The whistle that arrives sorted

A plasma reflects everything below its own plasma frequency, which is why long-wave radio bounces off the ionosphere and short-wave does not. That is the first rung of this ladder, and it describes a plasma with no magnetic field in it.

Add a field and the description changes character completely, because the electrons acquire a second frequency of their own — the rate at which they gyrate about the field — and the response to a wave now depends on how the wave’s rotation compares with theirs.

The mode that gets through #

For propagation along the field, a circularly polarised wave has refractive index

$$n^2 = 1 - \frac{\omega_p^2}{\omega(\omega \mp \omega_c)}$$

with the upper sign for the rotation matching the electrons’ gyration. That right-hand mode is the one that matters: below the gyrofrequency the denominator is negative, the whole term is positive, and $n^2 > 1$ — so the wave propagates, and with an index that grows without limit as the frequency falls.

The physical picture is a resonance. An electron gyrating at $\omega_c$ and a wave rotating the same way at $\omega < \omega_c$ stay in step for a long time, so the electron responds strongly and the medium’s index is large. The left-hand mode, rotating the wrong way, sees no such enhancement and is reflected exactly as an unmagnetised plasma would reflect it.

That is the whole of why a whistler exists: the magnetic field opens a channel below the plasma frequency for one sense of circular polarisation and not the other. A plasma that reflects everything becomes a plasma that is transparent to half of it.

The speed, and why the curve has a nose #

For $\omega \ll \omega_p$ the index simplifies to $\omega_p/\sqrt{\omega(\omega_c - \omega)}$, and the group velocity follows from differentiating $n\omega$:

$$v_g = \frac{2c\sqrt{\omega}\,(\omega_c-\omega)^{3/2}}{\omega_p\,\omega_c}$$

It is zero at $\omega = 0$ and zero again at $\omega = \omega_c$, so it has a maximum in between. Searching the drawn function finds it at exactly $\omega_c/4$, and the figure checks that against the closed form to a part in a thousand.

The consequence is the shape of the arrival curve. Everything either side of $\omega_c/4$ travels more slowly, so the delay against frequency is a curve with a minimum — a nose — with a descending branch below it and a rising branch above.

The classic whistler is the descending branch: a broadband click arrives with its high frequencies first and its low frequencies up to a second or two later, sliding downward in pitch. The rising branch above the nose is observed too, and a recording showing both joined at the nose is how the gyrofrequency along the path is read off directly — which means the magnetic field at the top of the path is being measured from the ground.

What a factor of a hundred in index means #

The refractive index of the medium a whistler travels through is not a number near one. At four kilohertz in the figure’s plasma it is around twenty; at four hundred hertz it is nearer seventy.

So the wave is travelling at a few per cent of the speed of light, through a medium containing about one particle per ten cubic centimetres. That is a striking combination: a near-vacuum by any laboratory standard, slowing light by a factor of seventy.

The reconciliation is that the index is large not because the medium is dense but because the response is resonant. Each electron is being driven close to its own gyrofrequency, so its displacement is enormous compared with what the same field would produce in a free electron, and a small number of very responsive charges does what a large number of unresponsive ones would. It is the same mechanism by which a dielectric’s index rises near an absorption, with the resonance supplied by the magnetic field rather than by a binding force.

The consequence is worth stating because it is unintuitive: the wavelength of a four-kilohertz whistler in the magnetosphere is a few kilometres, not seventy-five. A wave whose free-space wavelength is comparable with the Earth’s radius is, inside the plasma, small enough to be guided by a structure a few hundred kilometres across — which is what makes ducting possible at all.

The measurement it makes #

For frequencies well below the nose the delay simplifies to $t = D/\sqrt{f}$, and the constant $D$ — a few tens of seconds times the square root of a hertz for a typical magnetospheric path — is called the dispersion.

That single number is an integral along the path:

$$D \propto \int \frac{n_e^{1/2}}{f_c^{1/2}}\,ds$$

so it measures the electron content of a field line weighted by the local gyrofrequency. It is a remarkable thing to be able to measure from a receiver on the ground, and for two decades it was essentially the only probe of the outer magnetosphere.

The discovery it produced is worth recording. In 1963 Carpenter, analysing whistler dispersions, found a sharp discontinuity: field lines out to about four Earth radii carried a dense plasma and those beyond carried a hundred times less. That boundary — the plasmapause — had not been suspected, and it was found by measuring the pitch of a whistle.

The same gyration that makes the medium transparent to a whistler is what eventually takes the energy back. A wave is absorbed by the electrons it resonates with — those whose Doppler-shifted gyration matches its frequency — and an accelerated charge radiates into a pattern set by its own motion. So the mode that exists only because the electrons are turning is damped by the same turning, and the damping is strongest for exactly the electrons the wave is most in step with.

The path #

The wave does not go where it likes. Below the gyrofrequency the whistler mode is strongly guided by the magnetic field: its energy travels within about nineteen degrees of the field line whatever the wavevector’s direction, so the field line is a duct.

Where the plasma density has a field-aligned enhancement — and it often does, in ducts a few hundred kilometres across — the guiding is stronger still and the wave follows one line from one hemisphere to the other and back. That is why whistlers arrive in trains, each echo having made another trip, and why a single stroke can be heard several times with the dispersion multiplying each time.

The guidance is the same statement as a field line being something a conducting fluid cannot cross, one level up: the plasma is tied to the field, the density structure is therefore field-aligned, and the wave follows the density structure.

Flux frozen into a moving conductor is why the magnetosphere’s plasma is organised into field-aligned structures at all: the plasma cannot cross field lines without dissipation, so density enhancements stretch along them rather than across. The ducts a whistler follows are made by that constraint — a tube of enhanced density along a field line acts as a waveguide, and without one the signal would spread and never return an interpretable trace.

The two frequencies, and the regimes they divide #

A magnetised plasma has two intrinsic frequencies and the ordering between them decides everything about how it behaves.

The plasma frequency $\omega_p = \sqrt{ne^2/\varepsilon_0 m}$ is about restoring: displace the electrons and the charge separation pulls them back. It depends on density alone.

The gyrofrequency $\omega_c = eB/m$ is about turning: it is the rate at which an electron circles the field, and it depends on the field alone.

In the magnetosphere $\omega_p$ is an order of magnitude above $\omega_c$, and the whistler band sits below both. In a strongly magnetised laboratory plasma the ordering can be the other way round, and the modes are different. In the solar corona, in a fusion device, in the interstellar medium, the ratio takes different values and the wave physics is correspondingly different — so “a magnetised plasma” is not one medium but a family, indexed by that ratio.

The figure’s check enforces the ordering rather than assuming it: the generator refuses to draw a whistler if the plasma frequency is not well above the gyrofrequency, because outside that ordering the group velocity written down is simply wrong. That is the sort of refusal worth building in, since the formula would happily produce a plausible curve for parameters it does not describe.

Where else the mode turns up #

In the ionosphere, as the medium that makes VLF navigation and submarine communication behave oddly. The same right-hand mode below the gyrofrequency propagates in the ionosphere and is why very-low-frequency signals reach much further than a simple ground-wave calculation suggests.

In a laboratory plasma, where it is a heating and current-drive tool. A wave launched in the whistler branch — usually called the helicon in this context — deposits its energy in the plasma efficiently and is the basis of a family of plasma sources.

In the interstellar medium, as the pulsar dispersion measure. A pulsar’s pulses arrive later at lower frequencies for the same reason, with the unmagnetised $1/f^2$ law rather than the whistler’s $1/\sqrt f$ because the frequencies are far above the gyrofrequency there. The delay measures the integrated electron column, and that integral is how distances in the galaxy are estimated when nothing else is available.

And in a solid. A metal’s conduction electrons in a magnetic field are a magnetised plasma, and the same right-hand mode propagates through them. Helicon waves in metals were observed in the 1960s and are used to measure carrier densities: a solid whistler, at kilohertz frequencies, in a block of sodium.

That last case is the strongest argument that the effect belongs to magnetised electron gases rather than to space. The astrophysical and the solid-state versions differ in density by twenty-five orders of magnitude and obey the same dispersion relation.

A packet whose envelope and crests move at different speeds is the ordinary case, and a whistler is the extreme of it in the classical repertoire. The group velocity varies by a factor of several across an audible band, so a click — which is all frequencies at once — is stretched into a glide lasting a second or more, arriving high first and low last. Nothing about the source is musical; the tune is the medium’s.

The sound of it #

The frequencies are audible, which is the reason the phenomenon has the name it does and the reason amateurs discovered most of what is known about its variety.

A whistler-mode receiver is a long wire, an amplifier and a pair of headphones. The signals are between about three hundred hertz and ten kilohertz, which is squarely in the range of hearing, so the output is not a graph but a sound — and the descending glide of a whistler is unmistakable once heard.

The catalogue of natural emissions in that band was built by listening. Chorus is a rising tone repeated many times a second, said to resemble a dawn chorus of birds, and is generated by the electrons themselves rather than by lightning. Hiss is a broadband roar. Tweeks are short chirps, ionospheric rather than magnetospheric, with a sharp cutoff audible as a metallic ring. Each name is a description of a sound, and each corresponds to a distinct physical process.

That is unusual and worth appreciating: a branch of space physics whose objects were classified by ear before they were understood, and whose classification has largely survived the understanding.

Attenuation along a path is an integral of a local rate, and a whistler’s dispersion is a quantity of exactly that kind: one number standing for tens of thousands of kilometres of path through a density that varies along all of it. The inverse problem has the same character too — the measurement constrains an integral, and turning an integral into a profile needs either many paths or an assumed shape.

The name, and how long it took #

Whistlers were heard on telephone lines during the First World War — long wire circuits act as antennas at these frequencies — and were described as descending tones of unknown origin. Barkhausen reported them in 1919 and could not explain them.

Eckersley proposed in 1935 that they were dispersed radio waves and derived the $1/\sqrt{f}$ law, but the explanation required a plasma with a magnetic field in it, and the electron densities implied were far higher than anybody believed existed above the ionosphere.

Storey’s 1953 thesis settled it: the paths were tens of thousands of kilometres long, went from one hemisphere to the other, and required electron densities of a hundred per cubic centimetre out to several Earth radii — the first evidence that the space around the Earth is not empty. The magnetosphere was discovered by listening to it.

What the dispersion is worth as an instrument #

Reading a whistler is an inverse problem, and it is worth being explicit about how much can honestly be got out of it.

One well-recorded whistler gives one number. The dispersion $D$ is a single integral along a path, and no amount of care with a single trace separates the contribution of one part of the path from another.

A nose whistler gives two. If the trace shows both branches joined at the nose, the nose frequency gives the minimum gyrofrequency along the path — which is at the top of the field line, since the field is weakest there — and that fixes which field line the wave travelled on. With the path known, the dispersion becomes a measurement of the density along it.

A train of echoes gives the path length independently. Successive echoes have dispersions in the ratio 1:3:5, since each has made an extra round trip, and departures from that ratio say the path changed.

And a network gives the structure. Receivers at several latitudes catch waves that travelled on different field lines, and the set of dispersions constrains the density as a function of distance — which is how the plasmapause was mapped.

Each step buys more by adding a constraint rather than by measuring more carefully, which is the usual shape of an inverse problem: precision on one number does not substitute for a second number. The same lesson appears wherever a line-of-sight integral is the observable, and it is why an interferometer needs many baselines rather than one careful one.

The other thing two circular modes produce #

The essay’s central expression gives the two circular polarisations different refractive indices, and one consequence has been left implicit. A linearly polarised wave is the sum of the two circular ones, so if they travel at different speeds their relative phase changes as the wave goes, and the plane of polarisation turns.

That is Faraday rotation. The angle accumulated is proportional to the square of the wavelength and to an integral of the electron density times the component of the magnetic field along the path — so it depends on the field’s direction as well as its size, and on nothing else.

The wavelength-squared dependence is what makes it usable. Measure a distant source’s polarisation angle at several frequencies, fit a straight line against $\lambda^2$, and the slope is that integral with no assumption about what the source did at the start. Every polarised radio source in the sky therefore carries a measurement of the magnetised plasma between it and the Earth.

The pairing with this page’s dispersion is the powerful part. A pulsar’s dispersion measures $\int n_e\,\mathrm{d}s$ and its Faraday rotation measures $\int n_e B_\parallel\,\mathrm{d}s$, so the ratio of the two is a density-weighted average of the magnetic field along the line of sight — a number in microgauss, for a path of thousands of light-years, extracted from the timing and the polarisation of the same pulses. That is essentially how the Galaxy’s magnetic field is mapped, and the same trick applied to distant quasars probes the field between galaxies.

The laboratory version of the effect is a component. A transparent crystal in a magnetic field rotates polarisation the same way, and because the sense of rotation is fixed by the field rather than by the direction of travel, light coming back through it rotates further rather than unwinding. Put a polariser at each end at forty-five degrees and the device passes light one way and blocks it on return: an optical isolator, which every high-power laser needs to keep its own reflections out of it, and which is possible only because a magnetised medium is one of the few that does not treat the two directions alike.

Lightning on planets with no view of the ground #

Whistlers are made by lightning, so hearing one is evidence that lightning happened — which matters on worlds where nobody can watch for a flash.

Venus is the case in point. Its cloud deck is unbroken and opaque, and whether it has lightning was argued about for decades on the strength of ambiguous optical searches. Spacecraft carrying magnetometers and electric-field probes found bursts with the descending-tone signature of whistler-mode propagation in the Venusian ionosphere, and the argument moved considerably in favour of lightning being present.

The same mode has been recorded at Jupiter and at Saturn by every spacecraft equipped to listen, and in each case it does double duty: it says lightning occurred somewhere below, and its dispersion says something about the plasma density and magnetic field along whatever path it took.

That is an unusually direct kind of remote sensing. A pulse that nothing saw, travelling through a medium nothing has sampled, arriving as a note whose shape reports both.

What the picture cannot show #

The dispersion relation used is the low-frequency approximation. Dropping the ion contribution and taking $\omega \ll \omega_p$ is excellent through the whistler band and fails near the gyrofrequency and near the ion cyclotron frequency, where different modes exist.

The path is treated as uniform. A real field line runs from a few hundred kilometres above the ground to several Earth radii and back, and both the density and the field vary by orders of magnitude along it. The delay is an integral, and treating it as a length times a local group velocity is a summary of that integral rather than a calculation of it.

Damping is absent. Whistlers lose energy to the electrons they resonate with — the same resonance that makes them propagate makes them absorbable — and that interaction is what scatters energetic electrons into the atmosphere. It is one of the main loss processes for the radiation belts and none of it is in the arithmetic here.

And the source is idealised as instantaneous. A lightning stroke is not a delta function; it has its own spectrum and duration, and the received signal is the convolution of that with the propagation. The clean nose of the figure is what a perfect click would give.

The ladder from here #

Later rungs on this anchor: the full magnetised dispersion relation and the modes it contains, of which the whistler is one branch of one; cyclotron resonance and the pitch-angle scattering that empties the radiation belts; chorus and hiss, the naturally generated whistler-mode emissions that are not lightning at all and are amplified by the electrons themselves; ducting and the density structures that produce it; and the inversion problem — recovering the density profile along a field line from a set of measured dispersions.

The neighbouring ladders are the plasma cutoff, which is what a magnetic field opens a hole in, and the group velocity, which is the quantity a whistler makes audible. Flux freezing is what organises the medium the wave travels through.