One critical-point calculation, the velocity sign reversed; and what happens when it is asked to describe the Sun and a black hole.
Module 3 put a star in hydrostatic balance and left it there. Module 2 derived the equation that governs what happens when the balance fails, and applied it to a flow whose radial structure was measured rather than predicted. This module does the thing neither of them did: it takes the steady spherical flow seriously as an equation to be solved, discovers that the equation has a singular point, and finds that almost everything worth knowing about stellar winds and about accretion onto compact objects follows from the behaviour of solutions near that point.
The claim in the title is literal and is worth stating before any algebra. Write continuity and Euler for a steady, spherically symmetric flow. Eliminate the density. What is left is one ordinary differential equation for the speed $v(r)$, and that equation contains no information about which way the gas is going: $v$ enters it through $v^2$ and through $\mathrm{d}\ln v/\mathrm{d}\ln r$, and a solution with $v \gt 0$ everywhere maps onto a solution with $v \lt 0$ everywhere. Take the outward branch and it is E. N. Parker's solar wind, published in 1958. Take the inward branch and it is H. Bondi's spherical accretion flow, published in 1952. They are the same calculation. Fig. 1 is that sentence drawn.
The singular point is the reason this is not a routine exercise. At one particular radius the coefficient of $\mathrm{d}v/\mathrm{d}r$ vanishes, and unless the right-hand side vanishes there too, the solution has infinite gradient and stops existing. That is a strong constraint, and §3 shows it selects exactly one value of an integration constant out of a continuum. The solution that satisfies it is the only one that starts subsonic at a stellar surface and ends supersonic at infinity — and a wind that does not end supersonic decelerates as $v \propto r^{-2}$, which is a qualitatively different object with a qualitatively different signature in the data. §7 uses exactly that: the sign of a measured velocity exponent separates the two, with no fitting.
The module then does what this book always does, and asks whether the calculation describes anything. It asks twice, at the two extremes of the mass range, and gets two different kinds of answer.
The anchor is the solar wind, §7. Venzmer & Bothmer (2018) fitted four power laws to the combined Helios 1 and 2 data between $0.29$ and $0.98$ au, each exponent fitted independently of the others and with no fluid equation imposed on the fit. Those four numbers confirm two predictions of this module and refute a third, and the sharpest of them is not a confirmation or a refutation at all but a measured shortfall: steady Euler, fed the measured pressure gradient and given no wind model whatever, accounts for about one third of the acceleration the wind is observed to have at 1 au. The same table was used in Module 2 §10 to confirm an exact conservation law and refute the adiabatic closure. §7.1 says in print what is new here and what is not, because a reader is entitled to ask whether the book is checking one table twice.
The astrophysical check is Sagittarius A*, §8, and it produces the largest ratio in the book so far. The Bondi rate, computed from a measured black-hole mass and Chandra's measured gas density and temperature, exceeds the rate that Faraday rotation permits by a factor of $40$ — and the module then spends as much space on the conditions attached to that $40$ as on the number itself, because one of them cuts it to $4$. Two further comparisons, which do not depend on those conditions, survive at factors of $401$ to $4011$ and $2\times10^{7}$. The refutation is of the physical premises, not of the algebra.
One thing must be said at the outset, because the module's first draft got it wrong and the correction is instructive. It is often written that the steady equations determine the accretion rate, and that the coefficient Bondi computed is an eigenvalue forced by the mathematics. Bondi says the opposite, in his own summary, and §5 quotes him. The steady equations bound the rate; choosing the transonic solution is a physical argument about which flow nature realises, and it is his argument rather than a theorem. That distinction does not weaken §8 — two of its three legs never touch the coefficient — but a refutation built on a misattributed theorem would not be worth having.
Parker's 1958 paper does not begin with a wind. It begins by assuming there is none, and showing that the assumption cannot be maintained. The argument occupies his Sect. II, it needs nothing beyond Module 3, and it is the cleanest demonstration in this book that an equilibrium can be internally consistent and still be impossible.
Contrast this with the plane-parallel isothermal atmosphere of Module 3 Proposition 3, where $g$ is constant and $P \propto \mathrm{e}^{-z/H}$ falls to zero. The difference is entirely the geometry: $\int^\infty g\,\mathrm{d}z$ diverges in a uniform field and converges in an inverse-square one. That convergence is the whole argument. An atmosphere whose pressure tends to zero can sit in a vacuum forever; an atmosphere whose pressure tends to a positive constant needs something outside it pushing back with at least that much pressure, and if nothing does, it is not in equilibrium at all.
At $r_0 = 1.03\,R_\odot$ the Baumbach–Allen formula gives an electron density $n_{\rm e} = 3.161\times10^{8}\ \mathrm{cm^{-3}}$, hence $n_{\rm H} = 2.874\times10^{8}$ and $n_{\rm tot} = 6.179\times10^{8}\ \mathrm{cm^{-3}}$ under (2.1). At $T = 2$ MK that is a base pressure
$$ P_0 \;=\; n_{\rm tot}k_{\rm B}T \;=\; 1.706\times10^{-1}\ \mathrm{dyn\,cm^{-2}}, \tag{2.5}$$and (2.3) evaluates to $P(\infty)/P_0 = 1.908\times10^{-3}$, so that
$$ P(\infty) \;=\; 3.256\times10^{-4}\ \mathrm{dyn\,cm^{-2}}. \tag{2.6}$$A round interstellar medium of one particle per cubic centimetre at $10^4$ K has $nk_{\rm B}T = 1.381\times10^{-12}\ \mathrm{dyn\,cm^{-2}}$. The ratio is
The sensitivity to $T$ is severe and should be seen rather than asserted, because it is the same exponential that §6 will find governs the mass-loss rate. At 1 MK the ratio $P(\infty)/P_0$ is $3.642\times10^{-6}$ and at 3 MK it is $1.539\times10^{-2}$ — four orders of magnitude across a factor of three in temperature. Even at 1 MK, however, the absolute floor is $3.11\times10^{-7}\ \mathrm{dyn\,cm^{-2}}$, still $2.25\times10^{5}$ times the interstellar value, so the conclusion does not depend on the corona being as hot as 2 MK.
The base density is an input and it is extrapolated beyond its source's stated range. Allen (1947) fitted $n_{\rm e}(R) = 10^{8}\,[\,1.55\,R^{-6} + 2.99\,R^{-16}\,]\ \mathrm{cm^{-3}}$, with $R$ in solar radii, to eclipse brightness measurements and stated the range explicitly: "in the range $p = 1.2$ to $2.6$, the intensity which is scattered by electrons follows the formula (2)". (His radius variable is $p$; it is written $R$ here because $p$ is pressure throughout this book.) This module evaluates the formula at $R = 1.03$, a fifth of a solar radius below the fitted range, in the steep $R^{-16}$ term where extrapolation is most violent. Nothing in §2 depends on the precise value — the margin is $2\times10^{8}$ — but §7.7 builds a mass-flux comparison on it, and that comparison is labelled a weak test partly for this reason.Parker made this calculation in 1958 with different inputs, and the comparison is worth printing — not as a test of nature, but as a check that a 68-year-old argument has been reconstructed and not merely cited. His corona is not isothermal: it conducts, and its temperature declines outward, which makes his pressure fall faster than (2.2). From $N_0 = 3\times10^{7}\ \mathrm{cm^{-3}}$ and $T_0 = 1.5\times10^{6}$ K he gets $p_0 = 2N_0k_{\rm B}T_0 \approx 1.3\times10^{-2}\ \mathrm{dyn\,cm^{-2}}$ and $p(\infty) = 0.55\times10^{-3}p_0 = 0.6\times10^{-5}\ \mathrm{dyn\,cm^{-2}}$, against an interstellar pressure he takes as $1.4\times10^{-13}\ \mathrm{dyn\,cm^{-2}}$ from "10 hydrogen atoms/cm³ at 100° K". His ratio is $4.3\times10^{7}$.
His conclusion, in his own words, is the sentence this module rebuilds: "Since we know of no general pressure at infinity which could balance the $p(\infty)$ computed from equation (9) with the expected values of $n$, we conclude that probably it is not possible for the solar corona, or, indeed, perhaps the atmosphere of any star, to be in complete hydrostatic equilibrium out to large distances."
The corona must move. The question is how, and the answer is more constrained than it has any right to be.
Note what $r_c$ is not. It is not put in by hand and it is not a fitting parameter: it is the radius at which the escape speed $\sqrt{2GM/r}$ equals $2c_T$, equivalently where $c_T^2 = GM/(2r)$, and it is fixed the moment $T$ is fixed. Problem C1 checks that identity numerically.
Equation (3.5) shows the mechanism directly and is worth reading before the dimensionless form. The bracket on the left changes sign at $v = c_T$. The right side changes sign at $r = GM/2c_T^2 = r_c$. If the two sign changes do not occur at the same place, $\mathrm{d}v/\mathrm{d}r$ is infinite where the bracket vanishes and the solution terminates. There is no freedom here: a flow that is to go from subsonic to supersonic must do it at $r_c$ and nowhere else.
A note on the names. These six families are often numbered I to VI, and that numbering is usually attributed to Parker. It is not his. Parker (1958) was fetched and read for this module, and it contains no such enumeration: he writes the equations, imposes the transonic condition, and plots $v(r)$. The numbering is a later textbook convention, and because nothing here rests on it the families are named in words and no numeral is cited to anyone.
Fig. 2 draws (3.3) as a contour plot in the $(x,u)$ plane, which is the honest way to present it: the families are level sets of one function, not six separate constructions, and the classification is visible rather than asserted. The transonic pair is the saddle-crossing contour, and it is the only contour that passes through the corner point $(1,1)$.
Both limits of (3.3) are needed later and both are one line.
The wind. As $x\to\infty$ on the supersonic branch, $u^2 \gg 2\ln u$ and $4\ln x \gg 4/x$, so $u^2 \to 4\ln x$ and
$$ v \;\to\; 2c_T\sqrt{\ln(r/r_c)}, \qquad \frac{\mathrm{d}\ln v}{\mathrm{d}\ln r} \;\to\; \frac{1}{2\ln(r/r_c)}. \tag{3.6}$$The numbers confirm it: at $x = 30$ the computed $u$ is $3.651$ against $2\sqrt{\ln x} = 3.688$; at $x = 10^{3}$, $5.288$ against $5.257$; at $x = 10^{6}$, $7.503$ against $7.434$. The logarithmic slope falls only logarithmically. That single fact is what makes §7.4's refutation unavoidable: raising the coronal temperature moves $r_c$ in proportion, but it buys only a logarithm of slope, and a logarithm cannot be made to fall fast enough.
The breeze. On a subsonic branch with $C \gt -3$, $u\to 0$ as $x\to\infty$, so $h(u)\to -2\ln u$ and (3.3) gives $-2\ln u \approx 4\ln x$, that is $u \propto x^{-2}$. Computing the logarithmic slope between $x = 30$ and $x = 100$ on the $C = -2$ solution gives $-1.9612$, approaching $-2$ from above. Hence:
The isothermal assumption of §3 is a strong one and §7 refutes it. Before that, it is worth asking what survives without it, because the answer is a bound that the data can be held against — and a bound derived before any data is looked at is worth more than a fit made after.
Repeating the derivation of Proposition 2 with $P = K\rho^\gamma$ replaces $c_T^2$ by $c^2(r)$ in (3.5) and leaves the structure intact: the bracket still vanishes at $v = c$, the right side still vanishes where $2c^2/r = GM/r^2$, and the two must still coincide. What changes is that $c$ is now a function of position, so the critical radius is determined implicitly. The new content is in the slope there.
| $\gamma$ | $s_+$ | $s_-$ | product | accelerating transonic solution? |
|---|---|---|---|---|
| $1.0000$ | $+1.0000$ | $-1.0000$ | $-1.0000$ | yes — the isothermal case of §3 |
| $1.1000$ | $+0.7828$ | $-0.9733$ | $-0.7619$ | yes |
| $1.2000$ | $+0.5788$ | $-0.9424$ | $-0.5455$ | yes |
| $1.3000$ | $+0.3840$ | $-0.9058$ | $-0.3478$ | yes |
| $\mathbf{1.3940}$ | $+0.2051$ | $-0.8634$ | $-0.1771$ | yes — the measured value of §7.2 |
| $1.4000$ | $+0.1937$ | $-0.8604$ | $-0.1667$ | yes |
| $\mathbf{1.4360}$ | $+0.1250$ | $-0.8409$ | $-0.1051$ | yes — the median-fit value of §7.2 |
| $1.4900$ | $+0.0199$ | $-0.8071$ | $-0.0161$ | yes, barely |
| $\mathbf{1.5000}$ | $0.0000$ | $-0.8000$ | $0.0000$ | no — the positive root has just closed |
| $1.5500$ | $-0.1033$ | $-0.7595$ | $+0.0784$ | no |
| $1.6000$ | $-0.2183$ | $-0.7048$ | $+0.1538$ | no |
| $1.6667$ | $-0.5000$ | $-0.5000$ | $+0.2500$ | no — the two roots have merged |
| $1.7000$ | complex | complex | $+0.2963$ | no — no real transonic slope exists |
The bound is fixed before any measurement is consulted, and that is what gives §7.2 its force. Nothing about the Sun entered (4.1); it is a statement about the local structure of a differential equation at a singular point. A gas whose effective index happened to exceed $3/2$ would have no accelerating transonic solution, and the exponents that Venzmer & Bothmer fitted had no reason whatever to respect that.
The solution can also be integrated numerically from the critical point outward, which tests the integrator rather than the physics. Taking $\gamma = 1.394$, $T_0 = 6.0$ MK at $r_0 = 1.03\,R_\odot$ — which places $r_c$ at $0.769\,R_\odot$ — and integrating to $3.00$ au gives $v$ rising from $352.2$ to $505.3\ \mathrm{km\,s^{-1}}$. At $\gamma = 1.6$, with critical slopes $-0.2183$ and $-0.7048$, the integrator finds no accelerating branch and refuses, exactly as (4.1) predicts.
Reverse the arrow in Fig. 1. The gas is now at rest far away, at density $\rho_\infty$ and sound speed $c_\infty$, and falls in. Equations (3.5) and (4.1) are unchanged, and by Proposition 4 the relevant branch for $\gamma \gt 3/2$ — including the adiabatic $5/3$ — is a decelerating-outward, that is accelerating-inward, transonic solution. This is Bondi's 1952 problem.
Two checks on (5.2), one of them requiring no algebra. At $\gamma = 1$ the sound speed is constant, so the critical radius is $GM/2c_\infty^2$ while $R_B = 2GM/c_\infty^2$, a ratio of exactly $1/4$ — and $(5-3)/8 = 1/4$. At $\gamma = 5/3$ the ratio is zero: the adiabatic flow has no interior critical point at all, which is precisely why $\lambda$ is a limit there rather than a solution property, and why §8's Sgr A* calculation quotes $\lambda = 1/4$ without a sonic radius.
$R_B \equiv 2GM/c_\infty^2$, with the factor of 2. Some authors write $R_B = GM/c_\infty^2$ and every derived ratio then shifts by a factor of 2. The definition with the 2 is used here because it is the one Baganoff et al. (2003) use in the sentence §8 quotes, and mixing the two conventions across a comparison would be a silent factor-of-two error.| $\gamma$ | $\lambda_c$ | $r_c/R_B$ | note |
|---|---|---|---|
| $1.000000$ | $1.120422$ | $0.2500$ | isothermal; $\mathrm{e}^{3/2}/4 = 1.120422$ exactly |
| $1.000001$ | $1.120421$ | $0.2500$ | the numerical limit $\gamma\to1$, ratio $0.999999$ to the exact value |
| $1.100000$ | $0.995128$ | $0.2125$ | |
| $1.200000$ | $0.871158$ | $0.1750$ | not checked against the source — see below |
| $1.300000$ | $0.748069$ | $0.1375$ | |
| $1.400000$ | $0.625000$ | $0.1000$ | exactly $5/8$ |
| $1.500000$ | $0.500000$ | $0.0625$ | |
| $1.600000$ | $0.366950$ | $0.0250$ | |
| $1.666667$ | $0.250000$ | $0.0000$ | adiabatic monatomic; exactly $1/4$, and no interior critical point |
The first verification is internal. Assembling $\dot M = 4\pi r_c^2\rho_c c_c$ directly from the critical-point quantities, for a test case, gives $1.383290\times10^{11}\ \mathrm{g\,s^{-1}}$; evaluating the closed form (5.3) gives $1.383290\times10^{11}\ \mathrm{g\,s^{-1}}$. The ratio is $1.00000000$, eight figures. That verifies the algebra and says nothing whatever about nature, and §8 depends on the distinction.
Everything so far is structural. Put the Sun's numbers in.
With $M = M_\odot$ and the base at $r_0 = 1.03\,R_\odot$, where the escape speed is $608.6\ \mathrm{km\,s^{-1}}$, the transonic solution of (3.3) gives:
| $T_0$ [MK] | $c_T$ [km s⁻¹] | $r_c$ [$R_\odot$] | $r_c$ [au] | $v(0.29\,\mathrm{au})$ | $v(1\,\mathrm{au})$ | chord $\mathrm{d}\ln v/\mathrm{d}\ln r$ over $0.29$–$0.98$ au |
|---|---|---|---|---|---|---|
| $0.50$ | $86.0$ | $12.899$ | $0.05999$ | $209.0$ | $283.7$ | $0.2477$ |
| $1.00$ | $121.6$ | $6.449$ | $0.02999$ | $357.6$ | $451.3$ | $0.1885$ |
| $1.50$ | $148.9$ | $4.300$ | $0.02000$ | $478.2$ | $585.8$ | $0.1645$ |
| $2.00$ | $172.0$ | $3.225$ | $0.01500$ | $583.2$ | $702.3$ | $0.1505$ |
| $3.00$ | $210.6$ | $2.150$ | $0.01000$ | $764.9$ | $902.9$ | $0.1343$ |
Speeds of a few hundred kilometres per second at 1 au, from a calculation containing no free parameter but the temperature. That is the result which made the paper famous, and it is qualitatively right. §7 asks whether it is quantitatively right, and the last column is where the answer lives.
With the Baumbach–Allen base density $\rho_0 = 5.7685\times10^{-16}\ \mathrm{g\,cm^{-3}}$, the base speeds and rates are $77.82\ \mathrm{m\,s^{-1}}$ and $2.897\times10^{11}\ \mathrm{g\,s^{-1}}$ at 1 MK; $2753.71\ \mathrm{m\,s^{-1}}$ and $1.025\times10^{13}$ at 1.5 MK; $14\,468\ \mathrm{m\,s^{-1}}$ and $5.385\times10^{13}$ at 2 MK. A factor of two in temperature moves the rate by 186. (6.1) is the reason, and §7.7 is where it matters: any agreement between a predicted and an observed mass-loss rate, to within a factor of a few, is a very weak statement about the temperature.
This is a bound the model places on itself, derived from its own premise, and it is what turns §7.4's statement from "no temperature I tried worked" into "no temperature can work". The measured corona at 2–3 MK is a factor $2.1$ to $3.1$ below (6.2), so a real corona is comfortably subsonic at its base and the model is not in trouble there. It is in trouble elsewhere.
Helios 1 and Helios 2 flew inside Mercury's orbit between 1974 and 1986 and measured the solar wind's density, speed, temperature and magnetic field in situ over $0.29$ to $0.98$ au. Venzmer & Bothmer (2018) fitted a power law $x(r) = d\,r^{e}$ to each of the four quantities over the combined data set, with $r$ in au and $d$ the value at 1 au, and published both mean and median fits with errors on every coefficient and exponent. Their Table 3, in full:
| quantity | median $d$, $e$ | mean $d$, $e$ | yearly scatter in $e$ |
|---|---|---|---|
| density [cm⁻³] | $5.61(27)$, $-2.093(46)$ | $7.57(30)$, $-2.010(38)$ | $0.072$ |
| velocity [10² km s⁻¹] | $4.107(28)$, $+0.058(13)$ | $4.356(24)$, $+0.049(10)$ | $0.012$ |
| temperature [10⁴ K] | $7.14(23)$, $-0.913(39)$ | $9.67(21)$, $-0.792(28)$ | $0.050$ |
| magnetic field [nT] | $5.377(92)$, $-1.655(17)$ | $6.05(10)$, $-1.546(18)$ | $0.11$ |
Each exponent was fitted independently of the others, and no fluid equation was imposed on any of them. That is what makes the table usable as a test rather than as an illustration: nothing this module predicts was put into the fit. The yearly scatter, from their Fig. 9, is the weighted standard deviation of exponents fitted year by year across the solar cycle, and it is quoted alongside the formal error throughout because it is the larger and more honest measure of how repeatable an exponent is.
A reader who has come this far will notice that Module 2 §10 anchored on this same table, and is entitled to ask whether the book is checking one measurement twice and counting it as two results. It is not, and the distinction is worth setting out before any number is computed.
Module 2 used the density and velocity exponents to test continuity, which forces $\alpha-\beta = 2$ exactly for a steady spherical flow; the mean fits give $1.961$ and the median fits $2.035$, bracketing the prediction. It then used the density and temperature exponents to test the adiabatic closure, which predicts a temperature exponent of $-\alpha(\gamma-1) = -1.340$ against a measured $-0.792\pm0.028$, a $19.6\sigma$ refutation recorded in its §10.5.
This module uses the same four numbers to test three different things: the critical-point bound $\gamma \lt 3/2$ of Proposition 4, which is a statement about a singular point and not about a closure; the sign of the velocity exponent, which distinguishes the solution families of Proposition 3; and the momentum equation itself, fed the measured pressure gradient, which neither earlier test touched. §7.4(i) does re-use Module 2's refutation of a closure, and it cites that result rather than re-deriving it.
One further caveat, inherited from Module 2 and not repaired here. The fits describe the mean or median of a distribution of streams at each radius — slow dense wind and fast tenuous wind mixed — not any single stream. Module 2's §10 set out the condition under which power-law exponents carry across such a mixture, namely radial self-similarity of the mixing weights, and the same condition is assumed here. It is an assumption, and §7.6 names it again where it bites hardest.
For a polytropic flow, $P \propto \rho^\gamma$ with an ideal gas $P \propto \rho T$ gives $T \propto \rho^{\gamma-1}$, so with $\rho \propto r^{-\alpha}$ and $T \propto r^{e_T}$,
$$ \gamma_{\rm eff} \;=\; 1 + \frac{-e_T}{\alpha}. \tag{7.1}$$The mean fits give $\alpha = 2.010\pm0.038$ and $e_T = -0.792\pm0.028$, hence $\gamma_{\rm eff} = 1.3940\pm0.0158$. The median fits give $\alpha = 2.093\pm0.046$ and $e_T = -0.913\pm0.039$, hence $1.4362\pm0.0210$.
The same number separates the wind from both idealised closures: $\gamma_{\rm eff}$ is $24.9\sigma$ above the isothermal $1$ and $17.3\sigma$ below the adiabatic $5/3$ for the mean fits ($20.8\sigma$ and $11.0\sigma$ for the median). The solar wind is neither, which is the physical content of Module 2 §10.5 restated as an index.
Proposition 3 gives three qualitatively different objects and §3.1 gives their signatures: a transonic wind accelerates, so $\beta \gt 0$; a breeze decays as $r^{-2}$, so $\beta \lt 0$; a static corona has $v = 0$ and no exponent at all. The measurement is $\beta = +0.049\pm0.010$ (mean) and $+0.058\pm0.013$ (median).
The strength of this check is that it requires no model to interpret and no parameter to be chosen: one measures a sign. Its weakness is the same thing — a sign carries one bit. What it establishes is which of Proposition 3's families the wind belongs to, and nothing about whether the family's quantitative predictions hold. The next three subsections show that they do not.
(i) The temperature exponent. An isothermal wind has $T$ constant, so $e_T = 0$. The measurement is $-0.792\pm0.028$, a departure of $\mathbf{28.3\sigma}$ against the formal error and $15.8\sigma$ against the yearly scatter. Module 2 §10.5 reported the adiabatic closure failing at $19.6\sigma$ against the same exponent; that result is cited here and not re-derived. Both idealisations fail, in opposite directions, which is why §7.2's intermediate index exists at all.
(ii) No coronal temperature reproduces the acceleration. The last column of §6's table is the logarithmic slope of the predicted $v(r)$ over the Helios chord $0.29$–$0.98$ au, which is the quantity $\beta$ measures. Scanning $T_0$ from $0.5$ MK to the ceiling $T_{\rm ceiling} = 6.26$ MK of Proposition 8:
| $T_0$ [MK] | $0.50$ | $1.00$ | $1.50$ | $2.00$ | $3.00$ | $4.00$ | $5.00$ | $\mathbf{6.26}$ |
|---|---|---|---|---|---|---|---|---|
| chord slope | $0.2477$ | $0.1885$ | $0.1645$ | $0.1505$ | $0.1343$ | $0.1246$ | $0.1179$ | $\mathbf{0.1119}$ |
Raising $T_0$ beyond the ceiling does keep lowering the slope, because of (3.6), and the price is worth printing. A decade in $T_0$ costs $52$ per cent of the slope from $0.5$ to $5$ MK, $46$ per cent from $1$ to $10$, $41$ per cent from $2$ to $20$ and $36$ per cent from $5$ to $50$ — the fall slows as $T_0$ rises, exactly as a $1/(2\ln x)$ dependence requires. The temperature that would actually reach $0.049$ is $1644$ MK: a factor $548$ to $822$ above the 2–3 MK corona, and $263$ times the temperature at which the model destroys its own premise. The isothermal wind is not slightly too fast to accelerate correctly; it is wrong by an amount no temperature can absorb.
(iii) Fit the temperature to the 1-au speed and look at what it implies. Choosing $T_0$ so that $v(1\ \mathrm{au}) = 435.6\ \mathrm{km\,s^{-1}}$ gives $T_0 = 0.947$ MK; matching the median $410.7$ instead gives $0.866$ MK. Venzmer & Bothmer's own paper cites a near-Sun coronal temperature of 2–3 MK (their Sect. 6, p. 11, quoting Billings 1959 and Liebenberg et al. 1975).
The obvious repair is to abandon $\gamma = 1$ and use the measured $\gamma_{\rm eff} = 1.3940$ throughout. It fails, in three independent ways, and each one is computed rather than argued.
First, a polytropic wind with $\gamma \gt 1$ has a minimum launch temperature — below it the gas runs out of enthalpy before reaching the critical point — and it follows from Proposition 5 in three lines. Bernoulli for a base at rest fixes the constant $B = c_0^2/(\gamma-1) - GM/r_0$, with $c_0^2 = \gamma k_{\rm B}T_0/(\mu m_{\rm u})$. Evaluating the same constant at the critical point, where $v = c_c$ and $c_c^2 = GM/2r_c$, gives $B = c_c^2(5-3\gamma)/[2(\gamma-1)]$. For $1 \lt \gamma \lt 5/3$ that bracket is positive, so a critical point exists if and only if $B \gt 0$: the gas must carry more enthalpy at the base than the potential well costs. That is $c_0^2/(\gamma-1) \gt GM/r_0$, or
$$ T_0 \;\gt\; \frac{\gamma-1}{\gamma}\,\frac{GM\mu m_{\rm u}}{k_{\rm B}r_0} \;=\; \frac{2(\gamma-1)}{\gamma}\,T_{\rm ceiling}, \tag{7.4}$$with $T_{\rm ceiling} = 6.26$ MK from (6.2) — the minimum launch temperature and Proposition 8's ceiling are the same quantity, scaled by $2(\gamma-1)/\gamma$. At $r_0 = 1.03\,R_\odot$ that minimum is $T_0 \gt 3.540$ MK for $\gamma = 1.3940$, and $3.804$ MK for the median $1.4362$; the same formula gives $1.1385$ MK at the $\gamma = 1.1$ of Problem D2. Both exceed the top of the measured 2–3 MK range, by factors of $1.18$ and $1.27$, and exceed its bottom by $1.77$ and $1.90$. The isothermal limit is the check on (7.4): as $\gamma \to 1$ the minimum goes to zero, which is right, because the isothermal wind never runs out of enthalpy.
Second, force the solution to reproduce the observed 1-au speed. With $\gamma = 1.3940$ that requires $T_0 = 5.39$ MK, and the resulting solution has a critical radius of $r_c = 1.020\,R_\odot$.
Third, that same forced solution gets the wrong temperature and the wrong acceleration. It predicts $T = 5.817\times10^{4}$ K at 1 au against the measured $9.670\times10^{4}$ K, a ratio of $0.602$; and $\mathrm{d}\ln v/\mathrm{d}\ln r = +0.00802$ at 1 au against the measured $+0.049\pm0.010$, too small by a factor of $6.1$.
Fig. 3 puts the preceding three subsections on one page. The upper panel plots $v(r)$ for three coronal temperatures with the Helios range shaded and the measured fit drawn over it, and on that axis the models look approximately right — which is the trap. The lower panel plots the residual: the logarithmic slope over $0.29$–$0.98$ au, predicted five different ways, against the measured value drawn both as a sixth bar and as a vertical band. On an axis running from 0 to 900 km s⁻¹ a slope of $0.049$ against $0.19$ is a difference of a few pixels; on the residual axis it is the whole panel. This is Module 2's lesson applied: where the disagreement is the result, plot the disagreement.
Every comparison so far has tested a wind model. The momentum equation can be tested without one. Rearranging (3.5) for the logarithmic slope and keeping the pressure gradient general,
$$ \frac{\mathrm{d}\ln v}{\mathrm{d}\ln r} \;=\; \frac{1}{v^2}\left[-\frac{r}{\rho}\frac{\mathrm{d}P}{\mathrm{d}r}-\frac{GM}{r}\right] \;=\; \frac{(\alpha-e_T)\,c_T^2 - GM/r}{v^2}, \tag{7.2}$$where the second form uses $P \propto \rho T \propto r^{-(\alpha-e_T)}$, so that $-\mathrm{d}\ln P/\mathrm{d}\ln r = \alpha-e_T$ is read directly off the fitted exponents. No closure is assumed — not isothermal, not polytropic, not barotropic. The pressure gradient is measured; the gravity is known; the speed is measured. Everything on the right of (7.2) comes from the data, and the left side is also measured, independently, as $\beta$.
At 1 au with the mean fits, $v = 435.6\ \mathrm{km\,s^{-1}}$, $T = 9.670\times10^{4}$ K and $n_p = 7.57\ \mathrm{cm^{-3}}$: $\alpha-e_T = 2.802$, $c_T^2 = 1.4301\times10^{13}\ \mathrm{cm^2\,s^{-2}}$ — that is $(37.8\ \mathrm{km\,s^{-1}})^2$ — giving a pressure term $+0.02112$ and a gravity term $-GM/(rv^2) = -0.00468$. Hence
$$ \left(\frac{\mathrm{d}\ln v}{\mathrm{d}\ln r}\right)_{\rm predicted} = 0.01644 \pm 0.00061 \qquad\text{against}\qquad \beta_{\rm measured} = 0.049 \pm 0.010. \tag{7.3}$$The prediction's own error, $\pm0.00061$, is small enough that it barely moves the significance, which is why it is printed: the uncertainty in this comparison lives almost entirely in the measured $\beta$.
And the composition does not rescue it. Venzmer & Bothmer treat the wind as a pure proton plasma — their Sect. 1, p. 2: "In the analyses, we treat the solar wind as a proton plasma — the average helium abundance is about 4.5 % and in slow wind at solar cycle minimum is even less than 2 %." Their density is therefore a proton density on a proton-plasma assumption, and the 5 per cent helium of (2.1) is this module's own choice and not theirs. Redoing (7.3) for a pure proton plasma, where $\rho = n_{\rm p}m_{\rm p}$ and $n_{\rm tot} = 2n_{\rm p}$, so that $\mu = m_{\rm p}/(2m_{\rm u}) = 0.50364$ against $m_{\rm u}$ — half of $m_{\rm p}/m_{\rm u}$, and not $1/2$, which would give $1.1244$ and a ratio of $2.57$ — raises $c_T^2$ by a factor $1.1163$ to $1.5964\times10^{13}\ \mathrm{cm^2\,s^{-2}}$ and the prediction to $0.01890\pm0.00068$, moving the ratio from $2.98$ to $\mathbf{2.59}$ and the shortfall from $3.2\sigma$ to $3.0\sigma$. The composition is worth about 12 per cent in $c_T^2$ and less than half a unit in the ratio. It is not the missing force either.
A careful reader will object that the predictor and the predicted are not fully independent: the pressure exponent uses the fitted $\alpha = 2.010$, and steady continuity ties $\alpha$ to $\beta$ by $\alpha = 2+\beta$, which Module 2 §10 verified. Imposing that tie exactly, $\alpha = 2.049$, moves the prediction from $0.01644$ to $0.01674$ — 1.8 per cent, against a measured-to-predicted ratio of $2.98$. The conclusion is unmoved.
Verscharen, Bale & Velli (2021) measured the radial mass flux per steradian, $F_m = r^2NMU_r$, from Ulysses: about $3.5\times10^{-16}$ au² g cm⁻² s⁻¹ sr⁻¹ over the poles during the first fast-latitude scan, $2.2\times10^{-16}$ during the third, and $1\times10^{-16}$ to $15\times10^{-16}$ in the equatorial region (their Sect. 3.1). Multiplying by $4\pi\,\mathrm{au}^2$ — a conversion computed here, not quoted from them, since they never state a total — gives $9.843\times10^{11}$, $6.187\times10^{11}$ and $2.812\times10^{11}$ to $4.218\times10^{12}\ \mathrm{g\,s^{-1}}$ respectively. Helios flew in the ecliptic, so the equatorial range is the one to compare against; the polar rung is quoted because $4\pi\,\mathrm{au}^2$ times it is the familiar $10^{12}\ \mathrm{g\,s^{-1}}$.
The Helios fits supply an independent cross-check. With $n_p = 7.57\ \mathrm{cm^{-3}}$ and $v = 435.6\ \mathrm{km\,s^{-1}}$ at 1 au, a pure proton plasma gives $\dot M = 4\pi\,\mathrm{au}^2 n_p m_{\rm p}v = 1.551\times10^{12}\ \mathrm{g\,s^{-1}}$, that is $F_m = 5.52\times10^{-16}$; carrying the $1+4y = 1.2$ helium mass of (2.1) — this module's choice and not Venzmer & Bothmer's, and one that Module 2 §10 did not make when it read the same table — gives $1.861\times10^{12}\ \mathrm{g\,s^{-1}}$ and $F_m = 6.62\times10^{-16}$. Both lie comfortably inside the equatorial range and both are derived without Ulysses, which is the point: the composition choice does not decide this comparison. (Module 2 also reads a different row of that table, $n = 5.61\ \mathrm{cm^{-3}}$ at $410.7\ \mathrm{km\,s^{-1}}$, so its $1.084\times10^{12}\ \mathrm{g\,s^{-1}}$ differs from the $1.551\times10^{12}$ here by the density and the speed, not by the helium.)
The Parker prediction with the Baumbach–Allen base density gives $2.897\times10^{11}\ \mathrm{g\,s^{-1}}$ at 1 MK — a ratio of $0.294$ to the polar rung — and $1.566\times10^{11}$ at the fitted $0.947$ MK, a ratio of $0.159$. At 2 MK it gives $5.385\times10^{13}$, a ratio of $54.7$.
The inverse is the more informative direction, because it exposes the base density rather than hiding behind it. To deliver the observed mass flux, an isothermal model needs a base electron density of $1.074\times10^{9}\ \mathrm{cm^{-3}}$ at 1 MK, $3.036\times10^{7}$ at 1.5 MK, or $5.778\times10^{6}$ at 2 MK — against the $3.161\times10^{8}\ \mathrm{cm^{-3}}$ that Baumbach–Allen supplies. Three temperatures within the plausible range demand base densities spanning a factor of $186$. Given that the base density is itself an extrapolation below its source's fitted range (§2.1), this comparison constrains very little, and saying so is more useful than quoting the one temperature at which the two happen to agree.
Proposition 5 needs three numbers: a mass, a gas density and a sound speed at the accretion radius. For the black hole at the centre of the Galaxy all three have been measured, which is rare, and a fourth quantity — the actual accretion rate — has been bounded independently, which is rarer. The comparison is therefore possible, and it is the largest ratio this book has produced.
The mass is the GRAVITY Collaboration's, from the orbits of individual stars: $M = 4.30\times10^{6}\,M_\odot$ to about $\pm0.25$ per cent, that is $8.5502\times10^{39}$ g. The gas is Chandra's, from Baganoff et al. (2003), their Sect. 11.1.2: $n_{\rm e} \approx 130\ \mathrm{cm^{-3}}$ and $kT_{\rm e} \approx 2$ keV $= 2.321\times10^{7}$ K, with the flow taken as adiabatic, $\gamma = 5/3$, and the gas fully ionised at twice solar abundance, $\mu \approx 0.70$.
Baganoff et al.'s density convention is $\rho = n_{\rm e}\mu m_{\rm H}$ — mass per particle multiplied by the electron density — and it is reproduced here deliberately, so that the comparison in §8.1 tests the same quantity they computed. Their $\mu \approx 0.70$ is quoted against their own $m_{\rm H}$ and is the one mean molecular weight in this module not converted to the $m_{\rm u}$ convention of (2.1), because converting a number inside a quoted formula would misrepresent the source. The consequences of the convention are computed and shown to be immaterial at the end of §8.1.Those inputs give $\rho = 1.5221\times10^{-22}\ \mathrm{g\,cm^{-3}}$ and $c_\infty = \sqrt{\gamma kT/(\mu m_{\rm H})} = 675.4\ \mathrm{km\,s^{-1}}$, against the $670\ \mathrm{km\,s^{-1}}$ the paper quotes — a ratio of $1.0080$, the difference being rounding in their printed inputs. Then $R_B = 2GM/c_\infty^2 = 0.0811$ pc. By (5.2) at $\gamma = 5/3$ the critical radius is zero: the adiabatic flow has no interior critical point, which is exactly why $\lambda_c = 1/4$ enters as a limit. Proposition 5 then gives
$$ \dot M_{\rm Bondi} \;=\; 5.0549\times10^{20}\ \mathrm{g\,s^{-1}} \;=\; 8.022\times10^{-6}\ M_\odot\ \mathrm{yr^{-1}}. \tag{8.1}$$Baganoff et al. computed the Bondi rate from their own numbers and got $\dot M_B \approx 3\times10^{-6}\,M_\odot\,\mathrm{yr^{-1}}$ with $\lambda = 1/4$. They used the black-hole mass standard in 2001, which their own paper states as $2.6\times10^{6}\,M_\odot$ and which this module also recovers, as $2.61\times10^{6}\,M_\odot$, by back-solving their quoted $R_B \approx 0.05$ pc and $c_\infty \approx 670\ \mathrm{km\,s^{-1}}$. Since $\dot M \propto M^2$, rescaling to the GRAVITY mass gives $8.15\times10^{-6}\,M_\odot\,\mathrm{yr^{-1}}$.
The density convention can be varied to see whether it matters. Taking instead $\rho = n_{\rm e}m_{\rm p}(1+4y)/(1+2y)$ at $y = 0.1$ — mass per electron, for twice-solar helium — gives $2.537\times10^{-22}\ \mathrm{g\,cm^{-3}}$, a factor $1.67$ larger, and $\dot M = 1.34\times10^{-5}\,M_\odot\,\mathrm{yr^{-1}}$. Every conclusion below moves by less than a factor of two and none changes.
A second consistency check, also not a test: Baganoff et al.'s Sect. 11.3 gives a local diffuse plasma at $kT \approx 1.3$ keV and $n_{\rm e} \approx 26\ \mathrm{cm^{-3}}$ and quotes $\dot M_{B,\rm local} \sim 1\times10^{-6}\,M_\odot\,\mathrm{yr^{-1}}$; this module's formula at their mass gives $1.13\times10^{-6}$, a ratio of $1.13$. (Their $26\ \mathrm{cm^{-3}}$ is a root-mean-square electron density at unity filling factor, from their Sect. 7.)
Marrone et al. (2007) detected Faraday rotation toward Sgr A*, with a ten-epoch mean rotation measure $\mathrm{RM} = (-5.6\pm0.7)\times10^{5}\ \mathrm{rad\,m^{-2}}$ — $8.0\sigma$ from zero. The rotation measure integrates $n_{\rm e}B_\parallel$ along the line of sight through the accretion flow, so it bounds how much plasma can be there. The bound depends on how far in the rotating medium extends:
| inner radius | Marrone et al.'s limit on $\dot M$ [$M_\odot\,\mathrm{yr^{-1}}$] | kind | $\dot M_{\rm Bondi}$ / limit |
|---|---|---|---|
| $\sim 30\ r_S$ | $2.0\times10^{-7}$ | upper | $40.1$ |
| $\sim 100\ r_S$ | $5.0\times10^{-8}$ | upper | $160.4$ |
| $\sim 10\ r_S$ | $1$–$2\times10^{-8}$ | lower | $802$ to $401$ |
| $\sim 3\ r_S$ | $2$–$4\times10^{-9}$ | lower | $4011$ to $2006$ |
Two further comparisons do not depend on the field at all, and the refutation rests on them.
The lower limits. Marrone et al. state explicitly that these "are not subject to the above caveats since the uncertainties act to raise the minimum accretion rate". They give each as a range and not as a single value — "$\dot M$ must be greater than $1$–$2\times10^{-8}\,M_\odot\,\mathrm{yr^{-1}}$ or $2$–$4\times10^{-9}\,M_\odot\,\mathrm{yr^{-1}}$, respectively" — so the excess is quoted here as a range too: $802$ to $401$ at $\sim10\,r_S$ and $4011$ to $2006$ at $\sim3\,r_S$. Take the end of each range that is kindest to the Bondi model, $2\times10^{-8}$ and $4\times10^{-9}$, and the excess is still $\mathbf{401}$ and $\mathbf{2006}$. An upper limit that might be soft is one thing; a rate that exceeds even the minimum plasma content the rotation measure demands, by two to three orders of magnitude on the most generous reading, is another.
The radiative efficiency. Baganoff et al.'s absorption-corrected 2–10 keV luminosity is $\approx 2\times10^{33}\ \mathrm{erg\,s^{-1}}$. If the Bondi rate were actually accreting, the available power would be $\dot M c^2 = 4.543\times10^{41}\ \mathrm{erg\,s^{-1}}$, so the implied efficiency is
$$ \eta_{\rm rad} \;=\; \frac{L_X}{\dot M_{\rm Bondi}c^2} \;\approx\; 4\times10^{-9}. \tag{8.2}$$ One significant figure and no more. Baganoff et al. give $L_X = (2.4^{+3.0}_{-0.6})\times10^{33}\ \mathrm{erg\,s^{-1}}$ and state that the 2–10 keV luminosity is "known only to within a factor of two"; they then round to $\approx 2\times10^{33}$ throughout their discussion, which is the figure used here. The interval on $L_X$ alone spans $\eta_{\rm rad} = 4\times10^{-9}$ to $1\times10^{-8}$. Quoting three digits for a quantity whose numerator is uncertain by a factor of two would be a false precision, and the conclusion needs none.Fig. 5 places all of these on one logarithmic axis, which is the only fair way to show a ratio of $4011$ and a ratio of $1.016$ in the same picture.
Not the algebra. §5.1 verified (5.3) against a direct assembly from the critical point to eight figures and against Bondi's own table to every legible digit. And note that the algebra was never doing as much work as is often claimed: Bondi proves $\lambda \le \lambda_c$, so even a perfect implementation does not fix the coefficient. A refutation resting on "the algebra is right, so the physics must be wrong" would be resting on a theorem that does not exist.
What fails are the three physical premises of Proposition 5, each of which Sgr A* violates in a way that is separately documented:
Each premise, on its own, would reduce the rate. All three act in the same direction, and together they span the factor of 40 — or 4, or 401, depending on which leg is being used — without difficulty. The value of the calculation is not that it predicts the rate. It is that it predicts a rate which is definitely wrong, by a definite amount, in a definite direction, and the size of the discrepancy is the measure of how much physics the spherical picture omits.
The equations of this module are steady, spherically symmetric continuity and Euler, with an isothermal or polytropic closure and no source terms. The table names each idealisation and where it is repaid.
| Idealisation made here | What it costs | Where it is repaid |
|---|---|---|
| No angular momentum | Premise (1) of §8.3. Sgr A*'s gas circularises near $100\,r_S$ instead of falling in, and no spherical calculation can describe what happens inside that radius | Module 11 (accretion discs) |
| No magnetic field | Larger than it looks. Marrone et al.'s limits are entirely statements about the field, and the factor 40 of §8.2 becomes 4 if the field is 3 per cent of equipartition. The solar wind's Alfvén surface near $12\,R_\odot$ is a structure this module cannot see at all. And Module 1 §8.1's magnetic rescue is what licenses a fluid description of the wind in the first place, at $\mathrm{Kn}\approx1.9$ | Module 12 |
| Radiation dynamically unimportant; $L_{\rm Edd}$ borrowed | (8.2) measures how badly this fails for Sgr A*: $\eta_{\rm rad} \approx 4\times10^{-9}$ against $0.1$. Problems K2 and K3 use the Eddington luminosity without deriving it | Module 13 (radiation) |
| One fluid, one temperature | False at 1 au, and it is where §7.6's shortfall is least trustworthy: $c_T^2$ uses a single $T$ for protons, electrons and helium. Module 1 gave the slow ion–electron equilibration that makes this generally wrong in hot plasma | Module 12; two-temperature flows in Module 11 |
| No heating term; the coronal temperature is an input | §7.4 refutes the isothermal wind but cannot say what heats the corona, and §7.6 measures a shortfall without naming the force that fills it. Module 2 §10.5 deferred the same question on the same data | Module 10 (turbulent dissipation), which is where Module 2 sent it |
| No thermal conduction | Parker's own 1958 corona conducts and cools outward; this module's does not, and that is most of the factor $5.5$ in Check 1. Conduction also plausibly sets the temperature profile that §7.5 finds no single polytrope can match | Not repaid anywhere in Modules 1–14. It is a genuine gap in this book's arc, and saying so is better than implying a later module closes it |
| Steady state | Excludes the solar cycle — which is why the yearly exponent scatter of §7 is quoted as a second error bar throughout — as well as coronal mass ejections and Sgr A*'s X-ray flares | Not repaid |
| Spherical symmetry | The real wind is fast and tenuous over the poles and slow and dense in the ecliptic. This is why §7.7 compares against the equatorial flux range, and why Problem D3 has to mix a polar mass flux with an ecliptic speed and say so | Not repaid |
| The critical-point condition itself | Proposition 3 shows the transonic solution is the only one that connects a subsonic base to a supersonic flow, but nothing here proves nature selects it. Bondi's own selection argument, quoted in §5, is a physical argument and not a theorem | Not repaid. §7.3 tests the selection observationally instead, which is the best this book does |
One limitation belongs to the checks rather than to the equations. Both anchors compare a steady, spherically symmetric, single-stream calculation against measurements that are ensemble averages over a structured flow: the Helios fits average slow and fast streams, and the Chandra measurement averages over a region five decades larger than the one where the accretion is decided. §7.1 states the self-similarity condition under which the first of these is legitimate. For the second there is no such condition, and the comparison in §8 should be read as a statement about orders of magnitude — which is all it needs to be, given that the smallest ratio in it is $4$ and the largest is $2\times10^{7}$.
Nine problems in the house set: three conceptual (C), three derivational (D), three computational (K). Every numerical answer below is produced by afd/figs/m09_numbers.py and independently verified by afd/figs/m09_problems_check.py.
$c_T = \sqrt{k_{\rm B}T/\mu m_{\rm u}} = 121.61\ \mathrm{km\,s^{-1}}$ with $\mu = 0.56220$. Then $r_c = GM_\odot/(2c_T^2) = 4.4868\times10^{11}\ \mathrm{cm} = 6.449\,R_\odot = 0.02999$ au. Integrating (3.3) on the transonic branch to 1 au gives $v = 451.3\ \mathrm{km\,s^{-1}} = 3.71\,c_T$.
The escape speed at $r_c$ is $\sqrt{2GM/r_c} = 243.2\ \mathrm{km\,s^{-1}}$, which is $2c_T$ to every digit. It must be: $r_c \equiv GM/2c_T^2$ rearranges to $2GM/r_c = 4c_T^2$, so $v_{\rm esc}(r_c) = 2c_T$ identically, for every temperature and every central mass. The critical radius is defined as the place where the escape speed is twice the sound speed, and the identity is a restatement of the definition rather than a result.
The physical reading is worth having. At $r_c$ the thermal energy per particle, $\tfrac12\mu m_{\rm u}c_T^2$, is one quarter of the binding energy $\tfrac12\mu m_{\rm u}v_{\rm esc}^2$. Inside that radius gravity dominates and the flow is subsonic; outside it, pressure does, and the flow can accelerate freely.
$10^{-4}M_\odot = 1.98841\times10^{29}$ g, so $t = 2.0201\times10^{17}\ \mathrm{s} = 6.401\times10^{9}$ yr, a factor $1.40$ longer than the Sun's $4.57\times10^{9}$ yr. Running the same rate over the Sun's actual age removes $7.14\times10^{-5}\,M_\odot$, seven parts in $10^{5}$.
The reconciliation with Module 2, which a reader should demand. Module 2 §10 computed $1.72\times10^{-14}\,M_\odot\,\mathrm{yr^{-1}}$ and $7.9\times10^{-5}\,M_\odot$ over 4.6 Gyr. The rate used here is $1.56\times10^{-14}\,M_\odot\,\mathrm{yr^{-1}}$, which is $10$ per cent smaller, and the two are not in conflict: they are different measurements. Module 2 took the Helios median fit at 1 au and assumed the ecliptic values hold over the whole sphere, which its own provenance note flags as false; this module takes Verscharen et al.'s polar rung, converted by $\dot M = 4\pi F_m\,\mathrm{au}^2$. A 10 per cent difference between an ecliptic extrapolation and a polar measurement of a flow known to be latitude-dependent is smaller than one would expect, not larger. Both give $\approx 8\times10^{-5}\,M_\odot$ over the Sun's life, and the conclusion — that the wind is irrelevant to the Sun's mass budget and dominant for the heliosphere — is the same in both.
$r_S = 2GM/c^2 = 1.2699\times10^{12}\ \mathrm{cm} = 0.085$ au. With $R_B = 0.0811$ pc from §8, $R_B/r_S = 1.970\times10^{5}$. The free-fall time from rest at $R_B$ is $t_{\rm ff} = (\pi/2)\sqrt{R_B^3/(2GM)} = 1.844\times10^{2}$ yr. The $\pi/2$ is what the substitution $r = R_B\cos^2\eta$ produces on integrating $\mathrm{d}r/\mathrm{d}t = -\sqrt{2GM(1/r - 1/R_B)}$, exactly as in Module 3's free-fall calculation; the bare $\sqrt{R_B^3/(2GM)} = 117.4$ yr is the free-fall scale and not the time. Either figure is short enough that the flow has had ample time to reach a steady state, which is one premise of Proposition 5 that Sgr A* does not obviously violate.
Baganoff et al. compute $R_B \approx 0.05$ pc and then adopt $0.06$ pc, and they attach the figure "$2\times10^{5}\,r_S$" to that pair at their black-hole mass of $2.6\times10^{6}\,M_\odot$. Both $R_B$ and $r_S$ carry exactly one factor of $M$, so at fixed $c_\infty$ the ratio is mass-independent outright: $R_B/r_S = c^2/c_\infty^2 = 1.970\times10^{5}$, with no mass in it at all. Their $2\times10^{5}$ is their computed $0.05$ pc — $0.05\ \mathrm{pc}/r_S(2.6\times10^{6}M_\odot) = 2.01\times10^{5}$ — and that is the figure which agrees with ours. Their adopted $0.06$ pc is $2.41\times10^{5}\,r_S$ at their own mass, and only $1.46\times10^{5}\,r_S$ at the GRAVITY mass. The lesson is that a ratio quoted in gravitational radii and a length quoted in parsecs carry different mass dependences, and a comparison across two papers with different assumed masses has to convert one of them — which is exactly what §8.1 does with the $(M_{\rm new}/M_{\rm old})^2$ rescaling of the rate.
The ratios are $1.0000$, $0.7155$, $0.3536$, $0.0894$ and $0.0075$, matching $(1+v^2/c_\infty^2)^{-3/2}$ exactly, since that is the formula. At $v = 5c_\infty$ the rate is down by a factor $133$: motion suppresses accretion far more effectively than temperature does, because $v$ enters where $c_\infty$ does and the exponent is $-3/2$ in the sum.
What kind of result it is. It is Bondi's own conjecture, not a derivation — his Sect. 8, in his own words: "it seems likely that it represents the order of magnitude of the accretion rate in the intermediate case, in which a star of mass $M$ moves with relative velocity $V$ in a uniform cloud of gas." He constructed it to interpolate between two cases he had solved: the velocity-limited Hoyle–Lyttleton rate, his eq. (21), and the temperature-limited rate at $\gamma = 3/2$, his eq. (22). Nothing in this module derives it, and the flow it describes is not spherically symmetric, so Propositions 2–6 do not apply to it.
One detail of the prefactor. Bondi writes $A = 2\pi(GM)^2(V^2+c^2)^{-3/2}\rho_\infty$, and his $2\pi$ is $4\pi\lambda_c(3/2)$, the coefficient for the single $\gamma$ his eq. (22) used. This module writes $4\pi\lambda_c(\gamma)$ so that $v \to 0$ reproduces (5.3) at any $\gamma$. That generalisation is a reasonable one and it is not his; it is recorded here so that no reader attributes it to him.
From (4.1) at $\gamma = 1.1$ the roots are $s_+ = +0.7828$ and $s_- = -0.9733$; the positive root exists because $1.1 \lt 3/2$. The minimum launch temperature at $1.03\,R_\odot$ is $1.1385$ MK, from (7.4). At $T_0 = 2$ MK: $c_c = 128.4\ \mathrm{km\,s^{-1}}$, $r_c = 5.785\,R_\odot$, $v(\infty) = 529.4\ \mathrm{km\,s^{-1}}$ and $v(1\ \mathrm{au}) = 371.8\ \mathrm{km\,s^{-1}}$.
Note that $v(1\ \mathrm{au})$ is well short of $v(\infty)$: at $\gamma = 1.1$ the wind is still accelerating at the Earth's orbit, having reached 70 per cent of its terminal speed. Compare the isothermal case at the same temperature, $702.3\ \mathrm{km\,s^{-1}}$ at 1 au from §6's table. A modest departure from isothermality — $\gamma$ from $1$ to $1.1$ — nearly halves the speed, because the gas cools as it expands and has less enthalpy left to convert.
$\lambda_c(1.1) = 0.9951$, from (5.3). The pair is the module's title in one problem. The same index, the same equation and the same critical-point condition describe a wind leaving the Sun at $372\ \mathrm{km\,s^{-1}}$ and an accretion flow arriving at a compact object with a coefficient of $0.995$. Only the sign of $v$ differs.
With $\dot M = 9.843\times10^{11}\ \mathrm{g\,s^{-1}}$ and $v = 435.6\ \mathrm{km\,s^{-1}}$: $\tfrac12\dot Mv^2 = 9.3384\times10^{26}\ \mathrm{erg\,s^{-1}}$, which is $2.440\times10^{-7}\,L_\odot$. The gravitational term $GM_\odot\dot M/R_\odot = 1.878\times10^{27}\ \mathrm{erg\,s^{-1}} = 4.91\times10^{-7}\,L_\odot$ — twice as large as the kinetic term, so most of the energy invested in the wind goes into lifting it out of the potential well rather than into its final speed. Verscharen et al.'s measured $F_E \approx 1$ au² g s⁻³ sr⁻¹ over the poles converts to $2.812\times10^{27}\ \mathrm{erg\,s^{-1}} = 7.35\times10^{-7}\,L_\odot$. The kinetic and gravitational terms sum to $2.81\times10^{27}$, and the agreement to three digits is a coincidence of a source figure quoted as "about $1$": the energy flux should exceed that sum by the enthalpy flux, $\tfrac52 c_T^2\dot M \approx 3.5\times10^{25}\ \mathrm{erg\,s^{-1}}$, about $1.3$ per cent, plus a conductive flux this module does not compute.
All three are within a factor of three of one another and all are $\sim10^{-6}$ of $L_\odot$. The solar wind is energetically negligible for the Sun and structurally decisive for the heliosphere, which is the same conclusion C2 reaches about mass. It also tells us why the coronal heating problem is hard: the energy required is so small a fraction of the luminosity that almost any mechanism could supply it, and the smallness gives no leverage for choosing between mechanisms.
A mixture that must be declared. $\dot M$ here is Verscharen et al.'s polar rung and $v$ is Venzmer & Bothmer's ecliptic fit. They describe different wind streams at different latitudes. The mixture is defensible for an order-of-magnitude energy budget — the two differ by less than a factor of two in every relevant quantity — but the answer is an order of magnitude and not a measurement, and it should not be quoted to three digits elsewhere.
Taking the cloud as fully ionised with $\mu \approx 0.6$ against $m_{\rm H}$, $c_\infty = \sqrt{k_{\rm B}T/\mu m_{\rm H}} = 9.81\ \mathrm{km\,s^{-1}}$, so $v/c_\infty = 2.65$ and motion dominates. The density is $\rho = n_{\rm H}m_{\rm p}(1+4y) = 4.683\times10^{-25}\ \mathrm{g\,cm^{-3}}$: $n_{\rm H}$ is the hydrogen density and $1+4y = 1.4$ is the helium mass, so this $\rho$ is $2.33$ times the $n_{\rm H}\mu m_{\rm H}$ that the sound speed is built from, and the distinction is worth $0.37$ dex in the answer. Then $R_B = 2GM/(c_\infty^2+v^2) = 3.437\times10^{13}\ \mathrm{cm} = 2.3$ au, and the rate from D1's formula at $\gamma = 1$, where $\lambda_c = \mathrm{e}^{3/2}/4 = 1.1204$, is $5.411\times10^{9}\ \mathrm{g\,s^{-1}} = 8.59\times10^{-17}\,M_\odot\,\mathrm{yr^{-1}}$.
Against the outward wind at $9.84\times10^{11}\ \mathrm{g\,s^{-1}}$, the ratio is 182: the Sun sheds mass $182$ times faster than it gathers it, so accretion is irrelevant to its mass budget by more than two orders of magnitude.
And the accretion does not happen at all, for a second and independent reason: $R_B = 2.3$ au lies deep inside the heliosphere, whose boundary is near $120$ au. Interstellar gas never reaches that radius; the solar wind sweeps it aside first. Two separate arguments kill the same effect, which is worth noticing — the ratio 182 would be the answer even if the heliosphere did not exist.
The ionisation assumption. $\mu = 0.6$ takes the cloud as fully ionised, which it is not; it is partly ionised, and a neutral cloud would give $\mu = 1.27$ and $c_\infty = 6.75\ \mathrm{km\,s^{-1}}$. Because $v^2 \gg c_\infty^2$ the denominator is dominated by the motion, so $\dot M$ moves by only 11 per cent and no conclusion changes. The assumption is stated rather than buried because a reader checking the sound speed would otherwise find it unexplained.
With $\mu = 2.33$ against $m_{\rm H}$ for molecular gas, $c_\infty = 0.2662\ \mathrm{km\,s^{-1}}$ — cold gas is very quiet. Taking $\gamma = 1$ so $\lambda_c = \mathrm{e}^{3/2}/4 = 1.1204$, $R_B = 2GM/c_\infty^2 = 3.746\times10^{18}\ \mathrm{cm} = 250\,413$ au $= 1.2140$ pc, and $\dot M = 5.124\times10^{22}\ \mathrm{g\,s^{-1}} = 8.133\times10^{-4}\,M_\odot\,\mathrm{yr^{-1}}$.
The Eddington luminosity is $L_{\rm Edd} = 4\pi GMm_{\rm p}c/\sigma_{\rm T} = 1.257\times10^{39}\ \mathrm{erg\,s^{-1}}$, and at $\eta_{\rm rad} = 0.1$ the corresponding rate is $\dot M_{\rm Edd} = 1.399\times10^{19}\ \mathrm{g\,s^{-1}}$. The Bondi rate exceeds it by a factor of $\mathbf{3664}$, and at that rate the hole would double its mass in $1.23\times10^{4}$ yr.
What it implies is that the calculation has refuted itself. A flow accreting three thousand times the Eddington rate would radiate far above $L_{\rm Edd}$, and the resulting radiation pressure would drive the gas away — violating the premise, used in Proposition 5, that radiation is dynamically unimportant. The honest reading is not "black holes in molecular clouds double in ten thousand years"; it is that spherical Bondi accretion is self-inconsistent in this regime, and a calculation that includes radiation is required. Module 13 supplies it.
$n = 10^{4}\ \mathrm{cm^{-3}}$ is taken as the total particle density here, not the H₂ density. Reading it as H₂ instead would raise $\rho$ by 20 per cent and $\dot M$ with it. $L_{\rm Edd}$ is quoted from Module 13 and not derived in this module; its $m_{\rm p}$ is a genuine proton mass and not a mean molecular weight.
$L_{\rm Edd} = 5.405\times10^{44}\ \mathrm{erg\,s^{-1}}$ at $4.30\times10^{6}\,M_\odot$, so $L_X/L_{\rm Edd} = 3.70\times10^{-12}$. At $\eta_{\rm rad} = 0.1$, $\dot M_{\rm Edd} = 9.545\times10^{-2}\,M_\odot\,\mathrm{yr^{-1}}$, giving $\dot M_{\rm Bondi}/\dot M_{\rm Edd} = 8.405\times10^{-5}$ and, for Marrone's equipartition upper bound, $2.095\times10^{-6}$.
The pattern is the point. The Bondi rate is already $10^{-4}$ of Eddington, so Sgr A* is not accretion-starved in any sense that radiation pressure could explain — nothing is pushing the gas away. Yet the luminosity is $10^{-12}$ of Eddington: $8.405\times10^{-5}$ against $3.700\times10^{-12}$ is a factor $2.27\times10^{7}$, which is $7.4$ orders of magnitude, and at $\eta_{\rm rad} = 0.1$ the two columns would have been equal. Splitting that factor isolates the discrepancy: $40.1$ of it lives in the rate itself, between the Bondi value and Marrone et al.'s equipartition upper limit, and the remaining $5.7\times10^{5}$ — $5.75$ orders of magnitude — lives in the radiative efficiency, which (8.2) puts at $4\times10^{-9}$ against a thin disc's $0.1$. Sgr A* is faint mainly because the gas that does arrive radiates almost nothing, not because little gas arrives.
Every symbol used in this module, with its units and the place it is defined. Seven symbols collide with earlier modules, and every collision is resolved here rather than left to the reader. This is the largest notation load of any module so far, because the same letters serve a wind, an accretion flow and two different sound speeds.
| symbol | meaning | units | defined |
|---|---|---|---|
| $v(r)$, $\dot M$ | radial flow speed, positive outward and negative inward; mass flux $4\pi r^2\rho|v|$, always quoted positive with its direction in words | $\mathrm{cm\,s^{-1}}$, $\mathrm{g\,s^{-1}}$ | Definition 1; Module 2 (10.1) |
| $\mu$, $m_{\rm u}$ | mean mass per particle in units of the unified atomic mass unit. Measured against $m_{\rm u}$, this book's convention from Module 3 onward, the unsubscripted glyph from Module 6; the same composition gives $0.55814$ against $m_{\rm p}$, and every product $\mu m$ is unchanged. The one exception is Baganoff et al.'s $\mu\approx0.70$, kept against their $m_{\rm H}$ inside their own formula | —, g | Definition 1, (2.1) |
| $c_T$ | isothermal sound speed $\sqrt{k_{\rm B}T/\mu m_{\rm u}} = \sqrt{P/\rho}$. Written $c_T$ and never $a$, because $a$ is the cosmological expansion factor of Module 5 book-wide; it does not appear in this module | $\mathrm{cm\,s^{-1}}$ | Definition 2, (3.1) |
| $c(r)$, $c_c$, $c_\infty$ | polytropic sound speed $\sqrt{\gamma P/\rho}$ and its values at the critical point and at infinity. This is the adiabatic sound speed of Module 4, written $c$ there too; $c_T$ is its $\gamma\to1$ limit | $\mathrm{cm\,s^{-1}}$ | Definition 3 |
| $r_c$ | the critical (sonic) radius, $GM/2c_T^2$ in the isothermal case. One symbol for the wind and for the accretion flow alike, because Proposition 6 shows they are the same point; the difference goes in words. Never written $r_s$, because $r_S$ below differs from it only by case | cm | Definition 2, (5.1) |
| $x$, $u$ | the dimensionless radius $r/r_c$ and speed $v/c_T$. $u$ here is a scalar dimensionless ratio, not the bulk velocity $\mathbf{u}$ of Modules 1–2, and not Module 8's similarity profile; the bulk velocity is written $v(r)$ throughout this module | —, — | Definition 2 |
| $h(u)$, $R(x)$, $C$ | the two halves of the first integral (3.3), always written with their arguments and confined to §3; $C$ is its integration constant. Module 8's $h$ is a mixing width and its $w$ an enthalpy; Module 3's $R$ is a stellar radius; Module 2's $C[f]$ is a collision term. None appears here | —, —, — | Proposition 3 |
| $s$ | the logarithmic slope $\mathrm{d}\ln v/\mathrm{d}\ln r$ at the critical point. Module-local, as are Module 4's entropy and acoustic coordinate and Module 5's growth rate, which share the glyph and nothing else | — | Proposition 4, (4.1) |
| $\gamma$, $\gamma_{\rm eff}$ | the polytropic index of the flow, and the value $1+(-e_T)/\alpha$ inferred from the measured exponents. Not the ratio of specific heats, which is $5/3$ for the gas throughout | — | Definition 3, (7.1) |
| $\alpha$, $\beta$, $e_T$ | the fitted exponents of density, velocity and temperature, $x(r) = d\,r^{e}$. The same meaning as Module 2 §10, deliberately, because both modules read the same table. Module 5's $\alpha$ is a length scale, Module 6's a mixing-length parameter, Module 7's $\alpha_B$ a mixing coefficient and Module 3's $\beta$ the gas-pressure fraction $P_{\rm gas}/P$, whose radiation share is $1-\beta$; none appears here | — | §7 |
| $\lambda_c$ | Bondi's accretion coefficient. Bare $\lambda$ is the coefficient the true flow realises; $\lambda_c$ is the maximum the steady equations permit, and Proposition 5 states the inequality $\lambda \le \lambda_c$ between them, so both glyphs are needed and neither is dropped. Neither is Module 1's mean free path nor Module 8's similarity variable. Parker's own escape parameter, also written $\lambda$ in 1958, is not used here | — | Proposition 5, (5.3) |
| $R_B$, $r_S$ | the Bondi radius $2GM/c_\infty^2$, with the factor 2, and the Schwarzschild radius $2GM/c^2$. Some authors omit the 2 in $R_B$; the convention here is Baganoff et al.'s, so that §8 compares like with like | cm, cm | (5.2), Problem C3 |
| $F_m$, $F_E$ | radial mass and energy flux per steradian, $r^2N M U_r$ and its energy counterpart. The totals $4\pi F\,\mathrm{au}^2$ are computed here, never quoted | au² g cm⁻² s⁻¹ sr⁻¹; au² g s⁻³ sr⁻¹ | §7.7 |
| $\eta_{\rm rad}$, $\epsilon$ | radiative efficiency $L/\dot Mc^2$; and the ratio of the magnetic field to its equipartition value in Marrone et al.'s scaling. The efficiency is written $\eta_{\rm rad}$, never bare $\eta$, because bare $\eta$ is the dynamic viscosity in the convention declared at Module 2's notation table and in force from Module 6 on — and because two other modules have already borrowed the bare glyph for module-local quantities, Module 7's interface displacement and Module 5's density-profile exponent, which is precisely why a bare $\eta$ here would carry no meaning at all. The subscript is the same device $\lambda_c$ uses. $\epsilon$ is Marrone et al.'s own symbol and is confined to §8.2 | —, — | (8.2), §8.2 |
| $L_{\rm Edd}$, $\dot M_{\rm Edd}$ | the Eddington luminosity $4\pi GMm_{\rm p}c/\sigma_{\rm T}$ and the rate $L_{\rm Edd}/\eta_{\rm rad} c^2$ at $\eta_{\rm rad} = 0.1$. Stated, not derived; its $m_{\rm p}$ is a genuine proton mass | $\mathrm{erg\,s^{-1}}$, $\mathrm{g\,s^{-1}}$ | Problems K2, K3; derived in Module 13 |
| $\mathrm{RM}$ | rotation measure, the line-of-sight integral of $n_{\rm e}B_\parallel$ | $\mathrm{rad\,m^{-2}}$ | §8.2 |
One symbol borrowed from a source and renamed on the way in: Allen (1947) writes his radius, in solar radii, as $p$. It is written $R$ in §2.1, because $p$ is a pressure everywhere in this book.
Every published value this module compares against, with what was taken from it and from where. Each was read from the text of the paper, extracted locally from the PDF; none was taken from a summary or from memory. One convention, carried over from Module 8: a § on its own is a section of this book and is a link; a section of a cited paper is written "their Sect. N", with no § and no link.
Every number in this module is produced by afd/figs/m09_numbers.py and every figure by afd/figs/m09_build_figs.py, which imports the first so that a figure cannot drift from the prose. The problem-set answers are independently verified by afd/figs/m09_problems_check.py, which recomputes them without importing either, and which rebuilds every modelling-choice input — $\mu$ from the composition statement, $r_0$ from the prose, the Baumbach–Allen density from the quoted formula, Baganoff's $\rho = n_{\rm e}\mu m_{\rm H}$ from their own sentence — rather than sharing a helper with the generator.