The classic derivation of the steady-state spherically symmetric accretion rate onto a point mass moving through a uniform gaseous medium: Ṁ = 4πλ(GM)²ρ∞/c∞³, where λ depends on the gas adiabatic index, ρ∞ and c∞ are the ambient density and sound speed. This is the “Bondi accretion” formula Giddings-Mangano apply (alongside the earlier Bondi & Hoyle 1944 treatment) once a hypothetical stable micro black hole has grown past the subatomic/electromagnetic-capture regime into the macroscopic gravitational-accretion regime, to derive how long it would take to consume a star or planet. relevance_note: The primary physics behind the macroscopic-accretion leg of the growth-timescale calculation that Premise B’s “no danger” conclusion depends on.
§§2-6 The steady-state solution
A-17 - Steady spherical adiabatic accretion admits solutions only up to a maximal rate 4πλc(GM)²ρ∞ c∞^-3
Reasoning (exact closed-form analysis; no data, no free parameters beyond γ):
- Setup: continuity 4πr²ρv = Ṁ and Bernoulli ½v² + ∫dp/ρ - GM/r = 0 with p/p∞ = (ρ/ρ∞)^γ. Non-dimensionalize with r = x·GM/c², v = y·c, ρ = z·ρ∞ (c = sound speed at infinity), giving x²yz = λ (with Ṁ = 4πλ(GM)²c⁻³ρ∞) and ½y² + (z^{γ-1}-1)/(γ-1) = 1/x.
- Substituting the local Mach number u = y z^{-(γ-1)/2} reduces the system to f(u) = λ^{2(γ-1)/(γ+1)} g(x), where f and g are each the sum of a positive and a negative power, hence each has a positive minimum (f at u=1, g at x = (5-3γ)/4).
- Since the physical range of x (from the stellar surface, x ~ 1e-5 for the Sun, to infinity) contains g’s minimum, a solution exists only if λ ≤ λc = [(γ+1)/2]^{-(γ+1)/2(γ-1)} · [(5-3γ)/4]^{-(5-3γ)/2(γ-1)}; Table I: λc = 1.12 (γ=1), 0.625 (γ=1.4), 0.5 (γ=1.5), 0.25 (γ=5/3). For λ > λc no steady solution exists at all.
- Solution taxonomy: for λ < λc the flow is everywhere subsonic (Type I); exactly at λ = λc a smooth monotonic transonic solution exists (Type II, subsonic outside xm, supersonic inside - the case whose limiting γ→5/3 form makes Types I and II coincide). u → 0 at infinity by the boundary conditions.
- Hence the derived scaling Ṁ ∝ M²ρ∞c∞⁻³ is forced by the equations; only the O(1) factor λ within (0, λc] is left undetermined by the steady-state analysis. This is the macroscopic gravitational-capture rate formula later applied (e.g. by Giddings-Mangano) to hypothetical stable black holes accreting stellar or planetary matter.
Validity verdict (step 6)
Reconstruction: long but elementary closed-form fluid mechanics - continuity + Bernoulli + polytrope, nondimensionalized, reduced via the Mach variable to f(u) = lambda^(2(gamma-1)/(gamma+1)) g(x) with f, g each a sum of a positive and negative power. Traced the load-bearing existence step: f and g each have positive minima, the physical x-range contains g’s minimum, so a steady solution exists only for lambda ⇐ lambda_c; the scaling Mdot ∝ M^2 rho_inf c_inf^-3 is forced by the nondimensionalization itself. The lambda_c values follow from the stated closed form. Checked; no defeater conditional on the steady spherical polytropic premises.
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§§7-8 Which rate is realized; the general case
A-18 - Energy comparison selects the maximal-rate transonic solution as the state actually realized
Reasoning:
- The steady-state equations leave λ ∈ (0, λc] free, so a selection principle outside them is needed (boundary conditions at the stellar surface impose nothing - the star swallows whatever falls in; a time-dependent entry calculation à la Bondi-Hoyle 1944 is intractable here).
- The gas energy per unit mass is constant (Bernoulli with zero constant), so comparing states reduces to comparing densities. From eqs. (13), (15), (16): for u < 1, z decreases as λ increases at fixed x - every shell has lower energy in the λ = λc state than in any smaller-λ Type I state; and at λ = λc the Type II (transonic) branch has lower density than Type I for x < xm.
- Expecting the lowest-energy available steady state to be the stable one, the system should sit in the Type II state with λ = λc. Bondi notes the caveats explicitly: the system is not isolated (the star is outside it) and is infinite in extent, so the energy-minimization principle is “not quite assured” - but the shell-by-shell agreement makes it “very likely”, and it matches the intuition that nothing stops accretion, so it proceeds at the greatest possible rate.
Validity verdict (step 6)
Reconstruction: premises = the shell-by-shell density comparison (z decreases with lambda at fixed x; Type II below Type I inside the sonic radius) plus the hidden premise, surfaced charitably, that an open steady system settles into the lowest-energy available steady state. Conclusion is already hedged to “expected to be physically realized”. The hidden premise is a heuristic, not a theorem - Bondi flags exactly this (system not isolated, infinite extent) - so the argument is valid-but-weak: conditional on the premises the hedged conclusion follows, and its limited strength is for step 7 to price, not a validity failure. The density-comparison step itself is traced from the cited equations.
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H-16 - Physical accretion realizes the maximal transonic rate, with an interpolation formula covering the moving-star case
Bondi flags both parts as plausible rather than proven: the steady-state equations do not select λ (an energy-comparison argument does), and the interpolation formula for the intermediate velocity-and-pressure-limited regime is an order-of-magnitude conjecture. He also notes the pressure limitation is likely somewhat overestimated if the gas can radiate compression heat (effective γ closer to 1, raising the rate).
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