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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.

Assumes Synchrotron radiation and Polarimetry.

18 min read 10 figures Light is the only messengerMeasuring the unreachable

A radio galaxy’s lobes are among the largest coherent structures in the universe and among the least directly measurable. They are made of two things — relativistic electrons and a magnetic field — and they are seen through one observable, which is the light the first emits while spiralling in the second.

The trouble is arithmetic. The synchrotron emissivity depends on the electron density multiplied by a power of the field, so a given brightness can be produced by many electrons in a weak field or by few in a strong one. There is no second equation.

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. 1 The two costs of an assumption, against the field assumed. Choosing a weak field is expensive in particles, because the particle energy required rises as the field to the minus three halves; choosing a strong one is expensive in the field itself, which rises as the square. Their sum has a minimum, and at the minimum the particle energy is exactly four thirds of the field energy.

What the observation actually constrains

A relativistic electron in a magnetic field radiates at a frequency proportional to the square of its Lorentz factor and to the field strength, and at a rate proportional to the square of both. Integrating over a power-law population of electrons gives the familiar power-law spectrum, and its slope is the electron index halved and shifted — a genuine measurement of the particle distribution’s shape.

The shape is measured; the normalisation is not. What the luminosity fixes is a product of the electron number density and the field raised to a power near one and a half. One equation, two unknowns, and no prospect of another from the radiation itself.

That is the whole difficulty, and it is worth insisting on it, because the number that gets quoted in every paper — “the equipartition field” — is often read as a measurement when it is a choice.

The situation would be different if the source were resolved into a region where the two could be separated, or if a second radiative process depending differently on the two were detected. Both are possible in a few cases and neither is generally available, which is why the substitute below is so entrenched.

It is worth comparing this degeneracy with the ones the collection has already met, because it is a different species. A distance and a luminosity are degenerate until a standard candle breaks them; a mass and an inclination are degenerate until a transit breaks them. Both of those are broken by finding a second observable of the same object. Here the second observable — the spectral shape — carries no information about the normalisation at all, so the degeneracy is not merely unbroken but structurally unbreakable within the synchrotron measurement itself.

The polarisation does not help either, though it looks as though it should. A synchrotron source’s polarisation fraction depends on the electron index and on how ordered the field is, and neither of those is the field’s strength. A source at thirty per cent polarisation has a field coherent over a third of its volume and could have any strength whatever. That is the same limitation an aligned-grain map runs into, and for the same reason: polarisation is a vector average, and an average of directions is a measurement of order.

The substitute: pick the cheapest total

Suppose the field is weak. Then to produce the observed luminosity there must be a great many electrons, because each radiates feebly, and their total energy is large. Suppose instead it is strong. Then few electrons suffice, but the magnetic energy filling the volume is itself large.

The particle energy required goes as the field to the minus three halves; the field energy goes as the field squared, times the volume. Their sum falls, turns and rises, and has a single minimum.

Differentiating gives the field at the minimum, and it depends on the luminosity divided by the volume, raised to the power two sevenths. Two sevenths is a strange exponent and it is worth seeing where it comes from: setting the derivative of a minus-three-halves power against a plus-two power to zero gives an exponent of two over seven halves, which is four sevenths in the ratio and two sevenths in the field.

At that minimum the two energies are not equal. The particle energy is exactly four thirds of the magnetic energy, and the factor of four thirds is the ratio of the two exponents’ magnitudes rather than a physical statement.

That near-equality is the entire origin of the word “equipartition” in this context. It is a property of where a curve turns, not a claim that the universe arranges for two energy reservoirs to be comparable.

The distinction is not pedantry, because there are physical arguments for genuine equipartition in some systems and they are separate arguments. A field in a turbulent medium saturates when its energy approaches the turbulent energy, which is a dynamical statement with a mechanism behind it; a cosmic-ray population streaming through a magnetised medium is limited by the instabilities it drives, which is another. Neither of those arguments applies to a radio lobe filled by a jet, and neither is what the minimum-energy calculation says.

The word has done real damage by conflating them. A reader who takes “equipartition field” to mean “the field is in equipartition with the particles, as physics requires” has acquired a belief with no support, and one that the independent measurements below contradict.

Three slopes: 2.5, −0.90 and −1.40, 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 = 2.8 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 −0.90, 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. 2 The same source with a steeper electron population — p=2.8p = 2.8 rather than 2.5. The straight middle section steepens from a slope of −0.75-0.75 to −0.90-0.90 and the section above the break from −1.25-1.25 to −1.40-1.40, and both differences are exactly (p−1)/2(p-1)/2 and p/2p/2 evaluated at the new index. Nothing else in the picture moves: the self-absorption turnover is set by the source’s own brightness temperature and the break by its age, and neither knows what the electron index is. Three slopes, and only the middle one is about the electrons — which is the whole reason a spectral index has to be measured between the bends and not across them.

Why the estimate is so hard to shake

The exponent is what makes the number quotable, and it cuts both ways.

An order of magnitude of error in the luminosity — which is generous, since fluxes are usually good to twenty per cent — changes the field by a factor of ten to the two sevenths, which is about 1.9. An order of magnitude of error in the volume changes it by the same. Two orders of magnitude of ignorance about the filling factor, which is genuinely unknown, gives a factor of four.

So the estimate is robust in the sense that it is hard to be very wrong. It is uninformative for exactly the same reason: a quantity that barely responds to its inputs barely measures them.

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. 3 The one thing the radiation does measure well. The spectral index gives the electron energy index directly, with no normalisation in it, and the values cluster where strong-shock acceleration puts them. Everything about the shape is measured; everything about the amount is assumed.

There is a second reason it persists, which is comparability. Two sources analysed the same way can be compared, and most of what has been learned about radio galaxies is comparative — lobes are more strongly magnetised than the diffuse medium around them, hotspots more than lobes, jets more than either. Those statements survive a common systematic that a single absolute value does not.

The measurement that does separate them

There is a way to break the degeneracy, and it is the reason this subject has moved at all in thirty years.

The same relativistic electrons that radiate synchrotron light also scatter any photons passing through them up to much higher energies. The microwave background is everywhere, its energy density is known exactly from its temperature, and it is not something a source can be shielded from. So the inverse-Compton X-ray luminosity depends on the electron content times the photon energy density, while the synchrotron luminosity depends on the electron content times the magnetic energy density.

Two equations, two unknowns, and the photon energy density is known independently. Dividing one by the other gives the field with no assumption at all. The measurement is hard — the X-ray emission is faint and has to be separated from thermal gas and from unrelated sources — but it has been made for several dozen radio galaxies. The answer is that the true field is typically a factor of a few below the minimum-energy value, meaning the sources are particle-dominated rather than sitting at the minimum.

That is a mild result and it is the right kind of result. The minimum-energy estimate is not badly wrong; it is systematically high by a factor between two and five, and the direction is consistent across sources.

The direction is also the physically sensible one. A source below the minimum-energy field contains more total energy than it needs to, which requires something to have put it there — and a jet delivering particles continuously into a lobe is exactly such a thing. A source sitting exactly at the minimum would require a coincidence between the injection history and the field’s decay.

There is a second and independent route in a handful of cases: a synchrotron spectrum that steepens at a break frequency where the electrons’ radiative lifetime equals the source’s age. If the age is known from another argument, the break gives the field directly. The two methods agree where both are available, which is reassuring and rare.

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. 4 The break that supplies the second route. Electrons radiate away their energy at a rate proportional to the square of the field, so the highest-energy part of the population dies first and the spectrum develops a break whose frequency falls with time — an age, if the field is known, or a field, if the age is.
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. 5 The same map from a measured slope to an electron index, marked at p=2.2p = 2.2 instead of 2.5 — the value diffusive shock acceleration predicts for a strong shock, and the one a great many sources are close to. The relation α=(p−1)/2\alpha = (p-1)/2 is a straight line, so an uncertainty in the measured slope is half that uncertainty in pp, and the polarisation fraction the same population would show is a third number nobody measures. Two of the three quantities on this page are inferred from the first, and the minimum-energy estimate then treats all three as known.

What the energy actually is

Once a field is adopted, the total energy follows, and the numbers are worth stating because they are the reason the exercise is done.

A large radio galaxy’s lobes at minimum energy contain something like ten to the sixtieth ergs. That is the rest energy of about a million solar masses, delivered over ten to a hundred million years by a jet, into a volume a megaparsec across.

Dividing by the time gives a power of ten to the forty-fourth or forty-fifth ergs a second, sustained for tens of millions of years. That is comparable to the total radiative output of the host galaxy’s stars, and it is being delivered mechanically rather than as light — which is why the energy content of radio lobes is one of the standard inputs to arguments about how a cluster’s cooling gas is kept from cooling.

The energy has to come from accretion onto a central black hole, and dividing gives an accreted mass — typically a few tens of millions of solar masses, which is a substantial fraction of the black hole itself. So the minimum-energy calculation, for all its softness, sets the scale of what an active nucleus does to its surroundings over its life. And because the estimate is a minimum, the energy quoted is a floor. A source that is particle-dominated, as the inverse-Compton measurements say most are, contains more energy than the minimum — by a factor that is the third power of the field ratio in one term and the inverse three-halves in the other, so being a factor of three below the minimum field costs a factor of about five in total energy.

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 0.5, 2, 8, 32 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 ageing calculation over a wider range of field — half a microgauss to thirty-two, a factor of sixty-four. Every curve still has a log-log slope of exactly −2-2, and the field that maximises an electron’s life is still 1.9 microgauss, because that optimum is set by the competition between synchrotron losses and inverse-Compton scattering off the microwave background rather than by anything about the source. A source in a weak field is not aged more slowly beyond that point; it is aged by a different photon bath, and the age read off its break is wrong in a direction that depends on redshift.

The same problem in three other places

The degeneracy is not peculiar to radio galaxies, and seeing it elsewhere clarifies what is and is not being assumed.

Supernova remnants have the same structure and a better handle on it, because their ages are known. A remnant expanding into a medium of measurable density has a shock speed, a compression ratio and therefore a post-shock field that can be predicted from the interstellar field alone — and the observed synchrotron brightness then requires far more field than that prediction gives, which is the main evidence for magnetic field amplification at shocks — the amplification the cosmic-ray spectrum’s knee also requires.

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. 7 The particle population the amplification exists to accelerate. Reaching the knee at three petaelectronvolts requires a field in the remnant far above the interstellar value, and the requirement is one of the arguments that the amplification the brightness implies is real rather than a failure of the estimate.

The galactic disc has it too, and there the answer is checked. A minimum-energy field for the Milky Way’s own synchrotron emission gives about six microgauss, and Faraday rotation and Zeeman measurements of the same volume give two to six. Agreement within a factor of two, between an estimate and two measurements, in the one galaxy where all three can be done.

Pulsar wind nebulae are the case where the estimate is worst. Their fields are known independently from the sharpness of their X-ray structure and from their spectral breaks, and the minimum-energy value can be off by an order of magnitude — because the pair content is large, the filling factor is small, and the injection is continuous. A source with a known age and a known injection is the one place the assumption behind the estimate can be checked and found badly wrong.

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 1.5e-4 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. 8 The same minimum with a hundred protons for every electron rather than one. The total energy at the minimum scales as the proton fraction to the four-sevenths power, so an assumption nobody can test moves the answer by a factor of seven — and the field strength by rather less, which is why field strengths from this method are quoted with more confidence than energies. The minimum is real; what is uncertain is which quantity it is a minimum of.

Where the assumptions are hiding

Four inputs go into the estimate and none of them is measured.

The volume. A source’s projected extent is measured; its depth is assumed, usually by taking it to be comparable to the width. For a jet seen at a small angle to the line of sight that assumption fails badly.

The filling factor. The emitting plasma may fill the apparent volume or may occupy filaments within it, and the polarisation fraction gives only a weak handle. A filling factor of a tenth raises the minimum-energy field by about a factor of two. The proton content. Synchrotron radiation is produced by electrons; if the source also contains relativistic protons, as the galactic cosmic-ray population certainly does at a hundred times the electron energy, then the particle energy is larger by that factor and the minimum shifts. The ratio is written as a parameter and set to one or to a hundred according to taste, and the resulting fields differ by a factor of three.

The low-energy cut-off. The integral over the electron population diverges weakly at the bottom end, so the total particle energy depends on where the population stops — an energy far below anything the observed spectrum reaches.

Three slopes: 2.5, −0.60 and −1.10, 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 = 2.2 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 −0.60, 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 300 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 40 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. 9 A flatter spectrum with the self-absorption turnover an octave higher. Below the turnover the source absorbs its own radiation and the spectrum rises as ν5/2\nu^{5/2} regardless of the particle distribution — so the turnover frequency is a measurement of the source size and the field together, and it is the one place a synchrotron spectrum constrains the energy content without the minimum-energy assumption. Where a source is optically thick, the argument this essay is about is not needed.

The jets, where the same numbers decide a mechanism

The lobes are the easy case because they are large and old. The jets that feed them are the interesting case, and there the energy budget decides between two pictures of what a jet is made of.

A jet carrying its energy mainly as magnetic field is a Poynting flux: an electromagnetic structure with relatively little matter in it, launched by a rotating magnetosphere and collimated by its own hoop stress. A jet carrying its energy mainly as particles is a hydrodynamic flow that happens to be magnetised. The observational discriminator is the same one as before: an inverse-Compton measurement of the particle content, set against the synchrotron measurement of the product. Where it has been done on resolved jets the answer is that the inner regions are closer to magnetically dominated and the outer regions are not, which is consistent with a jet that starts as a Poynting flux and converts.

The conversion is where the acceleration happens, and it is the least understood part. But the framing — a ratio of two energy densities, measured by two processes with the same electrons — is exactly the framing of this essay, applied where the answer changes a mechanism rather than a number.

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 1 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 155 million years. The oldest radio galaxies are limited by a photon field rather than by their own.
Fig. 10 The same four fields read at a break of one gigahertz rather than five. Every age moves up by the same factor — the break frequency goes as the inverse square of the age, so a fifth of the frequency is a bit over twice the age — and the ordering between the fields does not change at all. The break frequency is a clock whose calibration is the field strength, and the field strength is the quantity the minimum-energy argument produced by assuming the answer. The circularity this essay is about closes here, in a number quoted as an age.

What the exercise is worth

It would be easy to conclude that a quantity depending on four unmeasured inputs is not worth computing, and that would be wrong for two reasons.

The first is the exponent again. Because everything enters to the two sevenths, the accumulated uncertainty from all four inputs together is a factor of a few rather than orders of magnitude — which is unusual for a quantity nobody can measure, and is why the estimate has survived seventy years. The second is that the alternative is not a better number but no number. Radio sources have to be assigned an energy content in order to be compared with the thermal energy of the gas they are displacing, with the binding energy of the galaxy, and with the output of the nucleus that fed them. Every argument about whether radio galaxies regulate the cooling of cluster gas rests on such a comparison, and the argument is not sensitive to a factor of three. The comparison with the cluster gas is made against a hot atmosphere whose own energy content is measured directly, which is the asymmetry that makes the estimate tolerable: one side of the balance is known well.

There is one more use, and it is the one that matters most outside radio astronomy. The same minimisation applied to the interstellar medium of a normal galaxy gives its field from its radio luminosity and its size, which is how fields are quoted for the hundreds of galaxies too faint or too distant for anything better. Those numbers underpin the statement that spiral galaxies generally carry fields of a few to a few tens of microgauss, and hence the claim that a dynamo has saturated in all of them. A statistical result built on an estimate is still a result, provided the estimate’s bias is the same everywhere — and here it is.

What should be resisted is the drift from estimate to fact. “The equipartition field” reads like something that was measured, and it is a choice with a rationale — the choice that makes one quantity least, on a curve whose position depends on four things nobody knows. The correct reading is that a source’s field is somewhere within a factor of a few of a value that a minimisation supplies, and that where the check has been made independently the true value sits below it.

That is a considerably weaker statement than the literature’s phrasing suggests, and it is the honest one.

There is a broader habit worth taking from it, and it applies well outside radio astronomy. A quantity obtained by minimising something is a quantity whose value depends on what was held fixed during the minimisation, and the thing held fixed here is the observed luminosity — which is an observation — together with the volume, the filling factor, the proton content and the low-energy cut-off, which are not.

Minimisation is a powerful device because it converts an underdetermined problem into a determined one at no apparent cost, and the cost is always the same: the answer is now a property of the family of models being minimised over rather than of the object. That is a legitimate move when the family is well motivated and a trap when it is a family of convenience.

The test that distinguishes the two cases is whether an independent measurement, when it eventually arrives, lands near the minimum. Here it does — the inverse-Compton fields sit a factor of two to five below, which is close enough that the estimate was useful and far enough that it was not a measurement. That is about the best outcome a minimisation argument can have, and it is worth recording as the reason the practice survived rather than as a vindication of it. A device that gets within a factor of three of an unmeasurable quantity, and admits that it might not, is doing more work than most of what replaces it. What it must not do is lose the qualifier on the way into a table, and that is exactly what happened to the word equipartition. A qualifier that survives the paper and dies in the review article is the usual route by which an estimate becomes a fact. It is worth checking, whenever a field strength is quoted for a radio source, which of the two it is.

Where this goes next

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. The energy at which the sky begins to point. Gyroradius against energy for a singly charged particle, in three field strengths, with the thickness of the galactic disc and the size of its halo marked. A cosmic ray is not an image of anything: its path is a helix about a field line, and by the time it arrives the direction it came from has been erased. The erasure is quantitative. At 3 µG a proton's gyroradius equals the disc's half-thickness at 4.2·10¹⁷ electronvolts and the halo's radius at 4.2·10¹⁹, so below the first the particle is stored and stirred for tens of millions of years, and only above the second does it travel in something like a straight line. The measured spectrum has a break — the knee — at about 3×10¹⁵ eV, which is within a factor of a few of the first of those; the sky only starts showing structure above 10¹⁹, which is the second. Two features of a spectrum measured on the ground, both located by one straight line on this plot. The energy at which a sky begins to point A cosmic ray arrives from a direction that has nothing to do with where it came from. Its path is a helix about a galactic field line, and only above the energy at which that helix is bigger than the galaxy does the arrival direction start to mean anything — which is one particle per square kilometre per century. A 20-day delay, recovered at 20 days from two curves that share no wavelength. Above: a quasar's continuum, drawn as a damped random walk with a 90-day damping time, and the broad emission line responding to it. The line curve is the continuum convolved with a top-hat response of half-width 20 days, so it is later and smoother — it varies only 51 per cent as much, because at any instant it is an average of the continuum over a range of light-travel times. Below: the cross-correlation of the two, which peaks at 20 days. That number is a length: 20 light-days is 5.2·10¹⁴ metres, or 3463 astronomical units, and it has been measured for an object that subtends 3.5·10⁻⁵ arcseconds at a hundred megaparsecs — some thirty times finer than the best optical interferometry has ever resolved anything, and reached here with a photometer and a clock. The irregularity of the continuum is what makes this work. A periodic source would give a cross-correlation with many equal peaks and no way to choose; a random one gives a single peak, and the whole method rests on active nuclei being erratic. A size measured from a delay An active nucleus at redshift two is a point source in every telescope ever built, and its central mass is measured anyway. The continuum varies, the broad lines follow days later, and the lag is a light-travel time — which is a length, recovered from two light curves that share no wavelength.

About the same objects

Not linked from either essay — found by the objects both name.

The objects this essay names

Each one links to every other essay that touches it.

Cosmic raysDegeneracyEnergy budgetEquipartitionInverse ComptonJetsMagnetic fieldMinimum-energyRadio galaxyRelativistic electronSynchrotron radiation