12. Magnetohydrodynamics

1. When is a fluid magnetised?

"Magnetised" is not one question. It is two, and the same gas can answer them differently: can the field push the gas? — which is the plasma beta — and does a particle remember the field between collisions? — which is the magnetisation $\omega_{\rm c}\tau$. The first is settled for all five census rows in this section; the second is a number this module does not need until §8, where it is computed for the intracluster medium and comes out at $10^{12.71}$, and it is defined here so that §8's $(\omega_{\rm c}\tau)^{-2}$ arrives with its symbol already known.

This is why $\beta$ is the census variable and $v_{\rm A}$ is not. The Alfvén speed needs a mean mass per particle and therefore a composition; $\beta$ needs neither. The table below prints both, and prints (1.2) beside the ratio of two separately computed speeds, so that if a $\mu$ is ever changed in one place and not the other the last column stops reading $1.0000$.

Two things in that figure are worth stating in words. First, $\beta$ is a ratio and not a field strength: the corona at $10$ G and the cloud at $10\ \mu$G — six decades apart in $B$ — are equally magnetised in this sense. Second, as Fig. 1 shows, four of the five rows are far from $\beta = 1$ and the fifth is not. Where $\beta \gg 1$ the gas drags the field; where $\beta \ll 1$ the field steers the gas; the solar wind does both, at different radii, and §6 finds the radius where the balance tips.

1.1 Every speed in this module, and how many components each one has

A one-dimensional quantity compared with a three-dimensional one has been the defect found in four consecutive modules of this book. Module 10 §6.4 adopted the remedy that works, which is not a resolution to be careful but a printed table. This module inherits it, and it is load-bearing here: the factor $\sqrt{3/2}$ that appears in §8.1 is nothing but a component count.

No two of the three thermal speeds are compared anywhere in this module before a reader has been given this table. The two bulk speeds, $c_s$ and $v_{\rm A}$, are compared once already — at Proposition 1 and in the census table above — and that comparison is (1.2), which is an identity and needs no component count. The three thermal speeds stand in the fixed ratios $v_\perp : v_{\rm rms} : \bar v = 1 : 1.2247 : 1.1284$, and each is correct for its own job.

2. The induction equation, and why flux freezing is a number

The field obeys Maxwell's equations; the gas obeys Module 2's. What couples them is Ohm's law, and what comes out is a single equation for $\mathbf B$ with exactly two terms.

Equation (2.3) is the reason astrophysical plasmas are good conductors, and it is not the reason a reader expects. A large plasma is a good conductor through its size, not through its density. Two gases at the same temperature have the same resistivity whether one is a million times denser than the other; what differs is $L$, and $L$ enters $\mathrm{R_m}$ linearly. The table makes the point, and its last column is (2.4) used as a self-check on the column above it.

The result of Proposition 4 is Alfvén's, in a one-page note of 1942. That paper was not fetched for this book and is cited for the result only, in the same way Module 1 cites Maxwell (1867); its volume, pages and date were confirmed at Crossref and nothing else about it is claimed here. The proof above is this module's own.

3. Magnetic pressure and magnetic tension

The field acts on the gas through the Lorentz force on the current it carries. What that force is, mechanically, is the content of this section, and it is what Module 2 §11, Module 3 §11, Module 4 §8, Module 5 §8 and Module 7 §5 have each been promised.

The last column is $1/\beta$ by construction and carries no new information; it is printed so that a reader can see the two pressures as pressures. Note that the photosphere's $p_{\rm mag}\approx1\ \mathrm{dyn\,cm^{-2}}$ is the dynamic pressure $\tfrac12\rho v^2$ of sea-level air moving at $0.41\ \mathrm{m\,s^{-1}}$ — a draught, not a wind —, against a gas pressure comparable to the Earth's atmosphere — which is the sense in which a $5$ G field is, for the photosphere, negligible.

4. Three wave speeds, and what one of them costs Module 7

Module 4 derived sound from a gas whose only restoring force was its own pressure, and its captures-and-misses table names magnetic pressure and tension as the restoring forces it left out. Adding them turns one wave into three.

All speeds in km s⁻¹. Note the photosphere row: with $v_{\rm A}$ three hundred times smaller than $c_s$, the field changes the wave speeds of Module 4 by three parts in a million, which is why Module 4 was entitled to ignore it and says so.

4.1 Module 7's one-degree bound, recomputed — a validation, not a check

Module 7 §9.5 states the debt in one line: "A purely hydrodynamic calculation that needs a one-degree statement about a field it never mentions is a calculation waiting for magnetohydrodynamics." Its Proposition 9 says that Kelvin–Helmholtz growth at the observed rate requires $$ B\cos\theta \;\lt \; \Delta U\sqrt{\frac{2\pi\rho_h\rho_l}{\rho_h+\rho_l}} \tag{4.2}$$ for equal fields on both sides of the interface, with $4\pi$ in place of $2\pi$ if the field is one-sided. That inequality is recomputed here from Proposition 6's dispersion relation rather than copied, so that the comparison below is able to fail.

And here is the gap this module cannot fill. The book's shape is: derive the terrestrial result, change the regime, check against a measured astronomical number. Module 4 had the speed of sound in air, Module 6 the dry adiabat, Module 10 the Kolmogorov constant. Magnetohydrodynamics has terrestrial measurements — Alfvén waves were first seen in liquid mercury, and every tokamak is an MHD experiment — and none of them was read for this book. What occupies the terrestrial slot above is a reproduction of Module 7, which is a check against this book and not against a laboratory. The four checks of §§5–8 are all astronomical, and a reader should weigh them knowing that no bench-top number stands behind them.

5. The anchor: the Parker spiral

Module 9 built a wind that leaves the Sun radially and is still accelerating at 1 au. The Sun rotates. A field line is frozen to the gas by Proposition 4 and its foot is carried round by the rotation while its body streams outward — so the line is drawn into a spiral, and the shape of that spiral is fixed by two numbers this book already has: the solar rotation rate and the wind speed. Nothing is fitted.

The observable consequence is the total field's radial falloff. Writing $x \equiv \Omega(r-r_0)\sin\theta/v$, $$ |\mathbf B| = B_r\sqrt{1+x^2}, \qquad \frac{\mathrm{d}\ln|\mathbf B|}{\mathrm{d}\ln r} = -2 + \frac{x^2}{1+x^2}\left(\frac{r}{r-r_0} - \frac{\mathrm{d}\ln v}{\mathrm{d}\ln r}\right). \tag{5.2}$$ Fig. 2 draws both halves of this. At small $r$ the field is nearly radial, $x\ll1$, and the index is $-2$. At large $r$ it is nearly azimuthal, $x\gg1$, and the index tends to $-1$. Between the two, over the range a spacecraft actually flew, the index is something in between — and that is a measurement.

5.1 Check 1 — the anchor

Venzmer & Bothmer (2018) fitted power laws $x(r) = d\,r^{e}$ to combined Helios 1 and 2 data over $0.29$–$0.98$ au and published both mean and median fits, with errors on every coefficient. Module 9 reads the mean fits; Module 2 reads the median row. This module prints both, as equals, and never mixes them inside one comparison — which is the rule an earlier draft of the generator broke by quoting the median field against the mean density, two denominators from one table.

The control is what makes this a measurement and not a coincidence, and it is Module 2's preferred shape: one exact conservation law confirmed and one modelling assumption tested, from the same table. Mass conservation through a steady spherical flow forces $\rho\propto r^{-2-\alpha_v}$, where $\alpha_v$ is the measured velocity exponent of the same fit set — which predicts $-2.0580$ (median) and $-2.0490$ (mean). The measured density indices are $-2.093\pm0.046$ and $-2.010\pm0.038$, gaps of $0.0350$ and $0.0390$. These are divided by the density exponent's own year-to-year scatter, $0.072$, and not by the field's $0.110$ — Venzmer & Bothmer's Fig. 9 gives one scatter per fitted quantity, and using the field's here would be the two-denominators error this section refuses two paragraphs above. The gaps are therefore $0.49$ and $0.54$ scatter units. So this data set can see a near-inverse-square falloff when there is one. The field does not show one, and (5.1)'s second half says why: the field is wound, and the density is not.

5.2 Sensitivity, priced rather than hidden

The largest shift in that table is $0.0194$, smaller than the year-to-year scatter $0.110$ by a factor of $5.7$. The verdict turns on none of them. That sentence is the point of the table.

6. The Alfvén surface, bracketed by two measurements

Close to the Sun the field is strong and the wind slow, so $v \lt v_{\rm A}$ and the field controls the flow: it can enforce corotation, and it carries away angular momentum. Far out the wind is fast and the field weak, so $v \gt v_{\rm A}$ and the flow drags the field, which is what §5 just measured. In between there is a surface where the two are equal, and it is the boundary between those two regimes.

6.1 Check 2 — and the result is a bracket

Fig. 3 shows why the bracket is a result rather than an accident of the headline choice: the band of eight variants is narrower than the gap between the two published surfaces. Two extrapolations are priced here and neither is hidden. First, $v\propto r^{+0.058}$ is a fit over $0.29$–$0.98$ au used at about $0.08$ au, a factor $3.5$ outside its own range; it moves $r_{\rm A}$ by $6.9$ per cent against a constant speed, which is small, and this book prices that kind of thing rather than assuming it. Second, the real wind is slower close in than any power law fitted beyond $0.29$ au says — which is the physical reason a Weber–Davis radius comes out smaller than this extrapolation does, and it is the direction of the discrepancy with Verscharen.

7. Magnetic support against collapse

Module 5 §8 is owed this: magnetic pressure and tension add support against collapse, and the statement that results is not an extra pressure term but a critical mass-to-flux ratio. Below it no field strength is enough; above it the cloud is held for as long as the flux stays with the gas.

Applied to this module's $10$ pc cloud — $n_{\rm tot} = 100\ \mathrm{cm^{-3}}$ at $\mu = 2.33$, $R = 5$ pc, $B = 10\ \mu$G — the two geometries give:

The cloud's mass is $2.993\times10^{3}\,M_\odot$. Both answers are supercritical: on this census field the cloud is not held. The two differ by the coefficient ratio $1.2657$.

7.1 Check 3 — the coefficient, and the sign

Troland & Crutcher (2008) observed 34 dark cloud cores for about $500$ hours with Arecibo in OH Zeeman, and converted each measurement to a mass-to-flux ratio with their equation (2), $$ \lambda \;=\; 7.6\times10^{-21}\,\frac{N({\rm H_2})}{B_{\rm los}}, \qquad N \text{ in cm}^{-2},\ B \text{ in }\mu\text{G}, \tag{7.3}$$ stating in the line above it that their critical ratio is Nakano & Nakamura's — the same constant (7.2) computes. So their coefficient and this module's can be checked against each other rather than merely quoted together.

8. Transport across a field, and the answer that is wrong by twenty-three decades

Module 2 §9 put two problems together and pointed at this module for both: the intracluster medium's Reynolds number of about $600$, which is not large enough for the turbulence that is observed, and the thermal-conduction problem it names in the same sentence and attributes to Module 1. Neither module puts a number on the conduction half: Module 1's Problem K3 compares conduction with the sound-crossing time and finds it slow, and says in the same solution that the comparison the cooling-flow problem needs is against the radiative cooling time, which this book has not made. So the second problem arrives here as a scaling and a suspicion, not as a measured discrepancy. It said the two "have the same suspected resolution — magnetic suppression of transport across field lines — and Module 12 treats both together." This section is that treatment, and it ends by refuting the classical form of the resolution.

Braginskii (1965) does the calculation properly and supplies the coefficients. For $Z = 1$ his equations (2.12) and (2.13) on p. 217 give the electron conduction, $$ \kappa_\parallel = 3.16\,\frac{n_{\rm e}T_{\rm e}\tau_{\rm e}}{m_{\rm e}}, \qquad \kappa_\perp = 4.66\,\frac{n_{\rm e}T_{\rm e}}{m_{\rm e}\omega_{\rm e}^2\tau_{\rm e}}, \qquad \frac{\kappa_\perp}{\kappa_\parallel} = \frac{4.66}{3.16}\frac{1}{(\omega_{\rm e}\tau_{\rm e})^2}, \tag{8.2}$$ and his equations (2.22) and (2.23) on p. 218 give the ion viscosity, which Module 10 §6.2 already uses, with $\eta_0^{\rm B} = 0.96\,n_iT_i\tau_i$ and $\eta_1^{\rm B} = (3/10)\,n_iT_i/(\omega_i^2\tau_i)$.

8.1 Two magnetisations for one plasma, and the exact account of their ratio

The intracluster medium yields two different numbers for $\omega_i\tau_i$, and a module that printed both without explanation would leave a reader with two magnetisations for one gas.

The ratio of the two is $1.2015$, and it decomposes exactly: $$ \frac{\lambda/r_g}{\omega_i\tau_i^{\rm B}} \;=\; \sqrt{\tfrac32}\times\frac{\ln\Lambda_{\rm e}({\rm derived})}{\ln\Lambda({\rm census})} \;=\; 1.224745 \times \frac{37.081}{37.8} \;=\; 1.224745\times0.980985 \;=\; 1.2015. \tag{8.3}$$ The $\sqrt{3/2}$ is exact and it is a component count. A Spitzer mean free path is $v_{\rm rms}\tau$ with the three-dimensional rms speed $\sqrt{3k_{\rm B}T/m}$, while the first row of the table divides it by $\sqrt{2k_{\rm B}T/m}$, the rms of the two components perpendicular to $\mathbf B$ — which is the speed a gyroradius takes. Read §1.1's table and the factor is not a discrepancy but an identity. The second factor is the electron-branch Coulomb logarithm derived here against Module 1's census value; the difference is $1.9$ per cent and is not a correction to either module.

The $\tau_i/\tau_{\rm e}$ row is a self-check: Braginskii's two collision times differ by $\sqrt{2m_{\rm p}/m_{\rm e}}$ exactly, so if the row ever stops reading $1.000000$ one of the two has been edited alone.

8.2 Check 4 — the classical answer is refuted, and the direction is named

The direction of the failure is the physical content, and Fig. 4 is where to read it. The measurement lies far nearer the unsuppressed Coulomb value than the classical perpendicular one. That is not a small correction to (8.1); it says the premise of (8.1) is wrong for a real cluster. The field in a cluster is tangled, so heat and momentum follow field lines that wander, and a particle that has travelled one mean free path along a line has also moved a long way across the mean field direction. The step is therefore neither $\lambda$ nor $r_g$, and it is set by the field-line random walk and by pressure-anisotropy microinstabilities.

9. The magnetorotational instability

Module 8 §6.3 names the magnetorotational instability in its list of what this module builds, so it is not optional. It is done locally: the shear is assumed, not derived from a disc model, because disc structure — the $\alpha$ prescription, the thin-disc solution — is Module 11.

The problem it solves is stated in one line. A Keplerian disc has angular velocity $\Omega\propto R^{-3/2}$, so its specific angular momentum $\ell = \Omega R^2 \propto R^{1/2}$ increases outward. By the Rayleigh criterion that is hydrodynamically stable: displace a ring outward and it has too little angular momentum to stay there, and it falls back. A purely hydrodynamic Keplerian disc does not transport angular momentum, and therefore does not accrete. Adding the weakest imaginable magnetic field destroys that stability.

10. Anisotropy, critical balance, and Module 10's debt

Module 10 §7.2 confirmed Kolmogorov's $-5/3$ most cleanly on the magnetic spectrum of the solar wind, and said plainly what was uncomfortable about that: the confirmed exponent belongs to a magnetised plasma, which is not the fluid its Proposition 2 describes, and the same exponent follows from a quite different argument about magnetised turbulence. Its own sentence is the one to keep: "agreement of an exponent does not select between two derivations that give the same exponent."

Note also that the bare exponent $2/3$ is not dimensional. Their relation is $k_\parallel \approx k_\perp^{2/3}L^{-1/3}$, and the outer scale $L$ is what makes the two sides agree in units. A module that printed $k_\parallel \sim k_\perp^{2/3}$ alone would be printing a relation between quantities of different dimensions.

11. What this theory captures, and what it misses

Every row below is an idealisation this module makes, what it costs, and where — if anywhere — it is repaid. Where nobody repays it, the row says so, rather than pointing at a module that will not deliver.

11.1 The point Module 1 asked this module to return to

Module 1 §8 observed that the magnetic rescue of the fluid description is partial, and partial in a measurable way. A field constrains motion across itself and does nothing to motion along itself, so a magnetised collisionless plasma is not an ordinary fluid with a scalar pressure: it has $p_\parallel$ along the field and $p_\perp$ across it, and the two relax toward each other only through collisions the plasma is not having. The resulting description is the Chew–Goldberger–Low or double-adiabatic theory. Pressure anisotropy is routinely measured in the solar wind, and it drives the firehose and mirror instabilities, which scatter particles and so act as the plasma's substitute for collisions.

That is the return, and it is all of it. Single-fluid MHD as built in §2 and §3 assumes a scalar pressure and therefore cannot contain any of it. No instability threshold is printed here, because no CGL source was read for this book, and a threshold quoted from memory is the failure this project keeps finding in its own drafts. What can be said without a source is the structural point: the same anisotropy that Module 1 named is the leading candidate for the microinstabilities that §8 needed and could not compute, so two of this book's unpaid debts are probably one debt.

12. Problems

Every modelling input a solution uses is stated in its own problem. Where an input is a census value with no source, or an assumption rather than a measurement, the statement says so — because a problem that hides its assumptions teaches the wrong habit. All nine are rebuilt from their statements alone by afd/figs/m12_problems_check.py, which imports nothing from the numbers script.

Notation

Every symbol used in this module, with its units and where it is defined. Every glyph this module gives a meaning that an earlier module gave a different meaning is resolved here — $\eta$, $\lambda$, $\beta$, $\sigma$, $\kappa$, $\Omega$ and $\mu$ — rather than left to the reader, and one cross-module constant differs from an earlier module's by a stated amount.

One cross-module constant, and the size of the difference. §8 uses the derived electron-branch Coulomb logarithm $\ln\Lambda = 37.081$ where Module 1's census value is $37.8$ and Module 10 reuses that census value. The difference is $1.9$ per cent, it is not a correction to either module, and it is one of the two factors in (8.3). §1's table of speeds is the companion device for the other factor.

Sources

Every published value this module compares against, with what was read and how. Where a paper was not read, the entry says so, and nothing in this module rests on it.

• Venzmer, M. S. & Bothmer, V. (2018), "Solar-wind predictions for the Parker Solar Probe orbit", A&A 611, A36, arXiv:1711.07534. Equation (10) and Table 3, with Fig. 9 for the year-to-year exponent scatter. The anchor of Check 1. Verified for Module 9 by fetching the arXiv PDF and reading it locally; Table 3 is on page 7. Both the mean and the median fits are used here, always labelled.
• Verscharen, D., Bale, S. D. & Velli, M. (2021), "Flow deflections and Alfvén radii in the solar wind", MNRAS 506(4), 4993–5004. Table 3: $r_{\rm A} = 12.080\pm0.236\,R_\odot$ for the first fast-latitude scan and $9.504\pm0.221\,R_\odot$ for the third. One half of Check 2. Verified for Module 9 and quoted in its own sources list. Their sonic radius, $0.309\pm0.004\,R_\odot$, is a lower limit on a different surface and is named here only so that the two cannot be confused.
• Kasper, J. C. et al. (2021), "Parker Solar Probe enters the magnetically dominated solar corona", Phys. Rev. Lett. 127, 255101. The other half of Check 2. Read from the APS open-access PDF. Their Table I gives three sub-Alfvénic intervals in encounter 8 with median Alfvén Mach numbers $0.79$, $0.49$ and $0.88$, and their summary sentence gives the range as $16$–$20\,R_\odot$ — not the $19$–$20$ that circulates, which is the start of the first interval alone. Their PFSS solution uses a source surface at $2.0\,R_\odot$, "selected to accurately reproduce low latitude coronal holes observed in STEREO", which is evidence that the parameter is fitted per event.
• Braginskii, S. I. (1965), "Transport processes in a plasma", Reviews of Plasma Physics 1, 205–311, Consultants Bureau, translated by H. Lashinsky. All seven coefficients used here were read from the original scan, which has no text layer, by rendering its pages as images: (2.5e) on p. 215 for $\tau_{\rm e}$ with its $4\sqrt{2\pi}$, (2.5i) for $\tau_i$ with its $4\sqrt\pi$, (2.7)–(2.8) on p. 216 for $\sigma_\parallel = 1.96\,\sigma_\perp$, and (2.12), (2.13), (2.15), (2.16) on p. 217 for $3.16$, $4.66$, $3.9$ and $2$. The ion viscosity came in at Module 10. Every one is for $Z = 1$; his Table 1 on p. 216 gives the $Z$-dependence, quoted in §8.
• Zhuravleva, I. et al. (2019), "Suppressed effective viscosity in the bulk intergalactic plasma", Nature Astronomy 3, 832–837, arXiv:1906.06346. The measured half of Check 4. Verified for Module 10. A one-sided bound on the effective viscosity of the Coma cluster, not a value, and their own two caveats are carried in this module's §8.2.
• Mouschovias, T. Ch. & Spitzer, L. (1976), "Note on the collapse of magnetic interstellar clouds", ApJ 210, 326–327. Read from the ADS scan at 400 dpi, because the OCR is unusable. Their equation (2) and their equation (4): the paper prints $c_1 = 0.53$ and never prints $0.13$. Their $c_2 = 0.60$ applies to the maximum external pressure, and their footnote 1 distinguishes the critical mass at fixed flux from the critical mass at fixed density.
• Nakano, T. & Nakamura, T. (1978), "Gravitational instability of magnetized gaseous disks", PASJ 30, 671–679. Read from the ADS scan. Their abstract, p. 671: the disc is stable when $\sigma_0/B_0 \lt (4\pi^2G)^{-1/2}$, which is $1/(2\pi\sqrt G)$ — a column density over a field, which is why it is the flattened-cloud value.
• Troland, T. H. & Crutcher, R. M. (2008), "Magnetic fields in dark cloud cores: Arecibo OH Zeeman observations", ApJ 680, 457–465, arXiv:0802.2253. Check 3. Thirty-four cores in about $500$ h. Their Section 3.2 defines $\lambda$ against Nakano & Nakamura's critical ratio; their equation (2) is (7.3) here. Their geometry-corrected $\lambda_c = 1.4$–$2.1$ and $\lambda_{1/2,c} = 1.7$–$2.6$ are the numbers to quote, and their raw $4.2$ and $5.2$ are not.
• Goldreich, P. & Sridhar, S. (1995), "Toward a theory of interstellar turbulence. II. Strong Alfvénic turbulence", ApJ 438, 763–775. Read from the ADS scan as images. Their abstract gives $k_z \approx k_\perp^{2/3}L^{-1/3}$ and a one-dimensional spectrum $\propto k_\perp^{-5/3}$. It also states, in its third and fourth sentences, that they treat only the symmetric case and that this "precludes application to the solar wind" — quoted in full in §10, because that restriction is what stops this module citing them as a prediction for the solar-wind indices.
• Parker, E. N. (1958), "Dynamics of the interplanetary gas and magnetic fields", ApJ 128, 664–676. Section II was verified for Module 9; the spiral is his equations (24), (26) and (27) on p. 673, read at this module's §5. Note the $(r-b)$ in his (26), which this module carries as $(r-r_0)$.
• Snodgrass, H. B. & Ulrich, R. K. (1990), "Rotation of Doppler features in the solar photosphere", ApJ 351, 309–316. Read from the ADS scan. Their $\omega(\theta) = 14.71 - 2.39\sin^2\theta - 1.78\sin^4\theta$ deg day⁻¹ sidereal is where $24.47$ d comes from, and their own abstract states that their rate is about $2$ per cent faster than the magnetic one — which is why §5.2 treats it as a different tracer and not a different latitude.
• NASA NSSDC Sun Fact Sheet. Read as HTML. Sidereal rotation period $609.12$ hours $= 25.380$ d, adopted at $16^\circ$ latitude, with the law $14.37 - 2.33\sin^2L - 1.56\sin^4L$ deg/day, which reproduces it there. It also prints the field census that §1's table did not take its numbers from.
• Alfvén, H. (1942), "Existence of electromagnetic–hydrodynamic waves", Nature 150, 405–406. Cited for the result only; the paper was not fetched, exactly as Module 1 cites Maxwell (1867). Title, volume, pages and date confirmed at Crossref. The proof of Proposition 4 is this module's own.
• Balbus, S. A. & Hawley, J. F. (1991), "A powerful local shear instability in weakly magnetized disks. I", ApJ 376, 214–222. Cited for the result only; the paper was not fetched. Volume, pages and year confirmed at Crossref. The derivation of Proposition 11 is this module's own, so no number here comes from an unread paper.
• Crutcher, R. M. (2012), "Magnetic fields in molecular clouds", ARA&A 50, 29–63. NOT READ — paywalled, with no open-access copy known to OpenAlex or Semantic Scholar, both queried by DOI. Listed because a reader wanting the full Zeeman compilation should be sent to it. Nothing in this module rests on it; Troland & Crutcher (2008) is the primary survey behind the part of it §7 needed.
• Podesta, J. J., Roberts, D. A. & Goldstein, M. L. (2007), ApJ 664, 543–548, Table 2. Verified for Module 10. Used here only to restate Module 10's anchor in §10, so that the two modules quote one set of indices. No check in this module is built on it.

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