Astronomy & Celestial Mechanics
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Starlight

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.

Assumes Spectra, Polarimetry and Magnitudes.

18 min read 8 figures Light is the only messengerMeasuring the unreachable

A blackbody spectrum has a peak, the peak has a wavelength, and the wavelength is a temperature. Everything this collection has done with starlight rests on that: a colour is a thermometer, a spectral class is a temperature sequence, and the shape of the continuum under the lines is the first thing a spectrum says.

A radio lobe’s spectrum has no peak. Over four or five decades of frequency the flux density falls as a straight line in log–log — a power law, Sν∝ν−αS_\nu \propto \nu^{-\alpha} — and a power law is scale-free. There is no characteristic frequency in it, so there is no characteristic energy, so there is nothing a thermometer could read.

What such a spectrum carries instead is a different quantity entirely, and it is one no instrument can otherwise reach: the energy distribution of the electrons that produced it.

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.
Fig. 1 The whole shape, computed for a population of electrons with energy index p=2.5p = 2.5 in a uniform field. The straight middle section is the measurement: its slope is −(p−1)/2-(p-1)/2, and reading it off gives the particle spectrum somewhere no particle detector will ever be. The two bends are two further measurements — the low-frequency turnover is where the source becomes opaque to its own radiation, which gives an angular size without resolving anything, and the high-frequency break is where the fastest radiators have already died, which gives an age.

Taken as given

The single-particle physics is not derived here. A relativistic electron spiralling in a magnetic field radiates, the radiation is beamed into a cone of half-angle 1/γ1/\gamma about its instantaneous velocity, and the spectrum from one electron peaks at a frequency proportional to γ2B⊥\gamma^2 B_\perp. That is a consequence of electromagnetism and special relativity, it belongs to the field that owns that derivation, and this essay treats it the way an observational essay treats an equation: as a stated relation whose consequences can be measured against.

The two relations used below are

νc  ∝  γ2B⊥,(dEdt)  ∝  γ2B⊥2,\nu_c \;\propto\; \gamma^2 B_\perp, \qquad \left(\frac{dE}{dt}\right) \;\propto\; \gamma^2 B_\perp^2 ,

and the second of them, integrated, gives an electron’s lifetime as γ−1B−2\gamma^{-1}B^{-2}.

The transfer

Suppose the electrons have a power-law energy distribution, N(E) dE∝E−p dEN(E)\,dE \propto E^{-p}\,dE, over a wide range. Each electron of energy EE radiates near one frequency ν∝E2\nu \propto E^2, so the emission at frequency ν\nu comes from electrons of energy E∝ν1/2E \propto \nu^{1/2}, and the flux is the number of them times the power each radiates. Working the substitution through gives

Sν  ∝  ν−(p−1)/2,α=p−12.S_\nu \;\propto\; \nu^{-(p-1)/2}, \qquad \alpha = \frac{p-1}{2}.

A power law in energy becomes a power law in frequency, and the index survives the transfer with a factor of two in it. That is the whole basis of the technique, and it is a strong claim: it says that measuring a flux ratio between two frequencies is a measurement of how particles a million times more energetic than anything in a laboratory plasma are distributed in energy.

A flux ratio, an electron index, and a polarisation nobody measures. Above: the map from what a radio telescope measures to what it means. The observation is a flux ratio between two frequencies, which is a slope α; the quantity of interest is the electron energy index p, which no instrument can reach; and the relation between them is α = (p − 1)/2, a straight line of slope exactly one half. The five sources are placed at their measured indices — Cassiopeia A at 0.77, the Crab at 0.3, Cygnus A's lobes at 0.74 — and the range they span is not scatter about one number: a shock accelerating particles gives p near 2.4, and the Crab's much flatter spectrum is the signature of something else entirely, a wind fed continuously by a pulsar rather than a population left behind by a blast. Below: the fractional polarisation the same electrons should produce, (p + 1)/(p + 7/3), which is between 78 and 66 per cent everywhere and is one of the most nearly constant predictions in astrophysics. Not one source approaches it. The dots below the curve are the measured values, a third of the prediction or less, and the shortfall is the figure's most useful content: polarisation is a vector average, so it measures the order of the field rather than its strength, and a source at 30 per cent has a field that is coherent over a third of its own volume.
Fig. 2 The map, drawn with real sources on it. Above, the relation between what is measured and what it means, a straight line of slope one half. The measured indices are not scattered about one value: a shock accelerating particles gives pp near 2.4, and the Crab’s much flatter spectrum is a signature of something structurally different — a wind fed continuously by a pulsar rather than a population left behind by a blast. Below, the polarisation those same electrons should produce, which is between 69 and 75 per cent for every index a real source has. Not one source approaches it, and the shortfall is the disorder of the field.

Why a power law at all

The observation is that the index is nearly the same in a great many unrelated objects: supernova remnants, radio galaxies, the diffuse emission of the Milky Way’s disc, the jets of active nuclei. Something is producing p≈2.4p \approx 2.4 nearly everywhere.

The standard answer is diffusive shock acceleration, in which a particle crosses and re-crosses a shock front, gaining a fixed fractional energy each time and having a fixed probability of escaping. A process with a constant gain per crossing and a constant escape probability produces a power law by construction — the exponent is the ratio of the logarithm of one to the logarithm of the other — and for a strong non-relativistic shock the arithmetic gives p=2p = 2 with no adjustable parameter.

The measured indices are a little steeper than that, and the excess is where the interesting physics is: energy losses during acceleration, the finite lifetime of the shock, and the back-reaction of the accelerated particles on the shock they are crossing.

Three slopes: 2.5, −0.50 and −1.00, 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 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 −0.50, 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.
Fig. 3 The theoretical case drawn: p=2p = 2 exactly, which is what a strong non-relativistic shock delivers with nothing fitted. The straight section has slope one half, so the flux falls by a factor of three across a decade of frequency rather than the hero’s factor of five, and the difference between this drawing and that one is the whole of what the excess above is. It is not a small difference in a spectrum and it is a very small one in an index — 0.50 against 0.74 — which is the reason radio spectral indices are quoted to two decimals and the reason a survey with two frequencies rather than five is not usable for this.

The turnover, which is a size

At low enough frequency the same electrons that emit also absorb, and the source becomes optically thick to its own radiation. Below that frequency the emergent spectrum is no longer a power law: it rises as ν5/2\nu^{5/2}, a slope fixed by the geometry and not by anything about the electrons.

The reason it is 5/25/2 rather than the 22 of a blackbody is worth a sentence, because it is one of the more elegant facts in the subject. An opaque thermal source has a Rayleigh–Jeans tail going as ν2T\nu^2 T with TT fixed. An opaque synchrotron source has an effective temperature set by the electrons that radiate at that frequency, and those have E∝ν1/2E \propto \nu^{1/2} — so the temperature is not fixed, it rises as ν1/2\nu^{1/2}, and the product goes as ν5/2\nu^{5/2}.

The consequence is a measurement. At the turnover the brightness temperature equals the electrons’ own, and the brightness temperature of a source of flux SS and angular size θ\theta at frequency ν\nu is ∝Sν−2θ−2\propto S\nu^{-2}\theta^{-2}. So a turnover frequency and the flux there give θ\theta — an angular size for a source that no telescope has resolved, from two numbers on a spectrum.

Three slopes: 2.5, −0.79 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 −0.79, 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 1.5 GHz 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 60 GHz the spectrum steepens by 0.46: 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.
Fig. 4 The same source made twenty times more compact, so its turnover moves from eighty megahertz to one and a half gigahertz. That is the measurement running backwards: the turnover frequency goes up as the source shrinks at fixed flux, because a smaller source has a higher brightness temperature and reaches its own electrons’ temperature sooner. The practical consequence is which instrument sees which sources. A metre-wave array finds the extended, aged, low-turnover population; a centimetre-wave one finds the compact cores. Neither is choosing — the turnover decides, and it is a size.

The break, which is a clock

The radiative lifetime of an electron goes as γ−1B−2\gamma^{-1}B^{-2}, so the most energetic electrons die first. A population injected all at once and left to radiate therefore develops a cut-off in energy that moves downwards with time, and the spectrum acquires a break at the corresponding frequency.

If instead particles are being supplied continuously — a jet still running — the break is softer: the high-energy end reaches a steady state in which injection balances loss, and the spectrum steepens by exactly half a power. That factor of a half is a strong prediction, it is very nearly what is seen in the lobes of active radio galaxies, and it is the reason the break can be interpreted at all.

αhigh−αlow=0.5.\alpha_{\rm high} - \alpha_{\rm low} = 0.5 .

The break frequency then gives an age, through the lifetime expression, provided the magnetic field is known.

A break frequency that dates a source, and a field strength that maximises its life at 1.9 μG. Above: where the spectral break falls, against the time since the electrons were accelerated, for fields of 1, 3, 10, 30 microgauss. Every curve has log-log slope exactly −2 — the electrons that radiate at frequency ν have a lifetime going as ν^−1/2, so the frequency above which the population has emptied falls as the inverse square of the age — and a break measured anywhere on this diagram is therefore an age, provided the field is known. Below: the lifetime at a fixed break of 50 GHz, plotted against the field. It is not monotonic, and the turnover is the part worth carrying away. A weaker field radiates less, so a naive reading makes a weak-field source arbitrarily old; but the same electrons also scatter microwave-background photons, and that loss does not care about the field at all. Below 1.88 microgauss — the microwave background's own equivalent field of 3.25 μG divided by √3 — inverse-Compton losses dominate and the lifetime falls again. There is a maximum age any synchrotron source can have and still show a break where one is seen, and at redshift 0 it is about 22 million years. The oldest radio galaxies are limited by a photon field rather than by their own.
Fig. 5 The same clock read at a break of fifty gigahertz rather than five. Every age drops by the square root of ten, because the break frequency goes as the inverse square of the elapsed time — so an order of magnitude in the observed break is only a factor of three in age. That insensitivity cuts both ways. It means a break measured to a factor of two gives an age good to forty per cent, which is unusually forgiving; and it means the ages of two sources differing by a factor of three in break frequency are barely distinguishable, so this clock sorts sources into decades rather than ranking them.
A break frequency that dates a source, and a field strength that maximises its life at 1.9 μG. Above: where the spectral break falls, against the time since the electrons were accelerated, for fields of 1, 3, 10, 30 microgauss. Every curve has log-log slope exactly −2 — the electrons that radiate at frequency ν have a lifetime going as ν^−1/2, so the frequency above which the population has emptied falls as the inverse square of the age — and a break measured anywhere on this diagram is therefore an age, provided the field is known. Below: the lifetime at a fixed break of 5 GHz, plotted against the field. It is not monotonic, and the turnover is the part worth carrying away. A weaker field radiates less, so a naive reading makes a weak-field source arbitrarily old; but the same electrons also scatter microwave-background photons, and that loss does not care about the field at all. Below 1.88 microgauss — the microwave background's own equivalent field of 3.25 μG divided by √3 — inverse-Compton losses dominate and the lifetime falls again. There is a maximum age any synchrotron source can have and still show a break where one is seen, and at redshift 0 it is about 69 million years. The oldest radio galaxies are limited by a photon field rather than by their own.
Fig. 6 The clock and its ceiling. Above, the break frequency against elapsed time for four field strengths: every curve has log–log slope exactly −2-2, because the lifetime goes as ν−1/2\nu^{-1/2}. Below, the age a fixed observed break implies, plotted against the field — and it is not monotonic. A weaker field radiates less, so a naive reading makes a weak-field source arbitrarily old; but the same electrons also scatter microwave-background photons, and that loss does not care about the field. Below about 1.9 microgauss the photon losses dominate and the lifetime falls again. There is a maximum age any synchrotron source can have and still show a break where one is seen.

What cannot be measured

The emissivity depends on the number of electrons and on the field, and only through their product. Nothing in the spectrum separates them: a source with twice the field and a quarter the particles radiates identically at every frequency.

The universal workaround is to assume that neither dominates — that the energy in particles and the energy in the field are comparable. Minimising the total energy required to produce the observed luminosity gives a field close to that equipartition value, so the assumption and the minimisation give nearly the same answer and are often conflated. Either way, every magnetic field strength quoted for a radio source in the literature is the output of an assumption rather than a measurement, and every derived quantity that depends on it — the age from the break above, the total energy content, the pressure of a lobe against its surroundings — inherits that.

There are two escapes and both are indirect. The inverse-Compton X-rays from the same electrons scattering the microwave background depend on the particle density and on the photon field, whose energy density is known exactly from the background’s temperature; combining that with the synchrotron flux separates BB from NN. And Faraday rotation measures the line-of-sight field in the material in front of the source, which is a different field in a different place but constrains the same order of magnitude.

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.
Fig. 7 The assumption itself, drawn as the curve it is. Total energy against assumed field strength for a fixed observed luminosity: the particle term falls as the field rises, because a stronger field needs fewer electrons to produce the same flux, and the field term rises. The sum has a minimum, and the minimum is close to where the two terms are equal — which is why “equipartition” and “minimum energy” are used interchangeably even though they are different statements. What matters for reading any published number is the flatness near the bottom: the total varies by less than a factor of two over an order of magnitude in field, so the minimum is well defined in energy and poorly defined in the quantity everybody quotes from it.

What was actually measured

The observations behind everything above are flux densities: a source’s brightness in janskys at a handful of frequencies, from a handful of telescopes, calibrated against a small set of sources whose spectra are themselves defined by convention.

For Cygnus A — the archetype, at a distance of 230 megaparsecs — the lobes have α=0.74\alpha = 0.74 over the decade from 100 MHz to 1 GHz, steepening above a few gigahertz. Turning that into a statement about electrons requires only the transfer relation, and gives p=2.5p = 2.5. Turning the break into an age requires the equipartition field of about 50 microgauss and gives a few million years, which is comfortably shorter than the light-travel time across the source and is therefore a statement that the lobes are being actively resupplied.

For the Crab, α=0.30\alpha = 0.30 from radio to infrared, implying p=1.6p = 1.6 — flatter than any shock produces, which is the observation that the pulsar wind rather than the supernova blast is the particle source.

For the Milky Way’s own diffuse emission, α≈0.8\alpha \approx 0.8 at metre wavelengths, which is the electron population that also arrives at the Earth as cosmic rays and can be measured directly. That comparison is the only place in the subject where the inferred electron spectrum and a directly measured one can be put side by side, and they agree to within the uncertainties of the local field.

The one population that can be caught

The inference throughout has been from a spectrum to a particle distribution nobody can sample. There is exactly one place where the particles arrive and can be counted, and the comparison is the technique’s only direct calibration.

Cosmic rays reach the Earth continuously, and a small fraction of them are electrons rather than nuclei. Their energy spectrum is measured directly by detectors above the atmosphere, and over the range that matters here — a few to a few hundred gigaelectronvolts — it is a power law with an index near three.

That is steeper than the index inferred from the Galactic radio emission, and the difference is not a discrepancy but a prediction. The electrons arriving here have been propagating through the Galaxy for millions of years, radiating as they go, and radiative losses steepen a spectrum by exactly one power in the regime where losses dominate. Correcting for that recovers an injection index consistent with the radio one.

The agreement is worth what it costs to state carefully. The radio emission comes from the whole Galactic volume, weighted by the magnetic field; the direct measurement comes from a cubic metre near the Earth, at one moment. That the two agree after a propagation correction is evidence both that the emission mechanism is understood and that the local sample is representative — neither of which could be established from either measurement alone.

There is one further feature of the direct spectrum that the radio cannot see. At about three thousand teraelectronvolts the cosmic-ray spectrum steepens abruptly — the knee — and the break is thought to mark the maximum energy Galactic shock acceleration can reach. Electrons at that energy radiate synchrotron in the X-ray rather than the radio, and the corresponding break is looked for in the X-ray spectra of young supernova remnants.

A spectrum measured on the ground and a spectrum measured on the sky are the same population seen at two ends of a propagation problem, and reconciling them is the one place this essay’s inference chain closes on itself.

Ten decades of one power law, with two places where it bends. The cosmic-ray spectrum, multiplied by energy to the 2.7 so that its features can be seen at all — undivided, it falls by thirty orders of magnitude across this plot and every bend in it is invisible. The knee at 3·10¹⁵ electronvolts is where the spectrum steepens from E^−2.7 to E^−3.1, and it sits within a factor of a few of the energy at which a proton's gyroradius in a microgauss field becomes comparable to the thickness of the galactic disc: above it, confinement begins to leak. The ankle at 3·10¹⁸ is where it flattens again, which is read as a galactic population running out and an extragalactic one taking over. Above 5·10¹⁹ the spectrum is cut off, because a proton that energetic loses energy to the microwave background within about fifty megaparsecs and cannot have come from further. Three features, each of them a statement about a magnetic field: one about the galaxy's, one about where the galaxy's ends, and one about the fact that empty space is not empty.
Fig. 8 The directly measured spectrum, over ten decades of energy, with the two places it bends. Nothing here is inferred from a radiation mechanism — these are particles counted one at a time by detectors above the atmosphere and, at the top end, by the air showers they produce. The knee near three thousand teraelectronvolts is where Galactic shock acceleration is thought to run out, and the ankle is where an extragalactic population takes over. Set against the radio inference, this is the only calibration the whole technique has: a power law measured by counting, and a power law measured by reading a slope off a spectrum, agreeing after a propagation correction that neither of them was tuned to produce.

What a flat spectrum is

A power law with an index near 0.7 is the ordinary case and there is a class of source whose spectrum is nearly flat — index near zero, so that the flux density is almost the same at every frequency over two or three decades. Those are the compact cores of active nuclei, and the flatness is not a property of their electrons.

A single homogeneous region that is self-absorbed has a spectrum that rises steeply, turns over, and falls. Stack many such regions of different sizes along a jet — the innermost small and dense, turning over at high frequency; the outermost large and rarefied, turning over at low — and each contributes a peak at its own frequency. The sum of a series of overlapping peaks, if the sizes and densities are distributed in the right way, is a flat spectrum.

The condition that makes it flat is a particular scaling of the magnetic field and the particle density along the jet, and it happens to be the scaling a conical jet in equipartition produces. So a flat spectrum is a statement about the jet’s geometry rather than about its particles.

Two consequences matter observationally. A flat-spectrum source is a superposition, so the position of its apparent centre depends on the observing frequency — the higher the frequency, the closer to the true base of the jet. That frequency-dependent position shift is measurable by interferometry and is the reason a jet’s core is not a fixed point on the sky.

And a flat-spectrum source cannot be aged by the method of this essay’s break, because the break belongs to whichever component dominates and the components are at different ages. The technique that works on a lobe fails on a core, and the failure is structural rather than a matter of precision.

The generalisation

The distinction this essay turns on recurs wherever a spectrum is read.

A thermal spectrum is a statement about equilibrium: the emitting particles have exchanged energy until their distribution is the one with the most ways of happening, and everything about the history that got them there has been erased. One number remains.

A non-thermal spectrum is a statement about a process: the particles have not equilibrated, their distribution is the fossil of whatever accelerated them, and the shape is a record rather than a state. More is recoverable, and more has to be assumed to recover it.

Which regime an emitter is in is decided by whether the collision time is shorter than the loss time, and the answer is nearly always the same at high density and the opposite at low. That is why the interiors of stars are thermal to extraordinary precision and the near-vacuum of a radio lobe is not, and it is why the same electron energy of a few gigaelectronvolts is unremarkable in a cosmic-ray detector and unreachable in any laboratory: the lobe has no walls to thermalise against.

What happens below the turnover, and what is doing it

The low-frequency end of the spectrum was described above as a self-absorption turnover, and for a compact source that is what it is. For an extended source at metre wavelengths there is a second mechanism producing a very similar bend, and distinguishing them is the standing difficulty of low-frequency radio astronomy.

Ionised gas along the line of sight absorbs by free–free absorption, whose optical depth rises steeply towards low frequency — as roughly the inverse square. So a source seen through a foreground of warm ionised gas shows a turnover whose position depends on how much gas is in front of it rather than on anything about the source.

The two bends look alike on a plot and differ in three ways that can be exploited.

Self-absorption produces a rise as the frequency to the power five-halves below the turnover; free–free absorption produces an exponential cutoff, which is much sharper. Distinguishing them therefore needs measurements at several frequencies below the bend, which is exactly where the sky is brightest and the ionosphere worst behaved.

Self-absorption depends on the source’s own angular size, so a resolved source turns over at different frequencies in different parts of it. Free–free absorption depends on the foreground, so it produces a turnover that follows the distribution of ionised gas rather than the source’s structure.

And self-absorption is intrinsic, so it moves with redshift; free–free absorption by Galactic gas does not.

The practical consequence is that the low-frequency sky contains a population of sources whose spectra bend for reasons that have to be established object by object. Since a steep-spectrum source is brightest at low frequency, the surveys that find the oldest and most aged populations are exactly the ones where the interpretation is hardest — the regime with the most sources is the regime with the least certainty about what a spectrum means, which is the ordinary situation whenever a survey pushes into a new band.

Where this ladder goes next

Later rungs on this anchor: Faraday rotation, and how a rotation measure separates the field along the line of sight from the field in the plane of the sky; the inverse-Compton counterpart of every synchrotron source, which breaks the field–density degeneracy at the cost of an X-ray detection; the beaming of a relativistic jet, and why the same object looks like two different classes depending on its orientation; the flat-spectrum cores of active nuclei, which are flat because they are the superposition of many self-absorbed components rather than because their electrons are; and the low-frequency sky, where whole populations of aged, steep-spectrum sources exist that no gigahertz survey has ever detected.

Where this goes next

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. 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. Two measured numbers, and everything else on the page derived from them. 25 pulsars in the plane of period against period derivative, at their catalogued values. Only the two axes are measurements; the three families of contour are models. Constant surface field runs at slope −1 because B ∝ √(PṖ), constant characteristic age at slope +1 because τ = P/2Ṗ, and the two families cross the population at right angles — which is why a single dot fixes both. The Crab sits at 3.8·10¹² G and 1257 years, and its true age is 972; the millisecond pulsars at the lower left have fields ten thousand times weaker and characteristic ages of billions of years, because they were spun back up by a companion long after they died. The line at the lower right is the death line, B/P² below which the model says no pair production and therefore no radio emission — and J2144−3933 is drawn below it, an 8.5-second pulsar that is radiating anyway. A clock that runs down and says what it is A pulsar hands over two measured numbers, a period and its derivative. Everything else usually quoted about it — an age, a magnetic field, a luminosity — is derived from those two through a model, and the one measurement that tests the model refutes it.

What this makes readable

Essays that name this one as a prerequisite.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Brightness temperatureCooling breakElectron energy distributionEquipartitionMinimum-energy fieldNon-thermal emissionPolarisationPower lawRadio lobeSelf-absorptionSpectral indexSynchrotron radiation