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Sandbox Physics
Follow a radio pulse across the plasma

AC06 · L2

From electron motion to a radio delay

Sandbox Physics ·

The relative arrival time measures the electron column because each small piece of plasma contributes a delay proportional to its electron density. Adding those pieces produces DM. We will derive that relation, then ask why aligning the frequencies can leave a pulse broad.

Define the wave and the medium

Let the wave travel along a coordinate toward the observer. The electron number density is measured in inverse cubic meters for the SI derivation. A positive time delay means a later arrival. We assume cold, nonrelativistic electrons, stationary ions and negligible collisions. The wave frequency is far above the plasma and gyro frequencies; we first omit the magnetic field.

An electric field gives each electron a force opposite to the field because its charge is negative. For sinusoidal motion, taking two time derivatives multiplies a displacement by minus the angular frequency squared. Solving the force equation gives a displacement in phase with the electric field and a polarization opposite to it. Here polarization means electric dipole moment per unit volume, distinct from the orientation of the radio wave discussed in the next chapter.

mex¨=−eE,x=eEmeω2,Pe=−neex,ϵr=1+Peϵ0E=1−ωp2ω2,ωp2=nee2ϵ0mem_e\ddot x=-eE,\quad x=\frac{eE}{m_e\omega^2},\quad P_e=-n_e e x,\quad \epsilon_r=1+\frac{P_e}{\epsilon_0E}=1-\frac{\omega_p^2}{\omega^2},\quad \omega_p^2=\frac{n_e e^2}{\epsilon_0m_e}
The elementary charge e is positive; the electron charge is minus e. Angular frequency is measured in radians per second. Relative permittivity describes how the induced electric field changes wave propagation.

Reading: Condon & Ransom, Essential Radio Astronomy, Appendix D

Time the envelope, not a crest

The plane-wave relation connects angular frequency and spatial wave number. Rearranging it and differentiating with respect to wave number gives the envelope speed, called group velocity. A derivative here compares how the frequency changes between neighboring wave components that form a pulse.

The group velocity is below the vacuum speed. The phase velocity of a crest is above it, and neither result sends a signal faster than light. We calculate the excess travel time of a short path segment by subtracting its vacuum travel time.

The fractional plasma correction is very small at the frequencies used here. The first-order expansion replaces the square-root expression with one half of that correction. The approximation fails near the plasma cutoff.

ω2=c2k2+ωp2,vg=dωdk=c1−ωp2ω2,dtexcess=(1vg−1c)dl≃ωp22cω2dl\omega^2=c^2k^2+\omega_p^2,\qquad v_g=\frac{d\omega}{dk}=c\sqrt{1-\frac{\omega_p^2}{\omega^2}},\qquad dt_{\rm excess}=\left(\frac1{v_g}-\frac1c\right)dl\simeq\frac{\omega_p^2}{2c\omega^2}dl
Wave number k has units of radians per meter. The approximation uses (1 minus a small number) to the power minus one half, expanded to first order.

Reading: Condon & Ransom, Essential Radio Astronomy, Appendix D

Add the small delays along the path

Substitute the plasma frequency into the segment delay and add all segments. The integral is the electron column in SI units. Astronomers usually put density in inverse cubic centimeters and path length in parsecs. One parsec equals about thirty-one quadrillion meters; converting both units produces the familiar coefficient.

Only a difference of arrival times is normally accessible because the emission epoch is unknown. Subtract the delay at a chosen reference frequency. Every time and frequency here is measured by the observer. For a compact host screen at redshift z, the contribution expressed in the observed DM is the host-frame column divided by one plus z; a distributed cosmological path requires the corresponding redshift integral.

DM=∫ne dl,texcess=e28π2ϵ0mecν2∫ne dl\mathrm{DM}=\int n_e\,dl,\qquad t_{\rm excess}=\frac{e^2}{8\pi^2\epsilon_0m_ec\nu^2}\int n_e\,dl
Use SI density and length in the second formula. The astronomical unit of DM is parsecs per cubic centimeter.
Δt=4.149 ms DMpc cm−3[(νGHz)−2−(νrefGHz)−2]\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]
The coefficient is rounded. A lower frequency has a positive delay relative to a higher reference frequency.

Reading: Condon & Ransom, Essential Radio Astronomy, Appendix D · Condon & Ransom, Essential Radio Astronomy, Chapter 6 · Pandhi et al. (2024), CHIME/FRB polarimetry, arXiv v2

Worked example: infer a column from a sweep

A pulse at four hundred megahertz arrives 389 milliseconds after its eight-hundred-megahertz counterpart. Assume the source emitted both frequencies together and that cold-plasma dispersion dominates the difference. Choose the relative-delay equation because the unknown emission time then cancels.

The two squared inverse frequencies, expressed in gigahertz, differ by 4.6875. Dividing 389 by 4.149 and by 4.6875 gives a DM of about twenty parsecs per cubic centimeter. If all the electrons occupy a uniform one-kiloparsec screen, dividing the column by one thousand parsecs gives 0.020 electrons per cubic centimeter. A different screen length changes the density estimate while leaving the measured DM unchanged.

The default synthetic detector uses this column, a six-millisecond intrinsic Gaussian width and a twelve-millisecond tail at six hundred megahertz. With seed 606 its sample spacing is 1.081 milliseconds. The selected tail grows to 60.75 milliseconds at four hundred megahertz under the chosen inverse-fourth-power law. These are inputs to the model; the next chapter lists the fitted polarization outputs.

DM=3894.149(0.4−2−0.8−2) pc cm−3≃20.0 pc cm−3,ne=201000 cm−3\mathrm{DM}=\frac{389}{4.149(0.4^{-2}-0.8^{-2})}\,\mathrm{pc\,cm^{-3}}\simeq20.0\,\mathrm{pc\,cm^{-3}},\qquad n_e=\frac{20}{1000}\,\mathrm{cm^{-3}}
The positive delay gives a positive electron column. The density step additionally assumes a known uniform screen length.

Reading: Condon & Ransom, Essential Radio Astronomy, Chapter 6

Scattering is a distribution of extra paths

Write the output as a sum of delayed copies of the input. A response function tells us how much weight to assign to each extra delay. Causality requires zero weight for a negative extra delay. In the infinite isotropic thin-screen approximation with a Gaussian angular distribution, the response is a one-sided exponential.

Its integral is one, so collecting the entire pulse preserves fluence, the time integral of intensity. Its mean extra delay is the tail timescale. A finite observation window can lose part of that tail. The saved detector samples retain that loss; the program never rescales a cropped tail to restore its area.

Dedispersion shifts channels by their frequency-dependent delay. It cannot undo the collection of delays already mixed inside each channel. The Lab prescribes scattering independently from DM: the electron column alone gives neither the density-fluctuation spectrum nor the screen geometry needed to predict a tail.

h(u)={τ−1e−u/τ,u≥0,0,u<0,Iout(t)=∫0∞Iin(t−u)h(u) duh(u)=\begin{cases}\tau^{-1}e^{-u/\tau},&u\ge0,\\0,&u<0,\end{cases}\qquad I_{\rm out}(t)=\int_0^\infty I_{\rm in}(t-u)h(u)\,du
The integration variable u is the extra path delay. Intensity is in arbitrary units, h in inverse milliseconds, and the output retains intensity units.
∫0∞h(u)du=1,∫0∞uh(u)du=τ,τ(ν)=τ600(ν600 MHz)−4\int_0^\infty h(u)du=1,\quad \int_0^\infty uh(u)du=\tau,\quad \tau(\nu)=\tau_{600}\left(\frac{\nu}{600\,\mathrm{MHz}}\right)^{-4}
The last relation is the selected screen scaling. Different geometry or turbulence can give a different frequency dependence.

Reading: Geyer & Karastergiou (2016), revisiting pulsar scattering · Condon & Ransom, Essential Radio Astronomy, Chapter 6

Exercises and explanations

  1. A screen gives the same DM but occupies one hundred parsecs instead of one thousand. What changes?

    Read the explanation

    Its mean electron density is ten times higher, 0.20 per cubic centimeter in the example. The far-above-cutoff dispersion delay is unchanged because the electron column is unchanged. Predicting its scattering additionally requires the fluctuation spectrum and screen geometry.

  2. A dedispersed pulse remains broad and nearly symmetric. Is an exponential scattering tail the first model you should choose?

    Read the explanation

    Inspect the intrinsic duration, channel smearing and time resolution. The exponential model predicts an asymmetric late tail. A symmetric profile could be dominated by the emitted pulse or the instrument. Compare several frequency subbands before assigning a scattering timescale.

Try the prediction

Separate a delay from a tail

  1. Before changing anything, calculate the default delay between four hundred and eight hundred megahertz. Compare it with the delay readout.
  2. Double only the electron column. Check the frequency sweep and the dedispersed profile. Restore the column, then change only the tail. Explain the difference.

Sources and calculation scope

  1. Pandhi et al. (2024), CHIME/FRB polarimetry, arXiv v2

    Version 2, 2 May 2024. Sections I.1, II.1–II.2: Stokes data, redshift weighting and calibration. The text of equation 2 lists kpc alongside coefficient 0.812; this course uses the dimensionally consistent pc normalization from NRAO.

  2. Condon & Ransom, Essential Radio Astronomy, Appendix D

    D.1–D.2, equations D.14–D.27: cold-plasma group velocity and magnetic rotation. Consulted 28 October 2026. The text uses Gaussian units; the chapter derives the SI delay coefficient separately.

  3. Condon & Ransom, Essential Radio Astronomy, Chapter 6

    Section 6.2, equations 6.44–6.46 and Figure 6.5: dispersion and pulse broadening. The inverse-fourth-power scattering law is a model choice here.

  4. Geyer & Karastergiou (2016), revisiting pulsar scattering

    Section 2.1, equations 1–3: the one-sided exponential assumes an infinite isotropic thin screen with a Gaussian angular distribution. Finite and anisotropic screens alter the response.

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