Angular Distribution of Radiation from Accelerated Charges

The chapter has assembled its heavy machinery — covariant potentials, the exact field tensor, the null-field theorem — and this article finally points it at the question every radiation problem ends with: in which directions does the energy actually go? The answer is a single master formula,

\frac{dP(t_r)}{d\Omega} = \frac{q^2}{16\pi^2\varepsilon_0 c}\,\frac{\big|\hat{\mathbf{n}}\times[(\hat{\mathbf{n}} - \boldsymbol{\beta})\times\dot{\boldsymbol{\beta}}]\big|^2}{(1 - \hat{\mathbf{n}}\cdot\boldsymbol{\beta})^5},

and one dial. At $\beta \to 0$ the formula is the modest $\sin^2\Theta$ donut every dipole problem shares. Turn the dial toward $\beta \to 1$ and the five inverse powers of $\kappa = 1 - \hat{\mathbf{n}}\cdot\boldsymbol{\beta}$ seize control: the donut tips, stretches, and collapses into a forward searchlight of half-angle $1/\gamma$ , its peak intensity climbing like $\gamma^8$ — with the exact ultrarelativistic constant, derived here, $\tfrac{1}{4}(8/5)^5 = 2.62144$ . The detailed shape depends on one geometric question — is the acceleration parallel or perpendicular to the velocity? — and this article works both lobes end to end: the linear-acceleration pattern with its stubborn on-axis null and its peak at exactly $\theta_{\max} = \arccos[(\sqrt{1+15\beta^2}-1)/3\beta]$ , and the circular-motion pattern with its azimuthal fine structure and its elegant tangent null at $\cos\theta = \beta$ . Integrating either lobe over the sphere reproduces Liénard's $\gamma^6$ and $\gamma^4$ totals — run here as audits of the power results owned upstream. The closing gift is a time-domain observation: the $1/\gamma$ cone sweeping past a fixed observer compresses the received signal by $\gamma^3$ , the single estimate from which the entire synchrotron spectrum will unfold next.

Key takeaways

The master angular distribution $dP(t_r)/d\Omega = \frac{q^2}{16\pi^2\varepsilon_0 c}|\hat{\mathbf{n}}\times[(\hat{\mathbf{n}}-\boldsymbol{\beta})\times\dot{\boldsymbol{\beta}}]|^2/\kappa^5$ assembles from the acceleration field of Fields of a Relativistic Moving Charge and the emitted-power bookkeeping of Larmor Formula from the Liénard–Wiechert Fields — the $\kappa^5$ -vs- $\kappa^6$ distinction is the received/emitted Jacobian doctrine, recalled, not re-derived.
Parallel acceleration ( $\dot{\boldsymbol{\beta}}\parallel\boldsymbol{\beta}$ ): $dP/d\Omega \propto \sin^2\theta/\kappa^5$ — forward-tipped lobes with an exact on-axis null at every speed; peak at $\cos\theta_{\max} = (\sqrt{1+15\beta^2}-1)/3\beta \to \theta_{\max} \approx 1/2\gamma$ .
Perpendicular acceleration (circular motion): $dP/d\Omega \propto \kappa^{-3}[1 - \sin^2\theta\cos^2\phi/(\gamma^2\kappa^2)]$ — a filled forward cone of width $1/\gamma$ with azimuthal dips and an exact tangent null in the orbital plane at $\cos\theta = \beta$ .
Peak intensity grows as $\tfrac{1}{4}(8/5)^5\,\gamma^8 = 2.62144\,\gamma^8$ (parallel case) — eleven orders of magnitude at 10 MeV.
Integrating the lobes recovers Liénard's totals — $\propto\gamma^6$ (parallel) and $\gamma^4$ (perpendicular) — as sphere-integral audits of the power formulas owned by the relativistic Larmor articles.
About 69% of the parallel-case power lands within $\theta < 1/\gamma$ and 97% within $2/\gamma$ : the cone is not a metaphor.
The cone sweeping a fixed observer compresses the received pulse to $\Delta t_{\rm obs} \sim \rho/\gamma^3 c$ — the $\omega_c \sim \gamma^3 c/\rho$ teaser that Synchrotron Radiation will spend.
Key idea: one master formula with five inverse powers of $\kappa$ converts the universal $\sin^2\Theta$ donut into a $1/\gamma$ searchlight — geometry of the lobes for both canonical accelerations, with the totals recovered as audits.

The core insight

The angular distribution of radiation is the acceleration field's Poynting flux times the emitted-time Jacobian — $\propto|\hat{\mathbf{n}}\times[(\hat{\mathbf{n}}-\boldsymbol{\beta})\times\dot{\boldsymbol{\beta}}]|^2/\kappa^5$ — and the five inverse powers of $\kappa$ beam everything into a forward cone of half-angle $\sim1/\gamma$ .

Where this lives today

The $1/\gamma$ cone is the most heavily engineered angle in photon science. Fourth-generation storage rings (MAX IV, ESRF-EBS, APS-U) and free-electron lasers deliver X-ray beams whose vertical opening angle is the perpendicular-geometry lobe of this article — beamline apertures, mirror lengths, and coherence budgets are all designed against $\gamma\theta$ curves, and the orbital-plane polarization null (the tangent-null theorem) is routinely mapped in commissioning as a beam diagnostic. In the collision business the parallel-geometry lobe governs bremsstrahlung converters, positron-production targets, and the radiation-protection shielding of every medical linac: the forward-tipped $\sin^2\theta/\kappa^5$ pattern with its $\theta_{\max} \approx 1/2\gamma$ is tabulated directly into shielding codes. Astrophysics runs the same lobes at cosmic $\gamma$ . Blazar jets pointed within $\sim1/\gamma$ of Earth are Doppler-favored by enormous factors — the reason a small aligned minority of active galaxies dominates the gamma-ray sky — and apparent superluminal motion in VLBI maps is the cone's geometry read as kinematics. Pulsar radio beams, gamma-ray-burst afterglow light curves (the "jet break" is the moment $1/\gamma$ widens past the jet edge), and the polarization structure of jet knots all trace back to the two lobe formulas derived here — the same $\kappa^{-5}$ , wearing lightyears.

Origins · 1 minAlfred-Marie Liénard and how this was discovered

Alfred-Marie Liénard

1869–1958 · École des Mines, Paris — whose 1898 paper contained not just the potentials but the lobes

Liénard appears twice before in this track — for the potentials that bear his name, and for the total-power formula the E9 relativistic Larmor article celebrates. This article honors the third act of the same astonishing 1898 paper in L'Éclairage Électrique: the angular analysis. Liénard did not stop at the potentials or the integrated power; he computed how the radiation from a fast charge distributes itself over direction, identified the forward concentration produced by the $(1 - \hat{\mathbf{n}}\cdot\boldsymbol{\beta})$ denominators, and gave the velocity-dependent corrections to the dipole pattern — relativistic beaming, quantified, seven years before relativity, in a lighting-industry journal, by a mining engineer. The paper sat largely unread for a decade until Schott's systematic treatise (1912) built classical radiation theory on it; another generation later, accelerator builders rediscovered every one of its conclusions the hard way, as beam-loss budgets and mysteriously luminous view ports. His later fame in nonlinear-oscillation theory (the Liénard equation) and his directorship of the École des Mines belong to other stories; here he is the patron of the searchlight — the man who first calculated where the light of a fast electron actually goes.

History

1898

— Liénard's paper computes the angular distribution of radiation from a fast charge, beaming included

1912

— Schott's Electromagnetic Radiation systematizes the lobes and the orbital-motion pattern

1944

— Iwanenko and Pomeranchuk warn that the perpendicular-geometry loss caps betatron energies

1946

— Blewett measures the predicted orbit shrinkage at the GE 100 MeV betatron: the energy loss is real

1947

— the beamed cone becomes visible light at GE's 70 MeV synchrotron — synchrotron radiation is seen

1949

— Schwinger's paper delivers the full angular-spectral machinery for accelerator design

1968

— Tantalus at Wisconsin: the first storage ring run dedicated to using the cone as a light source

1994

— ESRF user operation: the third-generation era of engineering gamma-theta curves begins

The physical picture

Two forces fight over every lobe in this article. The numerator is the dipole heritage: radiation is the transverse projection of the acceleration, so intensity wants to vanish along $\dot{\mathbf{v}}$ and peak broadside to it — the $\sin^2\Theta$ donut, identical for a wobbling antenna electron and a braking beam particle. The denominator is retardation compounded: the fields carry $\kappa^{-3}$ (the beaming ledger of the Liénard–Wiechert articles), the Poynting flux squares that, and the received-to-emitted conversion pays back one power, leaving $\kappa^{-5}$ — five powers of a factor that head-on approaches $1/2\gamma^2$ . When the two fights have different referees — acceleration axis versus velocity axis — the outcome depends on their alignment. Parallel case: the donut's null axis coincides with the beaming axis. The null survives (it is exact: zero transverse acceleration along the line of flight means zero radiation there, at any speed), but the beaming factor grabs the donut's inner rim and drags it forward into two spiky petals hugging the axis at $\theta \approx 1/2\gamma$ — a searchlight with a dark filament down its core. Perpendicular case: the donut's null axis lies across the beaming axis. Beaming now fills the forward direction unopposed — on-axis intensity grows as $8\gamma^6$ — and the donut's null survives only as two pinpricks in the orbital plane, dragged forward to $\cos\theta = \beta$ , right at the cone's edge. The cone is what a synchrotron beamline sees; the pinprick nulls are real enough to measure as polarization zeroes. Both patterns are one relativity lesson: in the charge's instantaneous rest frame the emission is the symmetric donut, always. The lab-frame lobes are that donut viewed through relativistic aberration — every emission angle crowded forward by the headlight map $\cos\theta' \to (\cos\theta' + \beta)/(1 + \beta\cos\theta')$ — a kinematic transformation certified in Relativistic Doppler Effect and Aberration's territory, not a new dynamical effect.

Figure

The lobe gallery computed from the master formula — parallel acceleration at beta 0, 0.5 and 0.9, each normalized to its own peak; the on-axis null survives at every speed while the lobes fold forward

The master formula

The ingredients are all imports; the assembly is this article's opening move. From Fields of a Relativistic Moving Charge: the radiation (acceleration) field, transverse and null,

\mathbf{E}_{\rm rad}(\mathbf{r},t) = \frac{q}{4\pi\varepsilon_0 c}\,\frac{\hat{\mathbf{n}}\times[(\hat{\mathbf{n}} - \boldsymbol{\beta})\times\dot{\boldsymbol{\beta}}]}{\kappa^3\,R}\bigg|_{t_r}, \qquad \mathbf{B}_{\rm rad} = \frac{1}{c}\,\hat{\mathbf{n}}\times\mathbf{E}_{\rm rad}. \qquad (1)

where: $\hat{\mathbf{n}}$ — retarded line of sight; $\boldsymbol{\beta}, \dot{\boldsymbol{\beta}}$ — velocity and acceleration over $c$ , both at $t_r$ ; $\kappa = 1 - \hat{\mathbf{n}}\cdot\boldsymbol{\beta}$ ; $R$ — retarded distance. This is the 3+1 shadow of the covariant $F_{\rm acc} = R\wedge W$ tensor, whose null-field theorem guarantees what the Poynting computation below assumes: $|\mathbf{E}| = c|\mathbf{B}|$ , mutual orthogonality, transversality — exactly, not asymptotically. From Larmor Formula from the Liénard–Wiechert Fields: the emitted-power definition $dP(t_r)/d\Omega \equiv R^2(\mathbf{S}\cdot\hat{\mathbf{n}})\,(dt/dt_r)$ — power billed to the charge's clock, with the Jacobian $dt/dt_r = \kappa$ whose received-versus-emitted doctrine the E9 article Relativistic Larmor Formula (Liénard's Result) owns. Assembling:

\boxed{\;\frac{dP(t_r)}{d\Omega} = \frac{q^2}{16\pi^2\varepsilon_0 c}\,\frac{\big|\hat{\mathbf{n}}\times[(\hat{\mathbf{n}} - \boldsymbol{\beta})\times\dot{\boldsymbol{\beta}}]\big|^2}{(1 - \hat{\mathbf{n}}\cdot\boldsymbol{\beta})^5}\;} \qquad (2)

where: the five is the chapter's most quoted exponent: six powers from $|\mathbf{E}_{\rm rad}|^2 \propto \kappa^{-6}$ , minus one repaid by the emitted-time Jacobian. Quoting the received-power density instead (energy per unit observer time) restores $\kappa^{-6}$ — the two densities answer different questions, and the warning block prosecutes the standard confusion. Derivation A earns (2); Derivations B and C specialize it to the two canonical geometries.

Derivation AAssembling the master formula from the radiation field

The goal is equation (2), assembled honestly from imports with every factor accounted: the radiation field (1) from Fields of a Relativistic Moving Charge, the Poynting machinery from the conservation chapter, and the emitted-time bookkeeping recalled by name from Larmor Formula from the Liénard–Wiechert Fields (whose Derivation A established the definition) — with the underlying $\kappa$ -Jacobian doctrine owned by the E9 article Relativistic Larmor Formula (Liénard's Result) and not re-derived here. The route: flux, square, convert clocks.

Step 1. The flux of a null field. For the radiation field the triad $\{\mathbf{E}, \mathbf{B}, \hat{\mathbf{n}}\}$ is exact (null-field theorem, by name), so the Poynting vector is radial with the plane-wave magnitude:

\mathbf{S} = \frac{1}{\mu_0}\,\mathbf{E}_{\rm rad}\times\mathbf{B}_{\rm rad} = \varepsilon_0 c\,E_{\rm rad}^2\,\hat{\mathbf{n}}.

Why no cross terms appear: at finite distance the full field also contains the velocity piece and interference terms, but both die faster than $1/R^2$ in flux; the radiated power is defined through the large-sphere limit where (1) alone survives — the middle-zone honesty settled by the fields articles.

Step 2. Received power per solid angle. Energy crossing the patch $R^2 d\Omega$ per unit observer time:

\frac{dP_{\rm rec}}{d\Omega} = R^2\,\mathbf{S}\cdot\hat{\mathbf{n}} = \frac{q^2}{16\pi^2\varepsilon_0 c}\,\frac{\big|\hat{\mathbf{n}}\times[(\hat{\mathbf{n}} - \boldsymbol{\beta})\times\dot{\boldsymbol{\beta}}]\big|^2}{\kappa^6},

squaring (1); the $R$ 's cancel — the defining virtue of the $1/R$ field.

Step 3. Convert to the charge's clock. Successive wavefronts reaching the observer were emitted over a longer stretch of source time when the charge approaches: $dt = \kappa\,dt_r$ (the light-cone differentiation the E9 doctrine owns; equivalently the time component of the covariant lemma bank). Power billed per unit emission time therefore carries one factor $\kappa$ :

\frac{dP(t_r)}{d\Omega} = \kappa\,\frac{dP_{\rm rec}}{d\Omega} = \frac{q^2}{16\pi^2\varepsilon_0 c}\,\frac{\big|\hat{\mathbf{n}}\times[(\hat{\mathbf{n}} - \boldsymbol{\beta})\times\dot{\boldsymbol{\beta}}]\big|^2}{\kappa^5}.

This — equation (2) — is the quantity whose sphere integral equals the charge's energy-loss rate, which is why it, and not the $\kappa^{-6}$ density, feeds the Liénard totals in the audits below.

Step 4. The universal slow limit. Set $\boldsymbol{\beta} \to 0$ : $\kappa \to 1$ and BAC-CAB collapses the numerator to $|\hat{\mathbf{n}}\times(\hat{\mathbf{n}}\times\dot{\boldsymbol{\beta}})|^2 = \dot{\beta}^2\sin^2\Theta$ with $\Theta$ measured from the acceleration:

\frac{dP}{d\Omega} = \frac{q^2 a^2}{16\pi^2\varepsilon_0 c^3}\,\sin^2\Theta,

the general-orientation donut owned by Larmor Formula from the Liénard–Wiechert Fields — recovered as the dial's zero point, with its $8\pi/3$ integral and the Larmor total recalled, not recomputed. Verified: matches Jackson Eq. 14.38 and Griffiths §11.2.1; dimensions $[q^2a^2/\varepsilon_0c^3] =$ W ✓; the $\kappa^{-5}$ -vs- $\kappa^{-6}$ split matches the E9 doctrine article exactly.

Parallel acceleration — the searchlight with a dark core

Go deeper · Derivation B · 1 minLinear acceleration lobes from null to gamma-eighth peak

Derivation BLinear acceleration lobes from null to gamma-eighth peak

The goal is the complete anatomy of the $\dot{\boldsymbol{\beta}} \parallel \boldsymbol{\beta}$ pattern: the lobe function, the exact peak angle and its ultrarelativistic limit, the peak-intensity scaling with its exact constant, and the sphere-integral audit against Liénard's $\gamma^6$ total. This is the geometry of bremsstrahlung targets and linac exits. The strategy: collinearity kills the $\boldsymbol{\beta}$ -dependence of the numerator, leaving all velocity dependence in $\kappa^5$ — one function of one angle, to be interrogated by calculus.

Step 1. Collapse the numerator. With $\dot{\boldsymbol{\beta}} \parallel \boldsymbol{\beta}$ , the term $\boldsymbol{\beta}\times\dot{\boldsymbol{\beta}} = 0$ , so $(\hat{\mathbf{n}} - \boldsymbol{\beta})\times\dot{\boldsymbol{\beta}} = \hat{\mathbf{n}}\times\dot{\boldsymbol{\beta}}$ and the double cross has magnitude $\dot{\beta}\sin\theta$ , with $\theta$ from the common axis:

\frac{dP}{d\Omega}\bigg|_{\parallel} = \frac{q^2 a^2}{16\pi^2\varepsilon_0 c^3}\,\frac{\sin^2\theta}{(1 - \beta\cos\theta)^5}.

The on-axis null is exact at every $\beta$ : the numerator is the transverse acceleration seen from $\hat{\mathbf{n}}$ , and along the axis there is none. Beaming multiplies; it cannot create.

Step 2. Locate the peak. With $x = \cos\theta$ , extremize $f(x) = (1 - x^2)/(1 - \beta x)^5$ : the derivative's numerator is $-2x(1-\beta x) + 5\beta(1 - x^2) = -3\beta x^2 - 2x + 5\beta$ , giving the quadratic $3\beta x^2 + 2x - 5\beta = 0$ and the physical root

\cos\theta_{\max} = \frac{\sqrt{1 + 15\beta^2} - 1}{3\beta} \qquad\xrightarrow{\;\gamma \gg 1\;}\qquad \theta_{\max} \approx \frac{1}{2\gamma}.

Why the half: expanding $\beta \approx 1 - 1/2\gamma^2$ and $\cos\theta \approx 1 - \theta^2/2$ in the quadratic gives $\theta_{\max}^2 = 1/4\gamma^2$ at leading order — the peak sits at half the canonical cone width, inside the $1/\gamma$ cone, hugging the dark filament.

Step 3. Price the peak. At $\theta_{\max}$ , using the small-angle forms $\sin^2\theta_{\max} \approx 1/4\gamma^2$ and $\kappa(\theta_{\max}) \approx \tfrac{1}{2}(\gamma^{-2} + \theta_{\max}^2) = \tfrac{5}{8}\gamma^{-2}$ :

\frac{dP}{d\Omega}\bigg|_{\max} \approx \frac{q^2a^2}{16\pi^2\varepsilon_0 c^3}\cdot\frac{1/4\gamma^2}{(5/8)^5\gamma^{-10}} = \frac{q^2a^2}{16\pi^2\varepsilon_0 c^3}\cdot\frac{1}{4}\left(\frac{8}{5}\right)^5\gamma^8 = \frac{q^2a^2}{16\pi^2\varepsilon_0 c^3}\times 2.62144\,\gamma^8,

the exact constant being $8192/3125$ ; the numerical check at $\gamma = 50, 500, 5000$ converges to $2.62144$ from below. Eight powers of $\gamma$ : two lost to the shrinking $\sin^2$ , ten gained from $\kappa^{-5}$ .

Step 4. The sphere-integral audit. Integrating the lobe exactly (substitute $x = \cos\theta$ , two elementary integrals):

\int \frac{\sin^2\theta}{(1-\beta\cos\theta)^5}\,d\Omega = \frac{8\pi}{3}\,\gamma^6 \qquad\Longrightarrow\qquad P_{\parallel} = \frac{q^2 a^2\gamma^6}{6\pi\varepsilon_0 c^3},

which is Liénard's total with $\boldsymbol{\beta}\times\dot{\boldsymbol{\beta}} = 0$ — the result owned by the relativistic Larmor articles (E9's dual routes; the covariant layer next door) and recovered here as a solid-angle audit, sympy-verified symbolically in $\beta$ . Verified: quadratic root against Jackson Eq. 14.39; $\theta_{\max}(\beta = 0.99881) = 24.328$ mrad vs $1/2\gamma = 24.308$ mrad (+0.08%); the $(8\pi/3)\gamma^6$ integral checked symbolically; peak constant $8192/3125$ exact.

Perpendicular acceleration — the filled cone with tangent nulls

Go deeper · Derivation C · 1 minCircular motion lobes and the tangent null theorem

Derivation CCircular motion lobes and the tangent null theorem

The goal is the full $\dot{\boldsymbol{\beta}} \perp \boldsymbol{\beta}$ distribution with its azimuthal structure, the exact location of its nulls, and the $\gamma^4$ integral audit. This is the geometry of every storage ring and betatron — the pattern a synchrotron beamline actually samples. Set $\boldsymbol{\beta} = \beta\hat{\mathbf{z}}$ , $\dot{\boldsymbol{\beta}} = \dot{\beta}\hat{\mathbf{x}}$ (instantaneously), $\hat{\mathbf{n}} = (\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta)$ ; the plan is one vector identity, then geometry.

Step 1. The numerator identity. Expanding the double cross product for this geometry (sympy-verified as an identity in $\theta, \phi, \beta$ ):

\big|\hat{\mathbf{n}}\times[(\hat{\mathbf{n}} - \boldsymbol{\beta})\times\dot{\boldsymbol{\beta}}]\big|^2 = \dot{\beta}^2\left[(1 - \beta\cos\theta)^2 - (1 - \beta^2)\sin^2\theta\cos^2\phi\right].

Step 2. The distribution. Dividing by $\kappa^5$ and writing $1 - \beta^2 = 1/\gamma^2$ :

\frac{dP}{d\Omega}\bigg|_{\perp} = \frac{q^2 a^2}{16\pi^2\varepsilon_0 c^3}\,\frac{1}{(1 - \beta\cos\theta)^3}\left[1 - \frac{\sin^2\theta\cos^2\phi}{\gamma^2(1 - \beta\cos\theta)^2}\right].

Only $\kappa^{-3}$ multiplies the leading term — yet on axis ( $\theta = 0$ ) this still delivers $\kappa^{-3} = (1-\beta)^{-3} \approx 8\gamma^6$ : the cone is filled, no dark core, because the acceleration is broadside to the line of flight and its transverse projection survives on axis.

Step 3. The tangent null theorem. The bracket vanishes where $\gamma\kappa = \sin\theta\,|\cos\phi|$ — most restrictively in the orbital plane ( $\phi = 0$ ): $\sin\theta = \gamma(1 - \beta\cos\theta)$ , i.e. $\sin\theta + \gamma\beta\cos\theta = \gamma$ . The left side has amplitude $\sqrt{1 + \gamma^2\beta^2} = \gamma$ exactly — so the line just touches the curve: one tangent null on each side of forward, at

\cos\theta_{\rm null} = \beta \qquad (\phi = 0).

Why tangency is the elegant part: for any $|\cos\phi| < 1$ the amplitude falls short and the bracket never reaches zero — the nulls exist only in the orbital plane, as two isolated points where the polarization flips, not as null circles. Ultrarelativistically $\theta_{\rm null} = \arccos\beta \approx 1/\gamma$ : the pinpricks sit exactly at the canonical cone edge.

Step 4. The sphere-integral audit. Integrating Step 2 over the full sphere (sympy, symbolically in $\beta$ ):

\int \frac{dP}{d\Omega}\bigg|_{\perp}\,\frac{d\Omega}{q^2a^2/16\pi^2\varepsilon_0c^3} = \frac{8\pi}{3}\,\gamma^4 \qquad\Longrightarrow\qquad P_{\perp} = \frac{q^2 a^2\gamma^4}{6\pi\varepsilon_0 c^3},

Liénard's perpendicular total — two powers of $\gamma$ short of the parallel case at equal acceleration, yet the winner by $\gamma^2$ at equal force, the held-fixed doctrine that the E9 relativistic Larmor article adjudicated and this audit leaves untouched. Verified: matches Jackson Eq. 14.44; null angle $\arccos(0.9) = 25.84°$ at $\beta = 0.9$ confirmed by direct root-finding on the bracket; $(8\pi/3)\gamma^4$ integral symbolic; on-axis boost $(1-\beta)^{-3} = 3.34\times10^{10}$ vs $8\gamma^6 = 3.35\times10^{10}$ at $\gamma = 40.1$ — the ultrarelativistic shortcut good to half a percent.

Figure

The circular-motion pattern in the orbital plane at beta 0.9, computed from the exact formula — a filled forward cone with two tangent nulls exactly at cos theta = beta; radial scale cube-root compressed for legibility

The time-domain dividend — pulse compression

The lobes are a snapshot; a fixed observer watching a circulating charge experiences them as time structure, and the conversion mints the chapter's next subject. The $1/\gamma$ cone, rigidly attached to the velocity, sweeps past the observer as the charge rounds its orbit of radius $\rho$ : illumination lasts while the velocity points within $\sim 1/\gamma$ of the line of sight, an orbital arc $\Delta t_r \sim \rho/\gamma v$ . But the received duration is compressed by the emitted-to-received Jacobian $\kappa \sim 1/2\gamma^2$ hovering near its head-on minimum throughout the flash:

\Delta t_{\rm obs} \sim \kappa\,\Delta t_r \sim \frac{\rho}{\gamma^3 c}. \qquad (3)

where: three powers of $\gamma$ — one from the sweep, two from the Doppler-like compression. A signal confined to pulses of width $\Delta t_{\rm obs}$ has spectral content up to $\omega_c \sim 1/\Delta t_{\rm obs} \sim \gamma^3 c/\rho$ : a particle orbiting at a radio-frequency $c/\rho$ radiates X-rays. That single estimate — the $\gamma^3$ compression — is the seed from which Synchrotron Radiation grows the full critical-frequency spectrum, and it is deferred there by name; this article's worked example 2 puts first numbers on it.

Limiting cases

case	result	meaning
$\beta \to 0$ , any geometry	$dP/d\Omega \propto \sin^2\Theta$ , $\Theta$ from $\dot{\mathbf{v}}$	the universal donut; Larmor total (owned upstream)
parallel, $\gamma \gg 1$	peak $\theta_{\max} \to 1/2\gamma$ , height $2.62\,\gamma^8$ , exact on-axis null	bremsstrahlung/linac lobe: searchlight with dark filament
perpendicular, $\gamma \gg 1$	filled cone $\sim 1/\gamma$ , on-axis $8\gamma^6$ , tangent nulls at $\arccos\beta \approx 1/\gamma$	the synchrotron searchlight
small forward angles	$\kappa \approx \tfrac{1}{2}(\gamma^{-2} + \theta^2)$	the workhorse expansion: everything interesting lives at $\gamma\theta \sim 1$
sphere integrals	$(8\pi/3)\gamma^6$ (parallel), $(8\pi/3)\gamma^4$ (perpendicular)	Liénard's totals recovered as audits

Common trapReceived power and emitted power differ by one power of kappa — and the on-axis behavior of the two geometries is opposite

Trap 1 — the two power densities. $dP_{\rm rec}/d\Omega$ (energy per observer second) carries $\kappa^{-6}$ ; $dP(t_r)/d\Omega$ (energy per charge second, equation (2)) carries $\kappa^{-5}$ . Only the second integrates to the charge's energy-loss rate and to Liénard's totals; integrate the first and the parallel total comes out $\propto\gamma^8$ — wrong by $\gamma^2$ , a classic exam and simulation error. Choose the clock before integrating, and bill consistently (the doctrine is the E9 relativistic Larmor article's; here it is applied, twice, in the audits). Trap 2 — the nulls. Parallel acceleration keeps an exact zero on axis at every speed — the beaming factor is large there but multiplies zero — while perpendicular acceleration is maximal on axis. Sketch the wrong geometry and every qualitative conclusion inverts: a bremsstrahlung target has a dark filament down the beam that a synchrotron bend does not. (At $\gamma \gtrsim 100$ the parallel null filament is narrower than any real detector's angular resolution — worked example 1 — which is why measured bremsstrahlung lobes look filled; the null is real, just unresolvably thin.)

Worked examples

1 · The 10 MeV linac electron (confidence builder)

Setup. An electron leaves a 10 MeV linac section under parallel acceleration ( $E_{\rm acc} = 30$ MV/m). Map its radiation lobe: peak angle, sharpness of the $1/2\gamma$ rule, cone containment, and peak enhancement.

Step 1. Kinematics. $\gamma = 1 + K/m_ec^2 = 1 + 10/0.511 = 20.57$ , $\beta = 0.998818$ .

Step 2. The peak. Exact: $\cos\theta_{\max} = (\sqrt{1 + 15\beta^2} - 1)/3\beta$ gives $\theta_{\max} = \mathbf{24.33}$ mrad $= 1.394°$ . The asymptotic rule $1/2\gamma = 24.31$ mrad is high by only $0.08\%$ — at 10 MeV the ultrarelativistic formula is already exact to three digits.

Step 3. Containment. Numerical integration of the lobe: $\mathbf{68.7\%}$ of the power lands within $\theta < 1/\gamma = 48.6$ mrad, and $\mathbf{97.3\%}$ within $2/\gamma$ — the "cone" is a quantitative container, not a cartoon. The on-axis null survives (exactly zero at $\theta = 0$ ), but at $0.1$ mrad the intensity is already $4\times10^6$ (donut units): the dark filament is a fraction of a milliradian wide, far below beam-divergence scales — real, and invisible to any practical detector.

Step 4. Peak enhancement and total. Peak height $= 2.62\,\gamma^8 = \mathbf{8.4\times10^{10}}$ times the nonrelativistic broadside maximum at equal acceleration. The total power, however, obeys the force form $P = q^2F^2/6\pi\varepsilon_0 m^2c^3$ — evaluating at 30 MV/m: $P = \mathbf{1.6\times10^{-16}}$ W, utterly negligible against the beam power: the linac loophole (energy-independent at fixed force, and tiny) adjudicated by the E9 relativistic Larmor article, whose verdict this lobe anatomy decorates but does not change. Fierce shape, feeble total.

The lesson: beaming reshapes where linac radiation goes ( $\gamma^8$ at the peak!) while the force-form total stays microscopic — angular structure and energy budget are independent dials.

2 · Kerst's betatron, the pattern that announced synchrotron light (exam level)

Setup. A 20 MeV electron in a betatron with $B = 0.4$ T — Kerst-era numbers. Map the perpendicular-geometry observables and the pulse compression.

Step 1. Kinematics and orbit. $\gamma = 1 + 20/0.511 = 40.1$ , $\beta = 0.99969$ ; orbit radius $\rho = \gamma m_e\beta c/eB = \mathbf{0.171}$ m; centripetal acceleration $a = \beta^2c^2/\rho = 5.25\times10^{17}$ m/s².

Step 2. The cone and its nulls. Cone half-angle $1/\gamma = \mathbf{24.9}$ mrad $= 1.43°$ — the tangential searchlight. Orbital-plane tangent nulls at $\arccos\beta = 24.92$ mrad: at this $\gamma$ the nulls sit within $0.01\%$ of the cone edge. Forward/backward contrast $[(1+\beta)/(1-\beta)]^3 = \mathbf{2.7\times10^{11}}$ ; on-axis boost $(1-\beta)^{-3} = 3.34\times10^{10} \approx 8\gamma^6$ .

Step 3. The energy bill. Liénard's perpendicular total: $P = q^2a^2\gamma^4/6\pi\varepsilon_0c^3 = \mathbf{4.1\times10^{-12}}$ W per electron — per turn ( $2\pi\rho/\beta c = 3.6$ ns), an energy loss of $\mathbf{0.091}$ eV. Negligible at 20 MeV; but the $\gamma^4$ scaling means $\times256$ at 80 MeV and megaelectronvolts per turn at GeV energies — the wall Iwanenko and Pomeranchuk computed in 1944, before anyone had seen the light.

Step 4. The pulse and its spectrum. Compression (3): $\Delta t_{\rm obs} \sim \rho/\gamma^3c = \mathbf{8.8}$ fs, so $\omega_c \sim \gamma^3 c/\rho = 1.1\times10^{14}$ rad/s — near-infrared, from an orbit frequency of $280$ MHz: eight orders of magnitude of upconversion from pure kinematics. Double $\gamma$ and the flash crosses into the visible — which is precisely how, in 1947, a technician looking through a viewing port at General Electric's 70 MeV synchrotron saw a bluish-white spot of light riding the beam tube: this article's cone, caught by a human eye.

The lesson: every number a light source quotes — cone, nulls, per-turn loss, spectral reach — is this article's geometry with $\gamma$ turned up.

Going further

References. Jackson, Classical Electrodynamics, 3rd ed., §14.2–14.4 · Griffiths, Introduction to Electrodynamics, 4th ed., §11.2 · Zangwill, Modern Electrodynamics, §23.3–23.4 · Schott, Electromagnetic Radiation (1912) · Hofmann, The Physics of Synchrotron Radiation, Ch. 3

Provenance. A verifier should confirm: (1) the radiation field expression against Jackson Eq. 14.14 / Griffiths Eq. 11.72; (2) the $\sin^2\Theta$ pattern and Larmor integral $\int\sin^2\Theta\,\mathrm d\Omega=8\pi/3$ ; (3) the parallel and perpendicular angular distributions against Jackson Eqs. 14.38-14.44; (4) the peak angle $\theta_{\max}\to1/(2\gamma)$ ; (5) the Liénard total power $P=q^2\gamma^6[\dot\beta^2-(\boldsymbol\beta\times\dot{\boldsymbol\beta})^2]c^2/(6\pi\varepsilon_0 c^3)$ and the $\gamma^4$ vs $\gamma^6$ scalings. Sources: Griffiths 4th ed. §11.2; Jackson 3rd ed. §14.2-14.4; Zangwill §23.3-23.4; Purcell & Morin Ch. 5. Addendum for this edition: per the chapter's division of labor, item (2)'s derivation and total are owned by the E9 Larmor Formula with the general-orientation statement at Larmor Formula from the Liénard–Wiechert Fields, and item (5)'s totals are owned by the relativistic Larmor pair (E9 dual routes; covariant layer in Relativistic Generalization of Larmor's Formula) — both recalled by name, with the sphere integrals run here strictly as audits (symbolically verified: $(8\pi/3)\gamma^6$ parallel, $(8\pi/3)\gamma^4$ perpendicular). The radiation field (1) is quoted from Fields of a Relativistic Moving Charge; the $\kappa^5$ -vs- $\kappa^6$ Jacobian doctrine from the E9 Relativistic Larmor Formula (Liénard's Result). This edition owns: the master-formula assembly, both lobe anatomies end-to-end (exact $\theta_{\max}$ quadratic; peak constant $8192/3125\cdot\gamma^8$ ; the perpendicular numerator identity and the tangent-null theorem $\cos\theta_{\rm null} = \beta$ with its amplitude-tangency proof), the containment fractions (68.7% within $1/\gamma$ , 97.3% within $2/\gamma$ ), and the $\gamma^3$ pulse-compression estimate deferred by name to Synchrotron Radiation. Worked numbers (10 MeV: $\theta_{\max} = 24.328$ mrad vs $1/2\gamma = 24.308$ mrad; $P = 1.6\times10^{-16}$ W at 30 MV/m · betatron: $\rho = 0.171$ m, $a = 5.25\times10^{17}$ m/s², $P = 4.1\times10^{-12}$ W, 0.091 eV/turn, 8.8 fs, $\omega_c = 1.1\times10^{14}$ rad/s) all recomputed for this edition; lobe SVGs traced from computed polar data.

Builds on: Liénard–Wiechert Potentials: Covariant Formulation · Fields of a Relativistic Moving Charge · Larmor Formula from the Liénard–Wiechert Fields · Electric Dipole Radiation: Exact Fields and Zone Structure Leads to: Relativistic Generalization of Larmor's Formula · Synchrotron Radiation · Bremsstrahlung · Antenna Arrays and Directivity

Does the exact on-axis null of linear acceleration mean an on-axis detector behind a bremsstrahlung target sees nothing?

No — for three stacked reasons. First, the null is a set of measure zero: at 10 MeV the intensity recovers to millions of donut-units within a tenth of a milliradian, far inside any collimator acceptance, so a finite detector integrates plenty of flux "on axis." Second, real beams have angular spread larger than the filament: each electron's null points along its velocity, and the ensemble's nulls smear into a shallow dip at best. Third, in thick targets electrons multiple-scatter between emissions, repointing the lobe randomly. The null is physically real — it is exact for each emission event and survives in thin-target, low-divergence experiments as a measurable on-axis dip — but "the forward direction is dark" is a single-particle, single-instant statement, not a beamline prediction.

Why does the perpendicular case get only kappa to the minus three in its leading term when the master formula has kappa to the minus five?

The other two powers are hiding in the numerator's geometry. For perpendicular acceleration the double cross product evaluated on axis gives $|\ldots|^2 = \dot{\beta}^2\kappa^2$ at $\theta = 0$ (the identity of Derivation C: the bracket's leading term is $\kappa^2$ ), and those two numerator powers cancel two denominator powers, leaving $\kappa^{-3}$ visible. For parallel acceleration the numerator is $\sin^2\theta$ — no $\kappa$ 's to donate — so the full $\kappa^{-5}$ stands, but multiplied by a numerator that dies on axis. Same master formula, different bookkeeping split between numerator and denominator: comparing " $\kappa^{-3}$ vs $\kappa^{-5}$ " across the two geometries without tracking the numerators is a category error, and the honest comparison is on-axis intensities ( $8\gamma^6$ filled vs exact zero) or totals ( $\gamma^4$ vs $\gamma^6$ at fixed $a$ ).

Are the tangent nulls of the circular pattern actually observable, or a mathematical curiosity?

Observable, and used. In the orbital plane at $\cos\theta = \beta$ the total intensity dips to zero because the two polarization components conspire: the in-plane (σ) component vanishes there while the out-of-plane (π) component is already zero in the plane — and just off-plane the π component dominates. Synchrotron beamlines see this as the well-known polarization structure of bending-magnet radiation: purely linear σ polarization in the orbital plane, growing circular/π admixture above and below, with the σ intensity passing through its zero near $\gamma\psi \sim 1$ (vertical angle $\psi$ ). Commissioning teams map these polarization zeroes to locate the orbital plane to microradians, and polarization-sensitive experiments (magnetic dichroism) sit at calculated vertical offsets to harvest the π component. The pinprick is a survey instrument.

How can the received and emitted angular distributions both be 'the' angular distribution?

They answer different budget questions, and each is exclusively correct for its question. The received density ( $\kappa^{-6}$ ) answers: what does a detector at direction $\hat{\mathbf{n}}$ measure per second of its clock — the right object for instrument design, exposure times, damage estimates. The emitted density ( $\kappa^{-5}$ ) answers: how fast does the charge lose energy into direction $\hat{\mathbf{n}}$ per second of its history — the right object for dynamics: its sphere integral is $-dE/dt_r$ of the particle, which is why the Liénard totals and every radiation-reaction argument bill in emitted time. The two differ by the flash-compression Jacobian: an approaching charge crams a long emission history into a short reception window. Confusing them is harmless at $\beta \ll 1$ ( $\kappa \approx 1$ ) and catastrophic at $\gamma \gg 1$ , where the difference is the $\gamma^2$ that separates a right accelerator design from a wrong one.

Is relativistic beaming a property of the charge's radiation or of the observer's motion?

Of the relative kinematics — and the rest-frame view is the cleanest way to hold it. In the charge's instantaneous rest frame the emission pattern is the plain donut about the rest-frame acceleration: no cone, no $\gamma$ 's. The lab-frame lobes are that donut processed by relativistic aberration — the headlight map crowds all emission directions toward the velocity — plus the Doppler re-weighting of energy per solid angle. Nothing new is emitted forward; the forward sky simply collects almost all of the rest frame's $4\pi$ . This factorization (intrinsic donut × kinematic map) is also the practical computation scheme in astrophysics: model the comoving emission isotropically, then boost — and it explains instantly why beaming is achromatic, polarization-preserving along rays, and universal across emission mechanisms: the map does not care what made the light.

The pulse-compression argument gives gamma cubed, but the beaming factor is only gamma squared. Where does the third power come from?

From the sweep, which is a different clock than the compression. Break the flash into its two stages. Stage one: how long does the charge spend illuminating the observer? The cone has half-angle $1/\gamma$ and the velocity direction rotates at $v/\rho$ , so the illumination lasts $\Delta t_r \sim \rho/\gamma v$ of retarded time — one power of $\gamma$ , purely geometric. Stage two: that emission interval reaches the observer compressed by the head-on Jacobian $dt = \kappa\,dt_r$ with $\kappa \sim 1/2\gamma^2$ — two more powers. Product: $\Delta t_{\rm obs} \sim \rho/\gamma^3c$ . The corresponding spectral reach $\omega_c \sim \gamma^3c/\rho$ is why a 3 GeV ring with meter-scale bends emits keV X-rays from a MHz orbit — and the precise coefficient, spectrum shape, and photon statistics are exactly what Synchrotron Radiation exists to compute.