The fainter polarisation belongs to the lobe further away

A powerful radio galaxy is a pair of lobes, each tens to hundreds of kiloparsecs from a central galaxy, fed by jets from the black hole at its centre. Seen on the sky it is flat. The two lobes are symmetric about the core to within a factor of two in distance, and their brightness in total intensity says nothing about which one is on the near side of the host and which on the far. The axis could lie in the plane of the sky or point nearly at the observer, and a map of the radio emission cannot distinguish most of those orientations.

The orientation matters. If the jets are relativistic, the approaching one is brightened and the receding one dimmed, and whether a given source is seen as a radio galaxy or a quasar may depend on nothing more than the angle at which it is viewed. Testing that requires a way to tell which lobe is nearer that does not depend on the jets.

The lobes’ polarisation supplies it. Synchrotron emission from a lobe is strongly linearly polarised, and on its way out of the host system the light passes through the hot, magnetised gas that surrounds the host galaxy. That gas rotates the plane of polarisation by an amount proportional to the wavelength squared, and where the rotation varies across the telescope’s beam, the polarisation from different parts of the beam is rotated by different amounts and partly cancels. The two lobes’ light does not cross the same amount of that gas.

A screen that varies across the beam #

The hot gas around a massive elliptical galaxy or a group of galaxies is well described by a density falling with radius as a smooth power of $1 + r^2/r_c^2$, the β-model used for X-ray haloes, with a core radius $r_c$ of some tens of kiloparsecs. The figures use a central density of $3\times10^{-3}$ electrons per cubic centimetre, a core radius of 40 kiloparsecs and $\beta = 0.6$, typical of a rich group. The gas carries a magnetic field of a few microgauss, tangled on scales of a few kiloparsecs, whose strength falls outward with the density.

The rotation measure along one line of sight is $0.812\int n_e B_\parallel\,dl$ in $\mathrm{rad\,m^{-2}}$, with the density in electrons per cubic centimetre, the field in microgauss and the path in parsecs. With a tangled field, the parallel component changes sign from one cell to the next, and the rotation measure is a random walk: its mean over many lines of sight is near zero, and its dispersion grows as the square root of the number of cells crossed. For cells of size $\Lambda$,

$$\sigma_{\mathrm{RM}}^2 \approx 0.812^2\,\Lambda \int n_e^2\,\frac{B^2}{3}\,dl .$$

A telescope beam a few kiloparsecs across at the source covers many cells, and each one rotates its share of the polarisation by a different angle. Burn’s law for a turbulent external screen gives the result: the polarised fraction falls as $\exp(-2\sigma_{\mathrm{RM}}^2\lambda^4)$. The fourth power of the wavelength makes the fall abrupt.

The dispersion along the source #

The integral runs from each lobe towards the observer. For a lobe on the near side, the path starts in front of the host’s centre and heads away from it; for a lobe on the far side, at the same projected distance, the path starts behind the centre and has to cross the densest part of the halo before leaving.

The two curves agree at the core, where both paths start at the same point, and separate immediately. At a projected distance of 40 kiloparsecs, about one core radius, the far side’s dispersion is three times the near side’s. Further out both fall, because both paths begin further from the halo’s centre, but the ratio keeps growing: the near side’s path clears the halo’s core entirely, while the far side’s still passes behind it at a smaller radius than its starting point. The asymmetry is a direct consequence of geometry, and it needs no assumption about the lobes themselves, which could be identical.

Two wavelengths, one lobe lost #

The fourth power of the wavelength turns a factor of three in dispersion into a factor of nearly twenty in surviving polarisation. At 6 centimetres the rotation across the beam is small whatever the dispersion, a few degrees, and both lobes keep their intrinsic polarisation. At 21 centimetres the rotations are twelve times larger, the far lobe’s spread of angles exceeds a radian, and its polarised fraction collapses to a few per cent while the near lobe keeps most of its own.

Observers summarise this as a depolarisation ratio: the polarised fraction at the long wavelength divided by that at the short. A ratio near one means no depolarisation; a ratio near zero means the long-wavelength polarisation has gone. In the figure the near lobe’s ratio is 0.71 and the far lobe’s 0.04. A survey that measures both lobes of many sources at two frequencies, and asks which lobe in each has the smaller ratio, is asking which lobe lies further away.

The measurement is made in the source’s frame and read in the observer’s. For a source at redshift $z$ the observed rotation measure is smaller than the rest-frame one by $(1+z)^2$, because the rotation happened at a shorter wavelength than the one received; a quasar at redshift one shows a quarter of its halo’s dispersion. The distant sources for which the effect was discovered are depolarised at observed wavelengths of 20 centimetres or so, rest-frame wavelengths of ten.

A screen in front, not a source that rotates itself #

The argument depends on where the rotating gas is. If the magnetised plasma that rotates the polarisation were mixed into the lobes themselves, each lobe would depolarise its own emission, light from its far side rotated more than light from its near side, and the depolarisation would depend on the lobe’s internal density and field rather than on its position relative to the host. Two similar lobes would depolarise similarly whichever side they were on, and there would be no asymmetry to measure.

The two cases leave different signatures. Internal rotation makes the polarisation angle turn with wavelength squared at half the rate of an external screen and the polarised fraction fall as a sinc function, with nulls and weak recoveries; an external screen with a random dispersion leaves the angle turning linearly and the fraction falling as a Gaussian in $\lambda^2$, with no recovery. Observations across several wavelengths of lobes that show the asymmetry favour the second: the lobes are mostly depolarised by gas outside them. The lobes of powerful radio sources are, on this evidence, filled with relativistic plasma and field and little thermal gas, which is consistent with how they are thought to form, as cavities inflated by the jets in the surrounding medium.

That same fact makes the lobes useful as backlights. The rotation measure in front of a lobe is the rotation measure of the halo along that line, and the density of the gas along it can be measured independently from its X-ray emission. The ratio of the rotation measure to the integrated density is a field strength, averaged along the path. The same division, with the electron column supplied by a pulsar’s dispersion measure instead of X-rays, is how the field of the Milky Way is measured.

The discovery #

The asymmetry was found in 1988, in two papers published side by side. Laing measured the depolarisation of the lobes of radio galaxies and quasars with one-sided jets; Garrington, Leahy, Conway and Laing measured it for a sample of quasars. In both, the lobe on the side of the visible jet was the less depolarised in the large majority of sources. Neither team had selected sources by their polarisation, and the jet’s side was known before the polarisation was measured.

That was a test of beaming with a prediction that could have failed. If one-sided jets are one-sided because the jet material moves at a large fraction of the speed of light, the visible jet is the approaching one, on the near side, and should point at the less depolarised lobe. If they are one-sided because the source itself is lopsided — more power ejected on one side, or a denser medium impeding one jet — there is no reason for the jet side to coincide with the side facing the observer, and the depolarisation asymmetry should be random with respect to it. The observed correlation, strong and one-directional, is what beaming predicts.

How the angle matters #

In the plane of the sky the two paths are mirror images and the effect vanishes. As the axis turns towards the line of sight, the far lobe sinks behind the halo and the near lobe rises in front of it, and the ratio grows steadily. At 60° it is already two, which is enough at 21 centimetres to make a clear difference in depolarisation for any source whose halo dispersion is large enough to depolarise at all.

That dependence is what makes the effect useful statistically and ambiguous individually. In a sample with random orientations, most sources lie between 30° and 90° of the line of sight, and nearly all of them show an asymmetry pointing the right way. For one source, the size of the asymmetry depends on the angle, on the halo’s density and core radius, and on the field, none of which is known well enough to invert for the angle. The effect says which lobe is nearer, reliably; it says how much nearer only roughly.

Which sources show it #

The asymmetry also depends on the size of the source, in a way that the halo picture predicts and that a lopsided-source picture would not. A small source, its lobes well inside the halo’s core, is behind a thick screen on both sides; both lobes are depolarised almost completely at 21 centimetres, and the asymmetry is invisible because there is nothing left to compare. A large source, its lobes far outside the halo, is behind a thin screen on both sides; neither lobe is depolarised, and again there is no asymmetry. In between, where the near lobe has emerged from the halo and the far lobe’s light still crosses it, the difference is large. For the halo used here the peak is at about 68 kiloparsecs, a little beyond the core radius.

Surveys that extended the original samples found this dependence: the asymmetry is strongest in sources a few tens to a hundred kiloparsecs across and weakens in giant sources. They also found that depolarisation overall correlates with the density of the environment, stronger in sources in rich groups and clusters, as a halo screen requires. Both results tie the effect to gas around the host rather than to anything in the lobes.

Why the jet points at the survivor #

Relativistic beaming comes from two effects of special relativity on a moving emitter. Aberration crowds the emission towards the direction of motion, and the Doppler factor raises the frequency and the rate of arrival of photons from the approaching side and lowers them from the receding side. For a continuous jet with a power-law spectrum of index $\alpha$, the brightness ratio of approaching to receding jet is

$$R = \left(\frac{1+\beta\cos\theta}{1-\beta\cos\theta}\right)^{2+\alpha},$$

with $\beta$ the jet speed as a fraction of that of light and $\theta$ the angle to the line of sight. At half the speed of light and 60°, the ratio is about five; at nine-tenths, fourteen; at nine-tenths and 30°, several hundred.

Jets in powerful radio galaxies are faint beside their lobes, and a receding jet dimmed by a factor of ten or more falls below the sensitivity of most maps. So one jet is seen, and it is the approaching one. The depolarisation asymmetry says independently which lobe is approaching, and the two agree. That agreement, together with the apparent faster-than-light motions seen in jets on parsec scales, established that the jets are relativistic out to kiloparsec distances from the core. It also supported the idea that radio galaxies and radio-loud quasars are the same objects seen at different angles, with quasars being those whose axis is nearer the line of sight.

One of several ways to read an orientation #

The depolarisation asymmetry is one of a small family of measurements that try to recover the angle at which an active galaxy is seen, and each has its own weakness. The ratio of the compact core’s radio brightness to the extended lobes’ grows as the axis nears the line of sight, because the core is beamed and the lobes are not, but it also varies intrinsically from source to source by a large factor. Apparent superluminal motion in the parsec-scale jet gives a lower limit on the jet speed and an upper limit on the angle, but only for sources with bright, moving components. The time delay of a quasar’s broad emission lines behind its continuum can reveal which way the gas nearest the black hole is moving, and its geometry is tied to orientation too.

The angle is not a curiosity. Many quantities measured for quasars depend on it, from the apparent brightness of the jet to the width of the broad lines — which is why the factor that converts a line width into a black-hole mass is thought to depend on the viewing angle and is calibrated only on average. Each orientation indicator is weak alone. The depolarisation asymmetry’s particular strength is that it is independent of everything the jet does, since it reads the orientation from gas the jet never touched.

What the model leaves out #

The figures use a smooth, spherical halo and a field tangled on a single scale. Real haloes are neither: the gas is disturbed by the radio source itself, which inflates cavities, drives shocks and compresses the gas around each lobe, and the field has a spectrum of scales rather than one. A lobe that has swept up a dense sheath of gas can depolarise itself regardless of orientation, and a source in a lopsided environment can have one lobe behind a denser screen for reasons unrelated to its angle. Each of these weakens the correlation with the jet side without removing it, and the observed fraction of sources in which the jet points the other way, a minority, is consistent with them.

The Burn law also assumes the beam covers many field cells. When the cells are resolved, as they are in the best modern maps of nearby sources, the rotation measure can be mapped directly across each lobe, and the dispersion computed from the map rather than inferred from a depolarisation. Such maps, of radio galaxies in nearby groups and clusters, show rotation-measure structure on the scales of a few kiloparsecs, with larger dispersions over the receding lobe, and fitting them has become one of the main ways the magnetic fields of hot intracluster gas are measured.

Still open: a halo that is also the source’s #

The effect was interpreted as a property of a static halo, and it is used in reverse to measure haloes’ fields. A radio source, though, is not a passive probe. The lobes push the halo gas aside, and the boundary between lobe and gas — where the gas is compressed and its field amplified and ordered along the boundary — may contribute much of the Faraday rotation. If it does, the rotation measure in front of a lobe measures the interaction between the lobe and its environment as much as the environment, and the field strengths inferred for group and cluster gas from radio sources would be biased high. Separating the undisturbed halo’s contribution from the sheath’s requires sources seen through the same halo from behind — background sources, not the cluster’s own — and enough of them to map the halo’s rotation measure independently. The next generation of radio surveys will find tens of background sources behind each nearby cluster, and whether their rotation measures agree with those inferred from the cluster’s own radio galaxies is not yet known.