough for the seasonal window to separate them. The dashed line is the power a pure-noise series would exceed once in 1000 trials, computed from 586 independent frequencies rather than from the 4691 grid points searched; using the grid count instead would put the line 2.08 higher and reject a real detection.

A period found in the gaps

Astronomical time series are sampled when the sky is dark and clear and the target is up, which is a schedule with a spectrum of its own. That spectrum is convolved with the real one, so a single sinusoid produces several peaks — and the tallest is not always the true one.

 A 3-gauss field moves the line by 0.33% of its width and is measured anyway. Above: the Fe I 6173 Å line, Landé factor 2.5, at a Doppler width of 0.041 Å. The solid curve is the unmagnetised profile; the dashed curve is the same line in a longitudinal field of 3 gauss, which splits it by 1.3·10⁻⁴ Å — 0.33 per cent of its own width — and is drawn on top of it because the two are not distinguishable. The double-lobed curve underneath them is the Stokes V profile of that same field, magnified 100 times: the two σ components are circularly polarised with opposite handedness, so what is lost in the sum survives in the difference, and the difference of two profiles a hair apart is the derivative of one of them. Below: what that buys. The V amplitude is linear in the field, because it is a first derivative; every signature of the same field in the intensity is quadratic, because a symmetric splitting can only broaden. At a polarimetric precision of 10⁻⁴ the first reaches 0.1 gauss; at a line-width accuracy of 0.001 the second reaches 38, a factor of 420 worse. A quantity far too small to resolve is measured because it is the only thing in the signal that carries a sign — and the same argument run the other way says what polarimetry cannot do: a field of mixed polarity inside the resolution element cancels in V and does not cancel in I, so the 3000-gauss field of a sunspot, which does resolve, is measured the other way round.

A field strength read off a line that will not split

A few hundred gauss splits a spectral line by a ten-thousandth of its own Doppler width, which no spectrograph will ever resolve. The two components are circularly polarised with opposite handedness, so the split survives in the difference of two polarisations — where it is linear in the field rather than quadratic.

 Three slopes: 2.5, −0.75 and −1.25, and only the middle one is about the electrons. A radio source's spectrum across five decades of frequency, computed for an electron population with index p = 2.5 in a uniform field. Nothing in this curve is a temperature, because a power law has no scale and therefore nothing a thermometer could read. The straight section between the two bends has slope −0.75, measured here off the drawn curve, and the electron index follows from it and from nothing else: α = (p − 1)/2, so a flux ratio between two frequencies is a measurement of the energy distribution of particles in a place no detector will ever visit. The two bends are the other two measurements. Below 80 MHz the source is opaque to its own radiation and rises as ν^2.5 — a slope fixed at 5/2 by the geometry alone, whatever the electrons are doing — and the frequency at which that happens gives an angular size for a source nothing has resolved, because the turnover is where the brightness temperature meets the electrons' own. Above 12 GHz the spectrum steepens by 0.50: the electrons that radiate at high frequency lose their energy fastest, so the top of the distribution has already emptied, and the frequency of the break is a clock. What no part of this curve gives is the magnetic field. Only the field and the particle density together enter the emission, and every field strength ever quoted for a radio source comes from assuming the two share the energy equally.

A spectrum with no temperature in it

Every spectrum in this collection so far has been a thermometer. A radio lobe's is a power law, and a power law has no scale — so there is nothing for a thermometer to read, and what the shape carries instead is the energy distribution of the particles that made it.

 Two means of one opacity, 75 times apart, and only the smaller one is in the equation. Above: a synthetic opacity across the frequencies that carry a star's flux, drawn against x = hν/kT, with a continuum falling as ν⁻³ and a forest of 6 lines per unit x on top of it. The shaded curve is the Rosseland weighting function, ∂B_ν/∂T, which peaks at x = 3.83 and is what decides which frequencies matter. The two horizontal lines are the two ways of averaging. The Planck mean is an ordinary average and lands high, among the lines, because that is where most of the opacity is. The Rosseland mean is a harmonic average — it averages 1/κ rather than κ, because what carries the flux out of a star is transparency and transparencies add — and it lands 75 times lower, close to the continuum, because a harmonic mean is dominated by the smallest values in it. In other words the opacity that appears in the equation of radiative transport is a measurement of the gaps between the lines. Below: what that means for a table. The Planck mean rises as the first power of the line density, slope 0.76 as drawn — every line added is another contribution to an ordinary average. The Rosseland mean does almost nothing at first, slope 0.13, and then turns up sharply, slope 0.99, once the lines are close enough to blanket the windows. That is why adding several million atomic transitions to an opacity table in the early 1990s changed nothing for decades and then changed stellar structure: the new lines were not the first lines, they were the ones that finally closed the gaps.

A mean dominated by the gaps

The opacity in the equation of radiative transport is not an average of the opacity. It is a harmonic average weighted by the temperature derivative of the Planck function, which makes it a measurement of the transparent windows between the lines rather than of the lines — and that single fact decides what adding a million spectral lines to a table does.

 What the error bar is made of, on a 1 m in 60 s. The four contributions to a photometric error, against the brightness of the star, for a 1-metre aperture, a 60-second exposure and a sky of 21 magnitudes per square arcsecond. The star's own photons give a line of slope exactly 0.2 — σ ∝ N^−1/2 and N ∝ 10^−0.4m, so a magnitude of extra faintness costs a fifth of a magnitude of precision, and no instrument changes that. The sky and the read noise are fixed counts, so their lines have slope 0.4, twice as steep, and they overtake the star at V = 18.25 — that crossing is the faint limit of the night, and it moves when the Moon rises rather than when the telescope changes. Scintillation is flat, because the atmosphere modulates a bright star and a faint one by the same fraction: at 4.09e-4 relative it is 0.44 millimagnitudes here and it is what caps the bright end, up to about V = 10.8. Below all of them is the systematic floor at 0.3 millimagnitudes, which is flat-fielding and colour terms and does not integrate down at all.

The faint star is measured against a brighter sky

For anything at the edge of detection the dominant source of noise is not the object. It is the sky in the same aperture, which is brighter than the star and is subtracted rather than measured — and once that is true, every rule of thumb about apertures, exposure times and image quality changes.

 The light dust removes at 0.56 µm comes back at 124 µm. One energy budget drawn twice, on a wavelength axis spanning four and a half decades. The upper curve is starlight from a 6500 K photosphere; the shaded region under it is the part removed by a magnitude of visual extinction, computed from the same CCM law the other modes here draw, and weighted towards the ultraviolet exactly as that law says. The curve on the right is what the grains do with it: a modified blackbody at 20 K with an emissivity index of 1.8, normalised so that the energy under it equals the energy under the shaded region. Integrating the two curves actually drawn returns a ratio of 1.000. The starlight peaks at 0.56 microns and the re-emission at 124, a factor of 220, so nothing about the two is recognisable as the same photons and everything about them is the same joules. The practical consequence is a rule about arithmetic: a galaxy's ultraviolet luminosity and its far-infrared luminosity are not two independent measurements of how many young stars it has. One is the light that escaped and the other is the light that did not, and adding them without noticing counts part of the population twice.

The dust is not lost light, it is moved light

An extinction curve says how much starlight dust removes. It does not say the light is gone. Every photon a grain absorbs heats the grain, which radiates it back out at twenty kelvin — so an extinction measurement and a far-infrared spectrum are two halves of one energy budget, and the area under them is the same area.

 A wind that writes its own terminal speed on the blue edge, at 2017 km/s. A P Cygni profile, computed by integrating a spherical wind on a grid rather than drawn. The wind accelerates outward as one minus the inverse radius to the power 0.8, reaching 2000 kilometres a second, and its density falls as the inverse square of the radius over its own speed; the horizontal axis is the Doppler shift in units of that terminal speed, with blue to the left. The column directly in front of the stellar disc is moving toward the observer at every speed from nearly nothing near the surface up to the terminal value far out, so it takes light out of the beam across that whole range and the absorption trough runs from the rest wavelength to the blue edge at -1.01 of the terminal speed. That edge is the measurement: no model of the star, its distance or its mass enters it, only the geometry of a column seen end-on. The emission comes from everywhere else, where the wind moves across the line of sight and can only add photons; it is Infinity times stronger to the red than to the blue, because the part of the shell receding directly away is hidden behind the star and its blueshifted counterpart is not. Two features, one shell, and between them a velocity and a mass-loss rate.

A speed read off an edge

A hot star drives material off itself at thousands of kilometres a second, and the profile that wind prints on the star's own spectrum has a sharp blue edge. That edge is the terminal velocity, measured with no model of the star, no distance, and no calibration — one of the few numbers in stellar astrophysics obtained from geometry alone.

 A better measurement that made the model worse: 0.9 per cent in the sound speed. The fractional difference between the Sun's sound speed as its own oscillations measure it and as a structural model predicts it, against fractional radius. Zero would be agreement. The lower curve is the model built on the solar abundances used until the mid-2000s, and it hugs the axis: a part in a thousand across most of the interior, which was for a long time the best-tested piece of stellar physics anybody had. The upper curve is the same model with the abundances re-measured using three-dimensional atmospheres and without assuming local thermodynamic equilibrium — better measurements by every methodological standard, which lowered carbon, nitrogen and oxygen by around thirty per cent. The disagreement grows to 0.9 per cent, and it is not spread through the star: it peaks at 0.683 of the radius, just beneath the base of the convection zone at 0.713. The same substitution moves the model's own convection-zone base from 0.715 to 0.729, against a seismic value known to about a thousandth. What is being tested here is not really the abundances but what converts a composition into a structure, which is the opacity: the metals whose abundances fell are exactly the ones whose bound–free absorption dominates at those temperatures, and an opacity larger by some fifteen per cent near that boundary would restore the agreement. Laboratory measurements of iron at those conditions have since come in high by about that much, which is a satisfying result to have arrived at by way of a discrepancy in the sound speed of the Sun. The curves are published inversions and model differences rather than anything computed here; what the figure adds is where they peak and by how much.

A better measurement that made the model worse

The Sun's composition was re-measured with better atmospheres and better physics, and the carbon, nitrogen and oxygen abundances fell by about thirty per cent. The improved model then disagreed with the Sun's own oscillations by ten times as much as the model it replaced, and it still does.

 A transit depth of 1.200 per cent for a planet of area 1.000 per cent. Three transits of the same planet across the same star, differing only in how the star's brightness falls toward its edge. A planet of radius ratio 0.1 covers 1.000 per cent of the stellar disc's area, and if the disc were uniformly bright that would be the depth. It is not uniformly bright: a sight line near the limb leaves the photosphere at a shallow angle and therefore from a cooler layer, so the edge is dimmer than the centre, and a planet crossing near the middle blocks light that is brighter than average. The transit drawn with realistic coefficients is 1.200 per cent deep — 20 per cent deeper than the area — and it is also rounder, because the covered brightness changes through the crossing instead of staying flat. The consequence is stated in the numbers beside the curves. Each is a least-squares fit of the radius ratio to the realistic curve, performed with a different assumed limb-darkening law, and the recovered radius moves by up to 3.6 per cent depending on which law is assumed. Fitting with the law the curve was made from returns the input to five figures, which is the control: the bias is the mis-specification and not the fitter. Since the coefficients come from a model atmosphere rather than from the light curve, every published planetary radius carries a systematic from stellar physics that no amount of photometric precision removes — and it is the dominant one for the best-measured planets. The picture holds the impact parameter fixed; a grazing transit is worse, because it samples only the limb, where the disagreement between laws is largest.

The depth is not the area

A planet covering one per cent of its star's disc does not make a transit one per cent deep. The star is brighter in the middle, so a planet crossing the middle blocks more than its share — and the correction depends on coefficients that come from a stellar atmosphere model rather than from the light curve.

 A star 1.24 times wider than it is tall, and 19 per cent brighter pole-on. Left, the meridional section of a star rotating at ω = 0.9257 of its critical angular velocity, computed from the Roche potential rather than sketched: the equator sits at 1.245 polar radii, and at the critical rate that ratio is exactly 1.5 whatever the star is made of. The same rotation expressed as a fraction of the critical equatorial speed is 0.768, and the two conventions differ by the distortion itself — a figure that prints one under the other's name is wrong by an amount that looks like rounding. Effective gravity at the equator is 0.329 of its polar value, so von Zeipel's flux law makes the pole hotter than the equator by a factor 1.320 at the theoretical exponent 0.25 and 1.232 at the 0.188 that interferometric imaging actually fits. Right, the apparent bolometric brightness against viewing inclination, integrated over the visible gravity-darkened surface: pole-on the star is 1.19 times brighter than edge-on, and the apparent temperature falls with it. The consequence is that a rapid rotator's place on the Hertzsprung–Russell diagram is partly a statement about the observer's position, which no spectrum taken alone can undo.

A temperature that depends on where the observer stands

A star turning near its break-up rate is half again as wide as it is tall, and its equator is thousands of degrees cooler than its poles. Neither of those is a small correction to a spectrum — the effective temperature and the luminosity such a star appears to have are partly statements about which way its axis happens to point.

 A slope, not an angle. Above: the polarisation angle of a background source against the square of the observing wavelength, at the five wavelengths a radio survey actually uses. The plane rotates as it passes through magnetised plasma, by an amount proportional to the integral of the electron density times the field along the path — and to λ². The intercept is the angle the source emitted at, which nobody knows, so a measurement at one wavelength contains no information whatever; the slope is the rotation measure, here 42 radians per square metre, and it needs no knowledge of the source at all. Below: the fractional polarisation that survives. A telescope beam covers many lines of sight with slightly different rotation measures, their angles disagree by more at longer wavelengths, and the vector sum collapses — which is why the useful band has a long-wavelength edge that has nothing to do with sensitivity. Divide the rotation measure by the dispersion measure of the same path, 26.8 in the usual units, and the electron density cancels: the mean line-of-sight field is 1.93 µG, obtained without knowing the distance, the density, or where along the path the field was.

A slope that needs no source

Polarised light passing through magnetised plasma has its plane rotated, by an amount proportional to the square of the wavelength. A single measurement of the angle is worthless, because nobody knows the angle the source emitted at. A measurement of the slope against wavelength squared does not need to know.

 The field nobody measured, and the reason the number is quoted anyway. The energy content of a radio source against the field strength assumed for it, both logarithmic. A synchrotron luminosity constrains only the product of the relativistic electron content and the field: a bright source can be many particles in a weak field or few in a strong one, and no observation of the radiation distinguishes them. The two curves are the two costs. Assuming a weak field is expensive in particles, because the particle energy needed rises as the field to the minus three halves; assuming a strong one is expensive in the field itself, which rises as B². Their sum has a minimum at 5.0e-5 gauss, and at that minimum the particle energy is exactly four thirds of the field energy — which is where the word equipartition comes from, and it is a property of where a curve turns rather than a statement about nature. The estimate is quoted because it is robust: the minimum-energy field goes as luminosity to the two sevenths, so an order of magnitude of ignorance about the luminosity is a factor of 1.93 in the answer. It is also, for the same reason, nearly uninformative — a number that hardly moves is a number that hardly measures.

A minimum that was mistaken for a principle

A radio source's brightness fixes the product of its particle content and its field, and nothing about the radiation separates them. What is quoted instead is the field that makes the total energy least — and at that field the particles happen to carry four thirds of what the field does, which is where the word equipartition comes from and why it is not a physical assumption at all.

 Two effects that are blind in opposite directions. The sensitivity of two magnetic diagnostics against field strength, on a logarithmic axis spanning five decades. The Zeeman effect measures the line-of-sight component and adds it up along the path, so a field tangled into 100 independent cells inside one resolution element averages down by a factor of 10 and reports almost nothing — which is exactly the situation in a chromosphere or a turbulent cloud. The Hanle effect is a different device altogether: a field precesses the atom between absorption and re-emission, so a scattering line's polarisation is rotated and reduced, and the amount depends on how far the precession gets in one radiative lifetime. That makes it sensitive around 1 gauss for a 100-nanosecond level, and — the useful part — it does not care about sign, so a tangled field does not cancel. Above the crossing at 27.7 gauss the Hanle signal has saturated and carries no strength information, and the Zeeman effect is the instrument. Neither is a measurement of the field; each is a measurement of what the field did to something else.

Two instruments blind in opposite directions

The Zeeman effect measures the component of a field along the line of sight and adds it up, so a field tangled into a hundred cells reports a tenth of one cell's strength. The Hanle effect measures how far an atom precesses between absorbing and re-emitting, does not care about sign, and saturates just where the Zeeman effect becomes useful.

 4000 lines averaged into one profile, and 2.3 m/s out of it. A cross-correlation function: the average absorption profile obtained by shifting a mask of 4000 line positions across a spectrum and summing what falls under it. The faint curves behind are individual lines, each with its own depth, its own width and its own small offset; the heavy curve is what averaging them produces. The velocity is the position of the peak, and its precision is the width divided by the contrast, the signal-to-noise and the square root of the number of lines — 2.3 metres a second here. Nothing about this construction is a measurement of any one line. It is a measurement of where a weighted average of thousands of them sits, and the weights are a choice: a mask built for one spectral type applied to another weights the disagreement between the lines differently, and moves the peak.

A velocity that is an average of lines that disagree

A radial velocity measured to a metre a second is not measured from a line. It is the position of the peak of a cross-correlation against a mask of thousands of lines, and those lines do not agree with each other by hundreds of metres a second — because each one forms at a different depth in an atmosphere that is boiling.

 A dome flat that is 3.5 per cent wrong leaves 0.042 magnitudes across the field. Two things called the flat field. The left panel is the true illumination of the focal plane, falling by 12 per cent from centre to corner because of vignetting and the filter's own radial transmission. The middle panel is what a dome flat measures, which is the illumination produced by a screen at a finite distance lit by lamps — a different angular distribution, and therefore a different fall-off by a few per cent. The right-hand plot is what survives dividing one by the other: a smooth radial gradient of 0.042 magnitudes from centre to edge. Pixel-to-pixel response scatter, which is what most people mean by a flat field, is 1.8 per cent per pixel and averages down to 0.255 per cent inside a photometric aperture. The term everybody removes is the one that does not matter, and the term that matters is smooth, is different for every flat-fielding method, and looks exactly like a real gradient in the sky.

A response measured pixel by pixel

Two completely different quantities are called the flat field. One is the detector's pixel-to-pixel response, which everybody removes and which averages away anyway. The other is the illumination pattern of the optics, which is smooth, is different for every method of measuring it, and survives into every magnitude the instrument produces.

 A continuum drawn 4.3 per cent below the real one. A short stretch of spectrum with one strong line in it and 150 weak ones scattered across the same interval. The upper dashed line is the true continuum — the flux the star would emit with no lines at all — and it is not observable. The lower one is what a fit through the highest points of the spectrum returns, which is 4.3 per cent lower, because the weak lines have depressed the gaps between the strong ones. Measuring the strong line's equivalent width against the apparent continuum instead of the real one makes it 11.2 per cent too small. The error has a sign, it is worse in spectra with more lines, and it therefore correlates with metallicity — which is exactly the quantity being measured.

A continuum that was never observed

An equivalent width is an area measured relative to the continuum, and the continuum is not in the data. It is drawn — a curve through the highest points of the spectrum — and in any spectrum with many weak lines those highest points are already below the true continuum, because the weak lines have eaten the gaps.

 Four spectra, and 3.9 magnitudes between them at the same redshift. The K-correction — the magnitude that has to be added to compare a redshifted object with a nearby one through the same filter — against redshift, for four power-law spectra. At zero redshift every correction is zero by construction. Beyond that they diverge, because a filter at a fixed observed wavelength samples a different part of the source's own spectrum at every redshift, and how much flux is there depends on the spectrum. A flat spectrum needs no correction at all at any redshift, and the two extremes drawn differ by 3.9 magnitudes by z = 1.2. The circularity is the point: applying the correction requires the spectrum, and the spectrum is what a magnitude is being used to constrain. The dashed line is the way out — observe in a band chosen so that it lands on the rest-frame band of interest, and the spectral term cancels, leaving only the bandwidth stretch.

A magnitude in a band the source never had

A filter passes light at a fixed observed wavelength, and a redshifted source emitted that light at a shorter one. Comparing a distant galaxy with a nearby one through the same filter therefore compares two different parts of two spectra — and the correction between them needs the spectrum, which is what the magnitude was going to be used to find out.

 One over a noisy parallax, at three precisions. The distribution of the distance obtained by inverting a parallax, for a star truly at 100 parsecs measured with fractional errors of 5, 10, 20 per cent. At five per cent the distribution is nearly symmetric and inverting is harmless. At twenty per cent it is strongly skewed: the mean sits at 105 parsecs rather than 100, and the tail runs to distances several times the truth, because a parallax scattered a little towards zero is a distance scattered a long way outward. The asymmetry is a Jacobian and nothing else — the parallax measurement is unbiased and symmetric throughout. Above about twenty per cent the mean of the distribution stops existing at all, because the density falls only as the inverse square of the distance and the integral of d times that diverges.

The distance is not one over the parallax

A parallax is measured with symmetric errors and a distance is one over it. Inverting a noisy positive quantity is not a change of units — it is a change of distribution, and the one that comes out is skewed, biased outward, and above about twenty per cent error has no mean at all.

 A sample that gets brighter with distance because the faint ones drop out. The mean absolute magnitude of a magnitude-limited sample, relative to the population it is drawn from, against distance. The population has a spread of 0.5 magnitudes about a mean of -4, and the survey stops at apparent magnitude 20. Nearby, everything is detected and the sample is unbiased. Beyond about 316228 parsecs the faint end of the distribution starts falling below the limit and the survivors are brighter than average; further out the bias deepens without limit, because eventually only the extreme tail is detectable. The horizontal line is the classical Malmquist value of 1.382 times the square of the spread, which is what the bias averages to over a magnitude-limited sample as a whole — it is a property of the sample rather than of any one object, and using it as a correction for an individual star is a common and specific mistake.

A sample brighter than the population it came from

Every survey stops at some apparent brightness. At any distance it therefore contains only the objects luminous enough to make the cut, so the average object in it is brighter than the average object in the universe — by an amount that grows with distance and that has been shortening every distance in astronomy since 1920.

 A peak worth 10.8 in a narrow search is worth nothing in a wide one. The probability that noise alone produces a peak at least as tall as a given power, for searches over four different numbers of independent frequencies. A single frequency examined in isolation gives a one-per-cent chance at a power of 4.6; searching fifty thousand frequencies for the same one-per-cent chance requires 15.4. The threshold rises as the logarithm of the width of the search, which is why the penalty is survivable — but it is a penalty, it is often not applied, and the number of independent frequencies in an unevenly sampled time series is not the number of frequencies on the grid. Overestimating that count is conservative and underestimating it is not, which is the one asymmetry worth remembering.

The tallest peak in nothing at all

A periodogram of pure noise has peaks in it, and the tallest is not small. How tall it has to be before it means something depends on how many frequencies were searched and on what the noise actually is — and astronomical noise is almost never the white noise the standard formula assumes.

 Two diagnostics, two bands, and a crossing to 49 kelvin. The plane of effective temperature against surface gravity, with the constraints from two spectroscopic diagnostics drawn as bands. The wings of a hydrogen line are broadened by collisions, so they respond steeply to the gravity and weakly to the temperature: a narrow, steep band. An ionisation balance — requiring that the same element give the same abundance from its neutral and its singly ionised lines — responds to both, and its band is much shallower. Neither diagnostic determines either quantity on its own. Where the two cross is the answer, and the size of the crossing region is set by the band widths divided by the difference of the slopes — so two diagnostics that respond similarly give a long, thin, nearly useless error region however precise each one is. Choosing diagnostics that disagree in their sensitivities is the whole of the art.

A temperature and a gravity that trade against each other

A stellar spectrum contains the star's temperature, its surface gravity and its composition, and no single feature in it contains only one of the three. Every diagnostic is a band in the parameter plane rather than a point, and the answer is where the bands cross — which makes choosing diagnostics that disagree in their sensitivities the whole of the art.

 An instrument 1.4 times as polarised as the sky it is measuring. The Stokes plane, with the two linear polarisation parameters as axes. The tight cluster near the centre is a set of stars known to be unpolarised, observed through the same instrument: they should sit at the origin and do not, and their mean is the instrumental polarisation — 0.77 per cent here, which is 1.4 times the real polarisation of the field. The other cluster is the field stars, whose measured values are the sum of their own polarisation and the same instrumental offset. Subtracting one mean from the other recovers 0.59 per cent at the right angle. Two things make this worth doing carefully. The offset is a vector, so leaving it in rotates the measured position angle as well as changing its magnitude — by 31 degrees here — and a calibration that only fixes the scale does not touch that. And the offset depends on where the telescope was pointing, because the reflection angles do, so the standards have to be observed at the same place in the sky and at the same instrument rotation.

An instrument more polarised than the sky

Every oblique reflection polarises. A telescope is a stack of oblique reflections, so it adds a fraction of a per cent of polarisation to everything it looks at — which for most astronomical sources is more than they have themselves, and which is a vector rather than a scale error, so it rotates the answer as well as changing its size.

 A few per cent of diameter, hidden in the second lobe. Visibility against baseline in the natural variable πθB/λ, for stars with linear limb-darkening coefficients of 0, 0.3, 0.6, 0.9, each rescaled to the uniform disc that best fits its own first lobe. In the first lobe the four curves are within 0.72 per cent of one another; past the first null they differ by up to 4.2 per cent. That is the whole difficulty of measuring a stellar diameter. A limb-darkened star has a faint edge, so a uniform-disc fit returns a diameter 9.7 per cent too small at u = 0.9 and 2.5 per cent too small at u = 0.3 — and the information needed to tell which is in a region where the visibility is under five per cent and the calibration errors of a real interferometer are comparable to the signal. The correction from what is measured to what is wanted is taken from a model atmosphere, because the observation that would supply it is the hardest one there is. The fit here is done by least squares on the drawn curves rather than read from a conversion table, and the zero-coefficient case returns the uniform disc to 0.00 per cent, which is the fit checking itself.

A diameter that depends on a model atmosphere

An interferometer measures fringe visibilities and somebody fits a disc. A uniform disc and a limb-darkened one agree to within a per cent across the whole of the first lobe and differ by five in the second — where the visibility is under five per cent and the calibration errors are the same size.

 742 kelvin between a dwarf and a supergiant, at the same ionisation. The fraction of calcium still neutral against temperature, for three surface gravities: a dwarf, a giant and a supergiant. The three curves are the same curve slid sideways. The Saha equation carries the electron pressure in its denominator, and the photospheric pressure follows the surface gravity as its square root — hydrostatic equilibrium gives a gas pressure of order g over the opacity, and the opacity in a cool star is set by the electrons themselves. So the pressure runs from 19 newtons a square metre in the dwarf down to 0.48 in the supergiant, a factor of 40, and the half-ionisation point moves from 4427 kelvin to 3685 — 742 kelvin apart. At fixed temperature the ionisation ratio goes as g^-0.50, exactly the square root the algebra requires. A star's luminosity class is a measurement of the density of its photosphere, read off which stage of an element the lines belong to.

A luminosity class is a density measurement

The Saha equation has an electron pressure in its denominator, and a supergiant's photosphere is forty times less dense than a dwarf's at the same temperature. So the same element sits in different ionisation stages in the two, the spectrum says which, and the second axis of stellar classification is a barometer.

 Where the axes of rapid rotators point, in a sample chosen by brightness, at a darkening exponent of 0.19. The distribution of rotation-axis inclinations — 0° pole-on, 90° equator-on — for gravity-darkened stars whose axes point at random in space, surveyed down to a limit in apparent brightness, with the surface temperature following the local gravity to the power 0.19. The dashed curve is random orientation, which puts 13.4 per cent of stars within 30° of pole-on. At 0.9 of the critical rotation rate a star looks 1.093 times as luminous pole-on as equator-on, the survey reaches correspondingly further for the pole-on ones, and 14.4 per cent of the sample lies within 30° of pole-on; the nearly pole-on stars are 1.09 times as common as random orientation would make them. At 0.98 of the critical rotation rate a star looks 1.188 times as luminous pole-on as equator-on, the survey reaches correspondingly further for the pole-on ones, and 15.3 per cent of the sample lies within 30° of pole-on; the nearly pole-on stars are 1.19 times as common as random orientation would make them. Averaged over random orientations the apparent luminosity of each star equals its true luminosity to better than half a per cent, as it must; the tilt towards pole-on comes entirely from choosing stars by how bright they look.

The pole-on stars a brightness limit prefers

A rapidly rotating star looks brighter and hotter from its pole than from its equator, so a survey that picks stars by how bright they look should pick more of them pole-on. It does — and the surprise is how little. Averaged over random orientations, a gravity-darkened star's apparent luminosity is exactly its true one, because every photon goes somewhere; a brightness limit restores only a fraction of a per cent of bias, while the error in any single star is ten times larger and of either sign.

 How fast the circulation a rotating star drives would stir it, against its lifetime. The classical Eddington–Sweet circulation time — the star's Kelvin–Helmholtz time divided by the ratio of centrifugal to gravitational acceleration at its surface — as a multiple of its main-sequence lifetime, against rotation rate as a fraction of critical, for stars of 2, 5, 15 solar masses, on logarithmic axes. Below the line at one, the circulation would turn the star over within its life. For 2 solar masses that happens above 0.101 of critical; for 5 solar masses that happens above 0.123 of critical; for 15 solar masses that happens above 0.115 of critical. The dot is the Sun, turning at 0.0084 of its critical rate, whose circulation would take 150 times its main-sequence life. By this estimate almost every star turning at more than a tenth of its critical rate should be stirred from core to surface; the measured surface compositions of such stars show that it is not, because the composition gradient left by core burning resists the circulation, and how strongly it resists is what the nitrogen at their surfaces measures.

The circulation that should have stirred every fast rotator

A rotating star in radiative equilibrium cannot be balanced in pressure and in heat at the same time, and the mismatch drives a slow circulation from pole to equator. The classical estimate of its speed says that any star turning at more than a tenth of its break-up rate should be stirred from core to surface within its life, bringing the nitrogen of hydrogen burning up with it. Some fast rotators show that nitrogen and some do not, and some slow rotators show it when they should not — including more than the angle of their axes can explain.

 Where titanium oxide comes apart, and what the pressure does to it. The fraction of titanium oxide dissociated into Ti + O, against temperature, for a gas of equal parts at 10, 300, 6000 N/m² of total pressure. The equilibrium is Saha's with a free atom in place of the free electron and the pair's reduced mass in place of the electron's — the two atoms' partial pressures multiplied together, divided by the molecule's, equal an equilibrium constant Kₚ(T) — which for equal abundances gives a dissociated fraction of √(Kₚ/(Kₚ+P)) exactly. A molecule is one particle becoming two, so pressure suppresses the reaction exactly as it suppresses ionisation — the half-dissociation point moves from 3145 K at 10 N/m² to 4164 K at 6000. The bond energy is 6.87 eV against hydrogen's ionisation potential of 13.6, which is why this happens at a third of the temperature. Rotational and vibrational partition functions are included, classically and harmonically; the atoms keep their ground-term weights.

A spectrum with no continuum left in it

Below about four thousand kelvin a photosphere stops being a gas of atoms. Titanium oxide, water and carbon monoxide take over, the same equilibrium settles how much of each survives with a bond energy where an ionisation potential used to be — and the bands are so crowded that the level every line depth is measured against is nowhere on the plate.

 Where a photosphere's free electrons come from. The share of the free electrons donated by metals rather than by hydrogen, against temperature, at a gas pressure of 12000 N/m² and at solar abundance, 1 dex below solar, 2 dex below solar. The electron pressure is not assumed here — it is solved for, as the value at which the ionisation of the gas supplies exactly the electrons the gas contains. Below about 5897 K at solar abundance every free electron in the gas comes from an element present at one part in ten thousand, because hydrogen's 13.6 eV keeps it neutral while magnesium's 7.6 does not. The handover is fast: solar crosses a half at 5897 K, 1 dex down crosses a half at 5088 K, 2 dex down crosses a half at 4467 K. At 5,772 K and this pressure the solved electron pressure is 1.78 N/m².

The continuum is made by one atom in ten thousand

The Sun's light leaves through an ion that exists only because a hydrogen atom will hold a second electron by three-quarters of an electronvolt. The electrons it holds come almost entirely from magnesium, silicon and iron — so the level against which every solar line depth is measured is rationed by elements present at one part in ten thousand.

 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.

 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.

 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.

 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.

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.

 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.

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.

The ladders in this field

21 anchors · one idea each

ParallaxMagnitudesStellar colourThe Doppler effectSpectraDistance ladderOpacityExtinctionLine formationPhotometric systemsAngular diameterIonisationLimb darkeningInterferometryPhoton noisePolarimetryPeriodogramsSynchrotron radiationStellar windsGravity darkeningFaraday rotation

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