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A rotation measure the lobe writes on its own boundary

The magnetic fields of hot group and cluster gas are mostly measured through the Faraday rotation of the radio galaxies that live in it, on the assumption that a lobe is a passive lamp behind a screen. But a lobe made its own room. The gas that filled it is now a shell around it, compressed if the lobe grew faster than sound, and the shell rotates the light too — so the inferred field comes out too low for a lobe that grew gently and up to three times too high for one that drove a shock. Only sources behind the halo, which carry no shell, can say which.

Assumes Faraday rotation and Faraday rotation.

20 min read 7 figures Measuring the unreachableLight is the only messenger

Most of what is known about the magnetic fields in the hot gas of galaxy groups and clusters comes from radio galaxies. A lobe is a bright, strongly polarised source lying inside the gas, and its polarised light, on the way out, is rotated by an amount proportional to the wavelength squared. Map the rotation across a lobe, measure how much it scatters from one beam to the next, fit a model of the gas density taken from X-ray images, and the field strength falls out — a few microgauss at the centre of a rich cluster, about one in a group. The same picture explains why the lobe further from the observer loses its polarisation first: its light crosses more of the halo.

The method treats the lobe as a lamp. The gas in front of it is the screen, the screen belongs to the halo, and the lamp is assumed to have done nothing to it. That assumption is not obviously safe. A lobe tens of kiloparsecs across is a volume of relativistic plasma and field that used to be filled with the halo’s gas, and that gas has gone somewhere. Wherever it went, it is still between the lobe and the observer, and the field it carries went with it.

The gas the lobe pushed aside

A jet inflating a lobe does work on the gas around it. If the lobe grows slowly compared with the speed of sound in the halo, the gas is pushed aside gently and spreads out over a region larger than the lobe; if it grows faster than sound, a shock runs ahead of the boundary and the gas piles up into a thin, dense shell. A shock in a gas of monatomic particles compresses it by the Rankine–Hugoniot factor

C=4M2M2+3,C = \frac{4M^2}{M^2 + 3},

where MM is the Mach number of the expansion. The factor is 1 for a lobe growing at exactly the sound speed, about 2.3 at Mach 2, and approaches 4 however fast the lobe grows — a shock cannot compress a monatomic gas more than four times. The mass that filled the lobe is conserved, so a shell of compression CC around a lobe of radius RR is R/3CR/3C thick.

A lobe, the shell it pushed out, and two sight lines through it. A radio lobe of radius 20 kpc drawn in the plane containing the line of sight, the observer to the right, 60 kpc from the host in the same halo as before. Growing at Mach 2, it swept the gas that filled its volume into a shell compressed 2.29 times and 2.9 kpc thick, and the field in that shell was amplified along the boundary. A sight line through the lobe's centre crosses the shell head-on, where the amplified field lies across the line of sight and contributes nothing: the shell there turns the light by about 6 rad m⁻². A sight line at 85 per cent of the radius crosses it obliquely, along 4.8 kpc of shell with the amplified field partly along the path, and the shell turns it by about 18 rad m⁻². The rotation the halo model attributes to the gas in front of the lobe is partly written by the lobe itself, and it is written most strongly around its edge.
Fig. 1 A lobe of radius 20 kiloparsecs, 60 kiloparsecs from the host galaxy, that grew at Mach 2: its gas sits in a shell 2.9 kiloparsecs thick and 2.3 times denser than the halo around it. A sight line through the lobe’s centre crosses the shell head-on and is rotated by about 6 rad m−2\mathrm{rad\,m^{-2}}; one at 85 per cent of the radius runs obliquely through it and is rotated by about 18.

The field is carried with the gas. A plasma this hot conducts so well that field lines are frozen into it, and a compression across a field line squeezes the lines together. So the two components of the field lying along the boundary are amplified by the same factor CC as the density, while the component perpendicular to the boundary, along which the gas was squeezed, is not. A shell compressed 2.3 times carries 2.3 times the density and, in its tangential components, 2.3 times the field. The rotation measure is 0.812∫neB∥ dl0.812\int n_e B_\parallel\,dl, the product of the two along the path — and both have gone up.

The model drawn here keeps everything else from the halo used for the asymmetry between the two lobes: a β-model density with a central value of 3×10−33\times10^{-3} electrons per cubic centimetre, a core radius of 40 kiloparsecs and β=0.6\beta = 0.6, a field of 3 microgauss at the centre falling outward as the square root of the density, tangled in cells of 5 kiloparsecs. The lobe is 20 kiloparsecs in radius and sits 60 kiloparsecs from the host. Nothing about the halo is changed by the lobe except the gas the lobe moved.

A shell that rotates the edge and spares the middle

Where the sight line crosses the shell decides how much the shell contributes. Through the centre of the lobe, the line crosses the boundary along its normal. The compressed gas there is only one shell-thickness deep, and the amplified field lies in the plane of the shell, which is across the line of sight: its component along the path is the unamplified normal one. Towards the limb the line meets the boundary obliquely, so it runs through more of the shell, and the amplified tangential field now has a component along it.

Across the lobe's face: what the shell adds and what the hole removes. Two rotation-measure dispersions across the face of a lobe of radius 20 kpc, 60 kpc from the host, against distance from the lobe's centre as a fraction of its radius. The first is what the compressed shell contributes, for a lobe that grew at Mach 2 and compressed its gas 2.29 times. The second is what the gas the lobe displaced would have contributed had it stayed where it was — the column a halo-only model still counts. At the centre the shell gives 6 rad m⁻² against 9 rad m⁻² for the displaced gas, because the sight line crosses the shell along its normal and the amplified field lies across it. The shell overtakes at 49 per cent of the radius and at the limb gives 30 rad m⁻² against 6 rad m⁻². Averaged over the face the shell gives 15 rad m⁻² and the displaced gas 8 rad m⁻², so the sheath adds a net dispersion of 13 rad m⁻² in quadrature to whatever the halo contributes — concentrated in a ring around the lobe's edge.
Fig. 2 Across the face of the same lobe, the dispersion in rotation measure contributed by the compressed shell, and that which the gas would have contributed had it stayed where the lobe now is. At the centre the shell gives 6 rad m−2\mathrm{rad\,m^{-2}} against 9 for the displaced gas; the shell overtakes at half the radius and at the limb gives 30 against 6. Averaged over the face, the shell adds a net 13 rad m−2\mathrm{rad\,m^{-2}} in quadrature.

The second curve is the one that makes the problem subtle. A model of the halo integrates the gas density from the lobe’s position towards the observer, and that integral runs through the lobe’s own volume and its shell as if they were still full of ordinary halo gas. It is not: the lobe’s interior is relativistic plasma with almost no thermal electrons, and so contributes almost nothing to the rotation. So the halo model already counts a column of gas that is no longer there, and what the shell adds has to be set against that missing column rather than added on top of the model.

In the middle of the face the missing column wins. The line through the centre crosses 20 kiloparsecs of evacuated lobe and a thin shell whose amplified field lies across it, and the shell does not make up for the hole: 6 rad m−2\mathrm{rad\,m^{-2}} against 9. Halfway out they balance. At the limb the line skims the lobe, crosses very little of it, and runs nearly tangentially through the dense shell with the amplified field partly along it, and the shell contributes five times what the missing gas would have.

The net effect, averaged over the emitting face of the lobe and weighted by area — more of a disc lies near its edge than its centre — is an extra 13 rad m−2\mathrm{rad\,m^{-2}} added in quadrature to whatever the halo contributes, concentrated in a ring around the lobe’s edge. In the same halo at this distance the halo’s own dispersion in front of a lobe in the plane of the sky is also about 13. The sheath, for a lobe growing at twice the sound speed, is as large as the thing being measured.

A bias whose sign depends on the speed of the lobe

The fit of a halo-only model to a lobe’s rotation measures attributes all of the dispersion to the halo, and the model’s dispersion scales in proportion to the central field. So the ratio of the dispersion actually measured to that the halo alone produces is the factor by which the inferred field is wrong.

How far a halo-only model overstates the field, by how fast the lobe grew. The central field a halo-only model infers from a lobe's rotation-measure dispersion, divided by the true one, against the Mach number at which the lobe grew, for a lobe of radius 20 kpc at 60 kpc from the host. Three lobes are drawn: the near lobe of a source at 45° to the line of sight, a lobe in the plane of the sky, and the far lobe at 45°. At Mach 1 the lobe has only displaced its gas, and the hole makes the model underestimate the field: by factors of 0.64, 0.84, 0.97 respectively. The displacement and the compression cancel at about Mach 1.3. At Mach 2 the factors are 1.75, 1.43, 1.11, and at Mach 8 3.30, 2.43, 1.44. The bias saturates because a shock cannot compress gas more than four times, and it is largest for the near lobe because the halo in front of it is thinnest, so the same sheath is a larger share of what is measured.
Fig. 3 The central field inferred by a halo-only model, divided by the true one, against the Mach number at which the lobe grew, for the near and far lobes of a source at 45° to the line of sight and for a lobe in the plane of the sky. At Mach 1 the factors are 0.64, 0.97 and 0.84; they pass through one at about Mach 1.3, and at Mach 2 are 1.75, 1.11 and 1.43.

At the sound speed the shell has no compression, and the lobe has only hollowed out the halo. The rotation measured is less than the halo model predicts, and the inferred field comes out too low: by a factor of 0.84 for a lobe in the plane of the sky and 0.64 for the near lobe of a tilted source. As the Mach number rises, the compression grows and the shell’s contribution overtakes the missing column at about Mach 1.3. Above that the field is overestimated, by 1.75 for the near lobe at Mach 2 and by up to 3.3 as the compression saturates at four.

The near lobe is the most affected in both directions, and the far lobe the least. The sheath’s contribution is about the same for both lobes, because they sit at the same distance from the host in the same gas; what differs is what it is compared with. The near lobe sees only the outskirts of the halo, and its halo dispersion is small, so the same sheath is a large fraction of what is measured. The far lobe looks through the halo’s dense middle, and the sheath is a small addition to a large number. The lobe whose light is best for the measurement — the near one, polarised enough at long wavelengths to map — is the one whose map is most contaminated by itself.

What the shocks around real lobes say

The Mach number is therefore the number that decides the sign. X-ray observations have measured it around a handful of sources, by finding the jump in density and temperature at the edge of a shell of gas around a lobe. Around the radio sources in the cores of several nearby clusters, the shocks found are weak, at Mach numbers of about 1.2 to 1.7 — about where the figure’s curves cross one. Around the south-western inner lobe of Centaurus A, the nearest radio galaxy, a shell of shocked gas has been found expanding at about eight times the local sound speed, where the bias saturates.

The speed of a lobe’s head can also be estimated from its age. A lobe’s length divided by the time since its electrons were last accelerated gives advance speeds of around a hundredth of the speed of light for large sources — some 3,000 kilometres a second, against a sound speed of several hundred in group gas, so the head drives a shock of Mach several. But the two estimates of age disagree by factors of several, and the advance of the head is not the expansion of the sides, which push sideways into the gas far more slowly and make most of the boundary a sight line crosses. Both widen the range of Mach numbers rather than narrowing it.

Older lobes, and those that have stopped being fed, have generally slowed to below the sound speed and have pushed the gas aside without compressing it; these are the cavities seen in X-ray images of cluster cores, holes in the gas with no bright rim. For them the figure says the halo’s field is underestimated, and by up to a third for a near lobe.

So the bias has no single direction. It depends on each lobe’s history, and the history of the lobe being used to measure the field is exactly what is not known from its rotation measures.

Two lobes that look more alike than their geometry

The sheath affects the depolarisation asymmetry too. That asymmetry is read as orientation: the far lobe has the larger dispersion because its light crosses more of the halo, and the ratio of the two lobes’ dispersions grows with how far the axis is tilted from the plane of the sky.

The sheath makes two lobes look more alike. The ratio of the far lobe's rotation-measure dispersion to the near lobe's, including each lobe's own sheath, against the Mach number at which the lobes grew, for sources at 30°, 45°, 60° to the line of sight. Without any sheath the ratios are 4.65, 3.04, 2.04; at Mach 1, where the lobes have only hollowed out the halo, they are 7.84, 4.58, 2.70; at Mach 2, 2.68, 1.92, 1.47; and at Mach 8, 1.64, 1.33, 1.16. Both lobes carry a sheath of the same strength, because they sit at the same distance from the host in the same halo, and a common term added to two unequal ones shrinks their ratio. The asymmetry survives — the far lobe is still the more depolarised at every Mach number drawn — but its size no longer measures the halo alone.
Fig. 4 The ratio of the far lobe’s dispersion to the near lobe’s, with each lobe’s own sheath included, against the Mach number at which the lobes grew, for sources at 30°, 45° and 60° to the line of sight. Without a sheath the ratios are 4.65, 3.04 and 2.04; at Mach 2 they fall to 2.68, 1.92 and 1.47, and the far lobe stays the more depolarised throughout.

A common term added in quadrature to two unequal ones shrinks their ratio. At Mach 2 the ratio for a source at 45° falls from 3.04 to 1.92, which a halo-only model would read as an axis at a little more than 60° to the line of sight rather than 45°. A lobe that only hollowed the halo goes the other way: removing the same column from both lobes leaves the far lobe’s dispersion little changed and cuts the near lobe’s, and at Mach 1 the ratio for 45° rises to 4.6.

The asymmetry’s direction survives at every Mach number in the figure — the far lobe is always the more depolarised. That is what was used to establish that the visible jet points at the nearer lobe, and it does not depend on the sheath. What does is any attempt to read an angle from the size of the asymmetry, and any statistical argument that sums asymmetries over many sources at once. Sources with fast, shocked lobes will look as though they lie closer to the plane of the sky than they do.

Sources behind the halo carry no sheath

The way out is to measure the halo with light that did not come from a lobe inside it. Behind every group and cluster lie unrelated radio sources at much larger distances, many of them polarised. Their light crosses the whole halo, from back to front, and carries no sheath that belongs to the halo: whatever surrounds them lies at their own distance, in their own environment, and adds scatter but not the halo’s signature.

Background sources against the cluster's own lobes. Rotation-measure dispersion against projected distance from the host, for three kinds of source seen through the same halo. A background source far behind the group sees the whole path through the halo. A lobe in the plane of the sky sees only the half in front of it, so with no sheath its dispersion is exactly 1/√2 of the background value, 13 rad m⁻² against 18 rad m⁻² at 60 kpc. The same lobe with a sheath from growing at Mach 2, radius 20 kpc, measures 18 rad m⁻² at 60 kpc and 5 rad m⁻² at 120 kpc, where the background value is 5 rad m⁻². Across the whole range drawn the lobe with its sheath measures between 0.86 and 1.12 of the background value, where the halo alone would give 0.71: the sheath has put back almost exactly the half of the halo the lobe does not look through. Because the sheath's gas and the halo's scale together, the ratio barely changes with distance, and so a single profile of background sources is the reference for every lobe in the group.
Fig. 5 The rotation-measure dispersion against projected distance from the host, for a background source behind the whole halo, for a lobe in the plane of the sky with no sheath — exactly 1/21/\sqrt{2} of the background value — and for the same lobe with the sheath it would have from growing at Mach 2, which lies at 0.86 to 1.12 of the background value across the range drawn.

A lobe in the plane of the sky looks through half the halo, from its own position outward, and a background source at the same projected distance looks through all of it. Without a sheath the two variances differ by exactly a factor of two, and the dispersions by 2\sqrt{2}: 13 against 18 rad m−2\mathrm{rad\,m^{-2}} at 60 kiloparsecs. With a Mach 2 sheath the lobe measures 18 there as well. The shell has added, almost exactly, the half of the halo the lobe does not look through.

That coincidence is particular to Mach 2 in this halo, but the shape is general. Both the shell’s gas and the halo’s scale with the local density, so the ratio of the lobe’s dispersion to the background’s barely changes with distance from the host — it stays between 0.86 and 1.12 over a factor of five in distance. A single profile of background sources, measured once, is therefore the reference for every lobe in the group. Comparing the two is a test a lobe’s own map cannot make of itself.

This is not a new idea, and it has been applied in pieces. In 2001 Clarke, Kronberg and Böhringer compared the rotation measures of sources seen through a sample of nearby clusters — background and embedded together — with those of sources seen beside them, found the ones seen through the clusters systematically larger within about half a megaparsec of the centres, and deduced fields of several microgauss. In the Coma cluster, Bonafede and colleagues fitted the rotation measures of seven sources, some inside the cluster and some behind it, with one field model. What neither could do with the numbers of sources available was compare the two kinds against each other closely enough to measure a sheath.

Why one source cannot settle it

The comparison is not as simple as checking whether a lobe exceeds the background. The lobe’s dispersion depends on its depth in the halo, and that depends on the angle of its axis, which is rarely known.

The test is a distribution, because a lobe's tilt is unknown. A lobe's rotation-measure dispersion divided by that of background sources at the same projected distance, against the angle between the source's axis and the line of sight, for lobes 60 kpc from the host. The lower band spans the near and far lobes with no sheath: the ratio runs from 0.15 for a near lobe seen nearly end-on to 0.99 for a far lobe, and never exceeds one, because a source inside the halo cannot see more of it than a source behind it. The upper band is the same lobes with a sheath from growing at Mach 2: 0.27 to 1.07. At 45° the near lobe moves from 0.31 to 0.55 and the far lobe from 0.95 to 1.05. The bands overlap, so one source cannot be assigned to either, and its tilt is rarely known well enough to say where in its band it sits. What separates the two cases is a population: lobes with ratios above one exist only if sheaths do, and the share of lobes above the sheath-free band's upper edge counts them.
Fig. 6 A lobe’s dispersion divided by background sources’ at the same projected distance, against the angle of the source’s axis to the line of sight. With no sheath the near and far lobes span 0.15 to 0.99 and never exceed one. With a Mach 2 sheath they span 0.27 to 1.07; at 45° the near lobe moves from 0.31 to 0.55 and the far lobe from 0.95 to 1.05.

Without a sheath, the ratio runs from 0.15 for a near lobe seen almost end-on to 0.99 for a far lobe, which looks through nearly the whole halo. It cannot exceed one: a source inside the halo cannot see more of it than a source behind it. With the sheath, both edges of the band move up, and the band overlaps the first almost everywhere. A near lobe measuring 0.5 of the background value could be a sheathless lobe at 60° or a sheathed one at 45°, and with no independent knowledge of the angle there is no way to say which.

Only the region above one is free of the ambiguity, and at Mach 2 the far lobe enters it by a few per cent at most. Strong shocks push further — at Mach 8 the far lobe at 45° reaches about 1.4, and the band’s top edge 1.7 — and gently grown lobes do not enter it at all. So the test is statistical. Over many lobes at random orientations, a halo-only model predicts a distribution of ratios bounded by one, with a known shape set by geometry. Sheaths shift the distribution upward and fill in a tail above the boundary. The fraction of lobes above one, and how far above, counts the sheaths; the fraction below the sheathless band’s lower edge counts the cavities.

The orientation can be constrained independently for some sources. Where the jet is visible on one side only, the jet side identifies the approaching lobe through relativistic beaming, and the ratio of jet to counter-jet brightness gives a rough angle if the jet speed is assumed. That converts a band into a narrower one for those sources, at the cost of an assumption about the jet.

How many sources the test needs

The halo’s dispersion from background sources is a variance estimated from a sample, and the sample is small. A group’s halo covers a few square arcminutes on the sky at a distance of 100 megaparsecs, and the density of background sources bright enough to measure a rotation measure is what limits the test.

How many background sources the test needs. The fractional uncertainty on the halo's rotation-measure dispersion, measured from background sources between 30 and 90 kpc of the host in projection, against the density of polarised sources a survey detects, for a group like the one drawn throughout at 20, 50, 100 Mpc. Each source is taken to carry 10 rad m⁻² of scatter from the Milky Way's own foreground and its own interior, which has to be subtracted, against a halo dispersion of 18 rad m⁻² at 60 kpc. A Mach 2 sheath raises a sky-plane lobe's dispersion 43 per cent above the halo's, and the horizontal line is half that — the precision needed to see the excess at two standard deviations. A curve starts where the annulus holds three sources. The densities needed are 110 per square degree at 20 Mpc, 661 per square degree at 50 Mpc, 2754 per square degree at 100 Mpc. A group is small on the sky, so the test needs either a group close enough to be large or a survey dense enough to find tens of sources behind a few square arcminutes.
Fig. 7 The fractional uncertainty on the halo’s dispersion measured from background sources between 30 and 90 kiloparsecs of the host, against the density of polarised sources a survey detects, for the same group at 20, 50 and 100 megaparsecs. To reach half the 43 per cent excess a Mach 2 sheath adds needs about 110 sources per square degree at 20 megaparsecs, 660 at 50 and 2,800 at 100.

Each background source carries its own scatter, independent of the halo: rotation from the Milky Way’s gas along that line of sight, which varies on scales of a degree and has to be modelled and subtracted, and rotation inside the background source and its own environment. The figure takes 10 rad m−2\mathrm{rad\,m^{-2}} for the two together. The halo’s dispersion is then the measured variance minus that scatter, and the uncertainty on a variance from NN samples falls only as 2/N\sqrt{2/N}.

To see a Mach 2 sheath at two standard deviations, the halo’s dispersion must be known to about 21 per cent. The only all-sky catalogue of rotation measures has about one source per square degree, and gives no sources at all behind most groups. Surveys now under way aim for a few tens per square degree, which falls short even for a group 20 megaparsecs away, covering half a degree of sky: it needs about a hundred. Densities of several hundred to a thousand per square degree, planned for the deepest surveys of the next decade, would reach groups at 50 megaparsecs.

A rich cluster is easier: its halo is ten times larger, so it covers a hundred times more sky and holds a hundred times more background sources, and the test has been attempted there first. Groups, where the fields are weakest and the lobes most likely to have stirred and shocked the gas they sit in, are where it is hardest and most needed.

What has been seen

The idea that some of the rotation belongs to the source is older than the means to test it. Rudnick and Blundell argued in 2003 that the rotation measures of cluster radio sources might be produced largely in their immediate surroundings, which would make them useless as probes of the cluster as a whole. Enßlin and colleagues replied that the rotation-measure maps showed the statistics expected of a field in the cluster gas rather than at the lobe boundaries, and that background sources showed comparable excess rotation. The exchange did not settle the question so much as show what would.

Since then the best maps have resolved structure in the rotation across individual lobes. In a few, the rotation measure varies in bands aligned with the lobe’s edges rather than randomly across it — the signature of an ordered field draped around the boundary, which is what a shell of compressed field would produce. Guidetti and colleagues described such bands in several nearby radio galaxies and attributed them to gas compressed by the lobes. Other lobes in the same studies show no bands at all, and the random structure the halo model expects.

Both are consistent with the figures. A ring of extra rotation concentrated near the limb is what a shocked shell produces, and a lobe that has slowed below the sound speed shows none. What the maps cannot do on their own is separate a modest sheath from a slightly stronger halo field, because both raise the dispersion, and the sheath’s ring blends into the halo’s random structure unless it is resolved. That separation is the work for the background sources.

What the model leaves out

The shell drawn here is uniform, the same thickness all round the lobe. A real lobe is elongated and advances fastest at its head, so the shell is thickest and most compressed at the tip and thinnest along the sides, and the sight lines that matter cross it at a range of compressions. The halo’s own dispersion varies across a lobe’s face, which the figures take as uniform at its value for the lobe’s centre. And the field in the shell is taken as tangled on the same scale as the halo’s, amplified but not reorganised. Compression stretches the field’s cells along the boundary, and an ordered field draped around the lobe, as the banded maps suggest, would produce a larger and more structured rotation than the random cells assumed.

The cavities have their own complication: the gas they push aside need not form a shell at all at low Mach numbers, but can spread out as a broad region of slightly raised density. The model puts it in a shell anyway, at the halo’s density, which is the most conservative assumption about its effect on the rotation. Each would move the Mach number at which the bias changes sign, and the size of the bias at a given Mach number, by amounts this model cannot fix. None changes the conclusion that the sign of the bias depends on how fast the lobe grew.

The field inferred for the halo also enters other questions. The strength of the fields in groups and clusters is used to argue about where those fields came from and how far they have been amplified, and the energy carried by the hot gas is compared with the field’s pressure to see whether the gas is in equilibrium. A systematic factor of two in the field, of either sign, is a factor of four in the field’s pressure.

Still open: whether the cavities and the shocks average out

A population of lobes in every stage of growth contains both kinds — young, shocked ones that bias the field high and old, quiet ones that bias it low. Averaged over every radio source used to measure fields in groups and clusters, the two biases could cancel, or one could dominate, and which depends on how long lobes spend expanding supersonically compared with how long they remain visible afterwards. The ages read from lobe spectra disagree with their dynamical ages by factors of several, so that ratio is not known to better than a factor of a few.

The background-source test answers it without needing the ages. A survey dense enough to find tens of sources behind each of a sample of nearby groups and clusters measures each halo’s dispersion directly, and comparing it with the dispersions of all the lobes in the same haloes gives the distribution of the ratio — whether its tail above one is filled, and whether the band below is depleted. If the two cancel, fields measured from lobes are unbiased on average and scattered by a factor of about two around the truth; if they do not, every field inferred for group gas from its own radio sources carries a shift that no individual measurement can reveal. Which of these holds is not yet known.

Where this goes next

Two lobes, one halo, and two different paths out. A double radio source drawn in the plane containing the line of sight, the observer far to the right. The host galaxy sits at the centre of a hot halo — the rings mark 20, 40, 60, 100 kpc, in a β-model of core radius 40 kpc — and its two lobes lie 60 kpc out along an axis 45° from the line of sight. Light from the near lobe leaves the halo through its outskirts; light from the far lobe has to cross the halo's middle. With a tangled field of 3 µG at the centre in cells of 5 kpc, the rotation measure in front of the near lobe varies across the beam with a dispersion of 9 rad m⁻² and in front of the far lobe 28 rad m⁻². On the sky the two lobes are the same distance from the core, 42 kpc, and nothing in a total-intensity image says which is which. The fainter polarisation belongs to the lobe further away The two lobes of a radio galaxy lie at the same distance from the core on the sky, and nothing in a picture of them says which is nearer. Their polarisation does. Light from the far lobe crosses the magnetised hot halo of the host on its way out and light from the near lobe mostly does not, so at long wavelengths the far lobe loses its polarisation first — and the one-sided jet, it turned out, points at the lobe that keeps it. A source that erases its own polarisation. The polarised fraction of a uniform slab that both emits and Faraday-rotates, relative to its intrinsic value, against wavelength, for total Faraday depths of 10, 40, 150 rad m⁻². Light from the back of the slab is rotated by the whole depth and light from the front by none, so the polarisation vectors from different depths fan out and cancel. The fraction falls as |sin(Φλ²)/(Φλ²)| and vanishes altogether at λ² = π/Φ — 56.0 cm for 10, 28.0 cm for 40, 14.5 cm for 150 — then returns weakly and vanishes again. A source measured only at wavelengths beyond its first null looks unpolarised, and a source measured across the null shows a polarised fraction that depends on the wavelength in a way no foreground screen can produce. The shape of the fall is a measurement of how the rotating plasma is arranged along the line of sight. A source that rotates its own light The rotation-measure method assumes the plasma that turns the polarisation sits in front of the source. When it is mixed into the source instead, light from the far side is turned more than light from the near side, the contributions cancel, and the straight line in wavelength squared bends, halves its slope and then vanishes. The cure is to stop measuring in wavelength at all. A spectral age with a ceiling, against the time the lobe took to grow. The spectral age of electrons whose spectrum breaks at 2 GHz, against the magnetic field in microgauss (curve), beside the dynamical age of a 400-kpc lobe whose head has advanced at 0.01c, 0.03c, 0.1c (horizontal lines: 130, 43, 13 million years). The spectral age has a maximum, 109 million years, at a field of 1.9 microgauss — the microwave background's equivalent field divided by √3 — because a weaker field leaves the electrons cooling on the background's photons instead. A lobe that advanced at 0.01c is 1.2 times older than any field can make its spectrum, so the disagreement cannot be removed by choosing a different field. Either the advance speed is overestimated, or the electrons at the break are younger than the lobe — re-accelerated, or freshly mixed in from the hotspot. Two clocks for one lobe A radio galaxy's lobe can be dated twice: by the break in its spectrum, where the oldest electrons have radiated away their energy, and by its length divided by the speed its head advances. The two clocks disagree, the spectral age usually several times younger, and no choice of magnetic field can make them agree for the slowest lobes. What the disagreement measures is that the electrons are not as old as the lobe.

The objects this essay names

Each one links to every other essay that touches it.

Background sourcesDepolarisationFaraday rotationFlux freezingIntracluster mediumMagnetic fieldRadio galaxiesRotation measureShock compression