A lobe that reads the time since it was stirred

A radio lobe’s spectrum carries a clock: the electrons that make it lose energy by radiating, the most energetic fastest, so the spectrum steepens above a break frequency that falls as the population ages. Read off the spectrum and combined with a magnetic field, the break gives an age. For many lobes that age comes out several times younger than the lobe’s length divided by the speed its head advances, and the disagreement has narrowed to one question: whether the electrons are accelerated once, at the hotspot where the jet ends, and then left to age, or whether they are re-energised again and again as they flow back through the lobe.

The second possibility is not exotic. A lobe is a turbulent, magnetised bag of plasma, stirred by the back-flow from the hotspot, by instabilities at its boundary and by the buoyant rise of the lobe through the gas around it. Electrons scattering off moving magnetic irregularities gain energy on average, the process Fermi proposed in 1949 for cosmic rays, and the same turbulence that tangles the field can feed the electrons that radiate in it. This essay asks what that does to the clock — and the answer is that it does not slow the clock, it stops it.

Gain against loss #

Each electron’s energy, written as a Lorentz factor γ, changes at a rate set by two terms. Radiation removes energy at a rate proportional to $\gamma^2$: synchrotron emission in the lobe’s field and inverse-Compton scattering of the microwave background, which drains the same electrons at a rate equivalent to a field of 3.25 microgauss today. Stirring adds energy at a rate proportional to $\gamma$, with a characteristic time τ — the time in which the turbulence multiplies an electron’s energy by e — taken here, in the simplest form of the process, to be the same at every energy.

At low energy the gain wins and the electron is boosted; at high energy the loss wins and it cools. Between them there is an energy at which the two balance, $\gamma_{\rm eq} = 1/(b\tau)$ in terms of the loss coefficient b, and every electron is driven towards it, from above or below. For a 3-microgauss field and a stirring time of 30 million years, $\gamma_{\rm eq}$ is about 42,000, and electrons at that energy radiate most strongly at a few gigahertz.

A population of electrons left alone has no such floor. Its most energetic members cool fastest, and the energy above which nothing is left — the cut-off that produces the break — falls in inverse proportion to the time since injection, so the break frequency falls as the square of the age. That is the clock. In a stirred population the cut-off falls only until it reaches $\gamma_{\rm eq}$, and then stays there. The opening figure shows it: after ten million years the stirred and unstirred spectra barely differ, after thirty the stirred one has fallen behind, and after a hundred the unstirred spectrum has collapsed below a gigahertz while the stirred one sits where it was at about thirty.

The model behind every figure here is exact within its assumptions. The equation for an electron’s energy has a closed-form solution, so each electron is followed from its injected energy through any history of stirred and quiet stretches, and the spectrum is the sum of their emission. The spectral age every figure reports is not a formula but a fit: the unstirred population whose spectrum best matches the computed one between 0.1 and 10 gigahertz, with the normalisation free — the procedure an observer applies to a measured spectrum.

An age that stops at the stirring time #

The reading follows the true age while the lobe is younger than the stirring time, because over that interval the stirring has barely acted, and then it saturates at almost exactly the stirring time. The coincidence is not one: the age an unstirred population needs to bring its cut-off down to $\gamma_{\rm eq}$ is $1/(b\gamma_{\rm eq})$, which is τ. A stirred lobe’s spectral age is the stirring time and nothing else, and it says nothing at all about how long the lobe has existed.

That changes what the disagreement between the clocks means. A lobe whose spectral age is a third of its dynamical age is usually read as a lobe whose electrons are a third as old as the lobe, freshly supplied by back-flow or mixing. On the stirring reading, the electrons can be as old as the lobe; the spectral age has measured the turbulence’s time-scale, and the two clocks were never measuring the same quantity. The field-dependent calibration of the spectral clock — the weakness that the minimum-energy argument papers over — becomes irrelevant to the question, because no field turns a measurement of τ into an age.

A shape that cannot tell #

The obvious defence is to look at the spectrum’s shape. Different histories of the electrons produce different curvature: a single burst cut off sharply, continuous injection gently, a mixture of ages in between. Spectral-ageing studies distinguish these models by measuring the spectral index in two frequency ranges at once and plotting one against the other.

The stirred population does not leave the track. Its spectrum at every moment sits on the curve traced by unstirred populations of increasing age, and when the stirring holds the break steady, the point simply stops moving. At a hundred and twenty million years, the stirred lobe’s two indices are within half a hundredth of an unstirred lobe aged twenty-nine.

The reason is that stirring and ageing act on the same electrons in the same way at the top of the distribution. In both, the high-energy electrons above some energy have lost their energy and the electrons below it are untouched in shape; what differs is only whether that energy keeps falling. A spectrum is a snapshot of that distribution and has no record of how it came to be. The gain does change the low-energy electrons, boosting them all by the same factor, but a uniform boost shifts a power law without changing its slope, so the low-frequency spectrum keeps its injected index. Stirring of this kind is therefore not a different curvature that a better fit could discover. It is a frozen point on the ageing curve.

That separates stirring from the other explanation for a lobe reading too young. A mixture of electron ages curves more gently than a single burst, because several breaks overlap, and that gentler curvature is exactly what the colour–colour plot was devised to detect. Mixing therefore leaves a mark on the shape that a good enough spectrum can find. Stirring at a rate the same for every electron leaves none, because it lands on the single-burst track and stays there.

A map that can tell #

A lobe is not one spectrum but a map of them, and the map has a geometry that the integrated spectrum lacks.

An unstirred lobe whose electrons stay where they were deposited reads older the further back from the hotspot one looks, in proportion to the distance, and the slope of that gradient is the head’s speed. A stirred lobe reads the same near the hotspot, where the electrons are younger than τ, and then flat: everywhere further back than the head travels in about one stirring time, the break sits at the equilibrium frequency.

Mixing flattens the gradient as well, which is what the comparison of the two clocks found. But mixing flattens it without stopping it: the oldest electrons at each position are still older further back, and the mixture’s break still falls with distance, only more slowly. Stirring gives a plateau. A map that keeps ageing all the way back, however gently, argues for mixing; a map that stops ageing at some distance and stays flat argues for stirring, and the distance at which it stops, divided by the head’s speed, is the stirring time measured a second way.

Low-frequency surveys make such maps possible at the resolution that matters. Arrays operating below two hundred megahertz see electrons whose break has fallen far below a gigahertz, which in an unstirred lobe means the oldest material at the far end. A lobe whose low-frequency map shows no such electrons — no region whose spectrum is steeper than the plateau — has either been stirred, or has had its oldest material mixed away, and the shape of the transition between the steep region and the flat one distinguishes the two.

What the stirring costs #

The energy in a population of relativistic electrons with a power-law spectrum steeper than γ to the minus two is carried mostly by the low-energy electrons, which radiate hardly at all, and that is why an unstirred population loses only a few per cent of its energy in a hundred million years even as its radio spectrum collapses. A stirring that acts at the same rate on every electron boosts those low-energy electrons too, and their energy grows as $e^{t/\tau}$. Keeping the break frozen for four stirring times costs the turbulence about forty times the electrons’ initial energy.

A lobe does not have that to give. The minimum-energy argument puts the particles’ energy and the field’s within a factor of order one of each other, and the turbulent kinetic energy of the lobe’s plasma is a fraction of its pressure, which is itself the sum of the two. If the turbulence holds as much energy as the electrons, it can double their energy once and is spent; if it holds three times as much, it lasts for about one and a half stirring times. Either way, stirring at an energy-independent rate is self-limiting. It can hold a break steady for something like a stirring time, not for many.

There are two ways out, and both leave the conclusion intact in different form. The turbulence may be continually replenished by the jet, which converts its own power into lobe turbulence through the hotspot and back-flow; in that case the stirring lasts as long as the jet does, and the spectral age measures τ for as long as the jet is on. Or the acceleration may be weak for low-energy electrons and strong only for high-energy ones, which is how resonant scattering off small-scale magnetic waves behaves for some spectra of turbulence; then the low-energy electrons are not boosted, the bill falls by orders of magnitude, and the stirring can run indefinitely at the top of the distribution while doing nothing to the bottom. Which of these holds is a question about the electrons a radio telescope never sees: those below a Lorentz factor of a few hundred, whose emission falls below the ionosphere’s cut-off at ten megahertz.

The clock restarts when the stirring stops #

When the stirring stops, the electrons resume ageing from where the stirring held them, at the equilibrium cut-off, and the break begins to fall again as though the population had been injected one stirring time before the stirring ended. The reading climbs at exactly a year per year from the plateau, and at any later moment it equals the time since the stirring stopped plus τ. The seventy-one million years of stirred life before that leave no trace.

This is the reading that the radio galaxies whose jets have switched off require. A lobe whose jet has stopped supplying it loses both its fresh electrons and the turbulence the jet was driving, and its spectrum ages from the moment it was abandoned. Those remnant lobes, now found in their hundreds at low frequencies, have steep, curved spectra and spectral ages that are taken as the time since the jet switched off. The stirring reading adds one correction: the time since switch-off is the spectral age minus the stirring time that preceded it, not the spectral age itself. On a remnant whose spectral age is fifty million years, and whose stirring time was thirty, the jet has been off for twenty.

And it gives a remnant a way to return. A remnant lobe that is stirred again — by a merger, by the passage of a shock through the cluster around it, by the buoyant lobe being sheared — will have its break driven back up towards the equilibrium, and its spectrum will become younger without any fresh electrons arriving. Such revived lobes are seen in galaxy clusters, where they are called radio phoenixes, and their spectra are steep and strongly curved, with breaks consistent with an age of tens of millions of years in a population whose host galaxy has been quiet for far longer.

What a sample of lobes would show #

One lobe cannot settle the question, because its stirring time is unknown and its plateau could be mistaken for a lobe that happens to be young. A sample can. If lobes are left alone, their spectral ages should grow with their size, because a larger lobe has taken longer to grow at a given head speed; the spread in head speeds blurs the relation but does not remove it. If lobes are stirred, the spectral age of every lobe larger than its head travels in one stirring time is that stirring time, so spectral age should stop growing with size above some length, and the length at which it stops is the head speed times τ.

The stirring time is not a universal constant, so the plateau will not be one number across a sample. But it should depend on the lobe’s plasma — its field, its turbulent velocity, its density — and not on its size or its age, and that is a testable distinction. A spectral age that correlates with the lobe’s length argues for ageing; one that correlates with the lobe’s equipartition field or its pressure, and not with its length, argues for stirring.

Comparisons of spectral and dynamical ages across samples of radio galaxies have generally found the discrepancy largest in the biggest and oldest sources: the largest lobes are the ones whose spectral ages fall furthest short. That is the pattern a stirring time would produce, because a small lobe is younger than τ and its spectral age is honest, while a large one is many stirring times old and its spectral age has stopped. It is also the pattern produced by back-flow mixing, which matters more in long lobes, and by head speeds that were faster early on. The size dependence alone does not decide it; the plateau in the maps, and the absence of curvature beyond the single-burst track, are the two observations that would.

The same physics elsewhere #

The electrons in a radio lobe are cosmic rays, and the competition between gain and loss that sets $\gamma_{\rm eq}$ is the same one that limits cosmic-ray electrons everywhere. A positron at a teraelectronvolt cannot have come far because its synchrotron and inverse-Compton losses act in a few hundred thousand years; a lobe electron at $\gamma_{\rm eq}$ loses its energy in one stirring time, and is held there only because the turbulence restores it as fast. The microwave background’s light that cuts off the highest-energy cosmic-ray protons by pion production also enters here, as the inverse-Compton term in b that sets a ceiling on any spectral age, stirred or not.

And the stirring’s energy dependence is the same unknown as the energy dependence of cosmic-ray transport in the Galaxy, where how fast particles escape scales as a single power of energy that is measured two ways and disputed. In both settings, the turbulence’s spectrum on scales comparable with a particle’s gyration radius decides how each energy is treated, and in both, that spectrum is not measured directly but inferred from the particles it shapes.

What the model leaves out #

The stirring here is systematic: every electron gains at the same rate. Real second-order acceleration is stochastic, each electron’s gain fluctuating, so the population is spread around $\gamma_{\rm eq}$ rather than piled onto it, and the cut-off is softer. Pitch-angle scattering is assumed to keep the electron distribution isotropic, the magnetic field is uniform and steady, and the lobe does not expand; expansion would add an adiabatic loss at every energy that partly offsets the boost at low energies. The injected spectrum is a single power law, the single-electron emission is approximated by a standard fit to the synchrotron kernel, and inverse-Compton losses are from the microwave background alone.

The central result does not depend on these. Any process that adds energy to the electrons faster than radiation removes it below some energy will freeze the cut-off at that energy, and any spectrum read as an age will then read the time the process takes rather than the time it has been running.

Still open: what the stirring time is #

If a lobe’s spectral age is the stirring time, the useful question is what sets it. Turbulence of velocity v on scale L, with magnetic irregularities moving at the Alfvén speed, accelerates electrons on a time-scale of order $(c/v_{\rm A})^2$ times the scattering time, which can range from millions to billions of years for plausible lobe conditions. The equilibrium break frequency is a direct measurement of it, and two independent handles exist: the plateau’s onset distance along the lobe, and the polarisation of the emission, which falls as turbulence tangles the field. A population of lobes whose plateau ages agree with the stirring time inferred from their field and turbulence would confirm that radio spectral ages measure the plasma rather than the clock — and would turn the synchrotron spectrum, which has no temperature in it, into a thermometer for turbulence.