Astrophysical Fluid Dynamics

One field, four equations, and a different regime in every chapter. Final-year undergraduate to first-year graduate level; derivations shown in full.

Foundations

1. Why a Fluid at All?ready Mean free path, the Coulomb logarithm, the Knudsen number, and the systems where the continuum picture fails.
2. Continuity, Euler, Navier–Stokes, energyready Moments of the Boltzmann equation; the closure gap; Lagrangian against Eulerian derivative. Continuity checked against the measured solar wind to 2%.
3. Hydrostatic equilibriumready Scale height, polytropes, the Lane–Emden equation. The solar convection zone confirmed as an n = 3/2 polytrope to 1.8%; a standard solar model's convection-zone base refuted at 29σ by helioseismology.
4. Sound waves and linear perturbation theoryready Linearisation, Newton against Laplace, the acoustic cutoff, the large separation. Laplace agrees with the measured speed of sound in air to 0.04%; an isothermal Sun refuted by a bound, 3.25% below the measured solar large separation.

Instability

5. Jeans instabilityready Gravity against pressure; the Jeans mass; the Jeans swindle stated openly, then repaired by an expanding background and by the Bonnor–Ebert sphere. The Jeans criterion at mean density calls every isothermal equilibrium sphere stable, while Barnard 68 sits 2.25σ on the unstable Bonnor–Ebert branch; its measured outer density slope, 3.24, exceeds the 2.5176 ceiling of every isothermal sphere.
6. Convection and thermal instabilityready Schwarzschild, Ledoux and Field criteria; the collapse of ∇_ad in an ionisation zone; mixing-length theory and its price; the two-phase interstellar medium. Mixing-length theory over-drives the solar photosphere by a factor 76 in flux, and survives to 25% once it is given the measured flux; the predicted two-phase pressure window contains the measured median of 89 sight lines, and the same measurement refutes a static medium by putting 29% of the cold gas below a floor that allows 0%.
7. Rayleigh–Taylor and Kelvin–Helmholtzready One dispersion relation with four terms carries both instabilities: an upside-down glass of water picks a 3.0 cm wave and turns it over in 27 ms, and clear-air turbulence becomes possible at 2.6 m s⁻¹ of wind change per 100 m of altitude. A laser experiment at the National Ignition Facility confirms the self-similar mixing law h = α_B A g t² and refutes its own abstract's claim to have settled the coefficient, leaving an astrophysical mixing width uncertain by a factor 2. Kelvin–Helmholtz vortices on the flank of a coronal mass ejection yield no confirmation at all, but do bound the magnetic field along the wavevector at 0.0675 gauss for equal fields on both sides, or 0.0954 gauss for a one-sided field — which in a 10-gauss active region means k must lie within 0.39° to 0.55° of perpendicular to a field nobody measured.

Shocks and flows

8. Shocks and the Sedov–Taylor blast waveready A gas whose equations contain no length can hold a discontinuity, and the three conservation laws fix everything across it without a word about the interior: compression stops at 4 for a monatomic gas while the temperature has no ceiling at all, the entropy jump forbids the rarefaction shock that the conservation laws alone would allow, and a shock that radiates compresses by 2802 instead of 4. A point explosion has no length either, so R ∝ t²ᐟ⁵ before any dynamics — a prediction Taylor published in 1941 and tested in 1950 against twenty-five photographs of the Trinity fireball. The exponent survives to 2.4 per cent over a factor 620 in time; the model that produced it does not, refuted at 7.3σ by the residuals of its own fit, and the energy recovered falls 3.8σ below the modern radiochemical yield. Voyager 2 confirms the ceiling of 4 at Neptune's bow shock and refutes the single-fluid closure at the termination shock, where 80 per cent of the energy goes to pickup ions and the measured layer is a million times thinner than a mean free path. Carried to Tycho, Module 7's mixing law is short by a factor 11 with the Atwood number and the expansion index both already at their ceilings.
9. Bondi accretion and the Parker windready One critical-point calculation, and the velocity sign reversed: Bondi's accretion and Parker's wind are the same solution of the same equation. A corona in a 1/r potential cannot end in hydrostatic balance — its pressure at infinity floors at 2.4×10⁸ times the interstellar value — so it must flow. Eleven checks: the isothermal wind is refuted three ways, the single-polytrope repair puts its critical point below its own base, and the steady Euler equation with the measured pressure gradient still under-predicts the observed acceleration by 2.98. At Sagittarius A* the Bondi rate exceeds what Faraday rotation permits by 40 — or by 4 if the field is 3 per cent of equipartition — and exceeds the field-independent lower limits by at least 401. Bondi proves only λ ≤ λ_c; what is refuted is the physics assumed, never the algebra.
10. Turbulenceready Kolmogorov 1941 from two units and no fitting, the dissipation scale and the Re³ cost of resolving it, and the steeper supersonic exponent of the interstellar medium — the spectrum is an inference, through the bridge δv ∝ ℓ^h ⇒ E(k) ∝ k^−(1+2h), and is measured nowhere here. Four checks, of which two refute. The terrestrial constant C_K = 0.53 ± 0.055 holds across more than 100 flows and 4.60 decades in Reynolds number. The anchor is Podesta's solar wind at 1 au: the mean magnetic exponent confirms 5/3 to 0.70 per cent, t = −0.41 on 3 dof, while the velocity exponent is refuted in every one of the four intervals, 3.8 to 8.3 half-widths below and all on one side. Hitomi prices Module 3's "static" idealisation at 3.98 per cent of the Perseus mass, 7.66 if the large-scale shear doubles it. And 273 molecular clouds give a size–linewidth exponent 0.50 ± 0.05, which lies 3.33 systematic envelopes from Kolmogorov's 1/3 and exactly on Burgers' 1/2. The intracluster medium's collisional Reynolds number is 0.90 at Kn = 0.375, a formula outside its own domain, so only the magnetic field supplies a cascade for the cluster check to measure.

Where the field gets hard

11. Accretion discsready Angular momentum, not gravity, is what stops gas falling in — and the molecular viscosity that could carry it away is 12.01 decades too slow, so the rest of the module is about what replaces it. A Keplerian disc is Rayleigh-stable at every radius, because ℓ = ΩR² rises outward and d(ℓ²)/dR = GM everywhere; the shear q = 3/2 that Module 12 §9 assumed is derived here. Four checks, of which two refute. The temperature profile T ∝ R⁻³ᐟ⁴ contains no viscosity at all — ν, α and Σ cancel — and comes out at −0.74902 at 10⁵ R_g, confirmed as a shape and not a digit because the data side carries no error bar. The radiative efficiency is exactly 1/12 at the Schwarzschild ISCO, which is 45.7 per cent above the relativistic 0.057191 computed from Bardeen, Press & Teukolsky’s own orbit formula — and both ends agree to 5 per cent with the 0.06 and 40 per cent Shakura & Sunyaev printed in 1973. And α is refuted as a constant of nature: one review gives 0.1–0.4 for ionised discs against 0.01 for protostellar ones, and at least a factor of 5 between what observations need and what simulations of Module 12’s own instability produce.
12. Magnetohydrodynamicsready One extra term in Module 2's momentum equation and one extra equation from Maxwell, and four measurements the pair reaches. Flux freezing is a number, not an idealisation: the smallest magnetic Reynolds number in the census is 1.02×10¹², which is 12.0 decades of the system's own dynamical time. The anchor is the Parker spiral against Venzmer & Bothmer's Helios fits, and it CONFIRMS that the field is wound while leaving the amount unresolved: on both fit sets the prediction beats an unwound r⁻² field by 3.43 and 1.98 in year-to-year scatters, and on neither is it inside the formal fit error. The control is the exact half of the same table — the density index sits 0.49 scatters from what mass conservation forces, so the data CAN see an inverse square when there is one. The Alfvén surface comes out BRACKETED: all eight extrapolated variants land between 15.76 and 19.35 R_⊙, above Verscharen's 12.08±0.236 by 1.305–1.602 and below Parker Solar Probe's lower bound of 19.8 by 1.023–1.256, matching neither, because the two published numbers are 1.64 apart and are not the same quantity. Zeeman measurements of 34 dark cloud cores confirm the critical mass-to-flux coefficient to the two figures it is printed with, and refute the practice of quoting a mass-to-flux ratio without its geometry — the sphere-to-sheet factor 1.2657 spans 0.381 of the log width of the whole measured range. And Braginskii's classical perpendicular transport is refuted by 22.8 to 24.8 decades against the only measured bound, in the direction that says the field is tangled and the step is neither the mean free path nor the gyroradius — what it is, this book does not compute.
13. Radiation hydrodynamicsready Photons carry momentum, and this module derives the Eddington limit that five shipped pages had been quoting: the r² cancels, so the limit is a luminosity for a mass and not a radius. Four checks. CHECK 1 reproduces every Eddington number the book prints, and finds that Module 9's proton mass is a composition — pure ionised hydrogen; for protosolar gas the limit is 1.1587 times higher. The anchor, CHECK 2, turns Module 6's photospheric pressure into an opacity and CONFIRMS it as a column mean: 0.15231 cm² g⁻¹ against ATLAS9's own column mean 0.17487, a factor 1.1482 inside the inputs' spread of 1.2958. Read as a local opacity it is 2.361 to 2.920 low, which measures how fast the opacity rises with depth and not an error. CHECK 3 evaluates the one computation the book deferred by name: at the photosphere ωχ/c_s² is 0.3445 to 0.4468, so the adiabatic assumption fails there, as Module 4 predicted. CHECK 4: Eddington's grey atmosphere holds against ATLAS9 to 3.0 per cent in T. The adiabatic index of gas plus radiation reproduces Chandrasekhar's 1939 table to 3.44×10⁻⁴.
14. Numerical methodsready Every astrophysical code solves a problem that is not the one written down, and this module says which. The Courant condition that Module 10 used is derived, and shown to be necessary and not sufficient. The upwind scheme solves the advection equation plus a diffusion D = aΔx(1−C)/2, exact for the variance of a profile to the ninth decimal, so a grid sets its own Reynolds number, 2N/(1−C). The exact Riemann solver reproduces the five printed Sod star values to every decimal. Four checks. The anchor runs Godunov’s method on Sod’s shock tube from 100 to 6400 cells: the error at the contact is predicted with no fitted parameter, 1.0659 times the prediction at 6400 cells, and falls as Δx^0.511, while the total falls as Δx^0.646, so a first-order scheme does not converge at first order on a flow with a contact. Module 8’s Sedov blast puts the shock at r = 1.00663 on 400 cells, with the peak density 4.4010 against the ceiling 6. Particle hydrodynamics carries a pressure error of 0.0893 to 0.0942 at a contact that does not fall with resolution; artificial conductivity reduces it to 0.0216. A census of Modules 1–13 counts 29 simulation, 71 numerical-model and 30 own-integration inputs, and lists eight places where a shipped page calls a simulation or a model “measured”.

Figure data and every quoted number are regenerated by the scripts in figs/.