AST 1440 – Radiation (2026)

• Intensity, source function, black body, diffusion [RL 1]
• Wave-plasma interaction [RL 8]
• Accelerated charges, radiation reaction [RL 3]
• Free-free radiation and opacity [RL 5]
• Cyclotron and synchrotron radiation [RL 6]
• Compton and inverse Compton scattering [RL 7]
• Thermal distributions, Boltzman & Saha [RL 9]
• Atomic transitions; more complicated systems; collisions [RL 10,11]
• Applications! (of all of the above)

The primary course text book is Radiative Processes in Astrophysics (RL) by Rybicki & Lightman, Wiley (online access via UofT library).

Two secondary book are Theoretical Astrophysics I: Astrophysical Processes (Pad) by Padmanabhan, Cambridge Univ. Press (online access via UofT library; more theoretical) and Astrophysical processes (Bradt) by Bradt, Cambridge Univ. Press (online access via UofT library; more applied).

A great resource, which includes links to mini-lectures by Aaron Parsons (UCB), is astrobaki:Radiative_Processes_in_Astrophysics.

• Three problem sets (30% total), due two weeks after posting: set 1 (due Oct 12 15)
• Long presentation (20%) on a more advanced topic. Can be on any topic (with approval, but ideally directly relevant to your own research).
• Final exam (40%; oral).

Define the specific intensity of a beam; then all bulk properties can be derived from it: flux \(F\), radiation density \(U\), radiation pressure \(P\), etc.

Basic picture is a beam of many photons going through sequence of layers, where the beam can interact with the matter, and energy can also be added to it. Ignoring scattering for now, one has \[ \frac{dI_{\rm \nu}}{ds} = -\alpha_{\rm \nu}I_{\rm \nu} + j_{\rm \nu} \] Often divide by the absorption coefficient \(\alpha_{\rm \nu}\), and define optical depth \(d\tau_{\rm \nu} = \alpha_{\rm \nu}ds\) and source function \(S_{\rm \nu} = j_{\rm \nu}/\alpha_{\rm \nu}\), \[ \frac{dI_{\rm \nu}}{d\tau_{\rm \nu}} = -I_{\rm \nu} + S_{\rm \nu} \]

Basic ideas:

• Charges are accelerated by E, B (produced by other charges or otherwise);
• Acceleration produces EM waves, i.e., photons and thus E, B to interact with.

Classically, total power emitted (with acceleration determined in particle frame) is \[ \frac23 \frac{q^{\rm 2}a^{\rm 2}}{c^{\rm 3}}\qquad\hbox{(cgs, $\times1/4\pi\epsilon_{\rm 0}$ for SI)} \] Often, one can get surprisingly far with the classical picture, especially if taking into account special relativity. But in detail need quantum mechanics (often captured by correction factors).

Note: In class, I stated without much explanation that for a charge moving at constant velocity, the E field always points to the instantaneous position of the charge, even though it can only know about where the charge was at the retarded time. This comes about because the field depends on both the scalar potential of the charge and the vector potential due to the current corresponding to the charge's motion (which also leads to a magnetic field; you can play with this by inspecting \(B\) after running the animation code linked above). See RL 3.1-3.2, as well as the topical video at astrobaki:Larmor_Formula.

Example of estimates:

From force balance, \[ f_{\rm grav}= g\rho = f_{\rm rad} = \frac{F}{c}\kappa\rho \qquad\Leftrightarrow \frac{GM\rho}{r^{\rm 2}} = \frac{\kappa{}\rho{}L}{4\pi{}r^{\rm 2}c}. \] For fully ionized material with fractional abundance \(X_{\rm H}\) of Hydrogen by mass, \[ L_{\rm edd} = \frac{4\pi{}GcM}{\kappa} = 4\pi{}GcM \frac{m_{\rm H}}{\sigma_{\rm T}\frac12(1+X_{\rm H})}. \]

Used through-out astronomy, for studies of accreting sources (X-ray binaries, AGN, etc.), to the superwind of AGB stars (though with opacity due to dust instead of electron-scattering).

For a source with given flux \(F\) at distance \(d\), it allows setting a mass constraint, \[ M \geq 4\pi{}d^{2 }F \frac{\sigma_{\rm T}\frac12(1+X_{\rm H})}{4\pi{}Gcm_{\rm H}}. \]

I followed Pad §1.4.4 in my derivation of \(\sigma_{\rm T}\) and Pad §6.4 for the extension to Rayleigh scattering (or scattering by an electron in a strong field). RL §3.4 and §3.6 (latter starting at p.99) are similar.

Consider an observation of an X-ray emitting cloud with radius \(R\) at distance \(d\), observed with a detector with acceptance opening half-angle \(\Delta\theta\) (and thus \(\Delta\Omega_{\rm det}=\pi\Delta\theta^{\rm 2}\)), effective detector area \(A_{\rm eff}\) (geometric area times detection efficiency), integration time \(\Delta{}t\) and bandwidth \(\Delta\nu\).

Unlike in the problem in the textbook, let's for simplicity assume that the cloud is optically thick, so \(I=C\) for \(\theta\leq{}R/d\) and 0 otherwise. We also define \(\Delta\Omega_{\rm src}=\pi(R/d)^{\rm 2}\).

The brightness temperature is often used in radio astronomy. In the problem set, one first calculates \(I_{\rm \nu}\) and then uses it to calculate \(T_{\rm b}\). Combining the two, \[ T_{\rm b} = \frac{f_{\rm \nu}}{\Delta\Omega}\frac{\lambda^{\rm 2}}{2k} \] In the problem, the source is resolved, but what if it is unresolved? Then, one can use that \(\Delta\Omega A=\lambda^{\rm 2}\) (plausible with \(\Delta\Omega\simeq(\pi/4)(1.22\lambda/D)^{\rm 2}\) and \(A=(\pi/4)D^{\rm 2}\), but exact; e.g., https://www.cv.nrao.edu/~sransom/web/Ch3.html). So, \[ T_{\rm b} = \frac{f_{\rm \nu}A^{}}{2k} = \frac{P_{\rm \nu}}{2k} \] where \(P_{\rm \nu}\) is the power received by the telescope. Often, the noise in a telescope system is characterized by an equivalent system temperature \(T_{\rm sys}=P_{\rm noise}/2k\).

For further radio telescope terminology, an effective area can be defined by \(A_{\rm eff}=P_{\rm \nu}/f_{\rm \nu}\), or, in terms of the brightness temperature, one has a forward gain \(G=A_{\rm eff}/2k=T_{\rm b}/f_{\rm \nu}\), typically given in units of [K/Jy]. More directly relevant for the sensitivity is the ratio \(G/T_{\rm sys}\). One can estimate this using measurements of the power on and off a source with known flux \(f_{\rm \nu}\), \(P'_{\rm on}\) and \(P'_{\rm off}\). Here, the primes indicate that these do not directly measure the power, but rather something proportional to it (due to amplification, digitization, etc.), i.e., \(P_{\rm off}=f_{\rm scl}2kT_{\rm sys}\) and \(P_{\rm on}-P_{\rm off}=f_{\rm scl}A_{\rm eff}f_{\rm \nu}\). But the scale factor drops out in ratios, so one can calculate \(G/T_{\rm sys}=(P_{\rm on}-P_{\rm off})/P_{\rm off}f_{\rm \nu}\). The inverse of this is known as the system equivalent flux density, \({\rm SEFD} = f_{\rm \nu}P_{\rm off}/(P_{\rm on}-P_{\rm off})\), the flux of a source for which the received power would equal that of the system noise.

\(S=\frac43aVT^{\rm 3}\), so in an adiabatic process, \(T\propto1/R\), in contrast for an ideal gas for which \(T\propto{}V^{-(\gamma-1)}\propto1/R^{\rm 2}\). Hence, in an expanding universe, the radiation temperature scales as \(1/a=1+z\). It also means that supernovae would not be all that luminous if there was no reheating by radioactive decay: by the time the ejecta are large enough for photons to be able to diffuse out rapidly, they have cooled down a lot.

The problem discusses what happens in the absence of stimulated emission, or for neutrinos instead of photons. There is little to add to the solutions. But for the regular case, the most useful consequence arguably is just that it helps to understand why, if matter is roughly in Local Thermal Equilibrium (LTE), the source function \(S_{\rm \nu}=B_{\rm \nu}(T)\).

The easiest way to get the observed flux is to calculate the luminosity by adding up the optically thin emission, and then dividing by \(4\pi{}d^{\rm 2}\). But of course, the integral over specific intensity should give the same answer (and can be done correctly for all optical depths). One has, \[ F_{\rm \nu}=\int_{\rm \Omega} I_{\rm \nu} \cos \theta d\Omega = \int_{\rm 0}^{\rm \infty}I_{\rm \nu}(b)\frac{2\pi{}bdb}{d^{\rm 2}}, \] where we used cylindrical symmetry around the line of sight, and that \(\cos \theta\simeq1\) and \(d\Omega=2\pi{}bdb/d^{\rm 2}\) since the source is far away. With \(I_{\rm \nu}=2R(1-(b/R)^{\rm 2})^{1/2}j_{\rm \nu}\) and \(I_{\rm \nu}=B_{\rm \nu}\) for the optically thin and thick cases, respectively, this reproduces the solutions.

For the more general but thermal emission case, one would start with the solution to the radiative transfer equation along the line of sight to the observer (\(z\) axis, with object at origin and observer at \(z=+\infty\)), \[ I_\nu(b)=\int_{-\infty}^{\rm \infty} S_{\rm \nu}(b,z)(1- e^{-\tau_{\rm \nu}(b, z)} \] (this is just RL, Eq. 1.29, but without a \(I_{\rm \nu}^{\rm 0}\) term since we start at \(z=-\infty\)), where we can often use \(S_{\rm \nu}=B_{\rm \nu}(T)\). The optical depth is given by (RL, Eq. 1.26), \[ \tau_{\rm \nu}(b,z)=\int_{\rm z}^{+\infty} \alpha_{\rm \nu}(b, z') dz'. \]

If one looks to a star through a shell transparent to the continuum but opague to a line, then in the line the flux will be lower if the shell is cooler than the star (absorption line) and higher if the shell is hotter (emission line). To the side of the star, one will always see an emission line, since there is no continuum.

For a star like the sun, the photosphere, where the continuum is formed, is surrounded by layers that are cooler, leading to the absorption spectrum seen in visible light. But further out, the temperature increases again, leading to the strongest lines to be in emission. See apod:230114 and apod:180409.

Seen at grazing incidence, one will reach \(\tau=1\) at a higher altitude than seen face on. Since the temperature at these higher laters is (generally) cooler, this leads to limb darkening. See apod:240513.

If the star has a wind, then generically absorption in the wind will be blue shifted, while emission will be over a large range of velocities. One gets a P Cygni profile. (If a star is accreting, e.g., because it is still forming, one can get a reversed P Cygni profile, where the absorption is on the redshifted end of the profile.)

Discussed the implications of Eq.~7 in the RL solution (page 320), \(F_{\rm \nu}=(4\pi/\sqrt{3})B_{\rm \nu}[\sqrt\epsilon/(1+\sqrt\epsilon)]\), where \(\epsilon=\alpha_{\rm \nu}/(\alpha_{\rm \nu}+\sigma_{\rm \nu})\), in particular how even for the case that all properties (\(B_{\rm \nu}\), \(\alpha_{\rm \nu}\), and \(\sigma_{\rm \nu}\)) do not depend on depth inside the layer, one can get line features.

First, one sees that the more scattering dominates over absorption, the smaller \(\epsilon\) and thus the smaller the flux. I found it most useful to think about this as the large scattering opacity effectively slowing down the photons (since they have to random-walk towards the edge), thus reducing their rate and therefore the flux.

For resonant lines, photon absorption looks more like a scattering process, since after absorbing one photon, an ion will typically emit another one at the same frequency before it can be collisionally de-excited. Hence, \(\epsilon\) will be lower than in the continuum and thus such lines will appear as absorption lines (even though the temperature is constant throughout).

Conversely, for weak non-resonant lines, where de-excitation is dominated by collisions and the scattering thus dominated by continuum processes, \(\alpha_{\rm \nu}\) and thus \(\epsilon\) will be slightly larger than in the continuum and thus one expects (weak) emission features.

I siscussed how to estimate line profiles both directly, as well as by calculating optical depths and using those to integrate the radiative transfer equation (assuming thermal emission). For both cases, the basic simplification we make is that natural and thermal line broadening will be substantially smaller than \(v_{\rm w}/c\).

Ignoring the contribution from the star or continuum, we first discussed how a line profile in the range \(\lambda_{\rm 0}(1\pm{}v_{\rm w}/c)\) arises, where we assume this is sampled at some (small) fixed wavelength spacing \(\Delta\lambda\).

For a given angle \(\theta\) relative to the direction to the observer, a constant velocity wind will have a radial velocity component \(v_{\rm rad}=-v_{\rm w}\cos \theta\), which will map to \(\lambda=\lambda_{\rm 0}(1+v_{\rm rad}/c)\). To get the range \(\Delta\theta\) corresponding to \(\Delta\lambda=\lambda_{\rm 0}\Delta{}v_{\rm rad}/c\), we use that \(dv_{\rm rad}/d\theta=v_{\rm w}\sin\theta\), so that \(\Delta\theta=\Delta{}v_{\rm rad}/(v_{\rm w}\sin \theta)=c(\Delta\lambda/\lambda_{\rm 0})/(v_{\rm w}\sin \theta)\propto1/\sin \theta\).

For optically thin emission, the total line luminosity at given frequency will be the emissivity \(j_{\rm \nu}\) times the volume \(V\). The volume for a given velocity \(v_{\rm rad}\) is a cone with opening angle \(\theta\) and width \(\Delta\theta\). For some range in radius \(\Delta{}R\), one thus has \(F_{\rm \nu}=\Delta{}R2\pi{}\sin \theta \Delta\theta/4\pi{}d^{\rm 2}\). Inserting the dependencies of \(\theta\) and \(\Delta\theta\), one has \(F_{\rm \nu}\propto\sin \theta / \sin \theta\), i.e., one expects a top-hat line profile, constant between \(-v_{\rm w}\) and \(+v_{\rm w}\).

For optically thick emission, at each velocity, the conical section is optically thick, so one observes flux from the area of that section, i.e., \(F_{\rm \nu}=\pi{}(\Delta{}R\sin \theta)^{\rm 2}B_{\rm \nu}/4\pi{}d^{\rm 2}\propto \sin^{\rm 2}\theta \propto 1-(v_{\rm rad}/v_{\rm w})^2\).

Of course, an actual integration of specific intensities should give the same result. We started by writing out the general expression for the flux, \[ F_{\rm \nu} = \int_{\rm \Omega} d\Omega\,I_{\rm \nu}(\Omega)\cos \theta = \int_{-\pi/2}^{\pi/2}d\theta\int_{\rm 0}^{\rm 2\pi}d\phi\sin \theta I_{\rm \nu}(\theta, \phi) \cos \theta. \] Definiting a cylindrical coordinate system centred on the star with the observer at \(z=d\) (\(d\) the distance), then for a line at given impact parameter (cylindrial radius) \(p\), with \(p/d\ll1\), one has \(\sin \theta\simeq{}p/d\), \(\cos \theta\simeq1\) and \(d\theta\simeq{}dp/d\). Assuming a spherical symmetric wind, i.e., no \(\phi\) dependence, \[ F_{\rm \nu} = \int_{\rm 0}^{\rm \infty} \frac{2\pi p dp}{d^{\rm 2}} I_{\rm \nu}(p). \] (Here, the integral is taken to \(\infty\), but it is assumed that \(I_{\rm \nu}(p)\rightarrow0\) while one still has \(p/d\ll1\).)

To see that this is tractible, I rederived RL Eq. 1.30 in class, which found that if the source function \(S_{\rm \nu}\) is constant, one has, \[ I_{\rm \nu} = S_{\rm \nu} (1 - e^{-\tau_{\rm \nu}^{\rm max}}) \] where the total optical depth, \[ \tau_{\rm \nu}^{\rm max} = \int_{-\infty}^{+\infty} dz\, \alpha_{\rm \nu}(r) = \int_{-\infty}^{+\infty} dz\,\alpha_{\rm o}(r) \phi(\nu-\nu_{\rm 0}(1-v_{\rm rad}/c)), \] where we used that for a line, \(\alpha_{\rm \nu} = \alpha_{\rm 0}\phi(\nu-\nu_{\rm 0})\), with \(\phi\) the profile function (defined such that \(\int_{\rm \nu}\phi(\nu-\nu_{\rm 0})d\nu=1\)). To make progress, one changes the integral to be over frequency, substituting \(\nu_0^{\rm \prime}=\nu_{\rm 0}(1-v_{\rm rad}/c)=\nu_{\rm 0}(1+(z/r)(v_{\rm w}/c))\). Hence, \(d\nu_{\rm 0}^{\rm \prime}/dz = (\nu_{\rm 0}v_{\rm w}/c)(1/r - z^{\rm 2}/r^{\rm 3}) = (\nu_{\rm 0}v_{\rm w}/c)(p^{\rm 2}/r^{\rm 3})\), and \(z=\pm\infty\) correspond to \(\nu_{\rm min,max}=\nu_{\rm 0}(1\pm{}v_{\rm w}/c)\), so that, \[ \tau_{\rm \nu}^{\rm max} = \int_{\nu_{\rm min}}^{\nu_{\rm max}} d\nu_{\rm 0}^{\rm \prime}\,\alpha_{\rm 0}(r) \phi(\nu-\nu_{\rm 0}^{\rm \prime}) \frac{c}{\nu_{\rm 0}v_{\rm w}}\frac{r^{\rm 3}}{p^{\rm 2}} = \alpha_{\rm 0}(r) \frac{c}{\nu_{\rm 0}v_{\rm w}}\frac{p}{\sin^{\rm 3}\theta}, \] where for the second equality we approximated the line profile as a delta function and used that \(r=p/\sin \theta\).

If we now take \(S_{\rm \nu}=B_{\rm \nu}\) and assume an isothermal wind so it is constant, and furthermore (more unrealistically) assume that the wind only emits between some \(R_{\rm low}\) and \(R_{\rm high}\), having \(\alpha_{\rm 0}\) constant within that range and zero outside, we can rederive the optically thin and thick cases. For optically thin, \[ F_{\rm \nu} \simeq \int_{p_{\rm low}}^{p_{\rm high}} \frac{2\pi p dp}{d^{\rm 2}} B_{\rm \nu}\alpha_{\rm 0}^{\rm \prime}\frac{p}{\sin^{\rm 3}\theta}, \] where \(p_{\rm low,high}=R_{\rm low,high}/\sin \theta\) and \(\alpha_{\rm 0}^{\rm \prime}=\alpha_{\rm 0}(c/\nu_{\rm 0}v_{\rm w})\) (which has units of inverse length). For given \(v_{\rm rad}\), \(\theta\) is constant, so the integral is simple, and one recovers that \(F_{\rm \nu}\) does not depend on \(\theta\), i.e., a top-hat line profile.

For an optically thick line, everything after \(B_{\rm \nu}\) disappears, so the integral again is simple and one recovers \(F_{\rm \nu}\propto\sin^{\rm 2}\theta\) as before.

Of course, these cases are not realistic, but they set the extremes, i.e., real line profiles should be intermediate. In particular, in general \(\alpha_{\rm 0}\) will not be constant; e.g., for lines driven by recombination, the opacity will scale as the product of the electron and ion densities, i.e., for a constant velocity wind one will have \(\alpha\propto{}n^{\rm 2}\propto1/r^{\rm 4}\). But it turns out this case is also tractable, as we will see in the problem set (which is partially inspired by the analytic calculations of profiles for Wolf-Rayet stars first done by Hillier et al. 1983ApJ...271..221H).

Use that \(F_{\rm \nu}=\int_{\rm \Omega}I_{\rm \nu}\cos \theta d\Omega\) should be the same just above and below a plane separating a change in index of refraction. I found it easiest to consider is a source directly above the plane (i.e., \(\theta=0\) and \(\cos \theta=1\)), with opening \(\delta\theta\) and thus \(d\Omega=\pi(\delta\theta)^{\rm 2}\), and then apply Snell's law.

For a source at some arbitrary \(\theta\) not close to 0, \(\cos \theta d\Omega = \cos \theta d\theta\sin \theta d\phi\), and one finds the same result. This is easiest to see by realizing that the \(\cos \theta d\theta = d\sin \theta\) term and the \(\sin \theta\) both give a factor of \(n_{\rm r}^{\rm \prime}/n_{\rm r}\), while \(d\phi\) is the same on both sides.

One advantage of SI units is that it makes it easier to keep track of where the ε and μ go. In class, I showed that for this problem, one finds \(m^2=\mu\epsilon(1+i\sigma/\epsilon\omega)\).

In class, I followed the derivation of problem 2.2 with one for an electron in vacuum (microscopic prescripton), then identifying its effect with a different permeability \(\epsilon\). I noted that the derivation assumes an electron accelerated by the EM wave, but ignores its radiation, i.e., is incomplete in that sense. But for practical purposes, it does not matter too much; it just corresponds to a (very slight) reduction in intensity.

The final time delay due to dispersion is, \[ \Delta{}t(\nu) = \frac{e^{\rm 2}}{8\pi{}^{\rm 2}m_{\rm e}\epsilon_{\rm 0}c}\frac{DM}{\nu^{\rm 2}} = {\cal D}_{\rm 0}\frac{DM}{\nu^{\rm 2}} \] In pulsar timing, \({\cal D}_{\rm 0}\equiv(10^{\rm 4}/2.41){\rm s\,MHz^{\rm 2}\,cm^{\rm 3}/pc}\), with the value what one would find from the physical constants as known in the 1970s. This is not updated, since then DM values (dispersion measures, typically given in pc/cm³) could no longer be easily compared.

Generally, one should treat measured DM strictly as a fit of \(\Delta{}t(\nu)=D/\nu^{\rm 2}\), with an interpretation of \(D\) as a (scaled) electron column density. For instance, scattering may bias the measurements (it scales as \(1/\nu^{\rm 4}\), but may not be easily recognizable because of noisy measurements or a small bandwidth). See Kulkarni 2020arXiv200702886K.

I mentioned the Poincaré sphere as a nice way to visualize polarization. See wiki:Unpolarized_light#Poincar%C3%A9_sphere.

We discussed how for weak magnetic fields, with observing frequency \(\nu\gg\nu_{\rm B}\), the two natural modes in the plasma are those of left and right cicrular polarization, which have different speeds, leading to regular Faraday rotation for linearly polarized waves.

But this changes when the magnetic field is very strong, or is nearly perfectly perpendicular to the direction of propagation. In that case, the two relevant modes are the O and X modes (with \(E\) parallel and perpendicular to the background \(B\), resp.). These differ in whether the electron can interact with the wave (by sliding along the field line) or cannot (since the electron would have to move perpendicular to B). Also in this case, waves that are not exactly O or X will be rotated around the relevant axis.

The relative importance of the two types is set by \(2(\nu/\nu_{\rm B})\cos(\alpha)\) (where \(\cos(\alpha)=\hat{k}\cdot\hat{B}\)). I showed examples from Li et al. 2019MNRAS.484.5723L (see, in particular, their Fig. 2), but the methods section of Li et al. 2023Natur.618..484L has more detail (and the paper itself presents a nice example in nature). See Pad §9.5 for detailed derivations.

RL 8.1 briefly discusses refraction in a cold plasma, mentioning that the index of refraction \(n_{\rm r}=(1-(\nu_{\rm p}/\nu)^{\rm 2})^{1/2}\) (Eq. 8.16) and that light will get bend as \(dn_{\rm r}\hat{k}/dl=\nabla{}n_{\rm r}\) (Eq. 8.18). Integrating that over a not-too-large lens bending radiation by a small amount, one finds a bending angle \[ \alpha=\int_{\rm lens}\nabla n_{\rm r} dl = \frac{e^{\rm 2}}{8\pi^{\rm 2}m_{\rm e}\epsilon_{\rm 0}\nu^{\rm 2}}\nabla N_{\rm e} = \frac{r_{\rm e}}{2\pi}\lambda^{\rm 2}\nabla N_{\rm e}, \] where we used that \(r_{\rm e}=e^{\rm 2}/4\pi\epsilon_{\rm 0}m_{\rm e}c^{\rm 2}\). The constant terms are \(8314{\rm\,mas\,MHz^{\rm 2}cm^{\rm 2}cm}\) and \(0.0925{\rm\,mas\,m^{-2}cm^{\rm 2}cm}\), respectively.

We also briefly discussed applications to pulsar scintillation, and we found that typical values of \(\alpha\) around a few mas are required, which implies strangely large \(\nabla{}N_{\rm e}\simeq10^{\rm 3}{\rm\,cm^{-2}cm^{-1}}\). For the suggested physical picture of sheets seen at grazing incidence, see Pen & Levin 2014MNRAS.442.3338P, and for the paper showing the Toronto sky line, Liu et al. 2016MNRAS.458.1289L. For a more general overview of how scintillation works and how we can use it to learn about pulsars, see the background sections in the screens package documentation.