AC06 · A pulse crosses the plasma

Interstellar Radio Propagation

Why does one cosmic radio flash arrive at different times and with a rotated polarization? Free electrons delay lower radio frequencies more strongly. A magnetic field along the path also rotates linear polarization. Unequal paths through density fluctuations spread the pulse into a tail, so its timing and polarization reveal different parts of the intervening plasma.

Drag an electron screen and reverse its magnetic field. Watch a pulse disperse, grow a scattering tail and rotate its polarization, then recover the rotation from recorded detector samples.

400 to 800 MHz delay $389\,\mathrm{ms}$

Fitted rotation measure $32.5\,\mathrm{rad\,m^{-2}}$

Recovered linear polarization $80.0\,\%$

Prescribed tail at 600 MHz $12.0\,\mathrm{ms}$

Send a pulse through space

Electron column

20

\mathrm{DM}\;[\mathrm{pc\,cm^{-3}}]

Field toward observer

2

B_\parallel\;[\mu\mathrm G]

Scattering tail at 600 MHz

12

\tau_{600}\;[\mathrm{ms}]

Drag the cloud up and down; drag the gold field sideways. Each change acquires a new synthetic detector record.

Channel response and model assumptions

Frequency channel width

0.390625

\Delta\nu\;[\mathrm{MHz}]

A one-kiloparsec uniform screen sets the electron density from the column. The field points toward the observer when positive. Scattering strength is a separate screen parameter. The simulation averages I, Q and U over rectangular frequency channels before adding independent Gaussian noise.

Read the propagation course and measured-data tutorial

Model assumptions

The detector records synthetic Stokes intensity and linear polarization in an observer-frame cold-plasma screen model. A prescribed causal exponential tail represents an isotropic thin scattering screen, with a chosen inverse-fourth-power frequency scaling. Electron column and signed field set dispersion and Faraday rotation independently of that scattering prescription. Gaussian detector noise and rectangular frequency-channel averaging are included. Plasma cutoff, collisions, intrinsic spectral evolution, instrumental calibration, absorption and cosmological propagation are omitted. The scene compresses travel distance and enlarges differential delays. The companion CHIME tutorial uses measured intensity only.

Receiver brightness and polarization marks use the same saved synthetic I/Q/U record as these plots.

A synthetic pulse at the receiver

Align the frequency channels

Recorded linear polarization

Find the rotation measure

Gray profiles precede correction and mint profiles use the selected DM; Q is blue, U pink, and the response-aware RM fit gold, with secondary peaks from sampling aliases.

Readout and fit scope

The scene compresses distance and enlarges differential delay; blue, violet and pink mark high, middle and low frequencies. Receiver brightness uses saved intensity and its direction marks use integrated Q/U; all panels share one frozen synthetic record.

Waterfall brightness uses recorded intensity above zero with a fixed asinh stretch and a blue-white display palette. Replaying the scene preserves the acquired noise samples.

The delay and 600-MHz tail are prescribed model quantities. RM and linear fraction are recovered from time-integrated recorded Q/U, using equal Gaussian channel noise, one Faraday-thin component, and a common intrinsic polarization spectrum. The search spans minus 330 to plus 330 radians per square meter in steps of a quarter. Integration includes the saved window; its exponentially small missing tail is not rescaled. The finite-step scattering kernel gives sub-sample timing uncertainty. The display sample interval adapts to the pulse window.

Start with the essentials

Focus question: Which changes when the field reverses: arrival time, polarization or both?
One-sentence intuition: Dispersion measures electron column; signed Faraday rotation measures the field weighted by that column.

Core mathematical model

Relative group delay

\Delta t=4.149\,\mathrm{ms}\,\frac{\mathrm{DM}}{\mathrm{pc\,cm^{-3}}}\left[\left(\frac{\nu}{\mathrm{GHz}}\right)^{-2}-\left(\frac{\nu_{\rm ref}}{\mathrm{GHz}}\right)^{-2}\right]

Lower frequencies arrive later. The coefficient is rounded, and frequencies are observed frequencies.

Faraday rotation

P=Q+iU=L e^{2i\chi},\qquad \chi=\chi_0+\mathrm{RM}\lambda^2,\qquad \mathrm{RM}=0.812\int n_e B_\parallel\,dl

Electron density is in inverse cubic centimeters, field in microgauss and path length in parsecs. Positive field points toward the observer; RM is in radians per square meter.

A causal scattering tail

h(t)=\frac{e^{-t/\tau}}{\tau}H(t),\qquad I_{\rm out}(t)=\int I_{\rm in}(t-u)h(u)\,du

The normalized response redistributes fluence toward later times. Its shape assumes an isotropic thin screen.

Common difficulties

A polarization angle repeats

Typical misconceptionA straight line through the displayed angles always gives the rotation measure.

Better mental modelAn angle repeats after a half-turn. Fitting Q and U together avoids choosing an angle unwrap, though finite frequency sampling still creates competing RM peaks.

Run the experiment

01
Change one sign
Predict the result, then choose Reverse the field.
What to observe: The rotation-measure peak changes sign. Dispersion follows the same electron column.
02
Spread the arrival paths
Choose A long scattering tail and compare the dispersed and dedispersed profiles.
What to observe: Removing the frequency delay aligns the leading pulse, but a tail remains within each channel.
03
Average a rotating vector
Choose Coarse channels lose polarization. Reduce only Frequency channel width.
What to observe: Opposing polarization vectors cancel within wide channels. Narrowing them restores the observed Q and U oscillations.

Interstellar Radio Propagation

Three changes to one pulse

Core mathematical model

Relative group delay

Faraday rotation

A causal scattering tail

Common difficulties

A polarization angle repeats

Run the experiment

Change one sign

Spread the arrival paths

Average a rotating vector

Related experiments and articles

Measuring the sky

Radio pulses through interstellar space