The brightness that is not the same for everyone

A moving source’s light is shifted in frequency, and the shift survives even at right angles, where there is no approach to produce it. That is the effect everybody knows about, and it is much the smaller of the two.

The larger effect is on the brightness, and it goes as the fourth power of the same factor. A source at ninety-nine per cent of the speed of light is brighter approaching than receding by a factor of $1.6\times10^9$.

The one quantity nobody argues about #

The result comes from an invariance rather than from a chain of transformations, which is what makes it clean.

The specific intensity — power per unit area, per unit solid angle, per unit frequency — is not the same for every observer. Nor is the frequency. But the combination

$$\frac{I_\nu}{\nu^3}$$

is a Lorentz invariant: every observer computes the same number for it.

That is a statement about counting states rather than about light. Photons live in a phase space, the volume element of that space is invariant, and $I_\nu/\nu^3$ is proportional to the number of photons per phase-space cell — the occupation number, which is a count and cannot depend on who is doing the counting.

Everything follows in one step. Define the Doppler factor

$$\delta = \frac{1}{\gamma(1 - \beta\cos\theta)},$$

which is the ratio of observed to emitted frequency. Then $I_\nu = \delta^3 I'_{\nu'}$ per unit frequency, and integrating over frequency adds a fourth power because the frequency interval is stretched by $\delta$ too.

So the flux carries $\delta^4$, and the whole subject is that exponent.

Where the invariant comes from #

The claim that $I_\nu/\nu^3$ is the same for everyone is doing all the work, and it is worth seeing why it is true rather than taking it.

The invariance behind every number here is a counting statement: how many photons occupy each cell of phase space. A count is the one thing that cannot depend on who is doing the counting, so the photon occupation number is the same in every frame — and everything else in this essay is that single invariant expressed in whichever variables an observer happens to use.

Consider a small bundle of photons: those within a volume $dV$ of position and a volume $d^3p$ of momentum. The number of them in that bundle is a count, so every observer agrees on it. And the phase-space volume $dV\,d^3p$ is itself invariant — one factor of $\gamma$ from the contraction of the spatial volume and one from the expansion of the momentum volume, which cancel.

So the occupation number, photons per unit phase-space volume, is invariant. Writing it out in terms of what an astronomer measures gives $I_\nu/\nu^3$, because the specific intensity carries three powers of the momentum in the way it is defined.

That is a much better foundation than transforming intensities directly, and it makes the exponent unarguable. It also explains a fact that would otherwise be a coincidence: a thermal spectrum stays thermal under a boost. A Planck spectrum is a particular occupation number as a function of $h\nu/kT$; multiplying every frequency by $\delta$ and leaving the occupation number alone gives a Planck spectrum at temperature $\delta T$. No other spectral shape is preserved, and that is why the microwave background’s dipole is a temperature shift rather than a distortion.

Reading the numbers off #

Three features of the curve are worth extracting, and the figure computes each.

The front-to-back ratio is $\big((1+\beta)/(1-\beta)\big)^4$. For $\beta = 0.5$ that is $81$; for $0.9$ it is $1.3\times10^5$; for $0.99$ it is $1.6\times10^9$.

Side-on is fainter than at rest. At $\theta = 90^\circ$ the Doppler factor is exactly $1/\gamma$, so the flux is $1/\gamma^4$ — a factor of $5.8\times10^5$ down at $\beta = 0.99$. Motion does not brighten a source; motion towards the observer does, and motion in any other direction dims it.

Half the light arrives inside a cone of one over $\gamma$. A source radiating isotropically in its own frame sends half its photons into the forward hemisphere; aberration crushes that hemisphere into a cone whose half-angle satisfies $\cos\theta = \beta$, which for a fast source is $1/\gamma$. At $\beta = 0.99$ that is eight degrees.

The three together explain the appearance of a great many astronomical sources without any astrophysics at all. A jet pointed nearly at the observer is enormously bright; its counterpart pointed away is enormously faint; and a survey of the sky finds mostly the first kind.

It is worth putting the everyday size of the effect beside the astronomical one. The Earth moves round the Sun at thirty kilometres a second, so a star seen ahead of that motion is brighter than the same star seen behind it by a factor of about $1.0004$ — four parts in ten thousand, from the fourth power of a Doppler factor differing from one by a part in ten thousand. That is measurable and is measured, and it is the same expression that gives a factor of a thousand million for a jet. Nothing switches on; there is one formula and one number in it.

The most precisely measured instance has no jet in it #

The largest and best-measured example of the effect is in the microwave sky, and it is about the observer rather than about anything observed.

A thermal spectrum Doppler-shifted by $\delta$ is a thermal spectrum at a temperature $\delta$ times the original — the shape is preserved exactly, which is a special feature of the Planck curve and not a general one. So an observer moving through the photon gas that fills the universe sees a sky whose temperature varies as $\delta(\theta)$.

At the Solar System’s speed of $369$ kilometres a second the variation is $\pm3.35$ millikelvin about $2.7255$ kelvin, and the measured dipole is $3.362$. That agreement is how the velocity is known.

It is worth being careful about what has been measured. There is no absolute frame and none has been found. What exists is a particular gas of photons, and there is a frame in which that gas is isotropic; the dipole measures the Solar System’s velocity relative to it, in the same way a wind speed is measured relative to the air rather than relative to space.

The dipole is about a hundred times larger than every other feature of the microwave sky, and it is subtracted before anything cosmological is done with the map. It is also the only part of the map that would be different if the measurement were made from somewhere else in the galaxy.

What it does to a survey #

Beaming is a selection effect of a severity that is easy to underestimate, and correcting for it is most of what is done with a catalogue of such objects.

Beaming multiplies the flux by a factor depending only on orientation, which does something awkward to any survey. Selecting sources by brightness selects on orientation as strongly as on distance, so a flux-limited sample of a beamed population is not a sample of the population at all — it is a sample of the ones pointed this way, and the ones pointed elsewhere are missing in a way no amount of observing time repairs.

Suppose jets point in random directions. The fraction pointing within $1/\gamma$ of the line of sight is about $1/2\gamma^2$, which at $\gamma = 10$ is one in two hundred. Those few are brighter by up to $\delta^4 \approx 1.6\times10^5$, so in a flux-limited survey they are visible to a distance four hundred times greater and over a volume $6\times10^7$ times larger, since flux falls as the inverse square.

The two factors nearly cancel in the counts, which is why beamed sources are neither rare nor dominant in such catalogues — and it means a survey’s population is a wild mixture of a few nearby unbeamed objects and a great many distant beamed ones, with no way to tell them apart from a flux alone.

The one-sidedness follows from the same arithmetic. A two-sided jet at a small angle has a flux ratio of $\big((1+\beta\cos\theta)/(1-\beta\cos\theta)\big)^4$ between its two halves, which at any plausible speed is large enough to put the receding side below the noise. So most jets are drawn as one-sided because most jets are seen as one-sided, and the second half is there.

The ceiling this puts on a source, and how it is exceeded #

There is a limit on how bright a self-luminous radio source can be, and beaming is how it is broken.

Brightness temperature is the temperature a thermal emitter would need in order to be that bright at that frequency, and it is a useful number precisely because it is not a temperature. When it comes out at $10^{12}$ K for a source that cannot possibly be that hot, the excess is a measure of about the beaming rather than about the heat — which is how the Doppler factor of a jet is estimated in practice.

A synchrotron source cannot have a brightness temperature above about $10^{12}$ kelvin. Above that the electrons scatter their own radiation to higher energies faster than they emit it, the source cools catastrophically, and the excess is radiated away in a different band. That is the inverse-Compton catastrophe, and it is a hard ceiling on what a source can do in its own frame.

Many compact radio sources are observed with brightness temperatures far above it — sometimes by four orders of magnitude. The resolution is that the observed brightness temperature is $\delta$ times the intrinsic one, so a source at $\delta = 20$ can appear at $2\times10^{13}$ while remaining ordinary at home.

That turns the limit into a measurement. An observed brightness temperature above the ceiling gives a lower bound on the Doppler factor, independent of any timing and independent of any jet geometry, and the bounds obtained that way agree with those from apparent superluminal motion.

Three independent routes to one number — the apparent speed, the brightness temperature, and the ratio between the two sides of a jet — is what makes the beaming picture believed rather than merely available.

The unification the exponent made possible #

An effect that strong turns a catalogue of apparently different objects into one object seen from different angles, and that is what happened to the classification of active galaxies.

What reaches a survey depends on orientation as strongly as on distance, and the two are not separable from the counts alone. That is the honest limitation: a population whose members are beamed can be characterised only if the beaming pattern is assumed, and the assumption is doing as much work as the data.

By the 1970s there were a dozen classes of active galactic nucleus with different names, different spectra and different variability, and no obvious relation between them. The beaming exponent supplied one: several of those classes are the same kind of object seen at different angles to its jet.

A source seen nearly down the jet has its continuum brightened by $\delta^4$, which swamps the emission lines and produces a featureless, violently variable spectrum. The same object seen at thirty degrees shows the lines, because the continuum is no longer beamed into the line of sight. At ninety degrees the central region is hidden behind the surrounding material altogether and only the extended structure is seen.

The classification therefore turned into a geometry, and the test is a counting one: if the classes are one population seen at different angles, their relative numbers must match the solid angles involved. They do, roughly, and where they do not is where the model is currently argued about.

That is a large claim to rest on an exponent, and it is worth noticing what makes it testable. Orientation is not observable directly, so the model is tested by consistency — the same Doppler factor has to explain the brightness, the variability, the apparent speed and the counts at once. A model with one free number per source explaining four measurements per source is a model that can fail, and largely has not.

There is one further number worth having, because it makes the survey argument concrete. The probability that a randomly oriented jet points within one over γ of the line of sight is about $1/2\gamma^2$, so for γ of ten it is one chance in two hundred. Two hundred sources, one of which is a hundred and sixty thousand times brighter than the others: a flux-limited catalogue is then a catalogue of the one, and the hundred and ninety-nine are somewhere below the noise being classified as something else.

The energy that was never there #

The exponent is usually met as an effect that makes things bright. Its most consequential use is the reverse: it removes an energy that a class of objects appeared to have and did not.

Gamma-ray bursts are detected as brief flashes, and the natural way to state a flash’s energy is to assume it was radiated equally in all directions and multiply the measured fluence by the area of a sphere at the source’s distance. That isotropic-equivalent energy reaches $10^{47}$ joules for the brightest bursts.

The rest energy of the Sun is $1.8\times10^{47}$ joules. So the brightest bursts appeared to be radiating, in a few seconds and in gamma rays alone, a substantial fraction of a solar mass converted with perfect efficiency. No mechanism produces that.

The resolution is that the emission is not isotropic. A burst comes from a jet moving at a Lorentz factor of a few hundred, its radiation is beamed into a cone, and an observer inside the cone sees an enormously brighter source than the true energy warrants. The correction is geometric: if the outflow is confined to two cones of half-angle $\theta_j$, the true energy is smaller than the isotropic-equivalent one by the fraction of the sky the cones cover, which for a five-degree jet is about four parts in a thousand.

That takes $10^{47}$ joules down to $4\times10^{44}$ — an ordinary supernova’s kinetic energy, from a mechanism that exists.

What makes this more than an excuse is that the opening angle is measurable, and the measurement is a consequence of the same beaming. As the jet ploughs into the surrounding gas it decelerates, so its Lorentz factor falls and the cone $1/\gamma$ widens. While that cone is narrower than the jet, the observer sees only a patch of the jet’s face and has no way of knowing the jet is not a sphere. Once $1/\gamma$ exceeds the jet’s own opening angle, the edge comes into view, there is no more emitting material to be revealed, and the light curve steepens.

That steepening is the jet break. Its timing gives $\theta_j$ directly, and its signature is that it is achromatic — the same break at every wavelength at the same moment — which distinguishes it from any spectral feature.

Applying the correction to the bursts with measured breaks produced one of the more striking results in the subject: isotropic-equivalent energies spanning three orders of magnitude collapsed, once corrected, into a distribution a factor of a few wide. The bursts are not wildly different explosions; they are similar explosions seen at different angles into differently-shaped jets.

The claim has weakened somewhat since it was first made — later samples show more scatter, and some bursts have no clean break to measure — but the central point stands. An energy budget that made no sense was repaired by a purely kinematic factor, and the factor was then measured independently.

The machine built to exploit it #

The most direct engineering use of the exponent fills a building and is visited by thousands of people a year.

Send electrons round a storage ring at three billion electronvolts and their Lorentz factor is about six thousand. Every time the beam is bent, the electrons radiate — and because they are moving at very nearly the speed of light, that radiation is not emitted over the sphere it would fill in the electrons’ own frame. It is compressed into a cone of half-angle $1/\gamma$, which is under two tenths of a milliradian.

That collimation is the whole point. The same power spread over a full sphere would be useless; concentrated into a pencil a fraction of a milliradian wide it is a beam that can be taken down a pipe to an experiment a hundred metres away and focused onto a crystal a few micrometres across.

The time compression that goes with it is what supplies the wavelength. An observer sees a given electron only while its $1/\gamma$ cone sweeps past, which is a fraction $1/\gamma$ of the time the electron takes to turn through that angle — and there is a further factor of $\gamma$ because the electron is chasing its own light. The pulse an observer receives is therefore shorter than the orbital period by $\gamma^3$, which turns a megahertz circulation into a flash lasting picoseconds and a spectrum extending into hard X-rays. That $\gamma$-cubed compression is the same one that turns three powers of the Doppler factor into four in the flux.

The numbers are worth stating because the gain is not marginal. A laboratory X-ray tube with a rotating anode delivers a brightness — photons per second per unit area per unit solid angle per unit relative bandwidth — of about $10^{10}$. A bending magnet on a storage ring gives $10^{15}$. An undulator, which wiggles the electrons within the $1/\gamma$ cone so that the emission from every period adds in phase, reaches $10^{20}$ or more.

Ten orders of magnitude, and the whole of it is the beaming exponent plus the coherent addition the collimation makes possible. Protein crystallography, X-ray microscopy, high-pressure mineral physics and a great deal of materials science exist in their present form because of it.

Where the model runs out #

The exponent is not always four. Four is right for a discrete blob whose emission is followed as it moves. A continuous, steady jet has one power less — the emitting material is spread along the line of sight and the time compression does not apply — so its flux carries $\delta^3$, and for a spectrum that is a power law of index $\alpha$ the exponents become $3+\alpha$ and $2+\alpha$. Which case applies is a modelling decision, and it changes the inferred Doppler factor substantially.

Every quantity in this essay is a function of one number, and near the top of the speed range a small change in that number is a large change in all of them. That sensitivity is why jet speeds are quoted with wide uncertainties even when the observations are good: the observable depends on the speed so steeply that inverting it amplifies every error.

The source is treated as a point moving at one speed. Real jets have velocity gradients across and along them, so different parts have different Doppler factors, and the observed brightness is an integral over a distribution rather than a single factor. A “the Doppler factor is 12” is always shorthand for a fit.

The invariance assumes empty space between. Absorption, scattering and gravitational redshift all change the intensity along the way, and none of them respects the invariant. For a cosmological source the expansion contributes its own factor, which is why the quantity that is conserved along a ray in an expanding universe is the intensity divided by the fourth power of one plus the redshift — the same bookkeeping that cools a photon gas as it expands.

And nothing here says the source’s own luminosity is unchanged. The transformation is of what an observer receives. How much energy the source is emitting in its own frame is a separate question, and inferring it from a beamed flux requires the Doppler factor — which is exactly the quantity that is hardest to measure and that this whole subject is arranged around estimating.

The ladder from here #

Later rungs on this anchor: the Doppler factor as the single unknown that beaming, variability and apparent superluminal motion all constrain, and how the three are combined; the brightness temperature limit and what exceeding it implies; inverse Compton scattering, which puts a ceiling on how far the limit can be exceeded before the source destroys itself; and the beaming of a spectral line, where the shape as well as the size is transformed.

The neighbouring ladders are the shift that survives at right angles, which is the Doppler factor with no approach in it, the sky that crowds into a cone, which is the aberration that concentrates the light, and the motion that measures faster than light, which measures the same two unknowns by timing rather than by brightness.