Bondi accretion in the finite luminous region of elliptical galaxies

M. Samadi; S. Abbassi; S. Zangeneh

REVIEW 3 major objections 6 minor 14 references

Bondi accretion in the finite luminous region of elliptical galaxies

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Galaxy gravity boosts Bondi accretion rate

desk verdict A sound, modest extension of Bondi accretion to finite radii and Hernquist galaxy radiation; the algebra holds, but the saddle claim and radiation-force assumptions need sharper treatment. read the letter →

arxiv 1908.07356 v1 pith:D5A2S5JN submitted 2019-08-19 astro-ph.HE

M. Samadi , S. Zangeneh , S. Abbassi This is my paper · ORCID

classification astro-ph.HE

keywords BondiaccretionfiniteradiusboundaryellipticalgalaxiesmassiveblackholesHernquistgalaxyradiationpressureThomsonscatteringrate

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks how spherical accretion onto a central massive black hole changes when the inflow starts not at infinity but at the outer edge of a real galactic gas reservoir, and when the gravity and light of the host elliptical galaxy also push on the gas. It establishes two things. First, in the classical Bondi problem with a finite outer boundary, the kinetic energy of the gas at that boundary is negligible once the boundary is more than about two semi-Bondi radii $GM/c_{sf}^2$, and the maximum dimensionless accretion rate $\lambda_c$ returns to Bondi's classical values ($0.625$ for $\gamma=1.4$, $0.367$ for $\gamma=1.6$) once the boundary is two to three orders of magnitude beyond that radius. Second, for a Hernquist galaxy the galaxy's gravity raises $\lambda_c$ roughly linearly in the dimensionless linear density $m_g/R_g$, while radiation from the central source and from the galaxy, parameterized by $l$ and $l_g$, lowers it. Getting these scalings right matters because real observations of inflowing gas in elliptical galaxies sit at finite radii, so using them to estimate black hole accretion rates depends on correcting for boundary and galactic radiation effects.

What carries the argument

The machinery is the finite-radius Bernoulli integral in dimensionless form. The paper writes the steady spherical flow equations in units where $r = R r_e$, $v = V c_{sf}$, $c_s = C_s c_{sf}$, and $\rho = Z \rho_f$, with $r_e = GM/c_{sf}^2$; mass conservation becomes $R^2 V Z = \lambda$, where $\lambda = \dot M / (4\pi G^2 M^2 \rho_f/c_{sf}^3)$ is the dimensionless accretion rate. The Bernoulli equation separates into $f(u)$, unchanged from Bondi, and $g(R)$, which encodes all finite-boundary and galactic terms; the maximum of $g(R)$, at the critical radius where the Mach number is one, determines $\lambda_c$. For the Hernquist galaxy the function is $g(R) = R^{4(\gamma-1)/(\gamma+1)}[(1-l)/R + m_g(1-l_g)/(R+R_g) - (1-l)/R_f - m_g(1-l_g)/(R_f+R_g) + V_f^2/2 + 1/(\gamma-1)]$; its derivative gives the critical radius $R_c$ and its value at $R_c$ gives $\lambda_c$ (Eqs. 18 and 32). The paper also checks the topology of the transonic solutions and finds saddle-type critical points.

What would settle it

Re-solve the sonic-point condition retaining the boundary kinetic term $V_f^2/2$ in the Bernoulli integral for a grid of $R_f$ and $m_g/R_g$ values; if the resulting $\lambda_c$ differs from the values produced by Eq. (32) by more than the paper's quoted few-percent accuracy for $R_f \gtrsim 2$, or if $\lambda_c$ fails to approach $0.625$ ($\gamma=1.4$) and $0.367$ ($\gamma=1.6$) as $R_f$ reaches about $10^3$, the central finite-radius claim is falsified.

Watch

Extended reading notes

Core claim

On its own terms, the paper claims that Bondi's steady spherical accretion solution is robust to replacing the boundary at infinity with a finite radius $r_f$, provided $r_f$ is large enough. The Bernoulli integral retained at the finite boundary shows that the boundary kinetic term $V_f^2/2$ is negligible for $R_f = r_f/(GM/c_{sf}^2) \gtrsim 2$, so the approximate critical accretion parameter $\lambda_{c0}$ and the exact $\lambda_{c1}$ agree to within a few percent. At large $R_f$, $\lambda_c$ asymptotes to the classical Bondi constants $0.625$ ($\gamma=1.4$) and $0.367$ ($\gamma=1.6$). In the Hernquist galaxy case, the galaxy's potential adds a term $m_g/(R+R_g)$ to the Bernoulli function, and optically thin Thomson-scattered radiation from the central source and the stellar population adds repulsive terms $(1-l)/R$ and $m_g(1-l_g)/(R+R_g)$; maximizing the resulting function $g(R)$ (Eq. 32) gives a critical radius and a critical $\lambda_c$ that increase with the galaxy's linear density $m_g/R_g$ and decrease with both luminosity parameters, with a saddle-type transonic solution at the critical point.

Load-bearing premise

The load-bearing premise is that every gas parcel is optically thin with opacity set only by Thomson scattering and that both the central black-hole luminosity and the galaxy luminosity are constant with radius, so the radiation force acts as pure inverse-square repulsion; if absorption, multiple scattering, or radiation-driven outflows become significant, the derived dependence of $\lambda_c$ on $m_g/R_g$, $l$, and $l_g$ no longer follows.

Editorial extensions

If this is right

For observed elliptical galaxies, Bondi estimates of black hole accretion rates made from gas density and temperature at a finite radius remain reliable whenever that radius is more than about $10^2$-$10^3 GM_{\mathrm{BH}}/c_{sf}^2$; closer in, boundary corrections should be applied.
In a Hernquist host, a denser, more compact stellar distribution (larger $m_g/R_g$) raises the maximum possible accretion rate, so galaxy mass profiles push the inferred rate above the classical Bondi value.
Radiative feedback suppresses accretion: higher Eddington-scaled central luminosity $l$ and galactic luminosity $l_g$ both decrease $\lambda_c$, with the central luminosity having the stronger effect.
The critical point is always a saddle, so the analytic maximum-accretion solution is the physically selected transonic solution and can serve as the reference state for time-dependent or multi-dimensional work.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The near-linear dependence of $\lambda_c$ on $m_g/R_g$ hints at a ready correction factor of the form $1 + \alpha\, m_g/R_g$ for Bondi-based black hole accretion estimates in ellipticals, with $\alpha$ set by $\gamma$, $l$, and $l_g$; fitting that factor to observed galaxies would be a direct test of the model.
Because the radiation terms appear as $(1-l)$ and $m_g(1-l_g)$ multiplying the two gravitational terms, the same equations imply an Eddington-scaled threshold near $l=1$ or $l_g=1$ at which the effective gravity reverses and outflow rather than accretion should set in, a regime the paper does not explore.
The optically thin, Thomson-only treatment is best suited to hot, diffuse coronal gas; at smaller radii where absorption and line driving matter, the derived scalings would need to be replaced, so for bright systems the present $\lambda_c$ likely acts as an upper bound rather than a full prediction.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know this paper before going deeper: it's a sound but modest extension of Bondi's spherical accretion problem to a finite outer radius in a Hernquist galaxy, including radiation pressure from the central source and the galaxy. The central derivation is internally consistent and reduces to the classical Bondi value in the right limit. If you work on SMBH accretion estimates from X-ray data, the finite-radius correction here is worth knowing, but it's not a breakthrough.

The genuinely new pieces are the general polytropic index (earlier work by Korol et al. 2016 and Ciotti & Pellegrini 2017, 2018 was isothermal), and the separate treatment of central and galactic luminosity in the radiation force. The paper also makes a clean quantitative statement: once the outer radius R_f is about 2-3 orders of magnitude larger than GM/c_sf^2, the critical accretion parameter λ_c is essentially the Bondi value. That is useful and testable.

The soft spots are in proportion to the claim. First, the comparison with Korol et al. and Ciotti & Pellegrini is qualitative. The reader is told these exist but never shown a direct numerical comparison, so it's hard to see exactly what the polytropic generality adds in practice. Second, the saddle-type conclusion in Section 5 is asserted from visual inspection of Mach-number plots. No local stability analysis is given, so that conclusion is under-supported. Third——and this is the most physically fragile piece——the radiation force is assumed to be optically thin Thomson scattering, which makes the (1-l)(1-lg) factorization in Eq. (32) exact. If absorption or multiple scattering matters, the monotonic trends in Fig. 6 are not guaranteed. The authors acknowledge this in Section 6 and defer it to future work, so it's not a hidden flaw, but it does bound the applicability of the main result.

Presentation issues: the abstract in the version I have is truncated ("e.g.:"), and some figure captions are garbled. Minor, but worth fixing before publication.

Overall, this paper deserves a serious referee. It's a coherent, self-contained derivation that extends known results in a meaningful way. A specialist in spherical accretion or galaxy-scale feedback will get value; a general reader can skip. My recommendation: send it to review, but require the authors to add a quantitative comparison with the related isothermal papers and to either prove the saddle claim or soften it.

Referee Report

3 major / 6 minor

Summary. The paper revisits steady, spherically symmetric polytropic Bondi accretion with boundary conditions imposed at a finite outer radius r_f rather than at infinity. It derives the modified Bernoulli equation and the critical accretion parameter λ_c in dimensionless form (Eqs 12-19), showing numerically that λ_c approaches the classical Bondi value when r_f is 2-3 orders of magnitude larger than GM/c_sf^2 and that the kinetic energy at r_f is negligible for r_f > about 2 GM/c_sf^2. It then specializes to a Hernquist galaxy, adding the galactic gravitational potential and, in the optically thin Thomson-scattering approximation, the radiation force of the central source and the galaxy (Eqs 20-32). Parameter scans indicate that λ_c increases roughly linearly with the dimensionless galaxy linear density m_g/R_g and decreases with both Eddington-scaled luminosities l and l_g (Figs 3-6). The final section argues from Mach-number plots that the transonic critical point is a saddle.

Significance. If the central claims hold, the paper provides a simple, self-contained extension of Bondi's formula to finite, luminous galactic environments. The dimensionless formulation is clean, the algebra is internally consistent, and the recovery of the classical Bondi limit for large r_f is a reassuring external benchmark. The paper contains no fitted parameters and the parameter scans cover the relevant regime, so the qualitative scaling of λ_c with m_g/R_g, l, and l_g is useful for interpreting X-ray observations of elliptical galaxies. However, the dependence on l and l_g is only as robust as the inverse-square Thomson radiation force assumed in Eq. (29), and the paper's quantitative statements would benefit from a local stability analysis and numerical error estimates.

major comments (3)

[§2, equation after Eq. (19), Fig. 1] The 'exact' solution λ_c1 is not actually defined. The displayed algebraic equation involves two unknowns, λ_c1 and R, and is only the sonic-point condition u=1; it is not accompanied by the critical-point condition dg/dR=0 that would determine R_c. Please state the complete system, for example R_c from dg/dR=0 with A = -1/R_f + λ_c1^2/(2 R_f^4), together with λ_c1^{2a} = g(R_c,λ_c1)/f(1), and describe the numerical solution method. Without this, the comparison in Fig. 1 that is used to justify neglecting V_f^2/2 is not reproducible.
[§5, Eq. (33), Figs 7-8] The conclusion that the transonic critical point is of saddle type is not established by the plotted Mach-number curves. Saddle, node, and center are distinguished by the eigenvalues of the linearized autonomous system at the critical point, or equivalently by the signs of the two slopes of the integral curves in the (R,u) plane. The observation that 'the slope of u's curve completely changes and becomes positive at R=R_c' is not a local stability analysis. Please either add the eigenvalue calculation or soften the claim to 'consistent with a saddle'.
[§4, Eq. (29), Fig. 6, §6] The central claim that λ_c decreases monotonically with l and l_g is derived solely from the optically thin Thomson radiation force, where the luminosity enters through exact inverse-square terms and factorizes into (1-l) and (1-l_g). The manuscript lists absorption as future work in §6, but the abstract and summary present the dependence without this caveat. In a real galactic nucleus, dust absorption, line driving, or multiple scattering would modify both the magnitude and the radial dependence of f_rad, so the factorization in Eq. (32) and the linear trends in Fig. 6 are not guaranteed. Please state the optically thin, Thomson-dominated regime as a scope condition in the abstract and conclusion, and if possible give a rough column-density or dust-to-gas criterion for where the approximation breaks down.

minor comments (6)

[Abstract] The abstract contains the typo 'kinitic' for 'kinetic', and the text has several other typographical errors ('accrection', 'centeral') that should be corrected.
[Figure 2] The caption and text say Figure 2 shows R_c, but the rendered ordinate label appears to be λ_c; please verify that every panel in Figs 2 and 3 is labeled with the quantity actually plotted.
[Eq. (25)] The symbol L is used both for the central source luminosity and for the total galaxy luminosity in the same expression; please use distinct notation, for example L_BH and L_gal.
[§5] The statement that 'there are two solutions for u per each λ < λ_c' is imprecise, because for a fixed radius there may be zero, one, or two solutions depending on R; please rephrase as a statement about the topology of the solution curves.
[Figs 1-6] The paper does not report numerical convergence or error bars for the parameter scans; please state the root-finding tolerance, grid resolution, or other numerical accuracy used to generate the figures.
[Eqs. (15) and (17)] The critical-radius formulas are valid only for γ < 5/3; the paper treats γ = 1.4 and 1.6, but this domain of validity should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper derives λ_c from algebraic critical-point equations with all galaxy and luminosity parameters scanned as inputs, and the Bondi limit is a genuine consistency check.

full rationale

The derivation chain is self-contained. Starting from the steady, spherically symmetric polytropic Bondi problem, the paper integrates the momentum equation between a finite outer radius r_f and r, yielding the Bernoulli-type equation (5) and its dimensionless form (7). The critical accretion parameter λ_c is then obtained from the algebraic condition that f(u) and g(R) are simultaneously extremal, as stated in Eqs. (13)-(18). No parameter is fitted to reproduce the central claim. The galaxy mass ratio m_g=M_g/M_BH, scale-length ratio R_g=r_g/r_e, Eddington-scaled luminosities l and l_g, and outer radius R_f are all scanned inputs whose values are chosen by the authors, not inferred from the output. The result that λ_c approaches the classical Bondi value for large R_f is a limiting consistency check of the same equations: when r_f→∞ and the galaxy terms are absent, the function g(R) reduces to Bondi's, so the asymptotic values λ_c ≈ 0.625 for γ=1.4 and 0.367 for γ=1.6 are consequences of that limit, not an independently fitted endpoint. The radiation-force model of Eq. (29), f_rad = G M_BH [l/r^2 + (M_g/M_BH) l_g/(r+r_g)^2], is an explicit physical assumption (optically thin, Thomson scattering, constant luminosity) that determines the parameterization of l and l_g; it affects the astrophysical applicability of the result under absorption or multiple scattering, but it is not a circular step because the dependence of λ_c on l and l_g is derived from that assumed force law, not defined in terms of the conclusion. There is also no load-bearing self-citation: the cited galaxy-potential and radiation-density results (Hernquist 1990; Dehnen 1993; Tremaine et al. 1994) are external, and the paper does not invoke a uniqueness theorem or prior work by its own authors to force its choice of equations. Overall, the central derivations are independent of their conclusions, so the circularity score is 0.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

No new entities are introduced. The radiation force and gravitational potential are standard physical inputs. All dimensionless parameters are scanned model inputs, not fitted to data, so circularity burden is low.

free parameters (6)

γ (polytropic index) = 1.4 and 1.6 used in plots
Ratio of specific heats; chosen as representative values for the gas equation of state; the central results depend on it.
m_g = M_g/M_BH
Ratio of galaxy mass to black hole mass; scanned in Figs 2, 3, 8; the claim of increasing λ_c with m_g/R_g depends on this parameter.
R_g = r_g/r_e
Dimensionless galaxy scale length; scanned; affects the critical radius and accretion rate.
l = L/L_Edd
Eddington ratio of the central accreting source; scanned from 0 to 0.8; higher l reduces λ_c.
l_g = L_gal/L_Edd,g
Eddington ratio of the galaxy luminosity; scanned from 0 to 0.8; reduces λ_c but increases R_c.
R_f = r_f/r_e
Dimensionless outer boundary radius; scanned; the paper claims λ_c approaches the Bondi value for large R_f.

assumptions (5)

domain assumption Steady state, spherical symmetry, and a polytropic equation of state p = K ρ^γ for the accreting gas.
Basis of the Bondi model; stated in Sections 1-2.
domain assumption The gas is optically thin and opacity is dominated by Thomson electron scattering.
Used in Section 4 to write the radiation force as σ/c times the local flux; the authors state this holds for low accretion rates.
domain assumption The galaxy follows the Hernquist (1990) density and luminosity profile.
Adopted in Section 3 for the potential and in Section 4 for L_g(r).
domain assumption The central source luminosity L is constant with radius (Cassinelli & Castor 1973).
Used in Section 4 to treat L as a constant in the radiation force.
domain assumption Gas self-gravity is neglected.
Standard Bondi assumption; the gravitational potential includes only the central black hole and the Hernquist galaxy.

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Cite this review

Pith. "Pith review of Bondi accretion in the finite luminous region of elliptical galaxies." pith.science (2026). https://pith.science/paper/D5A2S5JN

@misc{pith2026190807356,
  author       = {Pith},
  title        = {Pith review of: Bondi accretion in the finite luminous region of elliptical galaxies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D5A2S5JN}},
  note         = {Machine review of arXiv:1908.07356}
}

abstract

The classical Bondi model is adopted to study accretion onto the finite luminous region around the central massive black hole (MBH) in an elliptical galaxy. Unlike Bondi (1952), we define the boundary conditions at a certain finite radius ($r_f$) instead of at the infinity and examine the variation of solutions for a simple case. In the following, we consider the special case of an MBH at the center of a Hernquist galaxy and involve the gravity and luminosity of its own galaxy. Our results in the first part show that kinetic energy at the final radius is ignorable even for not so far away from the center. Moreover, the mass accretion rate will be approximately equal to its Bondi value if the final radius ($r_f$) becomes about 2-3 orders of magnitude larger than semi-Bondi radius, i.e. $GM/c_{sf}^2$ (where $M$ and $c_{sf}$ are the mass of the central object and the sound speed at $r_f$). In the second part, adding the two extra forces of gravity and radiation in the momentum equation let us know that the maximum possible of accretion rate increases with a greater characteristic linear density of galaxy and lower radiation. e.g.:

Figures

Figures reproduced from arXiv: 1908.07356 by the authors.

**Figure 1.** Variation of critical dimensionless mass accretion rate, λc, critical radius, Rc, and their error percentages, ∆λ/λc0, ∆Rc/Rc0, with respect to the final radius of a certain region of accretion, Rf . These quantities have been obtained in two ways of approximate and exact solutions. The dashed lines are related to exact solutions that we have included the kinetic energy of particles at the end of accretion region. h… view at source ↗

**Figure 2.** Variation of critical radius, Rc with respect to the final radius of a certain region of accretion, Rf . Here the polytropic index, γ is equal to 1.4 and we have separately examined the effects of mg(= Mgalaxy/MBH) and Rg(= rg/re, rg is the characteristic scale-length of galaxy, re = GMBH/c2 sf ). Notice that for these plots, we have neglected the kinetic energy of particles at the end of selected accretion region. … view at source ↗

**Figure 3.** Variation of critical accretion parameter, λc with respect to the final radius of a certain region of accretion, Rf . Here the polytropic index, γ is equal to 1.4 and we have examined the effects of mg and Rg, separately. Notice that for these plots, we have assumed Vf ∼ 0, means that particles on the sphere of Rf are approximately at rest. 100 200 500 1000 2000 5000 1 ´ 10 0.100 0.050 0.200 0.030 0.150 0.070 Rf Rc … view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Variation of critical radius, Rc with respect to the final radius of a certain region of accretion, Rf . Here we have γ = 1.4 and we have used several values of l and lg which show the fraction of the luminosity of accretion process to the Eddington limit and the ratio…

**Figure 5.** Figure 5: Variation of critical dimensionless mass accretion rate, λc with respect to the final radius of the selected accretion region, Rf . Here γ = 1.4 and we have examined several values of l and lg. In these plots, particles are assumed to be at rest at the final point. The…

**Figure 6.** Figure 6: Variation of critical accretion parameter, λc and critical radius, Rc with respect to the characteristic dimensionless linear density of galaxy, i.e. mg/Rg. Here, γ = 1.4 and we have examined several values of l and lg. Following Bondi’s model, We have assumed that bot…

**Figure 7.** Figure 7: Mach number, u, as a function of the dimensionless radius, R. Here, γ = 1.4 and we have examined several λ’s. One of them is equal to the critical value of accretin parameter, λc = 0.8621. − (1 − l) Rf − mg(1 − lg) Rf + Rg + 1 γ − 1 (33) We can solve this equation with…

**Figure 8.** Figure 8: Mach number, u, as a function of the dimensionless radius, R. Here, γ = 1.4 and we can see how changing each parameter in the potential energy affect Mach number. creasing l whereas lg made Rc larger. However, both l and lg had a negative influence on λc and decreased …

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