Two clocks for one lobe

A synchrotron spectrum has no temperature in it, but it has a clock. The relativistic electrons that make a radio lobe’s emission lose energy as they radiate, and the most energetic lose it fastest, so the population’s spectrum steepens above a frequency that falls as time passes. The break frequency, read off the spectrum, is an age — provided the magnetic field the electrons have been radiating in is known. That proviso has been the method’s weakness since it was introduced, and the field is exactly what the observation does not fix.

A lobe also has a second clock that needs no field at all. A radio galaxy’s jets drive their lobes outward through the surrounding gas, and the head of each lobe advances at a speed that can be estimated — from the asymmetry between the two sides, from the pressure balance with the surrounding gas, or from the lobe’s shape. The lobe’s length divided by that speed is how long it has been growing. For the same lobe the two clocks should agree. They do not, and the way they disagree is informative.

A clock with a ceiling #

The spectral age depends on the field in a way that is not monotonic, and that is the first thing the figure shows. A strong field makes electrons radiate quickly, so a given break means a young population. A weak field makes them radiate slowly, so the same break would mean an old one — but only down to a point. Electrons also lose energy by scattering the photons of the microwave background, a loss that does not depend on the lobe’s field at all, and below a field of about three microgauss, the background’s equivalent field today, that loss dominates. The electrons’ lifetime at a given break is therefore longest at an intermediate field, the background’s equivalent divided by $\sqrt{3}$, and it falls on either side.

So the spectral age has a ceiling. For a break at two gigahertz it is about 110 million years; for higher breaks, younger. No choice of field can make a spectrum say its electrons are older than that. And the dynamical age of a large, slowly growing lobe can exceed it: a lobe four hundred kiloparsecs long whose head has advanced at a hundredth of the speed of light is 130 million years old.

That is the sharpest form of a long-standing discrepancy. Across samples of radio galaxies, ages from spectra are typically several times younger than ages from dynamics, and the usual first response — that the field was overestimated by the equipartition assumption — runs into the ceiling. Lowering the field raises the spectral age only until the background takes over; past that, lowering it further makes the spectrum say the electrons are younger again. For the largest and slowest lobes no field reconciles the clocks.

Electrons younger than their lobe #

The resolution is that the two clocks are timing different things. The dynamical clock times the lobe: how long the structure has been growing. The spectral clock times the electrons at the break: how long since they were last given their energy. The two agree only if every electron in the lobe was accelerated once, at the hotspot where the jet ends, and then left to age where it was deposited.

That is the simple picture, and it predicts a clean gradient: the break should fall steadily from the hotspot back towards the core, as the inverse square of the time since the head passed each point.

Observed lobes show gradients, but they are usually flatter than the simple picture predicts and broken by regions where the spectrum is younger than its position would allow. Two processes can do that. Material flowing back from the hotspot along the lobe — back-flow, seen in simulations of jets and in the lobes’ own structure — carries younger electrons past older ones and mixes them. And turbulence and shocks inside the lobe can re-accelerate electrons that have already aged, resetting their clocks without moving them. Either way each position holds a mixture of ages, and the spectrum of a mixture is not the spectrum of its average.

The speeds involved make the mixing plausible. Back-flow in simulated lobes runs at a few tenths of the speed of light near the hotspot, slowing as it spreads, which carries plasma across a hundred kiloparsecs in a few million years — a small fraction of the lobe’s age. So any position in a mature lobe can hold plasma that left the hotspot at very different times, and an electron’s age need have little to do with where it is found. A lobe is less a strip chart recording the head’s progress than a reservoir being stirred while it fills.

What a mixed spectrum says #

The mixture is where the spectral age loses its meaning as an age.

Each age contributes its own break, and above the oldest electrons’ break the younger ones are still radiating. The combined spectrum curves gently rather than turning over sharply, and a break fitted to it lands at the frequency where the younger electrons begin to fade. The age read from that break is younger than the oldest electrons by a factor that grows with the spread of ages — half again younger for a modest spread, half the true age for a wide one — and nothing in the spectrum’s shape alone says how wide the spread is. A spectral age is a lower bound on the time since the oldest electrons were energised, and in a lobe where electrons are mixed or re-accelerated it can be a loose one.

The detailed shape of the turnover can in principle distinguish the cases. Electrons injected once and left alone, with their directions kept random, give one shape; electrons injected continuously give a break that steepens by exactly half a power rather than turning over; populations in a field that varies through the lobe give a third. Fits of these models to spectra measured at many frequencies with the low-frequency arrays now operating have begun to separate them in individual lobes, and they tend to favour models in which the electrons see a range of fields — which is a mixing of conditions rather than of ages, and lengthens the inferred age in the right direction.

The ceiling falls with redshift #

The microwave background was denser in the past, and its share of the electrons’ cooling was larger.

The background’s energy density grows as the fourth power of one plus the redshift, its equivalent field as the square, and the ceiling on the spectral age falls as the 3.5 power. At redshift three the oldest electrons a lobe can show above a gigahertz are about a million years old. Lobes take tens to hundreds of millions of years to grow, so at high redshift most of their volume holds electrons that have already given their energy to the background — scattering its photons up to X-ray energies — and are dark in the radio.

That has a consequence for what radio surveys find. The radio galaxies seen at high redshift are dominated by their young parts, the hotspots and inner jets, and their extended lobes fade into an X-ray glow from the scattered background photons. Lobes that would be bright at low redshift are missing at high redshift not because they are absent but because the background has made them radiate in another band — and the X-ray emission from the lobes of distant radio galaxies, detected around some of them, is the background’s own photons, reflected off electrons that the radio cannot see. Brightness temperature limits in compact cores involve the same inverse-Compton process acting on the source’s own photons; in old lobes it is the universe’s photons that do the work.

Where the dynamical clock is weak #

The disagreement is not entirely the spectral clock’s fault. The dynamical age divides a length by a speed, and the speed is the harder number. Hotspot advance speeds are inferred from the brightness asymmetry between approaching and receding sides — which assumes the two sides are intrinsically identical — or from pressure balance with the surrounding gas, which needs the gas density measured in X-rays and the lobe’s own pressure, which depends on the field again. A lobe whose head moved quickly early and slowly later has a length that over-represents its early speed. And the length is projected on the sky, so a lobe pointed partly towards the observer is longer than it looks and older than its apparent length implies.

The two clocks’ errors therefore run in opposite directions for the largest sources: the dynamical age tends to be overestimated by slow late advance and the spectral age underestimated by mixing. Some of the discrepancy is each clock’s bias, and how to divide it is argued source by source.

Which lobe is further away #

The projection that biases the dynamical age can sometimes be measured, and the measurement comes from polarisation. A radio galaxy’s two lobes lie at roughly equal distances from its core on the sky, and nothing in a picture of them says which points towards the observer. But light from the far lobe crosses more of the host galaxy’s magnetised hot halo on its way out, and the far lobe loses its polarisation first at long wavelengths. That fixes which side is approaching, and together with the brightness asymmetry of the jets it constrains the angle to the line of sight — and so how much longer each lobe really is than it appears.

The same Faraday rotation complicates the field measurement that the spectral age needs. Where the lobe’s own magnetised plasma rotates the light it emits, the field’s structure inside the lobe — ordered or tangled, uniform or concentrated in filaments — changes how its electrons age. A lobe whose field is concentrated in filaments, with weak fields between them, has electrons that age quickly where they cross the filaments and slowly elsewhere, which is one of the mixtures of conditions the spectral fits are now finding.

A jet that switched off and on again #

Some radio galaxies show two pairs of lobes along the same axis: a large, faint outer pair and a smaller, brighter inner one. These double-double radio galaxies are jets that stopped and restarted. The outer lobes, no longer fed, age passively, and their spectral age measures the time since the jet turned off; the inner lobes are growing from the restarted jet, and their dynamical age measures the time since it turned on again. The difference is the interruption, and it comes out at a few million to tens of millions of years — a measurement of how long a black hole’s activity can pause, made with the same two clocks this essay has set against each other.

In these systems the clocks are used where they are most trustworthy: the spectral clock on lobes with no hotspot re-accelerating anything, the dynamical clock on lobes young enough that their advance speed has not changed much. The ages come out consistent, which supports the reading that the discrepancy in ordinary radio galaxies comes from mixing and re-acceleration in lobes that are still being fed, rather than from a failure of either clock.

The same clock in the Galaxy #

The clock is not peculiar to radio galaxies. The cosmic-ray electrons filling the Milky Way’s disc and halo radiate synchrotron emission in the Galaxy’s microgauss field and scatter starlight and the microwave background, and their energy spectrum steepens at high energies for the same reason a lobe’s does: the most energetic electrons lose their energy before they can travel far from the sources that made them. The electron spectrum measured directly near the Earth steepens sharply around a teraelectronvolt, and electrons above that energy cannot have travelled more than a few hundred parsecs in their lifetime — so the highest-energy electrons arriving at the Earth come from a handful of nearby sources, and their spectrum is a map of the local supernova remnants and pulsars, read through the same losses. Protons and nuclei do not suffer them, which is why the highest-energy particles’ arrival directions are a different problem.

What is measured #

The measurements are spectra at many frequencies across a lobe’s area, from tens of megahertz to tens of gigahertz, and the inferred break frequency at each position. The low-frequency arrays built in the last decade, sensitive below 200 megahertz, see the oldest electrons — those whose break has fallen lowest — and have found lobes far larger and older than were known, including remnant radio galaxies whose jets have switched off, leaving lobes that are only ageing. Those remnants are the cleanest test of the spectral clock, because nothing is re-accelerating their electrons and the jet that fed them has stopped: their spectra are steep and curved everywhere, and their spectral ages, read with a field from equipartition or from X-ray inverse-Compton measurements, give the time since the jet turned off.

The X-ray measurements are the second route to the field. The same electrons that make the radio emission scatter background photons into X-rays, and the ratio of the two luminosities is the ratio of the background’s energy density to the field’s, which the background fixes. Where both are measured the field comes out somewhat below equipartition, typically by a factor of two or three — which lengthens spectral ages, though not past the ceiling.

A third clock in the gas around the lobe #

Where a radio galaxy sits at the centre of a cluster of galaxies, its lobes push aside the hot gas that fills the cluster, and the X-ray images show the result: cavities, dark bubbles in the gas’s X-ray glow, filled with the lobes’ radio-emitting plasma. A cavity gives a third age. A bubble of light plasma in heavy gas rises by buoyancy, and the time it takes to rise from the centre to its present height is set by its size and the cluster’s gravity, which the gas’s own temperature measures. Its volume times the pressure of the gas around it is the work the jet did to inflate it.

The buoyancy ages of cavities are typically tens of millions of years, consistent with the dynamical ages of lobes of similar size, and longer than their spectral ages, which agrees with the other two clocks’ disagreement rather than resolving it. But the energy they give is the quantity that matters most: in many clusters the work done inflating the cavities is enough to offset the cooling of the gas around the central galaxy, which would otherwise form stars at hundreds of solar masses a year and does not. The radio lobe’s age is one factor in that budget, and the clock the budget needs is the dynamical one — how long the jet has been doing work — rather than the spectral one, which times only the electrons.

Still open: how often a lobe’s electrons are re-energised #

The disagreement between the two clocks has narrowed to a question about the electrons’ history inside the lobe: whether they are accelerated once at the hotspot and age passively, or repeatedly re-energised by turbulence and shocks as they flow back. The first makes the spectral age a clock with a field-dependent calibration; the second makes it a measure of how recently the lobe’s internal processes last stirred the electrons, which is a different quantity. Resolved spectral maps at low frequencies, combined with X-ray maps of the inverse-Compton emission that fix the field at each position, are beginning to separate the two in individual sources. The answer bears on how much energy a radio galaxy deposits in the gas around it over its life, which the minimum-energy argument and the dynamical age both enter, and which is one of the main ways black holes are thought to regulate the growth of their galaxies.