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), set 3 (due Dec 8).
• 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 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.

\(S=\frac43aVT^{3}\), 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 \(0.023\,R_{\rm \odot}\), 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, \(\omega_{\rm min}=\omega_{\rm p}=(e^{2}n_{\rm e}/m_{\rm e}\epsilon_{\rm 0})^{1/2}\). 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 the 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 an accelerated 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 leant 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.

For non-relativistic scattering, one generally has \[ \frac{\Delta\epsilon}{\epsilon} = \frac{4kT - \epsilon}{m_{\rm e}c^{2}}. \] So, if \(\epsilon\ll{}kT\) as assumed in this problem, one expects the photon energy initially to increase exponentially, with a factor \(4kT/m_{\rm e}c^{2}\) per scattering, and eventually to asymptote to \(4kT\). Writing this as a diffential equation, solving it with sympy, and then simplifying gives, \[ \frac{d\epsilon}{dn} = \frac{\epsilon}{m_{\rm e}c^{2}}(4kT-\epsilon) \quad\Rightarrow\quad \epsilon(n) = \frac{4kT}{1-\exp(-n4kT/m_{\rm e}c^{2})\left(1-(4kT/\epsilon_0)\right)}. \]

So, this starts to asymptote roughly when \(\exp(\ldots)=1/(1-(4kT/\epsilon_{\rm 0}))\simeq\epsilon_{\rm 0}/4kT\), or when \[ \tau^{2}=n\simeq\frac{m_{\rm e}c^{2}}{4kT}\log\left(\frac{4kT}{\epsilon_{\rm 0}}\right). \]

Basically inserting numbers. The main message is that one has to be careful in checking that a given calculation is self-consistent: the same electrons producing bremsstralung (or synchrotron) can also upscatter photons.

Assuming the electron initially at rest, energy and momentum conservation give,

\begin{align} \varepsilon_{\rm i} + m_{\rm e}c^{2} &= \epsilon_{\rm f} + \gamma{}m_{\rm e}c^{2},\nonumber\\ \frac{\epsilon_{\rm i}}{c}\hat{n}_{\rm i} &= \frac{\epsilon_{\rm f}}{c}\hat{n}_{\rm f} + \gamma{}m_{\rm e}\vec{\beta}c\nonumber \end{align}

Moving the final electron terms to the left and the rest to the right, multiplying by \(c\) on each side for the momentum equation, and squaring both gives,

\begin{align} \gamma{}^{2}m_{\rm e}^{2}c^{4} &= \left(\epsilon_{\rm i} - \epsilon_{\rm f} + m_{\rm e}c^{2}\right)^{2},\nonumber\\ \gamma{}^{2}\beta^{2}m_{\rm e}^{2}c^{4} &= \left(\epsilon_{\rm i}\hat{n}_{\rm i} - \epsilon_{\rm f}\hat{n}_{\rm f}\right)^{2}.\nonumber \end{align}

Using that \(\gamma^{2}\beta^{2}=\gamma^{2}-1\), equating the two yields, \[ \epsilon_{\rm i}^{2} - 2\epsilon_{\rm i}\epsilon_{\rm f}\cos \theta + \epsilon_{\rm f}^{2} = \epsilon_{\rm i}^{2} - 2\epsilon_{\rm i}\epsilon_{\rm f} + \epsilon_{\rm f}^{2} +2(\epsilon_{\rm i}-\epsilon_{\rm f})m_{\rm e}c^{2} \quad\Rightarrow\quad \epsilon_{\rm f} = \frac{\epsilon_{\rm i}}{1+(\epsilon_{\rm i}/m_{\rm e}c^{2})(1-\cos \theta)}. \] At noted in RL, the result is nicer in terms of wavelengths: \[ \lambda_{\rm f}-\lambda_{\rm i} = \lambda_{\rm C}(1-\cos \theta) \quad\hbox{with}\quad \lambda_{\rm C} = \frac{h}{m_{\rm e}c} = 2.426{\rm\;pm}. \]

• Non-relativistic case: \(\Delta\epsilon=(\epsilon/m_{\rm e}c^{2})(4kT-\epsilon)\).
• Relativistic case: \(\Delta\epsilon\simeq\frac34\gamma^{2}\epsilon\), i.e., \(\gamma^{2}\) change in energy (ignores \(\epsilon_i\), since now much smaller).
• Limit from energy conservation, \(\epsilon_{\rm f}<\epsilon_{\rm i}+\gamma{}m_{\rm e}c^{2}\), so of order \(\gamma{}m_{\rm e}c^{2}\), since \(\epsilon_{\rm i}\) must be smaller than \(\gamma{}m_{\rm e}c^{2}\) too (otherwise the electron would, on average, gain enery; also the cross-section drops steeply for photons with \(\epsilon>m_{\rm e}c^{2}\) in the electron rest frame).
• Compton \(y\) factors for thermal distribution \[ y=\max(\tau_{\rm es}, \tau_{\rm es}^{2}) \left\{\begin{array}~ \displaystyle \frac{4kT}{m_{\rm e}c^{2}} & \quad\hbox{non-relativistic}\\ \displaystyle \left(\frac{4kT}{m_{\rm e}c^{2}}\right)^{2} & \quad\hbox{relativistic}\\ \end{array}\right. \]
• Total emitted power \(P=\frac{4}{3}\sigma_{\rm T}c\gamma^{2}\beta^{2}U_{\rm ph}\). Calculating cooling times, one finds \(t_{\rm cool,rel}=\gamma{}m_{\rm e}c^{2}/P\simeq4{\rm\,ms\,}\gamma^{-1}T_{\rm 6}^{-4}\) (\(\gamma\gg1,~\beta=1\)) and \(t_{\rm cool,nr}=\frac12{}m_{\rm e}v^{2}/P=2{\rm\,ms\,}T_{\rm 6}^{-4}\) (\(\gamma\simeq1,~\beta\ll1\)), where \(T_{\rm 6}=T/10^{6}{\rm\,K}\). See Pad, §6.6.
• Ratio of synchrotron vs inverse compton power: \(P_{\rm sync}/P_{\rm compt}=U_{\rm B}/U_{\rm ph}\) (electrons wiggled by EM fields in both cases!).

Found before that \(\nu_{\rm cyc}\simeq10^{3}{\rm\,Hz}\). Since the peak of the synchrotron emission is at \(\sim10^{15}{\rm\,Hz}\), or \(\sim4{\rm\,eV}\) (near UV), there are ample electrons with \(\gamma\simeq10^{6}\). Letting these scatter the near-UV photons, one gets \(4{\rm\,TeV}\) ones, which in Earth's atmosphere interact with electrons, causing them to travel at high Lorentz factor, faster than the speed of light in air, and emit Cherenkov light. See Bradt Fig. 8.2 (and p. 302 for synchrotron, p. 340 for synchotron self-Compton).

(Also showed Figs 9.3, 9.5, and 9.6 from Bradt, about inverse compton in black-hole X-ray binaries and blazars.)

Clusters of galaxies are filled with hot gas, \(kT\simeq10{\rm\,keV}\), observed directly through its bremsstralung emission with X-ray telescopes. As recognized early on by Sunyaev and Zeldovich, CMB photons will be scattered in and out of our line of sight by this gas (preserving total number), with the scattered photons having a typical energy gain of \(4kT/m_{\rm e}c^{2}\simeq0.1\). Given a typical optical depth \(\tau=\int_{\rm cluster}n_{\rm e}\sigma_{\rm T}dl\simeq10^{-3}\), one should thus see effects of order \(10^{-4}\). In detail, one needs to take into account the angle dependence of the boost, with some photons downscattered in energy, and more upscattered. This leads to a distorted shape of the spectrum; see Bradt §9.5 or the review by Carlstrom et al. 2002ARA&A..40..643C.

For understanding the clusters, a nice aspect of the S-Z effect is that it depends linearly on \(\langle{}n_{\rm e}\rangle\), while the X-ray emission scales as \(\langle{}n_{\rm e}^{2}\rangle\) and thus can be biased by clumping.

For cosmology, an advantage is that the S-Z effect does not depend on redshift, so it can be used as a probe of structure formation (at the largest mass scales).

A smaller effect is due to the peculiar motion of clusters, in particular the component \(v_{\rm rad}\) along the line of sight. This leads to a temperature change \(\Delta{}T/T=(v_{\rm rad}/c)\tau\). Since typical velocities are \(v\simeq10^{3}{\rm\,km/s}\), this kinetic S-Z effect is about a factor 30 weaker than the thermal S-Z.

We discussed how it is possible that a Doppler shift can look like a temperature increase: e.g., in the Rayleigh Jeans tail, a blueshift would by itself lead one to see fewer photons at a given frequency, but this is being compensated by seeing a higher rate (time dilation) and the photons being emitted slighly in the forward direction (beaming). The latter effects are why \(I_{\rm \nu}/\nu^{3}\) is Lorentz invariant (RL 4.9). (I also mentioned how I used this subtle variant of Doppler boosting to understand Kepler light curves; see 2010ApJ...715...51V).

RL problem 9.4 introduces the thermal De Broglie wavelength, \[ \lambda_{\rm th} = \frac{h}{(2\pi m_{\rm e}kT)^{1/2}}. \] This is like the regular De Broglie wavelength for particles, \(\lambda=h/p\), but for a typical thermal momentum. It is easiest to see where that comes from by considering that for a single particle, the probability distribution for its momentum in a single dimension is the normal one, \[ f(p_{\rm 1d}) \sim \exp(-E/kT) ~ \exp(-p_{\rm 1d}^{2}/2mkT). \] Requiring that \(\int_{\rm P}f(p)dp=1\), then gives a normalization constant \(1/(2\pi{}m_{\rm e}kT)^{1/2}\).

One can thus see \(\lambda_{\rm th}\) as a typical size of the particle for a thermal distribution, and \(\lambda_{\rm th}^{3}\) as the typical volume occupied by it. With that, \(n\lambda_{\rm th}^{3}\) gives the typical occupation rate of units of momentum phase space; if it goes beyond unity, degeneracy becomes important.

With the thermal De Broglie wavelength, the Saha equation can be written as, \[ \frac{N_{j+1}}{N_{\rm j}} = \frac{U_{j+1}}{U_{\rm j}}\frac{2}{N_{\rm e}\lambda_{\rm th,e}^{3}} e^{-\chi/kT}. \] Note that this does look like the usual Boltzmann equation. What makes its effect different is that \(2/N_{\rm e}\lambda_{\rm th,e}^{3}\) is typically very large: there are many more ways for an electron to be free than there are for it to be bound.

Writing \(\gamma=-\ln(N_{\rm e}\lambda_{\rm th,e}^{3})\), and taking the natural log,

\begin{equation} \ln \left(\frac{N_{j+1}}{N_{\rm j}}\right) = \ln \left(\frac{2U_{j+1}}{U_{\rm j}}\right) + \gamma - \frac{\chi}{kT}.\nonumber \end{equation}

Thus, ignoring the partition functions, one gets \(N_{j+1}/N_{\rm j}\simeq1\) at \[ \gamma - \frac{\chi}{kT} \simeq 0 \quad\Rightarrow\quad kT \simeq \frac{\chi}{\gamma}. \]

To estimate the fractional temperature change \(\Delta{}T/T\) over which the ionized fraction changes significantly, say, \(\Delta\ln(N_{j+1}/N_{\rm j})=1\), calculate \(d\ln(N_{j+1}/N_{\rm j})/d\ln T=\frac{3}{2}+\chi/kT\simeq\gamma\) (since at a transition, \(\chi/kT\simeq\gamma\gg\frac{3}{2}\)). So, \(\Delta{}T/T\simeq1/\gamma\), i.e., transitions are rapid. This implies one typically has to deal with only a few ionization stages of any given element.

For an excited state of an atom, near the transition one will have, \[ \frac{N_{\rm i,j}}{N_{\rm 0,j}} = \frac{g_{\rm i,j}}{g_{\rm 0,j}}\exp(-\chi_{\rm ij}/kT) \simeq \frac{g_{\rm i,j}}{g_{\rm 0,j}} \exp(-\gamma\chi_{\rm ij}/\chi_{\rm j}). \] Since typically the energy difference between levels \(\chi_{\rm ij}\) is smaller but of the same order as the ionization potential, the fraction of an element in an excited state will generally be small. Indeed, often it is easier to calculate it relative to the number in the next ionized state, \[ \frac{N_{\rm i,j}}{N_{0,j+1}} = \frac{g_{\rm i,j}}{2g_{0,j+1}}N_{\rm e}\lambda_{\rm th,e}^{3} \exp((\chi_{\rm j}-\chi_{\rm i,j})/kT). \]

For pure hydrogen, the ratio of interest is \(N_{\rm p}N_{\rm e}/N_{H^{0}}=\delta^{2}N_{\rm H}\). Using that for the Saha equation, \[ \delta^{2} = \frac{g_{\rm p}g_{\rm e}}{U_{H^{0}}} \frac{1}{N_{\rm H}\lambda_{\rm th,e}^{3}}\exp(-\chi_{\rm H}/kT), \] where the ratio of spin states is unity. For the problem, the rest is a matter of rearranging.

In class, we did a simple application to estimating the epoch of recombination, using that the temperature \(T_{\rm CMB}=2.73{\rm\,K}/a\) (where \(a\) is the scale factor), \(N_{\rm H}\simeq\Omega_{\rm B}(H_{\rm 0}^{2}/8\pi{}G)/m_{\rm H}a^{3}=0.037{\rm\,m^{-3}}/a^{3}\), and thus \(\xi=N_{\rm H}\lambda_{\rm th,e}^{3}=3.4\times10^{-24}/a^{3/2}\). To get \(\delta=\frac12\), one then requires, \[ 2\ln(\frac12) = \frac{\chi_{\rm H}}{kT_{\rm 0}}a - \ln\xi_{\rm 0} - \frac32\ln a. \] Since \(\chi_{\rm H}/kT_{\rm 0}\simeq58000\) and \(\ln\xi_{\rm 0}\simeq-54\), \(a\simeq10^{-3}\). Using that for \(\ln a\) and evaluating, one finds \(a=1/1370\), or \(1+z\simeq1370\) and \(T_{\rm rec}\simeq3700{\rm\,K}\), which is in good agreement with what one finds from detailed calculations.

Another way of looking at why recombination is so late, is that there are many photons and relatively few electrons, so that any recombination \(p+e\rightarrow H^{0}+\gamma\) is quickly followed by an ionization, even just by photons far in the Wien tail. For electrons, we found the occupation number \(n_{\rm e}\lambda_{\rm th,e}^{3}\ll1\), but for photons, it is \(n_\nu=1/(\exp(h\nu/kT)-1)\) (RL, eq. 1.49), which is of order unity for \(h\nu\simeq{}kT\).

(Different from what we discussed in class, which was based on RL problem 9.4. I was inspired by Baierlein 2001AmJPh..69..423B, hence the discussion of chemical potential further down.)

Consider a fixed volume V at a fixed temperature T (or, equivalently, constant ρ and T). In thermal equilibrium, systems go to their most probable state, i.e., one maximizes entropy, \(S=k \log Z\), where \(Z\) is the partition function, a sum over all possible states \(i\), weighted by \(\exp(-E_i/kT)\). Usually, one can split contributions, e.g., for non-interacting photons, ions, and electrons, one has \(Z=Z_{\rm \gamma}\times{}Z_{\rm e}\times{}Z_{\rm i}\) (and thus \(S=k\sum\log Z\)).

In the volume, for one particle at some momentum \(p\), the number of phase space elements available is \((V/h^3)\times4\pi{}p^{2}dp\), with a probability \(\exp(-\epsilon_p/kT)\). The total number of phase space elements is thus \(\sim{}(V/h^3)p_{\rm th}^3\), where \(p_{\rm th}\) is some typical momentum associated with the temperature. Doing the integral gives the Maxwellian and \(p_{\rm th}=\sqrt{2\pi{}mkT}\), and one is led back to the typical size, \(\lambda_{\rm th}\equiv{}h/p_{\rm th}\), the ``thermal De Broglie'' wavelength. Then, the number of possible states is simply \(V/\lambda_{\rm th}^3\). For a set of N identical particles, the contribution to the partition function is thus \[Z_N=\frac{[g(V/\lambda_{\rm th}^3)\exp(-\epsilon/kT)]^N}{N!},\] where \(g\) is the number of internal states, the factorial \(N!\) ensures we do not overcount states where two particles are swapped, and \(\epsilon\) is an energy cost beyond thermal kinetic energy there may be for having this particle.

Let's apply this to pair creation, assuming some mix of photons, ions, electrons and electron-positron pairs. Assuming a dilute plasma, their contributions to \(Z\) can be split, i.e., \(Z=Z_{\rm \gamma}\times{}Z_{\rm e}\times{}Z_{\rm i}\times{}Z_{\rm \pm}\) (of course, the physical picture is that there is a formation rate from the interactions of two photons, balanced by an annihilation rate; for the statistics, we are only concerned about the final equilibrium ). Since the electrons and positrons are independent, \(Z_{\rm \pm}=Z_{+}\times{}Z_{-}\), with both given by the above equation with \(\epsilon=m_{\rm e}c^2\), but with \(N_{+}=N_{-}=N_{\rm \pm}\) (we also assume here that the electron density is sufficiently low that we do not have to worry that we are counting regular electrons and pair ones separately). Hence, \(Z_{+}=Z_{-}\), and to find the number of particles, we can just find the maximum of \(S_{+}=k\log{}Z_{+}\), i.e., \[\frac{\partial{}S_{+}}{\partial{}N_{+}} = \frac{\partial{}k\log Z_{+}}{\partial{}N_{+}} = \frac{\partial}{\partial{}N_{+}}kN_{+}\left[\log\left(g\frac{V}{\lambda_{\rm th}^3}\right)-\frac{m_{\rm e}c^2}{kT} -\log N_{+}-1\right]=0,\] where we used that for large \(N\), \(\log N!=N\log N - N\). Solving this for \(N_{+}\), one finds \[N=g\frac{V}{\lambda_{\rm th}^3}\exp(-m_{\rm e}c^2/kT).\] Equivalently, one has \(n\equiv{}N/V=g\exp(-m_{\rm e}c^2/kT)/\lambda_{\rm th}^3\), which has the nice implication that for classical particles, the probability for one with given internal state to exist in a given volume element \(\lambda_{\rm th}^3\) is simple \(\exp(-\epsilon/kT)\). Thus, for this very small volume, the probability becomes significant for \(kT\approx{}m_{\rm e}c^2\). But when does the number of pairs become significant on larger scales? One measure to use is when \(n_{\rm \pm}=n_e\), i.e., when \(\exp(-m_{\rm e}c^2/kT)=_{}\lambda_{\rm th}^3 n_e/g\). For electrons (\(m=m_e\)), one has \(\lambda_{\rm th}=2.4\times10^{-10}T_9^{-1/2}\) cm, and \(n_e=\rho/\mu_e{}m_H=6\times10^{23}(\rho_2/\mu_e)\) cm^-3, so it requires \(T_{\rm 9}\approx{}m_{\rm e}c^{2}/k(11.7+\log{}gT_9^{1/2}/\rho_2)\approx0.6\). This is comparable to what one finds in stellar evolution books (e.g., Fig. 36.1 in Kippenhahn, Weigert and Weiss).

Now to the chemical potential \(\mu\) (and Baierlein 2001AmJPh..69..423B). It is a quantity that enters all thermodynamic potentials (internal energy U, enthalpy H, Helmholtz free energy F, Gibbs free energy G), as an additional term \(\dots+\mu{}dN\), i.e., the energy required to add one particle. For our example of constant T, V, Helmholtz is handiest: \(F(T,V,N)=PV+\sum_{\rm i}\mu_{\rm i}N_{\rm i}\) and thus \(dF=PdV+\sum_{\rm i}\mu_{\rm i}dN_i\), i.e., the \(PdV\) term is zero). For pair plasma, minimizing \(F\) for \(N_{+}=N_{-}\) (holding \(T\), \(V\), other \(N\) constant), one requires \(\mu_{+}+\mu_{-}=0\). With the above microscopic definition, one has \(\mu=\epsilon+kT\log(g\lambda_{\rm th}^3/n)\), and thus one recovers the solution. Doing a similar procedure for ionisation, one finds \(\mu_0=\mu_p+\mu_e\). In general, for any reaction left↔right, one expects that in equilibrium, \(\sum_{\rm left}\mu=\sum_{\rm right}\mu\). (In that sense, the above are missing photons – but these have \(\mu_\gamma=0\).)

All the above was for classical particles, but the same holds for non-classical ones (except of course that one cannot assume a Maxwellian once particles start to overlap, \(\lambda_{\rm th}\approx{}d=n^{-1/3}\)). For instance, for completely degenerate neutron gas, where \(\mu=\epsilon_F\), one now trivially finds that there will be a contribution of protons and electrons such that \(\mu_n=\mu_p+\mu_e\). (Here, there is no \(\mu_\nu\), since the neutrinos escape; for a hot proto-neutron star, where the neutrino opacity is still high, one does need to include it.) Remember, however, that above we derive a final, equilibrium state. The process to get there can be slow – not all baryons are in the form of iron yet!

Pad §1.2.2 presents a nice, simple way to estimate size and energy, starting from the total energy (Hamiltonian), which is given by (non-relativistic, cgs), \[ H = \frac{p^{2}}{2m_{\rm e}} - \frac{Ze^{2}}{r} \simeq \frac{\hbar^{2}}{2m_{\rm e}L} - \frac{Ze^{2}}{L}, \] where in the approximate equality a typical length scale \(L=\hbar/p\) was assumed. Either using the Virial Theorem or finding the minimum by setting \(dH/dL=0\), one finds, \[ L = \frac{\hbar^{2}}{Zm_{\rm e}e^{2}} \equiv\frac{a_{\rm 0}}{Z} \quad\hbox{and}\quad H_{\rm min}=-\frac{Z^{2}e^{2}}{a_{\rm 0}} = Z^{2}\,\hbox{Ryd}, \] where \(1{\rm\,Ryd}\simeq13.6{\rm\,eV}\).

In class, I made an analogy with two distributions of random numbers, including some correlation between the two. The case for equal spin then amounts to some amount of anti-correlation (expected for fermions, for bosons, there would be positive correlation).

The structure of the orbitals is described by \(n,l,m\), with

\(n-1\geq0\)

Total number of nodal surfaces;

\(0\leq{}l\leq{}n\)

Number of nodal surfaces crossing the origin;

\(0\leq{}|m|\leq{}l\)

Number of nodal surfaces crossing the pole;

Hence, the number of nodal spheres is \(n-l-1\).

Time evolution adds a \(e^{-itE/\hbar}\) term. Since all spherical harmonics have an \(e^{im\phi}\) term, one sees that for \(m=0\), as a function of time just the phase of the pattern changes, while for \(m\neq0\), the pattern rotates. So, while there are fixed patterns around the \(z\) axis, there are no fixed ones along the \(x\) and \(y\) axes.

What one takes as the \(z\) axis is arbitrary for a spherical potential, and hence states with the same \(m\) must be degenerate in their energy level. An external electric or magnetic field can break this degeneracy, as does coupling with the spin.

For a \(1/r\) potential like the Coulomb one, the different \(l\) levels are also degenerate, but again the levels are split when the potential deviates from \(1/r\), e.g., because of the presence of other electrons.

The angular momentum of the orbitals does not have a nice classical analogue, as one can only know the total orbital angular momentum, \(L=\sqrt{l(l+1)}\hbar\) and its z-projection, \(L_{\rm z}=m\hbar\). For a bit more explanation than given in RL, see, e.g., wiki:Angular_momentum#Angular_momentum_in_quantum_mechanics.

I presented a physical picture for the selection rules in class, drawing a photon with different polarizations affecting a ground state atom. These drawings were OK, but my interpretation of angular momentum more than sloppy. A better description is in Feynman's lectures, specifically https://www.feynmanlectures.caltech.edu/III_18.html (This includes a lovely discussion of how positronium often decays via a three-photon process, because the two-photon path can be forbidden.)

I find that selection rules are a bit easier to understand by specifically considering absorption, where the photon is disturbing the initial state, causing it to partially overlap with the final one, and thus giving some probability for it to change. Selection rules identify states where the mixing can happen.

I found the description in Andrew Cummin's lecture notes in terms of spherical harmonics quite helpful. Specifically, we get something like, \[ \langle\psi_{\rm f}| \vec{I}\cdot\vec{r} |\psi_{\rm i}\rangle = \int_{\rm 0}^{\infty} r^{2} dr R_{n_{\rm f}l_{\rm f}}^{\ast}(r)rR_{n_{\rm i}l_{\rm i}}(r) \int d\Omega Y_{l_{\rm f}m_{\rm f}}^{\ast} \vec{I} \cdot \hat{r} Y_{l_{\rm i}m_{\rm i}}. \]

Here, \(\vec{I}\) is the polarization of the photon, which we can write in terms of the spherical coordinates as, \[ \vec{I} = I_{\rm x}\sin\theta\cos\phi + I_{\rm y}\sin\theta\sin\phi + I_{\rm x}\cos\theta = \sqrt{\frac{4\pi}{3}}\left( I_{\rm z}Y_{\rm 1,0} + \frac{I_{\rm x}+iI_{\rm y}}{\sqrt{2}}Y_{1,-1} + \frac{-I_{\rm x}+iI_{\rm i}}{\sqrt{2}}Y_{\rm 1,1} \right). \] Thus, the angular integral is one of the form \(\int d\Omega Y_{l_f,m_f}^{\ast}Y_{\rm 1,m}Y_{l_i,m_i}\), where \(m=0, \pm1\). The \(\phi\) dependence of spherical harmonics always has an \(e^{im\phi}\) term, so for the integral to be non-zero requires \(-m_{\rm f} + m + m_{\rm i} = 0\), i.e., \(\Delta m = 0, \pm1\). The \(\Delta l=\pm1\) selection rule is a bit less obvious, but can at least be seen to be true when the initial state is the ground state (i.e., \(l_{\rm i}=m_{\rm i}=0\)). Then, \(Y_{\rm 0,0}=C\) and because the spherical harmonics are all orthogonal to each other, one requires \(l_{\rm f}=1\) and \(m_f=m\).

The selection rules forbid a transition from the \(n=2,l=0\) state to the ground state (\(n=1,l=0\)). I discussed how there are possible transitions (magnetic dipole directly, etc.), but these are all very slow (days to years), and faster and thus most common is two-photon emission, in which two photons with energies summing up to \(10.2{\rm\,eV}\) are emitted.

This process is important for producing continuum ultraviolet emission, and indeed constributes to the overall ultraviolet background, see Kulkarni 2022PASP..134h4302K. Physical details in Duncan, Kleinpoppen and Scully https://doi.org/10.1016/S1049-250X(01)80039-2.

I found wiki:Electron_configuration a big help to understand the relatively concise RL text. The basic idea is to give configurations of electrons over the various \(n,l\) orbitals.

L-S coupling then describes terms using the total orbital (\(L\)), and spin (\(S\)) angular momenta. These may be further split in levels based on the total angular momentum (\(J\)), among which the values of \(m_{\rm J}\) determine the final states.

For ground-state atoms, typically the configurations are filled in order of increasing \(n+l\), using increasing \(n\) in case of ties. Within an orbital, first fill with one spin (conventially \(m_{\rm s}=+\frac12\)), from one side of \(m_{\rm l}\) (conventially the positive one) to the other, then the same for the other spin state.

For these, it is helpful to know how angular momenta are added in quantum mechanics; see wiki:Angular_momentum_coupling.

These are basically applications of the selection rules for electric dipole transitions:

• For the jumping electron, \(\Delta{}l=\pm1\), \(\Delta{}m_{\rm l}=0,\pm1\), \(\Delta{}m_{\rm s}=0\);
• For the state, \(\Delta{}S=0\), \(\Delta{}L=0,\pm1\).
• \(\Delta{}J=0,\pm1\), but not \(J=0\) to \(J=0\).

The last rule also holds for magnetic dipole and electric quadrupole transitions.

For grotian diagrams for some elements, see http://hyperphysics.phy-astr.gsu.edu/hbase/Atomic/grotrian.html.

Helium

Note the separation in triplet and singlet states. The 2s states in both are metastable. The singlet one decays like hydrogen, via 2-photon decay, on a \(20{\rm\,ms}\) timescale. The triplet 2s state decays via magnetic dipole emission, on a timescale of \(7850{\rm\,s}\), making it the longest-lived metastable state known. It underlies the wiki:Helium-neon_laser. I also briefly mentioned it helping cause a glow at \(1.083{\rm\,\mu{}m}\) from the resonance scattering of sunlight in the \({}^{3}P\rightarrow{}^{3}S\) transition; see Kulkarni 2025arXiv250914499K.

Lithium

Note how the higher levels become like those of hydrogen, since the outer electron effectively sees a unit-charge nucleus (this can be seen for helium too). Because even the ground state is further from the nucleus, the electron is not very strongly bound. The \(2s\) and \(2p\) states have quite different binding energy, with \(2s\) lower as it penetrates the inner shell more. The difference in energy corresponds to the strongest line, which is used to determine whether stars have lithium.

Sodium

The transition between \(3p\) and \(3s\) is responsible for the yellow light emitted by sodium lamps.

Nitrogen

Mostly shown just because this has doublets and quartest instead of singlets and triplets.

RL Fig. 10.3 visualizes the effect of collisions as sudden phase shifts in a sinusoidal electric field, i.e., a multiplication with a function \(f(t)=e^{-i\phi(t)}\). In the Fourier domain, this would then correspond to a convolution of a delta function with the Fourier transform of \(f(t)\). Writing out, \[ \left|{\cal F}(f(t))\right|^{2} = \left| \int dt\; e^{i\phi(t)}e^{i\omega{}t} \right|^{2} = \int dt_{\rm 1}\int dt_{\rm 2}\; e^{i(\phi(t_{\rm 1})-\phi(t_{\rm 2})}e^{i\omega(t_{\rm 1}-t_{\rm 2})}. \] On average, the phase term only depends on the time difference \(\Delta{}t=t_{\rm 1}-t_{\rm 2}\):for small \(\Delta{}t\), this is unity, since no collision will have happened, while for very large \(\Delta{}t\), it approaches 0, as a collision is more or less guaranteed to have happened, causing a random phase difference; Poisson statitics gives that the average is \(\exp(-|\Delta{}t|\nu_{\rm coll})\). Hence, rewriting as two integrals over \(t_{\rm 1}\) and \(\Delta{}t\), and then realizing that the \(t_{\rm 1}\) part is just a delta function, one is left with, \[ \int d\Delta{}t\; e^{-|\Delta{}t|\nu_{\rm coll}} e^{i\omega\Delta t}, \] i.e., a Fourier transform of \(\exp(-|x|)\), which is a Lorentzian.

The standard Voigt profile is a convolution of a Lorentzian and a Gaussian,

\begin{eqnarray} \phi_{\rm L}(\nu) &=& \frac{\gamma/4\pi}{(\nu-\nu_{\rm 0})^{2}+(\gamma/4\pi)^{2}},\\ \Phi_{\rm G}(\nu) &=& \frac{1}{\Delta\nu_{\rm D}\sqrt{\pi}}e^{-((\nu-\nu_{\rm 0})/\Delta\nu_{\rm D})^{2}}. \end{eqnarray}

Since it is a convolution, if either is very narrow, the width is dominated by the other. For low T, the width is thus dominated by the Lorentzian, and independent of temperature. For high T, the Gaussian dominates, and the width scales as \(\sqrt{T}\). The transition will be when the widths match, i.e., \(kT=\frac18(\gamma/\omega_{\rm 0})^{2}mc^{2}\).

To get a rough sense of the transition temperature, one can use that the transition rate – and thus \(\gamma\) – is roughly \(10^{9}{\rm\,s^{-1}}\). Thus, \((Q/\omega)^{2}\simeq(10^{9}/10^{16})^{2}\simeq10^{-14}\), and \(kT\simeq10^{-15}{\rm\,GeV}\simeq{\rm\,\mu{}eV}\), and \(T_{\rm c}\simeq10^{-2}{\rm\,K}\). Thus, for an optically thin line, Doppler broadening almost always dominates. In astronomical context, this is usually true also taking into account collisional broading.

For optically thick lines, the situation changes. Generally, the expected intensity is \(I_{\rm \nu}=S_{\rm \nu,line} + e^{-\tau_{\rm \nu}}(I_{\rm \nu,phot} - S_{\rm \nu,line})\), where typically a line is formed further out and thus \(S_{\rm line}\) will be lower than \(I_{\rm phot}\), i.e., one gets an absorption line.

For small optical depths \(\tau=N\sigma_{\rm 0}\phi(\Delta\nu)\), the total number of photons absorbed (as measured by, e.g., the equivalent width) is proportional to \(N\), but once part of the line becomes optically thick, it becomes proportional to the width of the optically thick part of the line, \(\propto\Delta\nu_{\tau=1}\). As long as the Gaussian dominates the wings, one has \(\tau\simeq{}NC_{\rm G}\exp(-(\Delta\nu/\Delta\nu_{\rm D})^{2}\), which implies \(\Delta\nu_{\tau=1}\propto(\ln N)^{1/2}\). But when the Lorentzian wings start to dominate, it becomes \(\tau\simeq{}NC_{\rm L}/\Delta\nu^{2}\) (since in the wings, \(\Delta\nu\gg\gamma\)) and thus \(\Delta\nu_{\tau=1}\propto{}N^{1/2}\).

I presented basically Pad §1.4.2 and §1.4.1.

See Bradt §10.3 for a very nice description of how the interaction between the electron and hydrogon magnetic moment works.

Imagine inserting a hot star in neutral ism. Its UV photons will ionize the gas, and the generated photo-electrons, with \(\epsilon_{\rm el}-\chi\simeq{}kT_{*}\) will heat it up, until some equilibrium is reached, where ionizations are balanced by recombinations and heating by cooling. The line-dominated spectra that one typically observes reflects this: lines emitted during recombination, and by de-excitation following collisional excitation (with many excitations being to low-lying excited states, where there are no electric-dipole transitions to the ground state; hence, one often sees forbidden lines).

Different types of ionization nebulae are:

• H II regions around massive young stars, with typical temperatures \(T_{*}\) of 30 to 50 kK, leading to mostly singly ionized H and He, and singly or doubly ionized metals. The regions are often fairly messy, with cooler condensations, turbulence, expansion, and ionization fronts.
• Planetary Nebulae around hot proto-white dwarfs. Temperatures can exceed 50 kK, leading to doubly-ionized He and higher ionization stages of metals (e.g., O III, Ne III, Ne V). The structures often are a bit simpler (though many are bipolar, if they resulted from common-envelope evolution, where giant is stripped, exposing its core).
• Some supernova remnants, like the Crab nebula, where the ejecta are ionized by the UV synchrotron emission from the pulsar wind, which is a much harder spectrum and thus leads to even higher ionization stages. (Other supernova remnants are ionized mostly by shocks.)
• Active Galactic Nuclei, again having much harder ionizing spectra.

The basic constraint is that, in equilibrium, at any given point in the nebula, the ionization rate has to balance the recombination rate. The ionization rate looks like \(\int_{v_{\rm 0}}^{\infty}(4\pi{}J_{\rm v}/h\nu)\alpha_{\rm \nu}(H^{0})d\nu\), where \(\alpha_{\rm v}\) is the cross section and \(J_{\rm \nu}\propto(L_\nu/4\pi{}r^{2})e^{-\tau_{\rm \nu}(r)}\) the local mean intensity.

The recombination rate looks like \(\int_0^{\infty}\alpha_{\rm rec}(v)P(v)vdv\), where \(\alpha_{\rm rec}(v)\propto{}1/v^{2}\) is the recombination rate for electrons with velocity \(v\) – the velocity dependence is due to focussing: at a given impact parameter, slower electrons are bent more towards the ion, and hence have a bigger chance of recombining. The probability distribution \(P(v)\) is normally a Maxwellian to good approximation, and the extra factor \(v\) represents the velocity dependence of the rate of interactions. Combined, one finds that the total recombination rate is not very sensitive to temperature, \(\propto{}1/v\propto{}1/\sqrt{T}\). At a typical nebular temperature of \(T\simeq10^{4}{\rm\,K}\), the rate for hydrogen is \(\alpha_{\rm rec}(T)\simeq4\times10^{-13}{\rm\,cm^{3}/s}\).

One tricky bit is that recombinations directly to the ground state lead to emission of photons that can ionize neutral hydrogen elsewhere, i.e., they have no net effect (unless material is so optically thin that the photons can escape). Instead, one needs to consider only recombinations to the higher levels. Those are following by line transitions to the ground state. Even if those are forbidden, they are typically far faster than the ionization rate, so to good approximation all neutrals and ions will be in the ground state.

As a concrete example, taken from O&F, consider an O7.5 star with \(T_{*}=39700{\rm\,K}\), which emits ionizing photons at a rate \(Q(h\nu\geq\chi)\simeq10^{49}{\rm\,s^{-1}}\) in a nebula with hydrogen density \(n_{\rm H}=10{\rm\,cm^{-3}}\). At a distance of \(5{\rm\,pc}\), one finds that the ionization rate is \(10^{-8}{\rm\,s^{-1}}\), i.e., the timescale for a neutron hydrogen atom to ionize is a few years. In contranst, the timescale for a proton to recombine with an electron is almost 10,000 years, so in equilibrium the degree of ionization is very high, \(1-\xi\simeq4\times10^{-4}\).

Further out, this will change. And the moment the degree of ionization dips, the optical depth to ionizing photons starts to increase rapidly, leading to a transition zone of approximate width \(1/n_{\rm H}\alpha_{\rm \nu}\simeq0.1{\rm\,pc}\) – the precise width depending also on the spectrum: since the ionization cross section \(\alpha_{\rm ion}\propto1/\nu^{3}\), harder photons can penetrate more deeply. Thus, the typical structure of a nebula is that of a sphere in which everything is ionized, surrounded by a relatively thin transition zone, a so-called Strömgren sphere. For neutral helium, the shape will generally be similar to that of hydrogen, except it becomes neutral a little earlier, since the ionization energy is higher. But those photons can also penetrate further (partially as the abundance of helium is lower), so the transition region is wider. In contrast, doubly-ionized helium will typically occur only within its own, smaller zone.

Note that the above idea of a Strömgren sphere is based on the assumption of equilibrium, which may be out of reach is a source is not very luminous and moves (or varies) fast. As an example where that may be important, see my work on the very small cometary shaped nebula around the isolated neutron star RX J1856.5-3754, 2001A&A...380..221V.

Every ionization liberates an electron which carries excess energy, \(\epsilon_{\rm el}=h\nu-\chi\). On average one will get \(\int_{\nu_0}^{\infty}(h\nu-\chi)J_{\rm \nu}\alpha_{\rm \nu}d\nu/\int_{\nu_0}^{\infty}J_{\rm \nu}\alpha_{\rm \nu}d\nu\simeq{}kT_{*}\), where the approximate equality will be true if the spectrum is well in the Wien tail, where \(J_{\rm \nu}\propto{}e^{-h\nu/kT}\) varies much faster than \(\alpha_{\rm \nu}\propto1/\nu^{3}\).

Cooling processes are:

• Recombination, where the excess energy of an electron is carried off by the photon (plus subsequent line emission). Since the recombination cross section scales as \(1/v^{2}\), typically the energy carried off will be a bit less than the typical average energy, \(\frac32kT_{\rm neb}\). In any case, since \(T_{\rm neb}\) is well below \(T_{*}\), recombination cooling on its own cannot compensate for the heating by ionization.
• Free-free emission (typically only a small contribution, but does produce nebular continuum).
• Collisionally excited line emission. Most efficient would be on neutral hydrogen or helium, but those have high excitation levels, well out of reach for typical nebular temperatures (for temperatures high enough to excite these levels, cooling is very fast; indeed, this is why there is no thermally stable state for interstellar gas between the warm and hot ionised phases: gas either is about 10⁴ or \(10^{6}{\rm\,K}\)). Hence, excitation of lower levels in neutral or ionized metals dominate.

For the collisional excitation, the lower-lying levels typically have no electric dipole transitions to the ground state, so lead to forbidden line emission, which can escape the nebula directly (since interaction cross-sections are minimal too). This also means the excited state is relatively long lived, and for high enough electron density, collisional de-excitation becomes important; above the so-called critical density, levels will tend to those predicted by thermal equilibrium.

The cross-section for excitation is usually written as \((\pi\hbar^{2}/m_{\rm e}^{2}v^{2})\Omega(1,2)/g\), where one recognizes that the first part is just \(\pi\lambda_{\rm dB}^{2}\) (with \(\lambda_{\rm dB}\) the De Broglie wavelength), \(g\) is a statistical weight, and \(\Omega(1,2)\) is a function of energy that has clear bumps when the electron energy matches the excitation energy. Integrated over a Maxwellian, however, the cross sections become quite smooth.

The spectrum will be a combination of collisionally excited lines discussed above, recombination lines, and bits of free-free and bound-free continua.

For the recombination spectrum, one has to deal with the fact that recombination can be to any level, so one needs to consider the whole cascade of line transitions that follows. Special treatment is needed for transitions directly to the ground state, since the resulting photons can be absorbed by other ions or neutrals in the ground state.

For instance, for hydrogen, a Lyα photon will keep being absorbed and emitted until it is either absorbed by dust (i.e., no net cooling effect), or escapes the nebula. For higher Lyman lines, every time the photon is absorbed, there is a small chance that it will not decay as the same Lyman line, but rather make another transition (e.g., Lyβ might lead to Hα and then Lyα).

Typically, the above complication is treated by going to one or the other extreme:

• Case A: the nebula is optically thin to Ly transitions, i.e., one can ignore the above.
• Case B: the nebula is optically thick to Ly transitions, i.e., for purposes of calculating the line spectrum, one simply treats the cascade as ending at 2p and 2s (for cooling purposes, one then still has to decide what to do with the Lyα lines from 2p; the two-photon emission from 2s can of course simply escape).

In more detail, calculations of the nebular spectrum have to deal with some cute complications. For instance, Lyβ at 1025.72Å is nearly coincident in wavelength with a transition in OI from \(2p^{4}\,{}^{3}P\) to \(3d\,{}^{3}D\) at 1025.76Å. This "Bowen fluorescence" can lead to OI lines at 11286.9 and 8446.36Å.

Nebular temperatures can be measured using lines that arise from different collisionally excited states of the same ion, so that their strengths will be \(\propto{}e^{-\Delta{}E/kT}\). For instance, in many nebulae, one sees [OIII] lines at 5007 and 4959Å that arise from the collisionally excited \({}^{1}D_{\rm 2}\) singlet state directly to the \({}^{3}P\) ground state, as well as emission at 4367Å, which arises from a transition between a more excited \({}^{1}S_{\rm 0}\) state to \({}^{1}D_{\rm 2}\). Hence the ratio between the strength of this latter line relative to those of the former gives an estimate of the temperature.

For estimates of the density, one uses that at higher densities, collisional de-excitation becomes more important, changing the line ratios. This will happen around the critical density for the given atom or ion, which is different for different ions, so one can use different line ratios in different regimes.

Note that the permitted lines that arise from recombination (such as the hydrogen Balmer lines) are not helpful to infer temperatures, as their ratios are completely determined by the recombination cascade (and recombination to different levels has very weak temperature sensitivity), and thus are essentially fixed once one knows whether one is in case A or B (e.g., Hα is about 3 times stronger than Hα in case B). These fixed ratios are useful, however, to estimate the reddening towards a nebula.

Other temperature measurements use the continuum. One methods relies on the continuum depending differently on temperature than the lines. Another uses that at long enough wavelength the whole nebula becomes optically thick (recall that the free-free opacity scales as \(\lambda^{3}\)). In that case, it total flux must be \(\pi{}R^{2}B(T)\), and at long wavelengths \(B(T)\) will simply be that of the Rayleigh-Jeans tail, i.e., \(\propto{}T\).