AST 1440 – Radiation (2025)

• Basics – intensity, source function [RL 1.1-1.4]
• Relations – Blackbody, Kirchoff, Einstein coefficients [RL 1.5-1.6]
• Radiation diffusion – stars, supernovae [RL 1.7,1.8]
• Wave-plasma interaction [RL 8]
• Free-free radiation and opacity [RL 5]
• Cyclotron and synchrotron radiation [RL 6]
• Compton and inverse Compton scattering [RL 7]
• Atomic transitions; more complicated systems; collisions [RL 9,10,11]

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.

• Short on-the-spot presentations (10% total) explaining some of the reading or problems assigned for a given class.
• Three problem sets (30% total), due two weeks after posting: set 1 (due Oct 2), set 2 (due Nov 10).
• Long presentation (20%) on a more advanced topic (two each on 24 and 27 Nov). Can be on any topic (with approval, but ideally directly relevant to your own research). Format roughly that of TASTY (18 min + 7 min discussion).
• 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 \[ \frac23 \frac{q^{2}a^{2}}{c^{3}}\qquad\hbox{(cgs, $\times1/4\pi\epsilon_{\rm 0}$ for SI)} \] 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).

Example of estimates we made in class:

• Electron passing ion: Bremsstrahlung.
• Electron in atom: transition rates between atomic states.
• Electron in B: cyclotron, and its relativistic version, synchrotron.

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 about). See RL 3.1-3.2, as well as the topical video at astrobaki:Larmor_Formula.

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 \(\Delta\Omega_{\rm det}=\pi\Delta\theta^{2}\)), 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 \(\Delta\Omega_{\rm src}=\pi(R/d)^{2}\).

From force balance, \[ f_{\rm grav}= g\rho = f_{\rm rad} = \frac{F}{c}\kappa\rho \qquad\Leftrightarrow \frac{GM\rho}{r^{2}} = \frac{\kappa{}\rho{}L}{4\pi{}r^{2}c}. \] 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.

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, \[ T_{\rm b} = \frac{f_{\rm \nu}}{\Delta\Omega}\frac{\lambda^{2}}{2k} \] In the problem, the source is resolved, but what if it is unresolved? Then, one can use that \(\Delta\Omega A=\lambda^{2}\) (plausible with \(\Delta\Omega\simeq(\pi/4)(1.22\lambda/D)^{2}\) and \(A=(\pi/4)D^{2}\), 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 twe 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.

\(S=\frac43aVT^{3}\), so in an adiabatic process, \(T\propto1/R\), in contrast for an ideal gas for which \(T\propto{}V^{-(\gamma-1)}\propto1/R^{2}\). 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 most useful consequence arguably is just that 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 \(4\pi{}d^{2}\). But of course, the integral over specific intensity should give the same answer (and will be correct for all optical depths). One has, \[ F_{\rm \nu}=\int_{\rm \Omega} I_{\rm \nu} \cos \theta d\Omega = \int_{\rm 0}^{\infty}I_{\rm \nu}(b)\frac{2\pi{}bdb}{d^{2}}, \] where we used cylindrical symmetry around the line of sight, and that \(\cos \theta\simeq1\) and \(d\Omega=2\pi{}bdb/d^{2}\) since the source is far away. With \(I_{\rm \nu}=2R(1-(b/R)^{2})^{1/2}j_{\rm \nu}\) 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 write \(I_\nu=B_{\rm \nu}(1-\exp(-\tau_{\rm \nu}))\) and calculate \(\tau_{\rm \nu}(b)=\int_{-\infty}^{+\infty}\alpha_{\rm \nu}dz\).

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 \(F_{\rm \nu}=(4\pi/\sqrt{3})B_{\rm \nu}[\sqrt\epsilon/(1+\sqrt\epsilon)]\), where \(\epsilon=\alpha_{\rm \nu}/(\alpha_{\rm \nu}+\sigma_{\rm \nu})\).

For resonant lines, photon absorption looks more like a scattering process, since after absorbing one photon, they typically emit another one at the same frequency before they can be collisionally de-excited. Hence, \(\epsilon\) is low and 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.

Discussed how to estimate line profiles both directly and by calculating optical depths (under the assumption that line broadening will be substantially smaller than \(v_{\rm w}/c\)) and using those to integrate the radiative transfer equation (assuming thermal emission).

Ignoring the contribution from the star or continuum, we found that for a completely optically thin wind, one expects \(F(v)\sim{}C\), i.e., a top-hat line profile, while for an optically thick line, one gets \(F(v)\propto{}1-(v/v_{\rm w})^{2}\). Neither of these cases is realistic, but real line profiles should be intermediate (as is indeed the case for the profiles calculated for Wolf-Rayet stars in 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., \(\theta=0\)), with opening \(\delta\theta\) and thus \(d\Omega=\pi(\delta\theta)^{2}\), and then apply Snell's law. For a source at some arbitrary \(\theta\) not close to 0, \(d\Omega=d\theta\sin \theta d\phi\), and one finds the same result.

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

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 - just a reduction in intensity.

The final time delay due to dispersion is, \[ \Delta{}t(\nu) = \frac{e^{2}}{8\pi{}^{2}m_{\rm e}\epsilon_{\rm 0}c}\frac{DM}{\nu^{2}} = {\cal D}_{\rm 0}\frac{DM}{\nu^{2}} \] In pulsar timing, \({\cal D}_{\rm 0}\equiv(10^{4}/2.41){\rm s\,MHz^{2}\,cm^{3}/pc}\), with the value what one would find from the physical constants as known in the 1970s. This is not updated, since then DM values could no longer be easily compared. Generally, one should treat measured DM strictly as a fit of \(\Delta{}t(\nu)=D/\nu^{2}\), with an interpretation of \(D\) as a (scaled) electron column density. For instance, scattering may bias the measurements (it scales as \(1/\nu^{4}\), 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 also briefly discussed how polarized light and its interaction with media and instruments can be described with Jones vectors and matrices; see wiki:Jones_calculus.

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.), and other modes can be rotated around that 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, 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 \(n_{\rm r}=(1-(\nu_{\rm p}/\nu)^{2})^{1/2}\) (Eq. 8.16) and that light will get bend as \(dn\hat{k}/dl=\nabla{}n_{\rm r}\) (Eq. 8.18). Integrating that over a not-too-large lens, one finds a bending angle \[ \alpha=\int_{\rm lens}\nabla n_{\rm r} dl = \frac{e^{2}}{8\pi^{2}m_{\rm e}\epsilon_{\rm 0}\nu^{2}}\nabla N_{\rm e} = \frac{r_{\rm e}}{2\pi}\lambda^{2}\nabla N_{\rm e}, \] where we used that \(r_{\rm e}=e^{2}/4\pi\epsilon_{\rm 0}m_{\rm e}c^{2}\). The constant terms are \(8314{\rm\,mas\,MHz^{2}cm^{2}cm}\) and \(0.0925{\rm\,mas\,m^{-2}cm^{2}cm}\), 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 \(\nabla{}N_{\rm e}\simeq10^{3}{\rm\,cm^{-2}cm^{-1}}\). 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.

One finds that the resolution is about 0.023 R_o, i.e., in principle very precise measurements are possible. It becomes even better at longer wavelengths, scaling as \(\propto1/\lambda\). For an example where scintillation was used to see changes of 20 km in the emission location as a function of pulse phase, see Pen et al. 2014MNRAS.440L..36P.

The high spatial resolution also means the most objects other than pulsars and (some) AGN do not scintillate, as their emission regions are resolved (the same reason looking through Earth's atmosphere, stars twinkle but planets do not).

Main thing to remember is that E points at current position for a particle moving at constant velocity (or, if there is acceleration, where it would be if it had continued at the same velocity). B also exists for a moving particle (as it is a current), \(|B|=\beta|E|\) in cgs units.

The book has a nice explanation of Cherenkov radiation (aside: Pad excercise 4.2 asks for a derivation of the emitted spectrum, but it is non-trivial; see wiki:Frank-Tamm_formula). It uses that for media with \(n_{\rm r}>1\), the retarded potentials only have information within a radius \(R = (c/n)t < ct\), which can be smaller than \(v_{\rm part}t\) if \(v_{\rm part}>c/n\).

However, I found RL's analogy for the Razin effect obscure, as it seemed to suggest that for \(n<1\) the potentials would have information out to radii larger than \(ct\), which would break causality.

I found the discussion in Melrose 1972Ap&SS..18..267M more insightful (esp. §4). As noted in class, an electron emitting bremsstrahlung as it interacts with an ion, will emit low frequency radiation far from the ion where the acceleration is small (\(\omega\sim{}a/v\)) and at higher frequency closer to it, where the acceleration is larger (up to \(\omega_{\rm max}\sim{}a_{\rm max}/v\sim{}v/b\), where \(b\) is the impact parameter).

In a plasma, the minimum frequency at which radiation can propagate is the plasma frequency, \(\omega_{\rm min}=\omega_{\rm p}=(en_{\rm e}/m_{\rm e})^{1/2}\). For a particle travelling at high \(\gamma\) at large enough \(b\) that its trajectory is nearly a straight line, the perpendicular acceleration is most important. The latter is invariant, i.e., the same in its frame as estimated above. Also invariant is the plasma frequency (the particle sees a higher \(\gamma{}n_e\), but also higher masses \(\gamma{}m_{\rm e}\) for the electrons in the plasma). In the observer's frame, the emission will be beamed in a narrow angle \(\theta\sim1/\gamma\). If one is within this cone, one will thus see a break at the Doppler-shifted plasma frequency, \(\gamma{}\omega_{\rm p}\).

Since RL introduced magnetic dipole emission rather implicitly, I derived both electric and magnetic dipole emission in class. I followed the LibreText on classical electrodynamics, in particular sections 8.1 (retarded potentials), 8.2 (electric dipole radiation), and 8.9 (magnetic dipole radiation).

Given the formula for magnetic dipole emission, \[ P = \frac{2}{3}\frac{\left|\ddot{\vec{m}}\right|^{2}}{c^{3}}, \] the problem is fairly trivial.

Two notes made in class:

1. In general, for \(\dot{\omega}=-C\omega^{n}\), one finds a spin-down age \(\tau_{\rm sd}=P/(n-1)\dot{P}\), i.e., one does not have to rely on the dimensional estimate \(\omega/\dot{\omega}\). For magnetic dipole radiation, \(n=3\) and hence \(\tau_{\rm sd}=P/2\dot{P}\). This is used by all pulsar astronomers.
2. Numerical simulations show that even with an aligned magnetic field, a neutron star will slow down. Numerical simulations (Spitkovski 2006ApJ...648L..51S) give \[ P_{\rm sd} = \frac{m^{2}\omega^{4}}{c^{3}}\left(1+\sin^{2}\alpha\right), \] where the prefactor for the whole equation is \(1\pm0.05\), and that in front of the sin term \(1\pm0.1\).

Electron scattering will cause light to be polarized perpendicular to the direction to the source. For spherically symmetric, unresolved sources, the signal is zero, but some sources are not as symmetric: I mentioned Be stars with their decretion disks, and WR stars with (symmetric) winds with bright companions. Note that I misremembered how the technique for using this to estimate mass-loss rates worked (the risks of not looking things up…): the companion is used as a light source, not as an occulter, and relative to the companion, the scatterers in the WR wind are of course highly asymmetric. See St Louis et al., 1988ApJ...330..286S.

For the intensity of a bunch of incoherent emitters, one found \[ \left|\hat{E}(\omega)\right|^{2} = \left |\sum_{k=1}^{n}\hat{E}_{\rm 0}(\omega)e^{i\omega{}t_{\rm k}}\right|^{2} = \left|\hat{E}_{\rm 0}(\omega)\right|^{2}\left(\sum_{k=1}^{n}e^{0} + \sum_{i=1}^{n}\sum_{j\neq{}i}e^{i\omega(t_{\rm i}-t_{\rm j})}\right). \] For incoherent emission, the cross-term averages to zero for large \(n\), so \(|E|^{2}\propto{}n\), while for coherent emission, where \(\omega\Delta{}t\ll1\), they add, giving \(|E|^{2}\propto{}n^{2}\).

I noted in class that for small \(n\) one can see interference effects, as seen in, e.g., frequency banding in Crab giant pulses, which are composed of multiple nanoshots (e.g., Bij et al. 2021ApJ...920...38B, where bands in one pulse are observed to drift during the scattering tail, which is used to infer the plasma responsible for the giant pulses moves highly relativistically, with \(\gamma\simeq10^{4}\)).

Note that coherence can mean related things in different contexts. Here, it was the coherent emission of particles. We earlier discussed how scintillation relies on a source not being resolved, so that scattered rays can interfere coherently (but do not have to be produced by coherent emission – stars twinkle). And one can also coherently combine radiation, e.g., with a good enough mirror, or by applying appropriate delays to signals from multiple telecopes.

RL Figs 2.1 to 2.3 present nice examples of received signals and their Fourier transforms. The primary things to remember are that

• A delta function in one domain is flat in the other (special case of Gaussian goes to Gaussian, with widths inversely related).
• A time offset corresponds to phase shift.
• Multiplication in one domain corresponds to convolution in the other.
• Derivatives with time produce factors of frequency.

As a specific example, I mentioned a Fourier transform of a specific time segment. This corresponds to multiplication of the (presumably) infinite time stream with a top-hat function, corresponding to convolution in the frequency domain with a \(\rm sinc\) function. Since this mixes frequencies between different channels, by-channel dedispersion of spectra will produce artefacts, where the bits of signal from neighbouring channels are dedispersed with the wrong frequency. This is seen in any FRB or pulsar publication of data taken with the CHIME backend, e.g., in the Bij et al. 2021ApJ...920...38B paper discussed above.

The statement that the emission of a single electron passing an ion produces a flat spectrum at low frequencies is really true only for \(\omega\ll{}v/b\), and is difficult to appreciate except on a logarithmic scale for \(\omega\). As I was confused, I followed Pad §6.9.1 and calculated the expected spectrum explicitly for an electron accelerated by an ion, using examples/brems1p.py (as of this writing, a very poorly finished script, but it produces the figure I showed in class).

I mentioned how in a problem like this, I tend to try to work in sensible units, here perhaps \(L/L_{\rm \odot}\), \(M/M_{\rm \odot}\), \(R/R_{\rm \odot}\) and \(T/10^{4}{\rm\,K}\). The latter since that is a typical temperature in the warm, ionised phase of the interstellar medium. Arguably, the typical temperature in the hot phase, \(T/10^{6}{\rm\,K}\), would be more logical, since then Bremsstralung actually is important.

The problem itself is not that realistic, but something similar happens when a star is formed: at first, when in a molecular cloud core, gravity starts to overcome pressure and the core starts to shrink, the temperature stays roughly constant, at a few 10s of K, leading quickly to collapse on the free-fall time scale. This continues until the cloud becomes optically thick to its own radiation, when it will start to heat up inside. Eventually, it reaches hydrostatic equilibrium, by which time it will be fully convective. It then contracts and heats up further, at roughly constant effective temperature at its photosphere. Once it becomes radiative, the effective temperature stars to increase with further contraction, and the lumonisity varies much less, becoming roughly constant once electron scattering dominates the opacity.

Related to problem 5.1, the nice features of working in "useful" units. Here, for HE, \(R/R_{\rm \odot}=0.004(M/M_{\rm \odot})(T/10^{9}{\rm K})^{-1}\). One can check it is about right, since for \(T=10^{7}{\rm\,K}\), it gives \(R\simeq{}R_{\rm \odot}\).

Otherwise, this problem is mostly a question of writing down the constraints, the radius above and \(L_{\rm obs}=4\pi{}d^{2}F=3\times10^{4}L_{\rm \odot}\), \(L_{\rm thin}=1.4\times10^{25}L_{\rm \odot}(\rho/\rho_{\rm \odot})^{2}(R/R_{\rm \odot})^{3}(T/10^{9}{\rm\,K})^{1/2}\), then using that the bend in the spectrum is at roughly \(kT\simeq100{\rm\,keV}\), so \(T\simeq10^{9}{\rm\,K}\) (very hot, but not quite so hot that one has to worry about relativistic effects), and solving.

In the problem set, one found that the effective radius for free-free emission in the Rayleigh-Jeans tail scales as \(R\sim\lambda^{2/3}T^{-1/2}\), so the luminosity goes as \(L_{\rm \nu}\sim{}R^{2}B_{\rm \nu}(T)\sim(\lambda^{4/3}T^{-1})(T/\lambda^{2})\sim\lambda^{-2/3}\), i.e., the temperature dependence drops out. One also found that for weaker lines, the line profile goes as \((1-(v/v_{\rm w})^2)^{1/2}\). The line absorption coefficient scales with temperature as \(T^{-5/2}\), while free-free goes as \(T^{-3/2}\), so the line strength is proportional to \(T^{-1}\).

I discussed papers from my thesis on Cygnus X-3, where in the first paper we had discovered helium emission lines like those seen in Wolf-Rayet stars (1992Natur.355..703V), but in later observations the lines were much weaker, which I explained as being due to most of the wind being ionized by the X-ray source and thus much hotter, with the remaining line emission mostly from the colder part in the companion's shadow. This gave a prediction for how the velocity would change with orbital phase, being blueshifted when the colder part was pointed towards the observer. At that time, the colder, more opaque part, would also obscure the hotter part of the wind, causing a flux minimum, just as observed. See 1993A&A...276L...9V and 1996A&A...314..521V (plus my thesis, 1993PhDT........76V).

We discussed an alternative way of thinking about the relation between observed variation timescales and size. In particular, rather than a shell expanding relativistically as one in RL (which is a very useful model, use for GRBs e.g.), we considered a picture more like that in problem 4.7, with a blob travelling at some velocity \(v=\beta{}c\) at an angle \(\theta\) from the line of sight.

If the blob emits two pulses, separated by \(\Delta{}t_{\rm int}\) (as seen from the side), then in between the two, the light from the first pulse will travel \(c\Delta{}t_{\rm int}\), while the blob will travel \(v\Delta{}t_{\rm int}\cos(\theta)\) towards the observer. Thus, the pulses will arrive separated by \[ \Delta{}t_{\rm obs}=\Delta{}t_{\rm int}(1-\beta\cos(\theta))\simeq\Delta{}t_{\rm int}\left(\frac{1}{2\gamma^{2}}+\frac{\theta^{2}}{2}\right), \] where we used \(\beta=\sqrt{1-1/\gamma^{2}}\simeq1-1/2\gamma^{2}\) and \(\cos(\theta)\simeq1-\theta^{2}/2\), i.e., assumed \(\theta\) and \(1/\gamma\) were small. Since for relativistic motion, little emission is seen beyond \(\theta\simeq1/\gamma\), one derives \(\Delta{}t_{\rm int}\simeq\gamma^{2}\Delta{}t_{\rm obs}\).

For the motion projected on the sky, use that perpendicular to the line of sight the blob's velocity is \(v_{\rm \perp}=v\sin(\theta)\), so between the two pulses, it moves \(\Delta{}r_{\rm \perp}=v_{\rm \perp}\Delta{}t_{\rm int}\). The motion inferred by the observer, however, is, \[ v_{\rm sky}=\frac{\Delta{}r_{\rm \perp}}{\Delta{}t_{\rm obs}}=c\frac{\beta\sin(\theta)}{1-\beta\cos(\theta)}. \] One finds that at the maximum, for \(\beta=\cos(\theta)\), one has \(v_{\rm sky,max}=\gamma\beta{}c=\gamma{}v\). With beaming, one again expects not to see much beyond \(\theta=1/\gamma\). This is also the most probably value, since for random orientations, the probability to see a given θ scales as \(\sin(\theta)\simeq\theta\), so the maximum will be the largest θ one can have before the brightness starts to fall off.

The giant pulses from the Crab are brief, very bright flashes of radio emission. They last about \(1\mu{}s\) and are composed of multiple very short nano-shots. In observations at lower frequency, such as with CHIME, where the sampling time in each channel is \(2.56\mu{}s\), the nanoshots cannot be distinguished, but their interferance can be seen as banding in the spectra, typically of order MHz, but can be wider if the dominant nanoshots happen to be emitted closer together.

We discussed how during a scattering tail, one effectively looks at slightly different angles and hence \(\Delta{}t_{\rm obs}\) will be slightly different, leading to what will look like a Doppler shift in the band frequencies. We observed such shifting bands in Bij et al. 2021ApJ...920...38B, which indeed is what led us to the above picture.

It also implies that the nanoshots would appear to come from slightly different locations on the sky. For the pulse discussed by Bij et al., the inferred spatial difference \(\Delta{}r_{\rm \perp}\) is about 60 km. In later work, we found that more typically, the nanoshots in a giant pulse appear separated by of order 1000 km (Lin et al. 2023ApJ...945..115L). This again is much larger than the naive estimate of the light travel time, consistent with the high values of γ inferred by Bij et al. An interesting consequence of this is that while perpendicular to the line of sight, nanoshots are separated by about 1000 km, along the line of sight, their separtion is another factor γ larger, i.e., they do not really arise close together at all (in the frame of the blob, though, it would not appear to be that long of a distance).

For a particle moving relativistically, the total emitted power in the lab frame is, \[ P = \frac{1}{4\pi\epsilon_{\rm 0}} \frac{2q^{2}}{3c^{3}}\left(\gamma^{4}a_{\rm \perp}^{2}+\gamma^{6}a_{\rm \parallel}^{2}\right), \] where we simply used the expression for the amplitude of the acceleration in the lab frame.

A particle moving at some \(\gamma\) in a constant magnetic field will gyrate with a frequency \(\omega_{\rm B}=qB/\gamma{}m\equiv\omega_{\rm cyc}/\gamma\) (note the missing \(1/c\) in SI units; the speed comes in via the effective increase in mass), the acceleration \(a_{\rm \perp}=\omega_{\rm B}v_{\rm \perp}=\omega_{\rm B}c\beta_{\rm \perp}\) and hence the power is, \[ P_{\rm synch} = \frac{1}{4\pi\epsilon_{\rm 0}} \frac{2q^{4}B^{2}}{3m^{2}c}\gamma^{2}\beta_{\rm \perp}^{2}. \]

Basic idea is simple: equate the above power to \(\dot{e} = \dot{\gamma}mc^{2}\), i.e., \[ \dot{\gamma} = -\frac{P}{mc^{2}} = -A\gamma^{2}\beta_{\rm \perp}^{2},\quad\mbox{with}\quad A = \frac{1}{6\pi\epsilon_{\rm 0}}\frac{q^{4}}{m^{3}c^{3}}B^{2}=\frac{B^{2}}{16.4{\rm\,yr\,G^{2}}}. \]

RL assumed \(\beta_{\rm \perp}\simeq1\), which gives \(\gamma=\gamma_{\rm 0}/(1+A\gamma_{\rm 0}t)\). One can be find a solution for any orientation and speed by writing \(\beta_{\rm \perp}^{2}=\beta^{2}-\beta_{\rm \parallel}^{2}=1-\gamma^{-2}-\beta_{\rm \parallel}^{2}\), where one then uses that \(\beta_{\rm \parallel}\) is constant. With that, \[ \dot{\gamma} = -A\left(\gamma^{2}(1-\beta_{\rm \parallel}^{2}) - 1\right) = -A(\gamma_{\rm r}^{2}-1), \] where we defined \(\gamma_{\rm r}^{2}\equiv\gamma^{2}(1-\beta_{\rm \parallel}^{2})=\gamma^{2}/\gamma_{\rm \parallel}^{2}\). This has the solution, \[ \frac{\gamma_{\rm r}-1}{\gamma_{\rm r}+1} = \frac{\gamma_{\rm r,0}-1}{\gamma_{\rm r,0}+1}e^{-2At/\gamma_{\rm \parallel}} \] One can fairly easily show that for \(\gamma_{\rm r}\gg1\) and \(\gamma_{\rm \parallel}=1\), one recovers the RL solution, while for \(\beta\ll1\), one finds \(\beta_{\rm \perp}=\beta_{\rm \perp,0}\exp(-At)\).

Overall, one sees that the timescale scales with \(1/\gamma\), i.e., for ultrarelativistic particles it can become short. Still, for the interstellar medium, with \(B\simeq1{\rm\,\mu{}G}\), the timescales are shorter than the age of the universe only for \(\gamma\rlap{\raise0.3ex{>}}_{\rm \sim}10^{4}\).

Discussed how my approach to part (b) was simplier, using that for adiabatic processes, and wavelengths scale with the length, and that since the momentum for a particle is inversely proportional to the De Broglie wavelength, \(p\sim1/\ell\). Also discussed how for ultrarelativistic particles, one has \(T\sim\rho^{1/3}\sim1/\ell\) and thus \(p\sim{}e/c\sim{}kT/c\sim1/\ell\), while for non-relativistic ones, one has \(T\sim\rho^{2/3}\sim1/\ell^{2}\) and \(p\sim(2me)^{1/2}\sim{}T^{1/2}\sim1/\ell\).

I leaned very havility on Bradt, Chapter 8, for the description of how the wiki:Crab_Nebula led to the realization that synchrotron emission could be important, and showed his Fig. 8.2 (model for observed multi-frequency spectrum) and Fig. 8.3 (observations through a polarizer).

I also followed his Table 8.1 in inserting some real numbers. For the Crab, with \(B=0.5{\rm\,mG}\) (probably a little on the high side), the cyclotron frequency \(\nu_{\rm cyc}\simeq1.4{\rm\,kHz}\), corresponding to an enery \(h\nu_{\rm cyc}\simeq6{\rm\,peV}\). For optical radiation, with \(h\nu\simeq2{\rm\,eV}\), one thus finds one needs \(\gamma\simeq6\times10^{5}\). This means \(\nu_{\rm B}\simeq2{\rm\,mHz}\) (i.e., a cycle time of \(500{\rm\,s}\)), and a gyration radius \(r_{\rm gyr}=c/2\pi\nu_{\rm B}=0.17{\rm\,au}\). The required electron energy is a very large \(\gamma{}m_{\rm e}c^{2}=300{\rm\,GeV}\); the opening angle of the beam is very small, \(\theta=1/\gamma=0.3{\rm\,arcsec}\); and the cooling time is short, \(t_{\rm cool}\simeq100{\rm\,yr}\), substantially shorter than the lifetime of the nebula, proving the energy is replenished by the pulsar (this is even more obvious for keV X-rays for which the lifetime is a few years).

The solution explains the projection effect well enough, but does not explicitly mention that another factor of \(\cos(\alpha)\) comes from the requirement that the beam is emitted at an angle \(\alpha\) relative to the direction of motion, which means one must have \(v_{\rm \parallel}=v\cos(\alpha)\).

The reason that a particle in a varying magnetic field feels a force in the direction of lower field strength is because as it gyrates around a given field line, it is being affected by field lines that are not quite parallel to that field line, and thus \(\vec{v}\times\vec{B}\) does not point exactly to the centre.

More specifically, the magnetic moment is conserved, which is related to the angular momentum, \[ m = \frac{\gamma{}mv_{\rm \perp}^{2}}{2B} = \frac{qv_{\rm \perp}^{2}}{2\omega_{\rm B}} = \frac{q}{2}r_{\rm \perp}v_{\rm \perp} = \frac{q}{2\gamma{}m}L, \] where we used \(\omega_{\rm B}=qB/\gamma{}m\) and \(r_{\rm \perp}=v_{\rm \perp}/\omega_{\rm B}\).

Also conserved is the total energy, \(E=\gamma{}mc^{2}\). Thus as B increases, \(v_{\rm \perp}\propto{}B^{1/2}\) increases as well, and \(v_{\rm \parallel}\) decreases. At the same time \(r_{\rm \perp}\propto{}B^{-1/2}\) decreases, conserving the total flux through the loop. The acceleration increases, \(a_{\rm \perp}=\omega_{\rm B}v_{\rm \perp}\propto{}B^{3/2}\), and thus the emitted power will scale as \(P\propto{}B^{3}\). Hence, particles will preferentially loose energy while in the stronger field regions.

See Pad 3.8.4 for motion in a varying field, as well as Pad 3.8.5 for drifts induced by electric fields, by other forces such as gravity, or by curved magnetic fields.

We already discussed different cases where the spectrum does not scale as \(\nu^{2}\) in the Rayleigh-Jeans tail. For instance, we found deviations for free-free emission from a wind, where \(F_{\rm \nu}\propto\nu^{2/3}\). There, T is constant, but the opacity varied and thus the effective radius changed (and we had contributions from both optically thick and thin parts). Furthermore, we noticed that emission and absorption lines arise when we see layers with different temperatures inside and outside of of lines.

Here, at a given frequency \(\nu\), we see mostly emission from electrons for which that frequency is the cutoff, \(\nu=\gamma^{2}\nu_{\rm cyc}\). Those electrons have energy \(E=\gamma{}mc^{2}\). One can define a temperature for them, which would be proportial to that energy, i.e., \(kT\sim{}E\sim\gamma{}mc^{2}\). So, the radiation we see will scale as \(\nu^{2}T\sim\nu^{2}E\sim\nu^{2}\gamma\sim\nu^{2}(\nu/\nu_{\rm cyc})^{1/2}\sim\nu^{5/2}\). Of course, as one goes up in energy and there are fewer electrons, at some point it is becoming optically thin, and then the spectrum turns over.

The overall picture of having possible competing effects of free-free and synchrotron absorption is used for, e.g., the analysis of radio lightcurves of supernovae. The classic reference is Chevalier 1998ApJ...499..810C; I also showed how the filaments in the Crab nebula absorb some emission through free-free absorption, as seen in LOFAR observations at very low radio frequency by Arias et al. 2025A&A...699A.319A.

Solving for the effective mass change when a particle absorbs a photon was to me less trivial than the solution seemed to assume. One has momentum and energy conservation. With \(m_{\rm f}\) the post-absorption mass, \(v\) and \(\gamma\) its velocity, \[ \gamma{}m_{\rm f}v = h\nu/c \quad\mbox{and}\quad \gamma{}m_{\rm f}c^{2} = mc^{2} + h\nu. \] Dividing momentum conservation by \(c\) on both sides, energy conservation by \(c^{2}\) on both sides, and then squaring gives, \[ \gamma^{2}m_{\rm f}^{2}\beta^{2} = (\gamma^{2}-1)m_{\rm f}^{2} = (h\nu/c^{2})^{2} \quad\mbox{and}\quad \gamma{}^{2}m_{\rm f}^{2} = (m + h\nu/c^{2})^{2}. \] Subtracting the two then gives the desired answer, \[ \frac{m_{\rm f}}{m} = \left(1+\frac{2h\nu}{mc^{2}}\right)^{1/2}. \]

The derivation found that the angular momentum loss corresponding to absorbing one photon and then reimitting it isotropically in the moving particle's frame was, for non-relativistic velocities, \(\Delta{}l/l = -h\nu/mc^{2}\). Having the particle orbit at some separation \(d\) around a star with lumonsity \(L\), then gives \[ \frac{\dot{l}}{l} = \frac{1}{2}\frac{\dot{d}}{d} = -\frac{1}{t_{\rm decay}} \quad\hbox{with}\quad t_{\rm decay} = \frac{L\sigma}{4\pi{}d^{2}mc^{2}}. \] Solving this yields, \[ \frac{d}{d_{\rm 0}} = \left(1-\frac{4t}{t_{\rm decay,0}}\right)^{1/2}. \] For small particles like those considered in the problem, spiral-in is fast (for \(\sigma=10^{-8}{\rm\,cm^{2}}\) and \(m=10^{-11}{\rm\,g}\), I get \(5\) instead of \(50{\rm\,kyr}\); I suspect the mass should have been \(10^{-12}{\rm\,g}\) to give a mean density closer to unity in cgs). Hence, particles like those responsible for the zodaical dust must be continuously produced.

Note that an arguably easier derivation is to consider the particle's point of view: it sees starlight arriving at a slight angle, \(v/c\), thus giving not just radiation pressure outward, but also a small braking effect.

The Poynting-Robertson drag is more important for smaller particles, since \(\sigma\propto{}r^{2}\), while \(m\propto{}r^{3}\) and hence \(t_{\rm decay}\propto1/r\).

One also should take into account radiation pressure, which becomes more important for smaller particles too, \(F_{\rm rad}/F_{\rm grav} = (\sigma{}L/4\pi{}d^{2}c)/(GMm/d^{2}) \propto 1/r\). Hence, below some size, particles are just blown away. Generally, small particles feel, effectively, a reduced mass and hence orbit slower.

If larger particles rotate, their re-emission of the solar flux will not be isotropic: the warmed-up side just rotating out of the sun (dusk) will be hotter than that cooled-down side rotating in (dawn). If rotation is prograte, this will cause them to slowly spiral out, while if the rotation is retrograte, they will spiral in. See wiki:Yarkovsky_effect.

More amusingly, if they have a weird shape, the re-emission can be anisotropic in a way that it changes their rotation. This is thought to be one pathway to forming very close binary asteroids: the idea is that one body spun up until it broke into two parts! See wiki:YORP_effect.

Generally described using the jerk, as measured in the instantanous co-moving frame, and averaged over an appropriate timespan, \[ \vec{F}_{\rm rad} = \frac{1}{4\pi\epsilon_{\rm 0}} \frac{2e^{2}\ddot{\vec{u}}}{3c^{3}} = m\tau\ddot{\vec{u}}, \quad\hbox{with}\quad \tau \equiv \frac{1}{4\pi\epsilon_{\rm 0}} \frac{2e^{2}}{3m_{\rm e}c^{2}} = \frac{2r_{\rm e}}{3c} = 6.27\times10^{-24}{\rm\;s}. \] Here, \(r_{\rm e}=2.82{\rm\,fm}\) is the classical electron radius, for which \(e^{2}/4\pi\epsilon_{\rm 0}r_{\rm e}=m_{\rm e}c^{2}\).

It was noted that the equation is only valid if the timescale over which the particles momentum changes is much larger than \(\tau\), i.e., \(T\simeq{}v/a\gg\tau\), implying, e.g., photon energies well below the electron rest mass.

For the damped oscillator, it was found that the emission spectrum, integrated over all time (i.e., the oscillator has damped completely) is given by, \[ \frac{dW}{d\omega} = E_{\rm osc}\frac{\Gamma/2\pi}{(\omega-\omega_{\rm 0})^{2}+(\Gamma/2)^{2}}, \] where \(E_{\rm osc} = \frac12{}m\omega_{\rm 0}^{2}x_{\rm 0}^{2}=\frac12{}mv_{\rm 0}^{2}\) is the inital energy.

The line width \(\Delta\omega=\Gamma=\omega_{\rm 0}^{2}\tau\) ends up looking particularly nice in wavelength space: \(\Delta\lambda=2\pi{}c(\Delta\omega/\omega_{\rm 0}^{2}) = 2\pi{}c\tau=\frac{4\pi}{3}r_{\rm e}=11.8{\rm\,fm}\) for any oscillator. This is very narrow.

The cross section for absorption is, \[ \sigma(\omega) = \sigma_{\rm T}\frac{\omega^{4}}{(\omega^{2}-\omega_{\rm 0}^{2})^{2} + (\omega_{\rm 0}\Gamma)^{2}}, \] and RL discuss the three relevant regimes:

• \(\omega\gg\omega_{\rm 0}\), where photons are too small and fast to notice the oscillator, so \(\sigma(\omega)\simeq\sigma_{\rm T}\).
• \(\omega\ll\omega_{\rm 0}\), where photons are big and slow that that they only can polarize the oscillator, so \(\sigma(\omega)\simeq\sigma_{\rm T}(\omega/\omega_{\rm 0})^{4}\); and
• \(\omega\sim\omega_{\rm 0}\), where they interact resonantly, so

\[ \sigma(\omega) \simeq \frac{\pi\sigma_{\rm T}}{2\tau}\frac{\Gamma/2\pi}{(\omega-\omega_{\rm 0})^{2}+(\Gamma/2)^{2}}. \]

Integrated over all frequencies, one finds the classical cross section, \[ \sigma = \int_0^{\infty}\sigma(\nu)d\nu = \frac{1}{4\pi\epsilon_{\rm 0}}\frac{\pi{}e^{2}}{m_{\rm e}c} = \pi{}r_{\rm e}c = 2.65\times10^{-6}{\rm\;m^{2}\,Hz}. \] This is used generally, with quantum effects encoded by an extra factor \(f\) called the oscillator strength.

The maximum cross section is nice if written in terms of wavelength, \(\sigma(\omega_{\rm 0}) = \sigma_{\rm T}/\tau\Gamma = \sigma_{\rm T}/\tau^{2}\omega_{\rm 0}^{2} = 6\pi{}c^{2}/\omega_{\rm 0}^{2} = (3/2\pi)\lambda_{\rm 0}^{2}\).

As noted in RL, for \(\omega\ll\omega_{\rm 0}\), the oscillator becomes polarized, with the resulting small dipole moment oscillating with the incident field, leading to Rayleigh scattering.

If the oscillator is an electron in a molecule, and \(\omega_{\rm 0}\) corresponds to some electronic transition, there may be vibrational transitions in the molecule well below \(\omega\), i.e., one has \(\omega_{\rm vib}<\omega<\omega_{\rm 0}\). These transitions can get excited during the interaction, leading to re-emission at the difference frequency, \(\omega-\omega_{\rm vib}\). This is called Raman scattering; it is used to study, e.g., to study material composition.

A related process can happen in atoms. For instance, if a photon is close to Lyβ, i.e. has an energy close to that required to excite to \(n=3\), in the tail of the resonance, the atom can get temporarily excited to that not-quite-right state. Usually, it will just fall back to \(n=1\), i.e., the process is like Rayleigh scattering, but occasionally it will decay to \(n=2\), emitting a photon close to Hα, and only then decay back to \(n=1\) by emitting a Lyα photon.