Astronomy & Celestial Mechanics
the sky, computed
The collection

Every essay — page 6

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 101–120 of 514.

Starlight

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

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.

8 figures · Opacity 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.

8 figures · Extinction 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.

9 figures · Line formation 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.

10 figures · Photometric systems 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.

8 figures · Angular diameter 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.

10 figures · Line formation 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.

8 figures · Ionisation 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.

8 figures · Limb darkening 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.

8 figures · Spectra 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.

9 figures · Extinction 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.

8 figures · Interferometry 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.

8 figures · Photon noise 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.

8 figures · Polarimetry 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.

8 figures · Photometric systems 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.

8 figures · The Doppler effect 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.

8 figures · Interferometry 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.

8 figures · Periodograms 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.

9 figures · Polarimetry 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.

8 figures · Synchrotron radiation 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.

8 figures · Opacity

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