AST-12

Synchrotron critical frequency

ν_c = (3/2) γ² ν_L, ν_L = e B / (2π m). Peak of the synchrotron spectrum.

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RadiationSynchrotron

Governing equation

\nu_c=\tfrac32\gamma^2 eB/(2\pi m)

$\gamma \in \left[10, 1\times 10^{7}\right]$

where

\gamma: Lorentz factor (—)
B: Field (µT)
\nu_c: Critical frequency (MHz)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\nu_c=\tfrac32\gamma^2 eB/(2\pi m)

$\gamma \in \left[10, 1\times 10^{7}\right]$

$\gamma = 10$ $\nu_c = 4.199$

Along the curve

②

Domain of the variable — blue

\gamma_{\min} = 10,\quad \gamma_{\max} = 1\times 10^{7}

$\gamma \in \left[10, 1\times 10^{7}\right]$

$\gamma = 10$ $\gamma = 10$

Along the curve

③

Response curve — green

\nu_c=\tfrac32\gamma^2 eB/(2\pi m)

$\gamma \in \left[10, 1\times 10^{7}\right]$

$\gamma = 10$ $\nu_c = 4.199$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\nu_c}{\mathrm{d}\gamma}

$\gamma \in \left[10, 1\times 10^{7}\right]$

$\gamma = 62903$ $\nu_c^{\prime} = 5282.4$

Along the curve

Lecture brief

Historical brief

Hubble expansion, Jeans collapse, Eddington luminosity, Bondi accretion and Stefan–Boltzmann stars are the first astrophysical budgets. The sheets scale a star, a cloud and a horizon. This sheet (AST-12 — Synchrotron critical frequency) is the form associated with Synchrotron. Working symbols: $\gamma$ , $B$ $\rightarrow$ $\nu_c$ . Power ∝ γ² B². Radio galaxies and supernova remnants shine this way.

Purpose

Purpose: compute

\nu_c

from

\gamma

B

in Astrophysics via

\nu_c=\tfrac32\gamma^2 eB/(2\pi m)

ν_c = (3/2) γ² ν_L, ν_L = e B / (2π m). Peak of the synchrotron spectrum. Use it when a real astrophysics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\gamma = 10000.000\,\mathrm{—}

B = 1.000\,\mathrm{\mu T}

, the governing relation

\nu_c=\tfrac32\gamma^2 eB/(2\pi m)

yields

\nu_c = 4.20e+6\,\mathrm{MHz}

. A relativistic electron, a B field, a beamed glow. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Lorentz factor \gamma10000 —Field B1 µT

Outputs

Critical frequency \nu_c4198873.48 MHz

Reading speed

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Narration of this film

A relativistic electron, a B field, a beamed glow.

Power ∝ γ² B². Radio galaxies and supernova remnants shine this way.

Reading speed

Watch on YouTube