Module 6 — Convection and Thermal Instability

Hydrostatic balance is not stability. A stratified fluid can fail in two independent ways: buoyancy can stop restoring a displaced parcel, which is convection, and cooling can run away under compression, which is thermal instability. The Schwarzschild and Ledoux criteria, the collapse of the adiabatic gradient in an ionisation zone, mixing-length theory and its price, then Field's criterion and the two-phase interstellar medium. Two checks against measured data: mixing-length theory against solar granulation, and the predicted two-phase pressure window against 89 measured lines of sight.

1. Two fluids that will not sit still

Module 3 solved the hydrostatic equation and built stars out of it. Nothing in that module asked whether the solutions it found would survive being disturbed. A stratified fluid in hydrostatic balance has every force balanced at every point, and that is a statement about the fluid at rest; it says nothing about what happens to a parcel of it that is nudged. Balance is not stability.

This module asks the stability question twice, of two different restoring mechanisms, and gets two different instabilities.

The first is buoyancy. Displace a parcel upward a little and it expands to the lower ambient pressure. If it arrives denser than its new surroundings it sinks back and the layer is stable. If it arrives lighter it keeps rising, and the layer convects. §2 turns that sentence into an inequality between two logarithmic derivatives, one belonging to the star and one to the gas, and §3 turns the inequality into a frequency. §4 asks what a composition gradient does to it and §5 what partial ionisation does. §6 then asks the quantitative question the criterion cannot answer — how much heat a convecting layer carries — and §7 takes the answer to two measurements of the solar surface.

The second is cooling. Compress a parcel of interstellar gas a little, at the pressure of its surroundings. If the compression makes it radiate away energy faster than it gains it, the parcel cools, and at fixed pressure a cooler parcel is denser, so it contracts further. No gravity is involved at all. §8 derives Field's criterion for that runaway, §9 shows that it carves the interstellar medium into two stable phases separated by a narrow band of pressures, and §10 compares the predicted band with 89 measured sight lines. That comparison is the anchor of this module, and it confirms one prediction and refutes another from the same measurement.

A terrestrial case sets the difficulty of the first question immediately. The International Standard Atmosphere fixes the tropospheric lapse rate at $6.50\ \mathrm{K\,km^{-1}}$, against a dry adiabatic value $g/c_p = 9.761\ \mathrm{K\,km^{-1}}$ for dry air. The ratio is $0.6659$, which makes the standard troposphere a polytrope of index $n = 4.2558$, with $\nabla = 1/(n+1) = 0.19026$ against the diatomic adiabatic value $2/7 = 0.28571$. By the criterion of §2 the troposphere is stable, with $33$ per cent to spare. It nevertheless convects every afternoon. The resolution is that air is not dry: a saturated parcel releases about $2.5\times10^{10}\ \mathrm{erg\,g^{-1}}$ of latent heat as it rises and condenses, which warms it on the way up and lowers its effective lapse rate to $5$–$6\ \mathrm{K\,km^{-1}}$, below the ambient $6.5$. Moist convection is unstable where dry convection is not. Problem C1 works this through.

The Sun has no latent-heat term of that kind, so the solar criterion is the clean one, and it is derived first.

2. The displaced parcel and the Schwarzschild criterion

The whole of this section follows from the Euler equation of Module 2 and one assumption about the parcel. No transport theory is needed, and none is used.

Four premises were used and each will be paid for later. Premise (i), uniform composition, is lifted in §4. Premise (iii), instantaneous pressure balance, requires the parcel to move slowly compared with the sound speed, and §6 finds $v_c/c_s = 0.4271$ at the solar photosphere, where the premise is therefore spent. Premise (iv), adiabatic change, requires the parcel to lose no appreciable heat while it travels, and §6 and §7 find that at the photosphere radiation carries $98.7$ per cent of the flux, so it is spent there too. Only premise (ii) survives everywhere this module uses the criterion.

The criterion has one immediate consequence. For an ideal gas with a constant ratio of specific heats $\gamma$, an adiabatic change obeys $P\propto\rho^\gamma$ and the ideal gas law gives $P\propto\rho T$, so $T\propto P^{(\gamma-1)/\gamma}$ and

which is $2/5 = 0.400000$ for a monatomic gas and $2/7 = 0.285714$ for a diatomic one. These are the two numbers §3 and Problem C1 compare against. Everything later in the module either changes the left-hand side of (2.4), which §3 does by reading $\nabla$ off a solar model, or the right-hand side, which §5 does by breaking the constancy of $\gamma$.

3. The buoyancy frequency, and where the Sun's convection zone starts

Proposition 1 gives a sign. Its magnitude is a frequency, and the frequency is a measurable property of a star.

Equation (3.1) is the Schwarzschild criterion again, with a rate attached. It also identifies what the criterion costs when it is close: $N^2$ is a small difference of two numbers of order $0.4$, so a model table has to resolve $\nabla$ well to give $N^2$ at all inside a convection zone. That limit is stated below and respected.

3.1 Reading the criterion off a solar model

The stratification is the tabulated standard solar model BS2005-AGS,OP of Bahcall, Serenelli & Basu (2005), the same table Module 3 used. From its columns $\nabla$ is a sliding least-squares slope of $\ln T$ against $\ln P$ over $\pm25$ rows, $g = Gm(r)/r^2$ from the tabulated mass fraction, and $H = P/(\rho g)$.

Table 1 gives the result and Fig. 2, upper panel, plots it. $N$ is computed twice on the right: once with the ideal monatomic $\nabla_{\rm ad} = 0.4$, and once with the model's own measured plateau $0.3957$, which is what its equation of state — including Coulomb corrections and a little radiation pressure — actually delivers. The last column adds the composition term of §4.

The maximum of $N$ in the radiative interior, over $0.05$–$0.65\,R$, computed with the plateau $\nabla_{\rm ad} = 0.3957$ rather than $0.4$, is

and the corresponding buoyancy period $2\pi/N_{\max}$ is $38.8$ minutes. The maximum lies inside that window because $N\to0$ at the centre, where $g\to0$, and again at the base of the convection zone. Since a gravity mode of the star is a standing pattern of the oscillation of Proposition 2, and no such oscillation can be faster than the local $N$, every solar g mode must have a period longer than $38.8$ minutes. The Sun's famous five-minute oscillations are therefore acoustic, as Module 4 treated them, and the two mode families cannot be confused by their periods. This is a prediction of the model read through Proposition 2, not a check: no measured g-mode period is compared against it here.

3.2 Why the convection zone reaches down to $0.727\,R$

Read the $\nabla$ column of Table 1 outward, from the centre to the surface. It falls through $0.2116$ at $0.3016\,R$ to a shallow minimum — $0.1968$ at $0.4002\,R$ and $0.1962$ at $0.5018\,R$ — and then rises: $0.2074$ at $0.5979\,R$, $0.2616$ at $0.6804\,R$, $0.3259$ at $0.7099\,R$, and $0.3963$ at $0.7500\,R$. Across the whole interval $0.4002\,R$ to $0.7500\,R$, $\nabla$ rises by a factor $2.014$. The interval is not monotone, and the sampling matters: the rows printed above dip to $0.1962$ at $0.5018\,R$, while the full profile reaches its minimum of $0.1937$ at $0.4531\,R$, a radius Table 1 does not sample. Measured from that minimum the factor is $2.046$. Either way the rise carries $\nabla$ straight across the flat adiabatic plateau $0.3957$. The base of the convection zone is where the crossing happens, at $0.7269\,R$, which is $T = 2.02\times10^{6}$ K in the same table.

It is worth being exact about which side moves. In this star the convection zone opens because $\nabla$ rises to meet $\nabla_{\rm ad}$, not because $\nabla_{\rm ad}$ falls to meet $\nabla$: over the whole radiative interior $\nabla_{\rm ad}$ is the flat plateau value. A radiative layer carries its flux with $\nabla = \nabla_{\rm rad}\propto \kappa L P/(m\,T^4)$, so a rising opacity $\kappa$ and a falling temperature both push $\nabla$ up, and both are happening here. This module computes neither. It reads the rise off the model and does not apportion it between opacity and temperature; doing so would need the opacity tables the model was built with, which are not part of this module. §5 shows the other mechanism, the one that moves $\nabla_{\rm ad}$, and shows that it operates somewhere else entirely.

4. Ledoux: what a composition gradient does

Premise (i) of Proposition 1 was uniform composition. The Sun has burned hydrogen in its core for $4.6$ Gyr, so its $\mu$ is not uniform, and the premise has to be paid for.

The sign is the physical content. A $\mu$ that increases inward means $\nabla_\mu \gt 0$, since $P$ also increases inward; a parcel carried up from such a layer keeps its heavy composition and arrives denser than its new surroundings, which stabilises the stratification even where $\nabla \gt \nabla_{\rm ad}$.

Table 1's $\nabla_\mu$ column shows the Sun's. The $\mu$ used there is the table's own, $\mu = \rho k_BT/(P\,m_u)$, so no ionisation assumption enters. The share of $N^2$ contributed by the Ledoux term is $+84.05$ per cent at $0.050\,R$ ($\mu = 0.77684$, $\nabla_\mu = +0.37196$), $+74.96$ per cent at $0.100\,R$ ($\mu = 0.69264$, $\nabla_\mu = +0.22043$), $+50.74$ per cent at $0.150\,R$, $+16.53$ per cent at $0.200\,R$, $+3.07$ per cent at $0.251\,R$, $+1.37$ per cent at $0.302\,R$, $+0.67$ per cent at $0.400\,R$ and $+0.87$ per cent at $0.598\,R$. The gradient left by hydrogen burning is large where it exists and dies away outside the burning core.

The case where the choice of criterion does matter is a star with a receding convective core, which leaves a large $\nabla_\mu$ and a near-neutral layer in the same place. Such a layer is Schwarzschild-unstable and Ledoux-stable at once, and mixes on a thermal rather than a dynamical timescale — semiconvection. Problem D2 puts numbers on it and shows why the Sun escapes the ambiguity. This module does not develop semiconvection further.

The right-hand side of (2.4) has been treated so far as the constant $0.4$. It is not a constant.

5. The collapse of $\nabla_{\rm ad}$ in a partially ionised gas

Equation (2.5) assumed a constant $\gamma$. In a gas that is partly ionised, some of the energy a compressed parcel receives goes into pulling electrons off atoms instead of into raising the temperature, and the parcel therefore heats less for the same compression. That lowers $\nabla_{\rm ad}$, and it lowers it a long way.

The total particle density is $n_{\rm H}(1+x)$, so the gas pressure is $P = n_{\rm H}(1+x)k_BT$. Eliminating $n_{\rm H}$ between this and (5.1) gives a closed form with no iteration:

For a partially ionised pure-hydrogen gas, the first law with the ionisation energy included in the internal energy gives $$ \nabla_{\rm ad} \;=\; \frac{2+x(1-x)\Phi}{5+x(1-x)\Phi^2}, \qquad \Phi \;\equiv\; \frac{5}{2}+\frac{\chi_{\rm H}}{k_BT}. \tag{5.3}$$

The size of the effect is set by $\Phi$, and $\Phi$ is large. At the temperature where the depression is deepest on the photospheric isobar, $T = 12266$ K, $\chi_{\rm H}/k_BT = 12.86$ and so $\Phi = 15.36$; at the hotter temperature where the gas is half ionised, $T = 13136$ K, it is $12.01$. Either way the ionisation reservoir a parcel can pour heat into is about twelve times the thermal reservoir of one particle, which is why the depression is not a small correction. Table 2 evaluates (5.2) and (5.3) along an isobar at the photospheric pressure $P = 1.2\times10^{5}\ \mathrm{dyn\,cm^{-2}}$.

The minimum does not sit at $x = 1/2$, and the reason is worth one sentence. At fixed $\Phi$ the numerator and denominator of (5.3) both depend on $x$ only through $q = x(1-x)$, which peaks at $x = 1/2$; but $\Phi$ falls as $T$ rises, so moving along the isobar changes $q$ and $\Phi$ together, and the two effects put the minimum at $x = 0.3271$, on the cool side of half ionisation. Problem K1 follows the minimum across five decades of pressure and finds it drifts from $0.08097$ at $10580$ K to $0.12933$ at $16798$ K — Saha equilibrium needs a hotter gas to ionise a denser one.

5.1 What this explains, and what it does not

It would be natural to say that the collapse of $\nabla_{\rm ad}$ is what makes the outer part of the Sun convective. The model table says otherwise, and the numbers are unambiguous.

• The base of the convection zone is at $0.7269\,R$, where the table gives $T = 2.02\times10^{6}$ K. At that temperature hydrogen and helium are fully ionised, $x=1$, and (5.3) returns $\nabla_{\rm ad}=2/5$. The measured plateau, $0.3957$, is the evidence: it is the ideal value to within the model's equation-of-state corrections, right across the convection zone.
• The collapse of Table 2 sits between about $1.06\times10^{4}$ and $1.68\times10^{4}$ K, the band Problem K1 maps.
• The outermost row of the table, $0.98308\,R$, is already at $T = 8.12\times10^{4}$ K — above the whole collapse band.

So the hydrogen ionisation zone lies outside $0.983\,R$, in the outermost $1.7$ per cent of the radius. It does not open the convection zone, whose outer $27.3$ per cent by radius, from $0.7269\,R$ to the surface, is set the other way round by $\nabla$ rising, as §3.2 showed. What Proposition 4 explains is why convection reaches all the way up to the photosphere instead of stopping below it, and why the strongly superadiabatic layer of §6 sits where it does.

The criterion has now been derived, corrected for composition, and corrected for ionisation. It still says only whether convection happens. It says nothing about how much heat it carries.

6. Mixing-length theory, and how superadiabatic the Sun must be

A stability criterion cannot give a flux, because it contains no statement about how far a parcel travels before it merges with its surroundings. Mixing-length theory supplies exactly one such statement and derives everything else from it.

The factors $1/2$ in $\delta T$ and $1/8$ in $v_c^2$ are the two averaging conventions of the standard theory. A different convention would move both by a numerical factor, which is absorbed into a different calibrated $\alpha$; it changes no physics and no scaling. What the scalings say is that $\alpha$ is a weak handle on the velocity and a strong one on the excess, and Problem C3 exploits exactly that.

6.1 The excess through the solar convection zone

Inverting (6.3) for $\Delta\nabla$ at the flux the star must actually carry, $F = L(r)/(4\pi r^2)$ from the model's own luminosity column, gives Table 3. The local $L(r)$ is used rather than $L_\odot$; inside the convection zone the two agree to five figures.

That number repays a debt to Module 3. Module 3 fitted the solar convection zone with a polytrope of index $n = 3/2$ — the index of an adiabatic monatomic gas — and found $n_{\rm eff} = 1.5273$, agreeing to $1.8$ per cent. Table 3 says why such a fit can work at all: convection deep in the envelope is so efficient that the departure from an adiabat is six orders of magnitude below the adiabat itself. The polytrope is not an approximation that happens to be good; it is good because the transport is fast.

6.2 The photosphere, where the same theory is strained

The BS2005-AGS,OP table stops at $r = 0.98308\,R$, well below the photosphere, so the surface layer is computed from independent photospheric inputs: the IAU 2015 nominal effective temperature $T = 5772$ K, a gas pressure $P = 1.2\times10^{5}\ \mathrm{dyn\,cm^{-2}}$ at optical depth $\tau = 2/3$, and $\mu = 1.2250$ from the model's surface hydrogen and helium fractions, as Module 4 eq. (4.7) computes it. The gravity is the IAU nominal $g = GM_\odot/R_\odot^2 = 2.7420\times10^{4}\ \mathrm{cm\,s^{-2}}$, and $c_P = (5/2)k_B/(\mu m_u)$ is the monatomic ideal-gas value.

With those inputs, and requiring convection to carry the entire emergent flux $F = L_\odot/(4\pi R_\odot^2)$:

The range of turnover times is a separate result. Table 3 and Problem K2 give $\ell/v_c = 21.66$ d at $0.750\,R$, $8.38$ h at the last tabulated row $0.9831\,R$, and $1.38$ min at the photosphere: a factor $2\times10^{4}$ between the two ends of one convection zone. No single mixing length describes both.

7. Check A: mixing-length theory against solar granulation

Mixing-length theory makes two predictions about the solar photosphere that can be compared with measurements taken there. Both measurements are of the quiet Sun at disk centre, and neither used a stellar model to produce the quantity compared.

7.1 A1: the size of a granule

The overturning eddy of Definition 3 is one mixing length across. Seen from above, a convection cell has upflow in the middle and downflow at its edges, one mixing length each way, so the natural prediction for the diameter of a granule is $2\ell$. With $H = 143$ km and $\alpha = 2$, that is $\ell = 286$ km and $2\ell = 572$ km.

Abramenko et al. (2012) measured granule sizes with the New Solar Telescope and fitted the regular-granule population with a Gaussian, obtaining, in their section 3, a mean diameter $d_0 = 1050\pm22$ km and a width $\sigma_d = 480\pm11$ km.

Those two numbers are different quantities and this module prints both, with each denominator named, because they answer different questions.

The ratio of measured to predicted is $1050/572 = 1.84$; against $\ell$ alone it is $3.67$. Read the two departures together: $21.7\,\sigma$ says the mean granule is not $572$ km across, and $1.00\,\sigma$ says a $572$ km cell is an entirely ordinary member of the granule population. Abramenko's own Conclusions print "$1050\pm480$ km", which reads as an error bar and is not one; a reader who opens the paper will meet that line, and it is the population width.

7.2 A2: the speed of the upflow

Oba, Iida & Shimizu (2017) measured photospheric velocities with the Hinode spectropolarimeter, using a bisector analysis of Fe I $630.15$ nm with a subsonic filter applied to remove the five-minute oscillation. They find an upward speed falling from $0.65\ \mathrm{km\,s^{-1}}$ at a geometrical height $h = 40$ km to $0.40\ \mathrm{km\,s^{-1}}$ at $h = 163$ km, and an rms filtered convective velocity running from $0.6\ \mathrm{km\,s^{-1}}$ at the deepest level sampled to $0.3\ \mathrm{km\,s^{-1}}$ at the highest.

Against the $3.451\ \mathrm{km\,s^{-1}}$ of §6.2, the ratios are $5.3$ against the $0.65\ \mathrm{km\,s^{-1}}$ upflow and $5.8$ against the $0.6\ \mathrm{km\,s^{-1}}$ rms. Too fast by a factor $5.3$, against the factor $1.25$ the same theory returns in §7.3 once it is given the measured flux.

The right response is not to stop here. The calculation of §6.2 contains an assumption that is no part of mixing-length theory at all: that convection carries the whole luminosity at $\tau = 2/3$. That assumption can be tested against the same two measurements, and the independent warning $v_c/c_s = 0.43$ already says the theory has been pushed past its own premise.

7.3 A3, worked: what flux does the measured granulation actually carry?

This subsection uses no stellar model beyond the photospheric inputs already in use, and it rests on three assumptions that must be named before the numbers. The density $\rho$ at $\tau = 2/3$ is an input carried from Module 3, not a measurement. The observed intensity contrast is converted to a temperature contrast through a Planck source function, as if the contrast formed at a single depth. And the convective flux is formed as the product of a mean upflow speed with an rms temperature contrast, which assumes the two are perfectly correlated across the granulation pattern. A fourth mismatch is not an assumption but a fact worth stating: Oba et al.'s $0.65\ \mathrm{km\,s^{-1}}$ is measured at $h = 40$ km, $\tau_{500} = 0.55$, while $\rho$ and $c_P$ are taken at $\tau = 2/3$.

Step 1: from intensity contrast to temperature contrast. Abramenko et al. report an rms intensity contrast of $15.5\pm0.6$ per cent at $705.7$ nm. For a Planck function, $\delta I/I = (d\ln B_\lambda/d\ln T)\,(\delta T/T)$ with

At $\lambda = 705.7$ nm and $T = 5772$ K, $x = 3.5322$ and $d\ln B/d\ln T = 3.6386$. So $\delta T/T = 0.155/3.6386 = 0.04260$, that is $\delta T = 245.9$ K. The bandpass amplifies the contrast by a factor $3.64$: a $15.5$ per cent intensity contrast is only a $4.3$ per cent temperature contrast, and reading the intensity contrast as a temperature contrast directly would overstate the flux by that same factor.

Step 2: the convective flux the measurements imply. Feed the measured $\delta T$ and the measured velocity into the mixing-length flux expression $F = \rho\,v\,c_P\,\delta T$ of Proposition 5, which needs no $\alpha$ once $v$ and $\delta T$ are given:

Step 3: run the theory again, on the flux it should have been given. The calculation of §6.2 over-drove mixing-length theory by a factor $76$ in flux. Since $v_c\propto F^{1/3}$ at fixed $\alpha$ by (6.3), that is a factor $76^{1/3} = 4.2$ in velocity, which is most of the discrepancy of $5.3$ found in §7.2. Inverting (6.3) at $F = F_{\rm conv}$ instead gives $\nabla-\nabla_{\rm ad} = 0.0340$ and

What remains genuinely outside mixing-length theory is the split of the flux between radiation and convection, which needs an atmosphere model with radiative transfer that this module does not build. And the rest cannot be checked the same way: deep in the convection zone, where all four premises of Proposition 1 do hold, the excess is $10^{-6}$ and there is nothing to measure directly. The theory is checkable against data only in the one layer where it works worst.

Buoyancy is one way a stratified fluid loses its footing. The other needs no gravity at all.

8. Thermal instability: the loss function and Field's criteria

The question is the same one Proposition 1 asked — a parcel is perturbed, does the perturbation grow? — with a different restoring mechanism. Compress a parcel of interstellar gas slightly at the pressure of its surroundings. Its density rises, so it radiates more; if it also loses the ability to be reheated fast enough, it cools, and at fixed pressure a cooler parcel is denser still. Nothing in this argument involves gravity, buoyancy or a stratification.

Which criterion applies is not a matter of taste. A parcel held at fixed density but given a different temperature is at the wrong pressure for its surroundings, and the force equation will not let it stay there: it expands or contracts on a sound-crossing time, which in the interstellar medium is far shorter than the cooling time of the warm gas. The isobaric criterion is the one compatible with the momentum equation, and that was Field's point in 1965 against Parker's earlier isochoric treatment. Because (8.2) is the weaker condition, far more of the interstellar medium is unstable than the isochoric criterion would suggest.

8.1 The cooling function, tested before it is used

The interstellar gas of this section is heated by photoelectric emission from small grains and polycyclic aromatic hydrocarbons at a rate $\Gamma$ per hydrogen nucleus, taken independent of density and temperature, and cooled at a rate $n^2\Lambda(T)$ per unit volume. Koyama & Inutsuka (2002) fitted $\Lambda(T)$ over the range of the neutral medium with two terms: Lyman-$\alpha$ cooling, which switches on above about $8000$ K, and the [C II] $158\ \mu$m fine-structure line, whose $92$ K excitation temperature appears explicitly in the exponent and which is what cools the cold neutral medium.

The corrected function, used everywhere below, is

Read the two right-hand columns together, remembering that (8.1) and (8.2) are conditions for runaway while (8.5) is the condition for stability. The isochoric runaway fires nowhere in this table: $\Lambda$ increases with $T$ throughout, so $d\Lambda/dT\gt 0$ everywhere and (8.1) is never met. The isobaric runaway fires over a wide band, wherever the slope falls below $1$. That is the whole content of Field's 1965 correction, in one table.

The slope crosses $1$ twice: at $T = 184.0$ K, where $n_{\rm eq} = 8.682\ \mathrm{cm^{-3}}$ and $P/k = 1597.5\ \mathrm{K\,cm^{-3}}$, and again at $T = 5039.4$ K, where $n_{\rm eq} = 0.994\ \mathrm{cm^{-3}}$ and $P/k = 5007.0\ \mathrm{K\,cm^{-3}}$. Those two states bound the unstable branch, and §9 shows they are also the turning points of the equilibrium pressure.

9. The two-phase window

From the corrected fit (8.3), the turning points are

with geometric mean $2828.2\ \mathrm{K\,cm^{-3}}$ and width $P_{\max}/P_{\min} = 3.134$. Fig. 3 draws the curve, with the unstable branch between those two points marked separately.

9.1 The same window from a full calculation

Wolfire, McKee, Hollenbach & Tielens (2003) computed the same curve without a two-term fit: real gas-phase abundances of carbon and oxygen, grain and PAH photoelectric heating, [C II] and [O I] line cooling, and a radiation field appropriate to the solar neighbourhood. Their Table 3, at galactocentric radius $R = 8.5$ kpc with a cloud column $N_{\rm cl} = 10^{19}\ \mathrm{cm^{-2}}$, gives the row this module uses throughout.

A two-term analytic fit reproduces both edges of the window to better than $20$ per cent, and the coldest temperature the warm phase can reach to $0.01$ per cent. Wolfire et al. also quote the general result of Field, Goldsmith & Habing (1969) that $P_{\max}$ is at most about $3P_{\min}$: here the fit gives $3.13$ and the full calculation $2.45$. From §10 onward the full calculation's window $[1960, 4810]$ is the prediction under test, because it is the one whose inputs are traceable to independent measurements.

One property of the source has to be said rather than hidden: Wolfire's $P_{\rm th,ave}$ is the geometric mean of his own two edges to four figures, in all five variants of Table 6 — $\sqrt{P_{\min}P_{\max}}$ gives $3070.4$, $2216.8$, $3681.3$, $2488.7$ and $2410.6$ against the quoted $3070$, $2220$, $3680$, $2490$ and $2410$. So the "geometric mean" row of Table 5 restates the two rows above it rather than testing anything new, and the two readings of the anchor in §10.1 — the ratio to $P_{\rm ave}$ and the position in dex — are one statement written twice.

The window is narrow in pressure and wide in everything else: a factor $3.13$ in pressure, $8.7$ in density and $27.4$ in temperature (Problem C2). That asymmetry is what makes a pressure survey worth doing. Jenkins & Tripp had already made one before Wolfire et al. published: their 2001 survey of C I fine-structure excitation gave a mean pressure of $2240\ \mathrm{K\,cm^{-3}}$, which Wolfire et al. compare their own window against in their section 8.1. In 2011 the same two authors measured again, on 89 sight lines and with revised C I atomic data, and that second measurement is what §10 puts against the window.

10. The anchor: Jenkins & Tripp (2011) against the window

Jenkins & Tripp (2011) measured thermal pressures in the diffuse cold neutral medium along 89 sight lines, from the fine-structure excitation of neutral carbon observed with the STIS echelle spectrograph on the Hubble Space Telescope. Their mass-weighted lognormal fit, their equation (3), has mean $\log(p/k) = 3.58$ dex and an rms dispersion $\sigma_{\log P} = 0.175$ dex, which they state is a lower limit. The median is $3801.9\ \mathrm{K\,cm^{-3}}$ and the $\pm1\sigma$ range is $2541$ to $5689\ \mathrm{K\,cm^{-3}}$.

This module takes two predictions to that measurement, in the shape Module 2 used: one confirmed and one refuted, from the same source table.

10.1 Confirmed: the measured centre lies inside the predicted window

The content of that confirmation is that a factor-$2.5$ window exists at all, and sits where it does. Neither of its edges was fitted to a pressure value. Wolfire et al. computed $P_{\min}$ and $P_{\max}$ from grain photoelectric heating and [C II]/[O I] cooling, with gas-phase carbon and oxygen abundances fixed by independent Hubble absorption-line studies and a PAH abundance fixed by ISO infrared emission data. One heating parameter, $\varphi_{\rm PAH} = 0.5$, was calibrated to an observed C I/C II column-density ratio; Wolfire's own detailed working credits that ratio to the C I and H I data compiled by Welty & Hobbs (2001), although an earlier passage cites Jenkins & Tripp (2001) alongside it. Jenkins & Tripp (2001) is the predecessor of, and distinct from, the 2011 pressure measurement compared here.

10.2 How far the confirmation survives its own parameters

$\varphi_{\rm PAH}$ is a real lever on the window, not a formality. Wolfire's Table 4 tests five model variants, three of which vary $\varphi_{\rm PAH}$ discretely while two hold it at $0.5$ and vary other parameters. Raising $\varphi_{\rm PAH}$ raises all three pressures monotonically, and the authors state the size of the effect themselves: over $\varphi_{\rm PAH} = 0.25$ to $1$, $P_{\min}$ changes by a factor $1.5$ and $P_{\max}$ by $1.9$, with $P_{\max}$ decreasing by about $35$ per cent at $\varphi_{\rm PAH} = 0.25$.

What mitigates that must be stated as precisely as the defect, and it is stronger than one row. Wolfire et al. disfavour every non-standard variant, on evidence that is not a pressure. Their four-item summary reads, in their order: "Low $\varphi_{\rm PAH}$" models poorly match the C I*/C$_{\rm tot}$ ratio and the C$^+$ cooling rate, with values lower than observed; "High $\varphi_{\rm PAH}$" models produce temperatures and C I/C II ratios higher than observed; "Low PAH" — Table 4's "Low $n_{\rm PAH}/n$" — gives low C I*/C$_{\rm tot}$ ratios and C$^+$ cooling rate; "Low $G_0$" gives low C I*/C$_{\rm tot}$ and high C I/C II. Separately, they write that only "High $\varphi_{\rm PAH}$" and "Low $G_0$" "can be safely ruled out by the required C I/C II ratio".

The two tiers must be kept apart. Two rows are ruled out in those words. Two more, including the one that breaks the confirmation, are called a worse fit on two independent diagnostics without being ruled out. The only variant Wolfire et al. do not argue against is the standard one, and the standard one contains the measured median.

One number is stable across all five variants: the measured median exceeds every predicted average pressure, from $1.033$ times the "High $\varphi_{\rm PAH}$" value to $1.713$ times the "Low $\varphi_{\rm PAH}$" one. The measurement runs high against the central model value whatever the parameters. Only the window verdict moves.

10.3 Refuted: a static two-phase medium cannot put cold gas below $P_{\min}$

Below $P_{\min}$ the equilibrium curve has no cold branch: by Proposition 8 the only equilibrium state at such a pressure is the warm one. A strict, static two-phase medium therefore allows none of the cold neutral medium to sit below $P_{\min}$. Jenkins & Tripp measured the cold medium, and they report in their section 10.1.2 that $29$ per cent of it is there.

Fig. 4 puts both halves of the anchor on one page: the windows and the measured distribution in its upper panel, and in its lower panel the quantity that carries the disagreement, the fraction of cold gas below $P_{\min}$.

What is not refuted. Gas above $P_{\max}$ is allowed. Only the warm phase is forbidden there, and this survey measures the cold phase, so the $28$ per cent above $P_{\max}$ violates nothing and is not counted. The refutation rests on the low side alone, where a static cold medium is genuinely impossible.

The refutation survives the parameter sweep that the confirmation does not. Only the fitted fraction can be re-evaluated at another $P_{\min}$; the measured $29$ per cent is tied to $P_{\min} = 1960\ \mathrm{K\,cm^{-3}}$, the value Jenkins & Tripp themselves quote, and cannot be moved without their data. Across the five variants of Table 6, the fitted lognormal below $P_{\min}$ gives $5.0$ per cent (Standard, $P_{\min}=1960$), $1.4$ (Low $\varphi_{\rm PAH}$, $1560$), $10.0$ (High $\varphi_{\rm PAH}$, $2270$), $1.5$ (Low $n_{\rm PAH}/n$, $1580$) and $0.9$ (Low $G_0$, $1460$) — a range of $0.9$ to $10.0$ per cent, against the $0$ per cent a static medium allows, in every variant.

10.4 What fails, precisely: the word "static", not the criterion

Jenkins & Tripp offer two readings of their own excess and prefer the second: either the Wolfire curve does not apply to the gas they observe, or, in their words, "rarefactions caused by turbulence create momentary excursions below the curve". Their dispersion of $0.175$ dex is itself read as turbulence at Mach $1$ to $4$. On that reading the thermal-equilibrium curve is an attractor, and the gas is pushed off it faster than it relaxes back. Problem D3 makes that quantitative by putting the cooling time against the crossing time of a turbulent eddy, and finds the two phases differ by a factor $794$ in cooling length — which is why a cold excursion below $P_{\min}$ can only be momentary while a warm departure can be sustained.

The same failure seen from the warm side. Heiles & Troland (2003), from 21-cm absorption, find that at least $48$ per cent of the warm neutral medium by column density lies at $500$–$5000$ K, inside the thermally unstable band; a strict two-phase medium allows only the small fraction in transit between phases. Their other numbers: the warm medium is $61$ per cent of the H I column at $|b|\gt 10^\circ$, and the cold-medium spin-temperature histogram peaks at $40$ K with a column-weighted median of $70$ K. Both the $0.61$ and the $0.48$ are column-density weighted and both are lower bounds.

For reference, against the analytic-fit window $[1597, 5007]$ the same lognormal puts $73.7$ per cent of the mass inside, rather than the $67.0$ per cent quoted above for $[1960, 4810]$.

11. What this model captures, and what it misses

Each idealisation, and where it is repaid if it is repaid at all.

• No radiative transfer. The split of the flux between radiation and convection is not predicted anywhere in this module. §7 measures it at $1.3$ per cent at $\tau = 2/3$ and stops there. This is the single largest gap, because it is what makes the most direct test of mixing-length theory a test of something the theory does not claim. It is not repaid in Modules 1–10.
• Mixing-length theory is a one-parameter closure calibrated on the answer. $\alpha = 2$ was fitted to the solar radius. Problem C3 shows the freedom that leaves is small for the velocity, $\alpha^{1/3}$, and large for the superadiabatic excess, $\alpha^{-4/3}$. Not repaid.
• No rotation and no magnetic field. Both change convection in the Sun, and the magnetic case changes it qualitatively at the surface. Neither is treated in Modules 1–10.
• Linear stability only. Both criteria, Schwarzschild's and Field's, say that a perturbation grows. Neither says what it grows into. Module 5's Bonnor–Ebert analysis has exactly the same limit, and §10 shows the cost: the measured distribution of interstellar pressures is a statement about the non-linear, turbulent state, and the linear criterion can only say that the state it predicts is not the one observed.
• The two-phase medium is treated as static. The anchor identifies this as the assumption that fails. Turbulence is named as the mechanism, on Jenkins & Tripp's own reading of their data, and is not modelled here.
• The solar opacity is read off a model, not computed. §3.2 reads the rise of $\nabla$ from the table and deliberately does not apportion it between the opacity and the temperature, because this module computes neither.
• The parcel picture itself. Propositions 1 and 5 follow a single blob against a background that does not respond. Real convection is a connected flow with narrow fast downdrafts and broad slow upflows, which is why Oba et al. measure different upward and downward speeds at the same height. Nothing in this module distinguishes them.

12. Problems

Nine problems in the house set: three conceptual (C), three derivational (D), three computational (K). Every printed answer is reproduced independently by afd/figs/m06_problems_check.py.

Conceptual ↑ contents

Derivational ↑ contents

Computational ↑ contents

Notation

Sources

Every published value this module compares against, with what was taken from it.

1. Thermal instability. G. B. Field, Thermal Instability, The Astrophysical Journal 142, 531 (1965), ADS scan. Quoted from p. 532, equations (1)–(4b): $\mathcal{L}(\rho,T)$ defined as "energy losses minus energy gains, per gram of material per second exclusive of thermal conduction"; (4a) the isochoric criterion $(\partial\mathcal{L}/\partial T)_\rho\lt 0$; (4b) the isobaric criterion $(\partial\mathcal{L}/\partial T)_P = (\partial\mathcal{L}/\partial T)_\rho - (\rho_0/T_0)(\partial\mathcal{L}/\partial\rho)_T\lt 0$, and Field's statement that condensations are governed by (4b) rather than by Parker's (4a). The later form $(\partial(\mathcal{L}/T)/\partial T)_P\lt 0$ coincides with (4b) only on the equilibrium curve; this module uses (4b) as printed.
2. The two-phase window (the prediction under test). M. G. Wolfire, C. F. McKee, D. Hollenbach and A. G. G. M. Tielens, Neutral Atomic Phases of the Interstellar Medium in the Galaxy, The Astrophysical Journal 587, 278–311 (2003), arXiv:astro-ph/0207098. Quoted: Table 3, row $R = 8.5$ kpc, $N_{\rm cl} = 10^{19}\ \mathrm{cm^{-2}}$ — $P_{\min}/k = 1960$, $P_{\max}/k = 4810$, $P_{\rm th,ave}/k = 3070\ \mathrm{K\,cm^{-3}}$, warm-phase $T$ from $8310$ to $5040$ K and $n$ from $0.209$ to $0.860\ \mathrm{cm^{-3}}$, cold-phase $T$ from $258$ to $61.6$ K and $n$ from $6.91$ to $71.0\ \mathrm{cm^{-3}}$, cold-phase $T_{\rm ave} = 85.0$ K; Table 4, all five model variants of §10.2; the sentences ruling out "High $\varphi_{\rm PAH}$" and "Low $G_0$" and the four-item summary disfavouring all four non-standard variants; the $\varphi_{\rm PAH} = 0.5$ calibration on a C I/C II ratio, credited in their section 8.1 working to Welty & Hobbs (2001); their statements that $P_{\min}$ changes by a factor $1.5$ and $P_{\max}$ by $1.9$ over $\varphi_{\rm PAH} = 0.25$ to $1$, with $P_{\max}$ falling by about $35$ per cent at $0.25$; their section 8.1 post-hoc comparison against Jenkins & Tripp's (2001) mean pressure of $2240\ \mathrm{K\,cm^{-3}}$. Their gas-phase abundances come from HST absorption studies (Sofia et al. 1997; Cardelli et al. 1996; Meyer et al. 1998) and their PAH abundance from ISO emission (Tielens 1999); those papers were not opened here. Their photoelectric-heating formula was not re-derived, and Bakes & Tielens (1994) and Draine (1978, 1987) were not opened, so whether the heating physics carries a further pressure dependence beyond $\varphi_{\rm PAH}$ is an open question, not a closed one.
3. The measured pressures (the anchor). E. B. Jenkins and T. M. Tripp, The Distribution of Thermal Pressures in the Diffuse, Cold Neutral Medium of our Galaxy. II. An Expanded Survey of Interstellar C I Fine-Structure Excitations, The Astrophysical Journal 734, 65–86 (2011), arXiv:1104.2323. Quoted: abstract and eq. (3) — mass-weighted lognormal with mean $\log(p/k) = 3.58$, rms dispersion at least $0.175$ dex, normalisation $2.30\times10^{23}\ \mathrm{cm^{-2}}$ per dex, and the statement that outside $3.2\lt \log(p/k)\lt 4.0$ the fit understates the observed material; section 10.1.2 — "A substantial fraction of the material (29%) … is detected at pressures below those permissible for a static CNM, $(p/k)_{\min} = 1960\ \mathrm{cm^{-3}\,K}$, as defined by the standard model … presented by Wolfire et al. (2003)", read by them as turbulent rarefaction; abstract — $23$ per cent for the low-intensity subsample, about $0.05$ per cent above $\log(p/k) = 5.5$, turbulent Mach number $1\lt M\lt 4$; and their attribution of the rise from the 2001 value to updated C I atomic rates, revised optical pumping and a corrected temperature.
4. Paper I, cited but not compared against. E. B. Jenkins and T. M. Tripp, The Distribution of Thermal Pressures in the Interstellar Medium from a Survey of C I Fine-Structure Excitation, The Astrophysical Journal Supplement Series 137, 297–340 (2001), arXiv:astro-ph/0107177. Its mean pressure of $2240\ \mathrm{K\,cm^{-3}}$ enters only through Wolfire et al.'s section 8.1. Its text was searched for a C I/C II column-density ratio, a total carbon column and an N(C II): none appears, which is why §10.1 does not claim a shared technique between the $\varphi_{\rm PAH}$ calibration and the 2011 pressures.
5. The warm-side measurement. C. Heiles and T. H. Troland, The Millennium Arecibo 21 Centimeter Absorption-Line Survey. II., The Astrophysical Journal 586, 1067–1093 (2003), arXiv:astro-ph/0207105. Quoted from the abstract and sections 2.3.2 and 9.2.1: at least $48$ per cent of the warm neutral medium by column density lies in the thermally unstable range $500$–$5000$ K; $61$ per cent of H I is warm at $|b|\gt 10^\circ$; the cold-medium spin-temperature histogram peaks at $40$ K with a column-weighted median of $70$ K. Both fractions are column-density weighted lower bounds. Their $500$ K is a classification boundary, not a computed turning point, and §9 declines to compare it with the $184$ K of the fit.
6. The cooling function, and its two typographical errors. H. Koyama and S. Inutsuka, An Origin of Supersonic Motions in Interstellar Clouds, The Astrophysical Journal 564, L97 (2002), arXiv:astro-ph/0112420, equations (4) and (5), and their $140$ au grid cells on $2048\times512$ over $1.44\times0.36$ pc. The corrected form is printed as equations (3) and (4) of E. Vázquez-Semadeni, G. C. Gómez, A. K. Jappsen, J. Ballesteros-Paredes, R. F. González and R. S. Klessen, Molecular Cloud Evolution. II., The Astrophysical Journal 657, 870–883 (2007), arXiv:astro-ph/0608375, whose footnote 5 reads: "Note that eq. (4) in Koyama & Inutsuka (2002) contains two typographical errors. The form used here incorporates the necesary corrections, kindly provided by H. Koyama."
7. Granule velocities. T. Oba, Y. Iida and T. Shimizu, Height Dependence of the Velocity Structure of Photospheric Convection in Granules and Intergranular Lanes with Hinode/SOT, The Astrophysical Journal 836, 40 (2017), arXiv:1612.06175. Quoted from the abstract and section 4.4: upward speed falling from $0.65$ to $0.40\ \mathrm{km\,s^{-1}}$ with height and downward speed rising from $0.30$ to $0.50\ \mathrm{km\,s^{-1}}$ with depth; rms filtered convective velocity $0.6\ \mathrm{km\,s^{-1}}$ at $I/I_0 = 0.75$ falling to $0.3$ at $I/I_0 = 0.40$; stated velocity error $0.18\ \mathrm{km\,s^{-1}}$; Table 1, which places $I/I_0 = 0.75$ at $h = 40$ km, $\tau_{500} = 0.55$ and $I/I_0 = 0.40$ at $h = 163$ km, $\tau_{500} = 0.10$. Those heights are derived from the VAL81 model atmosphere, not measured.
8. Granule sizes and contrast. V. I. Abramenko, V. B. Yurchyshyn, P. R. Goode, I. N. Kitiashvili and A. G. Kosovichev, Detection of Small-scale Granular Structures in the Quiet Sun with the New Solar Telescope, The Astrophysical Journal Letters 756, L27 (2012), arXiv:1208.4337. Quoted from their section 3: the regular granules are Gaussian with $d_0 = 1050\pm22$ km and $\sigma_d = 480\pm11$ km; the mini-granules below $600$ km follow a power law of index $-1.82\pm0.12$; the area-contribution function peaks at $1080$–$1300$ km; and the rms intensity contrast $15.5\pm0.6$ per cent at $705.7$ nm.
9. The solar stratification. J. N. Bahcall, A. M. Serenelli and S. Basu, The Astrophysical Journal 621, L85–L88 (2005). The tabulated model BS2005-AGS,OP, at afd/data/bs05_agsop.dat, used here only as a stratification, exactly as Module 3 used it.
10. The textbook results. R. Kippenhahn and A. Weigert, Stellar Structure and Evolution (Springer, 1990), sections 6.1, 7.2 and 14.3: the Schwarzschild and Ledoux criteria, the mixing-length flux and velocity in the efficient limit with their $1/2$ and $1/8$ conventions, and $\nabla_{\rm ad}$ for a partially ionised gas, equation (5.3).
11. Cited through Wolfire et al., not read. D. E. Welty and L. M. Hobbs, The Astrophysical Journal Supplement Series 133, 345 (2001), for the C I and H I data behind the $\varphi_{\rm PAH}$ calibration; G. B. Field, D. W. Goldsmith and H. J. Habing, The Astrophysical Journal 155, L149 (1969), for the general result that $P_{\max}\lesssim 3P_{\min}$.
12. Transport coefficients. The electron conductivity $\kappa = 1.84\times10^{-5}T^{5/2}/\ln\Lambda$ was read from C. L. Sarazin, X-ray Emissions from Clusters of Galaxies (1988), eq. (5.37), which attributes it to L. Spitzer, Physics of Fully Ionized Gases; Spitzer's book was not opened, and its 1956 and 1962 printings are its first and second editions. Sarazin section 5.4 also supplies $\ln\Lambda = 37.8$ for the intracluster medium, carried over from Module 1. The neutral-gas conductivity $K = 2.5\times10^{3}\sqrt{T}$ is E. N. Parker (1953) as adopted by Koyama & Inutsuka below their eq. (3); Parker's paper was not opened.
13. Constants and standards. CODATA 2018 for $k_B$, $m_u$, $m_e$, $h$, $G$ and $\chi_{\rm H}$; IAU 2015 Resolution B3 for $GM_\odot$, $R_\odot$, $L_\odot$ and $T_{\rm eff,\odot} = 5772$ K; ISO 2533:1975 and the US Standard Atmosphere 1976 for the ISA lapse rate, $T_0 = 288.15$ K and the tropopause temperature $216.65$ K.
14. Earlier modules. The Euler equation (7.3) of Module 2; the hydrostatic equation, the temperature gradient $\nabla$ of Definition 5, the $n = 3/2$ polytrope and the photospheric inputs of Module 3; the composition $\mu = 1.2250$ of Module 4 eq. (4.7) and its photospheric scale height at module04.html:260; the Coulomb logarithm of Module 1, Definition 5.

Module 6 of Astrophysical Fluid Dynamics. Previous: Module 5 — The Jeans Instability