The frequency the sky returns only at a slant

The frequency below which nothing gets in found that a plasma has a negative permittivity below its plasma frequency, and that a medium with a negative permittivity is not an absorber but a perfect reflector: a wave cannot exist inside it and goes back the way it came. The ionosphere’s densest layer, the F2 layer about three hundred kilometres up, holds around $10^{12}$ free electrons per cubic metre and has a plasma frequency of about nine megahertz, which is why it turns back a medium-wave broadcast and lets an FM signal straight out to space. The whistle that arrives sorted followed a lightning flash’s radio crackle along the Earth’s magnetic field, and later essays went on to the sheaths a plasma builds and the biases it can be given.

Every one of those arguments sent the wave straight into the plasma, along the gradient of its density. Real radio waves rarely go that way. A short-wave transmitter in London aims its signal low, towards the horizon, and the signal is heard in Lagos. This essay asks what changes when the wave arrives at a slant, and finds that almost everything about long-distance radio follows from one cosine.

A layer that bends rather than bounces #

The ionosphere is not a sharp surface. Its electron density rises over tens of kilometres from almost nothing at the base to a peak, and the refractive index for a radio wave of frequency $f$ falls with it:

$$n^2 = 1 - \frac{f_p^2}{f^2},$$

where $f_p$ is the local plasma frequency, proportional to the square root of the electron density. A wave entering from below at an angle meets a medium whose index decreases with height, and the ray that bends without a surface found what a ray does in a medium whose index varies smoothly: it curves, continuously, towards the region of higher index. Going up into the ionosphere, the higher index is below, so the ray curves back down.

In a medium stratified in horizontal layers, the product of the index and the sine of the angle from the vertical is the same at every height: Snell’s law, applied between infinitely many infinitely thin layers. A ray launched from the ground at an angle $\varphi_0$ from the vertical, where the index is one, keeps $n \sin\varphi = \sin\varphi_0$ all the way up. It turns horizontal, and then starts down again, at the height where $\sin\varphi = 1$, which is where the index has fallen to $\sin\varphi_0$:

$$1 - \frac{f_p^2}{f^2} = \sin^2\varphi_0 \quad\Longrightarrow\quad f_p = f\cos\varphi_0.$$

That is the secant law. A wave of frequency $f$ arriving at $\varphi_0$ from the vertical turns back at exactly the height where a wave of frequency $f\cos\varphi_0$ sent straight up would. Equivalently, the highest frequency a layer with peak plasma frequency $f_c$ can return at that angle is $f_c\sec\varphi_0$.

Rays that come back and rays that do not #

The figure traces rays at fourteen megahertz into a layer whose peak plasma frequency, its critical frequency, is eight megahertz — a typical daytime value. A wave at fourteen megahertz sent straight up passes through the whole layer: at no height does the plasma frequency reach fourteen. Tilt the launch and the ray bends, but steep rays still escape, curving only a little on their way through. At an angle of 55.2 degrees from the vertical, the angle whose cosine is eight fourteenths, the ray reaches the layer’s peak exactly horizontal and turns back; every ray more oblique than that turns back lower down, before reaching the peak.

The rays that return come down at different distances, and they do not come down close to the transmitter. The steepest returning ray, the one that barely makes it, travels furthest within the layer before turning and lands about eleven hundred kilometres away; more oblique rays turn lower and travel further horizontally before they even reach the layer, so they land further out. Nothing lands nearer than the first. Inside that radius the transmitter’s sky wave never comes down: it is the skip zone.

The echo that comes from too high #

The ionosphere was first measured, and is still monitored, by sending pulses straight up and timing their echoes. Breit and Tuve did it in 1925, and stations around the world have recorded ionograms every few minutes since the 1930s. What the echo’s delay measures is not the height of the turning point. A wave in a plasma travels at a group velocity of $c\,n$, which falls to zero where $n$ does, so a pulse slows as it nears its turning point and spends a long time there. Half the round-trip delay times $c$ — the virtual height — is therefore larger than the true height, by a little at low frequencies and without limit near the critical frequency, where the pulse turns at the layer’s peak and creeps through a region in which the index is almost zero over a great depth.

The figure computes the ionogram for a smooth layer of the shape Chapman derived for a layer ionised by sunlight, peaking at three hundred kilometres. By day the trace rises slowly from about 190 kilometres and then shoots upward as the frequency approaches eight megahertz; above eight there is no echo at all. At night, with the sun no longer ionising the gas and the electrons recombining, the same layer has a critical frequency of about four megahertz and the cusp moves left. Reading off where the trace shoots upward is how the critical frequency is measured, and it is the number every short-wave forecast begins from.

There is a pleasing theorem hidden in this. The delay of an oblique pulse that turns back at some height is the same as the delay of a vertical pulse at the equivalent frequency $f\cos\varphi_0$, divided by $\cos\varphi_0$ — the path of the slowed-down wave through the layer is, for timing purposes, a straight line to a mirror at the vertical virtual height. Breit and Tuve’s theorem, and Martyn’s related one, mean that a vertical ionogram, measured at one place, predicts oblique paths in any direction from it.

Obliquity buys frequency #

Drawn as turning height against frequency, the secant law is a stretch. The vertical curve rises from the base of the layer at low frequencies to the peak at the critical frequency. The curve for 45 degrees is the same curve stretched along the frequency axis by $\sec 45° = 1.41$, and reaches the peak at 11.3 megahertz. At 70 degrees the stretch is 2.92, and the layer returns waves of up to 23.4 megahertz.

The reason is that only one component of the wave’s motion has to be stopped. The law that only asks about one component found that a boundary cannot change the part of a wave that runs along it, because both sides have to agree on the phase along the surface; in a stratified medium the horizontal wavenumber is conserved for the same reason. The plasma has only to turn round the vertical part, and a wave arriving at a slant carries only $\cos\varphi_0$ of its wavenumber vertically. It is total internal reflection in a medium whose index decreases smoothly: the ray turns back when its conserved horizontal component would need an index larger than the medium can supply.

The highest frequency to any distance #

On a flat Earth the secant could grow without limit as the ray approached the horizontal. On a round Earth it cannot. The most oblique ray that can reach the layer leaves the ground horizontally and meets the layer at an angle set by the Earth’s curvature, and that fixes both the longest single hop and the highest frequency that can make it.

Treating the layer as a mirror at its virtual height of three hundred kilometres, the geometry of a triangle gives the angle of incidence at the layer for any distance between two stations, and the secant law gives the highest frequency that will make the trip: the maximum usable frequency. By day, with an eight-megahertz layer, a link of five hundred kilometres can use up to 10.3 megahertz; a link of two thousand kilometres, 22.5. The longest single hop, with the ray leaving the ground tangentially, spans 3,836 kilometres and can use 27 megahertz, 3.37 times the critical frequency. At night every number halves.

Those numbers are the daily practice of short-wave radio. A broadcaster wanting to reach a city two thousand kilometres away by day picks a frequency a little below 22 megahertz, high enough that the ionosphere’s lower layers absorb little and low enough that the F2 layer returns it. After dark it moves down to seven or nine. Radio amateurs read the critical frequency from ionosonde reports to decide which band is open to which continent, and the eleven-year cycle of solar activity, which raises the layer’s density at solar maximum, opens the higher bands for years at a time and closes them for years at solar minimum.

The ring of silence #

The same arithmetic, turned round, gives the skip distance. Below the critical frequency a wave sent straight up comes back, so there is a sky wave at every distance down to zero. Above it, only rays more oblique than $\arccos(f_c/f)$ return, and they land no nearer than the distance at which the maximum usable frequency equals the frequency in use. With the layer as a mirror, that is 456 kilometres at ten megahertz, 909 at fourteen and 1,613 at twenty; above about twenty-seven megahertz no single hop returns anywhere.

The dashed curve traces the rays through the layer itself instead, and lands further out, eleven hundred kilometres at fourteen megahertz: the rays that land nearest turn back close to the layer’s peak, where they bend slowly and run a long way horizontally, so their effective mirror is well above three hundred kilometres. The gap between the curves is the cost of calling a layer a mirror, and it is largest exactly where the skip zone’s edge is decided.

Inside the skip zone a transmitter is not silent everywhere: its ground wave, which follows the curve of the Earth, carries it for tens of kilometres at these frequencies before the ground absorbs it. Between where the ground wave dies and where the sky wave first lands is a ring in which nothing is heard. Amateur operators in the early 1920s, given the short wavelengths nobody else wanted, found they could be in clear contact with a station two thousand kilometres away and unable to raise one at two hundred. Taylor and Hulburt explained the skip distance in 1926 with exactly this geometry, and it was one of the first results of ionospheric physics to be put to daily use.

Hops, ducts and the long way round #

A single hop reaches at most about four thousand kilometres, and signals are routinely heard on the far side of the world. They get there in several hops: down from the layer, reflected by the ground or the sea, up again, down again. Each ground reflection costs something — sea water reflects well, dry ground and ice poorly — and each pass through the lower ionosphere costs more, so a path of four or five hops is weaker than one of one, and the best frequency for it is set by the worst hop. Occasionally a signal goes all the way round the Earth and arrives a seventh of a second after the direct one, as a faint echo of itself, and operators have heard echoes that went round twice.

At much lower frequencies the geometry turns into a waveguide, a cousin of the channel with no walls that holds light in a fibre by bending it rather than by reflecting it. Below a few tens of kilohertz the ground and the bottom of the ionosphere, about seventy to ninety kilometres up, are both good reflectors, and the space between them guides waves round the planet as the pipe that will not carry a low note guided sound: there is a lowest frequency, about $c/2h$, near two kilohertz, below which the guide’s first mode will not propagate, and a lightning flash’s crackle heard through the guide arrives with its frequencies near the cutoff drawn out into a short descending chirp. The guide carries very-low-frequency navigation and time signals across oceans and lets submarines receive messages through a few metres of sea. Its lowest resonances, a standing wave round the whole circumference of the Earth at about eight hertz, are excited continuously by the world’s thunderstorms.

The secant law has a military use as well. Over-the-horizon radars send short-wave pulses up at a slant and listen for the echoes that come back down by the same route after bouncing off aircraft and ships a thousand to three thousand kilometres away, beyond any line of sight. Their operators retune them continuously to the frequency that the ionosphere of the moment will deliver to the range they want to watch, which is the maximum-usable-frequency curve read backwards.

The same cosine, turned against astronomers #

For a radio astronomer on the ground the ionosphere is a ceiling, and the secant law makes it a lower ceiling near the horizon. Radiation from a source in the sky arrives at the top of the layer at an angle and must cross it to reach a telescope; by the same reasoning as for a wave going up, it is turned back if its frequency is below the critical frequency times the secant of its angle from the vertical. A source overhead is visible down to the critical frequency, a few megahertz; a source low in the sky is shut out at frequencies several times higher. And below a megahertz or so the sky is closed to every ground-based telescope at all times, which is why the radio sky at the lowest frequencies is known mainly from spacecraft, and why a telescope on the far side of the Moon, shielded from both the ionosphere and the Earth’s own radio noise, has been proposed for decades. The cosine that lets a broadcaster reach across a continent is the cosine that hides the low-frequency universe from anyone standing on the ground.

What the pictures cannot show #

The figures use a single smooth Chapman layer over a flat Earth for the rays and a mirror at 300 kilometres over a round Earth for the maximum usable frequency; the real ionosphere has several layers — D, E, F1 and F2 by day — whose densities vary with latitude, season, time of day and the eleven-year solar cycle, and which tilt, ripple and drift. The Earth’s magnetic field splits every wave into two with different indices, the ordinary and extraordinary, which reflect at slightly different heights; the figures show only the ordinary wave without the field. The lower layers absorb, especially by day, which sets a lowest usable frequency on a path that the figures leave out. And the true maximum usable frequency on a long path includes a correction for the layer’s curvature, of order ten per cent, that the mirror model omits.

Still open: forecasting the layer a day ahead #

Short-wave communication, over-the-horizon radar and the corrections that satellite navigation applies for the ionosphere all depend on knowing the electron density above a place an hour or a day in advance. The ionosphere responds to the Sun’s ultraviolet output, to geomagnetic storms that can raise or wipe out the F2 layer within hours, and to waves rising from the weather in the lower atmosphere, and these drivers interact in ways that are only partly modelled. Data-assimilating models that combine ionosondes, satellite navigation signals and physics now forecast the critical frequency for the next few hours reasonably well in quiet conditions; forecasting the response to a storm a day ahead, which is when it is most needed, is not yet reliable.

The habit worth carrying away is to ask which component of a motion a boundary has to stop. In a medium stratified horizontally, a wave’s horizontal wavenumber is conserved, so a layer only has to turn round the vertical part: a ray at $\varphi_0$ turns back where $f_p = f\cos\varphi_0$, a layer with an 8 MHz critical frequency returns 23.4 MHz at 70°, and a single hop off it can use up to 27 MHz, 3.37 times what it reflects straight up. One frequency limit becomes a range of them, and a transmitter becomes inaudible near home.